BleaneyBleaney-ElectricityMagnetism2ndEd

LECTRICITY AND MAGNETISM BLEANEY AND BLEANEY SECOND EDITiON OXFORDElectricity and Magnetism SECOND EDITION B.I.BLEANEY and B.BLEANEY , I I I IELECTRICITY AND MAGNETISM BY B.I.BLEANEY FellowofSt.Hugh,'aCollege,Oxford AND B.BLEANEY Dr.Lell'aProf688OT of Experimental PhiloBOphy Univeraity ofOxford SECOND EDITION OXFORD ATTHEOLARENDON PRESS 1965 \ OxfordUniversity Pre88,AmenHou8e,LondonE.O.4 GLASGOW NEWYORKTORONTO MELBOURNE WELLINGTON BOMBAY CALOUTTA :MADRAS KARACHI LAHORE DAOOA OAPETOWNSALISBURY NAIROBI IBADAN AOCRA KUALA LUMPUR I10NGKONG ©OxfordUniver8ity Pres8,1965 PRESTON POLYTECHNIC 507 FIRSTEDITION 1957 REPRINTED LITIIOGRAPIIIOALLY INGREAT BRITAIN ATTIlEUNIVERSITY PRESS, OXFORD BYVIVIAN RIDLEIt, PRINTEIt TOTIlEUNIVERSITY FROMCORREOTED SHEE'IlS OFTHEFIRS'Il EDI'IlION 1959 SEOOND lIlDITION 1965\ I ! PREFACE TOTHEFIRSTEDITION INteaching fortheFinalHonour SchoolofPhysics inOxford.the authorshavelongfelttheneedforanup-to-date textonElectricity land Magneti8m whichwouldcoverthewholefield,boththetheoryandthe practice. Thisbookisanattempttosupplythisneed,andtomaketas comprehensive aspossiblechapters havebeenincluded whichmay partofagraduate courseratherthananundergraduate course.The ofthebookisasfollows:thefirsteightchapters coverthefundament sof thetheoryandincludeaccounts ofelectrical conductors andmagneism atanelementary level;chapters 9to11dealwiththetheoryofalteat­ ingcurrents andwaves;thenextfivechapters covertheexperime tal aspectsofgenerators, radio,andalternating currentmeasurements' the finalsectionisdevotedtofulleraccounts ofnoise,dielectrics, conducors, andmagnetism, andachapteronmagnetic resonance, withparticlar reference tomeasurements ofsomefundamental constants. Thewritingofanybookonelectricity andmagnetism isbedelIed bythequestion ofunits.Theauthors werebrought uponthetwo centimetre-gramme-second systems, andthepractical system. Tot~ese isnowaddedthemetre-kilogramme-second system, making four systemsatpresentinuse.Generaladoption ofthem.k.s.system w~uld reducethistoonesystem,whichissuchanobviousadvantage thatthe rationalized m.k.s.systemhasbeenadopted inthisbook.Unfortun­ atelynogeneralagreement onthedefinition ofmagnetization obtajned whilethisbookwasinpreparation; theauthorshavetherefore adopted thedefinition whichisclosesttothec.g.s.systems, whichhas,the advantage thatmagnetostatics iscloselyparalleltoelectrostatics. This choicehasbeenmadetosimplify asfaraspossiblethetransition £rom c.g.s.systemstothem.k.s.system,sincemanystudents whomayWish tousethisbookwillhavebeenbroughtupontheformer. Forthese students, achapter onunitshasbeenincluded wheremethods; are detailed fortranslating allthenumbered equations intotheir eq~iva­ lentsinoneofthec.g.s.systems. Thechoiceoftherationalized mi.k.s. systemmakesthistranslation morecumbersome, andtherathermlinor advantages of'rationalization' areoutweighed bythedisadvantltges ofchangesinthedefining equations ofanumberofthefundamE1ntal quantities. Thesewilldisappear whenthec.g.s.systemsfalloutofiuse, andtheauthorshavetherefore adoptedtherationalized m.k.s.system toconform withpresentpractice. vi PREFACE TOTHEFIRST EDITION Theauthorsaremuchindebted totheircolleagues-in particular Drs.D.M.S.Bagguley, A.H.Cooke,J.H.E.Griffiths, H.G.Kuhn, andG.W.Series-who havecriticized partsofthemanuscript andmade manyhelpfulsuggestions; toM.H.W.Gall,Esq.,ofMessrs.H. Tinsley &Co.Ltd.,Professor L.F.Bates,F.R.S.,andA.Hart,Esq.,ofNotting­ hamUniversity, andtoMessrs.L.J.ArundelandR.A.Kamperofthe Clarendon Laboratory, fortheconsiderable troubletheytookinobtain­ ingthephotographs forFigs.7.3,21.6,17.3, and22.7respectively; and toanumberofpupilswhohavereadvariouschapters andeliminated numerous errors.Theauthorsarenotsosanguine astobelievethat noerrorsremaininotherpartsofthebook,andtheywillbegrateful toreaderswhoinformthemofanyerrors. B.I.B. B.B. Clarendon Laboratory Oxford April1955 PREFACETOTHESECOND EDITION SINCEthefirsteditionofthisbookappeared in1957therehasbeen considerable progress, bothexperimental andtheoretical, inunder­ standing theelectrical andmagnetic properties ofmaterials. Muchof thisistooadvanced inapproach forabookintended primarily for undergraduates orgraduates starting research, buttheauthorshave attempted todistilasuitable fraction ofappropriate density for presentation atthislevelinthesecondedition. Thisisnotalways easy,andtheauthorsapologize bothtothosewhofindsectionslacking inthesimpleclaritywhichtheidealtextbook shouldpossess,andto thosewhofindthatover-simplification hasresultedinalackofaccuracy. Theplanofthebookissubstantially thesameasinthefirstedition. Thefirsteightchapters coverthefundamentals ofthetheoryand includeaccounts ofelectrical conductors andmagnetism atanelemen­ tarylevel;Chapters 9-11dealwiththetheoryofalternating currents andwaves;thenextfourchapters covertheexperimental aspectsof radioandalternating currentmeasurements; thefinalpartisdevoted tofulleraccounts ofnoise,dielectrics, conductors, andmagnetism, endingwithachapteronmagnetic resonance. Thispart,whichoverlaps withsolidstatephysics, hasbeenconsiderably rewritten; the·section onsemiconductors hasbeenexpanded intoaseparate chapter, with someaccount oftheprinciples ofjunctions butstopping shortof transistor circuitry; thatonanti-ferromagnetism hasbeenincorporated inanewchapterwhichalsoincludesferrimagnetism andtherare-earth metals. Thediscussion ofconduction inmetals,paramagnetism and ferromagnetism, andmagnetic resonance, hasbeenconsiderably revised andsomewhat enlarged. Material elsewhere hasbeenprunedwherever possibletominimize theincreaseinoverallsize;inparticular thechapter onelectrical machines hasbeenomitted, exceptforlowfrequency trans­ formers, whoseequivalent circuitisdiscussed attheendofChapter 9 onalternating currenttheory. Inthefirsteditiontheelectromagnetic dipolemomentofanelemen­ tarycurrentcircuitwasdefinedinsuchawayastoretainHastheforce (couple)vectoronamagnetic dipole,whileBistheforcevectorona current. Thiswasacompromise, intended toreducethegapbetween theolderc.g.s.systemandthenewm.k.s.system. However, the definition m=P-P-oIdShadobvious difficulties inferromagnetic media,andledtoinconsistencies between dia-andparamagnetism, viii PREFACE TOTHESECOND EDITION wheretheatomicformulae contained fl-ointhenumerator inonecase andinthedenominator intheother.Incommon withotherbooks usingthem.k.s.system, theauthors havetherefore adopted the definition m=IdSinthesecondedition,whichgivesamorelogical treatment inwhichBistheforcevectorbothforcurrents andmagnetic dipoles. Thishasnecessitated considerable changesinChapter 5,and theopportunity hasbeentakentorevisethetreatment togiveamore rigorous approach. Amongst otherminorchanges afullertreatment ofspherical harmonics isincluded inChapter 2,together withthe multipole expansion. Theauthorsaremuchindebted toDrs.B.V.Rollin,R.J.Elliott, andR.A.Stradling, whoreadpartofthemanuscript andmade suggestions forimprovements; alsotomanycolleagues inOxfordand readerselsewhere whotookthetroubletosendcomments onthefirst edition.Withtheirhelptheauthorshaveendeavoured toeliminate the errorsthatremained, butnodoubtafreshcrophasbeensowninthe secondedition,andtheauthorswillbegratefultoreaderswhoinform themofsucherrors. B.I.B. BoB. Clarerulon Laboratory Oxford April1964 ACKNOWLEDGEMENTS THEauthorsareindebted tothefollowing forpermission tousepub­ lisheddiagrams asabasisforfiguresinthetext:thelateSirK.S. Krishnan; G.Benedek; R.Berman; D.F.Cochran; S.Dresselhaus; G.Duyckaerts; R.D.Frauenfelder; S.A.Friedberg; M.P.Garfunkel; W.E.Henry; A.F.Kip;C.Kittel;J.F.Koch;B.T.Matthias; K.A.G.Mendelssohn; D.E.Nagle;H.M.Rosenberg; C.G.Shull; J.S.Smart;J.W.Stout;R.A.Stradling; W.Sucksmith; R.W. Taylor;P.Vigoureux; W.E.Willshaw; W.P.Wolf;American Institute ofPhysics; American Physical Society; Institution ofElectrical Engineers; Institute ofPhysicsandthePhysical Society(London); RoyalSociety(London) ;BellTelephone Laboratories; A.E.I.Research Laboratories; G.E.C.Research Laboratories; NorthHolland Publish­ ingCo. xii CONTENTS 5.THEMAGNETIC EFFECTS OFCURRENTS ANDMOVING CHARGES, ANDMAGNETOSTATICS 5.1Forcesbetween currents 126 5.2Magnetic shells 130 5.3Magnetostatics andmagnetic media 135 5.4Solution ofmagnetostatic problems 139 5.5Steadycurrents inmagnetic media 142 5.6Calculation ofthemagnetic fieldsofsimplecircuits 148 5.7Moving chargesinelectricandmagnetic fields 151 6.ELECTROMAGNETIC INDUCTION ANDVARYING CURRENTS 6.1Faraday's lawsofelectromagnetic induction 158 6.2Self-inductance andmutualinductance 161 6.3Transient currents incircuitscontaining inductance, resistance, andcapacitance 165 6.4Magnetic energyandmechanical forcesininductive circuits 172 6.5Magnetic energyinmagnetic media 175 7.DIRECT CURRENT MEASUREMENTS 7.1Galvanometers, ammeters, andvoltmeters; thewattmeter 179 7.2Galvanometer damping 184 7.3Theballistic galvanometer andfluxmeter 186 7.4Absolute measurements 190 8.MAGNETIC MATERIALS ANDMAGNETIC MEASUREMENTS 8.1Originsofmagnetism 195 8.2Diamagnetism 198 8.3Paramagnetism 201 8.4Ferromagnetism 204 8.5Production ofmagnetic fields 207 8.6Measurement ofmagnetic fields 214 8.7Measurement ofsusceptibility 216 8.8Experimental investigation ofthehysteresis curve 221 8.9Terrestrial magnetism 223 9.ALTERNATING CURRENT THEORY 9.1Forcedoscillations 9.2Useofvectorsandcomplex numbers 9.3Tunedcircuits 9.4Coupled resonant circuits 9.5Low-frequency transformers227 231 236 243 247 CONTENTS xiii. r10.ELECTROMAGNETIC WAVES 10.1Maxwell's equations oftheelectromagnetic field 256 10.2.Planewavesinisotropic dielectrics 260 10.3ThePoynting vectorofenergyflow 263 10.4Planewavesinconducting media 265 10.5Theskineffect 267 10.6Reflection andrefraction ofplanewavesattheboundary oftwo dielectrics 269 10.7Reflection fromthesurfaceofametal 277 10.8Thepressure duetoradiation 278 10.9Radiation fromanoscillating dipole 281 11.FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 11.1Elements offiltertheory 291 11.2Somesimpletypesoffilter 297 11.3Travelling wavesontransmission lines 302 11.4Terminated loss-free lines 306 11.5Attenuation onlossylines,andresonant lines 311 11.6Guidedwaves--propagation between twoparallel conducting planes 315 11.7Waveguides 321 12.THERMIONIC VACUUM TUBES 12.1Construction ofthethermionic vacuum tube 329 12.2Thediode 331 12.3Thethree·halves powerlaw 332 12.4Usesofthediode 335 12.5Thetriode 339 12.6Characteristics ofthetriode 340 12.7Equivalent circuitofthetriode 343 12.8Inputimpedance ofthetriode 344 12.9Thescreen-grid tetrode 347 12.10Thepentode 349 13.APPLICATIONS OFTHERMIONIC VACUUM TUBES 13.1Audio-frequency voltageamplifiers 351 13.2Negative feed-back amplifiers 354 13.3Audio-frequency poweramplifiers 355 13.4Radio.frequency amplifiers 359' 13.5Tunedanodeandtunedgridoscillators 364 13.6Poweroscillators 368 13.7TheKipprelayandthemultivibrator 370 13.8Amplitude modulation anddetection 375 13.9Frequency changing 380 13.10Frequency modulation 383 13.11Radioreceivers 386 xiv CONTENTS 14.THERMIONIC VACUUM TUBES ATVERYHIGHFRE- QUENCIES 14.1Effectsofelectrode impedanoe 14.2Effectoftransittimeoninputconduotanoe 14.3Modified oircuitsandtubesformetreanddeoimetre wavelengths 14.4Theklystron 14.5Themagnetron 14.6Crystaldiodes 14.7Travelling wavetubes 15.ALTERNATING CURRENT MEASUREMENTS 15.1Measurement ofvoltage, ourrent,andpower 15.2Measurement ofimpedanoe atlowfrequenoies 15.3Measurement ofimpedanoe atradiofrequenoies 15.4Measurement offrequenoy andwavelength 15.5Measurement ofdielectrio oonstant 15.6Measurement ofthevelocity ofradiowaves 16.FLUCTUATIONS ANDNOISE 16.1Brownian motionandfluotuations 16.2Fluotuations ingalvanometers 16.3Therelationbetween resistanoe noiseandthermal radiation 16.4Shotnoise 16.5Designofreoeivers foroptimum performanoe (minimum noise figure) 16.6Measurement ofreoeiver noise 17.THEORY OFTHEDIELECTRIC CONSTANT 17.1Moleoular struoture andthedielectric oonstant 17.2Dieleotrio oonstant ofnon-polar gases 17.3Statiodielectrio constant ofpolargases 17.4Dispersion ingases 17.5Statiodielectrio constants ofliquidsandsolids 17.6Statiodieleotric oonstants ofpolarliquids 17.7Radio-frequenoy dispersion inpolarliquids 17.8Scattering lEi.ELECTRONS INMETALS 18.1Kinetics offreeeleotrons inmetals 18.2Theenergybandapproximation 18.3Conduotors andinsulators onthebandtheory 18.4Specifioheatoftheoonduction electrons 18.5Electrical andthermal oonduotivity ofmetals 18.6TheHalleffeot 18.7Dis.-andparamagnetism ofoonduotion electrons390 393 395 400 405 411 412 41,4, 423 429 434 442445 452 454 458 462 466 470 475 477 480 483 488 491 493 498 504 506 515 517 521 528 529 ~ CONTENTS 19.SEMICONDUCTORS 19.1Intrinsic andextrinsic conductivity 19.2Elementary andcompound semiconductors 19.3Electron distribution andtheFermilevel 19.4Opticalproperties 19.5Transport properties 19.6Metal-semiconductor junctions 19.7Thep-njunction 19.8Thejunction transistorxv 536 538 543 548 553 560 565 569 20.THEATOMIC THEORY OFPARAMAGNETISM 20.1Ageneralprecession theorem 574 20.2Thevectormodeloftheatom 576 20.3Magnetic moments offreeatoms 585 20.4Themeasurement ofatomicmagnetic moments-the Stern- Gerlachexperiment 590 20.5Curie'slawandtheapproach tosaturation 591 20.6Susceptibility ofparamagnetic solids-the 4/group 593 20.7Susceptibility ofparamagnetic solids-the 3dgroup 598 20.8Susceptibility ofparamagnetic solids-strongly bondedcom- pounds 606 20.9Electronic paramagnetism-a. summary 609 20.10Nuolear moments andhyperfine structure 610 21.FERROMAGNETISM 21.1Exchange interaction between paramagnetic ions 618 21.2TheWeisstheoryofspontaneous magnetization 622 21.3Ferromagnetic domains 626 21.4Thegyromagnetic effect 634 21.5Thermal effectsinferromagnetism 637 21.6Measurement ofthespontaneous magnetization Moasafunc- tionoftemperature 640 21.7Foundations ofthetheoryofferromagnetism 644 21.8Spinwaves 648 21.9Mechanisms ofexchange interaction 650 22:ANTI-FERROMAGNETISM ANDFERRIMAGNETISM 22.1Anti·ferromagnetism 657 22.2Themolecular field-two sub-lattice model 659 22.3Ferrimagnetism 664 22.4Thelanthanide ('rareearth')metals 670 22.5Neutron diffraction 673 xvi CONTENTS 23.MAGNETIC RESONANCE 23.1Themagnetic resonance phenomenon 677 23.2Molecular beamsandnuclearmagnetic resonance 681 23.3Nuclear magnetic resonance inbulkmaterial 685 23.4Relaxation effectsinnuclearmagnetic resonance 689 23.5Applications ofnuclearresonance 692 23.6Electron magnetic resonance inatomicbeams 697 23.7Electron magnetic resonance insolids 703 23.8Cyclotron resonance withfreecharged particles 710 23.9Cyclotron resonance ofchargecarriersinsemiconductors 717 23.10Azbel-Kaner resonance inmetals 722 24:UNITS 24.1Unrationalized c.g.s.systems 24.2Practical units 24.3Therationalized m.k.s.system 24.4Conversion factorsfromrationalized m.k.s.system 24.5Equivalent equations inunrationalized c.g.s.systems729 733 733 736 737 ApPENDIX A.VECTORS A.lDefinition ofscalarandvectorquantities 744 A.2Vectoraddition andsubtraction 744 A.3Multiplication ofvectors 745 A.4Differentiation andintegration ofvectors 747 A.5Thedivergence ofavector 748 A.6Thecurlofavector 750 A.7Laplace's operator 751 A.8Stokes's theorem 751 A.9Thedivergence theorem 752 A.lOTransformation fromarotating coordinate system 753 A.llLarmor's theorem 753 APPENDIX B~THEUNIQUENESS THEOREM 755 AFPENDIX C.NUMERICAL VALUES OFTHEFUNDA· MENTAL CONSTANTS 756 APPENDIX D.SOMEATOMIC FORMULAE INM.K.S.UNITS 757 INDEX 7~ 1 1 ELECTROSTATICS I 1.1.Theelectrical natureofmatter THEfundamental lawsofelectricity andmagnetism werediscovered by experimenters whohadlittleornoknowledge ofthemoderntheoryofthe atomicnatureofmatter.Itshouldtherefore bepossibletopresentthese lawsinatextbook bydealingatfirstpurelyinmacroscopic phenomena andthenintroducing gradually thedetailsofatomictheoryasrequired. Inthiswaythesubjectmightbedeveloped almostinthehistorical order ofdiscovery, andtheseopeningsentences wouldtalkofamberandcat's fur.Itismoreinteresting, however, todiscusshereandtherethroughout thisbooktheinterpretation ofthemacroscopic lawsintermsofpresent atomictheory.Inthelaterchapters aconsiderable knowledge ofsuch theorywillbeassumed, sincetogiveanadequate accountofitwould greatlyincreasethesizeofthebook.Thiswillnotbeattempted, but inthefollowing paragraphs asummary ispresented ofwhatmaybe regarded almostascommon knowledge ofthenatureoftheatom. Onmoderntheorytheatomconsistsofacentralcore,ornucleus, of diameter about10-12cm,surrounded byanumberofelectrons. These electrons moveroundthenucleusinorbitswhosediameter isabout 10-8cm,andthesedetermine thesizeoftheatom.Thenucleuscontains twokindsofparticles: protons, whichareparticles roughly 1836times asheavyaselectrons, butwithapositive electriccharge+e,andneu­ trons,ofverynearlythesamemassasprotons, butwithnoelectric charge. Thenumberofelectrons surrounding thenucleusisequalto thenumberofprotons, andeachelectronhasanegative charge-e,so thattheatomasawholeiselectrically neutral. Thephysical and chemical properties oftheatomsaredetermined bythenumber of electrons theycontain,andhencethenumberofprotonsinthenucleus ischaracteristic ofaparticular element. Thenumber ofneutrons is roughly equaltothenumberofprotonsinlightelements butisover 1·5timesasgreatintheheaviest elements. Themassofthenucleusis determined bythetotalnumberofprotonsandneutrons, andagiven elementmayhaveseveralstableformsofdifferent nuclearmass,corre­ sponding tonucleiwithdifferent numbers ofneutrons, butthesame numberofprotons. Thesearecalledisotopes. Thustheoxygennucleus 851110 B 2 ELECTROSTATICS I [I.l has8protons,andtherearethreestableisotopes, oxygen16, 17,and18, with8,9,and10neutrons respectively, although thepercentage of isotopes 17and18occurring innatureisverysmall. Itisnowestablished thattheelectronic chargeisthefundamental unitofcharge,andallchargesareintegralmultiples of+eor-e.Itis therefore assumed thattheelectron isindivisible, andisafundamental particleofmatter; soalsoistheproton. Wemaysummarize thepro­ pertiesofelectron, proton,andneutron asfollows: Particle Electron Proton NeutronCharge -e +eoMass m 1836m 1838m e=1·602X10-19coulomb; m=0·911X10-27g. Sincechargesofopposite signattractoneanother, theelectrons are boundtotheatombytheelectrical attraction oftheprotonsinthe nucleus. Theforceswhichholdthenucleustogether areofdifferent character, andoperateonlyatveryshortranges,oftheorderofthe nucleardiameter. Oonductors andinsulators Forthepurposeofelectrostatic theoryallsubstances canbedivided intotwofairlydistinct classes:conductors, inwhichelectrical charge canfloweasilyfromoneplacetoanother; andinsulators, inwhichit cannot.Inthecaseofsolids,allmetalsandafewothersubstances such ascarbonareconductors, andtheirelectrical properties canbeexplained byassuming thatanumberofelectrons (roughly oneperatom)arefree towanderaboutthewholevolumeofthesolidinsteadofbeingrigidly attached tooneatom.Atomswhichhavelostoneormoreelectrons in thiswayhaveapositive charge,andarecalledions.Theyremainfixed inposition inthesolidlattice.Insolidsubstances ofthesecondclass, insulators, eachelectron isfirmlyboundtothelatticeofpositiveions, andcannotmovefrompointtopoint.Typical solidinsulators are sulphur, paraffinwax,andmica. Whenasubstance hasnonetelectrical charge,thetotalnumbers of positiveandnegative chargeswithinitmustjustbeequal.Ohargemay begiventoorremoved fromasubstance, andapositively-charged substance hasanexcessofpositiveions,whileanegatively-charged sub­ stancehasanexcessofelectrons. Sincetheelectrons canmovesomuch moreeasilyinaconductor thanthepositive ions,anetpositive charge 1.1] ELECTROSTATICS I 3 (1.1)isusuallyproduced bytheremovalofelectrons. Inachargedconductor theelectrons willmovetopositions ofequilibrium undertheinfluence oftheforcesofmutualrepulsion between them,whileinaninsulator theyarefixedinposition and any initialdistribution ofchargewill remainalmostindefinitely. Inagoodconductor themovement of chargeisalmostinstantaneous, whileinagoodinsulator itisextremely slow.Whilethereisnosuchthingasaperfectconductor orperfect insulator, suchconcepts areusefulindeveloping electrostatic theory; metalsformagoodapproximation totheformer,andsubstances such assulphurtothelatter. 1.2.Coulomb's lawandfundamental definitions Theforceofattraction between chargesofopposite sign,andofrepul­ sionbetween chargesoflikesign,isfoundtobeinversely proportional tothesquareofthedistance between thecharges(assuming themtobe locatedatpoints),andproportional totheproductofthemagnitudes ofthetwocharges. Thislawwasdiscovered experimentally byCoulomb in1785.Inhisapparatus thechargeswerecarriedonpithballs,andthe forcebetween themwasmeasured withatorsionbalance. Theexperi­ mentwasnotveryaccurate, andamodernmethodofverifying the inversesquarelawwithhighprecision willbegivenlater(§1.3).From hereonweshallassumeittobeexact. Ifthechargesareqlandq2'andristhedistance between them,then theforceFonq2isalongr.Ifthechargesareofthesamesign,the forceisoneofrepulsion, whosemagnitude is F-OQlq2-r2 ' Thevectorequation fortheforceis F-oql;2r. r HereF,rarecounted aspositivewhendirected fromQltoQ2'Equation (1.1)isthemathematical expression ofCoulomb's law. TheunitsofFandrarethosealreadyfamiliar frommechanics; it remainstodetermine theunitsof0andq.Heretherearetwoalterna­ tives:either0isarbitrarily givensomefixednumerical value,when equation (1.1)maybeusedtodetermine theunitofcharge,ortheunit ofchargemaybetakenassomearbitrary value,whentheconstant 0is tobedetermined byexperiment. Theelectrostatic systemofunits (e.s.u.)makestheuseofthefirstmethod. TheforceFisindynes,and 4 ELECTROSTATICS I [1.2 (1.2)thedistance rincentimetres (i.e.botharemeasured inthecentimetre­ gramme-second system), andtheconstant 0issetequaltounity.Then ql'q2aremeasured ine.s.u.ofcharge,theunitbeingdefinedasthat chargewhichrepelsanequalchargeatadistance of1cminvacuowith aforceof1dyne.Inthemetre-kilogramme-second-coulomb system (m.k.s.), whichwillbeusedthroughout thisbook,theunitofchargeis theooulomb, thestandard practical unitofcharge(equaltoone-tenth oftheunitofchargeintheelectromagnetic systemofunits).Forthe presentpurposeitmayberegarded asdefinedbythechargerequired todepositacertainmassofsilverinasilvervoltameter, beingthus definedinanarbitrary mannerinthesamewayasthestandard metre andstandard kilogramme. Equation (1.1)forCoulomb's lawisthen analogous tothatforgravitational attraction, exceptthatitdealswith electrical chargesinsteadofmasses,andtheunknown constant ofpro­ portionality 0mustbedetermined byexperiment. Inthe'rationalized' metre-kilogramme-second-coulomb system,theconstant 0iswritten asl!47rEO'thefactor 47Tbeingintroduced tosimplify certainequations whichappearlaterinthetheory.Equation (1.1)therefore becomes F=_1_qlq2r, 47rEOr3 whereFisinnewtons, rinmetres,andqincoulombs. Thequantity EOisknownasthe'permittivity offreespace'(see§1.5);itsexperi­ mentalvalueisfoundtobe(see§7.4)8'85X10-12coulomb2newton-1 metre-2(thisunitcanbemoreconveniently calledfaradmetre-1(see §1.6)).Since1newton=105dyne,and1metre=102cm,itmay readilybeshownthat1coulomb =2·998X109e.s.u. Electricfieldandelectricpotential Theforcewhichachargeq2experiences whenintheneighbourhood of anotherchargeqlmaybeascribedtothepresence ofan'electricfield'E produced bythechargeql'Sincetheforceonachargeq2isproportional tothemagnitude ofq2'wedefinethefieldEbytheequation F=EQ2' (1.3) Fromthisdefinition andCoulomb's lawitfollowsthatEdoesnot dependonQ2'andisavectorquantity, likeF.Fromequation (1.2)we findthat (1.4) istheelectricfieldduetothecharge Ql' • ------_._- ------------ 1.2] ELECTROSTATICS I 5 B V=-IE.ds. A Thisisascalarquantity knownastheelectricpotential.IfthefieldE isduetoasinglechargeqat0,asinFig.1.1,thentheforceonunitIfaunitpositive chargeismovedaninfinitesimal distance dsina fieldE,thentheworkdonebythefieldisE.ds,andtheworkdone againstthefieldis-E.ds.Thisfollowsfromthefactthattheforce onunitchargeisequaltotheelectricfieldE.Theworkdoneagainst thefieldinmovingaunitpositive chargefromapointAtoapointB willtherefore be o E A FIG.1.1.Calculation ofthepotential difference between pointsAandBduetothefield ofapointchargeqatO. chargeatanarbitrary pointPisalongOP,anddsisthevectorelement PIP2•NowE.ds=Ecos8ds=Edr,andhence B r. VB-VA= -JEdr=_--.!LJd;=--!L(~_~\. 41TEOr 41TEOr2r~ A r, Thusthedifference ofpotential between AandBdepends onlyonthe positions ofAandB,andisindependent ofthepathtakenbetween them. Thepotential atapointdistance rfromachargeqistheworkdone inbringing upunitchargetothepointinquestion fromapointatzero potential. Byconvention, thepotential istakenaszeroataninfinite distance fromallcharges, thatis,V=0forr=00.Therefore the potential atapointdistance rfromachargeqis V=qj(41TEor). (1.5) Thedifference inpotential dVbetween PIandP2(Fig.1.1)distance dsapartisdV=-E.ds =-(Exdx+Eydy+Ezdz). 6 ELECTROSTATICS I [1.2 Hence E=-gradV= -VV, (1.6) whereinCartesian coordinates gradV=i8V/8x+i 8V/8y+k8V/8z and i,i,kareunitvectorsparalleltothex,y,andz-axes.Thecomponents ofEalongthethreeaxesare oV oV 8VE=--, E=--, E=---.xox YBy Z8z Thenegative signshowsthatofitselfapositive chargewillmovefrom ahighertoalowerpotential, andworkmustbedonetomoveitinthe opposite direction. (Forvectorrelations, seeAppendix A.) c FIG.1.2.Theworkdoneintakinganelectricchargerounduclosed pathinanelectrostatic fieldiszero. Theworkdoneintakingachargeqroundaclosedpathinanelectro­ staticfieldiszero.ThiscanbeseenfromFig.1.2.Thework done in takingthechargeqroundthepathABOAis W=-qfE.ds=q(VB-VA)+q(Vo-VB)+q(VA-Vd =0, andisindependent ofthepathtakenprovided itbeginsandendsatthe samepoint.Therefore theelectricpotential isasingle-valued function ofthespacecoordinates inanystationary distribution ofelectriccharges; ithasonlyonevalueatanypointinthefield. Sincepotential isascalarquantity thepotential atanypointis simplythealgebraic sumofthepotentials duetoeachseparate charge. Ontheotherhand,Eisavectorquantity, andtheresultant fieldisthe vectorsumoftheindividual fields.Henceitisnearlyalwayssimplerto workintermsofpotential ratherthanfield;oncethepotential distribu­ tionisfound,thefieldatanypointisfoundbyusingequation (1.6). Units Fromequation (1.3)weobtainthedefinition ofelectricfield.An electricfieldof1unitexertsaforceof1newtononachargeof1cou­ lomb.Electric fieldscantherefore beexpressed innewton/coulomb. 1.2] ELECTROSTATICS I 7 Theunitofpotential isdefinedasfollows:When1jouleofworkis doneintransferring achargeof1coulomb fromAtoB,thepotential difference between AandBis1volt.Fromequation (1.6)Ecanbe expressed involts/metre, andthisistheunitwhichiscustomarily used. Itiseasilyverifiedthatthetwoalternative unitsforEareequivalent. (a) (b) FIG.1.3.(a)Linesofforcebetween equalchargesofopposite sign.(b)Linesofforce between equalchargesofthesamesign. Linesofforce Alinedrawninsuchawaythatitisparalleltothedirection ofthe fieldatanypointiscalledalineofforce.Figure1.3showsthelinesof forcefortwoequalcharges. Linesofforcedonotintersect oneanother sincethedirection ofthefieldcannothavetwovaluesatonepoint;they arecontinuous inaregioncontaining nofreecharges, andtheybegin andendonfreecharges..Thenumberoflinesofforcedrawnthrough unitareanormaltothedirection ofEisequaltothevalueofEatthat point. Ifaseriesofcurvesisdrawn,eachcurvepassingthrough pointsat agivenpotential, theseequipotential curvescutthelinesofforceortho­ gonally. Equipotential curvesaregenerally drawnforequalincrements ofpotential; thenEisgreatest wheretheequipotentials areclosest together. 8 ELECTROSTATICS I [1.3 1.3.Gauss's theorem LetSbeaclosedsurfacesurrounding achargeq,andletqbedistant rfromasmallareadSonthesurfaceSatA(Fig.1.4(a)).Theelectric intensity EatAhasthevalue E=-q-. 47T€Or2 E (a) (b) FIG.104.Illustrating Gauss'stheorem. ThenumberoflinesofforcepassingthroughanelementofareadSis E.dS=EcosfJdS=qcosfJdS, 47T€or2 wheretheoutward normaltothesurfaceelement makesananglefJ withE.Nowthesolidanglesubtended bydSat0isdw=cosfJdSjr2, andthevalueofEcosfJdS istherefore qdwj(47T€O). Hencethetotal numberoflinesofforcepassingthroughthewholesurfaceis JEcosfJdS =-.!LJdw='l.., 47T€O €o(1.7a) sinceaclosedsurfacesubtends atotalsolidangleof47Tatanypoint withinthevolumeenclosed bythesurface.Ifthereareanumberof charges ql>q2"'.'qninsideS,theresultant intensity ofEatanypoint isthevectorsumoftheintensities duetoeachseparate charge,andthe integration ofequation (1.7a)maybecarriedoutseparately foreach charge.InthiswayitisfoundthatJEcosfJdS=.2qj€o.Ontheother hand,thecontribution ofanychargeoutsideSiszero,asmaybeseen 1.3] ELECTROSTATICS I 9 fromFig.1.4(b),sinceinthiscase fEcos8dS =-.!L[fdSlC~S81_ fdS2C~S82] =O. 47TEO r1 r2 Wemaysummarize theseresultsintheform JEcos8dS =JE.dS=2,qIEo, (1.7b) wherethesummation istobetakenonlyoverthechargeslyingwithin theclosedsurfaceS.ThisisknownasGauss'stheorem. Weseethat theintegralofthenormalcomponent ofEoverthesurfaceisequal tothetotalchargeenclosed, dividedbyEO'irrespective ofthewayin whichthechargeisdistributed. Ifthereexiststhroughout avolumeenclosed byasurfaceSacharge distribution ofvaryingdensityp,wehave ~fpdT=fE.dS=fdivEdT, (1.7c) wheredTisanelementofvolume. Thetwovolumeintegrals mustbe equalwhatever thevolumeoverwhichtheintegration takesplace,and ittherefore followsthattheintegrands themselves mustbeequal. Hence divE=oEx+oEy+oEz=plEo. (1.8)oxoyOZ andthisistheexpression indifferential formofGauss'stheorem. The transformation fromasurfacetoavolumeintegral usedaboveisdue toGauss(seeAppendix A). Oneoftheconsequences ofGauss'stheorem isthattherecanbeno fieldwithinaconductor, norcantherebeanyvolumedistribution of chargewithinit.For,ifthereweresuchachargedistribution, afield wouldexistwithintheconductor, whichwouldactonthecharges. Since theyarefreetomoveinaconductor, theycannotthenbeinastate ofequilibrium. Thusnoelectrostatic fieldcanexistwithinthebodyof aconductor, andallpartsofitmustbeatthesamepotential.Ifthe conductor hasatotalchargedifferent fromzero,thenthischargemust resideentirelyinathinlayerontheoutersurface. Thefactthattherecanbenoelectricfieldwithinthebodyofaoon­ ductorhasanimportant consequence inthecaseofahollowclosed conductor. IfweapplyGauss'stheorem toasurfaceSlyingentirely withintheconducting substance, asinFig.1.5(b),thenJEcos8dS=0, sinceE=0everywhere overthesurface. Hencethenetchargeinside thesurfacemustbezero.Thiscanberealizedintwoways:(a)ifthere 10 ELECTROSTATICS I [1.3 isatotalchargeqinthehollowspacewithintheconductor, thelines offorcefromthechargescomprising qmustendonadistribution of chargeontheinnersurfaceoftheconductor, andthetotalchargein thislayermustbeequalto-q;(b)ifthereisnochargeinthehollow space,thentherecanbenofieldinthisspace.Thislastresultisim­ portant' formanyproofsoftheinversesquarelaw(seebelow)depend + (a) (b) FIG.1.5.Distribution ofchargeonahollowconductor. (a)Ahollowconductor withapointcharge+qinside,andinduced charges-qand+qontheinsideandoutsidesurfaces. (b)Thesameconductor withnochargeinside,andtotalcharge+q onthesurface. onit.Itmeansthatifweputaclosedconductor intoafield,acharge distribution ontheoutersurfacewillbesetupsuchthatthefieldinside remains exactlyzero. Experimental proofoftheinversesquarelaw Coulomb's attempts tochecktheinversesquarelawusingatorsion balancewerenotcapableofgreataccuracy, andmostsubsequent at­ temptshavereliedonthefactthatthefieldinsideaclosedconductor isonlyzeroiftheinversesquarelawholds.Weshallprovethisforthe specialcaseofaspherical conductor. InFig.1.6letanelementary coneofsolidangledwbedrawnwith vertexatthepoint0withinthesphere.Thisconeintersects thesurface ofthesphereintheelementary areasdSI,dS2atdistances rl,r2fromO. 1.3] ELECTROSTATICS I 11 Ifthechargeonthespherehasauniform density (Jperunitarea,then thefieldat0duetotheelements dB!anddS2willbe dE=....!!...-[dB!_dS2] %€o r~ r~ Solidangledw FIG.1.6.Thefieldinsideaspherical conductor atapointO.Distances OP=r1, OQ=rz.Fromthegeometry ofthecircle,LOPO=LOQO=e.Hence dw=dSl~ose =dSz~ose. 1 2 assuming thatthefieldofapointchargefallsoffasr-n•Butthesolid angledAn=dS!cos()jri=dS2cos()jr~,andwecantherefore write dE_(Jdw[11] -%€ocos()r~-2-r~-2• Thisgivesaresultant fieldtowards thenearerelementifn<2,and towardsthefurtherelementifn>2.Clearly,thewholesurfaceofthe spherecanbedividedintoelementary areasinthisway,andthevector resultant ofthefieldsat0willnotbezerounlessalltheindividual dE arezero,sincetherewillbearesultant towardsthenearerportionof thespherical surfaceifn<2,andviceversa.Thus,ifitisshown experimentally thatthereisnofieldinsideachargedsphere,itfollows thatthepowerofnintheinversepowerlawmustbeexactly2. Thisresultwasusedtotestthevalidityoftheinversesquarelawby Cavendish and,later,byMaxwell. Maxwell hadaspherical aircon­ denserconsisting oftwoconcentric insulated spherical shells.Theouter spherehadasmallholeinitsothattheinneronecouldbetestedfor chargebyinserting through theholeanelectrode connected toan 12 ELECTROSTATICS I [1.3 electrometer. Thetwosphereswereinitially connected byawireand chargedtoahighpotential, andtheninsulated fromoneanother. After earthing theoutersphere,theinneronewastestedandfoundtohave nocharge.InthiswayMaxwell foundthatthevalueofndidnotdiffer from2bymorethanonepartin21600. Theexperiment hasbeenrepeated byPlimpton andLawton (1936) withamoresensitive detector, theelectrometer beingreplaced byan Galvanometer 4-...------',~ Lightbeam Toalternating voltagegenerator Voltmeter FIG.1.7.Apparatus ofPlimpton andLawton forverifying theinversesquarelaw. amplifier andgalvanometer. Thedetecting apparatus wasplacedinside thesphereA(seeFig.1.7),anditsconducting case,together withthe hemisphere B,formedtheinnerconductor (itwasshownthatthis doesnotnecessarily havetobespherical inshape).Thegalvanometer deflexion wasobserved through asmallholeinthesphereA,covered byawiregridimmersed insaltsolution sothatAwaseffectively a closedconductor. Itwasfoundthatthiswasessential forthefield insidetoberigorously zerowhenn=2.Thegalvanometer wasun­ damped sothatitcouldswingatitsnaturalperiod (~sec),andan alternating voltageof3000V,whosefrequency wasadjusted tosyn-_ chronism withthegalvanometer, wasappliedtotheoutersphere.Nav potential difference between theinnerandoutersphereswasfound, thoughavoltageof10-6Vcouldhavebeendetected.Itwasfoundthat thiswasonlytrueiftheholeinAwascoveredwiththesaltsolution. Thegalvanometer deflexion observed ifthiswasremoved wasusedto checkthatthefrequency ofthealternating voltagewasequaltothe naturalperiodofthegalvanometer. Fromthisexperiment, Plimpton 1.3] ELECTROSTATICS I 13 andLawtonconcluded thatndidnotdifferfrom2bymorethanone partin109• p r+l~cosO}1.4.Electric dipoles Anelectricdipoleconsistsoftwochargesequalinmagnitude butof opposite sign,separated byasmalldistance. Figure1.8showssuchadipolewithcharges +qand-qseparated byadistance a.Then thepotential atapointPis V=4:€J~-k}' (1.9) andifa~r,sothatquantities oftheorder (ajr)2maybeneglected, V_q{1 -47T€Or-lacos0 FIG.1.8.Theelectricpoten­ tialandthefieldduetoan electricdipole. (1.10a) Heretheproduct qahasbeenwrittenasp,andisknownasthedipole moment.Ifrisavectordrawnfrom0asorigintoP,then pcosO=(p.r}jr, wherepisavectorwhosemagnitude isequaltothedipolemoment and whosedirection isfromthenegative chargetothepositivecharge.Then V=~. (1.10b)47T€or3 If0isafixedpointandPisregarded asavariable point,then rjr3=-grad(ljr), sothattheformulaforthepotential mayalsobewrittenas 1V=--{p.grad p(ljr)}. (1.11a) 47T€O Herethesubscript Pisaddedtotheoperator gradtodenotethat differentiation iswithrespecttoPasthevariable point.Ifweregard Pasfixed,andmovefromAtoB,thenequation (1.9)couldhavebeen writtenintheform 1V=-{p.grad o(1jr)}, 47T€o(1.11b) 14 ELECTROSTATICS I [1.4 sincethevectorrin(1.l0b)isnowdrawnintheopposite sense,where thesubscript 0denotesthat0isnowthevariable point,andthereis achangeofsignfromequation (1.11a)because risnowmeasured in theopposite direction. Thecomponents oftheelectric fieldatPcanbecalculated by -q•a+2q• p pa-q •+r:-q: a--J~q -q +q---:- p.. FIG.1.9.Twoquadrupole moments represented asanassembly ofchargesorapairof dipoles; notethatthenetchargeandnetdipolemoment arezeroineachcase. differentiating thepotential givenbyequation (1.lOa). Theradialand azimuthal components are Er= _(o!:,\=_1(2Pcos())}orJ047rEora (1.12) Eo=_!(OV) =~_(PSi~()) ro()r47rEor Theseequations showthattheelectricfieldofadipolefallsoffaslira, anditspotential asl/r2,whereas thecorresponding lawsforasingle polearel/r2andl/r.Thesignificance ofthisdifference isthatatlarge distances thefieldsofthetwoequalandopposite chargeswhichcom­ priseadipolecanceloneanotherinthefirstapproximation (thatis, termsvarying asl/r2vanish),buttermsinthenextorder(lirainthe field)remain. Similarly, iftwodipolesareplacedendtoend,givinga setofchargesasinFig.1.9(knownasaquadrupole), theirfieldsannul oneanotheratlargedistances, andthepotential ofaquadrupole falls offaslira(seeProblem 1.1),anditsfieldasl/r4• Ifadipoleconsisting oftwocharges-qand+qadistance aapart isplacedinauniform field,itspotential energyUpis(seeFig.1.10) Up=q(VB-m=-qacos()E =-qa.E=-p.E, (1.13) wherepisthedipolemoment. Thisshowsthattheenergydepends onlyontheanglewhichpmakeswithE,andnotonthepositionofthe 1.4] ELECTROSTATICS I 15 dipole.Hencethereisnotranslational forceactingonthedipole,but thereisacouple r=-(dUpjdfJ) =pEsinfJ =p/\E, (1.14) whichtendstoturnthedipoleintoaposition paralleltothefield. y E -----.. x z FIG.1.10.Dipoleformedbytwocharges-q,+qseparated bya distance a,inafieldE. Ifthedipoleisplacedinanon-uniform field,atranslational forceis exertedonit,andweshallderiveanexpression forthex-component ofthisforce.IfExisthevalueofthex-component ofthefieldatA,its valueatBmaybewrittenas E'=E+(oEx)+(oEx)+(oEx) xxoxaxoyayOZaz, whereax, ay,azarethecomponents ofaalongthethreeaxes.The x-component oftheforceonthedipoleistherefore Fx=-qEx+qE~ =qax(o~x)+qaye:x)+qaze~x) =Px(o~x) +py(o~x) +Pz(o~x). Now oEx_0(aD_a(aD_aEy8ii-oy-oxl-ox-oyl-ax' 16 ELECTROSTATICS I [1.4 sincetheorderofdifferentiation isimmaterial, andsimilarly oExoEIJaz=a;' Hencetheforcecomponent maybewrittenas F_oExoEyoEIJx-Pxax-+Pyax-+pIJ ax withsimilarexpressions fortheothercomponents.(1.15) 1.5.Thetheoryofisotropic dielectrics Faraday foundthatifaslabofinsulating material wasinserted between twometalplatesacrosswhichaconstant voltagewasapplied bymeansofabattery,thechargeontheplatesincreased.Iftheinsulator entirelyfilledtheintervening space,thechargeincreased byafactor €, where €iscalledthedielectric constant, relativepermittivity, orspecific inductive capacity oftheinsulator. Itvariesbetween 1and10formost solidsubstances, being1forvacuumand1·00057forairatroomtem­ perature andpressure. Tofindhow€isrelatedtotheintrinsic properties ofthematerial, ordielectric, itisnecessary toconsider whathappens insideadielectric whenanelectricfieldisappliedtoit. Dielectric substances areinsulators, andtherefore donotcontainfree electrons. Eachelectron isboundtotheioniclatticebytheelectro­ staticattraction betweenthenegative electronic chargeandthepositive chargesonthenuclei.Intheabsenceofanyexternal field,theelectrons aredistributed symmetrically withrespecttothenuclei,but.whena fieldisapplied,theelectrons aredisplaced inthedirection opposite to thatofthefield,whilethemoremassivenucleiareslightlydisplaced in thedirection ofthefield.(Thecentreofgravityremains fixed,since thereisnotranslational forceonthesystemasawhole.)Eachionthus acquiresanelectricdipolemoment whichisparalleltoandinthesame direction astheappliedfield.Ifaslabofdielectric isplacedbetween parallelmetalplatesasinFaraday's experiment andthevoltageacross theplatesisconstant, therewillbeaninduced negative chargeonthe dielectric surfacenearthepositiveplate,Fig.1.11,andasimilarpositive chargeonthesurfacenearthenegative plate.Therewillbenoresultant chargedensityatanypointwithinthedielectric asalltheindividual dipolesarealignedparalleltothefieldandhenceeachnegative charge ofonedipoleisnexttothepositive chargeofthenextdipole.Thesur­ facechargesonthedielectric willinducechargesofopposite signon theplates,andthechargeontheplatesisincreased whenthedielectric 1.5] ELECTROSTATICS I 17 isinserted, ifthevoltageiskeptconstant, asFaraday foundinhis experiments. Theactionoftheelectricfieldingivingeachatomofthedielectric an induced dipolemoment istermedpolarization. Thepolarization ofthe substance Pisdefinedastheelectricdipolemoment perunitvol~e, anditisproportional inmagnitude totheappliedfieldEatallordinaryrv + + + + + + dI~:0d;2/~1E=2Vjd--T~v-- FIG.l.l1.Effectofadielectric inincreasing thecapacitance between twoparallelplates. fieldstrengths. Pisavectorandinanisotropic substance itisparallel toEsothatwemaywrite P=X€oE. (1.16) HereXisaconstant foranygivensubstance, knownasthepolarizability ortheelectricsusceptibility. Theresultant moment foranelementofvolumedTisPdT,andthe potential ofsuchanelementadistance rawayis,byequation (1.11b), dV=_l_(PdT).grad(l/r), %€o v=I_l_{p.grad(l/r)}dT. 41T€O div(P/r) =!divP+P .grad(l/r).rButwherethedifferentiation iswithrespecttothecoordinates ofthevolume elementcontaining thedipoles. Thepotential duetoafinitevolumeof dielectric isthen Hence 851110v=I_l_div(P/r)dT_I_1_!divPdT %€o 41T€0r IIIIll.=--P.dS- --divPdT, 41T€Or 41T€0r c(1.17) 18 ELECTROSTATICS I [1.5 whereGauss'stheorem ofdivergence hasbeenusedinthetransforma­ tionfromavolumetoasurfaceintegral. Thesetwotermsinequation (1.17)showthattheresultant potential canbeattributed toanapparent surfacechargeofdensityPcos(J,where (JistheanglewhichPmakes withthenormaltothesurfaceofthedielectric, andanapparent volume distribution ofchargewhosedensityis -divP=-(oPx/ox+oPy/oy+o~/8z). IfXisuniform, isotropic, andindependent offield,andthereisno volumedistribution ofrealcharge,divP=XEodivE=0,andthereis novolumedistribution ofapparent charge. Thesurfacedistribution vanishes onlywhenthereisnoappliedfield.Theseapparent charges areoftencalled'polarization charges'. Notethatinthederivation of equation (1.17)ithasnotbeenassumed thatthedielectric isisotropic. Gauss's theoremindielectrics In§1.3itwasshownbyGauss'stheorem thattheintegralfE.dS ofthenormalcomponent ofEoveranyclosedsurfaceisequaltothe totalchargewithinthesurface, dividedbyEO'Ifthesurfaceiswithin adielectric medium, thetotalchargemustincludeboththefreecharges andthepolarization charges. Thevolume chargedensity isthus p-(divP), sothat fE.dS=fdivEdT=:0f(p-divP) dT sothat fdiv(EoE+P)dT=fpdT. Comparison ofthiswithequation (1.7c)showsthat,ineffect, EOEhas beenreplaced by(EoE+P). WemaydefineanewvectorD,suchthat D=EoE+P. (1.18) Disknownasthe'electric displacement', andequation (1.18)isvalid eveninananisotropic medium wherePisnotnecessarily paralleltoE. WemaynowwriteGauss'stheorem, inadielectric medium, intheform fD.dS=fdivDdT=fpdT. (1.19) Sincethevolumeintegrals mustbeequaloveranyarbitrary volume,it followsthattheirintegrands mustbeequal,i.e. divD=p, (1.20) whichisthedifferential formofGauss'stheorem ..Itiseasytoseethat theseequations reducetothoseof§1.3invacuo,whereP=O.An 1.5] ELECTROSTATICS I 19 alternative wayofderiving equation (1.20)directlyforthespecialcase ofCartesian coordinates isgiveninAppendix A. Inanisotropic dielectric, PisparalleltoE,andhencesoalsoisD. SinceP=XEOE,D=EO(1+X)E,andifwewrite E=I+X (1.21) wehave D=EEOE, (1.22) where Eisthe'dielectric constant' andXisthesusceptibility; forvacuum (orair,formostpurposes), X=0,andE=1.TheratioofDtoEis knownasthe'permittivity'; infreespaceDIE=EO'andthus EOisthe 'permittivity offreespace'.Whenamedium ispresentDIEisincreased bythefactor E,knownalsoasthe'relative permittivity'. Ifwehaveasinglepointchargeqinauniform dielectric ofconstant E, wemayapplyGauss'stheorem overasphereofradiusrwithcentreatq. Thenthesurfaceintegralreducesto471'r2D=471'r2EEoE=q,whence qrEEoE=D=--.471'r3(1.23) Itfollowsthattheforcebetween twochargesq1'q2adistancerapartis F=q1q2r %EEor3 andthepotential atadistance rfromapointchargeis v=-q-. 471'EEor(1.24) (1.25) Someproperties ofDandE Itisimportant todistinguish clearlybetweenthetwovectorquantities electricfieldEandelectricdisplacement D.Eisdefinedastheforce actingonunitcharge,irrespective ofwhether adielectric medium is presentornot.Thisdefinition isexpressed inequation (1.3).Thedis­ placement Disdefinedbyequation (1.18).Thequantity D.dSissome­ timesknownastheelectricfluxthroughtheelementofareadS.From (1.19)thetotalfluxisqthroughanareasurrounding achargeq,and thisfluxisunaltered bythepresence ofadielectric medium. Theunit offluxisthecoulomb, andtheunitofDiscoulomb/metre2• SinceDisavectorwemaydrawlinesofdisplacement analogous to linesofforce,suchthatthenumberpassingthroughunitareaisequal tothedisplacement. AlsobyGauss'stheorem linesofdisplacement are continuous inaspacecontaining nofreecharges; theybeginandend onlyonfreecharges. Attheboundary oftwodielectrics E1andE2,ifno 20 ELECTROSTATICS I [1.5 freechargeresidesthere,linesofDarecontinuous butlinesofEare not,because linesofforceendonbothfreeandpolarization charges, whereas linesofDendonlyonfreecharges. Thereisapolarization chargeonthesurfaceseparating twodielectrics, sincetheinduced moment perunitvolumeisdifferent inthetwomedia.LinesofEbegin .. -~-~--~- 2Bt FIG.1.12.Boundary conditions atthesurfacebetween twodielectrics. andendonthissurfacecharge,butnotlinesofD.Therulesgoverning thebehaviour ofEandDatthesurfaceofadielectric, ortheboundary between twodielectrics, areembodied intwo'boundary conditions', whichwillnowbederived. Tofindtheboundary condition forD,weapplyGauss'stheorem to asmallcylinder whichintersects theboundary, asinFig.1.12,and whoseaxisisnormaltotheboundary.Iftheheightofthecylinder is verysmallcompared withitscross-sectional area,theonlycontribution tofD.dSoveritssurfacewillcomefromthecomponents ofDnormal totheboundary. Sincethereisnofreechargeontheboundary, fD.dS=0,andhence D-D (126)1n-2n' • wherethesymbols refertothenormalcomponents ofDonthetwo sidesoftheboundary. Theboundary condition forEisfoundbyconsidering theworkdone intakingunitchargeroundasmallrectangular circuitsuchasABODA inFig.1.12.IfthesidesBO,ADareverysmallcompared withAB,OD, thentheworkdonewillbelhEt-2Et) =O.Hence lEt=2Et' (1.27) wherethesymbols refertothetangential components ofEoneither sideoftheboundary. Thusequations (1.26)and(1.27)areourtwo 1.5] ELECTROSTA'l'ICS I 21 fundamental boundary conditions. Itfollowsfromthemthatlinesof Dwillingeneralberefracted attheboundary between twodielectrics. Atypicalexample isshowninFig.1.13. FIG.1.13.Refraction oflinesofdisplacement attheboundary between twodielectrics (£a>£1).Fromequations (1.26)and(1.27)wehave D1cos81=Dacos8a• E1sin81={D1/£1£o)sin81=Easin8a={Da/£a£o)sin8a• Hence £1cot81=£acot8a. Similarboundary conditions maybeappliedatthesurfaceofacon­ ductor.Sincetherecanbenofieldinsidetheconductor, thetangential component ofEjustoutsidetheconductor mustalsobezero,andany fieldatthesurfacemustbenormaltothesurface.IfweapplyGauss's theoremtoanelementary cylinderintersecting thesurface,asinFig.1.12 wehaveIDndS=adS(since.JJn =0withintheconductor), whereais thechargedensityontheconducting surface. SinceIDmustbenormal tothesurface,wehaveID=IDn'andhence(dropping thesubscripts) D=EEOE=a (1.28) atthesurfaceoftheconductor immersed inamedium ofdielectric constant E. 1.6.Properties ofcapacitors andsystems ofconductors IfachargeQisgiventoanisolatedconductor itsvoltageisincreased byanamountV.Foragivenconductor theratioQjVisindependent 22 ELECTROSTATICS I [1.6 ofQanddepends onlyonthesizeandshapeoftheconductor. The ratioQjViscalledthecapacitance oftheconductor, andisdenotedbyO. Ifasecondconductor whichisearthedisbrought closetothefirstone, achargeofopposite signisinducedonit,andthepotential falls.Since Qisconstant iftheconductor isisolated, thecapacitance hasincreased. Thetwoconductors together formacapacitor, andthecapacitance of thecapacitor isdefinedastheratioofthechargeoneitherconductor to thepotential difference between them.Acapacitor isaninstrument for storingcharge,andacapacitor oflargecapacitance canstoreacorre­ spondingly largequantity ofchargeforagivenpotential difference between theplates.Thecapacitance depends onthegeometry ofthe conductors andthedielectric constant ofthemedium separating them. Ingeneral,calculation ofthecapacitance ofaconductor oracapacitor isdifficultunlesssimplegeometrical shapesareinvolved. Theprinciple ofthecalculation maybeillustrated bythecaseofanisolatedsphere, ofradiusa,inaninfinitedielectric. Suppose thiscarriesachargeQ. Thenbyapplying Gauss'stheorem overaspherical surfaceofradiusr, concentric withthesphere,wehave 47rr2D=47rr2EEoE=Q, since,bysymmetry, DandEareconstant overthespherical surface andeverywhere normaltoit.Hence QE= , 47rEEor2 whichisthesameasequation (1.23)forapointchargeQ.Thepotential ofthesphereis a V=-fEdr=-Q-. 47rEEoa 00 (1.30) andthecapacitance isHence 0=QjV=,47rEEoa. (1.29) Ifinsteadofanisolated spherewehaveacapacitor formedbytwo concentric spheresasinFig.1.14ofradiia,b(b>a),wemayplacea chargeQontheinnersphereandacharge-Qontheoutersphere. Thentheformula givenaboveforEholdsinthedjelectric-filled space between thespheres, whileeverywhere elseEiszero.Hencethepoten­ tialdifference between thetwospheresis V=47r~EO(~-~) 0=47rEEoabj(b-a). 1.6] ELECTROSTATICS I 23 Another simpletypeofcapacitor isformedbytwoplaneparallel platesofarea8andseparation t.Ifthelateraldimensions oftheplates arelargecompared withtheirseparation (oriftheplatesaresurrounded by'guardrings'atthesamepotential), thenthefieldbetween themis -Q FIG.1.14.Aspherical capacitor. uniformandnormaltotheplanes,beinggivenbyequation (1.28)with a=Q/8.Sincethefieldisuniform, thepotential difference between theplatesissimplyV=Et,andthecapacitance isthus 0=Q/V=€Eo8/t: (1.31) Thisequation (andalsoequations (1.29)and(1.30))showsthatthe capacitance increases byafactor Eifthespacebetween theplatesis filledwithamedium ofdielectric constant E.Thisagreeswiththe originaldefinition ofdielectric constant byFaraday, mentioned atthe beginning of§1.5. Theunitofcapacitance iscalledthefarad(F);theplatesofacapacitor of1 Fcarryachargeof1coulombiftheirpotential difference is1V. Reference toequations (1.29)to(1.31)above,ormorefundamentally, toequation (1.25),showsthatacapacitance hasthedimensions ofEO multiplied byalength;hencetheunitofEOisthefarad/metre. The faradisaverylargeunit(aspherethesizeoftheearthwouldhave acapacitance of'about10-3F),andthesubdivisions microfarad (fkF)=10-6F,andmicromicrofarad (fkfkForpF)=10-12Farecom­ monlyusedinstead. Aresultoftenrequired isthenetcapacitance ofanumberofcapaci­ torsjoinedeitherinseries,orinparallel, asinFig.1.15.Ifncapacitors 01>O2,•••,Onarejoinedinseries,andavoltageVappliedacrossthem, 24 ELECTROSTATICS I [1.6 acharge+QappearsontheplateAand-QontheplateB.Theplate 2ofthefirstcapacitor willhavecharge-Q,andplate1ofthesecond capacitor musttherefore havecharge+Qsincethetwo,thoughcon­ nectedtogether, areotherwise isolated andtheirtotalchargemust remainzeroonconnecting thebattery. Thuseachcapacitor carries J- (a) (b) FIG.1.15.(a)Capacitors inseries.(b)Capacitors inparallel. thesamecharge,irrespective ofitssize,andthepotential acrossthe n kthcapacitor isQ/Ok'Hencethetotalpotential isL(Q/Ok)'andthis 1 equalsQ/O,where°isthenetcapacitance. Equating thesetworesults gives I I I I I°=01+O2+...+Ok+...+On' (1.32) Ifthecapacitors arejoinedinparallel, thevoltage acrosseach capacitor isequaltoV.Thetotalchargecarriedbyallthecapacitors isQ=Q1+Q2+ ...+Qk+...+Qn=V(01+02+ ...+0k+··.+On) =va, where°isthenetcapacitance. Hence (1.33) Thepotential energyofasystemofchargesandchargedconductors Asystemofelectricchargespossesses potential energy,sincework mustbedoneinbringing upanyparticular chargethroughtheelectro­ staticfieldoftheremaining charges. Theenergydepends onlyonthe finalstateofthesystemandnotonhowthechargesareestablished. Wemaytherefore supposethateachchargeisincreased fromzeroto itsfinalvalueininfinitesimal stepssothatatanygiveninstanteach chargeisOI.qk'whereqkisitsfinalvalueand 01.isanumber lessthan 1.6] ELECTROSTATICS I 25 (1.35)(1.34) tfpVdT=tfdiv(VD)dT-tfD.gradVdT =!fVD.dS+tfD.EdT.unitywhichisthesameforallcharges. ThenifJkisthefinalvalueof thepotential atthepointoccupied byqk'theinstantaneous valueofthe potential willbeo:Jk,andtheworkdoneinincreasing thechargeby qkdo:willbe(o:Jk)(qkdo:).Thustheworkdoneinincreasing allthecharges byacorresponding amountwillbe0:do:!qkJk.Thetotalworkdone k isequaltothestoredenergy,whichwilltherefore be 1 U=!qkJkf0:do:=t!qkJk. k 0 k Wemayapplythisresulttoacapacitor withtwoplatesatpotentials ~,~carrying charges+Qand-Qrespectively. Theenergyofthe capacitor willbe U=tQ~-IQ~ =tQV=tOV2=IQ2jO, whereV=~-~isthepotential difference between theplates. Theenergyofthesystemmaybeexpressed inadifferent waywhich impliesthatitisdistributed overthespacebetweenthechargesoccupied bytheirelectrostatic field.Consider twonearbyequipotential surfaces inthisspacewhichdifferinpotential byasmallamountV,andarea distance dsapart.Iftwoparallelconducting platesofareadSwere inserted soastocoincide withtheseequipotentials, theywouldnot alterthefielddistribution inanyway.Theywouldformaparallelplate capacitor ofcapacity°andenergytOV2.But0=€€odSjds and V=-Eds,whereEistheelectricfieldatthispoint.Thecapacitor occupies avolumedT=dSdsanditsenergyis!€€oE2(dSds) =IDEdT. Wemaytherefore regardtheenergyasdistributed throughout thefield, theenergydensityatanypointbeing!DE.Thisequation maybe derivedmorerigorously byvectoranalysis, asfollows. InFig.1.16suppose thereexistsavolumedistribution ofchargeof density pperunitvolumeandasurfacedistribution ofdensity aper unitarea.Thenfromequation (1.34),ifthesummation isreplaced by integrations, wehaveforthetotalenergy U=!fpVdT+!faVdS, wherethesurfaceintegralistakenoverthesurfacesofalltheconduc­ torspresent. ByGauss'stheorem, p=divD,andhence,usingavector transformation (seeAppendix A),pV=VdivD=div(VD)-D.grad V. Therefore 26 ELECTROSTATICS I [1.6 Thefirstintegral mustbetakenoveraclosedsurfacebounding the wholevolume,andalsooverthesurfaceofeachconductor. Thefirst surfacemaybetakenataninfinitely largedistance fromallthecharges, anditscontribution tothesurfaceintegralthenvanishes. For,atlarge distances, Vvariesasr-1andDasr-2,whiledSincreases withr2;thus p D FIG.1.16.Diagram toillustrate thecalculation oftheenergydensity ofasystemofsurfaceandvolumecharges. theintegralisproportional tor-1andtendstozeroasrtendstoinfinity. Thetotalenergymaynowbewritten U=!ID.EdT+!IVD.dS+!IaVdS, wherethesurfaceintegrals aretakenoverthesurfaces ofallthecon­ ductors. Sincetheintegration isoverthesurfaceofthemedium, dSis avectordrawnoutwards fromthemedium andhenceintothecon­ ductingsurfaceasinFig.1.16.Butthenormalcomponent ofDinthis direction is-a,fromequation (1.28),andhenceD.dS=-adS,so thatthetwosurfaceintegrals inourexpression forUcancel.Ourfinal expression forUbecomes U=tID.EdT. (1.36) SinceEiszerowithinanyconductor, wemayregardtheenergyas distributed throughout thesurrounding dielectric medium, withdensity !D.E.Thisexpression isvalidinanisotropic dielectrics, whereDis notnecessarily paralleltoE,butitassumesthatDisalwayspropor­ tionaltoE. 1.7.Stressintheelectrostatic field Ithasalreadybeenshownthatthechargeonaconductor residesin athinsurfacelayer.Thisisduetothemutualrepulsion between charges oflikesign,sothateachportionofthechargeontheconductor is 1.7] ELECTROSTATICS I 27 (1.37)tryingtogetasfarawayaspossiblefromtheremainder. Thisresultsin atensionactingonthesurfaceoftheconductor, whosemagnitude will nowbecalculated. Weshallassumethatthechargeofsurfacedensity cr isinathinlayerjustoutsidetheconducting surface,inamedium of dielectric constant 10,asinFig.1.17.Byapplying Gauss'stheorem to asmallcylinder withitsaxisnormaltothesurface,andwithoneend insidetheconductor andtheotherwithinthesurfacelayer,thefieldE' i Elementary cylinder FIG.1.17.Deduction ofthetensiononacharged conductor. atthislatterpointisfoundtobeE'=(crex)j€€o,wherecrexistheportion ofthesurfacechargedensitylyingbetween theconductor andtheend ofthecylinder. Theforceontheelementofchargedensity crdexatthe endofthecylinder istherefore E'(crdex) =cr2exdexj€€o,andthetotalforce perunitareais 1 cr2f cr2 T= - exdex=-. 10100 210100o Atfirstsighttheassumption thatthechargelayerresidesinthe dielectric mayappearratherartificial, butthesameresultmaybe obtained byapplication oftheprinciple ofvirtualworktospecialcases. Forexample, consider aparallelplatecapacitor withamedium of dielectric constant 10between theplates.Iftheseparation between the platesisx,andtheirareaS,thecapacitance°=10100Sjx,andthestored energyisiQ2jO=!cr2S2jO,wherecristhechargedensityontheplates. Ifcriskeptconstant, andtheseparation oftheplatesisincreased, the rateofchangeofthestoredenergyis ddU=dd(cr2Sxj2€€o) =u2S•x X 210100 ) 28 ELECTROST ATICS I [1.7 But(dU/dx)=-ST,wherel'isthetensionperunitareaontheplate, andhenceT=-a2/2EEo, wheretheminussigndenotesthatthetension actsintheopposite direction tothemovement oftheoneplate,i.e.in thedirection ofdiminishing x. Itisinteresting toderivethetensionontheplatesofthiscapacitor ifthevoltage,ratherthanthecharge,iskeptconstant. Inthiscasewe mustincludetheworkdonebythebatterymaintaining theconstant potential difference.Ifthecapacitance increases bydO,thechargein­ creasesbyVdO,andtheworkdonebythebatteryisV(VdO)=V2dO. TheincreaseinthestoredenergyisdU=d(iOV2)=!V2dO,andthe external workdWrequired istherefore iV2dO-V2dO=-iV2dO=-dU. Hencethetensionontheplatesis =_~(_~2)(_E~21= _EEo2E2= -2::~' whichisthesameasbefore,asweshouldexpect. Thefactthatthe workdonebythebatteryinmaintaining thesystematconstant poten­ tialisjusttwicetheincreaseinthestoredenergyisgenerally true,and ourexample isjustaparticular case.Itis,however, probably more instructive forthestudenttoremember toputintheworkdonebythe batteryinworking aparticular problematconstant potential rather thanavoiding theissuebymakinguseofageneraltheorem. Stressesindielectric media BothFaraday andMaxwell usedtheconceptoftubesofforce.Atube offorcecontains anarbitrary butverylargenumberofJinesofforce, andthenumber oftubescrossing unitareaisequaltotheelectric intensity; similarly, thenumberoftubesofdisplacement perunitarea isequaltoD.Theyimagined thesetubestobeinastateoftension, sothattheforceofattraction between twochargesofopposite sign, forexample, wastransmitted alongthetubesofforce.Itwasalso necessary tostipulate thattherewasaforceofrepulsion between tubes offorceinadirection normaltotheirlength,otherwise thetubeswould allcontract untiltheypassedstraight fromone charge toanother. Theseforcescanbeexpressed intermsofthe'Maxwell StressTensor', andweshallquotetheresults(acomplete treatment isgiveninPanofsky 1.7] ELECTROSTATICS I 29 andPhillips, 1955).Thex-component dFxoftheforcedFtransmitted acrossasurfaceelementdSis dFx=TxxdSx+TxydSy+TxzdSz (1.38) withsimilarequations fordFy,d~.Thequantities Txx' Txy,etc.,form theninecomponents ofatensorT,whichcanbewrittenas (!(E",D",-EyDy-E.D.) E",Dy E",D.) T= EyD", l(EyDy-E.D.-E",D",) EyD. • EzDI1J EZDy !(E.D.-E",D",-EyD y) (1.39) y Exy .to Ex (lAO)FIG.1.18.Asurfaceelement dS,anditsstresscomponents. Fromequation (1.40) thesearedFI1J=T",,,,dSI1J=tEDdScosO,dFJI=TYlldSlI=-tEDdSsinO,sothat dF=(dFi+dFi)t=lEDdS. Thistensorissymmetric (thatis,Txy=Tyx'etc.),andbychoosing a specialsetofaxes,theoff-diagonal termscanbemadezero.Ifwechoose thex-axistobeparalleltoE,Ttakestheform (tED00)T= 0-tED 0 o 0-tED whosesignificance isthatwehaveaforcecomponent =+tED(dSx) paralleltoE,andcomponents -tED(dSy), -tED(dSz) normaltoE. TheseareshowninFig.1.18forasurfaceelement forwhichdSis normaltothez-axis,andinFig.1.19forthespecialcaseswhereEis normalandparalleltothesurface. Itmustberealizedthatthesestressesmustberegarded aspresent inthefieldirrespective ofwhetherdSisanelementofarealboundary ornot.IfdSisnotarealboundary, therewillbeequalandopposite stressesontheotherside,sothatequilibrium ismaintained. If,how­ ever,dSispartofarealboundary, andthevectorsE,Daredifferent 30 ELEOTROSTATIOS I [1.7 onthetwosides,therewillbeanetforceactingatthebounding surface. Thecharged conductor considered aboveisaspecialcasewhereE,D arezeroononeside,andnormaltotheboundary ontheother,andit iseasilyseenthatthetensiongivenbyequation (1.37)agreeswiththat givenbytheMaxwell StressTensor. Thetensorrepresentation isof coursemoregeneral, andmakesitpossible tocompute thestresson f---------- ......E 1- +T=!E.D (a)E 1 ~-------_T =iE.D (b) FIG.1.19.Thestressatasurfacewhichis(a)normaland(b)paralleltoafieldE. adielectric boundary. Thisisequaltothedifference t1ToftheMaxwell StressTensorsonthetwosidesoftheboundary, anditiseasytoshow thattheresultant forceisalwaysnormaltotheboundary. For,ifthe latteristakentobenormaltothez-axis,sothattheonlycomponent ofdSisdSz'thenwehave dFx=(t1Txz)dSz=hExl~-2Ex2~)dSz =0, dFy=(t1Tyz)dSz=(lEyl~-2Ey2~)dSz =0, whichbothvanishbecausetheboundary conditions makeEx,Ey,~ continuous acrosstheboundary. REFERENCES PANOFSKY, W.K.H.,andPHILLIPS, M.,1955,OlassicalElectricity andMagnetism (Addison-Wesley Publishing Co.,U.S.A.). PLIMPTON, S.J.,andLAWTON, W.E.,1936,Phys.Rev.50,106H. PROBLEMS 1.1.Anelectricquadrupole isformedbyacharge-2eattheoriginandcharges +eatthepoints(±a,0, 0).Showthatthepotential Vatadistanco rlargecom­ paredwithaisapproximately givenbyV=+ea2(3cos28-1)f41TEor3,where8is theanglebetween randthelinethrough thecharges. ELECTROSTATICS I 31 1.2.Showthatthereisnotranslational forceorcoupleonsuchanelectricquadru­ poleinauniform field.Provethatthecoupleonthequadrupole atadistance 'I" fromapointchargeqisa=3eassin2!.q,where8istheanglebetween 'I"andthe 47rE"o'l" linethrough thecharges. (Assume r~a.) 1.3.Showthattheforceonanelementary dipoleofmoment p,distance rfrom apointchargeq,hascomponents F.__qpcos8 Fe=qpsin8 r-27rE"or3' 47rE"or3 alongandperpendicular torintheplaneofpandr,where8istheanglewhich pmakeswithr. Ar::PS3(sin81sin8s-2cos81cos8s), ~£IE"or where81,8saretheanglesmadebyPIandPsrespectively withthelinejoining theircentres.1.4.Showthatthepotential energyoftwocoplanar dipolesPIandPsadistance r apartis 1.5.Achargeqisplacedateachofthefourcorners(±a,0,0),(0,±a,0)ofa square. Showthatthepotential atapoint(x,y,z)neartheoriginis V=A_q[4+(xs+yS_2zS)/as+...]. ~IIE"Oa Verifythatachargeofthesamesignplacedatthecentreofthesquareisinstable equilibrium againstasmalldisplacement intheplaneofthesquare,butisunstable foradisplacement normaltothisplane.Thisisanexample ofEarnshaw 'stheorem, whichshowsthatachargecannotrestinstableequilibrium inanelectrostatic field. 1.6.Thevaluesofthevertical potential gradient oftheearthatheightsof100 and1000metresaboveitssurfacearenoand25V/metrerespectively. Whatis themeanelectrostatic chargepercubicmetreoftheatmosphere between these heights? (Answer: 0·835X10-12coulombs/metre3.) 1.7.Aparallelplatecapacitor withplatesofareaSandseparation dhasablock ofdielectric, ofconstant E",ofcross-sectional areaS,andthickness t(t<d), inserted inbetween theplates.FindthevaluesofEandDinthespacebetween theplates,inbothairanddielectric. Showthatthecapacitance ofthecapacitor is 0=d~E"E"O1),andcalculate thechangeinstoredenergyofthesystemwhen E"-E"-t thedielectric isinserted (a)iftheplateshaveaconstant chargeQ,and(b)ifthey areconnected toabatteryataconstant potential V. (QS(E"-I)t T E"oSV2(E"-I)t) Answer: (a)t:..U= 2E"E"oS; (b)t:..V=+2d{E"d-(E"-I)t}' 1.8.Findanexpression forthecapacitance perunitlengthofacylindrical capacitor consisting oftwoconcentric cylinders, radiiaandb,separated byamediumof dielectric constant E".(Neglect edgeeffects.) (Answer:a=27TE"E"o/log(b/a).) 32 ELECTROSTATICS I 1.9.Acapacitor isformedbytwocoaxialcylinders ofradiiaandb.Theaxes ofthecylinders areverticalandtheinnercylinder issuspended fromabalance sothatithangsonlypartlywithintheoutercylinder. Findanexpression forthe masswhichmustbeaddedtotheotherpanofthebalancetomaintain equilibrium whenavoltageVisconnected between thetwocylinders. (Answer: mg=WEEoV2flog(bja).) 1.10.Theelectrometer isaninstrument formeasuring voltages bymeansofthe forceonacharged conductor. Anattracted diskelectrometer hasamovingplate ofarea100cm2,separated byadistance of1romfromthefixedplate.Calculate theforcebetween theplateswhenthepotential difference acrossthemis100V. Calculate thesensitivity atthisvoltageinnewtons pervolt. (Answer: F=4·42X10-4newtons, dFjdV=8'85X10-6newtonjV.) 1.11.Theupperdiskofsuchanelectrometer issuspended byaspring.Inequili­ brium,theseparation between thetwodisksisxwhenavoltageVisappliedand awhenV=O.Showthattheequilibrium intheformercaseisstableprovided thatx>2aj3. 1.12.Aspherecarrying achargedensity aperunitareaisimmersed inaninfinite dielectric medium. Verifyequation (1.37)byusingtheprinciple ofvirtualwork andallowing theradiustochangeinfinitesimally. 1.13.Assuming thatthetotalchargeZeofanatomicnucleusisuniformly distri. butedwithinasphereofradiusa,showthatthepotential atadistancerfromthe centre(r<;a)is Showthattheelectrostatic energyofsuchanucleus is U=3(Ze)2. 20wEoa Thiselectrostatic energymustbeprovided attheexpenseofasmalldecrease3m inthemassofthenucleus, suchthatU=omc2,wherec=velocity oflight,by theEinstein relation.omispartofthe'massdefect'. 1.14.Anatomwithanelectron inans-statehasafinitedensity -pofelectronic chargeinsidethenucleus. Usingtheformula forthepotential insidethenucleus givenintheprevious problem, showthatthepotential energyassociated with theelectron density-pinsidethesphereofradiusais-2Zepa2!OEO'andthatthis isgreaterbyZepa2!10EOthanitwouldhavebeenifthenucleuswereapointcharge. Sinceisotopes ofthesameelement havedifferent nuclear radii,thisenergy formspartofthe'isotopeshift';thatis,thedifference infrequency ofspectrum linesfromdifferent isotopes. 2 ELECTROSTATICS II 2.1.Theequations ofPoissonandLaplace INaregionwherethereexistsachargedistribution ofdensity pperunit volume,thedifferential formofGauss'stheorem is(equation (1.20)) divD=p. NowE=-gradV,andinanisotropic dielectric D=€€oE.Hence V2V=div(grad V)=-divE=-p/€€o. (2.1) ThisisknownasPoisson's equation. Ifthereisnofreechargepresent, p=0,andwehaveLaplace's equation (2.2) Theoperator denotedbyV2isascalaroperator, whichhasitssimplest forminCartesian coordinates, wherePoisson's equation becomes (2.3) (2.6)(2.4)Twoothercoordinate systems willbeconsidered. Thesearespherical polarcoordinates, wherePoisson's equation becomes 18(28!J 18(.8!J 182V_L f28rrfir)+r2sin888sm887f}+r2sin288ep2=€€o' andcylindrical polarcoordinates, wherewehave ~;(r~B+~~:::+~~= -~o· (2.5) Inprinciple, equation (2.1)enablesustocalculate thepotential distri­ butionduetoanygivensetofchargesandconductors. Aformalsolu­ tionofPoisson's equation canbefound, V-IpdT-4rr€€or' butthisholdsonlyinavacuum oraninfinitedielectric medium.If thereareconductors presentweshouldhavetoallowfortheeffectof thechargedistribution ontheirsurfaces, butwedonotingeneralknow whatthisdistribution is.Wehavetherefore toresorttoanumberof specialmethods, butwemustbesurethatanysolution weobtainwhich 851110 D 34 ELECTROSTATICS II [2.1 satisfiestheboundary conditions isthecorrectandonlyanswer.That thisisthecaseisshownbyanimportant theorem, knownastheUnique­ nessTheorem. Thistheorem (seeAppendix B)showsthatiftwodifferent potential distributions areassumed tosatisfyLaplace's equation andthe boundary conditions, theirdifference iszero.Wenowdiscussanumber ofmethods forthecaseofnofreecharges, wherethesolutions needed areofLaplace's ratherthanPoisson's equation. Therequired solution maybeasumofanumberoffunctions, each ofwhichsatisfies Laplace's equation; for,ifthefunctionsli,~,...,~ areeachindividual solutions ofLaplace's equation, then V=alli+a2~+ ...+an~' whereal'a2,...,anareasetofnumerical coefficients, isalsoasolution. Aseriesoffunctions, eachofwhichisasolutionofLaplace's equation, maysometimes befoundbymakinguseofthefactthatifTiisasolution, soalsoareanydifferentials ofliwithrespecttothespacecoordinates. ThusinCartesian coordinates thefunctionsoli/ox,oTi/oy,aTiloz,o2li/ox2 o2li/oxoy, etc.,allsatisfyequation (2.2)iflidoes.Theproofofthiscan beseenfromasingleexample. Onpartialdifferentiation ofequation (2.3)withrespecttox,wehave(settingp-0) o=~{o2li+o2li+02li}oxox2oy2OZ2 =~(Oli) +~(oli) +~(oTi) =V2(oTi),ox2oxoy2oxOZ2ox.ox sincetheorderofdifferentiation isimmaterial whenx,y,zareindepen­ dentcoordinates. Thevalueofthismethodliesinthefactthatonce aseriesoffunctions whichsatisfyequation (2.2)isestablished, anylinear combination ofthesefunctions maybetaken,andiftheycanbechosen insuchawayastosatisfytheboundary conditions byadjustment of thecoefficients, theygivetheuniquesolutiontotheproblem. Intheory,anyproblem involving electrostatic fieldsmaybesolved byfindingasolution whichsatisfiesequation (2.2)andgivestheright boundary conditions. Inpractice theproblem isalmostinsoluble by ordinary mathematical methods exceptincaseswherethereisahigh degreeofsymmetry. Thesemaybehandledbytheuseofaseriesof knownfunctions, andsomeexamples ofthismethodaregivenbelow. Weshallconsider alsoanotherspecialmethodwhichcanbeappliedto thecaseofoneortwopointchargesneartoaconducting surfaceof simpleshape.Thoughanumberofotherproblems maybehandledby• 2.1] ELECTROSTATICS II 35 (2.7)mathematical methods whicharebeyondthescopeofthisvolume(see thegeneralreferences attheendofChapter 1),mostoftheproblems metwithinpractice, suchasthedesignofelectrongunstogiveafocused beaminacathoderaytube,aredealtwitheitherbyuseofapproximate solutions, orbyplottingthelinesofequalpotential usingascalemodel asdescribed inChapter 3. 2.2.Solutions ofLaplace's equation inspherical coordinates Weshallconsider firstthecaseofspherical coordinates, andassume initiallythatwehavesymmetry aboutthepolaraxissothatVisinde­ pendentof4>.ThenLaplace's equation reducesto ~(r2oD+~{(1_ 2)OV}=0,oror}Of-' f-'Of-' wheref-'hasbeenwrittenforcos8.Thishassolutions oftheform V=rl.Pz,where.Pzisafunction off-'=cos8only,andlisaninteger. Ifwesubstitute suchafunction inequation (2.7),anddividethrough byrl,weobtainLe-,'R"~ ., •fo.Pz ~{(l-f-'2)~}+l(l+1).Pz =o. Itisreadilyseenthatreplacing lby-(l+l) leavesthisequation un­ altered,sothat.Pz=p_(l+l);thatis,V=rlp"andV=r-(l+l).Pzareboth solutions ofequation (2.7). Solutions ofLegendre's equation maybeobtained bystandard methods, butaquickalternative methodisasfollows. Weknowthat V=Ijrisasolution, andhencesoisanypartialderivative ofthissuch as(oVjoz)undertheconditions x,yconstant. Sincer2=X2+y2+Z2, wehave2r(orjoz) =2zwhenx,yarekeptconstant, sothat (::t,u=~. Hence -(~)(~)=\(or)=za=r-2cos8oZx,ur rozx,u r ifwetakeztoliealongthepolaraxis,sothatz=rcos8.Thetwo functions V=r-landr-2cos8arethefirsttwotypesofsolution in theinversepowers r-<l+l),andcorrespond tovaluesofl=0and1 respectively. ThusPo=1,andPl=cos8.Further functions maybe generated bysuccessive differentiation; thus (ojoz)(zjr) =(1-3z2jr2)r-a=(1-3cos28)r-a1 j I 36 ELECTROSTATICS II [2.2 givesthenextfunction, whichisproportional toP2•Ageneralformula for.Pzis Pz=2:ll(:fJ-Y(fJ-2 -1)l, (2.8) wherethenumerical coefficients aresuchthatPz=1at11,=I,i.e.at (J=O.Thefirstfewfunctions aregiveninTable2.1,together with theradialfunctions r-(l+!)andrwithwhichtheycombine togive solu­ tionsofLaplace's equation. TABLE2.1 Somespherical harmonic functions Legendre function Function ofr Po=1 P1=C080 P,=t(3c08'0-1) P3=t(5C0830-3C080) P4=1(35C0840-30c08'0+3) P5=1(63C0850-70c0830+15) P6=1\-(231c0860-315c0840+105c08'0-5)r-11 ,,-'"r-3r" r-4,r3 1'-5r4 ,.-6r5 ,.-7 rG (2.9)Associated Legendre functions Table2.1clearlydoesnotcontainallpossible solutions ofLaplace's equation, sincewecanfindothersbydifferentiation withrespectto x,y.Forexample, (0)(Z)3Z(or) 3zx .- - :::ii=""4-=--:::ii=3r-3cos(JsmecoscpoxY,z1-roxy,z T- mustalsobeasolution. Thefactthatitcontains cpshowsthatitisnot asolution ofequation (2.7),butofthemoregeneralequation which includesthedependence oncp.Thisis !.-(r20D ~{(I_ 2)oV}102V_oror)+ofJ-fJ-ofJ-+(1-fJ-2)Ocp2-0, wherewehaveagainwritten fJ-forcos(J.Asbefore,weassumethat thereexistsasolutionoftheformV=rle<l>,where0,<Darefunctions onlyof(J,cprespectively. Thenthedifferential equation fore<I>is ~{(I-fJ-2)0(0~}+l(l+I)(0<D)+_I_ 02(0<D),=0,(2.10) ~ ~ l-~~ whichcanbewrittenintheform (2.11) 2.2] ELECTROSTATICS II 37 wherethevariables areseparated. Theright-hand sidehasthesolution 4>m~(21T)-teimc/> (2.12) andtheequation for8becomes !.-{(1-JL2)80}+{l(l+1)_~}8 =O. (2.13)oJL oJL I-JL2 Itisapparent thatthefunctions listedinTable2.1aresolutions ofthis equation forthespecialcasem=0,wherethereisnodependence oncp. Thesolutions ofequation (2.13)are -Prm=Sinmo(!.-)m P,=(I_JL2)lm(!.-)m Ii'of' oJL _(I-JL2)lm(a)m+l(2_1)1(2.14) -21l!oJL JL• Thefunctions 4>definedbyequation (2.12)are'normalized'; thatis 211' 211'f4>;4>mdcp=f(21T)-le-imc/>eimc/>dep=I (2.15) o 0 andtheyarealso'orthogonal'; thatis(m'=1=m) 211' 211'f4>~.4>mdep=f(21T)-le-im'c/>eimc/>dep=o. o 0(2.16) ByusingtheKronecker 8,whoseproperties arethat8(m',m)=Iif m'=m,but8(m',m) =0ifm'=1=m,wecanwritethesetwoequa­ tionsintheshortform 211'f4>~·4>mdep =8(m',m). o(2.17) Thefunctions It,mareorthogonal butnotnormalized, andhenceit isoftenconvenient toworkintermsof'spherical harmonics' definedby where 8=(_I)m{(2l+1 Hl-1ml)!}tn .t' ----07m.L7m ~orm~," 2(l+lml)!" ~ 81,m=(-I)me ',lml form<O. Thefunctions Yz,marebothorthogonal andnormalized; thatis 211'11'ffYf.,m·Yz,msinOdOdep =8(l',l)8(m',m), o0(2.18) (2.19) (2.20) 38 ELECTROSTATICS II [2.2 wheretheintegration isoverthesolidangle41T.Herethesignificance ofthe3functions isthattheintegral iszerounlessbothl'=land m'=m,inwhichcaseitisunity. Expressions forthefirstmembers oftheseriesofspherical harmonics arelistedinTable2.2(notethatthesignsofthefunctions usedby different authors sometimes differ;wehavefollowed thedefinition adopted byRamsey (1956)andBrinkandSatchler (1962)). Ageneral proofoftheorthogonality andnormalization relations istedious,but thereadermayverifythattheyarecorrectbyevaluating theintegrals forsomeofthefunctions giveninTable2.2. TABLE2.2 Somespherical harmonic functions Y10+(3/417)tcos8 010+cos8 Yl+1-(3/817)tsin8e+i</> 01+1-2-tsin8e+i¢ Y1-1+(3/817)tsin8e-i¢ °1-1+2-tsin8e-i¢ Y,o +(5/1617)t (3cos'8-1)°'0+t(3cos'8-1) Y'H-(15/817)t cos8sin8e+i</>°'+1-(3/2)tcos8sinfJe+i</> Y'-l +(15/81T)t cos8sin8e-i</>°'-1+(3/2)t COSfJsinfJe-1</> Y,+, +(15/3217)t sin'8e+i2</>°2+1+(3/8)tsin2fJe+;2¢ Y,-, +(15/3217)t sin'8e-12</> OS-I +(3/8)tsin2fJe-i2¢ Yso+(7/161T)t(5 coss8-3 cos8) °so+t(5coss8-3 cosfJ) Y2+1 -(21/647T)t(5 cos'8-1)sin 8e+i4> °S+l-(3/16)t(5 cos2fJ-l)sin 8e+i4> Ys-1+(21/6417)t(5 cos28-1)sin 8e-1</>°S-l +(3/16)t(5 cos2fJ-l)sin 8e-1</> YSH+(105/3217)t cos8sin'8e+i2</> °s+.+(15/8)t cosfJsin28e+i'¢ Ys-,+(105/3217)t cos8sin28e-1'</> OS-I+(15/8)t cosfJsin2fJe-i·</> Ys+s-(35/6417)t sins86+is</>°S+8-(5/16)t sin38e+i3¢ Ys-s+(35/641T)t sins8e-is</> °s-s+(5/16)tsin3fJe~i3¢ (417)tThefunctions 0lmarerelatedtoYlmby0lm=2l+1Ylm.Notethat010=Pl' Thespherical harmonics havemanyapplications, someofwhichare discussed inthenextsection. Fromtheatomicviewpoint, theirparti­ cularinterestisthatYi,mrepresents theangularvariation ofthewave­ function foranelectroninanatomwhichhasorbitalangularmomen­ tum,J{l(l+I)}(hj21T), andacomponent m(hj21T)ofangularmomentum alongthez-axis(thepolaraxis),wherehisPlanck's constant. 2.3.Thernultipole expansion Ifwehaveachargedistribution withdensity pinaregionwherethe potential Visvarying, thepotential energyis Up·JpVd'T. (2.21) 2.3] ELECTROSTATICS II 39 Ifthechargeextends onlyoverasmallvolume, wecanexpandVin aseries V="Vo+x(oV/ox)+y(oV/By)+z(oV/oz)+ +tX2(02V/ox2)+ly2(o2V/oy2)+lz2(o2V /OZ2)+ +lxy(02V/oxoY)+iYx(o2V /Byox)+etc. (2.22) Thepotential energythenbecomes Up=fp"VodT+fpx(oV/OX)dT+fpy(oV/Oy)dT+fpz(oV/oz)dT+etc. (2.23) Herethefirsttermissimply q"Vo,whereq=fPdTisthetotalcharge. SinceoV/ox=-Ex,etc.,wecanwritethenextthreetermsas Px(oV/ox)+plI(oV/oy)+Pll(OV/oz) =-(PxEx+plIElI+PllEll) =-p.E, (2.24) andbycomparison withequation (1.13)weidentify thequantities Px'PlI'Pllasthecomponents oftheelectricdipolemoment ofthecharge distribution. Thisgivesageneraldefinition ofthedipolemoment of achargedistribution, thecomponents being Px=fpxdT,Py=fpydT,Pll=fpZdT,orp=fprdT.(2.25) Ifwehaveanumberofpointchargesratherthanacontinuous distri­ bution,theintegrals canbereplaced byasummation (adipoleconsist­ ingoftwoequalandopposite chargesseparated byasmalldistance, asdefinedinChapter 1,isthusaspecialcase).Notethatinequation (2.24)wehaveimplicitly assumed thatthefirstdifferentials ofVare constant overtheregionoccupied bycharge,sinceonlythencanwe takethemoutsidetheintegration. Ifthechargedistribution hasreflection symmetry intheplanez=0, thatis,ifthechargedensity patthepoint(x,y,z)isthesameasthat atthepoint(x,y,-z),thecomponent Pllofthedipolemoment willbe zero,sinceintheintegralfpzdTthecontributions fromthepoints (x,y,z)and(x,y,-z)willbeequalandopposite. Thusinadiatomic molecule, anelectricdipolemoment canexistparalleltothelinejoining thetwonucleiiftheyaredifferent (i.e.ifthemolecule isheteronuclear, suchasHCI),butnotiftheyareidentical, asinahomonuclear molecule (H2,CI2)· Similarconsiderations applytoPx'PlIofcourse,andshowthata diatomic molecule canhavenoelectricdipolemoment perpendicular totheinternuclear axis.Themethodcanbeextended tomorecompli­ catedmolecules, suchasCHaCl,C2Ha,CaHa. 40 ELECTROSTATICS II [2.3 Inanatomornucleusthechargedistribution isexpected tohave reflection symmetry inthreemutually perpendicular planes,sothat therewillbenopermanent electricdipolemoment inanydirection. Notethatthreesuchreflections changethepoint(x,y,z)into (-x,-y,-z)andareequivalent toinversion through theorigin.The assumption wehavemadeaboutthechargedistribution isequivalent toassuming thatthesystemis'invariant undertheparityoperation', i.e.thatitsproperties areunaltered byinversion. o (a)B=(R,On,1'lJ) qn O""""--"'"------~--_+ (b) (2.26)FIG.2.1.Expansion ofthepotential atAduetoapointchargeqBatB(orofthepowntial atBduetochargeqAatA)usingspherical harmonics. In(b),thepointsA,Barenot necessarily intheplane'" =O. Expansion inspherical harmonics Similarconsiderations canbeappliedtothehighertermsinequa­ tion(2.23),butitisobviousthatthelargenumber oftermsmakes theexpansion inCartesian coordinates clumsytohandle. Wethere­ foreturntoanothermethodofexpanding thepotential ofapointcharge whichmakesuseofspherical harmonics. AssumethatwehaveachargeatthepointBandwishtoknowhow itspotential variesintheneighbourhood ofthepoint0,e.g.atthe pointA(Fig.2.1(a)).Thisrequires evaluation ofthequantity (R2-2Rrcos8+r 2)-t,whichistheinverseofthedistanceAB.Ifr<R, thisfunction canbeexpanded inpowersof(rIR): 1-R=(R2-2Rrcos8+r 2)-tI-rl 1r r2 il =R+R2P1+R3P2+...+Rl+l~+.... 2.3] ELECTROSTATICS II 41 Herethefunctions P1aretheLegendre functions definedbyequation (2.8),ascanreadilybeverifiedbydirectexpansion forthefirstterms. Amoregeneralformula canbefoundwherethepointsA,Bin spherical coordinates areat(r,()A'rPA)and(R,()B'rPB)respectively. In Fig.2.1(b),Ozisthepolaraxis,andthelinesOA,OB(whicharenot necessarily coplanar) makeangles ()A'()Bwithit;theangleAOBis denotedby()AB'ItcanbeshownthattheLegendre function .Pz(cos()AB) canbeexpressed intermsof()A'rPAand()B'rPBbytheformula 47T+1 .Pz(COS()AB) =2l+Im~-z (-l)lmIYi,m(()A,rPA)ll,-m(()B,rPB) +Z =!(-l)lmlG"m(()A,rPA)Oz,-m(()B,rPB)' (2.27) m=-Z wherethefunction G"m=(2~1)tYi,m (2.28) isalsolistedinTable2.2.Bytheuseofequation (2.27)thepotential atthepointAduetoachargeqBatBmaybewrittenas v-qB1 -47TEEOJR-rJ 00+Z rZ =i;,:E2:2:(_l)lmlRl+lOz,m(()A'rPA)Oz,-m(()B, rPB)' (2.29) °z=om=-Z ThisshowsthatifwetakeAasavariable point,thepotential atAdue tothechargeqBatBhasaseriesofcomponents, andthemagnitude ofthecomponent whichvariesasrOzm(8A'rPA)isdetermined bythe valueofR-<J+1lOz,_m(()B' rPB)atthepointB.Wemayequallywelltake Btobeavariable pointatwhichwewishtofindthepotential dueto achargeqAatA;thisisgivenbyequation (2.29)onreplacing qBbyqA' Thepotential atBthenhasaseriesofcomponents, wherethemagni­ tudeofthecomponent varyingasR-CZ+1lOz,_m(()B' rPB)isdetermined by thevalueofrOz,m(()A'rPA)atthepointA. IfwehaveanumberofpointchargesqBatlargedistances, themagni­ tudeofagivencomponent inthepotential atAnear0canbefound bythesummation! qBR-<J+llOz-m(8B'rPB)'anexampleofwhichisgiven B ' inProblem 2.18.Conversely, ifwehaveanumberofpointcharges qAcloseto0,themagnitude ofagivencomponent inthepotential atadistantpointsuchasBcanbefoundbythesummation !qArOzm(8A,rPA)'Ineithercasethesummation isreplaced byan A ' integration ifwehaveacontinuous distribution ofcharge.Notethat 42 ELECTROSTATICS II [2.3 wheretheseriesexpansion forthepotential doesnotassumethatr~R,but thetermswillonlyconverge rapidlyifthisisso. Theelectrostatic energyoftwocharges qA'qBatthepointsA,B maybewrittenbymeansofequation (2.29)as 1 00+Z Up=~II(-I)lml{qA rQ.m(OA,cPA)}{qB R-(l+llq._m(OB,cPB)}, ~EEOZ=O m=-Z (2.30) whichhastheadvantage thatthetermsinvolving thecoordinates of thetwochargeshavebeenseparated. Thusifwehavedistributed chargeswithdensities PA'PBatthepointsA,Btheelectrostatic energy is (2.31) Al•m=fPArq.m(OA' cPA)dTA' (2.32) Bl.-m=f(-I)lmlpBR-(l+l)q._m(OB' CPB)dTE' (2.33) Thequantities Az•mmayberegarded asdefiningthecomponents of themultipole moments ofdegreel,ofthechargedistribution near0, anditwillbe-seenthattheyinteract onlywiththeconjugate com­ ponentsBl.-mwhichhavethesamevalueofl,m.Themonopole com­ ponent(l=0)contains onlyoneterm,whilethedipole(l=1) components containthreetermswhichmayeasilybeshown(Problem 2.15)togivethesameinteraction asequation (2.24).Ingeneralthe interaction energyinvolving rlandR-(l+l)contains (2l+1)terms,but theadvantage ofthequantities AZ•m'Bl•misthattheycanbeexpressed intermsoffunctions Or,mwhicharetabulated. Weshallgonofurther thanthequadrupole terms(l=2),whichcanbewrittenoutusing Table2.2.Itcanthenreadilybeverifiedthat,fortheparticular case whereeitherchargedistribution issymmetrical abouttheaxis0=0, allthetermsinequation (2.31)vanishexceptthatwithm=o.IfPA hassuchsymmetry (i.e.itisindependent ofcp),itsquadrupole inter- 2.3] ELECTROSTATICS II 43 (2.36)(2.34) IprlO"mdT=0if1isodd. (2.37) Thusinvariance undertheparityoperation excludes thepossibility of electricmultipole moments ofanyodddegree. Theforminwhichtheinteraction energyisexpressed inequation (2.31)isverysuitabletoacasewhereonechargedistribution (suchas thatofanucleus) isconfined toasmallvolume,butinteracts with another chargedistribution which is comparatively faraway(suchas theatomicelectrons). Theseriesthenconverges veryrapidly, since (riB)<I;experimentally nointeractions withnuclearelectricmulti­ poleshigherthanthequadrupole havebeendetected. Theconvergence is muchlessrapidinatomiccases,suchastheinteraction betweenelectrons withinanatom,orbetween atomicelectrons andthesurrounding ions inasolid(whichstrongly affectstheirmagnetic properties, seeChapter 20).However, integrals involving allbutthefirstfewvaluesof1can beshowntovanishbymeansoforthogonality theorems, sincethewave functions arethemselves spherical harmonics.actioncanbeexpressed intermsofasinglecomponent A2,o=IPAIT2(3cos28A-I)dTA=!qQ, wherethequantity (writingrcos8A=z) Q=I~IfPA(3z2-r2)dTA (2.35) iscalledthe'quadrupole moment' ofthechargedistribution, andhas thedimensions ofanarea.Itmightbeexpected thatonewouldtake q=JPAdTA'thetotalchargeinthedistribution, butforanucleus, byconvention, qistakenasthechargeonasingleproton(notthetotal nuclearcharge),andQisexpressed intermsoftheunitofa'barn' =10-24cm2•Thisunitischosenbecauseitisofthesameorderasthe squareofthenuclearradius(foranatomthequadrupole moment would beoforder10-16cm2). Ithasalreadybeenshownthatinvariance undertheparityoperation excludes thepossibility ofpermanent electricdipolemoments inatoms andnuclei.Thismaybeexpressed moregenerally usingspherical har­ monics. Inversion through theoriginisequivalent tochanging the point(r,8,cp)into(r,7T-8,7T+CP). Since ~,m(7T-8) =(_l)l-m~,m(8), eim(1T+4»=(-I)meime,6, wehave C"m(7T-8,7T+CP) =(-I)'C"m(8,cp), anditfollowsthat 44 ELECTROSTATICS II [2.4 2.4.Someelectrostatic problems Oonducting sphereinauniformfield Supposeanearthedconducting sphereofradiusRisplacedinauni­ formfieldEo.Thenthefieldimmediately aroundthespherewillbecome distorted owingtotheinduced chargesonthesurfaceofthesphere (Fig.2.2),butthefieldatlargedistances willapproach thevalueEo. ----~-------=====-::-------~ L-----­t---- ..Eo ----~- FIG.2.2.Thelinesofforcenearaconducting sphereina uniform electricfield. Ingeneralthepotential distribution canbeexpressed asasumofterms ofthetypegiveninTable2.1,withthecondition thatV=0overthe surfaceofthesphere.Ifwetakeawholeseriesofterms,itwouldturn outthatthecoefficients ofmostofthemarezero.Itissimplertotry apossible solution withafewtermswhosenatureissuggested bythe symmetry oftheproblem; ifitisthenpossibletosatisfytheboundary condition V=0atr=R(takingthecentreofthesphereasoriginof coordinates), thenthisistheonlycorrectsolution. Ifthepolaraxisistakenparalleltotheuniform fieldEo,thenthe potential atlargedistances isthatofthisfieldalone,sothat V=-Eorcos() forr.-+oo. InordertomakeV=0atr=Rforallvaluesof8,itseemslikelythat wecanonlyaddtermswhichvarywiththesamepowerofcos().Hence wetryasasolution V=-Eorcos()+Ar- 2cos(). (2.38) Itisclearthatthissatisfiesourboundary condition ifEoR=AR-2; thatis,A=EoR3.Hencewehave V=-Eorcos()(1-R3jr3). Thisshowsthatthepotential outsidethesphereisthatduetotheuni­ formfieldtogether withthatofadipoleofmoment p=41T€€OEoR3 situatedatthecentreofthesphere.Insidethesphereasolutionofthe 2.4] ELECTROSTATICS II 45 type(2.38)isnotacceptable, sinceitwouldgiveaninfinitepotential attheorigin.InsteadwemustaddatermEorcos8,whichwilljust cancelthepotential oftheexternal fieldsothatV=0everywhere insidethesphere. Themagnitude oftheinduced chargeatanypointonthespherecan befoundfromthenormalcomponent ofthefieldatthesurface. Thisis Er=-oV/or=Eocos8+2Eo(R3/rS)cos8 =3Eocos8 atr=R. Hencethechargeaperunitareawillbe(fromequation (1.28» a=EEOEr=3EEOEocos8, whereEisthedielectric constant ofthemedium surrounding thesphere. ---~====== £2 (a)£1<£2-£z (b)£1>£z. FIG.2.3.Thelinesofelectricdisplacement Dduetoadielectric sphere, £1'inauniform electricfield,inamedium ofdielectric constant £z. Dielectric sphereinauniformfield Aslightlyharderproblem isthatofadielectric sphereofradiusR anddielectric constant El'surrounded byamedium ofdielectric con­ stantE2'andplacedinauniform fieldEo,asinFig.2.3.Twoseparate potential functions mustnowbetaken,oneforinsideandtheotherfor outsidethesphere(ineffect,thiswasalsorequired fortheconducting sphere,buttheninsidewehadjustV=0).Wemustalsosatisfythe boundary conditions atthesurfaceofthesphere,whichare,from equations (1.26)and(1.27), E1E1r=E2E2randEll=E2/,' wheresubscripts Iand2refertoinsideandoutsidethesurface,and subscripts randtrefertonormalandtangential components respec­ tively.Guidedbythepotential function forthecaseoftheconducting sphere,weshallassume f;=-Eorcos8+A 2r-2cos8, Ti=B1rcos8+B2r-2cos8 46~~~--~~----_ ...._- ELECTROSTATICS II [2.4 foroutsideandinsidethesphererespectively. Clearlywecannothave li-+00forr-+0,sothatthecoefficient B2mustbezero.Itisalso obviousthatVmustbecontinuous attheboundary, sinceadiscon­ tinuitywouldgiveaninfiniteelectricfieldthere.Thusv;.=Vzatr=R; thisautomatically satisfiesoursecondboundary condition since Et=Eo=_!(of\, r08J andgives B1Rcos()=-EoRcos()+A2R-2COS(), or B1=A2R-3_Eo• (2.39) Thenormalcomponents ofEatthesurfaceare E1r=-(oli/or)r=R =-B1cos() and E2'l'=-(oJi;/or)r=R =Eocos()+2A2R-3cos(). Hencethefirstboundary condition gives -B1=(E2/E1}{Eo+2A2 R-3). (2.40) Thesolutionofequations (2.39)and(2.40)is B1=-(3E22-\EoandA2=(E1-E2)R3Eo, E1+E;) E1+2E2 sothatthepotential functions insideandoutsidethesphereare li=-(~)EorCOS()' (2.41)E1+2E2 TT(R3 E1-E2)E. ()f2= -1-3---orcos. (2.42) rE1+2E2 Theseequations showthatthefieldE1insidethesphereisparallelto Eo,andofmagnitude E1=~Eo. (2.43) E1+2E2 IfE1>E2'D1>Do(seeFig.2.3),butE1<Eo,thereduction beingdue tothereversefieldofthepolarization chargesonthesurfaceofthe sphere;thisreversefieldisknownasthe'depolarizing field'.Thepoten­ tialdistribution outsidethesphereisthatofadipoleofmagnitude %E2£OR3Eo(E1-E2)/(E1+2E2)'situatedatthecentreofthesphere,super­ imposed onthatduetotheuniform field.If£2=1,thesizeofthis dipolemoment isjustequaltothevolumeofthespheretimesthe polarization P1induced bythefieldwithinthespheresincethen P1=Eo(E1-1)E1=3Eo(E1-1)Eo/(E1+2). NotethatasE1-+00,thesolu­ tionstendtothoseobtained fortheconducting sphere;thisfollowsfrom 2.4] ELECTROSTATICS II 47 thefactthattheboundary condition thenrequiresthatthefieldinthe spherebezero. Problems withcylindrical symmetry-eonducting cylinderinaunijorm field Insomethree-dimensional problems thepotential maybeindependent ofonecoordinate, andtheproblem thenreducestoatwo-dimensional one.Itisoftenconvenient tousecylindrical coordinates insuchacase, takingthezdirection asthatinwhichthepotential isinvariant. Then TABLE2.3 Somecylindrical harmonic junctions Oylindrical harmonic Oorresponding solution8 ofLaplace's equation Do logr 1 D1 r-1(Acos8+Bsin8)r(Acos8+Bsin8) DI rl(Acos28+Bsin28)rl(Acos28+Bsin28) DI r-3(Acos38+Bsin38)r(Acos38+Bsin38) puttingp=0and02Vjoz2=0inequation (2.5),wehaveforLaplace's equation 8(8D02V rorror)+002=o. (2.44) Thisissatisfiedbyafunction oftheformV=rnDn,whereDnisafunc­ tionof0alone(knownasacylindrical harmonic) whichmustsatisfy thedifferential equation 02Dn+2D-0 (2.45)802nn-· Thisequation isunchanged bythesubstitution of-nforn,sothatif V=rnDnisasolutionofLaplace's equation, soalsoisV=r-nDn.One solution isV=loger,andothersolutions maybeobtained eitherby partialdifferentiation withrespecttox=rcosO,orbydirectsolution ofequation (2.45). Anumberofthesimplest functions aregiveninTable2.3;notethat thegeneralformofDnwillbeAncosnO+BnsinnO,whereAnandBn areconstants. Thetypeofproblem towhichthesolutions maybeappliedisillus­ tratedbythecaseofaconducting circularcylinder, initiallyuncharged, lyingwithitsaxisatrightanglestoauniform fieldEo.Iftheaxisof thecylinder istakenasthez-axis,itisclearthatthepotential distri­ butionwillbeindependent ofz.Atlargedistances thepotential will tendtoV--EorcosO, andwewillassumethatothertermsrequired 48 ELECTROSTATICS II [2.4 mustalsovaryascosO.Thenthepotential outsidethecylinder willbe oftheform VE 0=-orcosO+Ar-1cos . Tosatisfytheboundary condition V=0atr=Rforallvaluesof0, wemusthaveEoR=AR-l,sothatthepotential is V=-EorcosO(I-R2/r2). (2.46) Thefirsttermisthepotential oftheexternal field,thesecondthatof anextended dipoleconsisting oftwoparallel linesofpositive and negative chargeclosetothez-axis. Intheseproblems, wehaveassumed apotential containing justthe required numberofterms.Thisisamatterofintelligent anticipation ratherthanguesswork orknowing theanswerbeforehand. Ifwehad takenanylessterms,wecouldnothavesatisfied theboundary condi­ tions.Ifwehadtakenmoreterms,thecoefficients oftheadditional termswouldhavebeenfoundtobezero.Inthecasejustconsidered, termssuchascosnOorsinnOcouldnotsatisfythecondition V=0at r=Rforallvaluesof0,becausethepotential oftheexternal field variesonlyascosO.Wearejustified inassuming thatthesolution we havefoundisthecorrectandonlysolution becauseoftheuniqueness theorem. Thistheorem alsojustifiestheuseofanotherspecialmethod, whichweshallnowconsider. 2.5.Electrical images Ifwehavetwoequalpointchargesofopposite signseparated by acertaindistance 2a,theplanepassingthrough themidpoint ofthe linejoiningthemandnormaltothislineisanequipotential surfaceat zeropotential. Therefore ifthenegative charge(say)isreplaced by aplaneconducting sheetABinFig.2.4,thefieldtotherightofAB willremainunaltered. Conversely, ifapointchargeisplacedinfront ofaninfiniteconducting plane,theresultant electricfieldtotheright ofABwillbethesameasthatproduced bytheoriginal chargeplus anegative chargeanequaldistance fromtheplaneontheopposite side. Thenegative chargeisthe'electrical image'oftheoriginal chargein theplaneAB. Themethodofimagesthusconsistsinreplacing aconductor bya pointchargesuchthattheconducting surfaceisstillanequipotential surface. ThenLaplace's equation isstillsatisfiedatallpointsoutside theconductor, andbytheprinciple ofuniqueness, theproblem ofa pointchargeanditsimageisidentical withthatofapointchargeandan infiniteconducting surfaceasregardstheregionoutsidetheconducting 2.5] ELECTROSTATICS II 49 surface. Electrical imagesareentirelyvirtual;afieldononesideofa closedequipotential surfacecannotberepresented byanimageonthe samesideofthesurface,sincethiswouldgiveasingularity atthepoint occupied bytheimagecharge.Thefieldontheonesideofthesurface isidentical withthatwhichwouldbeproducedifthesurfacewere replaced byanimagechargeontheothersideofit. v=o \ \ A \ \ ----\ \ /' \" \///---"\///-----""\I/"" "\I/-"""""\ \ II//''-"\I! -----~'-----. "-..",,\\ (;/---~ -----""-q/'--------..:::-==--e----1---_---------- ..----;.X'--/'-----///1//\',,--- /'///1\ \""-// //I\ \"'-./ /11\" / //\" '/ \'-. //\" //\"---.. / ! "/ I ""----- / I B FIG.2.4.Apointchargeqanditsimagecharge-qinaninfiniteconducting planeAB, showing thelinesofforcefromqontherightoftheplanewhichendonthesurfacecharge onAB.XY=2a. Pointchargeandinfiniteconducting plane Themethodofimageswillnowbeappliedtoanumberofspecialcases, thesimplest ofwhichisthatofapointchargeqplacedadistance afrom aninfiniteconducting planeatzeropotential. Inthiscaseitisobvious thattheimagemustbeacharge-qatadistance abehindtheplane, asinFig.2.5.Thepotential atanarbitrary pointPisthen v=47r:€J~- {r2+4a2;4arcoSO}l}' where €isthedielectric constant ofthemedium outsidetheconductor. Inordertocalculate thechargedensityatanypointontheplane,we 851110 E 50 ELECTROSTATICS II [2.5 mustfindthecomponent oftheelectricfieldnormaltotheplane.This is(seeFig.2.5) Ex=Ercose-Eesine = -aVcose+! aVesinBar ra q[COSB rcosB+2a ] =47TEEO~-{r2+4a2+4arcosB)i atthepointP.AtapointQontheplane,rcose=-a,sothatatQ Ex=-qaj27TEE or3 Dielectric aConductor - -q........-~-.""----~~~4>-~~~-----'::Ofi-c.-~L-~~--~ x v=o FIG.2.5.Imageofpointchargeinaconducting plane. (cf.Problem 2.6),whereristhedistanceofQfromthecharge+q.The induced chargeperunitareaatQisthen a=EEOEx=-qaj27Tr3• (2.47) Theforceexertedonthepointchargebytheinduced chargeonthe planeisjustequaltotheforceexertedonthechargebyitsimage. Thatis,F=-q2j167TEEoa2,wherethenegative signindicates thatthe chargeisattracted towardstheplane(seeProblem 2.6). Pointchargeandconducting sphere Amoredifficultproblem isthatofapointchargeqplaced(invacuo) adistance afromthecentreofaconducting sphereofradiusR(Fig.2.6). Weshallconsider firstthecasewherethesphereisearthedandatzero potential. Bysymmetry, theimagechargemustbeonthelinethrough qtothecentreofthesphere0,andwewillas~umethatitconsistsofa singlechargeq'atadistancebfromO.Thepotential atapointQon 2.5] ELECTROSTATICS II 51 thesurfaceofthesphereisthen 1(qqjV=--+-%EEOr r 1[q q'] =%EEO(R2+a2+2aRcosB)! +(R2+b2+2bRcosB)i • ItisonlypossibletomakeV=0overthewholesurfaceofthesphere (i.e.forallvaluesofB)ifthefunctions inthedenominators aresimilar q A FIG.2.6.Apointchargeqanditsimageq'inaconducting sphere. OA=a,OB=b. functions ofB.ThisrequiresthatwechoosebsothatbjR=Rja;that is,BistheinversepointtoAinthesphere.Thenthepotential atQis V_ q+(ajR)q' -4'1TEEo(R2+a2+2RacosB)t' andthiswillbezeroifwemakeq'=-q(Rja). Hencetheimagecharge isofmagnitude -q(Rja)attheinversepointinthesphere,andthe readermayverify,byintegrating thechargedensityonthesphere, thatthetotalchargeonthesphereisequaltotheimagecharge. Ifthesphereisinsulated andinitiallyuncharged, thetotalchargeon itmustremainzero.Itistherefore necessary toaddasecondimage charge-q'atsuchapointthatthesurfaceofthesphereremainsan equipotential surface. Thisisaccomplished byplacing.~ charge+q(Rja) atthecentreofthesphereinaddition tothecharge'-q(Rja) atthe inversepoint.Ifthesphereisinsulated butcarriesaninitialchargeQ thetotalchargeatthecentrewouldofcoursebeQ+q(Rja). 52 ELECTROSTATICS II [2.6 (2.48)2.6.Linecharges Justaswehaveconsidered themathematical abstraction ofapoint charge,sowemaypostulate a'linecharge'inwhichchargeisuniformly distributed alonganinfinitestraight line.Itsstrength isdenotedbyA, thechargeperunitlength.Tofindthefieldofsuchalinecharge,im­ mersedinamediumofdielectric constant €,weapplyGauss'stheorem toasectionoflengthtofacylinder ofradiusrwhoseaxiscoincides withthelinecharge. Thisgives €€oE(27Trt) =At, sincebysymmetry thefieldEiseverywhere normaltotheaxis.Hence E=A/(27T€€or), andthepotential atadistancerfromtheaxisis V=__A_fdr=--A-Ioger+Vo. 27T€€O r 27T€€O Heretheconstant "Vocannotbedefinedbyassuming V=0atr=00 sincethelinechargeitselfextends toinfinity. Ifwehavetwoparallel line charges ofequalstrength but opposite sign,asinFig.2.7,the potential atapointwhoseperpendi­ culardistances fromthelinecharges arerl,r2respectively is FIG.2.7.Twoparallellinechargesnormal totheplaneofthepaper,withcharge+A and-Aperunitlength. Theequipotential surfaces givenbythisequation areshowninFig.2.8. Theyhavetheformofcylinders whosecross-sections formasetof coaxialcircleswithlimiting pointsatthelinecharges. Thesurface whosepotential is"Voisthemedianplane(forwhichrl=r2)between thetwolinecharges. Fromthisitfollowsthattheproblem ofaline chargeparalleltoaconducting planecanbesolvedbythemethodof images,usingalinechargeofopposite signasimage.Weshallapply ourresultstoamorerealistic problem. Oapacitance betweentwoparallelinfinitecircularcylindex8 Consider firstaninfinitelinechargeofstrengthAwhichisparallelto aninfinitecylinderofradiusa.Inthecross-section showninFig.2.9, thelinechargeisatPandtheaxisofthecylinderatO.Weimagine 2.6] ELECTROSTATICS II 53 FIG.2.8.Thelinesofconstant potential fortwoparallelinfinitelinecharges ,\and--', normaltotheplaneofthepaper.Theyformsystems ofcoaxialcircleswithlimiting pointsatthecharges. +A P FIG.2.9.Theimageofaninfinitelinechargeinaninfiniteconducting cylinder. 54 ELECTROSTATICS II [2.6 animagelinechargeofstrength-,\tobeplacedatP',thepositionof P'beingchosensothatthecircleformedbythecross-section ofthe conducting cylinder coincides withoneofthefamilyofcoaxialcircles whicharetheequipotentials ofthelinechargeanditsimage.Then thepotential atanypointQonthesurfaceofthecylinder is ,\QP V=--2-logQP'+Yo.7TEEO 2d FIG.2.10.Twoinfinitely longparallelconducting cylinders. QPjQP' =OPjOQ =OPja, andthepotential atQis ,\OPV=---log-+Yo, 27TEEOa whichisindependent ofthepositionofQonthesurfaceofthecylinder, showingthatthisisanequipotential. Weturnnowtothecaseoftwoinfiniteparallelcylinders, eachof radiusa,whoseaxesareadistance 2dapart.TheninFig.2.10the distance00'is2d,andP,P'arethelimiting pointsofafamilyof coaxialcircles.PandP'arechosensothattwoofthecirclescoincide withthesurfacesofthecylinders, makingeachoftheseanequipotential iflinechargesofstrength'\ and-,\wereplacedatPandP'respectively. Thenthepotential atanarbitrary pointwhosedistance isr1fromP andr2fromP'isSincePandP'arethelimiting pointsofthefamilyofcoaxialcircles, itfollowsthattheyareinversepointswithregardtoanyoneofthese circles.Hence wheretheconstant Yoiszeroifwetakethemedianplane(r1=r2).. between thecylinders tobeatzeropotential. Thenthepotentials of thetwocylinders are ,\ ,\VQ=---log(OPja) andVQ,=+--log(OPja) 27TEEO27TEEO 2.6] ELECTROSTATICS II 55 respectively. ButOP+OP'=2d,andOP.OP'=a2,sinceP,P'are inversepointsinthecircleofradiusa,centreO.Hence OP=d+~(d2-a2), andthecapacitance perunitlengthbetween thetwocylinders is a=Aj(VQ'-VQ) =log[{d+~(d2-a2)}/ar Whend?>a,thisapproaches thelimiting value 0=7T€€o/log(2dja).(2.50) (2.51) Asimilarproblem isthecapacitance ofahorizontal telegraph wire withrespecttotheearth.Thismaybetreatedasaninfinitecylinder ofradiusaadistance daboveaninfiniteconducting plane.Itisclear thatthepotential distribution willbethesameasinthecaseofthe twoparallelcylinders ifweassumethattheconducting planecoincides withthemedianplanebetween thecylinders, whichistheequipotential surfaceV=o.ThenthechargeonthewireperunitlengthisA,and thepotential difference betweenitandtheplaneisjusthalfthatbe­ tweenthetwocylinders intheprevious problem. Hencethecapacitance perunitlengthwillbe(assuming d?>a) a=27T€€O. (2.52) log(2dja) Notethattheapproximation d?>aistantamount toassuming thatthe wirebehaves asifithadalinechargeAperunitlengthalongits axis, sinceas(ajd)approaches zerothepointP'movestowards0inFig.2.10 andOP-+2d. 2.7.Images indielectrics Thepotential distribution duetoapointchargenearadielectric surfacemaysometimes befoundbythemethodofimages. Weshall illustrate thistypeofproblembyconsidering thecaseofapointchargeq adistanceafromasemi-infinite dielectric bounded byaplanesurface. Thisproblem ismorecomplexthanthatofapointchargeandconduct- -I.ingplanesinceasecondimagesystemisrequired torepresent thefield withinthedielectric.Itisnotobviousthatthefieldcanberepresented bythatofasinglepointcharge,butweshallassumethatthisispossible (ifourassumption iswrongweshallnotbeabletosatisfytheboundary conditions). Wetaketherefore asinglechargeq2atapointB,asin (2.53)56 ELECTROSTATICS II [2.7 Fig.2.11,andthepotential atapointQinthedielectric willthenbe VQ=~. 47T€Orz Thefieldofthepointchargeqwillpolarizethedielectric andtherewill therefore beasurfacechargeonthedielectric whichaffectsthefield outside. Weassumethatthiscanberepresented byanimagechargeql FIG.2.11.Imagesystems forapointchargeqandasemi-infinite dielectric. Thefieldoutsidethedielectric isthatofqandql;thefield insidethedielectric isthatofq2. at0inthedielectric, andthepotential atapointPoutsidethedielectric isthen 1{qql}Vp= --+-. (2.54) 47T€Orr1 Bysymmetry, q,qvandqzwillalllieonanormaltothedielectric surface. Notethatthedielectric constant €doesnotappearinthese equations sincetheeffectofthedielectric isreplaced bytwoimage systemsinvacuo. Toavoidaninfiniteelectricfieldattheboundary wemustassume thatVp=VQattheboundary, andthisautomatically satisfiesourfirst boundary condition, thatthetangential components ofEmustbethe sameoneithersideoftheboundary. Itisclearthatthecondition 2.7] ELECTROSTATICS II 57 a(q-q1) £aq2 41T£Or3=41T£Or3. HencewehavetherelationsVp=VQeverywhere ontheboundary canonlybesatisfiedifr,rvandr2 varyatthesamerateaswemovealongtheboundary, andwemust therefore haver=r1=r2,sothatBcoincides withA,andaisasfar behindthesurfaceasAisinfront.Inaddition, q+q1=q2'Oursecond boundary condition isthatthenormalcomponents ofDmustbecon­ tinuousattheboundary; i.e.(oVp/oz) =£(oVQ/oz)atthesurface,which wetaketobetheplane Z=O.Nowatanarbitrary point(x,y,z), r2={x2+y2+(a+z)2}l, andr1={x2+y2+(a_z)2}l, sothat ~G)=~(:J= -a~z, ~(B=ar~z. Usingtheserelations forthecasez=0,oursecondboundary cpndition becomes (2.55)q+q1=q2andq-q1=£q2' whichgiveq2=2q/(£+I) andq1=-q(£-I)/(£+I) fortheimage charges. Theforceofattraction onthechargeqtowards thedielectric isthereforeF= _ qq1=q2(£_I) • 41T£O(2a)2161T£oa2(£+I) Thelinesofdisplacement forthecaseofapointchargeandaninfinite dielectric areshowninFig.1.13. REFERENCES BRINK,D.M.,andSATCHLER, G.R.,1962,Angular Momentum (O.U.P.). RAMSEY, N.F.,1956,Molecular Beams(O.U.P.). PROBLEMS , 3£ P=P2€+1'2.1.Thepolarization chargeonthesurfaceofaspherical cavityis-(70cos8,at apointwhoseradiusvectorfromthecentremakesanangle8withagivenaxisOz. Provethatthefieldatthecentreis(70/3€0,paralleltoOz. Ifthecavityisinauniform dielectric subjecttoafieldEoparalleltothedirection 8=0,showthat(70=3Eo£0(E-l)/(l+2€), where€istherelativeperrnittivityof thedielectric. Verifythatthisgivesthecorrectvalueforthefieldatthecentre ofthecavity(equation (2.43))andnotethat(70isnotsimply(€-I)€oE obecause ofthedistortion ofthefieldinthedielectric causedbythepresence ofthecavity. 2.2.Adipolepissituatedatthecentreofaspherical cavityofradiusainan infinitemedium ofdielectric constant €.Showthatthepotential inthedielectric medium isthesameaswouldbeproduced byadipolep'immersed inacontinuous dielectric, where 58 ELECTROSTATICS II andthatthefieldinsidethecavityisequaltoEd+E" whereEdisthefieldwhich thedipolewouldproduce intheabsenceofthedielectric, and E _2(€-I)_-R-,-2€+1417€oa3' E,isknownasthe•reaction field',Theseformulae areusedinthetheoryof dielectrics (seeChapter 17). 2.3.Showthatthefieldinsideacylindrical cavityinadielectric ofconstant €is €~1Eo,whentheaxisofthecylinder isatrightanglestoauniform fieldEo' 2.4..Findanexpression forthesurfacedensityofchargeonaninfinitely long conducting cylinder ofradiusa,placedwithitsaxisatrightanglestoauniform electricfieldEo,asafunction ofthepolarangle8. (Answer: a=2€oEocos8.) 2.5.Auniform electric fieldEissetupinaninfinite dielectric. Showthat (a)ifalongneedle-shaped cavity,whoselateraldimensions areverysmallcom­ paredwithitslength,iscutinthedielectric withitsaxisparalleltoE,thenthe fieldinthiscavityisE;(b)ifaflatdisk-shaped cavity,whoselateraldimensions areverylargecompared withitsthickness, iscutwithitsplanenormaltothe direction ofE,thenthefieldinthecavityisD/€o'whereDisthedisplacement inthedielectric. Verifythatthefieldinanintermediate shapeofcavity(suchasinProblems 2.1 and2.3)liesbetween theseextreme values, 2.6.Apointeharge qisplacednearaninfiniteconducting plane.Verifythatthe totalchargeontheplaneis-qbyintegrating equation (2.47),andcalculate the totalforceontheplanebyintegration ofthetensionperunitarea(equation (1.37» overtheareaoftheplane.Verifytheexpression giveninthetextforthefieldatQ (Fig.2.5)byvectoraddition ofthefieldsofthepointchargeanditsimage, 2.7.Showthattheworkdoneinbringing upachargeqfrominfinitytoadistance afromaconducting planeatzeropotential is-q2/167T€€oa. Verifythatthesame resultisobtained usingequation (1.34)(remember thattheinduced chargeonthe planeisatzeropotential). 2.8.Showthattheforceonachargeqdistance ainvacuofromthecentreofan insulated anduncharged conducting sphereofradiusRis(a:>R) F_L(!!'_ Ra) -417€0a3(a2_R2)2• 2.9.ApointchargeisplacedinahollowmetalsphereofradiusR.Ifthecharge isqandisadistance bfromthecentreofthesphereshowthattheforceonitis q2Rb 417€0(R2_b2)2' (Hint:Findtheimageofqoutsidethespheresuchthatthesphereisanequi­ potential surface.) 2.10.Apointchargeqisplacedatadistance 3Rfromthecentreofanisolated conducting sphereofradiusRwhichalreadyhasachargeequaltoq.Provethat thesurfacedensities atpointsonthespherenearesttoandfarthest fromthe pointchargeareintheratio8:29. ELECTROSTATICS II 59 2.11.Anelementary dipoleofstrength pisplacedatapointP,outsideandat adistance afromthecentreaofanearthedconducting sphereofradiusR.The axisofthedipoleisinthedirectionaP.Provethatitsimagesystemconsistsof apointchargepR/a2andadipoleofstrength pR3/a3,bothsituatedatthepoint P'whichisinversetoPinthesphere. 2.12.ShowthatifaninfinitelinechargeAperunitlengthisatadistancedfrom aninfiniteconducting planeinamedium ofdielectric constant €,thesurface densityofchargeintheplaneisu=-dA/Trr2,whereristheshortest distance fromthelineofchargetothepointinquestion. 2.13.Calculate theforceperunitlengthontheinfinitelinechargeofthelast question. 2.14.Electric chargeisdistributed overathinspherical shellwithadensity whichvariesinproportion tothevalueofasinglefunction Pz{cos8)atanypoint ontheshell.Show,byusingtheexpansions (2.26)and(2.27)andtheortho­ gonalityrelations fortheLegendre functions, thatthepotential variesasrlPz{cos8) atapoint(r,8)insidethesphereandr-(I+I>Pz{cos8) atapoint(R,8)outside. 2.15.Showthat,forl=I,thequantities definedbyequation (2.32)are Al•O=P.,Al•l=-2-1{P.,+iPll)' Al._l=2-1{P.,-iPll)' whileforapointchargeatBinFig.2.1(b), A_IBlO=-E.,-41Bll=-2-I{E.,+iE,,), 4~Bl_l=2-I{E.,-iElI),'rlT€€0• 1T€€0. 1T€€0. andverifythatthisgivesthesameinteraction energyasequation (2.24). 2.16.Foranatominap-state,thewavefunction isif1=f(R)1';..0'andthecharge densityis-eif12.Showthattheatomicquadrupole moment (asdefinedbyequa­ tion(2.35»ist<R2),where<R2)isthemeansquaredistance oftheelectron from thenucleus. 2.17.Theisotopeofmass35ofchlorinehasanuclearelectricquadrupole moment Q.Showthatifitwereinachlorine atomwhosewavefunction if1=f{R)1';..0' theenergyofinteraction between thenuclear electricquadrupole moment and theelectron is where<R-3)isthemeaninversecubeofthedistance oftheelectron fromthe nucleus, andetheelectronic charge. IfQ=-0,079barns=-7·9x10-30m2,and<R-3)=5x1031m-3,showthat theenergyofinteraction (Up/h)expressed infrequency units(hisPlanck's con­ stant)isabout27Mc/s. 2.18.Sixequalcharges qareplacedatthepoints(±R,O,O), (O,±R,O), (O,O,±R).Showthatthetermsoflowestdegreeinthepotential atapoint (x,y,z)=(r,8,</»neartheoriginare V=41T~R +(~€€)(;5)(~){a,.0+{5/14)I{a4.,+a4._1)}' 60 ELECTROSTATICS II where04•0=P4(whichisgiveninTable2.1),and 04.±4=(35jI28)t sin48e±i44>. [Hints: Sincethesystemhasinversion symmetry through theorigin,termsin oddpowersofrmustvanish. AlsoVmusthavefourfold symmetry aboutthe polar(z-)axis, sothatrotations changing 1>by!7TmustleaveVunchanged: only functions withm=0or4satisfythiscondition.] NotethatinCartesian coordinates 6q(q)(1 )(35) 3 V=----+--- - (X4+y4+z4-or4), 47T€€0R47T€€0R64 showing thatthex,y,zaxesareallequivalent (cubicsymmetry). 3 STEADY CURRENTS (3.1) 1=JJ.dS.3.1.Introduction INtheprevious chapters onelectrostatics wehavebeenconcerned with stationary electriccharges.Ifafreechargeisplacedinanelectricfield itwillbeactedonbyaforce,andwillmoveinthedirection ofthelines offorce.Thus,ifaninitialdifference ofpotential existsinaconductor, thechargeswillmoveuntiltheyreachpositions ofequilibrium, andthe wholeoftheconductor becomes anequipotential surface. Butbycon­ nectingabattery between twopointsofaconductor, apermanent difference ofpotential maybemaintained between thesetwopoints, andtherewillthenbeacontinuing flowofcharge.Thisconstitutes an electriccurrent,andthestrength ofthecurrentIisdefinedbytherate atwhichchargepassesanygivenpointinthecircuit.Ifwearedealing withacurrentextended inspace,thenwemaydefinethecurrentdensity Jasthequantity ofchargepassingpersecondthrough unitareaofa planenormaltothelineofflow.Thetotalcurrentflowingthroughany surfaceisfoundbyintegrating thenormalcomponent ofthecurrent density:thatis Ifthecurrentiscarriedbyparticles ofchargeewithdensitynperunit volumeandvelocity v,thenthecurrentdensityis J=nev. (3.2) ThusJisavectorwhosedirection isthatofthevelocityvofthecarriers. Inearlyexperiments onelectricity therewasnoevidence forthesign ofthechargesformingthecurrent, sincetherewasnomeansofdis­ tinguishing between aflowofpositive chargesinonedirection anda flowofnegative chargesintheopposite direction. Thepositivedirection ofcurrentflowwastherefore takenasthatinwhichapositive charge wouldmoveinanelectricfield.Thusinacircuit,theconventional direction fortheflowofcurrentisfromthehigherpotential tothelower potential; i.e.fromthepositive poleofabatteryroundtheexternal circuittothenegative pole.Itiscustomary toretainthisconvention, although themodern theoryofmetallic conduction showsthatthe positively charged ionsarefixed,whileacertainnumberofelectrons 62 STEADY CURRENTS [3.1 arefreetomoveaboutthebodyofthemetal.Sincetheelectrons are negatively charged, theirdirection ofmovement isopposite tothatof theconventional currentflow. Measurement ofejmforcarriersofelectriccurrentinametal Thefirstdirectexperimental evidence thatthecarriersofelectricity inametalareelectrons wassupplied bythemeasurements ofTolman andStewart (1917).Theprinciple ofthisexperiment depends ona comparison oftheelectriccurrentwiththemomentum carriedbythe particles.Ifthecurrentdensityisnev,andtheparticles havemassm, thenthemomentum associated withthecurrentcrossing unitareaof aplanenormaltothedirection offlowisnmv.Thustheratioofthe electriccurrentdensitytothemomentum 'current density' issimply equaltotheratioofchargetomass(ejm)ofthecarriers. Thesignof ejmisobtained fromcomparison ofthedirections offlowoftheelectric currentandmomentum current. Themethodweshallnowdescribe is thatofalaterexperiment byKettering andScott(1944). Acircularcoilissuspended byathinfibresothatitsplaneishori­ zontalanditformsatorsional pendulum withverysmalldamping. The coil,consisting ofNturnsofradiusr,carriesacurrentI.Ifthenumber ofelectrons perunitlengthofthewireisn,andtheymovewithamean velocityv,theirangularmomentum abouttheaxisofthecoilis r=mrvn(21TrN), sincethetotalnumberofelectrons inthecoilisn(21TrN). Thecurrent 1=nev,andhencewehave r=21Tr2N(mje)I =2AN(mje)I, whereAistheareaofthecoil.Intheexperiment, acurrentIismain­ tainedinthecoil,andthensuddenly reversed. Thisimpartsanimpulse 2rtothecoil,whoseangularmomentum isthusalteredby4AN(mje)I. Inpracticetheamplitude ofswingofthecoilisobserved withthecurrent flowinginonedirection, andthecurrentisreversed atthemoment whenthecoilpassesthroughitsequilibrium position. Thischangesthe amplitude ofswing80byanamount D..80=2r(Tj21T~), where2risthechangeofangular momentum duetothecurrent reversal, Tisthetimeofswing,and~themoment ofinertiaofthecoil. Thevalueofejmcanthusbedetermined bymeasurement ofthechange inamplitude ofoscillation foragivencurrentI. 3.1] STEADY CURRENTS 63 Although thetheoryoftheexperiment issimplethereweremany practical difficulties. Leadstothecoilhadtobebrought insothatthe freesuspension bythetorsionfibrewasnotaffected. Toeliminate vibra­ tionanddisturbance duetochanging magnetic fields,theapparatus was installed inanunderground vault.Theexperiment wasperformed both withcoilsmadeofcopperandaluminium. Thevaluesofmjeobtained were5'64,5,67,and5·79X10-9gjcoulomb forthreedifferent copper coils,and5·66X10-9glcoulomb foranaluminium coil.Themeanof theseresults,5·69x10-9,isinverycloseagreement withthereciprocal (5'68X10-9gjcoulomb) ofthemostaccurate determinations ofelmfor freeelectrons. Thesignofmje,obtained fromthedirection ofthechange ofamplitude, corresponded tothecarriersbeingnegatively charged. 3.2.Flowofcurrent inconductors Sinceelectricchargecanneitherbecreatednordestroyed, itfollows thattherateofincreaseofthetotalchargeinsideanyarbitrary volume mustbeequaltothenetflowofchargeintothisvolume. Wehave therefore f::dT= -fdivJdT f(divJ+::) dT=O. orwheretheintegrals aretakenrespectively overthevolumeandthe surfacebounding it.Ontransforming thesurfaceintegralintoavolume integral, wehave (3.3) divJ=_8p, 8tThisintegralmustbezerowhatever thevolumeoverwhichweintegrate, andthiscanonlybetrueiftheintegrand isitselfzero.Wemaythere­ forewrite whichisknownastheequation ofcontinuity. Inthesteadystate 8pjot=0,andtherefore divJ=0 (3.4) inanyregionofcurrentflowwhichdoesnotcontainasourceorsink ofcurrent. Suchasourceorsinkbywhichcurrentmaybeinjectedinto orwithdrawn fromaconducting regionisknownasanelectrode. The totalcurrentflowtoorfromanelectrode maybefoundbyintegrating thecurrentdensityoveranysurfacewhichtotallyenclosestheelectrode. 64 STEADY CURRENTS [3.2 Ohm'slaw Itisfoundexperimentally thatinametallic conductor atconstant. temperature thecurrentdensityislinearlyproportional totheelectric field.Thisisexpressed bytheequation J=uE. (3.5) Theconstant uisknownasthespecificconductivity, anditsreciprocal asthespecificresistance orresistivity. Thelatterisusuallydenoted byp,anditisgenerally clearfromthecontextwhether thissymbolis beingusedtodenotechargedensityorspecificresistance. Ifaconducting wireofcross-section AcarriesacurrentI,then I=JA;ifthecurrententersatapointwherethepotential isT-i andleavesatapointadistancelawaywherethepotential isVz,then E=-(Vz-~)/l. Hence 1=uEA=uA(~-Vz)/l =(~-Vz)jR, (3.6) whereR=lj(Au)=pljAisknownastheresistance ofthewire.Equa­ tion(3.6),whichexpresses thefactthatthevoltagebetween theends ofaconductor isproportional tothecurrentflowingintheconductor isknownasOhm'slaw.Inthem.k.s.system(seeChapter 24)Vis measured involts,Iinamperes, andtheunitofresistance istheohm. Itsreciprocal, theunitofconductance, iscalledthemho,orreciprocal ohm.Sincep=ARjl,thedimensions ofspecificresistance arethoseof resistance Xlength,andtheunitistherefore theohm-metre. Valuesoftheresistivity patroomtemperature aregiveninTable3.1 foranumberofmetals,alloys,andinsulators. Allmetalsaregoodcon­ ductors; silveristhebest,butisexpensive, sothatcopperisgenerally usedinstead. Thespecificresistance ofallmetalsisindependent ofthe current density overanextremely largerange,butincreases with thetemperature. IfRoistheresistance attheice-point To,theresistance Ratatemperature Tcanbewrittenas wherea,b,andcareconstants whichdecrease rapidlyinorderofmagni­ tudeasthepowersofthebrackets increase. Intherangebetween the ice-point andthesteam-point onlyaisappreciable exceptinvery accurate work.Itisknownasthetemperature coefficient ofresistance, andsomevaluesaregiveninTable3.1.Itwillbeseenthatmanganin andconstantan canbeconsidered tohavearesistance independent of temperature overtherangeforwhichbisnegligible. 3.2] STEADY CURRENTS 65 Thereisanotherclassofsubstances calledsemiconductors, examples ofwhicharecarbon,germanium, silicon,andsomecompounds suchas zincoxide.Theconductivity depends onthepurityofthespecimen, butcanberepresented inmanycasesbyanequation oftheform a=aoe-b/Torp=poeb/T, whereao,Po'andbareconstants. TABLE 3.1 Specificresistance ofsometypicalmaterials SpecificreBista'fIU Temperature at20°0 coejJicient Substance (ohm-metres) (°0)-1 {""vo,1·6xlO-s 3·8XlO--s PuremetalsCopper 1·72xlO-s3·9x10-8 Aluminium 2·83xlO....s3·9x10-8 Platinum lOxlO-s 3·9x10-8 Alloys{Constantan 44·2xlO-s ,...,lO-s Manganin 44xlO-s ,...,lO-s Semiconductors{Puresilicon ,....2x108 negative Puregermanium ,...,0·5 negativer-p ",,,2XlOll InsulatorsSealingwax ,...,1014 Sulphur ,...,1015 Fusedquartz >5x101S Bycombining equations (3.2),(3.5)wefindthat a=ne(v/E)=neu.(3.8) (3.9) Hereuisaquantity knownasthemobility, andisequaltothemean driftvelocity whichtheelectrons acquireinunitelectricfield.Itis generally expressed inunitsof(em/sec) per(volt/em), orcm2/volt-sec; inm.k.s.unitsitmustbeconverted tom2/volt-sec, andthenumberthen obtained forthemobility willbesmallerbyafactor10-4thanthat expressed inthemorecustomary units.Fromequation (3.9)weseethat theconductivity depends onn,thenumberofcarriersperunitvolume, andu,theirmobility. Inametalnisvirtually independent oftempera­ ture,butuvariesroughlyinversely astheabsolute temperature except atverylowtemperatures; inasemiconductor theveryrapidchangein theconductivity withtemperature is.mainlyduetothefactthatn variesexponentially with(l/T).Theproperties ofmetallic conductors andofsemiconductors arediscussed furtherinChapters 18and19. 851110 F 66 STEADY CURRENTS [3.2 (3.10)J=O"E} 0"div(grad V)=0 .fJ.dS=I Since0=QjV,and1{R=IjV,O{EEOisequivalent to(l{R){O";thatis,Ourrentflowinanextended medium Fromequations (3.4)and(3.5)itfollowsthatinamedium where 0" isconstant, O"divE=O.ButsinceE=-gradV,wehave V2V=0, sothatLaplace's equation holds,asinelectrostatics. Iftwoperfectly conducting electrodes areimmersed inaninfinitemedium offinitecon­ ductivity, thepotential distribution inthemedium isthesameasin acondenser, whoseplates are formedbythetwoconductors; forthe potential mustsatisfyLaplace's equation ineachcasewiththeboundary conditions V=constant onthesurfaceoftheconductors. Inthe medium, thelinesofcurrentflowareorthogonal tothelinesofconstant V,andcoincide withthelinesofE.Sincetheresistance Rofthesolu­ tionbetweentheconductors, andthecapacity0ofthecondenser formed whenthesolution isreplaced byadielectric, dependessentially onthe distribution ofthelinesofE,thereisasimplerelationbetween them. Theanalogous equations forthetwocasesare: n=EEOE EEOdiv(grad V)=0 fn.dS=Q R=EEO' -0"0(3.II) Thefactthatthepotential distribution inaconducting medium is thesameasintheelectrostatic casemaybemadethebasisofamethod offindingexperimentally thedistribution inacasewhereitisnot amenable tocalculation (see§3.8). 3.3.Thevoltaiccircuit Themechanism bywhichabatteryactsasthesourceofaconstant potential willbediscussed inChapter 4.Forthepresentpurposethe batterywillmerelyberegarded asmaintaining apotential difference between itstwoterminals. Fig.3.1illustrates theusualnotation for thecaseofabatteryBwhichcausesacurrentItoflowthrough the resistance Rconnected acrosstheterminals P,Q.Sincethebattery drivesthecurrentroundanycircuitattached toit,thepotential differ­ enceitproduces isoftencalledthe'electromotive force'ore.m.f.The totale.m.f.isequaltothelineintegralfE.dstakenroundthecircuit; 3.3] STEADY CURRENTS 67 sincenoworkisdonebyanyexternal agencywemusthave 0=V-fE.ds. Hence,usingequation (3.6), V=fE.ds=E1=I(ljaA)=IR (3.12) ifthebatteryofe.m.f.Visconnected toasingleconductor ofconduc­ tivityu,cross-section A,andlength1.Equation (3.12)isthebasisof allcalculations onresistance networks. P I FIG.3.1.AbatteryBsendinga currentIthrough aresistance R.Q..---------' FIG.3.2.Representation ofabattery byanopen-circuit e.m.f.Vandan internal resistance r.R RI p B Whenacurrentflowsthrough aconductor offiniteresistance, charge isbeingtransferred fromapointatonepotential toapointatadifferent potential. Thedirection ofpositivecurrentflowistoaplaceatalower potential sothatthereisalossofelectrical energywhichappears as heatintheconductor.IfachargedQflowsbetween twopointsdiffering inpotential byVinatimedt,theenergylostpersecond(power)is W=V(dQ/dt)=VI=V2jR=I2R, (3.13) whereRistheresistance between thetwopoints.Theunitofpoweris thevolt-ampere, knownasthewatt.Inanextended medium ofcon­ ductivity u,thepowerdissipated inanelementofcross-section dSand lengthdsisVI=(Eds)(aEdS)=aE2dT,wheredTisthevolumeofthe element. Thetotalpowerdissipated isthenfoundbyintegration over thewholevolumeoftheconductor. Inpracticeitisfoundthatthee.m.f.produced byabatteryisnot quiteconstant, butdropsslightlywhenacurrentisdrawnfromit.The variation isthesameaswouldbeproduced byanidealsourceofe.m.f.V equaltothatproduced bythebatteryonopencircuit,lessthepotential 68 STEADY CURRENTS [3.3 dropinaresistance r.Theequivalent circuitisshowninFig.3.2,and risknownasthe'internal resistance' ofthebattery.Ifthebatteryis usedtosupplypowertoaloadofresistance R,asinFig.:3.2,thecurrent Iwhichflowsisgivenbytheequation V=I(r+R). Thepowerdissi­ patedintheloadistherefore W=12R=V2Rj(r+R)2. Iftheloadcanbevaried,sothatRisadjustable, thenbydifferentiating thisexpression forWitisfoundthatithasamaximum valuewhen R=r.Thisisanexample ofthe'Maximum PowerTheorem', which statesthat,ifavariable loadistobematched toasourceofpowerso thatthemaximum poweristobedissipated intheload,itsresistance mustbeadjusted tobeequaltotheinternal resistance ofthesource. Thegreatest valueofWwhichcanbeobtained isthusV2j4r,andthisis knownasthe'available power'ofthesource.Itshouldbenotedthat withagivenload,andarangeofbatteries ofthesamevoltagebutof different internal resistance, maximum powerisobtained withthebat­ teryoflowestinternal resistance, sothatthemaximum powertheorem doesnotapplytotheconverse problem. 3.4.Resistance networks Inacomplicated network ofresistances containing manybranches, thecalculation ofthecurrents inthevariousbranches isbasedontwo lawsduetoKirchhoff. Theyare: (1)thealgebraic sumofallthecurrents meetingatapointiszero; (2)thealgebraic sumofthepotential differences acrosstheresistances inanyclosedcircuitisequaltothetotale.m.f.inthatcircuit. Thefirstlawfollowsfromtheequation ofcontinuity, sincetherecanbe noaccumulation ofchargeatanypoint.Itcanbewritten Llk=O. Thesecondlawisanextension ofequation (3.12),andcanbewritten LlkRk=Lfj, k j whereIkisthecurrentintheresistance Rk• Theselawscanimmediately beappliedtofindtheequivalent resis­ tanceofanumberofresistances RI,R2,•••,Rninseriesorinparallel, as inFig.3.3.Intheformercasethevoltageacrossalltheresistances is 3.4] STEADY CURRENTS 69 whereRistheequivalent resistance. Hence R=R1+R2+...+Rn=!Rk. k Whentheresistances areinparallel,thevoltageacrosseachisthesame. ThetotalcurrentIis V V V (111)1=R+jf+"'+F =Vjf+-R+"'+-R =VIR, 1 2 n 1 2 n I v (a) (b) v I FIG.3.3.Resistance arranged (a)inseries,(b)inparallel. whereRistheequivalent resistance. Henceinthiscase Ifanetwork hasmanybranches, theproblem offindingthecurrent ineachbranchisbestsolvedbythemethodofcycliccurrents. Fig.3.4 ispartofanetworkinwhichtherearencycliccurrents allflowingin ananticlockwise sense.Suchasystemofcurrents satisfiesthefirstof Kirchhoff's lawsautomatically. ThenthecurrentthroughR1is11,but thecurrentthrough R12is(11-12),Forcircuit(1),forexample, wehave li=11R1+(I1-13)R13+(I1-I2)R12=11Rn-I2R12-13R13' 70 STEADY CURRENTS [3.4 whereRll=R1+R12+R1S'Ingeneral,ifRqq=Rq+IRqp, p Ti=+RllIl±RlZlz±R1SIs±···±Rlqlq± ±Rlnln, l;=±RzII1+Rzzlz±RzsIs± ..·±Rzqlq± ±R2nln, FIG.3.4.Ageneralnetwork withcycliccurrents. "faisthetotale.m.f.intheqthcircuit,andispositiveifitactsinthe direction ofIq•Thesubscripts totheresistances denotethecurrents whichflowthrough them.Thenifdpqisthecofactor ofRpqinthe determinant ±Rn1 ±Rnq +Rnn dpq=dqp,sinceRpqisidentical withRqp• Ifwesolvetheseequations forthecurrentIqwhenthereisonlyone sourceofe.m.f.~inthecircuit,thenwefind Iq=~tlpq/tl. Similarly, thecurrentIp,whenthereisonlythee.m.f.v,;incircuit,is Ip="fatlqp/tl. Sincetlpq=tlqp,wehavetheReciprocity Theorem, whichstatesthat agivene.m.f.inthepthbranchwillproducethesamecurrentintheqth 3.4] STEADY CURRENTS 71 branchofacircuitasthesamee.m.f.intheqthbranchwouldproduce inthepthbranch. 3.5.Wheatstone's bridgeandthemeasurement ofresistance Asanexample ofnetwork analysis weshalltakethearrangement showninFig.3.5,consisting ofabatteryandfiveresistances. Itis required tofindthecurrentthroughtheresistance G.Bymakingthe substitutions showninthefigureforlaand12,thefirstofKirchhoff's B c (3.14) (3.15)D v FIG.3.5.Wheatstone's bridge.Thecurrent1=0ifRIR.=RIRI• (InCallendar's notation, R.=R,RI=mR',R.=nR',RI=nmR'.) lawsissatisfied andtwounknowns areeliminated fromtheequations atonce.ThenfromthesecondofKirchhoff's lawswehave v=laRa+14R4=(14+1)Ra+14R4 ) 0=l1R1-1G-laRa=l1R1-1G-(1 4+1)Ra. 0=12R2-14R4+1G=(11+1)R2-14R4+1G Elimination of11and14fromtheseequations gives I=V(R1R4-R2Ra) , D+G(R1+R2HRa+R4) whereD=R1R2Ra+R2RaR4+RaR4Rl+R4R1R2' Similarexpres­ sionscanbefoundforthecurrents throughtheotherresistances. 72 STEADY CURRENTS [3.5 (3.16)Inspection ofequation (3.15)showsthatI=0provided that R1R4=R2R3, andthisisthebasisoftheWheatstone's bridgemethodofcomparing resistances ormeasuring anunknown resistance R4intermsofthree knownresistances RvR2,R3•Agalvanometer (seeChapter 7)isin­ sertedinthearmBDtodetectwhenIiszero,andGthenrepresents thegalvanometer resistance. Theresistance inoneofthearmsRvR2,R3 isvarieduntilthebalancepointisfound. Wheatstone's bridgecanbeusedtomeasure anunknown resistance whosevaluecanhaveaverywiderange.Thecurrents usedmustnot belargeenoughtoheattheresistances andsoaltertheirvaluesappre­ ciably;thestandard resistances aregenerally madeofconstantan or manganin sothattheirvaluesdonotchangewiththeexternal tempera­ ture.Likeallbridgemethods, itisanullmethod,andthegalvanometer isrequired onlytodetectthebalancepointsothatitscalibration isnot necessary. Thebridgeisaccurate, andquicktouse;asimpleversionis theslidewireformdescribed below.Whenmeasuring smallresistances, caremustbetakentoensurethattheresistances ofleadsandcontacts arenotappreciable. Forresistances offrom10-1to10-3ohmsthe Kelvindoublebridge(seeProblem 3.5)ismoresuitable. Sensitivity ofabridge Theaccuracy withwhichthenullpointcanbedetermined inabridge depends onhowrapidlythegalvanometer currentchanges nearthe balancepointforagivenfractional changeoftheresistance inonearm ofthebridge. ThusifR2isvariable, thesensitivity isdefinedas R2(8I/8R2)atthepoint1=0;thismaybeevaluated fromtheequa­ tionsforWheatstone's bridge,buttheresulting expression isvery cumbersome. Weshallfollowatreatment duetoCallendar (1910). Theresistances inthefourarmsoftheunbalanced bridgeofFig.3.5 arewrittenintheformR4=R(theunknown), R2=rnR',R3=nR', R1=nmR',wheremandnaresimplenumbers andthecurrents are asbefore.WhenR=R'thebridgeisbalanced, I=0and14=13, Fromequations (3.14)itcanbeshownthattheratioofthecurrent throughthegalvanometer (resistance G)tothatthroughtheunknown resistance is I R-R' 14=G(I+I/n)+R'(1+m)" Theratio1/14isameasure ofthesensitivity whichisespecially useful whenthecurrentwhichcanbeputthrough theunknown resistance is 3.5] STEADY CURRENTS 73 limited;itisindependent ofthee.m.f.andresistance inthebattery circuit.Foragivenvalueof(R-R'), 1/14isamaximum whennis madelargeandmsmall,thelimitingvaluebeing 1/14=(R-R')/(O+R') (n=00,m=0). Thisis,however, onlytwiceasgreataswhenn=m=1,sothatlittle isgainedbyusingalargevalueofnandasmallvalueofm.Onthe otherhand,ifnismadesmallandmlarge,thesensitivity willbegreatly reduced (seealsoProblem 3.6). Toobtainthegreatest accuracy indetection ofthenullpoint,the galvanometer usedmustbeassensitive aspossible. Thesensitivity is proportional tothenumber ofturnsonthegalvanometer coil(see Chapter 7),butifthisnumber isgreatlyincreased theresistance may becometoohigh.Withagivensizeofcoil,thecross-sectional areaof thewireusedmustbedecreased ininverseproportion tothenumber ofturns,sothattheresistance willincrease withthesquareofthe numberofturns.Hencethegalvanometer deflexion willbepropor­ tionalto'I/Oforagivencurrent,andthesensitivity ofthebridgewill varyas1'1/0.Fromdifferentiation ofequation (3.16)(multiplied by'I/O) itisfoundthat1'1/0isamaximum atthebalancepointR=R'when 0=Rn(l+m)/(1+n) (3.17) if0istreatedasthevariable.Itisreadilyshownthatthisisjustequal tothenetresistance ofthebranches BADandBODinparallel (see Problem 3.3). 3.6.Thepotentiometer Ifapairofresistances RllR2areconnected inseriestoabatteryof potential T;.,thepotential acrossthefirstresistance isT;.R1/(R1+R2), andthispotential maybevariedbyadjusting R1orR2•Suchadevice iscalledapotential dividerandisthebasisofamethodofcomparing twopotentials bymeansofa'potentiometer'. Thebasiccircuitofthis instrument isshowninFig.3.6,whereabatteryofpotential T;.iscon­ nectedacrossaslidewireAB.Anunknown e.m.f. ~isconnected in serieswithagalvanometer between thepointAandapoint0which canbeslidalongthewire.Thetappingpointisadjusted untilthereis nocurrentthroughthegalvanometer, andthen,ifthewireisuniform, wehave Thepurposeofthelargeresistance Raistoprotectthegalvanometer fromexcessive currents duringtheinitialstagesoffindingabalance. 74 STEADY CURRENTS [3.6 Asthebalancepointisneared,Raisreducedtogivegreatersensitivity infindingthenull.Atthebalancepointnocurrentistakenfromthe battery~, sothatitsopen-circuit e.m.f.ismeasured. Currentistaken fromthebattery ~,anditse.m.f.willdependontheamountofcurrent andalsoonotherfactorssuchastemperature andthetimesince charging if~isanaccumulator. Theuncertainty duetothisiselimi­ natedbycomparison withastandard cell,whosee.m.f.ismeasured on opencircuitandisveryconstant withtime,etc.Thecircuitforthisis GAI-------------r------JB FIG.3.6.Theslide-wire potentiometer. AO=11,OB=la' illustrated inFig.3.7,whichshowsageneraltypeofpotentiometer. Theslidewireisreplaced byaresistance chainRl>R2,whosetotal resistance isconstant andhastappingpointsevery10ohms,say.This isinserieswithaslidewireDBwhosetotalresistance risexactly 10ohms.Thetotalpotential dropacrosstheresistances R1,R2and theslidewireisfirstadjusted toastandard valueinthefollowing way. R1ismadezero,sothatthetappingpoint01isatA.Thestandard cellYsisthenbroughtincircuitbyclosingthekeyKl>andtheresistance R4isadjusted untilthepotential acrossRisexactlyequaltothatofthe standard cell.Thecurrentsupplied by~andflowingthroughRisthen exactlyI=¥SfR.K1isnowopenedandK2closedtobringtheunknown voltage ~incircuitinstead. Thebalancepointisfoundfirstbyadjusting thepositionofq,givingacourseadjustment insteps,andthenvarying O2,thetappingpointontheslidewiretogivetheultimate balance. The 3.6] STEADY CURRENTS 75 potential ~isthengivenby(writingD02=11>O2B=12) l'z=I{R2+r11/(11+12}}' BysuitablechoiceofthevaluesofRandtheotherresistances, adirect reading potentiometer maybemade.Theaccuracy ofcommercial instruments ofthistypeisoftheorderof1partinlOs,sothatapotential of0·1Vcanbemeasured toamicrovolt. L...-----V-.of~ K"l--...L--" FIG.3.7.Generalpotentiometer circuit. DOJ=ll'OJB=lJ. Thepotentiometer hasmanyusesofwhichweshallmention here onlythecalibration ofavoltmeter andanammeter. Thecircuitsfor thesemeasurements areshowninFig.3.8.Tocalibrate avoltmeter avariable voltageisappliedtoitfromabatteryandpotential divider, andthevoltage ~onthevoltmeter ismeasured bythepotentiometer. Tocalibrate anammeter aknownresistance Rsisconnected inseries withit,andfromthepotential acrossthisresistance, measured bythe potentiometer, theactualcurrentthroughtheammeter isfound. 3.7.Electron optics Inaconducting solidorliquidelectrons makeveryfrequent collisions withtheatoms,andthemeanvelocitywhichtheyacquireinthedirection 76 STEADY CURRENTS [3.7 ofanelectricfieldisproportional tothefield;themotionissimilarto thatofaparticleinaviscousmedium (seeProblem 3.9).Inararefied gascollisions withatomsareinfrequent, andconduction phenomena are quitedifferent (see§4.8);atsufficiently lowpressures collisions cease toplayanyroleindetermining themotionoftheelectrons. Thisisthe (a) (b) FIG.3.8.Calibration of(a)avoltmeter, and(b)anammeter. Thevoltagev,.isapplied tothepotentiometer asinFig.3.7. positioninthermionic vacuum tubes,wheretheelectrons movedirectly fromoneelectrode toanother.Itisoftennecessary todirecttheelectrons toaparticular electrode, ortocauseabeamofelectrons topassthrough assmallanareaaspossible, asinacathode-ray tube,wherethebeam muststrikethefluorescent screeninasmallspot.Theprinciples em­ ployedinthedesignofsuchtubesareoutlined below,withoutreference toparticular applications. Inanevacuated field-free spaceelectrons travelinstraightlines,and abeamofelectrons leavinganelectrode willeventually diverge. Under theactionofasuitable electric(ormagnetic) fieldthepathoftheelec­ tronsisbent,andthebeammaybemadetoconverge. Thisiscalled -focusing' theelectrons: theuseoftheword'focus'isborrowed from opticsanditmaybeshownthatthereisaverycloseanalogybetween thebehaviour ofelectrons inanelectrostatic fieldandthatoflightin arefracting medium. Thebasisofgeometrical opticsisSnell'slaw: whenalightraypassesfromamedium ofrefractive indexn1through aplaneboundary toanother medium n2,theanglesofincidence and refraction obeytherelation 3.7] STEADY CURRENTS 77 Inelectron opticsthecorresponding caseisthatofanelectronina:field­ freespace(Le.aregionofconstant potential ~)crossing intoaregion atanotherpotential ~,asinFig.3.9.Attheboundary thereexistsan electric:fieldwhichaccelerates theelectron inthedirection normalto theboundary, whilethecomponent ofvelocity paralleltotheboundary remains unchanged. Iftheinitialandfinalvelocities areVIandV2,the A N------ VI BV2 (3.18)FIG.3.9.'Refraction' ofanelectron oncrossing aboundary AB between tworegionsofpotential 1';andVa.NNisthenormalto thisboundary. components paralleltotheboundary areVIsin(XlandV2sin(X2'sothat wehave Iftheelectronstartedfromrestatapointwherethepotential iszero, thismaybewritten sinceVisproportional to.JV.Notethattheelectron velocity playsthe sameroleastherefractive indexanddoesnotcorrespond tothevelocity oflightinthemedium. Theexample justgivenisaspecialcaseofamoregeneralcorrespon­ dencebasedonFermat's principle ofleasttimeinopticsandHamilton's principle ofleastactioninmechanics. Theformerstatesthatthepath (3.21)78 STEADY CURRENTS [3.7 ofalightrayissuchthatthetimetakenbetween anytwopointsofthe pathisanextremum (generally aminimum). Thuswehave t=fdt=f~=~fnds=minimum. (3.19) Hamilton's principle statesthatthepathofaparticleissuchthatthe lineintegralofitsmomentum isaminimum; i.e. Imvds=minimum. (3.20) Solongasthemassoftheparticleisconstant (i.e.solongasrelativity corrections arenegligible) theanalogy between electron velocity and refractive indexiscomplete. Although theformulaforthefocallengthofathinlenscanbecalcu­ latedquitesimplyinoptics,theequivalent calculation forelectronoptics isgenerally verydifficult. Weshallcontentourselves byshowing howa focusing actioncanbeobtained inasimplecase.Fig.3.10showsapair ofparallelconducting planeseachwithasmallcircularaperture. The planesareatpotentials"fi,~,andthepotentials outsidetheplanesaway fromtheaperture areconstant andequalto"fiand~.Neartheaperture theequipotentials arecurvedandbulgeoutasshown.Ifanelectron travelling paralleltothez-axis(i.e.normaltotheplanes)entersthe aperture, itfindsitselfinaregionwherethelinesofelectricfield,which arenormaltotheequipotentials, havearadialcomponent. Thisgives theelectronanacceleration normaltotheaxis,anditemergesintothe field-free regionwithacomponent ofvelocitytoorawayfromtheaxis. Iftheelectron isinthexzplaneasshowninthediagram, thexcom­ ponentofvelocity giventoitis t t V=-f!:...Edt=+!:...f(c3V)dt.x mXmc3x o 0 Iftheelectron enterstheaperture atasmalldistance hfromtheaxis, thenwemaymaketheapproximation (c3~_(c3V) +h(c32~. c3X}X=h-c3xx=ofJx"2) Sincethepotential satisfiesLaplace's equation c32Vc32Vc32V c3x2+c3y2+c3z2=0, where,bysymmetry, c32V/c3x2=c32V/c3y2,and(c3V/c3x)=(c3V/c3y)=0 3.7] ontheaxis,wehave fromwhichSTEADY CURRENTS 79 v= -f!~h(02Ddt= -!!'-hf(o2!]dZt, z 2mOZ2) 2m OZ2}VZ where Vz=dz/dtisthezcomponent oftheinstantaneous velocityat anypoint.Onemerging fromtheaperture theelectron hasavelocity V2=(2e~/m)1 whichisindependent ofh,anditmovesatanangle {} z y FIG.3.10.Asimpleelectron lens,consisting oftwoparallelconducting planes atpotentials ~,Vawithapertures. Thebulgingoftheequipotential surfaces neartheapertures isshown. withtheaxiswheresin{}=Vz/V2'SinceVzisproportional toh,forsmall valuesof{}(wherethedifference between sinOandtan0isnegligible) allelectrons willmovetowards (orappeartodivergefrom)aparticular pointontheaxis,whosedistance fromtheaperture is Astheelectrons wereassumed toenterparalleltotheaxis,thisisone focalpointofthelens.IfV;xisnegative whenhispositive wehavea 80 STEADY CURRENTS [3.7 converging lenswhosefocallengthisgivenby 1 Vxe/2mf(B2!Jdzh.= -hV2=(2e~/m)l BZ2)v =(2:~;)l f(~z~(2e:~m)l =4~~f~~(~:::)dz,(3.22) wherewehaveassumed thatVz=v,theactualvelocityatanypoint, andsubstituted (2eV/m)l foritinsidetheintegral. Iftheelectrons hadenteredthelensfromtheright,theywouldhave beenbroughttoafocusatapoint/1'where*= -4~Jif)V(~:~dz. (3.23) Hence11/12=-,.,;v,:/..J~, aformula whichisexactlyanalogous tothe opticalcaseofathinlenswithinitialandfinalmediaofdifferent refrac­ tiveindicesn1andn2• Inspection ofequations (3.22)and(3.23)forthefocallengthsshows thattheyrequireaknowledge ofthevariation ofthepotential onthe axisofthelens.Itisonlypossibleinverysimplecasestoderivean analytical expression forVintheaperture, andingeneralthevariation ofVmusteitherbecalculated bynumerical methods, bywhichan approximate solutionofLaplace's equation withtherequired boundary conditions maybefound,orexperimentally bytheuseofanelectrolytic tank(seebelow).Thevariation ofValwaysoccupies afinitedistance, andifitoccupies adistance comparable witheitherofthefocallengths theelectronlensisa'thicklens'ratherthana'thinlens'.Thebehaviour ofthesystemisagainsimilartothatofanopticalsystem,andisdefined ifthecardinal pointsaredetermined. Thesecanbefoundbytracing thepathsofanumberofelectrons through thesystem. Onemethod ofdoingthisistodividethepotential fieldintothinslicesalongthe equipotential surfaces, andtreateachsliceasathinlens. Theapproximations madeaboveinexpanding (BV/ex)neartheaxis andretaining onlythefirsttermareequivalent totheapproximations madein'Gaussian optics' intreating onlyraysneartheaxis.Itisto beexpected, therefore, thatelectronlenssystemswillsufferfromdefects similartothoseofopticalsystems, suchasspherical aberration, etc. Theequivalent of'chromatic aberration' ariseswhennotallelectrons enterthesystemwiththesamevelocity, sincethiscorresponds toa variable refractive index.Ifthespreadinvelocity isdueonlytothe Maxwellian distribution ofvelocity onemission fromthecathode, 3.7] STEADY CURRENTS 81 chromatic aberration issmall.Anadditional effectinelectron lenses, notpresentinopticalsystems, arisesfromthemutualrepulsion ofthe chargedparticles, whichwillcauseabeamofelectrons initially moving paralleltoeachothertodiverge. 3.8.Theelectrolytic tank Itwasshownin§3.2thattheequipotential linesina.conducting medium between twoconductors offixedshapeandposition remain unalterediftheconducting medium isreplaced byadielectric. The electrolytic tankisadeviceforplottingthelinesofconstant Vexperi­ mentally usingelectrodes immersed inaconducting solution (tapwater isusuallysufficiently conducting forthispurpose). Suchadeviceis oftenusedincaseswheretheoretical calculation ofthepotential distri­ butionisdifficult;itisnotusuallyfeasibletoconstruct ascalemodel forathree-dimensional problem, butoftentheproblem canbereduced toatwo-dimensional one.Thesimplest caseisonewheretheconductors extendindefinitely inonedimension (e.g.thez-axis)without changeof cross-section. Aslabofconducting solution withparallelplanefaces maythenbeusedtosimulate asectionnormaltothez-axis.Sinceno currentcanflowoutofthesidesoftheslab,theequipotential surfaces inthesolution willalwaysbenormaltothesides,anditisessential thatthesesidesbenormaltotheequipotential surfaces inthethree­ dimensional electrostatic problem ofwhichthesolution givesamodel. Inthecaseunderconsideration thewatercanbecontained inaninsu­ latingtankwhosebottomisplaneandhorizontal. Thetankmustbe sufficiently largeincomparison withtheregionoverwhichthepotential distribution isimportant sothatthedistortion oftheequipotentials at thesidesofthetankdoesnotaffecttheproblem. Theapparatus isshowninFig.3.11.Thetankcontains twoelectrodes A,0whicharescalemodelsofthoseintheelectrostatic problem where thepotential distribution isdesired. Theyareconnected toalow­ frequency alternating-current generator toavoidpolarization effects andelectrolysis ofthesolution (seeChapter 4),andalsototwovariable resistances RI,R2inseries.Adetector ofalternating currentsuchasa pairofheadphones isconnected between themid-point Boftheresis­ tancesandasmallverticalprobeDimmersed inthesolution. The resistances ofthesolution betweentheprobeDandtheelectrodes A,0 thenformaWheatstone's bridgewithRIandR2,sothatnocurrent flowsthroughthedetector whenthepotential atBisthesameasthat atD.If0isatzeropotential, andAatpotential V,thenthepotential 851110 G 82~~~-------_ ..._---~~-- STEADY CURRENTS [3.8 atBisVRs/(R1+R,.). Theequipotential lineinthesolutionwiththis potential canthenbetracedoutbymovingtheprobesothatthedetec­ torcurrentisalwayszero.Theprobeisfixedtoaframework which slidesalongtwoperpendicular guIdeswithscales,anditsposition can bereadintermsofthesetwocoordinates andplottedongraphpaper. r---------( rv}---------, D FIG.3.11.Theelectrolytic tankforplotting equipotential lines. REFERENCES CALLENDAR, H.L.,1910,Proc.Phys.Soc.22,220. KETTERlNG, C.F.,andSCOTT,G.G.,1944,Phys.Rev.66,257. TOLMAN, R.C.,andSTEWART, T.D.,1917,ibid.9,64. PROBLEMS 3.1.Deduceequation (3.17)directly fromequations (3.14)and(3.15). 3.2.InFig.3.5showthatifthegalvanometer isremoved (0=00)theopen circuitvoltage acrosstheterminals EDis v=V(R1R,-RaRa)/(Rl+Ra)(Ra+R,) STEADY CURRENTS 83 whileifalsothebatteryisshort-circuited, theresistance measured attheterminals BDwould be RiRsRaR,r=---+---.Ri+RsRa+R, Showthatwiththegalvanometer andbatteryinplace,thebridgebehavesasa generator ofe.m.f.equaltov,andinternal resistance r,thegalvanometer beingthe load.Thisisanexample ofThevenin's theorem, whichmaybestatedasfollows: r FIG.3.12.Kelvin's doublebridge. Iftheopencircuitvoltageacrossterminals B,Dofanetwork isv,andifwhen BandDareshort-circuited acurrentIflowsbetween them,thentheresistance ofthenetwork measured betweenBandDafterallsourcesofe.m.f.havebeen short-circuited isr=vII.Thustheeffectonanycircuitconnected acrossBD willbethesameasthatofagenerator ofe.m.f.equaltovandinternal resistance r. (Foraproofofthistheorem, seeW.R.Smythe, StaticandDynamic Electricity, McGraw-Hill.) 3.3.Deriveequation (3.17)fromtheresultsofthelastproblem andthemaximum powertheorem. 3.4.Resistances P,Q,Reachof10ohmsareplacedinthreearmsofaWheatstone's bridge,andaresistance Sisadjusted inthefourtharm80thatthebridgeis balanced. Theresistance Risnowreplaced byaresistance X,andthebalance isrestoredbyshunting Switharesistance of10123ohms.WhatisthevalueofX1 Discusstheadvantages anddisadvantages ofthismethodofmeasuring resis­ tanceswhenhighaccuracy isrequired. 3.5.InFig.3.12,showthatnocurrentflowsthroughthegalvanometer provided thatRilRs=RalR,=RalRe.ThisisKelvin's doublebridgeformeasuring smaIl resistances oftheorderof0·01ohm.Theresistance rrepresents thecontact resistance between thetwosmallresistances RiORs'anditsvaluedoesnotaffect thebalance. Ifreadings aretakenwiththecurrents flowinginbothdirections, sothaterrorsduetothermoelectric e.m.f.satthejunctions areavoided, an accuracy ofabout0·02percentcanbeachieved. 84 STEADY CURRENTS 3.6.ThetotalamountofpowerW, whichcanbedissipated inthofourarmsofa Wheatstone's bridgeisfixed.Showfromequation (3.16)thatclosetobalance {R-R') {mW}t 1=G(l+ljn)+R'(l+m) R'(l+m)(l+n) • Hence,bydifferentiation withrespecttomandton,andusingoquation (3.17), showthatthemostsensitive arrangement ofthebridgeiswhen n=m=1,G=R. 3.7.SixI-ohmresistances arejoinedtoformaregulartetrahedron, andapotential of1Vismaintained acrossoneoftheresistances. Findthecurrentflowingin eachconductor. (Answer: 1,t,t,t,t,and0A.) 3.8.Twolongparallelcopperrodsofradiusofcross-section 0·25emareplaced withtheiraxes20emapartwithinalargetankofcoppersulphate solution.If theconductivity ofthesolution is4·1ohm-1metre-l, findtheresistance perunit lengthbetween therods. (Answer: 0·34ohmjmetre.) 3.9.Aparticleofmassmandchargeemovesinaviscousmedium wherethereis uniform electricfieldEparalleltothex-axis.Showthattheequation ofmotion maybewrittenas di·1Ejdt+XT =em, andthatitssolution, foraparticlestarting fromrestatt=0is i=(eETjm){I-exp(-tjT)}. Foranelectron inametal,theeffectofcollisions issimilartothatofaviscous force,andthevalueofTis;::j10-14sec(see§4.1).Thustheexponential termin theequation forxabovequicklyfallstozeroandthevelocity isproportional to theelectricfieldstrength; Tisknownasthe'relaxation time',sinceitgivesa measure ofthetimerequired toreachthenewequilibrium velocity whenthefield strength isaltered. Themobility oftheelectron isu=ijE=(ejm)T,andhencetheconductivity ofametalcontaining nelectrons perunitvolumeisu=n(e2jm)7(seeequation 4.3). 4 PROPERTIES OFELECTRICAL CONDUCTORS (4.2)4.1.Freeelectrons inmetals-classical theory ASIMPLE explanation ofmetallic conductivity wasputforwardby Drudein1900,basedonclassicaltheory.Itwasassumedthatinametal someelectrons arefreetomoveaboutthewholevolumeofthemetal likethemolecules ofaperfectgasinacontainer. Intheabsenceofan electricfieldtheelectrons moveinrandomdirections, makingcollisions fromtimetotimewiththepositive ions(whicharefixedinthelattice) orotherfreeelectrons. WhenanelectricfieldEisappliedtothemetal, theelectrons areaccelerated inthedirection ofthefieldandacquirean averagedriftmomentum p,paralleltoE.Thevalueofpcanbecal­ culatedasfollows.Inatimedtanelectronofcharge-eacquires an additional momentum -eEdtthroughtheacceleration bythefieldE. Inthetimedtafractiondnofthetotalnumberofelectrons nperunit volumemakecollisions, where dnjn=dtjT. (4.1) Theparameter Tisthemeantimebetween collisions, ascanbeverified byreference totextbooks onkinetictheory,ortoShockley (1950).We nowmaketheassumption thatimmediately aftercollisions theelectron velocities arecompletely random, sothatthemomentum gainedunder theinfluence oftheelectricfieldislost.Themomentum gainedintime dtis-neEdt,whilethemomentum destroyed incollisions is pdn=npdtjT. Forequilibrium thesemustbalance, sothat npdtjT=-neEdt p=-eET. or Thecurrentdensityis J=n(-ejm)p =n(e2jm)ET andhencetheconductivity is a=JjE=n(e2jm)T. (4.3) Anestimate ofthevalueofTcanbefoundforcopperifweassumethat thereisjustonefreeelectronperatom;thenfromthespecificresistance giveninTable3.1,avalueofT~2 X10-14secondsisobtained at20°C. 86 PROPERTIES OFELECTRICAL CONDUCTORS [4.1 Asisobviousfromequation (4.3),theconductivity isindependent of thesignofthechargeonthecarriers, sincereversing thissignchanges thedirection oftheirdriftmotionbutnotthedirection ofthecurrent flow.Equation (4.2)showsthatthemeandriftvelocityv=plmis proportional totheappliedfield,andtheratioIv/EI(irrespective ofthe sign)isdefined(seeequation 3.9)asthemobility u.Thisquantity is ausefulparameter, sinceitisdirectlyproportional toT: u=Iv/E\=Ie/miT. (4.4) Theunitofuismetresperseconddividedbyvoltspermetre=m2/Vsec; howeveritisnearlyalwaysquotedincm2/Vsec.Forcopperat200C wefindu~.40cm2/Vsec,whereagainwehaveassumed onefreeelec­ tronperatom. Sincemetalsaremuchbetterconductors ofheatthanelectrical in­ sulators, wemayassumethat.thethermalconduction inametalisalso mainlyduetothefreeelectrons.Ifweapplytheordinary kinetictheory formulaforthethermalconductivity Kofagastothe'electron gas'in aconductor, wehave K=Incl(dW/dT), (4.5) wherecistherandomelectronvelocityandWitskineticenergy=imc2• TakingthemeanfreepathlasequaltoCT,wefindasimpleexpression fortheratioofthermaltoelectrical conductivity: Kmc2dW2WdW -;;-3e2dT=3e2dT' Iftheelectrons obeyclassical statistics, thenW=fmc2=!kT,and K/a=~(~rT. (4.6) Thisequation showsthattheratioofthethermalandelectrical conduc­ tivitiesshouldbeproportional totheabsolute temperature foragiven metal,andshouldbethesameforallmetalsatagiventemperature. Thisisinaccordance withanempirical lawdiscovered byWiedemann andFranzin1853,andthenumerical valueof(K/aT)givenbyequa­ tion(4.6)isingoodagreement withtheexperimental valuesforcopper, silver,andgoldoverthelimitedtemperature rangeoftheexperiments. Theexperimental valuesofKandathemselves andtheirvariation with temperature donot,however, fitwithDrude's theory.Fromequation (4.3),ifl=OTisfixed,ashouldvaryasT-!becauseofthevariation intheaverage velocity 0oftheelectrons. Similarly, Kshouldvary asTl.Inpractice,atordinary temperatures Kisfoundtobepractically 4,.1] PROPERTIES OFELECTRICAL CONDUCTORS 87... constant and0'variesroughly asP-l.Atlowtemperatures 0'varies morerapidlystill;Fig.4.1showstheresistivity ofsodiumatvarious temperatures asafractionofitsvalueat2730K.Thethermalconduc­ tivityalso variesatlowtemperatures butinadifferent waysothat 1// i /i / /I /I /I II i !/I I /i I / ~v "1·0 0·8 0·6 ...L P.?3 0·4 0·2 o 100 200 300 FIG.4.1.Theresistivity ofsodiumasafunction oftemperature. (X/uP)isnotconstant. Thechiefobjection toDrude's theoryarises, however, fromthefactthattheatomicheatofmetalsshouldbegreater thanthatofinsulators by3R/2,corresponding totheexpression forthe average energyWassumed above.Inpractice theatomicheatsof metalsatordinary temperatures arenotsignificantly greaterthanthose ofinsulators, showingthatthecontribution fromtheelectrons ismuch smallerthan3R/2.Thisdifficulty wasnotovercome untilitwasrealized thatelectrons shouldobeyquantum statistics ratherthantheMaxwell­ Boltzmann statistics assumed intheclassical modelofa'free-electron gas'.Infact,sincetheelectrons haveanintrinsic spinangularmomen­ tumofiii=l(h/21T),wherehisPlanck's constant, theymustbetreated 88 PROPERTIES OFELECTRICAL CONDUCTORS [4.1 bythetypeofquantum statistics associated withthenamesofFermi andDirac. 4.2.Freeelectrons inmetals-quantum theory Onthefreeelectron modeltheconduction electrons areconfined to thevolumeofthemetal,butarequitefreetomoveaboutinsidethis volume, likegasmolecules inabox.Onclassical theorythekinetic energyofsuchaparticlecanhaveanyvalue,andthereisacontinuous distribution ofvaluesoftheenergy,thoughsomearemoreprobable thanothers.Whenthetemperature fallstheaverage energyofthe particles decreases linearlywiththetemperature, becoming zeroat0°K; atabsolute temperature Tthetotaltranslational energyofNparticles isiNkT,andthedifferential ofthisgivesthecontribution 3RJ2tothe molarheat.Onquantum mechanics noteveryvalueoftheenergyis allowed, andthecontinuous distribution ofenergies isreplaced bya discretesetofallowedenergylevels.Thespacingisextremely small, however, andthedifference between thisandtheassumed classical continuous distribution produces noobservable effectforrealgases. Itwouldbecomeappreciable onlyattemperatures solowthatordinary substances havenegligible vapourpressure, andathighertemperatures itisalwaysmaskedbydeviations fromtheperfectgaslawsowingtothe vanderWaalsforces.Inthecaseofelectrons inametal,thespacing oftheenergylevelsisratherlargerbecauseofthesmallermassofthe electron, andthenumberperunitvolumeismuchlargerthaninany realgas.Theresultofthisistoemphasize theroleplayedbythePauli exclusion principle, whichstatesthatnotwoelectrons inagivensystem canhavethesamesetofquantum numbers. Whenallowance ismade fortheintrinsic spinangularmomentum oftheelectron, thismeans thatonlytwoelectrons canoccupyanygiventranslational energylevel. Hencethekineticenergyoftheelectrons cannotbezeroat0°K,since thiswouldmeanthatalltheelectrons wereinoneparticular energylevel. Infacttheelectrons occupythelowestpossiblesetofenergylevelscon­ sistentwiththePauliexclusion principle, andtheirmeanenergyisvery farfromzeroat0°K.Atafinitetemperature theenergydistribution canonlybefoundusingtheFermi-Dirac statistics; i.e.thequantum statistics whichtakeaccountofthePauliexclusion principle. Atordinary temperatures itturnsoutthattheenergydistribution differsverylittle fromthatat0°K;thelattercanbefoundwhenthevaluesoftheallowed energylevelsareknown,andasimplemethodofcomputing thelevels willnowbegiven. 4.2] PROPERTIES OFELECTRICAL CONDUCTORS 89 Thefactthatelectrons andotherparticles haveawave-like aspectis wellestablished fromexperiments onelectronandneutron diffraction; thewavelength associated withaparticleoflinearmomentum pisgiven bythedeBroglierelation 27Tjk=A=hjpork=P(27Tjh)=pjli, (4.7) (4.8)wherekisthe'wavevector'whichisparalleltothedirection inwhich thewaveistravelling, andhisPlanck's constant andli=hj27T.For example, thewavelength is12·3X10-8emforanelectron ofkinetic energyequaltoIeV.(OneeVistheenergyacquired byanelectronin fallingthrough apotential difference ofIvolt.)Freeelectrons confined withinametalrebound fromthesurfaceofthemetalwithout losing anyenergy;onthewaveaspectthismeansthatthewavesaretotally reflectedattheboundaries, andstanding wavesaresetup.Thesestand­ ingwavesaretheallowedsolutions ofthewaveequation foraparticle inabox,inthesamewaythatstanding wavesofcertainwavelengths areallowedinawaveguide resonator (see§11.7).Byanalogywiththe theoryofheatradiation, thenumberdiofallowedwavelengths inthe rangeAtoA+dAis whereVisthevolumeofthebox.Forparticles thewavelength isdeter­ minedbythemomentum, andfromequation (4.7)thenumberdiof possiblevaluesofthemomentum intherangeptop+dpisfoundtobe (4.9) !1Px!1x=h,Thisrelationmaybederivedinanotherwaybytheuseoftheun­ certainty relation. Themomentum ofaparticleiscompletely specified inmagnitude anddirection bythecomponents Px'Py,pzalongthree Cartesian axes,andthesquareofthemomentum isp2=p~+p~+p~. A'momentum space'maybeconstructed asinFig.4.2wheretheco­ ordinates arethecomponents ofthemomentum (Px,Py,Pz) insteadof thecomponents ofposition(x,y,z),andthemomentum ofaparticleis thenspecified byapointinthisspace;themagnitude anddirection of themomentum aregivenbythelengthanddirection ofaradiusvector drawnfromtheorigintothepoint.Allvaluesofthemomentum which liebetween pandp+dparerepresented bypointswhichliewithinthe spherical shellbounded bytheradiipandp+dp,andthevolumeof thisshellis47Tp2dp.Bytheuncertainty relation, themomentum com­ ponentPxcannotbedetermined moreprecisely thantoanamount!1px' where 90 PROPERTIES OFELECTRICAL CONDUCTORS [4.2 wheredxistheuncertainty initsposition coordinate; andsimilarrela­ tionsholdfortheothertwoaxes.Hence dPxdpydpz =h3j(dXdydz) =h3fV iftheparticle isonlyconstrained tobewithinthevolumeV.Now dpxdpydpzisanelement ofvolumein'momentum space',andthe Pauliexclusion principle maybestatedintheformthatonlyonepoint I•(P.,P.,P.) ..,.-----t--t-------'O:p::"'o .....--+P. ) ----~// FIG.4.2.Themomentum pofaparticleisspecified byitscom· ponents (p",.Pll'P.).corresponding toapointin•momentum space'. At0°Kallelectrons arespecified bypointswithinthesphereof radiusPo.sincethismakestheenergyaminimum. inmomentum spaceisallowedineachelementofvolumeofsizeh3jV. Henceinthevolume 4'7Tp2dpofmomentum spacetheallowednumber ofpointsis47Tp2dpj(h3jV),whichisthesameasequation (4.9).Inaddi­ tionwehavetoconsider theangularmomentum oftheelectronindue toitsintrinsic spin,whichcanhaveoneoftwocomponents ±Inalong anyaxis(see§20.2).ThePauliprinciple thenallowstwoelectrons, withopposite valuesoftheseangularmomentum components, tohave thesametranslational energy,i.e.wecanassignamaximum oftwo electrons toeachpointinthe(linear)momentum space. Thetranslational energyWofanelectronofmomentum pandmassm isp2j2m,andthetotalkineticenergywillbesmallest when~(p2j2m) hasitsminimum value.Itisreadilyseenthatthisoccurswhenthe pointsinmomentum spacejustfillasphereoftheleastvolumewhich willaccommodate alltheelectrons, sinceifwereplaceanypointwithin 4.2] PROPERTIES OFELECTRICAL CONDUCTORS 91 (4.10) (4.11) andhencethespherebyoneoutside,thevalueofPandhenceoftheenergywill belarger.IftheradiusofthissphereisPo,thenitsvolumeisi17P~and thenumberofpossiblepoints(i.e.allowedvaluesofthemomentum) is (Vjh3)(i17p~). Sincewecanhaveamaximum oftwoelectrons perpoint, ifthetotalnumberofelectronsinthevolumeVofmetalisN,wemust have (4.13) (4.14)(4.12)Heren=NjVisthenumberofelectrons perunitvolumeofthemetal andlVz;.istheenergyofthehighestleveloccupied at0°K;lVz;.isknown asthe'Fermienergy'. Itisconvenient toexpressthedistribution intermsofenergyrather thanmomentum. ByusingtherelationW=p2f2m,andequations (4.9-11), thenumber (2fV)diofstatesperunitvolumewhichhave kineticenergyintherangeWtoW+dWisfoundtobe g(W)dW=(2jV)di=817p2dpjh3 =~(2m)!W1dW 21721i2 3nW1dW-2Wi Ifthequantity g(W)isplottedagainstW,asinFig.4.3,aparabola is obtained; atT=0eachstateisoccupied byanelectron uptothe sharpcut-offattheFermienergy lVz;..ThemeanenergyWoftheelec­ tronsmaybefoundintheusualway: Wp Wp W=~IWg(W)dW=iWFiIWidW=ilVz;., o 0 andthetotalinternal energy ~ofnelectrons at0°Kisthen Uo=tnlVz;.. (4.15) Thegreatdifference between thistypeofenergydistribution andthe Maxwell-Boltzmann distribution canonlybeappreciated ifnumerical valuesareconsidered. Equation (4.11)showsthatlVz;.depends onthe metalonlyinsofarasn,thenumberofelectrons perunitvolume, varies.Ifwetakesodiumasatypicalexample andassumethatjust theonevalenceelectron peratomisreleased inthemetalasafree electron, then 92 PROPERTIES OFELECTRICAL CONDUCTORS [4.2 andWFisfoundtobe3·1eV.ThemeanenergyWisiofthis,or1·9eV. Thisisverylargeindeed. Ontheclassical Maxwell-Boltzmann statis­ tics,wherethemeanenergyis!kTatanabsolute temperature T,it isequivalent toatemperature of200000K.Thevaluesforothermetals calculated inasimilarwayaregiveninTable4.1. TABLE 4.1 ValuesofJVp.(the'Fermi'energy),theworkfunction eP(asdeducedfrom measurements ofthephotoelectric effectandthermionic emission), andthe thermionic emission constant A WorkJunction '" Photoelectric Thermionic WF effect emission A Substance (eV) (eV) (eV) (ampcm-2deg-2) Li 4·7 2·2 Na 3·1 1·9 K 2·1 1·8 Cu 7·0 4·1 4'5 UO Ag 5·5 4·7 4·3 107 Au 5·5 4·8 4·25 100 1\10 5·9 4·2 50-U5 W 5·8 4·49 4·5 20-60 Pt 6·0 >6·2 5·3 30 Ni 7·4 4·9 4·5 120 Thoriated tungsten 2·6 60 (BaO,SrO) mixture 1·8 3 Bywayofcomparison, wewillcalculate thevalueofTJ:Fforamon­ atomicgasoftherareisotopeofmass3ofhelium,atomsofwhichshould alsoobeytheFermi-Dirac statistics. Theboiling-point ofthisgasis 3·20K,andthenumberofatomspercm3atatmospheric pressure in thegasatthistemperature is2·3X1021•ThisgivesWF=1·15x10-4eV, whichisequivalent onlytoatemperature of1,30K.Thisissmallcom­ paredwiththeactualtemperature, indicating thatdeviations fromthe perfectgaslawsduetoquantum effectswillnotbelarge.Itisobvious thatthelowvalueofWFforthis,oranyotherrealgas,ascompared withtheelectron gasinametal,isdueprimarily tothedifference in themassoftheparticles (whichcomesinthedenominator ofequation (4.11)),andalsopartlytothesmallernumberperunitvolume. When WF~kT,theparticles haveallenergies uptothoseoforderkT,and thedensityofoccupied pointsinmomentum spaceisJaw;thatis,the chanceoffindinganoccupied pointinthefundamental volume(h3fV) issmall.Underthesecircumstances theclassical statistics areavalid 4.2] PROPERTIES OFELECTRICAL CONDUCTORS 93 approximation. Foranelectron gas,ontheotherhand,WF~kTatall ordinary temperatures, andeveryfundamental volume(h3jV)ofmo­ mentum spacecontains anoccupied point,onlyonesuchpoint(ortwo electrons, allowing forthespin)ineachsuchvolumebeingallowedby theexclusion principle.Ifwetriedtousetheclassical picture,withthe averageenergyoftheorderkT,thiswouldcorrespond toputtingalarge numberofelectrons ineachvolume(h3jV)ofmomentum space,andthe exclusion principle wouldbeviolated. Theenergydistribution atafinitetemperature isgivenbyFermi­ Diracstatistical mechanics, andwequotetheresults.Each'point'in momentum spacecorresponds toaquantized stateoftranslational motion,andinclusion oftheelectron spingivestwoquantum states toeachpoint.Thenumberofsuchstatesg(W)dWperunitvolumein theenergyrangeWtoW+dWisknownasthe'densityofstates',and fromequation (4.12) g(W)=_1_(2m)!Wi=OmiWi, (4.16) 2172li2 where0isaconstant.IfIistheprobability thatanelectron occupies agivenstate,thenthenumberdnofelectrons withenergybetween W andW+dW isdn=lg(W)dW. Thequantitylisafunction bothof energyandoftemperature, beinggivenby Hence1 1=exp{(W-~)jkT}+f dn= OmiWi dW exp{(W-~)jkT}+ 1 andthetotalnumberofelectrons perunitvolumenis(4.17) (4.18) (4.19)00 n=Omif Wi dW.exp{(W-~)jkT}+ 1o Thisintegralcanbeevaluated numerically, butattemperatures where kT<WFoapproximate methods canbeused.AtT=0thedenomina­ torisinfiniteforW>~,andunityforW<~,sothatI=0inthe formercaseand1inthelatter.Thiscorresponds tothesharpcut-off intheoccupation ofstatesalreadydiscussed. Atafinitetemperature I isstillveryclosetounitywhenW<WFandtozerowhenW>~ exceptfortherangeofenergies whichliewithinafewkTof~.The distribution appropriate toatemperature ofabout2000°Kisshown inFig.4.3. 94 PROPERTIES OFELECTRICAL CONDUCTORS [4.2 Fromequation (4.17)JJj,.maybedefinedastheenergyatwhichthe probability ofastatebeingoccupied byanelectron isf=t,sincethe denominator istheneO+1=2.TheactualvalueofUFcanonlybe oI I dnldW oI I I/g(W) I dnldW FIG.4.3.Thenumberoffreeelectrons dnwithkineticenergybetween WandW+dW inametal,asgivenbytheFermi-Dirac distribution. At0°Ktheelectrons haveenergies onlyuptotheFermilevel(WF)o;thedistribution at2000°Kisshownontheright. Atroomtemperature thedistribution ismuchclosertothatontheleft,sincednjdWis onlyalteredforelectrons whoseenergylieswithin ~kTofWF' foundfromequation (4.19);ifwedenotethevalueforT=0defined byequation (4.11)by(UF)o,itcanbeshownthat • 'lT2(kT)2.WF~(JJj,.)o-12 JJj,.• (4.20) Thedifference isoforder(kT)2jWF,andatordinary temperatures can beneglected formanypurposes, butnotincalculating differentials such asthespecificheat.ThebasicreasonisthatbecauseoftheExclusion Principle anelectron canonlymoveintoanunoccupied state,andas thetemperature increases theextraenergyavailable isoforderkT. Thusonlytheelectrons withenergies differing fromWFbyamounts of thisordercanmovetohigherlevels,andthisisafraction oforder kTjWFofthetotalnumber. Thustheincrease ininternal energyis ofordernkT(kTjWF),andthespecificheatissmallerthantheclassical 4.2] PROPERTIES OFELECTRICAL CONDUCTORS 95 value3RJ2permolebyafactoroforder kTJ~.Theelectronic specific heatwillbediscussed furtherin§18.4,butinthefollowing paragraphs wecanusuallyneglectthedifference between theactualFermi-Dirac distribution andthatat00K,sinceatroomtemperature kTisequiva­ lentto0·025electron voltswhileWFisseveralvolts. 4.3.Workfunction andcontactpotential Theenergyrequiredtoremoveanelectronfromthetopoftheenergy distribution outofthemetaltoinfinityiscalledtheworkfunction, ep, andvaluesofepfordifferent metalsarealsogiveninTable4.1.Thefact thatepvariesfrommetaltometalgivesrisetothephenomenon of'con­ tactpotential'.Ithaslongbeenknownthatwhentwometalsareplaced incontact, thereisapotential difference between them,butitwasnot atfirstgenerally accepted thatthiswasafundamental property ofthe metals.FromFig.4.4itwillbeseenthatifmetalAhasasmallerwork function thanmetalB,electrons fromthetopoftheenergybandinA canflowintometalBwhencontactismade,sincetheywillthenhave alowerenergy. Thisflowcreatesapotential difference betweenBand Awhichincreases untilthetopsofthetwoenergydistributions reach thesamelevel,whennomoreelectronswillbetransferred, andequi­ libriumisattained. Theactualnumberofelectrons transferred isonly aninsignificant fractionofthetotal,sothattheareasinFigs.4.4(a) and(b)areequal.Hencethecontactpotential isequaltothedifference oftheworkfunctions. InFig.4.4thedistributions areshownappropriate toT=0,butin facttheequilibrium condition atanytemperature is~A=~B;this makesthedistribution functions fA'fBmatchforeveryvalueofW, since 1 1 fA=exp{(W-~A)JkT}+I' fB=exp{(W-~B)JkT}+1 Adetailed statistical treatment showsthat~isequivalent tothe 'thermodynamic potential', whichmusthavethesamevalueforall systemswhentheyareinthermalequilibrium (see,forexample, Dekker, A.J.,1958,SolidStatePhysics(Macmillan)). Theworkfunction ofametal,andhencealsothecontactpotential between twometals,isverysensitive tothestateofthesurface. For thisreasonmeasurements ofepshowratherawidescatter,andthe bestdeterminations aremadewithmetallic filmsnewlydeposited by evaporation invacuo(formetalswithlowboiling-points), or(forahigh melting-point metalsuchastungsten) withasurfacecleanedby'flashing' PROPERTIES OFELECTRICAL CONDUCTORS 96 IV"~-X- II! <PA MetalA MetalA-. I <PH ! MetalB -J.F{-r ___.!F,;;Ht-_ ....._-t MetalB(It) (Ii)[4.3 FIG.4.4.Energydistribution ofelectrons intwometals,A,B. (a)Beforecontact: <PA'<PBaretheenergies required toremoveanelectron torestat infinity. (b)Aftercontact: electrons flowtometalB,changing itspotential relativetoAuntil thetopsofthetwoenergydistributions arelevel(thetransfer ofelectrons required isaninsignificant fraction ofthewhole). Thecontact potential difference thusset upispractically equalto<PB-<PA' andanelectron released fromB(e.g.bythephoto­ electriceffect)wouldgainthisamountofenergyinmoving fromapointjustoutside BtoapointjustoutsideA. themetalatatemperature closetothemelting-point invacuo.Each oftheseprocesses removes tracesofabsorbed gasfromthemetalwhich affecttheworkfunction. Thecontactpotential between twosurfaces ismeasured indirectly asfollows(see,forexample, MitchellandMitchell, 1951).Anarrowbeamofelectrons fromanelectron gunisdirected on 4.3] PROPERTIES OFELECTRICAL CONDUCTORS 97 tothemetallic surface,andthecurrentreaching thesurfaceisplotted asafunction oftheretarding potential appliedtothesurface. When asecondmetallic surfaceissubstituted, andthecurrenttoitplotted, a curveofsimilarshapeisobtained butdisplaced byanamountequalto thecontactpotential difference between thetwosurfaces. Theworkof Mitchell andMitchell gavethefollowing meanvaluesforthecontact potential relativetoacleantungsten surface: copper-0·05±0·02 V; silver+0·23±0·03 V;aluminium +0·31±0·03 V. 4.4.Emission ofelectrons frommetals Ifanelectroncanacquireanexcessenergyatleastequaltothework function, itcanescapefromthemetal,anditwillthentraveltoanearby electrode heldatapositive potential withrespecttotheemitting sur­ face.Acontinuous flowofsuchelectrons constitutes acurrent,andthe possibility ofproducing acontinuous emission ofelectrons isthebasis ofthermionic vacuum tubes.Electrons canacquiresufficient energyto escapeintwoimportant ways;(a)ifthemetalisheated,theenergy distribution amongst theelectrons changes, developing apronounced 'tail'asincurveBinFig.4.3,inwhichanappreciable numberofelec­ tronshaveenergygreaterthan(4)+~);theescapeofelectrons fromthe metalisthenknownas'thermionic emission': (b)iflightofasufficiently shortwavelength shinesonthemetal,electrons acquireenergyfrom collisions withthephotonsandcanescapeifthephotonenergyisgreater thantheworkfunction 4>;thisisknownas'photoelectric emission'. Electrons canalsobeemittedifanintenseelectricfieldisappliedat themetalsurface('fieldemission'), orifthesurfaceisbombarded by electrons ('secondary emission') .Weshalldiscusstheseeffectsseparately. Thermionic emission Ifametalisheatedinvacuoandanelectrode atapositivepotential withrespecttoitcollectstheemitted electrons, asinFig.4.5,acon­ tinuouscurrentofmicroamperes uptomilliamperes maybeobtained. Suchadeviceisknownasa'thermionic vacuumtube'.Themagnitude ofthecurrentdepends ontheworkfunction oftheemitter,andvaries veryrapidlywithtemperature; copiousemission canbeobtained from puremetalsonlyattemperatures oftheorderof2000°C(seeChapter 12).Inordertoescape,anelectron musthaveanenergygreaterthan (4)+WF),andatagiventemperature thenumber represented bythe hatched areainFig.4.3willbeabletoleavethesurface. Theclose parallelbetween thermionic emission andevaporation wasrecognized by 851110 H 98 PROPERTIES OFELECTRICAL CONDUCTORS [4.4 Richardson, whoshowedthatthecurrentemittedperunitareashould begivenbytheequation J=AT2e-rplkT, (4.21) where cPistheworkfunction, kisBoltzmann's constant, andAshould beauniversal constant forallmetals,equalto120ampcm-2deg-2• (Forthederivation ofequation (4.21),see,forexample, Slater,1939, E - "'=::""B1.- FIG.4.5. FIG.4.6. FIG.4.5.Athermionic vacuum tube.BBisabatteryofabout4 Vforheatingelectrode 0 (thecathode): electrons flowfromthecathodetotheanodeAmaintained atapositive potential bythebatteryB1(::::::100V).Thewholeisenclosed inanevacuated glassor metalenvelope. FIG.4.6.Plotoflog.(JITB) againstlOilTforthethermionic elnission fromcopper,silver, andgold(afterJainandKrishnan, 1953). orZemansky, 1957).Thisequation hasbeenverifiedexperimentally, thoughthevariation oftheexponential termwithtemperature isso muchmorerapidthanthatoftheT2termthatinearlyexperiments itwasdifficulttobecertainthatthelatterwascorrect. However, if log(JIT2) isplottedagainstliT,alineargraphisobtained, asshown inFig.4.6.Representative valuesoftheconstants AandcParegivenin Table4.1;thevaluesofAvaryconsiderably, aneffectgenerally attri­ butedtoapartialreflection ofelectrons attempting toleavethesurface oftheemitter, thoughtherewillalsobeaslightvariation incPwith 4.4] PROPERTIES OFELECTRICAL CONDUCTORS 99 temperature, similartothatin~(seeProblem 4.6),whichwillaffect thevalueofAobtained fromtheuseofequation 4.21).Thereisalso evidence thattheemissionisdifferent fromdifferent facesofasingle crystal,andthatthevaluesofAobtained fromapolycrystalline surface aretoolow.Thermionic emission willbeconsidered inmoredetailin Chapter 12. Fieldemi88ion Theemission ofelectrons fromametalundertheinfluence ofan appliedfieldiscloselyrelatedtothermionic emission, sothatweshall discussitnext.Asweshouldexpect,itoccursonlywhenthedirection oftheappliedfieldissuchthatelectrons areattracted outofthemetal, andwemustconsider theeffectofanappliedfieldonthepotential barrieratthesurfaceofametal. Thepotential energyjumpatthesurfaceis~+tP,sinceitrepresents theenergywhichweshouldhavetogiveanelectron ofzerokinetic energytoextractitfromthemetal.Thepotential jumpisnotinfinitely sharpbecauseofthe'imageforce'whichactsonanelectronjustoutside themetal(§2.5).Iftheelectronisatadistance xfromthesurfaceitwill havepotential energyW=-e2j(167TEoX)whenxislargecompared with atomicdimensions, butwilldeviatefromthisforsmallvaluesofx.The shapeofthepotential energycurveisshowninFig.4.7;whenafieldE isapplied,thepotential energyoutsidethemetalbecomes W=-e2j(167TEoX)-Eex, (4.22) andthishasamaximum asshown.Thustheapparent workfunction isreduced byalargeelectricfield,andattemperatures wherethe thermionic emission isappreciable itwillbecorrespondingly increased. Atanytemperature fieldemission willoccuralsothroughthequantum mechanical 'tunneleffect',bywhichanelectronwithinsufficient energy tosurmount thepotential barriercan'leak'through it;thisrequires fields f'Ooo.I108Vjmetretoproduce appreciable emission. Photoelectric emi88ion Theenergyassociated withaquantum ofradiation offrequency v (a'photon') ishv,wherehisPlanck's constant, andinmanyrespects thephotons behavelikecorpuscles withthisenergy.Inaninelastic collision thewholeofthisenergymaybetransferred toanelectron.If hv>tP,theelectronmaybeejectedfromametallicsurfacewithmaxi­ mumkineticenergywsuchthat hv=tP+w. 100 PROPERTIES OFELECTRICAL CONDUCTOHS [4.4 Measurement ofthelowestfrequency (longest wavelength) oflight whichcanjustcausephotoelectric emission therefore provides another methodofdetermining theworkfunction 4>,assuming Planck's constant tobeknown;thevaluesof4>giveninTable4.1wereobtained bythis method. Thecut-offwavelength liesinthevisibleregiononlyforthe (1)+Wp)- - - - - - - - - - - - - - - - - -_------A TV.. ...... ...... ...... ...... ...... ...... ......8 L O· FIG.4.7.Variation ofpotential energyWofaneleotron withdistance xfromaconduotor. CurveA.Noexternal field. CurveB.Withexternal field(equation 4.22). alkalimetalsandbariumandstrontium, whichhavelowworkfunc­ tions;forothermetalsitliesintheultraviolet. Adevicefordetecting visibleradiation (sometimes calleda'photo­ tube')makesuseofthephotoelectric current. Atypicalconstruction isshowninFig.4.8.Anelectrode atapositive potential (theanode)in theformofalongthinrodisplacedalongtheaxisofacylindrical cathode fromwhichonesectionisremoved sothatlightcanenterandfallonthe insideofthecylinder, whichisthephotoelectric surface. Thewholeis placedwithinanevacuated glassenvelope. Thecurrenttotheanode reachesavalueindependent oftheanodepotential whenthelatteris morethanabout15V,sincealltheemittedphotoelectrons thenreach theanode.Foragivenwavelength oflight,thissaturation currentis 4.4] PROPERTIES OFELECTRICAL CONDUCTORS 101 -- FIG.4.8.Aninstrwnent usingthepheno­ menonofphotoelectric emission forthe measurement oflightintensities. AAnode.oNegative electrode, whoseinner surfaceisthesourceofelectrons. MMicroammeter. LIncident light.strictlyproportional tothelightintensity overaverywiderangeof intensities, andthetubecantherefore beusedasanintensity meter. Thecurrents areoftheorder10-5A,andcanbereadeitheronamicro­ ammeter, asshowninFig.4.8,or,toobtaingreatersensitivity, the currentcanbepassedthrough a largeresistance andthevoltagede­ velopedacrossthisresistance ampli­ fiedbyavacuumtubeamplifier (see Chapter 13).Thismethodisparti­ cularlyusefulforrapidlyfluctuating lightintensities, sincetheresponse ofthephoto-tube isvirtuallyinstan­ taneous(thetimedelayinemission ofelectrons afterswitching ona sourceoflightislessthan10-9sec). Gas-filled photo-tubes arealsoused. Theygiveagreatercurrent, since theinitialphotoelectrons giverise toadditional electrons andionson collision withgasmolecules, but sufferfromthedisadvantages that thecurrentisstrongly dependent ontheanodevoltageandtheresponse isnotasrapidasinthevacuum type. Secondary emission Whenasolidsurfaceisstruckbyelectrons orionsofappreciable energy,secondary electrons areemittedfromthesurface;ifthebombard­ ingparticles (primaries) areelectrons, theymusthaveanenergyofat leastafewelectron voltstoejectanappreciable numberofsecondary electrons, and,athigherenergies, thenumberofsecondaries maybe greaterthanthenumber ofincident primaries. Thephenomenon of 'secondary emission' iscommonly encountered inthermionic vacuum tubeswithseveralelectrodes, andalsofromthefluorescent screenofa cathoderaytube.Itisverysensitive toimpurities andtocontamination ofthesurface;forpuremetalsthesecondary emission ratio o=(number ofsecondary electrons/number ofincident primaries) hasamaximum valuevaryingfrom0·5to1·6.Atlowenergiesthereis aninitialrisein0becausethenumberofsecondaries released within themetalincreases withtheenergyoftheincident primary. However, 102 PROPERTIES OFELECTRICAL CONDUCTORS [4.4 theproduction ofsecondaries ismostcopiousneartheendoftheprimary path,andtheirchanceofescapedeclinesrapidlyastheprimaries pene­ tratedeeperintothemetalathigherenergies, causing,) tofallagain afterpassingthrough amaximum atabout200-400 eVintheprimary energy. Forcomposite surfaces (suchasalayerofCs20onabaseof silver,withasurfacefilmofabsorbed caesium ontheC820)Dmayrise FIG.4.9.Aphoto·multiplier tube,showing one arrangement oftheelectrodes (alineararrangement isalsoused). GGrill. 0Finalelectrode. SShield. LIncident light. tovaluesashighas10.Thereisstillmuchuncertainty astotheme­ chanism ofsecondary emission, buttherearereasonsforbelieving that thesecondary electrons comefromthosetightlyboundtothemetallic ionsratherthanfromthefreeelectrons. Theshapeofthecurveof secondary emission ratioagainstenergyoftheincident electrons is similarforallsubstances, andresembles thatfortheprobability of ionization ofagaseousatom;highratioscanalsobeobtained from insulators. Thechiefuseofsecondary emission isforcurrentamplification. For example, thephotoelectrons fromaphoto-sensitive surfacemaybeac­ celerated tostrikeasecondary emitting electrode, soarranged thatthe secondary electrons areaccelerated toanothersecondary emitting elec­ trode,andsoon.Theelectrode arrangement insucha'photo-multiplier' tubeisshownveryschematically inFig.4.9.Thecurrentamplification thatcanbeobtained isverylargeindeed;withasecondary emission ratioof3ateachof10surfaces, theoverallamplification is310orabout 4.4] PROPERTIES OFELECTRICAL CONDUCTORS 103 6X10'.Themaindifficulty istheproduction ofstablesurfaceswitha highratio;inpractice thereisgenerally somechangeintheamplifica­ tionwithtime.Thepotential ofeachsuccessive electrode isabout75V higherthantheprevious one,andthenecessary voltages areobtained fromapotentiometer system. 4.5.Thermoelectricity Whentwometalsarejoinedtogether, acontactpotential difference issetupbetween them.Ifasecondjunction ismadebetween them, sothataclosedcircuitisestablished, thecontactpotential difference atthesecondjunction isjustequalandopposite tothatatthefirst junction, sothatthereisnonete.m.f.inthecircuitprovided thatthe twojunctions areatthesametemperature. Theworkfunction ofa metalvariesslightlywithtemperature, however (fortungstenitchanges attherateof6to7X10-5eVrK;compare Problem 4.6)andsoalso willthecontactpotential. Hence,ifthejunctions areatdifferent tem­ peratures, thecontactpotentials willbeslightlydifferent, andanet e.m.f.willexistwhichcandriveacurrentroundthecircuit. Thisis knownasthethermoelectric e.m.f.,andwasdiscovered bySeebeckin 1821;itisgenerally oftheorderofmicrovolts perdegreetemperature difference between thejunctions. Theconverse effectwasdiscovered byPeltierin1834.IftheSeebeck e.m.f.isfrommetalAtometalBat thehotjunction, anexternal e.m.f.appliedinthisdirectionwillproduce acoolingatthisjunction andaheatingattheotherjunction. Both effectsareentirelyreversible. Sincetheenergyofthefreeelectrons inametaldepends onthetem­ perature, thepresence ofatemperature gradient inametalproduces aregionatoneendwheretheelectrons havemoreenergythanthoseat theotherend.Owingtotheirhighervelocities, electrons fromtheend withhigherenergywilldiffusedownthemetalfasterthanthosefrom theotherend,andtheflowwillcontinue untilapotential difference is setupwhichisjustsufficient tocounterbalance thisflow.Thisisknown astheThomson effect.Ingeneralthee.m.f.isfromthelowertem­ perature tothehighertemperature, butitcanhaveeithersign.Like theSeebeckandPeltiereffects,theThomson effectisreversible. ThePeltiercoefficientnisdefinedasfollows:theheatabsorbed when achargeQpassesfrommetalAtometalBisnQjouleswhenQisin coulombs. TheThomson coefficient (jisdefinedassuchthatthepoten­ tialdifference is(jAdT(involts)between twopointsinthemetalA wherethetemperature difference isdT(OK).Intables (jisgivenas 104 PROPERTIES OFELECTRICAL CONDUCTORS [4.5 positive whenthedirection ofthee.m.f.isfromthecoldtothehot end. Thermoelectric theory WhenachargeispassedroundacircuitsuchasthatinFig.4.10, therewillbeanirreversible heatingthroughout thecircuitduetoits resistance, aswellasthermal changesduetothePeltierandThomson MetalA x MetalB FIG.4.10.Figuretoillustrate thethermoelectric effeet. effects.However, theirreversible heatingisproportional tothesquare ofthecurrent,andcanbeneglected forverysmallcurrents.Ifacharge Qispassed,theexternal workdoneisQV,whereVisthetotale.m.f. inthecircuit;thenetheatabsorbed atthejunctions is(IlTl-IlT.)Q; and T, theincrease ininternal energyis-QJ(UA-UB)dT, sincethisrepre- T. sentstheenergyrequired toheattheelectrons fromtemperature T2to T1inmetalAlessthatemittedinthereverseprocessinmetalB.Hence fromthefirstlawofthermodynamics (dividing byQ), T, V=IlT,-IlT.+f(uA-uB)dT. T.(4.23) Inordertodetectthise.m.f.,thecircuitmustbebrokenandavolt­ meterintroduced at,sayX,inFig.4.10.Although thevoltmeter may containmetalsdifferent fromAandB,thetotale.m.f.inthecircuit willbeunaffected provided thatbothjunctions totheinstrument are atthesametemperature. For,ifthemeterMhasjunctions atatem­ perature T, T, V=(IlT,-IlT.)A->-B+(IlT)B->-M+(IlT)M->-B+ f(uA-uB)dT T. T, =(IlT1-Il T.)+J(uA-UB)dT, T. whichisthesameasbefore.Thesameresultholdsifanynumberof metalsatdifferent temperatures areconnected inthecircuit,provided thateachpairofjunctions toagivenmetalareatthesametemperature. 4.5] PROPERTIES OFELECTRICAL CONDUCTORS 105 Sincenotemperature changesoccurinthecircuit,theexternal work QVisalsothefreeenergyofthesystem, sothatwehavefromthe secondlawofthermodynamics (4.26)(4.25) (4.27)Hence or wherethequantityQV=U+Td(QV),whereU=QIT(UA-UB)dT.dTo T dV V=I(UA-UB)dT+T dT o anddifferentiation withrespecttoTgives d2VUA-UB=-TdT2' (4.24) Thenelimination ofUA-uBwiththehelpofequation (4.23)yields IIT,-IIT•=[T(dVjdT)];:, dVII dT=T' Although thePeltierandThomson effectsarereversible, theapplica­ tionofreversible thermodynamics toathermoelectric circuitisopen toanumberofobjections, sinceirreversible effectssuchasresistive heatingandthermal conduction arealwayspresent. Abettermethod istouseirreversible thermodynamics (seeZemansky, 1957)ortocon­ siderthechangeofentropyatajunction (seeCusack, 1958),andto relatethistothenetheatabsorbed atthattemperature. Amore rigorous treatment ontheselinesconfirms thatequations (4.24)and (4.25)arevalid,andtheyhavebeentestedexperimentally overalarge temperature rangeformanysubstances. Thequantity dVjdTiscalledthe'thermoelectric power',andfrom equation (4.24)wehave dVjdT= -fUA;UB dT+constant. Bythethirdlawofthermodynamics (see,forexample, Wilks,1961) dVjdT-+0asT-+0,andwecantherefore eliminate theconstant by takingtheintegralfrom0toT(toavoidinfinities intheintegrand this requires UtovaryasleastasthefirstpowerofT).Then T T dVjdT=fa;dT-f;dT=BB-SA' o 0 T S=f;dT o 106 PROPERTIES OFELECTRICAL CONDUCTORS [4.5 (4.29)isknownasthe'absolute thermoelectric power'andisaproperty of T. eachsubstance separately. Ouridentification ofthequantity QJadT Tlasthechangeintheinternal energysuggeststhat a=(l/Q)(dU/dT) =(l/Q)Oe' TABLE4.2 Value8ofthethermoelectric coefficient8 exandf3(equation (4.29))withtin00 l¥ fJ SUbstance (pY/deg) (p,V/deg2) Aluminium . -0·76+0·0039 Bismuth (commercial) -43,7 -0'465 Copper +1'34+0·0094 COJUltantan (60%CU-40% Ni)-38,1 -0·089 Gold +2·80+0·010 Iron.. +17·2 -0·048 Palladium -7'4-0'039 Pla.tinum -3·04-0,033 90%platinum-l0% rhodium. +7·0+0·0064 whereOeisthespecificheatoftheelectrons. Henceusingequation (18.16a) wehave,sinceQ=-ne, T T T S=J:!..dT=J(O/QT)dT =-JTr2 k2 nTdTT e 2lJ'FneTo 0 . 0 =-(Tr2k2/2elJ'F)T =f3T (4.28) andthee.m.f.between apairofmetalsA,Bwhosejunctions areat ~,T2shouldtaketheform(t=~-~; f3=f3B-{3A) Tl T, V=fSBdT-fSAdT=!f3(T~-T~) =ext+!f3t2• T. T. Though thisrelation isobeyedbymanypairsofmetalsoveralarge temperature range,ourderivation notonlypredicts ex=f3T2,butalso thatf3isnegative (equation (4.28)),neitherofwhichholdsgenerally (seeTable4.2). Thequantity S(seeequation 4.26)isanalogous toentropyperunit chargesincethefreeenergyFperunitchargeisV,andtheentropy is -dFjdT. OursimpletheorypredictsthatSshouldbeproportional toTforametal,andattemperatures above1000Kthisistruetoalarge extent.Atlowertemperatures SceasestovarylinearlywithT,and thiscannotbeaccounted forbyconsidering theenergydistribution of theconduction electrons alone.Thevibrations ofthecrystallattice 4.5] PROPERTIES OFELECTRICAL CONDUCTORS 107 mustalsobeincluded, sincethesealterthepotential fieldsthrough whichtheelectrons moveandprovide amechanism bywhichenergy canbeexchanged between electrons andlattice. Thiseffectisparti­ cularlyimportant atlowtemperatures andinsemiconductors, where thermoelectric effectsaremuchlarger.Forgermanium atroomtem­ perature Sisabout1millivolt perdegree(metalshavevaluesoforder 10-6V/deg:seeProblem 4.7),andthesehighvalues,combined with valuesofthermal conductivity whicharemuchlowerthaninmetals, makerefrigeration usingthePeltiereffectinasemiconductor aprac­ ticablepossibility. Thermocouples Ifonejunction ismaintained atafixedtemperature, thethermo­ electrice.m.f.canbeusedtomeasure thetemperature oftheother junction. Thevaluesofex,pcanbefoundfromthedifferences (taking dueaccountofsign)between thevaluesgivenintables(suchasTable 4.2,wheretheonejunction isassumed tobeat0°C).Thevaluesare tabulated foreachsubstance againstlead,whichisusedasareference metalbecauseitsThomson coefficient isverysmall.Itisobviously desirable toworkonthesteeppartofthe(V,t)curve,inorderthatV shallvaryalmostlinearly withtandthatdV/dtshallbelarge.As dV/dt=0atthetemperature t=-ex/P,different pairsofmetalsare usedfordifferent temperature ranges. Acopper-constantan coupleis usefulintherange-200°to400°C,andaplatinum-rhodium against platinum coupleisusedupto1700°C.Foraccurate workeachthermo­ couplemustbecalibrated attwoknowntemperatures todetermine ex andp,andthenacalibration curveisdrawnfromwhichanyunknown temperature canbereadoffintermsofthee.m.f.V.Itispossibleto obtainmillivoltmeters whichreadoffthetemperature directly, butfor accurate workapotentiometer andstandard cellshouldbeused. Thermocouples havetheadvantage thatthejunction hasaverysmall heatcapacity, sothatitrapidlyreachestherequired temperature with­ outalteringtheexperimental conditions; theyaretherefore suitable formeasuring varyingtemperatures. Itisalsoeasytoensurethatthe junction isingoodthermal contactwiththesubstance whosetempera­ tureisrequired. Measurement oftheThomson coefficient TheThomson coefficient ofonemetalrelativetoanothermayreadily befoundbymakingathermocouple fromthetwometals,measuring 108 PROPERTIES OFELECTRICAL CONDUCTORS [4.5 thee.m.f.asafunction oftemperature, andapplying equation (4.24). TheThomson effectisacharacteristic ofeachmetal,however, anditis desirable toknowitsabsolute value.Ifthiscanbedetermined forone metal,thenalltheothersfollowfromthermocouple measurements. We shalldescribe amethodofBorelius, Keesom, andJohansson (1928) whichcanalsobeusedatlowtemperatures. l', po • Gol' FIG.4.11.Diagram oftheapparatus usedbyBorelius, Keesom, andJohausson (1928) tomeasure theabsolute valueoftheThomson coefficient ofametal. TPlatinum resistance thermometer formeasurement atlowtemperatures. GHighresistance galvanometer. Pi'PIThermocouple junctions ofplatinum (P)andconstantan (0). Athinwireofthemetalisstretched invacuobetween twoheavyleads maintained ataconstant temperature. Whenacurrentispassedthrough thewire,theJouleheatingraisesthetemperature atthecentresothat inonehalfofthewiretheThomson effectgivesavoltageinthesame direction astheappliede.m.f.,whileintheotherhalfitisopposedto it.Thetemperature ofthewireismeasured attwosymmetrical points PI'~asinFig.4.11byapairofthermocouples whichareelectrically insulated fromthewirebymeansofthinpaper.Twomeasurements aremade.First,thethermocouples areconnected inseries,sothatthe meanincreaseoftemperature atthetwopointsisfound;thisgivesthe temperature increaseduetotheJouleheatingalone.Second,thecouples areconnected inopposition togivethedifference intemperature between thetwopointsduetotheThomson effect.Temperature differences due toasymmetry inthepositionofthepointsPI'Pzcanbeeliminated by repeating themeasurement withthedirection ofcurrentflowreversed. 4.5] PROPERTIES OFELECTRICAL CONDUCTORS 109 (4.30)Inthetheoryitisassumedthatheatislostonlybyconduction along thewire.Thenetheatflowintoasectiondxofthewireadistance x fromthecentrebyconduction isthen -KA(oTjox)+KA{(oTjox)+(o2Tjox 2)dx}=KA(02Tjox 2)dx, whereKisthethermalconductivity ofthewireandAitscross-section. Theheatgenerated inthesectionisI(IRdx-adT), whereIisthe currentflowingandRtheresistance perunitlength.Hencethedifferen­ tialequation is Theexactsolutionofthisequation is T=c{e1x-coshfa-(xja)sinhfa}, wheref=IajKAandc-1=(ajlaR)sinhfa; thetemperature atthe endsofthewirex=±aistakentobezero.Ifthethermocouple read­ ingsaretwhenconnected inseries,andTwheninopposition, thenthese areproportional tothesumanddifference ofthetemperatures at+b and-brespectively. Hence T(Ijfb)sinhfb-(ljfa)sinhfa 1- = =-3whenfissmall.tfb coshfb-coshfa 3TKAThisgives a=~. (4.31) Inthistreatment theresistivity ofthewirehasbeenassumed indepen­ dentoftemperature, andinpractice asmallcorrection mustbemade fortheslightchangeintemperature distribution duetothevarying resistance. 4.6.Conduction ofelectricity through liquids Certainliquids,suchashydrocarbons, areextremely goodinsulators, whileothers,suchaswater,haveanappreciable conductivity. Solutions ofsomesaltsinwaterhaveaconductivity oftheorderof10-5timesthat ofmetals,andsuchsaltsareknownasioniccompounds. Anexample issodiumchloride, whichinasimplepictureisformedbythetransfer ofoneelectron fromthesodiumatomtothechlorine atom,sothatthe molecule consistsofapositively charged sodiumionandanegatively charged chlorine ion.Suchamolecule hasapermanent electricdipole moment, andiscalleda'polarmolecule'; inanon-polar molecule, such ashydrogen, theelectrons aresharedequallybetween thetwoatoms, andthereisnopermanent dipolemoment. Thewatermolecule isitself strongly polar,andwhenasubstance suchassodiumchlorideisdissolved '110 PROPERTIES OFELECTRICAL CONDUCTORS [4.6 init,theelectricfieldsofthewatermolecules arestrongenoughto dissociate thesolutemolecules intoseparate sodiumionsandchlorine ions.Therearealsocomposite ionsformedbygroupsofmolecules which havegainedorlostanelectron. Thesolution iscalledan'electrolytic solution', thesolutebeingknownasthe'electrolyte'. Thedegreeof dissociation oftheelectrolyte insolution isdetermined bythedynamic equilibrium between recombination anddissociation. With'strong electrolytes', suchasNaCI,thedissociation ispractically complete atall ordinary concentrations; for'weakelectrolytes', suchasaceticacid,the degreeofdissociation isgreatestathighdilutionandfallssteadilywith increasing concentration according tothelaw 01.2/(1-01.)=K/c, where 01.isthefractionofsolutemolecules dissociated, ctheconcentra­ tion,andKisaconstant whichdepends onthetemperature. Ifapotential difference isappliedbetween twoelectrodes inanelec­ trolyticsolution, acurrentwillflowthroughthesolution. Thecurrent iscarriedbybothpositiveandnegative ions,whichareproduced bythe decomposition oftheelectrolyte; thehydrogen ormetallicradicalalways travelstothecathode, ornegative electrode, andtheacidradicalstravel totheanode,orpositiveelectrode. Thistransferofchargedionsbyan electriccurrentiscalledelectrolysis. Forexample, iftwocopperelec­ trodesareimmersed incoppersulphate solution, copperisdissolved off theanode,andisdeposited onthecathode. Withcarbonelectrodes in abrinesolution, hydrogen appearsatthecathode, andchlorineatthe anode.Therearetwofundamental laws,discovered byFaraday, which areobeyedbyallelectrolytes. (4.32) m=(A/v)It/F,Faraday's lawsofelectrolysis (1)Themassofagivensubstance liberated atoneelectrode ispro­ portional tothetotalchargewhichhaspassed. (2)Themassofagivensubstance liberated atanelectrode byunit chargeisproportional tothechemical equivalent ofthatsubstance. Thesetwolawscanbecondensed intothefollowing form:Ifthemass liberated atanelectrode ismwhenacurrentIpassesfortseconds,then m=ZIt,whereZisaconstant foragivenelement, calledtheelectro­ chemical equivalent. Thechemical equivalent ofanionistheatomic weightdividedbythevalency, thevalencybeingequaltothenumber ofelectronic chargescarriedbytheion.ThenifAistheatomicweight, andvthevalency, 4.6] PROPERTIES OFELECTRICAL CONDUCTORS 111 V-V'=lR,whereFisauniversal constant knownastheFaraday. TheFaradayis thechargeofelectricity whichliberates onegramme equivalent (A/v)of anioninelectrolysis. IfN(Avogadro's number) isthenumberofatoms inagramme atom,thetotalchargecarriedbyagramme equivalent is (N/v)(ve), sinceeachatom(morecorrectly, eachion)hasachargeve, andthistotalchargeNemustjustbeequaltoF.Hencewehavethe important relation Na+H20-+NaOH+H. Intheseexamples thehydrogen atomsorionswillcombine toform hydrogen molecules whichappearasamassofsmallbubblesofgas covering theelectrode. Inacellwheregasappearsattheelectrodes, thephenomenon of'polarization' oftheelectrodes isgenerally observed. Forexample, intheelectrolysis ofacidulated waterusingpolished plati­ numelectrodes, noelectrolysis occursuntiltheappliede.m.f.Vexceeds acertaincriticalvalueV',knownasthedecomposition potential ofthe electrolyte. IfV>V',thecurrent1flowingobeysamodified Ohm's lawrelationF=Ne. (4.33) Basically, thevalueoftheFaraday isfoundbydetermining themass ofelectrolyte liberated whenaknowncurrentispassedthrough asolu­ tionforameasured lengthoftime.Someveryaccurate determinations arethoseofCraigandHoffman (1950).Usingsilverelectrodes inasolu­ tioncontaining silverperchlorate andperchloric acid,theyobtained F=96523·3±6·2 coulomb/g, whileinanotherexperiment inwhichoxalateionsareoxidizedtocarbon dioxide(C204--2e--+2C02)attheanodeinasolutionofsodiumoxa­ lateinsulphuric acidtheyobtained F=96519·3±2·6 coulomb/g. Although theprimary reaction inelectrolysis issimplytheflowof ionsofpositiveandnegative signtothecathodeandanoderespectively, secondary processes mayoccurattheelectrodes sothatdifferent pro­ ductsappeartherewhichdonotcorrespond totheprimary ions.The firstproductmaybeunstable, asinthecase NH4+H20-+NH40H+H, oritmayreactwiththesolvent,thesolute,ortheelectrode inachemical reaction suchas butifV<V',nocurrentflows.Thecelltherefore becomes irreversible owingto'polarization' oftheelectrodes. Thisisprobably duetothe presence ofpositively-charged hydrogen ionsinthegasbubbles, which 112 PROPERTIES OFELECTRICAL CONDUCTORS [4.6 repelotherpositiveionsandsogivetheeffectofabacke.mJ.Practical cellsincorporate a'depolarizer' consisting ofasubstance whichreacts chemically withthehydrogen ionsappearing attheelectrode andso prevents theformation ofgasbubbles. L-. ~---------------I FIG.4.12.Alternating currentbridgeformeasuring theconductivity ofanelectrolyte.oisasmallvariable capacitor tobalanceoutthecapacitance between theelectrodes. Conductivity Polarization effectsdonotappearinstantaneously, butaftercurrent hasbeenpassedinonedirection forafinitetime.Theycanbeavoided ifalternating currentisused,sincethedirection ofthecurrentisthen reversed beforesucheffectscanbeestablished. Theconductivity of electrolytic solutions istherefore measured withalternating currentand aWheatstone bridge,thedetector beinganamplifier withearphones or cathoderayoscilloscope (seeChapter 15).Thesolution iscontained in acellwithtwoplatinum electrodes; thegeometrical arrangement isnot important ifonlyrelativemeasurements ofconductivity arerequired, but shouldbesuchastominimize thecapacitance between theelectrodes. In generalthiscapacitance willrequirebalancing withavariablecapacitance inanotherarmofthebridgeinordertoobtainabalancewithalternating current, asinFig.4.12. Thecurrentdensityatanypointinthesolution willbe J=(niVIul+n2v2u2)!e/E=aE, wherenl,n2arethenumbers ofpositive andnegative ionsperunit U] PROPERTIES OFELECTRICAL CONDUCTORS 113 (4.34) FIG.4.13.TheDaniell cell. Pisaporouspotcontaining copper sulphate solution in whichthecopperrodisim­ mersed. Thezincrodisina solution ofdilutesulphuric acid.volume, vI>V2theirvalencies, andU1,U2theirmobilities; thatis,the meanvelocityofanioninafieldofunitintensity. Hencethespecific conductivity ais aisfoundtobeproportional totheconcentration forverydilutesolutions, butincreases lessrapidlyathigherconcentrations. Forweakelectrolytes, theconductivity isdetermined bythedegreeofdissociation, i.e.the numberofionspresent, indicating thatthemobility isindependent of concentration. Forstrongelectrolytes, wheredissociation ispractically complete atallconcentrations, themobility fallsathighconcentrations becauseeachionattractsrounditselfan'atmosphere' ofionsofopposite signwhichretarditsprogress throughthesolutionwhenanelectricfield isapplied. 4.7.1Voltaiccells Iftwometalelectrodes areputintoanelectrolytic solution,itisfound thatundercertaincircumstances apotential difference existsbetween them.Inthe'con­ centration cell',thetwoelectrodes areofthe samemetalbutareimmersed intwosolu­ tionsofthesameelectrolyte withdifferent concentrations, usually separated bya permeable membrane whichallowsionsto passfromonesolution totheother.The e.m.f.developed isnormally oftheorderof hundredths ofavolt,andsuchcellsarenot ofpractical importance. Inthe'chemical cell'theelectrodes areofdifferent metals, andane.m.f.issetupofthesameorderas thecontactpotential difference betweenthe twometals.Ifametallic contactisestab­ lishedbetweenthetwoelectrodes acurrent willflow,theenergyrequired forthiscurrentbeingderivedfromthe chemical reactions whichtakeplaceattheelectrodes. Theessential processes involved maybeillustrated byreference to thesimpleDaniellcell,consisting ofazincelectrode immersed indilute sulphuric acid(oracidulated zincsulphate solution) andacopperelec­ trodeincoppersulphate solution, withamembrane through which ionscanpassfromonesolutiontotheother(Fig.4.13).Attheformer electrode Znionspassintosolution, andatthelattercopperionsare 851110 I 114 PROPERTIES OFELECTRICAL CONDUCTORS [4.7 deposited; thustheeffective chemical changeisessentially Zn+++CuS04-+ZnS04+Cu++. Thephysical changeatoneelectrode isthedetachment ofanionfrom thesurfaceofthemetalanditspassageintosolution whereitissur­ rounded bywatermolecules andbecomes hydrated, andviceversaat theotherelectrode. Thismaybetreatedinthefollowing schematic way.Thepotential energycurveofapositive ionnearthesurface YI (Il)I IMl ~_ Metalsurface x}Y -V (Il) TV xWatermolecule lIf IV x (e)v (d)VI .If x (e)x FIG.4.14.Potential energycurvesforapositive ionintheregionofametallic surface orawatermolecule. TheminimamarkedMdenotetheequilibrium positions forthe ionatthesurfaceofthemetal;thosemarkedW,theequilibrium positions fortheion inthesolution. ofthemetalisrougWy asshowninFig.4.14(a);thesteeprisetothe leftoccurswhentheionoverlaps withotherionsinthemetallattice, whenstrongrepulsive forcesaresetup,whiletheslowrisetotheright isduetotheimageforceattracting anionjustoutsidethemetalback tothemetal.Thenormalequilibrium position oftheionwillbeinthe potential minimum.Ifwenowtakeanisolatedionitspotential energy curveasitapproaches awatermolecule willbeofthetypeshownin Fig.4.14(b);atlargedistances therewillbeanattractive forcedueto induced polarization ofthewatermolecule, whileatshortdistances repulsive forceswillbemoreimportant. Foranionatthesurfaceofa metalimmersed inasolutionthecombined potential energycurvewill 4.7J PROPERTIES OFELECTRICAL CONDUCTORS 115 haveoneofthethreeformsshowninFig.4.14(c),(d),and(e);in(c)ions canpassreadilyfrommetaltosolutionandviceversa,sincethetwo potential minimaareequal;in(d)anioncanonlypassfromtheelectrode intosolution, whilein(e)anioncanpassonlyfromthesolutiontothe electrode. Withasingleelectrode inthesolutionthepassageofcharged ionsinonedirection ortheotherwillquicklysetupareversepotential difference between electrode andsolution, whichdisplaces thetwoun­ equalpotential minimaofsituations (d)or(e)untilwereachsituation (c),theequilibrium situation. Ifnowtwoelectrodes areplacedinthesolutionandanexternal junc­ tionismadebetween themetals,theusualcontactpotential difference issetupatthemetal-metal contactbecause electrons canpassvery muchmorereadilyacrossthiscontactthanmetalionscanmoveinto andoutofthesolution. Electrons moveacrossthemetal-metal contact fromthezinctothecopper(inourexample), thereby lowering the electron energylevelsinthezinc,butraisingthepositive ionlevelsin thezincbecauseoftheiropposite charge. Thesituation atthezinc­ solution junction willtherefore beasin(d),andzincionswillpassinto solution. Thepositive ionlevelsinthecopperwillbelowered and situation (e)willprevailatthecopper-solution junction, copperions beingdeposited ontheelectrode. Atboththesetwojunctions themove­ mentofthemetalionistowards lowerenergy,andthismakesavailable energytodrivethecurrentroundthecircuit.Thusthecontactpotential difference playsanessential roleindetermining thedirection ofcurrent flow,buttheenergyisderivedfromthechemical processes ateachelec­ trode.Sinceheatsofreactionareadditiveitisnotnecessary toconsider eachchemical reaction indetail,andtheavailable energycanbecalcu­ latedfromtheheatofreaction oftheeffective chemical change,the displacement ofcopperbyzincinthesulphate. Itisimpossible tomeasure separately thepotential difference setup ateachelectrode, butitisconvenient tohaveonestandard electrode againstwhichthee.m.f.ofotherelectrodes canbemeasured. This standard isthe'hydrogen electrode', consisting ofapieceofplatinum coveredwithplatinum blacksaturated withhydrogen gasatatmospheric pressure. Eache.m.f.listedinTable4.3ismeasured forthehydrogen electrode againstthemetalelectrode immersed inastandard solution. Thee.m.f.developed inacellwithanytwometalelectrodes immersed inthestandard solutionisthealgebraic difference ofthepotentials listed. Thusthee.m.f.oftheDaniellcell,copperagainstzinc,isapproximately (0'345+0·762) =1·1V. 116 PROPERTIES OFELECTRICAL CONDUCT01{S [4.7 Application ofthermodynamics tovoltaiccells Sinceinareversible cellthechemical reactions takingplacewhena currentispassedthroughitinonedirection maybereversed bysending thecurrentthroughitintheopposite direction, thestandard equations ofthermodynamics maybeappliedtothecell.Inanidealcasewemay supposethecurrenttobeinfinitely smallsothatJouleheatlosses,which TABLE 4.3 Standard electrode potentials involts,withrespectto thehydrogen electrode, at25°0 Li+ -2,959 Sn+++ -0·336 Rb+ -2,926 Pb++ -0·12 K+ -2,924 Pt,H.,H+ 0·0000 Na+ -2,715 Cu++ +0·345 Zn++ -0,762 Hg++ +0·799 Fe++ -0,44 Ag+ +0·798 Cd++ -0,402 dependonthesquareofthecurrent, canbemadenegligibly smallin comparison withthechemical energychangeswhichvarywiththefirst powerofthecurrent. Thentheequatidn forthechangeinthefree energyFofacellwhenachargeQispassedatconstant temperature is (4.35)V=(UjQ)+T(oVjoT) =h+T(oVjoT),F=U+T(oFjoT)v, whereUisthechangeininternal energyandFistheenergyavailable forexternal workprovided thecellvolumeisconstant (i.e.nogasesare liberated attheelectrodes). IfVisthee.m.f.ofthecell,thenF=VQ, andwehave whereh=UjQistheheatofreaction whenunitchargeispassed. ThereasonwhyVQisnotjustequaltoUisbecauseitmaybenecessary forthecelltoexchange heatwithitssurroundings inordertoremainat constant temperature, andthisflowofheatmayberelated,bythesecond lawofthermodynamics, tothetemperature coefficient ofthee.m.f. 4.8.Conduction ofelectricity through gases Underperfectconditions agasconsistsofuncharged molecules, and therefore behaves asaninsulator sincetherearenocharged particles presenttocarryacurrent. Inpractice, duetocosmicraysandradio­ activebackground (especially inthewallsofthecontaining vessel),there arealwaysafewionspresent, whicharesufficient toinitiateaspark 4.8] PROPERTIES OFELECTRICAL CONDUCTORS 117 discharge atsufficiently highelectricfields(oftheorderof30000VIcm inairatatmospheric pressure), butatlowfieldsthecurrentpassingis negligibly smallunlessionsaredeliberately produced inthegas,orelec­ tronsareliberated atoneoftheelectrodes (thecathode). Theessential distinction between thetwocasesisthatatlowfieldsthecurrentis D B_-----------------0 oIC..-- ~ _ --_Voltage FIG.4.15.Current-voltage characteristic ofagas. OAOhm'slawisobeyed; mostionsformedarelostbyrecombination. BOallionsformedareswepttoelectrodes beforerecombination cantakeplace. CDfreshionsareformedbycollision whenelectrons canreachtheionization potential ofthegasmolecules between collisions (Townsend discharge). limitedbythesupplyofionsthrough external action(X-raysorultra­ violetlightreleasing electrons fromtheelectrodes orfromthegasmole­ cules)whileathighfieldsnewionsarecreatedbycollisions between charged particles (accelerated bytheappliedelectricfield)andneutral molecules. Atypicalcurrent-voltage characteristic isshowninFig.4.15. Atverylowvoltages thecurrentisproportional tothevoltage,butat highervoltagesitriseslessrapidlyandreachesaconstant value,inde­ pendent ofthevoltageoverawiderange. AtpointsontheinitialpartOAofthischaracteristic (corresponding tocurrents of10-13_10-14A)thesituation isanalogous tothatinan electrolytic solution, andequation (4.34)maybeapplied. Thedensity ofionsisdetermined bytheequilibrium between therateofformation (4.36) u=eTjM=eljMv,US PROPERTIES OFELECTRICAL CONDUCTOHS [4.8 bytheX~raysandtherateoflossbyrecombination withinthegasand diffusion tothewalls.Asthevoltageincreases theionsmovepropor­ tionately fasterandappreciable numbers arelosttotheelectrodes so thattheiondensitydecreases. Whentheelectricfieldreachesabout 20Vjcmatatmospheric pressure, theionsreachtheelectrodes soquickly thatpractically noneislostbyrecombination. Thecurrentthenbe­ comesindependent oftheappliedvoltage, beinglimitedsolelybythe rateofformation ofionsbytheX-rays.Thesaturation currentispro­ portional tothenumberofX-raysincident onthegas,andsoformsa convenient measure oftheX-rayintensity. Themobility oftheionismuchgreaterthaninaliquid,andmaybe estimated inthefollowing way.Aftercollision withagasmolecule the ionisinitially movinginarandom direction, andisthenaccelerated bytheexternal fieldE.Theaveragevelocity acquired inthedirection ofthefieldcanbedetermined inthesamewayasfortheelectrons in ametal(seeequation (4.2)).ItiseETjM,where Tisthemeantime between collisions, ethechargeontheionandMitsmass,sothatthe mobility is where1isthemeanfreepath,andvtherandom molecular velocity. Forionsofmolecular dimensions, 1andvhavetheusualvaluesgiven bykinetictheory:1=Ij.y2n7Ta2,wherenisthenumberofmolecules per unitvolumeofdiameter a;andv=(8kTj7TM)l, wherekisBoltzmann's constant andTistheabsolute temperature. Atatmospheric pressure, bothpositiveandnegative ionshavemobili­ tiesoftheorderofacmjsecperVjcm,infairagreement withvalues calculated usingequation (4.36).Asthepressure islowered, themo­ bilityincreases inversely withthedensityforpositive ions,correspond­ ingtotheexpected increase inmeanfreepath,butfornegative ionsit increases muchmorerapidly. Thisisduetothefactthatatlowpressures mostofthenegative ionsareelectrons ratherthanheavychargedmole­ cules.Atagivenpressure, theratioofelectrons toheavynegative ions variesmarkedly fromgastogas;somemolecules, suchasC12,readily attachelectrons toformnegative ions,whileotherssuchasH2donot. Themobility ofelectrons ismuchgreaterthanthatofheavyions,mainly owingtotheirsmallmass,butalsopartlyduetotheirlongermean freepaths.Sincethediameter ofanelectron isnegligible, itscollision diameter withagasmolecule isonlyla,andsinceitsvelocity ismuch greaterthanthatofthegasmolecules thefactor..,12introduced byMax­ welltoallowfortherelative velocities isabsent,sothattheelectron 4.8] PROPERTIES OFELECTRICAL CONDUCTORS 119 meanfreepathis4Jn7Ta2,or4"';2timesthatofaheavyion.Inaddition, theaverage lossofenergybyanelectron inanelasticcollision with amolecule isverysmall(seeProblem 4.4),andtheaverageenergyof theelectrons whenafieldisappliedismuchhigherthanthatofthegas molecules orheavyions.Wemayexpressthisbysayingthatthe'mean temperature' oftheelectrons ishigherthanthatofthegas.Asthe pressure isreduced, andthemeanfreepathincreases, theenergygained byanelectronfromtheappliedfieldincreases andtheeffective electron temperature rises:Theenergygainedisproportional totheproductof themeanfreepathandtheappliedfieldE,andsincethemeanfreepath isinversely proportional tothepressurepitfollowsthattheconditions areafunction ofEJp.AtlowvaluesofEJptheenergygainedbyan electron between collisions issmall,anditmakesonlyelasticcollisions withthegasmolecules, butathighvaluesofEJpthemeanelectron temperature risesandthenumberofelectrons inthehighenergytail oftheenergydistribution increases rapidly. Thosewhichhaveafew electron voltsofenergycanmakeinelastic collisions inwhichmostof theenergyistransferred tothecolliding molecule. Theeffectonthe molecule willnowbediscussed. Onquantum theorythetotalenergy,kineticpluspotential, ofan electron boundinanatomcanonlyhavecertainallowedvalues,and inthenormalstatetheelectrons inanatomareinthelowestallowed levels;thisisthe'groundstate'oftheatom.Thedifferent energylevels canbeplottedonan'energyleveldiagram' (suchasFig.20.2).The atomcannotexistwithintermediate valuesoftheenergy,andifitis inanexcitedstate(oneofthehigherenergylevels)itmayreturnto thegroundstatebyemitting itsexcessenergyasaquantum oflight whosefrequency visdefinedbytheequation lli-~=hv. (4.37) Foramolecule theenergyleveldiagram issimilartobutrathermore complicated thanthatofanatom. Ifanelectron withsufficient energycollideswithanatomormole­ culeinitsgroundstate,itmaytransfer someofitskineticenergyto themolecule andraiseittoanexcitedstate.Forthistobepossiblethe electron musthaveatleastasmuchenergyasthedifference between thegroundstateandthefirstexcitedstateofthemolecule, andthe potential through whichtheelectronmustbeaccelerated toobtainthis energyiscalledthe'resonance potential' ofthemolecule. Ingeneral themolecule willgetridofthisextraenergybyemitting aphoton(light 120 PROPERTIES OFELECTRICAL CONDUCTORS [4.8 quantum) withinabout10-8sec,andthegasthusbecomes luminous whentheelectrons gainsufficient energyfromtheappliedelectricfield toraisethemolecules intotheseexcitedstates.Astheenergyofthe electrons increases, themolecules areraisedintohigherexcitedstates, corresponding toaboundelectron beinginanorbitoflargerradius,and finallythemolecule maybeionized;thatis,anelectron iscompletely removed, leavingthemolecule asapositively-charged ion.Theenergy required todothis(expressed inelectron volts)iscalledtheionization potential ofthemolecule. Thisprocessofionization through electron impactincreases thenumberofcharged ionsandelectrons, andwhen thevalueofEjpislargeenoughforittooccur,thecurrentthrough thegasisgreatlyincreased. ThesteepriseincurrentODwithapplied voltageshownafterthesaturation plateauBOinFig.4.15isdueto theformation ofionsbycollision; itwasextensively investigated by Townsend, andisknownastheTownsend discharge. Withspecially designed electrodes thevoltageinthisregionbecom,9s almostindepen­ dentofcurrent,andsmallgas-filled tubesareusedasvoltagestabilizers. Ionization byelectron collision istheprimary processinproducing freshionsinthebodyofthegas.Experiments haveshownthatcollisions withpositive ionsaremuchlesseffective incausingionization (owing totheirshortermeanfreepaths,heavyionspickuplessenergyfromthe appliedfieldthanelectrons), andthisprocesscanbeneglected incom­ parison. Themostimportant secondary processes forproducing further chargedparticles occuratthecathode, fromwhichelectrons areemitted undertheactionof(a)bombardment bypositive ions,(b)thephoto­ electriceffectcausedbyphotonsemittedbyexcitedmolecules, (c)bom­ bardment byexcitedmolecules. Therelativeimportance ofthesethree processes varieswiththeconditions; ingeneral (a)ismoreimportant withcathode surfaces ofhighworkfunction, and(b)withsurfaces of lowworkfunction. Themaintypeofexcitedmolecules reaching the cathodearethosein'metastable states',i.e.molecules incertainexcited stateswhichcannotreturntothegroundstatebyemitting aphoton, andsohavemuchlongerlivesthanthe10-8secmentioned above. Secondary ionization processes occurring withinthebodyofthegas, which(exceptathighpressure) appeartobelessimportant thanthose atthecathode, are(d)photoionization, inwhichhighenergyphotons emittedbyonemolecule areabsorbed byanother, andmayhavesuffi­ cientenergytoionizeit.Thisoccursmostlywiththehighfrequency ultraviolet radiation whichisfoundinhighvoltagedischarge tubes; (e)asthetemperature ofthegasrisesowingtotheconversion ofelec- 4.8J PROPERTIES OFELECTRICAL CONDUCTORS 121 tricalenergytoheatenergythrough collisions between molecules and ions,neutralmolecules mayhavesufficient kineticenergyofrandom motiontoionizeothermolecules bycollision. Thisprocessissometimes calledthermal ionization. (4.39) (4.40)dn=0lndx, whichonintegration givesn=noe01X,andthecurrentattheanodeis thereforeTheTOUJnsend discharge Suppose wehavetwoplaneparallelelectrodes adistance dapartand anelectricfieldisappliedbetween them.Letthenegative electrode (thecathode) beilluminated withultraviolet lightwhichcausesnoelec­ tronstobeemittedpersecond. Theseelectrons areaccelerated, andif thevalueofE/pissufficiently high,theywillproduce furtherionsby collision.Ifnelectrons crossaplaneatadistancexfromthecathode persecond,thenthenumberformedbyionization inthenextelement ofdistance dxwillbeproportional bothtonanddx,sothatwecan write e01d=(°2+1)/°2, (4.41) Thisimpliesthatafinitecurrentwillpasswhenthiscondition issatis­ fied,evenif10iszero.Thevoltageatwhichthisoccursisknownasthe1=ne=noe.e01d=loe01d, (4.38) where10isthecurrentduetotheoriginal noelectrons alone.Forlow currents thisequation isingoodagreement withexperiment, butathigh currents thecurrentshootsuprapidlytowards infinity. Thisisdueto thesecondary processes, whichincrease thesupplyofions,principally bycausingtheemission ofmoreelectrons fromthecathode.Ifnowthe totalemission ofelectrons fromthecathode isno,thenumberofextra electrons produced inthegasbyprimary ionization mustbeno(e01d-I), andthiswillalsobethenumberofpositiveions.Thenumberofexcited molecules emitting photons, andthenumberofmolecules inexcited states,willalsobeproportional tothisnumber, andhencesoalsowill bethenumberofsecondary electrons emittedfromthecathode, what­ everthemechanism. Hence no=nO+C2nO(eOld-I), fromwhich no/no={1-02(e01d_l)}-1, andthetotalcurrentwillbe e01d I-n'e eO,d-L-o·-00.(ad)'1-2e1-1 Thetotalcurrentwillbecomeinfinitewhenthedenominator iszero, thatiswhen 122 PROPERTIES OFELECTRICAL CONDUCTORS [4.8 sparking orbreakdown potential~. Experimentally itwasdiscovered byPaschenthatforagivengas~depends onlyontheproductpdofthe gaspressurepandtheelectrode separation d.ThisisknownasPaschen's law,anditholdsuptoveryhighpressures; itfollowsfromtheTownsend theory(above), fortheconstant 01isthenumberofionsproduced by anelectroningoingunitdistance. Thisnumbermustbeproportional V 2,000 1,000 0·2 0·1ol-----,----r----,--­ 0·3pd FIG.4.16.Breakdown voltageVforairplotted againstpd.Vinvolts,pinemofmercury, and dinem. tothenumberofmolecules perunitvolume,andhencetothepressure, anditalsodepends ontheaverageenergygainedbyanelectronbetween collisions. ThisenergyvariesasEl,wherelisthemeanfreepath,and sinceE=Vfdandlisinversely proportional top,wehave Old=(pd)F(Vfpd), (4.42) whereF(Vfpd) issomesingle-valued function of(Vjpd). SinceO2is aconstant itfollowsfromequation (4.41)thatthesparking potential ~isafunction onlyofpdforagivengas. Inspection ofequation (4.39)showsthatwhenthecondition ofequa­ tion(4.41)isfulfilled, 02(ec•d-1)=1;thatis,thesecondary processes produce alltheelectrons leavingthecathode. Theseelectrons increase atanexponential rate,andthedischarge currentrisesveryrapidly(in atimeoftheorder10-7secatatmospheric pressure) andasparkpasses. Atypicalcurveforthevariation of~withtheproduct (pressure Xelec­ trodeseparation) isshowninFig.4.16.Thesharpriseatthelowpressure endisduetothelowdensity,whenthechanceofanelectronencounter­ ingamolecule issmallandfewionsareformedbycollision. Inthehigh pressure regioncollisions arefrequent, butthemeanfreepathissmallso thatfewelectrons gainsufficient energyfromthefieldbetween collisions 4.8].. PROPERTIES OFELECTRICAL CONDUCTORS 123 tocauseionization. Thusforanygivenelectrode separation d,thereis alwaysacertainpressureatwhichthesparking potential isaminimum. LaterworkontheTownsend discharge (seeLlewellyn Jones,1953) showsthatthetheorygivenaboveholdsoveraverywiderangeofvalues ofpd.Atthehigherpressures (pdgreaterthanabout20emHgXem andsparking potentials of10-100kV)positive ionscannotreachthe cathodeintheduration ~10-7secfoundexperimentally foraspark, andmostofthesecondary emission fromthecathodeisduetophotons; however, cathodeemission isthenprobably lessimportant thanioniza­ tioninthebodyofthegas.Atpressures oftheorderof100atm,and withgapsoftheorderofcentimetres, Paschen's lawbreaksdown.This isduetothehighfieldsatthecathode (---106Vjcm),whichcause appreciable fieldemission, aprocesswhichdoesnotdependonthenum­ berofionsformedinthebodyofthegas,asassumed inequation (4.39). Sincebothfieldemission andphotoelectric emission dependonthework function, thenatureofthecathode surfacebecomes increasingly im­ portantathighpressures. 4.9.Plasma oscillations Interest ingasdischarge physics, underthemodemtitleofplasma physics, hasbeenrenewed inthequestforthermonuclear power. Aplasmamaybedefinedasanassembly ofchargedandneutralparticles instaticordynamic equilibrium, butthisequilibrium maybedisturbed locally. Suppose thatatsomeinstantamomentary excessofcharge occursinoneregion;themutualrepulsi8n ofthechargedparticles pushes themapart,sothattheexcessquicklydisappears. However, thevelocity gainedbytheparticles through theirmutualrepulsion maycarrythem toofar,andtheexcessisreplaced byadefectinthechargedensity,and theparticles areattracted back.Repetition ofthisprocesssetsupa periodicdisturbance knownas'plasmaoscillations'; theseareacommon featureofgasdischarges (wheretheymaybevisibleasstriations), and electron orionbeams.Weshallnotdiscussthesephenomena (which areverycomplex), butcontentourselves withasimplederivation ofthe frequency ofsuchoscillations. Inagasatlowpressure therelativepermittivity maybetakenas unity,sothatfromGauss'stheorem (equation 1.20)wehave EOdivE=p. (4.43) Through themovement ofchargesadriftcurrentJ=pvoccurs,and theconservation ofchargeasexpressed inthecontinuity equation (3.3) gives -dpjdt=divJ=div(pv). (4.44) 124 PROPERTIES OFELECTRICAL CONDUCTORS [4.9 Differentiation ofequation (4.43)gives,together withequation (4.44), €odiv(dEldt) =-div(pv) or €oE=-pv=-J, (4.45) wheretheconstant ofintegration iszerosinceintheabsenceofparticle movement (v=0)thetimederivative oftheelectricfieldisalsozero. Inaplasmawherebothpositiveandnegative ionsarepresent, the negative ionsaremostlyelectrons whichmoveverymuchmorerapidly thanthemoremassive ionizedatoms,andwecanassumethatallthe currentiscarriedbyelectrons; i.e.byonetypeofparticle, ofmassm andchargeq.Theequation ofmotionoftheseparticles ismv=qE,so that(usingequation (4.45)) dJldt=pV=(pqlm)E, d2Jldt2=(pqlm)E=-(pqlm€o)J· Ifweassumethatoscillations areofvanishingly smallamplitude, the departure ofpfromitsmeanvaluefig,wherenistheaveragenumber perunitvolume, isnegligible, andwecanwrite d2Jldt2=-(nq2Im€o)J. (4.46) Thisistheequation ofsimpleharmonic motion,showingthatthecurrent willoscillateatafrequency 27Tfp=(fig2Im€o)l. (4.47) Thisisknownastheplasmafrequency. Wecanestimate itsmagnitude bytakingasanexample aplasmaoffullyionizedhydrogen atapressure of10-5atm(0'0076mmHg),forwhichn=2·7X1020percubicmetre. Then f-21 1p=(ne47T2€om)'- =g'On1 =1·5X1011. (4.48) Thisfreq~ency corresponds toawavelength of2mmforelectromagnetic waves,anditsmeasurement isanimportant toolinplasmaphysicssince itgivesthedensityofelectrons. Inordinary gasdischarges thedegree ofionization isrelatively low,andtheplasmafrequency maylieinthe region103to108cis.Inmetals,ontheotherhand,theelectron density isverymuchhigher,andtheplasmafrequency isabout1015cis. REFERENCES BORELIUS, G.,KEESOM, W.H.,andJOHANSSON, C.H.,1928,Proc.Acad.Sci. Amsterdam, 31,1046. CRAIG,D.N.,andHOFFMAN, J.1.,1950,Phys.Rev.80,487. CUSACK, N.,1958,TheElectrical andMagnetic Properties ofSolids(Longmans, Green). 4.9] PROPERTIES OFELECTRICAL CONDUCTORS 125 JAIN,S.C.,andKRISHNAN, K.S.,1953,Proc.Roy.Soc.A,217,451. LLEWELLYN JONES,F.,1953,Ann.Rep.Progr.Phys.(Physical Society, London), 16,216. MITCHELL, E.W.J.,andMITCHELL, J.W.,1951,Proc.Roy.Soc.A,210,70. SHOCKLEY, W.,1950,Electrons andHolesinSemiconductors (VanNostrand). SLATER,J.C.,1939,Introduction toChemical Physics (McGraw-Hill). WILKS,J.,1961,TheThirdLawofThermodynamics (O.U.P.). ZEMANSKY, M.W.,1957,HeatandThermodynamics (McGraw-Hill). PROBLEMS 4.1.Itisfoundthatthethermoelectric powerdV/dtofacopper-nickel thermo­ coupleintherange0°Cto100°Ccanbeexpressed as20·4+0·0450tp,V;aC,where tisthecentigrade temperature measured onaparticular hydrogen gasthermo­ meter.ThePeltiercoefficient canbeexpressed as1330+7·78t+0·0107t2p,cal/ coulomb. Verifythatinthisthermometer thehydrogen behaves asaperfectgas, anddeducetheabsolute temperature oftheice-point. 4.2.Findtheequation ofmotionofafreeelectroninanelectricfieldE=Eosinwt. Ifthefieldstrength Eois104V/metre, andthefrequency is100Mc/s,showthat theamplitude ofoscillation oftheelectron is0·0045mandthatitsmaximum energyis22·5eV. 4.3.Asparkpassesbetween twoelectrodes Icmapartinairatatmospheric pressure whenauniform fieldof10kV/cmisappliedacrossthegap.Ifthemean freepathofanoxygenmolecule inairis6x10-6cmshowthatthetimerequired forasinglyionizedoxygenmolecule tocrossthegapis4·5X10-5sec. 4.4.Anelectron ofmassmcollideswithamolecule ofmassM.Showthatifthe molecule isstationary thefraction oftheelectron energywhichistransferred to themolecule inahead-on collision is4Mm/(m+M)2 andevaluate thisforthe caseM=200Xmassoftheproton. (Answer: ~1·1X10-5.) 4.5.Calculate theneutral temperature (t=-0I./{3)forthethermocouples copper/iron andplatinum/platinum-rhodium. (Answer: +276and-255°C.) 4.6.Useequation (4.20)toshowthatthetemperature coefficient (dWF/dT) for tungsten (WF=5·8.eV)at3000°Kisabout-0'6x10-5eVperdegree.WFwill alsovarybecause ofthermal expansion, sincethenumber ofelectrons perunit volumechanges.If01.isthelinearcoefficient ofexpansion, showthat dWF/dT = -201.WF· Fortungsten at3000°K,OI.isabout6X1O-6,fromwhich dWF/dT =-7X10-5eV perdegree,showingthattheeffectofexpansion isconsiderably moreimportant thanthesecondterminequation (4.20). 4.7.Calculate thevalueoftheabsolute thermoelectric powerSforgoldfrom equation (4.28),takingWF=5·5eV,andcompare itwiththevalueobtained fromTable4.2,at273°K. (Answer: -1·9and+2·7p,V/deg.) (5.1)5 THEMAGNETIC EFFECTS OFCURRENTS AND MOVING CHARGES, ANDMAGNETO STATICS 5.1.Forcesbetween currents THEfirstexperimental investigation oftheinteraction between coils carrying electriccurrents wasperformed byAmpere duringtheyears 1820-5,andtheworkwascontinued byOersted, Biot,andSavart. Theyfoundthattwolongparallelwirescarrying currents inopposite directions repeloneanother, whereas whencarrying currents inthe samedirection theyattractoneanother, sothatthedirection ofthe forceisreversed whenthecurrentisreversed. Ampere usedcircular coils,theleadstothecoilsbeingtwistedtogether, andastheseleads eachcarriedequalcurrents inopposite directions theyexertednoforce onothercircuits,and any forcesobserved weredueonlytothecoils. Refoundthat,ifthedimensions ofthecoilsweresmallcompared with theirdistance apart,onecoilexertsaforceandacoupleonanother coilexactlysimilartotheforceandcouplewhichoneelectricdipole exertsonanother. Themagnitude ofthisforceandcoupleispropor­ tionaltothecurrentthroughthecoil,thenumberofturns,andthearea. Iftheplaneofeachcoilisnormaltothelinejoiningthecentreofeach coil,theforceisalongthisline.Itisfoundalsothatifacoilcarrying a currentisplacednearamagnetitexperiences bothaforceandacouple. Atdistances largecompared withthedimensions ofeithercoilormagnet, thisforceandcouplearesimilarinnaturetothoseduetoasecondcoil carrying acurrent. Thusbothamagnetandacurrent-carrying coilare saidtoproduce amagnetic induction B,whichexertsforcesonother coilsormagnets. Bisavectorquantity andlinesofBcanbedrawn whosedirection atanypointisthatofB,inthesamewayaslinesof electricforcearedrawninanelectricfield.Thestrength ofBisshown bymakingthenumberoflinesperunitareanormaltoBnumerically equaltothevalueofB. TheforceexertedonanelementofwiredS1carrying acurrent11at aplacewherethemagnetic induction isBcanbeexpressed inthesimple form(seeFig.5.1) Thisequation thendefinestheunitofmagnetic induction asthatamount 5.1] MAGNETIC EFFECTS OFCURRENTS 127 (5.2)ofinduction whichexertsunitforceonunitlengthofawirecarrying one unitofcurrent.Inthem.k.s.a. system,theunitofforceisthenewton, thatoflengththemetre,andtheunitofcurrenttheampere. Theunit ofBisthennewtons (ampere metre)-l; weshallseelaterthatthiscan beexpressed asweberfmetre2(see§5.5)andthisisthemoreusualterm fortheunit. FIG.5.1.Diagram toillustrate equation (5.l).dSlandB areintheplaneofthepaper,theanglebetween thembeing 6.dFisnormaltothepaper,towardsthereader,andhas magnitude dBlBsin8. Theexperiments ofAmpereandothersshowedthattheforceonan element dSlcarrying acurrentIIduetoanother element dS2carrying acurrent12is where1-'0isaconstant. risthevectorjoiningthetwoelements, being positive whendrawnfromdS2todsl,asinFig.5.2.TheforcedF2on theelement dS2duetodSlisgivenbyasimilarexpression withdSl anddS2interchanged, andrmustthenbetakenaspositivewhendrawn fromdSltods2•Thedirections oftheforcesforthespecialcaseoftwo coplanar elements areshowninFig.5.2,anditwillbeseenthatthey arenotequalandopposite unlessthecurrentelements areparallel. This apparent violation ofNewton's thirdlawofmotionhascausedmuch discussion, butPageandAdams(1945)haveshownthatthereisnoreal violation, sincetheelectromagnetic fieldofthecurrentelements possesses momentum whichischanging ataratejustequaltothedifference ofthe twoforces.Ampere's originalformulation ofthelawofforcebetween 128 THEMAGNETIC EFFECTS OFCURRENTS AND [5.1 twocurrentelements wasdifferent fromequation (5.2),butgavethe correctresultwhenintegrated overaclosedcircuitcarrying aconstant current. Comparison ofequations (5.1)and(5.2)showsthatwemaysaythat \ r dSl,..:::::=----------+-- L_~ dF1 FIG.5.2.Diagram showing thedirection oftheforcesbetween twocurrentelements. dSlandds.areintheplaneofthepaper.Thevector(ds./\r) isnormaltotheplane ofthepaper,andthevector{dsl/\(ds./\r)} isintheplaneofthepaper,normaltodsl. Themagnetic fielddBduetods.isparallelto(ds./\r), andthefor"edFlondSldue toitisparallelto{dsl/\(ds./\rl). thecurrent12intheelement dS2produces amagnetic induction dBat adistance rgivenbytheformula dB=(':;~3)I2(ds2/\r). (5.3) Theseequations maybeusedtocalculate thefieldBproduced by acurrentinaninfinitestraight wire,andhencetheforcebetween cur­ rentsintwoparallel infinitestraight wires.InFig.5.:3wehavetwo suchwiresadistance aapart,carrying currentsIv12;wechooseacoordi­ natesystemwherethefirstwireliesalongthez-axis,andthesecondis paralleltoitbutpassesthroughthepointx=a,y=0.Wefirstcalcu­ latethefieldBatthepoint(0,0,0)duetothecurrent12inthesecond wire,usingequation (5.3).Thentheelement dS2hascomponents (dx,dy,dz)=(0,0,dz)andrhascomponents (-a,0,-z)sinceitis definedbythecoordinates ofthepoint(0,0,0)relativetothepoint A(a,0,z)atwhichdS2isplaced.Thenthecomponents of(ds2/\r)are (0,-adz,0),showingthatdBat0isantiparallel tothey-axis,wherever thepointAliesalongthesecondwire.HenceB=JdBwillalsobe 5.1] MOVING CHARGES, ANDMAGNETO STATICS 129 antiparallel tothey-axis,sothatBa;=Bill=0,andintegration yields forBytheresult +00 +111 B=/Lo12f-adz_=_/LoI2fCOfl,OdO= _/LoI2;(5.4) Y41T(a2+z2)t 47Ta 21Ta- -00 -I". wherewehaveusedthesubstitution z=atanO.Equation (5.4)shows thatthefieldofacurrent12inaninfinitewireisproportional to12and (0,0,0) ~-_....I--...A(a;O,z,) 14------a------H FIG.5.3.Parallel wirescarrying currents. inversely proportional tothedistance afromthewire.Bisnormalto theplanecontaining thewireandtheradiusvectorr,sothatlinesof constant Bformclosedcirclescentredonthewire. Wecannowuseequation (5.1)tofindtheforcedFonanelement dS1ofthefirstwire.SinceBisinthey-direction, anddS1inthez-direc­ tion,theforceisinthex-direction, itsonlycomponent being dFx=/Lo1112ds1• (5.5) 21Ta Ifthecurrents areinthesamedirection theforceisoneofattraction, ifthecurrents areopposedtheforceisoneofrepulsion, asstatedabove. Thevalueoftheconstant /Lodependsonthesystemofunitsemployed. Intheelectromagnetic system /Lo/41Tistakentobeunity,theforcein dynes,andthedistanceincentimetres. Thisgivesac.g.s.system,where theunitofcurrentmaybedefinedbytheuseofequation (5.5),being thatcurrentwhich,flowinginastraight infinitewireatadistance of 1cmfromaparallelwirecarrying anequalcurrent,produces aforceof ~llW K 130 THEMAGNETIC EFFECTS OFCURRENTS AND [5.1 2dynespeteentimetrelength. Similarly, 1electromagnetic unit(1gauss) ofmagnetic fieldBexertsaforceof1dyneon1cmofaconductor carry­ ing1electromagnetic unitofcurrent (1gauss=10-4weber/metre2). Inthem.k.s.a.system, theforcesareinnewtons andlengthsinmetres, andthecurrents inabsolute amperes. Hitherto wehaveregarded the ampere(orcoulomb, since1Aisacurrentof1coulomb/sec) asastandard ofcurrent(orcharge)definedinsomearbitrary way,similartothekilo-. grammeandmetre.ThevalueofP-oisthenaconstant tobedetermined. Inpractice P-oisdefinedtobeexactly 47TX10-7(newton/ampere2)since thismakestheunitofcurrentexactlyequaltoone-tenth oftheold electromagnetic unit,andhenceequaltothepractical unit(theampere) asgenerally usedinthepast.Equation (5.5)thenshowsthatfortwo parallelwires1metreapart,eachcarrying 1ampereofcurrent,theforce permetrelengthofwireis2X10-7newtons. Thismayberegarded as aconvenient wayofdefiningtheampere. Thequantity P-oisknownas the'permeability offreespace'(see§5.4),andit~unitisgenerally called thehenry/metre (see§6.2)ratherthannewton/ampere2;thetwounits areequivalent. 5.2.Magnetic shells Theinvestigations ofAmpereoftheforcesbetween twosmallcoils showedthattheyweresimilartothosebetween twodipoles. Compari­ sonwithequation (1.14)showsthatweshouldexpectsuchadipole,if placedinauniformfield,toexperience acouple,andweshallnowderive thiscouplebyuseofequation (5.1).Asmallplanecoilisplacedina regionofuniform magnetic induction B.Wedividetheareaofthecoil intothinstrips,asinFig.5.4,bydrawing linesparalleltothex-axis, whichistakentobetheprojection ofBontheplaneofthecoil.The currentIflowingroundthecoilmayberegarded asmadeupofacurrent Iflowingroundeachoftherectangular stripsinthesamesense;forthere isthenflowingalongeachline,such asODtwocurrents, fromneighbour­ ingstrips,ofopposite signsothattheyannul,leavingonlythecurrent alongtheperiphery. Inordertocompute theforcesonthestripODEF weresolveBintoacomponent Bcos()normaltotheplaneofthestrip (where ()istheanglebetween Bandthisnormal)andacomponent Bsin()inthe.plane,paralleltothex-axis.Theforceoneachsideofthe stripduetothenormalcomponent Bcos()isintheplaneofthecoil, normaltothesideandproportional tothelengthoftheside.Itis readilyseenthattheyformasetofforcesinequilibrium, fortheycan bedrawnasasetofvectorsformingaclosedfiguresimilartothestrip, 6.2] MOVING CHARGES, ANDMAGNETOSTAT!CS 131 turnedthrough arightangle.Theforcesduetothecomponent Bsin() arezeroonODandEF,whilethoseonDEandFOareproportional totheprojections oftheseelements onthey-axis(beingthusequaland opposite), andnormaltotheplaneofthecoil.Theytherefore forma coupleofmagnitude IXODX8yXBsin()tendingtoturnthecoilabout r ..~===~: ----+---------/ '----------------- ...LJ--: f5x -F1G.5.4.Diagram showing thecoupleonacurrent circuitduetoafieldofmagnetic induction Bwhichmakesanangle8withthenormaltotheplaneofthecircuit,and whoseprojection ontheplaneisparalleltothe:v-axis. they-axis;but(ODx8y) istheareaofthestrip,whichcanberepre­ sentedbyavectordSnormaltotheplane,whosesenseisthatofaright­ handedscrewturnedinthedirection ofthecurrent. Thecouplecan thenbewrittenintheform wheredr=IdSI\B =dml\B. dm=IdS(5.6) (5.7) (5.8) (5.9) m=IS.isdefinedasthemagnetic dipolemomentofthestripODEF.Thecouple actingonthewholeplanecoilisproportional tothearea,andsoalso isthemoment oftheequivalent dipole.Thatis, r=ml\B, Comparison ofequation (5.8)withequation (1.14)showsthatitisQf 132 THEMAGNETIC EFFECTS OFCURRENTS AND [5.2 thesameform,themagnetic fieldBplayingthesameroleastheelectric fieldE.Wemaytherefore expectthatitcanbewrittenasthegradient ofascalarpotential cp,sothat (5.10) Although thederivation abovewasgivenforaplanecoil,itisclear thatacoilofanyshapecanbedividedupinthesameway(asinFig.5.5) by.using anysurfacewhichisbounded bythecircuitformedbythecoil. FIG.5.5.Alargecoildividedintoanumberofsmallmagnetic shells. EachelementofareadBmayberegarded ashavingacurrentIflowing rounditsedge,andsummation ofthecurrents inalltheelements com­ prisingtheentiresurfaceleavesonlythecurrentinthecircuitasthe resultant.IfdBistakenasinfinitesimal inbothdirections, ratherthan thenarrowstripassumed above,itcanberegarded asaplaneelement andwillhaveanassociated magnetic dipolemoment givenbyequation (5.7),andthecoupleonthewholecircuitisobtained byintegration of equation (5.6).Thesurfaceformsamagnetic 'doublelayer',or'mag­ neticshell',withacertaindipolemoment perunitarea.Thepotential duetosuchashellwill nowbecalculated usingequation (5.3). ThefieldatapointPduetothecurrentcircuitisfoundbyintegration ofequation (5.3)roundthecircuit.InFig.5.6ifthepointPisdisplaced adistance &sthechangeinpotential willbe Scp=-~B.&s= -I&s.f(dal\r) =_If&s.(dal\r), (5.11) /La 41Tr341Tr3 where&scanbetakeninsidetheintegralsignbecauseitisaconstant during"the integration. Itisclearthatweshouldobtainthesamechange inpotentialifthepointPwerekeptfixedandthecircuitweredisplaced byanamount-&s.Insuchadisplacement thecircuitelementdasweeps 5.2] MOVING CHARGES, ANDMAGNETO STATICS 133 r.(8s/\da) 88.(da/\r)=-r3 r3outanarea-(88/\da),andthisareasubtends asolidangleatthepoint Pof Hencethelineintegralinequation (5.11)isthetotalsolidanglesub­ tendedatPbytheareasweptoutbythecircuitwhenitisdisplaced p FIG.5.6.Displacement ofa.currentelement, inordertocalculate thepotential atPduetoa.magnetic shell. by-88,andthisisequaltothechange8winthesolidangleduetothe displacement ofPby88.Hencewemaywritethechangeinpotential as [~I~ Iwandthepotential atPis ep=--, (5.12) 47T wherewisthesolidanglewhichthecircuitsubtends atP.But w=fds.r,r3 wheredSisanelementofareaofanysurfacebounded bythecircuit, andhence rp=fIdS.r=fdm.r. (5.13) 47Tr3 47Tr Heretheintegration isoverthesurfaceofthemagnetic shell,andthe potential ofanindividual dipolemmusttherefore be rp=m.r. (5.14) 47Tr3 134 THEMAGNETIC EFFECTS OFCURRENTS AND [5.2 (5.17)~B.ds=fLoI. (5.16) Inaregionofdistributed currentflow,thetotalcurrentthreaded by thepathisIJ.dS,whereJisthecurrentdensityinanelementdSof asurfacebounded bythepath.Hence ~B.ds=fcurlB.dS =fLofJ.dS, wherethetransformation fromalineintegralofBtoasurfaceintegral ofcurlBisanexample ofStokes's theorem (seeAppendix A).Since theintegrals mustbeequaloveranysurface,theintegrands mustbe equal,andwehavefB.ds=-fLofgradep.ds =-fLoep· (5.15) Byanalogywithelectromotive force,whichisthelineintegralofE(see equation (3.12)),thequantity -epissometimes knownasthe'magneto­ motiveforce',orm.m.f. Intheelectrostatic case,theworkdoneintraversing aclosedcircuit iszero,andthiswouldalsobethecaseforatruemagnetic doublelayer. IfwetaketheintegralIB .dsfromapointPveryclosetoamagnetic shellroundtoapointP'justontheothersideofthemagnetic shell, thedifference inthesolidanglewhichtheshellsubtends atthesetwo pointsis-41T,andthem.m.f.between thesetwopointsis -D.ep=-ID.wj41T=I, fromequation (5.12).Witharealmagnetic shell,ifwenowmovefrom P'toPthroughtheshell,therewouldbeacontribution tothem.m.f. whichwouldjustmakethetotalzero,butwithacurrentcircuitthere isnosuchcontribution. Wehavetherefore animportant difference, thatthem.m.f.increases byIeverytimewegoroundaclosedpath whichthreadsthecoilpositively (i.e.inthesamedirection asthelines ofB).Thusthemagnetostatic potential isnotsingle-valued andcannot beusedinaregionwheretherearecurrents flowing. Ontheotherhand, ifthepathdoesnotthreadacurrentcircuit,thechangeinsolidangle iszero,andthepotential issingle-valued. Ifthepathdoesencirclea currenti,wehaveThisisofthesameform.asequation (1.10b),exceptthatfLo'whichwe mightexpecttoreplacetheconstant £0'doesnotoccurherebutin equation (5.10).Thereasonforthischoicewillappearlater(seeequa­ tion(5.28)). Thequantity episrelatedtothelineintegralofBbetween twopoints, since 5.2]----~~---- MOVING CHARGES, ANDMAGNETO STATICS 135 Equation (5.16)isknownasAmpere's law,andequation (5.17)is itsrepresentation indifferential form.Sinceanyfunction suchas curl(gradeP) isidentically zero(seeAppendix A),wenoteagainthatB canonlybederivedfromascalarpotential ePinaregionwhereJ=O. 5.3.Magnetostatics.and magnetic media Thetheorysofarhasbeenconcerned withthemagnetic effectsof currentsinvacuo,i.e.intheabsenceofanymagnetizable media.Itis foundexperimentally thatamaterial substance acquires amagnetic polarization whenplacedinamagnetic field,justasadielectric medium acquiresanelectricpolarization inanelectricfield.Themagnetic dipole perunitvolumeofthematerial iscalledtheintensity ofmagnetization (oroften,simplythemagnetization), andisrepresented byavectorM. Allsuchmagnetic effectsareproduced bycurrentloopsofatomic dimensions, whicharisefromthecirculation ofelectricchargewithin theatom.Therelation between themagnetic moment ofsuchaloop andthecirculating currentisgivenbyequation (5.9).Ifamagnetic medium hasamagnetization M,whichisnotnecessarily uniform throughout thesubstance, theequivalent currentflowcanbefoundby considering elementary currentloops,asinFig.5.5.Therethecurrents wereallequal,andcancelled oneanotherexceptattheperimeter, but ingeneralthiswillnotbethecase.InFig.5.7bothMandIarefunc­ .tionsofthespacecoordinates. Weconsider anelement ofvolume d'T'=dxdydzatthepoint(x,y,z),forwhichthemagnetic moment has acomponent Mzdxdydz inthez-direction. Thisisequivalent toacurrent flowingroundtheloop,thestrength ofthecurrentbeing I=(~dxdydz)/(dxdy) =~dz, sincetheareaoftheloopisdxdy.Theadjacent loopatthepoint (x+dx,y,z)hasacurrent l'=I+(aI/ox)dx=~dz+(o~/ox) dxdz. Hencethenetcurrentflowontheinterface between thetwoelements hasacomponent inthey-direction ofmagnitude I-I'=-(oj!z/ox)dxdz andifJisthecurrentdensity, thiscomponent mustequalJydxdz.By considering similarcurrentloopsintheyz-plane, wefindthereisanother currentcomponent inthey-direction ofmagnitude (oMx/oz) dxdzasso­ ciatedwiththecomponent ofmagnetization Mx•Hencethetotaly­ component ofthecurrentdensityisJ"y=(oMx/oz)-(o~/ox), withsimilar expressions fortheothercomponents ofJ.Thesearethecomponents 136 THEMAGNETIC EFFECTS OFCURRENTS AND [5.3 ofthevectorcurlM,sothatwecanwriteforthecurrentdensityJm associated withamagnetization M Jm=curlM. (5.18) ItfollowsthatifMisuniforminspace,Jm=0,sothatanequivalent currentflowexistsonlyinregionswhereMisvarying. y .--/./',/ ./'./' /1';) /1~/' JJJz JJJZ·[/ I I II xx+dx (5.20)FIG.5.7.Representation ofanon-uniform magnetization bycirculating currents. ,8M.M.=M'+a;; dx. Inamedium whichisbothelectrically conducting andmagnetizable, thetotalcurrentdensitywillbethesumoftherealcurrentdensityJ andtheequivalent magnetization currentdensityJwbothofwhich mustbecountedinAmpere's law.Henceequation (5.17),whichwas derivedforavacuum, mustbereplaced by curlB=lto(J+J m)=ltol.J+curlM), or curl(B-ltoM) =ltoJ. (5.19) ThisistheformwhichAmpere's lawtakesinthepresence ofamag­ netizable medium, andthequantity (B-l-'oM), beingdirectlyrelated totheflowofrealcurrent, isusedtodefineanewvectorsuchthat B-ltoM =ltoH,or --_._------ ------ 5.3] MOVING CHARGES, ANDMAGNETO STATICS 137 ThenAmpere's lawtakesthesimpleform curlH=J, (5.21) whichismoregeneralthanequation (5.17),sinceitholdsbothinvacuo andinamedium. Obviously, invacuoB=1-'0H,sothatequation (5.17)isaspecialcaseof(5.21),whichisthegeneraldifferential form ofAmpere's law.Similarly, equation (5.16)mustbereplaced bythe moregeneralequation (5.23)(5.22) fH.ds=fcurlH.dS =fJ.dS=I. Itisclearfromthisequation thatthedimensions ofHmustbeamperes permetre,sincethelineintegralofHroundacircuitisequaltothetotal currentthreading thecircuit.Fromequation (5.20),Mmusthavethe samedimensions asH,andthiscanbereadilyverified, sinceM=mag­ neticmomentperunitvolume,andmagnetic moment =currentXarea, fromequation (5.9). Theprocessbywhichwehaveintroduced anewvectorHinmodify­ ingourequations toallowforthepresence ofapolarizable medium is analogous tothatinelectrostatics, whereanewvectorDwasintroduced. There,thisvectorfollowed fromthemodification ofGauss'stheorem neededinthepresence ofapolarizable medium; theforcevectorEis relatedbyGauss'stheorem tothesumoftherealchargedensityand thepolarization chargedensity, andtheadvantage ofDisthatitis relatedonlytotherealchargedensity.Inthemagnetic case,theforce vectorBisrelatedbyAmpere's lawtothesumoftherealcurrent densityandthemagnetization currentdensity,andtheadvantage of Histhatitisrelatedonlytotherealcurrentdensity. Wereturnnowtoequation (5.3)toderiveanimportant relationfor divB.Foravolumedistribution ofcurrent, thisequation maybe written dB=/l-o(J1\r)dT=_!!:.-~(J1\grad!)dT. 47Tr3 47T r Thenusingtherelationforthedivergence ofavectorproduct(seeAp­ pendixA),wehave divdB=_1-'odTdiV(J1\grad!) 47T r =_1-'1:T{(grad~). curIJ-J. curl(grad~)}. Nowthedifferentiation iswithrespecttothespacecoordinates only, sothatcurldoesnotoperateonJandcurlJ=0,whilecurlgrad(1jr) =0 138 THEMAGNETIC EFFECTS OFCURRENTS AND [5.3 byanothervectoridentity. HencewehavedivdB=0,andifthisis trueforthecontribution dBfromanyvolumeelementaT,itmustbe truealsoforthesumofallsuchcontributions. Hencewecanwrite divB=O. (5.24) Thisrelation hasbeenderivedonlyforacurrentinvacuo.However, wehaveshownthatanymagnetization Mpresentcanbereplaced by FIG.5.8.Boundary conditions atthesurfacebetween twomedia.Isisthesurface currentperunitwidthnormaltotheplaneofthecircuitABGDA. anequivalent currentdensityJmforwhichitwillalsobetruethat divB=O.Henceequation (5.24)holdsalsoinamagnetizable medium. Thisequation issimilartothatderivedfordivDinelectrostatics, exceptthatdivD=Pe'wherePeisthedensityoftrueelectriccharge, whiledivB=0becausewehavenotruemagnetic charges. Again,as inelectrostatics, wecanuseGauss'stheorem appliedtoanelementary flatboxsurrounding theboundary between twomagnetic mediaasin Fig.5.8toshowthatJB.dS=0overthesurfaceofthebox.Ifthe heightoftheboxisverysmallcompared withitscross-section, theonly contributions toJB.dScomefromthecomponents ofBnormalto theboundary. Hencewehave IBn=",Bn" (5.25) Theboundary conditions forHarefoundbyusingAmpere's law appliedtoasmallrectangular circuitABODA whosesidesBO,ADare verysmallcompared withAB,OD.IfthereisasurfacecurrentIsper unitlengthofthesurfacenormaltothecircuit,thenequation (5.22) leadsimmediately totheresult IH,-2H, =1.. (5.26) 5.3] MOVING CHARGES, ANDMAGNETO STATICS 139 Ifthereisnosurfacecurrent, IH,-2H, =0, (5.27) showingthatthetangential components ofHarecontinuous, whilefrom equation (5.25)thenormalcomponents ofBarecontinuous. These boundary conditions aresimilartothoseinelectrostatics, butnotethat theformalequivalence hereisbetweenBandD,andbetweenHandE. Thisequivalence canbecarriedastagefurther, sinceifnocurrents are present, wehavecurlH=0,andwecantherefore write H=-grad1>, (5.28) whichisanalogous toE=-gradV.Equation (5.28)istruebothin vacuoandinamagnetizable medium, ourearlierequation (5.10)being aspecialcase. 5.4.Solution ofmagnetostatic problems Inmanymaterials itisfoundthatthemagnetization Mislinearly proportional tothefieldH,sothatwecanwrite M=XH. (5.29) HereXisknownasthemagnetic susceptibility; ifwewishtodistinguish itfromtheelectricsusceptibility (§1.5)wemaywritethemasXmand Xerespectively, butwherethereisnodangerofconfusion thesubscripts maybeomitted. Representative valuesofXfordifferent substances varywidely,andwillbediscussed inChapter 8.Atordinary tempera­ turesXissmallandindependent ofHformostsubstances, theexceptions beingferromagnetics, whereXislargeandverydependent onfield strength; Mmayevenbenon-zero whenH=0. Fromequation (5.20)wehave B=f'o(H+M)=f'oH(I+X) =f'f'oH, (5.30) wherethequantity f'=I+X (5.31) isknownasthemagnetic permeability ofthemedium, orsometimes, sincef'oiscalledthe'permeability offreespace',asthe'relative per­ meability'. Itisclearthatf'playsasimilarroleinmagnetostatics to thatplayedbythedielectric constant Einelectrostatics. SincedivB=0,wehave,whenf'isindependent ofH, divB=div(f'f'oH)= -f'f'odivgrad1> =-f'f'oV21>=0, or V21>=0, (5.32) 140 THEMAGNETIC EFFECTS OFCUltRENTS AND [5.4 showing thatthemagnetostatic potential obeysLaplace's equation. ThetheoryofChapter 2maytherefore beadapted tomagnetostatic problems, andweshallillustrate thisbytreating aspecialcase. Theproblem ofapolarizable sphereinauniform electricfieldwas solvedbymeansofspherical harmonics in§2.4.Thecorresponding magnetic problem maybeapproached inthesameway,butweshall ----------------+. lIo (1',0) cP1 M-t------..-tC--...J-----f---- ....z FIG.5.9.Amagnetizable sphereinauniform fieldHo=-8cP./oz. ~l=-Hlrcos(J (r<a), ~2=-Horcos(J+Ar-2cos(J(r>a). Asintheelectrostatic case,therecanbenoterminr~2coseinsidethe sphere,sinceitwouldbecomeinfiniteatr=0;thusthefieldinsideis uniformandequaltoHl.Outsidethespherethefieldatlargedistances isuniformandequaltoHo;wetakeHotobegiven,sothatHlandA aretheunknowns tobedetermined fromtheboundary conditions.extenditslightlybyassuming thatMisnotnecessarily proportional toH,thoughstillparalleltoit.Then,fromequation (5.20) divB=-p..odivgrad~+p..odivM =0, whence V2~=divM. (5.33) Inthecorresponding electrostatic casewefoundthatthespherewas uniformly polarized, andweshallassumethatthisistruealsointhe magnetic case.ThendivM=0,andthepotentials required aresolu­ tionsofLaplace's equation. InFig.5.9,thepotentials insideandoutsidethesphereareassumed tobe 5.4] MOVING CHARGES, ANDMAGNETOSTATICS 141 The:firstboundary condition, thatthetangential components ofH becontinuous attheboundary, isequivalent tomaking ep1=ep2at r=a,giving H-H.-A-31- 0a. Theradialcomponents ofBare JLoMcosO-JLo(8ep1/8r) =JLo(M+H1)cosO (inside) and -JLO(8ep2/8r) =JLo(Ho+2Ar-3)cos0(outside). (5.35)Hence,equating thetwoatr=a,wehave M+H1=Ho+2Aa-3, andelimination ofAbetween thisequation andthefirstboundary condition givesH1=Ho-IM, (5.34) sothatH1issmallerthanHobyanamount1M.Thusthemagnetiza~ tionproduces areversefieldinsidethesphereknownasthe'demag­ netizing field'whichisproportional toM;thefactor1isknownasthe 'demagnetizing factor'.Itsvaluedepends ontheshapeofthespecimen, aY{ditisonlyameaningful conceptforanumberofsimpleshapeswhere theinternal fieldisuniformandparalleltoHo• Afullsolutionoftheproblem ispossible onlyifweknowhowM depends onH1.IfM=XH1=(JL-1)H 1,wefindH1=3Ho/(JL+2), M=3Ho(JL-1)/(JL+2). Asintheelectrostatic case,thefieldoutside thesphereisequaltoHoplusthefieldofadipoleatthecentreofthe sphereofmagnitude equaltothetotalmoment ofthesphere.Inthe ferromagnetic casewecanhaveafiniteMevenwhenHo=O.Thisis aspherical permanent magnet, whoseexternal fieldisthatofapoint dipole,andwhoseinternal fieldisjustthedemagnetizing field H1=-1M. Weconclude thediscussion ofmagnetostatics byfindingageneral expression forthemagnetic potential duetoamagnetized substance. InanelementdTthedipolemoment isMdT,andthepotential equation (5.13)maybewrittenintheform ep=~fM.grad(l/r)dT, 41T wherethedifferentiation iswithrespecttothecoordinates ofthevolume element dT(cf.equation (1.11b».Then,byavectortransformation similartothatusedinderiving equation (1.17),wefind lfl()lfl. ep= - - M.dS---(divMdT), 41Tr 41Tr(5.36) (5.38)142 THEMAGNETIC EFFECTS OFCURRENTS AND [5.4 showing thatthepotential canbeattributed toanapparent surface distribution ofmagnetic chargeofsurfacedensityMcose,whereeis theanglewhichMmakeswiththenormaldStothesurface,andan apparent volumedistribution ofvolumedensity-divM,which,since divB=0,isequalto+divH.ThusthefieldlinesofHterminate on thepolarization charges, whilethefieldlinesofBareallclosedloops sincetherearenorealmagnetic charges.Ifthesubstance isuniformly magnetized, divM=0andtherearenovolumecharges,butthereis asurfacedistribution corresponding tothe'magnetic poles'ofclassical magnetic theory. 5.5.Steadycurrents inmagnetic media In§5.3theeffectsofthepresence ofamagnetizable medium were considered, anditwasshownthatAmpere's lawtakesthesimpleform curlH=J (5.21) orinintegralformfH.ds=I. (5.22) Itfollowsfromtheseequations thatinaninfiniteuniformmagnetizable medium ofpermeability fLthevalueofthefieldHisunaltered bythe presence ofthemedium, provided thecurrentflowisunaltered, andis independent offL.Returning toequation (5.3),whichholdsinvacuo whereB=fLoH,weseethatitmayberewritten intermsofHas 1dH=4rrrsI(ds/\r) (5.37) 1=47Trs(J/\r)dr, wherethefirstformreferstoacurrentIinanelementdsandthesecond toacurrentdensityJinanelementdr.Fromthepreceding remarks itisobviousthattheseequations areunaltered inamagnetizable medium, andareknownasthelawofBiotandSavart. Weconsider nowtheforcevectorB.Weknowfrom§5.4that B=,ufLoH (5.30) anditfollowsthatinamagnetizable medium thevalueoftheforce vectorduetoagivencurrentdistribution isproportional tofL.Thus theforcesbetween twocurrentelements arealsoproportional to,.",and themoregeneralformofequation (5.2)becomes dF1=,u,."o\I2{ds1/\(ds2/\rH. (5.39) 47Tf 5.5] MOVING CHARGES, ANDMAGNETOSTATICS 143 (5.8) (5.9) m=lS whereWeseefromthisthatinamagnetizable mediumtheforcebetween two currentelements isproportional tothepermeability /-"incontrast with theelectrostatic casewheretheforcebetween twoelectricpolesis inversely proportional tothedielectric constant €. Sofarwehavenotconsidered thepotential energyofacurrentcircuit inafieldB,butthismaybefoundinasimplemanner.Itwasshown in§5.2thatthecoupleonacircuitinafieldBmaybewrittenas r=ml\B, istheequivalent magnetic dipolemomentofthecircuit.Theequation forthecoupleissimilartothatforanelectricdipoleequation (1.14), whichwasfoundfromdifferentiation ofthepotential energy,equation (1.13).Theformalmathematical equivalence showsthatthepotential energyofamagnetic dipolemustbe Up=-m.B. (5.40) Hencethepotential energyofacircuitcarrying aninvariant current lis Up=-!dm.B=-I!dS.B=-IN, (5.41) where N=!B.dS (5.42) isknownasthetotalfluxofBthroughthecircuit.Fromequation (5.41) itsunitisequaltoonejouleperampere, andisknownastheweber. Thusfromequation (5.42),asalreadymentioned in§5.1,theunitof Bisweberjmetre2• Theenergyisexpressed inequation (5.41)intermsofasurfacein­ tegral,butitisusefultobeabletoexpressitasalineintegraltaken roundthecurrentcircuit. Thistransformation maybeeffected by meansofStokes's theoremifweintroduce anewvectorA,suchthat B=curlA.SinceAisessentially derivedfromBbyanintegration, thisdefinition isnotcomplete, forwecouldaddanotherterm(equivalent toaconstant ofintegration) suchasgradif1,andstillhave curl(A+gradif1) =curIA=B. Wetherefore addasupplementary condition, anddefineAbythe relations curlA=B,divA=O. (5.43) ThevectorAisknownasthe'magnetic vectorpotential', andwenote thatthedefinition inequation (5.43)isconsistent withdivB=0,since divcurlA=0(seeAppendix A). 144 THEMAGNETIC EFFECTS OFCURRENTS AND [5.5 Thepotential energyofacurrentcircuitmaynowbeexpressed as Up=-If(curIA).dS =-IfA.ds, (5.44) wherethelineintegralistakenroundthecurrentcircuit,or,foravolume currentofdensityJ,sinceIds= JdT, Up- -f(A.J)dT. (5.45) Inanisotropic magnetic medium, B =1-'1-'0H,andwecancombine equations (5.21)and(5.43)togive I-'I-'0J=I-'I-'0curlH =curlB=curl(curlA) =graddivA-V2A, whence, sincedivA =0, V2A=-l-'l-'oJ. (5.46) Thisequation issimilartoPoisson's equation, equation (2.1),except thattheoperand isavectorinsteadofascalarquantity. Thisshould notcauseanydifficulty ifweremember thatequation (5.46)implies thateachofthecomponents ofthevectorseparately mustsatisfythe equation. Thenaformalsolution similartoequation (2.6)canbefound foreachofthecomponents Ax=I-'I-'°fJxdT,etc., 47rr whichmaybeexpressed invectorformas A=1-'1-'0f~dT. (5.47) 47rr Thissolution maybeobtained directly fromequation (5.38),which gives H =Lf(J~r)dT= -~f{JI\grad(~)} dT, (5.48) wheretheintegration isovertheregionofcurrentflowandthegradient iswithrespecttoadisplacement ofthefieldpoint(cf.equation (1.11a)). Byavectoridentity (Appendix A) curl(~)=~CUrlJ-Jl\grad(~) = -Jl\grad(~) sincethecurloperator actsonlyonthefieldpointandJisinvariant in thisoperation. Henceinauniform medium B =I-'I-'0H=~fCUrl(~)dT=':0curlf(~)dT, wheretheorderofthecurloperation andtheintegration canbeinter­ changed becausetheintegration isoverthecurrentdistribution while 5.5] MOVING CHARGES, ANDMAGNETO STATICS 145 thecurloperation referstothefieldpoint.SinceB=curIA,wehave foundasolutionforAwhichagreeswithequation (5.47)above. Foracurrentcircuitcarrying acurrentI,thesolutionforAis A=1L1LOIfds (5.49) 417r andwecanusethistofindthevalueofthemagnetic vectorpotential foranelementary currentcircuitandhenceforapointdipole.Wetake z... rp(x,y,z) =-- l+- ~x FIG.5.10.Themagnetic vectorpotential duetoaplanecircularcoilcarrying a. current1. asmallcircularcurrentofradiusa,andcalculate thevalueofAat apointP.Forconvenience wetakeCartesian coordinates, whosez-axis isnormaltotheplaneofthecoilandwhoseoriginisatthecentreof thecoil.ThenPisatro=(x,y,z),andristhedistance ofPfroman elementdsofthecircuit.Intermsoftheazimuthal angleep(seeFig. 5.10),thecomponents ofdsare(-adepsinep,adepcosep,0)sothatAll=o. Since r2=(x-acosep)2+(y-a sinep)2+Z2=r8-2axcosep-2aysinep+a2, anda~r,wehave 851110!=.!..+axcosep-t;aysinep +... rro ro L 146 THEMAGNETIC EFFECTS OFCURRENTS AND [5.5 andhence 2.. A=I-'I-'0If-asinc/>dc/> IX417 r o Similarly, 2.. A_I-'I-'OIfacosc/>dc/> _+I-'I-'O{ 2I(1·3)}y--- - 17ax10•47T r 417 o Thedipolemoment equivalent tothecurrentcircuitcanberepresented byavectormofsize17a21directed alongthez-axis,andthecom­ ponents ofAarethenproportional tothoseofthevectorm1\roo Hence,dropping thesubscript onro,wecanwrite A=I-'I-'~(m1\r).47Tr(5.50) Itcanreadilybeverifiedthatthelinesofconstant Aarecirclesabout thez-axis,anditisgenerally truethatforsimplecurrentcircuitsthe linesofAaresimilarintheirgeometry tothoseofthecurrentflow,as inthecasejustdiscussed. Equation (5.50)maybecompared with eq~ation (5.14).Thevector potential isproportional tothevectorproduct(m1\r),thescalarpoten­ tialtothescalarproduct(m.r);inaddition thequantity 1-'1-'0appears inthevectorpotential butnotinthescalarpotential becausetheformer isconnected withBandthelatterwithH. In§2.3ageneralexpression wasfoundfortheequivalent electric dipolemoment ofadistributed charge,andsomeapplications onthe atomicscaleweregiven.Asimilarexpression maybefoundforthe equivalent magnetic dipolemoment ofacurrentdistribution, anditis convenient todothisfromtheformula fortheenergyinauniform fieldB,usingthemagnetic vectorpotential. Forauniform field, A=t(Bl\r) (5.51) ascanreadilybeverifiedbycalculating thecomponents ofcurlAin cartesian coordinates. Thenfromequation (5.45)thepotential energy ofadistributed currentis Up= -f(A.J)d'T =-tf{(B1\r).J}d'T =-tJB.(rI\J)d'T =-B.fl(rI\J)d'T, ---------- 5.5] MOVING CHARGES. ANDMAGNETOSTATICS 147 whereBcanbetakenoutoftheintegral becauseitisconstant and independent ofthespacecoordinates. Theequivalent magnetic dipole moment maybefoundbyequating thisexpression fortheenergyto thatinequation (5.40),Up=-m.B, giving m=It(rAJ)dT. (5.52) Wemaycheckthatthisagreeswithourearlierdefinition ofthedipole moment equivalent toacurrentcircuit,sinceforthelatterequation (5.52)becomes m=IIt(rAds)=IIdS=IS, inagreement withequation (5.9). TABLE5.1 Oomparison ofvariousformulae Electrostatics Magnetostatics Ourrent8 D=t"oE+P =t"t"oE divD=p curlE=0 E=-gradV V2V=-plt"Eo V-1fPd-r -411t"t"0-r- D=~4m-a p=Jprd-r V=~ 411t"t"0,.a Up=-poE Up=JpVd-r U=JID.Ed ...B=JLo(H+M) =JLJLoH divB=0 curlH=0 H=-grad</> V2</>=0 </>=~41Tr3 Up=-moBH =(BIJLo)-M =B/JLJLo divB=0 curlH= J B=curIA V2A=-JLJLoJ A=liP.ofJd... 411r H=Idsl\r 4m-a m=Jl(rI\J)d... A=JLJLo(ml\r) 41r'T3 Up=-IN Up=-J<AoJ)d ... U=JIBoHd ... Theformulae derivedinthischapteraresummarized inTable5.1in aformwhichgivesareadycomparison withelectrostatics. Itisassumed thatthepermeability JLisindependent ofHandisotropic, andthat thereisnospontaneous magnetization. Inaferromagnetic medium theseconditions arenotsatisfied, andformulae mustbederivedusing therelation B=JLo(H+M) ratherthanB=JLJLoH(seeProblem 5.1). 148 THEMAGNETIC EFFECTS OFCURRENTS AND [5.5 Theformula U=f!B.HdTforthestoredenergyisquotedforcon­ venience, andisderivedin§6.5. 5.6.Calculation ofthema~netic fieldsofsimplecircuits Themagnetic fieldofacircuitofsimpleshapemaybefoundina numberofways,thechiefofwhichare: (a)useofequation (5.22).Thisispossible onlywhenthefieldhas ahighdegreeofsymmetry asintherathersimilaruseofGauss's theorem inelectrostatics; (b)useofthepotential oftheequivalent magnetic shell,equation (5.12); (c)useoftheBiot-Savart law,equation (5.37); (d)useofthemagnetic vectorpotential, equations (5.43)and(5.49). Simpleillustrations willbegivenoftheuseofthevariousmethods. Thefieldduetoaninfinitestraight wirecarrying acurrentIwas calculated by(c)in§5.1,butisveryquicklyfoundbymethod (a).By symmetry, Hcanonlybeafunction oftheradialdistance fromthe wire,andbyapplying equation (5.22)toacircleofradiusraboutthe wirewefindfH.ds=27TrH=I.Hencetheazimuthal component ofHisI/27Tr;sincethisdepends onlyonr,thelinesofforceareconcentric circlesaboutthewire,andnoothercomponents ofHexist. Ifthewirehasradiusa,andthecurrentdensityisuniform, thefield insidethewirecanbefoundbyasimilarapplication ofequation (5.22). Inthiscasethecurrentthreading acircleofradiusrisI(r2/a2),sothatfH.ds=27TrH=I(r2/a2)andH=Ir/27Ta2,showingthatHincreases linearlyfromthecentretothesurfaceofthewire. Thecaseofastraightwireservesalsoasasimpleexample wherethe vectorpotential canbefoundbysolvingequation (5.46).Takingthe axisofthewiretobealongthez-axis,itisobviousthattheonly component ofcurrentdensityis~,andhencetheonlycomponent of AisAz'sothatthelinesofAareparalleltothewire.Insidethewire (permeability 11-1) V2Az=-11-111-0~=-11-111-oI/(7Ta2). Since~isindependent ofzande(making useofcylindrical coordinates T,e,z),soalsoisAzandthedifferential equation becomes !~(roAz)= _11-111-oI/(7Ta2).roror Integration gives r(oAz/or) =-11-111-0Ir2/27Ta2, 5.6] MOVING CHARGES, ANDMAGNETOSTATICS 149 wheretheconstant ofintegration vanishes, becauseoAz/or=0atr=0 (otherwise weshouldhaveadiscontinuity inBAz/oroncrossingtheaxis.) Asecondintegration gives A=fl-lfl-oI(1-r2 )(inside), (5.53)s47r a2 whereforconvenience wemakeAz=0atr=a. Outsidethewire(assuming amediumofpermeability fl-2) !~(roAz)=0,rBrBr whenceoAs/or=c/r,and A.z=cln(r/a) (outside), wherethesecondconstant ofintegration ischosentomakeAscontinuous attheboundary; i.e.Az=0atr=a.Theconstant cisdetermined bytheboundary condition forBAz/oratr=a.Bywritingr2=X2+y2 andfindingthecomponents ofcurlA,itcanbeverifiedthat Bx=(oAzlor)(y/r), By=-(oAs/or)(x/r), sothatB(J=-oAz/or. SinceB(Jispurelytangential (theothercom­ ponentsarezero),theboundary condition isthatD(Jmustbecontinuous atr=a,anditiseasilyshownthenthat As=_fl-2fl-oIIn(r/a) (outside). (5.54)27T Methods (b)and(c)maybecompared infindingthemagnetic fieldon theaxisofaplanecircularcoil.Wewillassumethatthecoilhasnturns eachcarrying acurrentI,andtheradiusofthecoilisa.Then,atapoint ontheaxisadistance zaway,thesolidanglesubtended bythecoilis 27T{1-(Z2:a2)i}(thisformula maybeverified bytheintegration w=fd~;r,takenovertheplanesurfacebounded bythecoil).Hence ~=-dep/dz=-(nI/47T)(dw/dz) =fnla2/(z2+a2)f.(5.55) Whena~z,thisformula isthesameasthatforthefieldofadipole ofmomentm=nI(7Ta2)atapointonitsaxis(cf.equation 5.9). Inapplying theformulaofBiotandSavartweconsider firstthefield dBduetoanelementofwireds,asinFig.5.lI.SincedBisnormal bothtodsandtor,itwillhavethedirection showninthefigure.On integrating roundthecoilitisclearthatthesumofallthecomponents 150 THEMAGNETIC EFFECTS OFCURRENTS AND [5.6 FIG.5.11.Themagnetic fielddue.toacircular coilatapointonitsaxis. A IIz o -----z II II !!:z..._---.... dz FIG.5.12.Magnetic fieldontheaxisofasolenoid. normaltotheaxiswillbezero.Thecomponents paralleltotheaxis r..J;;:;;i">willsumto _ .-'Hz=nIf(cosifl/47rr2)ds=27TanIcosifl/47Tr2={tla2/(z2+a2)f~-Thisformulamaybeextended tothecaseofasolenoid withmturns perunitlength,uniformly woundroundacylinderofradiusa(Fig.5.12). Iftheturnsarecloselywound,wemayregardthemasbeingequivalent toauniform currentflowingroundthecylinder, sothatanelement dz ofitformsaplanecoilwithacurrentmIdz.Atthepoint0thisgives afieldalongtheaxisequalto dHz=imla2dz/(z2+a2)i =-tmlsinepdep, 5.61 MOVING CHARGES, ANDMAGNETO STATICS 151 whereepistheangleAOZ.Hence ,J" Hz=-tm1fsinepdep=!m1(cosepl-COSep2)' (5.56) .p. Foraninfinitesolenoid, epl=0andep2=17,sothat Hz=m1 (5.57) andisuniform insidethesolenoid. Weshallendthissectionbycalculating theforcebetween twosmall planecircularcoils,eachofoneturnofradiusacarrying acurrentI, withacommon axis,andseparated byadistance z(a~z).From equation (5.55)thefieldontheaxisatthecentreofthesecondcoilis H,.=!la2jz3. Sincea~z,H,.willnotvaryappreciably forasmalldistance offthe axis,andthefluxthroughthesecondcoilistherefore N=fB.dS=f-tf-toC!Ia2jz3)(17a2)=!f-tf-to171a"jz3. Thepotential energyofthesecondcoilisUp=-N1,andhencethe forceonitisF_dTTjd_ 3"12j" (5.58)- -UpZ--2f-tf-t017a z . Itisinstructive toseejusthowthisforcearises.SincedivB=0in theregionawayfromthesecondcoil,wehave oB",+oBJL+oBz=0, oxoyOZ wherexandyarenormaltothecommon axisofthecoils.Bysym­ metry,oB",/ox=oBlI/oy,andhenceeachequals-!(oBz/oz). Atasmall distance afromtheaxistherewillbearadialcomponent offieldequal toa(oB",jox)=-!a(oBzjoz) =3f-tf-toa31j4z".Therewilltherefore bea forceI(ds 1\B)oneachelementofthecoil,ofwhichthecomponents due toBzareradialandsumtozerooverthewholecoil,whiletheforce components duetotheradialcomponent ofBallactinthenegative z-direction (assuming thecurrentsineachcoilflowinthesamesense). Thesesumto-1(217a)f-tf-to(3a31j4z"),whichgivesthesameresultasin equation (5.58)(seealsoProblem 5.4). 5.7.Movingchargesinelectricandmagnetic fields Thefundamental equations (5.1)and(5.37)fortheforceonandthe fieldproduced byacurrentelementbothinvolvethequantity Ids.This maybetransformed togivethecorresponding formulae foramoving charge,whichisequivalent toacurrent. Themagnitude ofthecurrent 152 THEMAGNETIC EFFECTS OFCURRENTS AND [5.7 I=dqldt,therateatwhichchargepassesagivenpoint.Ifthecharge moveswithvelocityv,wehaveIds=(dqldt)vdt =dqv,andtheBiot­ Savartlaw(equation (5.37))thusbecomes H=fI(dsi\r) =fdq(Vi\r). 47Tr341Tr3 Ifallthechargeislocatedatapoint,vandrareconstant intheinte­ grationoverdq,andforapointchargeqwehave H=q(vi\r),41Tr3 (5.59) whiletheforceonamovingchargebecomes F=q(vi\B). (5:60) IfanelectricfieldEisalsopresent,thetotalforceis F=q(E+vi\B). (5.61) Itmayberemarked thatthoughequation (5.60)hasherebeenintro­ ducedasanadditional postulate, itfollowsasaconsequence ofequation (1.3)whenweapplythespecialtheoryofrelativity. Anobserver in whosesystemachargeisatrestwillascribetheforcesonittoapurely electrostatic fieldE.Onapplying thelawsforthetransformation of mechanical forcewefindthatamovingobserver wouldmeasure aforce ofthetypegivenbyequation (5.61);thatis,hewouldascribetheeffects totheactionofbothelectricandmagnetic fields.Inasimilarmanner, equation (5.59)canbededuced fromtheelectrostatic formula forD, equation (1.23). Themotionofcharged particles, usuallyelectrons orpositive ions, undertheactionofelectricandmagnetic fieldsisthebasisofmany fundamental experiments inphysics, afewofwhichwillbeusedas illustrations here.Themotioninpurelyelectrostatic fieldshasalready beendiscussed (§3.7),andweshallbeginbyconsidering themotionof achargeinauniform magnetic induction B.Ifthechargeisinitially movinginaplanenormaltoB,thentheforceonit(assuming E=0) isalsointhisplaneandnormaltoitsdirection ofmotion. Thusnowork isdoneontheparticle, sinceF .v=qvi\(vi\B)=0,anditsvelocity remains constant inmagnitude. Thechargewilltherefore moveina circleinthisplane,theforcetowardsthecentrebeing Mv21r=qvB, whereMisthemassofthecharged particle, andrtheradiusofthe circle.Hencewehave r=MvlqB and We=vir=B(q/M), (5.62) 5.7] MOVING CHARGES, ANDMAGNETO STATICS 153 where Weistheangular velocity. Thisequation showsthatWe'and hencethetimetakentomakeonerevolution, isindependent ofthe velocity oftheparticle (solongastherelativistic changeofmass withvelocity canbeneglected). Thisfactismadeuseofinmany applications. G G F I p I ------------ GG FIG.5.13.Double solenoid encased iniron,formagnetic focusing. G,Garesmall annular gapsintheironcasing. Magnetic focusing IftheinitialvelocityofthechargeisnotnormaltoB,butmakesan angle0withthedirection ofB,thenwecanresolvethevelocity into acomponent vcos0paralleltoB,andacomponent vsin0normaltoB. Thevectorproduct (v1\B)hasnocomponent paralleltoB,andthe component ofvelocity vcos0willtherefore continue unaltered. The projection ofthemotiononaplanenormaltoBwillbeacircleofradius r=MvsinOjqB, andtheactualpathoftheparticle willbeahelix. Onerevolution ofthehelixiscompleted inatime27TjW=27TMjqB, anq. theparticlehasthenmovedadistance z=27TMvcosOjqBinthedirec­ tionofB.Forsmallvaluesof0,thisdistance isindependent of0inthe firstapproximation (sincecosO~l-l02),andthisistheprinciple used inmagnetic focusing. InFig.5.13electrons leaveapointF,anditisdesiredtofocusthem sothattheyallreachapointPadistance zaway.Iftheelectrons emergefromagunwithelectrostatic focusing, theyallhavecloselythe samevelocity v,butarenotmovingquiteparalleltothelineFP.By meansofasolenoid, amagnetic fieldisappliedinthedirection FP,and thecurrentinthesolenoid adjusted sothatthetimetakentoreachP isequaltooneormoreperiodsofrevolution inthehelicalmotioncaused 154 THEMAGNETIC EFFECTS OFCURRENTS AND [5.7 bythemagnetic field.Itisoftenimpracticable tousealongsolenoid, andoneormoreshortsolenoids, encasedinironwithasmallannular gaproundtheinnercircumference, asshowninFig.5.13,areusedin­ stead.Suchcoilsgivealocalized, non-uniform field,whichactslike athinlens;theirdesignislargelyempirical. -1-81 ---8 2 PIIP2 I/f....x y~I II•Iz c nl J FIG.5.14.Diagram illustrating theprinciple ofBainbridge's mass spectrometer. Measurement Ofspecificcharge Determination oftheratioofchargetomass(or'specificcharge')of atomicparticles isofprimeimportance inatomicphysics. Allsuch particles carryacharge(positive ornegative) equaltotheelectronic chargee,orasmallintegralmultiple ofit,andtheratioofthecharge tothemassfortheelectronandtheprotonarefundamental constants. Fromequation (5.62)itwillbeseenthatanaccurate measurement of WeandBforparticles movinginacirclesufficestodetermine qfM,and recentmethods basedonthisprinciple aredescribed inChapter 23. Forpositiveionsofheaviernuclei,themaininterestliesinthemeasure­ mentofthemass,andinstruments formeasuring thespecificcharge (qfM)forthispurpose areknownas'massspectrometers'. Ingeneral theymakeuseofbothelectricandmagnetic fields,andtwomeasure­ mentsarerequired sinceboththevelocityoftheparticleandthevalue ofqfMareunknown. Thepositiveionsareusuallyformedinagaseous discharge andmorethanonetypeofionwithvaryingvelocity maybe 5.7] MOVING CHARGES, ANDMAGNETOSTATICS 155 present; theinstruments aretherefore designed tosorttheseout,and bringallparticles withthesamespecificchargetoacommon focus. Manysuchinstruments havebeendesigned, butsinceourpurpose hereisjusttoillustrate theprinciples, weshalldescribe onlyone,due toBainbridge. Itmakesuseofa'velocity selector', formedbytheflat platesPI'P2inFig.5.14,whichhaveaverysmallseparation. Ionsenter theseplatesfromasourcethroughslits81'82,sothattheyaretravelling withvelocity vparalleltothez-axisofthecoordinate systemshownin thefigure.Avoltageismaintained between theplates,sothatthereis anelectricfieldEinthex-direction. BymeansofapairofHelmholtz coils(seeProblem 5.2)auniform induction Bismaintained inthey­ direction, andthetotalforceonanionbetweentheplatesistherefore q(E-vB) inthex-direction. Iftheplatesarelongandclosetogether, onlyionsforwhichthisforceiszerowillemerge,andtheirvelocitymust therefore bev=EjB.Thedevicetherefore selectsionsofaparticular velocity determined bythisratio.Onemerging fromtheplatestheions travelinasemicircular pathundertheinfluence ofthefieldBalone untiltheystrikeadetectoratO.Thedistance DOistwicetheradius oftheorbitandisthus 2r=2MvjqB =2(Mjq)EjB2. (5.63) Hencethedistance DOislinearlyproportional tothemassoftheion. InProblem 5.11itisshownthatthedistance DOisindependent (tothe firstorder)oftheangletothez-axisatwhichanionemerges fromthe plates,provided thisissmall,sothatwehave'firstorder'focusing of theionswithagivenvalueofqjM. REFERENCE PAGE,L.,andADAMS, N.I.,1945,Am.J.Phys.13,141. PROBLEMS 5.1.Showthatiftherelation B=f'o(H+M) isusedratherthanB=f'f'oH, thedifferential equations forthepotentials c/>andAbecome VIc/>=divM (inregionswhereJ=0), VIA=-f'oJ-f'ocurIM. Theseapplyinmediasuchasferromagnetics, whereMisnotlinearlyproportional toH,andmayevenbefinite(spontaneous magnetization) intheabsenceofan appliedfield. 9p,H II;=(2p,+1)(p,+2)-2(p,-1)2(ajb)3156 THEMAGNETIC EFFECTS OFCURRENTS AND 5.2.Twoidentical circular coils,eachofnturnsofradiusa,areplacedwiththeir planesparallelandnormaltothelinejoiningthem,adistance Tapart.Calculate themagnetic fieldontheaxismidway between thecoilsduetoacurrentIthrough eachcoil.Showthat,ifl'=a,thefieldmidway between thecoilsisuniform overaconSiderable region;thatis,oHjorforonecoilisequalto-oHjorforthe othercoil,and02Hjor2=O.Suchanarrangement ofcoilswasusedbyHelmholtz foragalvanometer, asmallmagnetic needlesuspended byatorsionfibreatthe centrebeingdeflected bythecurrentthrough thecoils. 5.3.Twoinfinitecylindrical conductors areplacedparallel tooneanotherata distance 2aapart.Theycarryequalandopposite currents. Showthatinthe equatorial planethegradient ofthemagnetic fieldisgreatest atadistance aj.J3 fromtheplanethrough theaxesofthecylinders. 5.4.Deduceequation (5.58)bytreating eachcoilasapointdipoleofmoment 1(1Ta2),andusingtheformula fortheforceonadipoleinanon-uniform field F=m(oBjoz) equivalent totheelectrostatic formula F=p(oEjoz). 5.5.Showthat,ifamagnetofmoment missuspended byatorsionless fibreso thatitisfreetoswinginahorizontal planeinahorizontal fieldB,theperiodof smalloscillations abouttheequilibrium position isT=2'1T(3jmB)t, where3is themoment ofinertiaofthemagnetabouttheaxisofrotation. 5.6.Twoshortmagnets areattached toacorksothattheyfloatonwaterwith theiraxeshorizontal. Onemagnet lieswithitscentreontheaxisoftheother, butwithitsownaxisperpendicular tothelinejoiningthem.Assuming that thedistance between themagnets islargecompared withtheirlengths, sothat theycanbetreatedaspointdipoles, calculate theforceandthecoupleoneach magnet, andsatisfyyourself thatthereisnoresultant forceorcoupleonthe systemasawhole. 5.7.Showthatthemagnetic fieldinsideaspherical airbubbleinaparamagnetic substance ofpermeability p,is3p,Hj(2p,+ 1),ifthefieldinthesubstance away fromthebubbleisH.Willanytranslational forceactonthebubble? 5.8.Aspherical shellhasradiiaandbrespectively (b>a),andismadeofa material ofpermeability p,.Itisplacedinauniform fieldH.Showthatthefield insidetheshellis andthatforlargevaluesofp"thisapproximates to Hi=9Hj{2p,(I-a3jb3)}. Thusifp,islarge,HiismuchsmallerthanH,andaninstrument canbeshielded fromstraymagnetic fieldsbyplacingitinanironcase.Magnetic shielding is muchlessefficientthanelectrostatic shielding (especially ifajbisclosetounity), fortheeffective valueofEintheequivalent expression foraconductor isinfinite. 5.9.Ahollwsphereofinternal radiusa,external radiusb,hasauniform spon­ taneous magnetization Mperunitvolume. Showthatthefieldintheinternal cavity (1'<a)iszero,andthattheexternal field(1'>b)isthesameasthatofa dipolemoment m=4'1TM(b3-a3)j3,thetotalmoment ofthehollowsphere. MOVING CHARGES, ANDMAGNETO STATICS 157 Showalsothatthesquareofthefieldoutsidethesphereatapoint(r,8),measured fromthecentreofthesphereandwithrespecttothedirection ofmagnetization, is H2=(3cos28+1){M(b;~a3>r. IftheangleofdipSisdefinedastheanglewhichthelinesofforceatapointon theexternal surfaceofthespheremakewiththetangentatthatpoint,showthat tanS=2cot8. 5.10.AparticleofmassMandchargeqisrotating inacircularorbitofradiusr withangular velocity w.Showthatamagnetic dipolemoment misassociated withthemotionofthecharge,suchthat m=(q/2M)G, whereG=Mr2wistheangularmomentum oftheparticle. (Theorbitmaybe regarded asasmallcircuitcarrying acurrentI=qXthefrequency atwhichthe chargepassesanypointintheorbitperunittime.) 5.11.InBainbridge's massspectrometer theionsemergeinawedge-shaped beam ofsmallsemi-vertical angle8(seeFig.5.14).Iftheresolution oftheinstrument asamassspectrometer isdefinedasthereciprocal ofthesmallest fractional change of:plasswhichwillproduce non-overlapping tracesontheplaneOD,showthat theresolution is2/82• 5.12.Acharged particlestartsfromrestattheoriginofcoordinates inaregion wherethereisauniform electricfieldEparalleltothex-axis,andauniform magnetic induction Bparalleltothez-axis.Showthatthecoordinates ofthe particleatatimetlaterwillbe x=(ElwB)(I-coswt), Y=(ElwB)(wt-sinwt), z=0, wherew=eB1m.(Thepathoftheparticle isacycloid.) Electrons areliberated withzerovelocity fromthenegative plateofaparallel platecondenser, towhichisappliedaninduction Bparalleltotheplates.Show thattheywillnotreachthepositiveplateiftheplateseparation disgreaterthan 2mEleB2, whereEisthefieldbetween theplates. \ 6 ELECTROMAGNETIC INDUCTION AND VARYING CURRENTS 6.1.Faraday's lawsofelectromagnetic induction THEexperiments ofOerstedandothersshowedthat'electricity can produce magnetism', andestablished thelawsgoverning themagnetic fieldsetupbyacurrent. Manyexperiments weredevisedtodetectthe inverseeffect,theflowofelectriccurrentduetoamagnetic field,without success,mainlybecauseasteadycurrentflowwaslookedfor.In1831 itwasfoundbyFaraday thatatransient flowofcurrentoccurred in aclosedcircuitwhenthefluxofmagnetic induction throughthecircuit waschanged. Thechangeoffluxcouldbebroughtaboutinanumber ofways:inhisfirstexperiment twocoilsofwirewerewoundonaringof softironasinFig.6.1.Thepresence ofacurrentinthesecondcoilwas detected byconnecting ittoanothercoilnearasmallsuspended magnet. Whenthefirstcoilwasconnected toabattery, amomentary oscillation ofthemagnetoccurred, afterwhichitsettledinitsoriginalposition. Asimilaroscillation, thoughwithaninitialkickintheopposite direction, wasobserved ondisconnecting thebattery.Inotherexperiments Fara­ dayshowedthatsimilareffectswereobserved ifapermanent magnetwas movednearthesecondcoil,orifthecoilwasmovedintheneighbour­ hoodofamagnet. Hisresultsweresummed upinthetwolaws: (a)whenthefluxofmagnetic induction through acircuitischang­ ing,anelectromotive forceisinducedinthecircuit; (b)themagnitude ofthee.m.f.isproportional totherateofchangeof theflux. Thesignofthee.m.f.isgivenbyLenz'slaw,whichstatesthatitis suchthatanycurrentflowisinthedirection whichwouldopposethe fluxchangecausingthee.m.f.Thus,inFig.6.2,ifthemagnetismoved towards theclosedloopofwiresothatthemagnetic fluxthrough the coilisincreased, theinducedcurrentwillflowinsuchadirection thatits ownfieldopposestheincreased fieldofthemagnetthreading theloop. Theselawsareexpressed intheequation dNV=-([t' (6.1) 6.1] VARYING CURRENTS 159 whereVistheelectromagnetic forceroundthecircuit,andNisthe instantaneous valueofthemagnetic fluxthrough thecircuit. Now N=JB.dSandV=JE.ds,wheretheformerintegral istakenover vrs FIG.6.1.Faraday's experiment onelectromagnetic induction. Vbattery, Sswitch, Msuspended magnetic needle, 1ironring. ~-- sNI=====------J~--I-____J~+----- \ (6.2)FIG.6.2.Currentinducedinaloopbyamovingmagnet(broken linesrepresent linesof magnetic fieldproduced bytheinduced currentwhenthemagnet movestowards the loop). anyareabounded bythecircuitandthelatterintegralistakenround thecircuit. HencewehavefE.ds= -;tfB.dS. 160 ELECTROMAGNETIC INDUCTION AND [6.1 UsingthetransformationfE.ds=fcurlE.dS, andthefactthatthe timeandspacecoordinates areindependent variables, thiscanbere­ writtenintheform fcurlE.dS = -fd:.dS, andsincethismustholdoveranysurfacearea,theintegrands mustbe equal,givingthedifferential form dBcurlE=-(]i' (6.3) FIG.6.3.Relation between Faraday's lawofinduction andtheforceonamoving conductor.x aDII-__..O b A,I----=-----.By zAtfirstsightitwouldhavebeenexpected thatequation (6.1)would havecontained amultiplying constant tobedetermined eitherexperi­ mentally orfromtheory.Thisconstant isin factunity,ascanbeseeninthefollowing way.Letusassumewehaveaverythin conductor carrying nocurrent, whichis movedwithavelocityvinauniform field ofmagnetic induction B.Sincethewireis aconductor, itcarriescharges (electrons) whicharefreetomovealongthewire;let thevelocityofachargeqintheconductor beurelativetotheconductor. Sincethe conductor isverythinumustbeparallel tothedirection ofthewireatanypoint. Thevelocityofthechargeqrelativetothe observer isv+u,andtheforceonitwill therefore beF=q(v+u);\B.Ifthecharge movesadistance dralongthewire,the workdoneisF.dr;since(U;\B).dr=0becauseu,drareparallel, thisisthesameasifthereexistedindrane.m.f. dV=(V;\B).dr. (6.4) Foraclosedcircuitinauniforminduction B,since(V;\B)isconstant, thetotale.m.f.V=f(V;\B).dr=O.Iftheinduction isnotuni­ form,thefluxthroughthecircuitwillchangeasthecircuitmoves,and wecanrelateVtotherateofchangeofflux.Consider asmallrect­ angularcircuitABOD(Fig.6.3)withsidesa,bparalleltothex,yaxes ofaCartesian system. Thee.m.f.inananti-clockwise direction (the senseinwhicharight-handed screwwouldturntoadvance alongthe 6.1] VARYING CURRENTS 161 (6.6)(6.5) whencez-axis)is V=a(VyBz-vzBy)+b(vz{Bx+i~x}-Vx{Bz+i~z})- -a(Vy{Bz+ba:.!}_Vz{By+ba:y})-b(vzBx-vxBz) =-ab(VaBz+vaBz_v{8BxaBy})xaxyayzax+ay _b(aBz+aBz+aBz) --av-v-v-xaxYayzaz sincedivB=O.ButabBz=N,thefluxthroughthecircuit,andsince Vx=dx/dt,etc., (aNaNaN)V= --vx+-vy+-v z=-dN/dtaxayaz inagreement withequation (6.1).Theunitofmagnetic fluxNisthe weber,andane.m.f.of1Visgenerated inacircuitwherethefluxis changing attherateof1weber/sec. Equation (6.3)maybecombined withtherelationB=curIA(equa­ tion(5.43»togive acurlE=--curIA=-curl(8A/at),at aA aAE=--+constant =---grad V.at at Thisisamoregeneralequation thanequation (1.6)whichappliesonly tosteadyfields.Foraparticleofchargeqtherateofchangeofmomen­ tumpis(assuming gradV=0) ap/at=qE=-q(aA/at), whichonintegration gives p=Po-qA, (6.7) assuming thatA=0whenp=Po.Thisrelationplaysanimportant roleinquantum mechanics, wheretheeffectofamagnetic fieldona chargedparticlecanbeintroduced byreplacing pbyPo-qA. 6.2.Self-inductance andmutualinductance IfacurrentIisflowinginacircuit,amagnetic fieldissetupand therewillbeafluxNofmagnetic induction throughthecircuitasso­ ciatedwithitsownmagnetic field.Themagnetic fieldatanypointis proportional tothecurrentI,andhencesoalsoistheinduction andthe 861110 M 162iii"!""'..;;;;·....., ELECTROMAGNETIC INDUCTION AND [6.2 fluxN.Wemaytherefore write N=LI, (6.8) whereLisaconstant whichdepends onthegeometry ofthecircuitand thepermeability ofthemediuminwhichitisimmersed. Liscalledthe self-inductance ofthecircuit,andisequaltothetotalfluxthroughthe circuitwhenunitcurrentisflowing. Acircuithasunitself-inductance (onehenry)ifitisthreaded byoneweberoffluxwhenoneampereof currentisflowing. Ifasecondcoilisbrought neartoacoilcarrying acurrentI,there willingeneralbeafluxN2ofmagnetic induction through thesecond coilduetothecurrentinthefirstcoil.SinceN2isagainlinearlypropor­ tionaltoII'wemaywrite N2=llf21Il> (6.9a) whereM21iscalledthemutualinductance between thetwocircuits. The unitofmutualinductance isagainthehenry.TherewillalsobeafluxN1 throughthefirstcircuitduetoacurrent12inthesecondcircuit,givenby N1=M1212• (6.9b) Thecoefficients M12andM21areequal,ascanbeseenfromenergycon­ siderations. Thepotential energyofthesystemcanbefoundfromthe fluxofeithercoilduetothefieldoftheother;fromequation (5.41) Up=-:~I2=-M21I112=-N1I1--~11212I1' showingthat M12=M21. (6.10) Byusingequations (5.44)and(5.47)wecanderiveaformula forthe mutualinductance, since Up= -IIfA21·ds1=-IIJr:J12~S2).dS1 /L/LoI L55ds1·ds2-MIL (6.11)= -4rr1 2 r- - 1212' where,bysymmetry, M-M-/L/LoJJd~.ds2 (6.12)12-21-4rrr' ThisresultisknownasNeumann's formula. Sincetheunitofmutual inductance isthehenry,thisformula showsthatthedimensiona of/Lo arehenryJmetre. Ifthetwocoilsarecloselywound,sothatallthefluxgenerated bythe firstcoilpassesthrough thesecond,andviceversa,thentheratioof thetwofluxes ~.andN2willjustbeequaltotheratioofthenumberof 602] VARYING CURRENTS 163 turnsnvnzonthetwocoils.ForacurrentIIinthefirstcoilwehave (writingMforM1Z=MZ1) N1/Nz=(L1I1)/(MI1)=L1/M=nl/nZ' whileforthefluxgenerated byacurrentIzinthesecondcoil Nz/N1=(LzIz)/(MI z)=Lz/M=nZ/n1• Hence L1/M=M/Lz=n1/nZandMZ=L1Lz.(6.13) Ifthefluxthroughthetwocoilsischanging, thevoltages inducedinthe twocoilswillbeintheratio li/~=(dN1/dt){(dNz{dt)=n1/nZ=l/n. Hencesuchadevicemaybeusedasatransformer, sinceifachanging voltageliisappliedtothe'primary' coil,achanging voltageofdifferent magnitude willbeinducedinthe'secondary' coil.Thevoltagetrans­ formation ratioisn=~/li,the'turnsratio'ofsecondary toprimary. Ingeneralnotallthefluxofonecircuitpassesthrough theother,and Mislessthan(L1Lz)i;itmaybewrittenas M=k(LlLz)l(0~k~1), (6.14) wherekiscalledthe'coefficient ofcoupling'. Thetheoryoftransformers isconsidered furtherin§9.5. Themagnitude ofaninductance maybecalculated fromfirstprin­ ciplesbycomputing thefieldproduced byagivencurrentinthecoil,and thenfindingthetotalfluxthroughthesameoranother coil,according towhethertheself-inductance ofthefirstcoilorthemutualinductance betweenthetwocoilsisrequired. Thecalculations areillustrated below foranumberofsimpleshapesofcoil. Long801enoid Foraninfinitely longsolenoid, woundwithmturnsperunitlength andcarrying acurrentI,themagnetic fieldinsideisuniformandgiven byequation (5.57): H=mI. Ifthecoreofthesolenoidhaspermeability fL,thefluxthrougheachturn isN'=fLfLoAmI, whereAisthecross-sectional areaofthesolenoid. Theself-inductance perunitlengthistherefore L'=mN'/I=fLfLomZA; forasolenoidoflength 1,largecompared withitsdiameter, thisformula isstillverynearlycorrect,andwemaywriteforthetotalself-inductance L=fLfLomzAl. (6.15) Ifasecondshortcoilofnturns,insulated fromthefirst,iswoundonthe 164 ELECTROMAGNETIC INDUCTION AND [6.2 solenoid asinFig.6.4,themutualinductance is M=f'f'omnA. (6.16) Twocoaxialcoils Another simplecaseisthatoftwoplanecoaxialcoilsAandBasin Fig.6.5,ofradiiaandb,andtotalnumbers ofturnsn1andnzrespec­ tively,whosecentresareadistancezapart,wherez~a,b.Thefield Secondary coil ~--------------l··-- Primary coil FIG.6.4.Solenoid withprimary andsecondary coils. ··-I---.T-- -8-- FIG.6.5.Mutualinductance between twoplanecoaxialcoils.AB=z. atthecentreofBduetoacurrentIinAis,fromequation (5.55), H=ilazn1/z3, andthetotalfluxthroughBis7TbZnZ(f'f'o H),sincethefieldthrough thecoilwillbeuniforminthefirstapproximation whentheinequality z~a,bholds.Hencethemutualinductance is M=f'f'o7Ta2bzn1nz. (6.17) 2z3 Pairofcoaxialcylinders Animportant methodofcarrying radio-frequency alternating cur­ rentsisbymeansofapairofcoaxialcylinders ofradiia,b(b>a),as 6.2] VARYING CURRENTS 165f", inFig.6.6.Atanypointthecurrentintheinnercylinder isI,while thatintheoutercylinderis-I;thatis,itisexactlyequalinmagnitude butflowingintheopposite direction. Themagnetic fieldatadistance r fromtheaxiswhenr<bisthesameasthatduetoastraightwire,so that H=I/277r. Application ofthesameequation showsthattherewillbenofieldoutside thelargercylinder, sinceanycircuitdrawnrounditisthreaded bytwo D FIG.6.6.Self-inductance ofcoaxialcylinders. equalandopposite currents. Tocompute theself-inductance ofalength I,wefindthefluxthroughacircuitsuchasABODinFig.6.6.Thisfluxis b b lJftftoIIJdrftfto(b)(ftftoH)dr =-- - =-lllog - •277r277 ea a a Thefluxandhencealsotheinductance areproportional tothelength. Hencetheinductance perunitlengthis ftfto10ge(~)henry/metre. (6.18)277a Foranothermethodofderiving thisformula, whichavoidstheuseof thehypothetical circuitABGD,seeProblem 6.1. 6.3.Transient currents incircuits containing inductance, resis­ tance,andcapacitance Ifacircuitcontaining abatteryVandaresistance Risconnected toacoilthrough whichthefluxNischanging, thetotalvoltageV' appliedtotheresistance RwillbethesumofthebatteryvoltageVand thee.m.f.developed inthecoil.HenceV'=V-dN/dt =RI,or V=RI+dN/dt. (6.19)<', 166 ELECTROMAGNETIC INDUCTION AND [6.3 Wecanapplythisequation toanumberofproblems, thefirstbeing acircuitcontaining aself-inductance Landaresistance R,asshownin Fig.6.7,whichisconnected attimezerotoabatteryofconstante.m.f. V. SincedNjdt=L(dljdt), wehave V=IR+L(dljdt), (6.20) R I FIG.6.7.Battery drivingcurrentthroughRandL. •t FIG.6.8.RiseofcurrentincircuitofFig.6.7. (6.21)V1=-(1-e-(R/Llt),Rwhichisthefundamental differential equation relatingthecurrentIto thevoltageV.Integration ofthisequation, withthecondition1=0 att-0,gives showing thatthecurrent approaches exponentially thevalueVjR whichitwouldhaveiftherewerenoinductance present(seeFig.6.8). Therateofapproach tothissteadyvaluedepends ontheratioofre­ sistance toinductance. IfR=0,thesteadystate,corresponding to infinitecurrent,isneverreached,butthecurrentriseslinearlyaccording 6.3] VARYING CURRENTS 167 totheequation I=(VjL)t,obtained bydirectintegration ofequation (6.20)withR=o.WhenRis finite, theinitialrateofriseofcurrent, givenbythetangentattheorigininFig.6.8,isd1/dt=V/L,butas thecurrentthroughtheresistance increases, thevoltageacrossthein­ ductance falls,withacorresponding decrease ind1jdt. Theconverse problem, inwhichabatteryhasbeenconnected tothe circuitforalongtimesothatasteadycurrent10isflowing,andthen attimezerothebatteryisreplacedbyashortcircuit,leadstothesame differential equation (6.20),butwithV=o.Itssolution is (6.22) showingthattheeffectoftheinductance istopreventthecurrentfrom fallinginstantaneously tozero.Ifthebatteryissuddenly open-circuited, R 8 1 I.. FIG.6.9.Battery charging capacitanceathrough resistance R. thesuddencessation ofthecurrentproduces alargeimpulse voltage -L(d1jdt) intheinductance, whichmaybesufficient tocauseaspark acrossthepointatwhichthecircuitisbroken. Withlargeinductances suchasarefoundinelectromagnets (see§8.5)veryhighvoltages may ariseinthiswaywhichcandamagetheinsulation ifthecircuitisbroken suddenly. Inboththecasesconsidered abovetheexponential isoftheform e-t/.,.,andtheexponential rateofchangeofthecurrentisthesame;the quantity T=LjRiscalledthetimeconstant ofthecircuit. Oircuitwithcapacitance andre8istance Ananalogous problem isthatofacapacitance 0inserieswithare­ sistanceR,towhichabatteryVisconnected attimezero(seeFig.6.9). Atanyinstantthechargeonthecapacitor isq,andthevoltageacross 168 ELECTROMAGNETIC INDUCTION AND (6.3 itisq/C.SincethecurrentI=dq/dt,wehave V=q/C+RI =q/C+R(dq/dt). (6.23) Thesolutionofthisequation gives q=CV(l-e- t/RO) (6.24) showingthatthechargeonthecapacitor buildsupinamannersimilar tothecurrentintheprevious problem. Thetimeconstant ofthecircuit isnowT=RC,andthecurrentatanyinstantis V VI=dq/dt=Re-tiRO=Re-t/'T. (6.25) 1 r -----..t FIG.6.10.CurrentincircuitofFig.6.9afterswitchisclosed. Ittherefore fallsexponentially fromitsinitialvalue(VIR)tozero,as showninFig.6.10.Thevoltageacrossthecapacitor increases from zerotoitssteadyvalueV,whennomorecurrentcanflowinthecircuit. Ifwehaveanisolated capacitor initiallyatavoltageYo,anda resistance Risthenconnected acrossitattimet=0,thedifferential equation forthechargeatasubsequent timeisgivenbyequation (6.23) withV=O.Thesolution is q=CYoe-t/RO=CVoe-t/'T showingthatthechargedecaysexponentially tozero.(6.26) Circuitcontaining L,C,R Aninductance, acapacitance, andaresistance areconnected inseries, asinFig.6.11,andthecircuitisclosedataninstantt=0whenthe chargeonthecapacitor isqo'Sincethetotalvoltageinthecircuitis 6.3] alwayszero,wehaveVARYING CURRENTS 169! al q Lat+IR+O=O. SinceI=aq/dt,qmaybeeliminated bydifferentiation, giving d2ldlI Ldt2+Rdt+0=O. (6.27) R s I +q -qG FIG.6.11.Discharge ofacapacitance Gthrough resistance Randinductance L. (6.28)Thisequation hasageneralsolutionoftheform 1=e-(R/2L)t{Ae nt+Be-nt}, where n2=(R/2L)2-(I/LO) andA,Bareconstants determined bytheinitialconditions. Since I=0attimet=0,wemusthaveB=-A,andthevalueofAcan befoundbyintegration of(6.28)andsettingq=qoatt=O. Thenatureofthesolution depends onwhether nisrealorimaginary. Threecasescanbedistinguished: (a)nreal,thatis,(R/2L)>1/(LOp. Thedischarge ofthecapacitor isaperiodic, asshowninFig.6.12. (b)n=0;thatis,(R/2L)=l/(LO)l. Inthiscasethesolution isof theform q=qo(l+tR/2L)e-(R/2L)t (6.29) andtheaperiodic discharge ismostrapid. (c)nimaginary; thatis,(R/2L)<1/(LO)l. Thedischarge isnow oscillatory, asinFig.6.13,andthecurrentmaybewrittenas 1=Ae-(RI2L)tsinwt, (6.30) wherew2=I/LO-R2/4L2. Iftheresistance issmall,sothat (R/2L) ~1/(LO)l (i.e.!R(O/L)l ~1), q,I 1 ----------. t FIG.6.12.Chargeoncapacitor andcurrent incircuitofFig.6.11(non-oscillatory discharge). I\ \ \ \ \ \ \ ;' / / / /Ae-(R/2L)! I FIG.6.13.Oscillatory discharge inacircuitcontaining acapacitance, aresistance, andaninductance. I=AIl-(B/IL)tsin 011. 1$.3] VARYING CURRENTS 171 theangularfrequency wwillbeclosetoIj(LO)i. Wemaycall 1 fo=21T.J(LO) thenaturalfrequency ofoscillation ofthecircuitintheabsenceof damping. Whenthedamping issmalltheamplitude oftheoscillations decaysslowly,andthemaxima, whichlieontheexponential curve 1=Ae-(R/aL)t, occurverynearlyatthepointswheresinwt=1.Thesepointsoccurat intervals oftimeP,whereP=21TjW=21T(LO)l (wherewehaveagain o I v FIG.6.14.Battery Vcharging capacitance 0through resistance Randinductance L. 10ge(lmjlm+1)=PR/2L=1TR(OjL)i =1TjQ (6.31) isknownasthe'logarithmic decrement' ofthecircuit.Thequantityassumed thatthedamping issmall),andtheamplitude ofsuccessive maxima (ofthesamesign)therefore decreases bytheconstant ratio exp(-PRj2L).Ifwedenotesuccessive maxima by1m'Im+1'the quantity isknownasthe'qualityfactor'ofthecircuit.Eitherofthesequantities servesasanimportant criterion fortheperformance ofanoscillatory circuit,butinelectricity itiscustomary tousethequalityfactorQrather thanthelogarithmic decrement. Further relations involving Qwillbe obtainedinChapter 9. Theconverse problem, whereabatteryofe.m.f.Visconnected at timezerotoacircuitcontaining L,0,Rinseries(Fig.6.14),isleftas anexerciseforthereader.Thedifferential equationisthesameas(6.27), andthesolutions aresimilartothoseabove,withanaperiodic oroscil­ latoryapproach totheequilibrium statewherenocurrentflowsandthe voltageacrossthecapacitance isequaltoV.Animportant pointis 172 ELECTROMAGNETIC INDue'rrON AND [6.3 (6.32)thattheapproach totheequilibrium stateintheL,0,Rcircuitismost rapidatthechange-over pointfromanaperiodic toanoscillatory condi­ tion;thatis,when(Rj2L)=(LO)-!orQ=(LjO)!jR=f. 6.4.Ma~netic ener~yandmechanical forcesininductive circuits InthecircuitofFig.6.14therelationbetween theappliedvoltageV andthecurrentisgivenbytheequation dI q V=Ldt+RI+O' whereqisthechargeonthecapacitance. Therateatwhichworkis doneisfoundbymultiplying byI,giving VI=LIdI+RI2+(dq)!1dt dt0 since1=dqjdt.Theworkdoneinatimeinterval t,inwhichthe currentchanges from11to12,andthechargeonthecapacitance from q1toQ2'willbe t t W=fVIdt=iL(I~-If)+fRI2dt+2~ (q~-qf)· o 0 NowWisthetotalworkdonebythebattery, andtheintegralofRI2 istheenergydissipated asheatintheresistance, whichisalwaysposi­ tive.Thelastterm, t(q~-qmO, represents thechangeinthestored energyofthecapacitance, andweinterpret thefirstterm tL(1~-I~) asthechangeintheenergystoredintheinductance. Wenotethat thischangeisreversible, sinceifthecurrentisfirstincreased to12and thenreturned toitsinitialvalue11,thechangeinthestoredenergyis zero.If11=0,andacurrent1isestablished intheinductance, an energytLI2willbeassociated withit. Anexpression fortheenergystoredinaseriesofinductances will nowbederivedinamoregeneralway.IfNkisthefluxthrough the kthcircuit,thevoltageinducedinitisJk=dNkjdt,andtherateofdoing workisIklie=Ik(dNkfdt). Weassumethatcurrents inallthecircuits wereinitially zeroandincreased proportionately withtime,sothatat anintermediate instantt'thevaluesareITc=r:i.lk,Nk=r:i.Nk'Then thetotalworkdoneis tII"V"dt'=Ik~Ir:i.(dotfdt')dt' o 0 1 =IkNkfotdot=ilkNk·o 6.4] VARYING CURRENTS 173 Onsumming overallthecircuits, wehave U=t2IkNk· k(6.33) Ifeachcoilhasbothselfandmutualinductance, Nk=Lklk+2MkjI;· (6.34)Uk Henceequation (6.33)canbewrittenintheform Uk U=t2Lk1i+t22MkjlkI;· (6.35) k jk SinceMkj=M;k'eachterminthesecondsummation occurstwice. Thisisclearlyseenifweconsider justtwocoils,forwhich(writingM forM12)~=L1I1+MI z,Nz=LzIz+MI 1, sothattheenergybecomes U=1l1(LIIl+M1z)+llz(Lz1z+M11) =t(Ll1~+Lz1~)+M111z. (6.36) Itisimportant torealizethedistinction between thisstoredenergyU andthepotential energyofonecircuitcarrying aninvariant currentII inthefieldofasecondcircuitcarryinganinvariant current1z,givenby equation (6.11).Thedifference liesinthefactthattheformula forUp assumesthatthecurrents areinvariant, andnoaccountistakenofany workdonebythebatteries inmaintaining thecurrents, whereas the storedenergyUincludes theworkdonebythebatteries insettingup thecurrentflow,starting fromzerocurrent. ThusUpisthepotential energyfunction fromwhichanymechanical forcecanbecalculated by theusualformulae suchas Fx=-dUp/dx (6.37) forthex-component oftheforce,underthecondition thatthecurrents arekeptconstant. Thedistinction between UandUpcanbeseenfromasimpleexample, thatoftworigidcircuitsasinFig.6.15,oneofwhichismovingwith velocity dx/dtwithrespecttotheother.Thentherateatwhichwork isdonebythebatteries is dW/dt=(dU/dt)+F x(dx/dt)+R 11~+Rz 1~, (6.38) wheredU/dtistherateatwhichthestoredenergychanges. Nowfrom equation (6.37)Fx(dx/dt) =-(dUp/dx)(dx/dt) =-dUp/dt andhence dW/dt=(dU/dt)-(dUp/dt)+RII~+Rz1~. .174 ELECTROMAGNETIC INDUCTION AND [6.4 Nowthefluxthrough eachcircuitis ~=LlIl+MI2, N2=L2I2+MI1, andtherateatwhichworkisdonebythebatteries is dW/dt=Il(RlIl+dNl/dt)+I2(R2I2+dN2/dt) =2(dM/dt)IlI2+RlI~+R2n ---- FIG.6.15.Twocircuitswithamutualinductance M.Circuit(2)ismovingrelative tocircuit(1)sothatMisvarying, butL1,L2and11.12areconstant. since11,12areconstant. Comparing thesetwoequations, weseethat dU/dt=dUp/dt+2(dM/dt)I lI2. Nowfromequation (6.36),sinceonlyMischanging, dU/dt=(dM/dt)IlI2 andhence dUp/dt=-(dM/dt)I lI2=-dU/dt. Thusthecomponent ofthemechanical forceis Fx=-dUp/dx =+dU/dx.(6.39) (6.40) Fromequation (6.39)weseethatthedifference insignbetween the twoexpressions fortheforcein(6.40)arisesfromthefactthatthebat­ teriesdoworkattherate2(dM/dt)I lI2(apartfromtheirreversible Joule heatingrepresented bythetermsRlI~,etc.)whichisjusttwicethe rateofincrease (dU/dt)ofthestoredenergy. Thissituation withregard tothemagnetic energy,andthecalculation ofthemechanical forces inasystemwherethecurrents arekeptconstant, issimilartothatin electrostatics whenchangestakeplaceinasystemwheretheconductors aremaintained atconstant potential.Itwasshownin§1.7thatina changewherethestoredenergyincreases bydU,thebatteries dowork 2dU,sothattheamountofexternal workdWdoneonthesystemis 6.4] VARYING CURRENTS 175 -dUoHencethecomponent oftheforceinthex-direction isgivenby Fx=-dUpjdx =+(dU/dx)v, wherethesubscript Vdenotesthatthisformulaistobeusedunderthe condition V=constant. Similarly, wemayemphasize thatinequa­ tionssuchas(6.40)thecurrentiskeptconstantbywritingtheminthe form Fx=+(dU/dx)z, etc. Asweshouldexpect,thetorqueonacoilcanbeshowntobe r=-(dUp/d8)=+(dU/d8)z =(dM/d8)1112,(6.41) where0istheanglewhichthecoilmakeswithsomefixedaxis. 6.5.Magnetic energyinmagnetic media Sinceforanycircuit N=fB.dS=fcurlA.dS=fA.ds (6.42) wecantransform equation (6.33)asfollows: U=1-~lkNk=tf:lkfA.ds. Forasystemofdistributed currents, wewriteIds=Jd'T,andtake theintegraloverallspace,sincecontributions ariseonlyfromregions whereJisfinite,giving (6.43) (6.44)wehaveU=tf(A.J)d'T. However, J=curlH,andusingthevectoridentity div(A/\H) =H.curIA-A.curIH, U=!f(H.curiA)d'T-IIdiv(A/\H) d'T =!f(H.B)d'T-!f(A/\H)dS. Ifthevolumeintegral isextended overallspace,thesurfaceintegral istakenoverthesphereatinfinity. Forafinitesystemofclosed circuits,thefieldsH,Batlargedistances willfalloffatleastasrapidly asthoseofdipoles;henceH""'oJr-3andA""'oJr-2,sothattheintegrand diminishes asr-oandtheintegralvanishes asr-700.Hencewehave U=tf(H.B)d'T. Thisformula showsthattheenergymayberegarded asdistributed throughout theregionoccupied bythefieldswithdensitytH.B, whichisclearlyanalogous totheresulttD.Eobtained inelectrostatics (1.36).Ineachcaseithasbeenderivedundertheassumption thatB(D) islinearlyproportional toH(E),i.e.thatthemediahavevaluesoff'(t-), 176 ELECTROMAGNETIC INDUCTION AND [6.5 whichareindependent offieldstrength, Wemayrelaxthisassumption byconsidering infinitesimal changesofflux,forwhichI)U=1lk'8Nk• k Thenbyapplying transformations exactlysimilartothoseabove,we find 8U=fJ.'8AdT=fH.'8BdT, (6.45) aresultwhichcanthenbeintegrated overanyfinitechangeinBifwe knowhowHandBarerelatedateverypointduringthechange.IfB islinearly proportional toH,thisresultclearlygivesagainequation (6.44). Weclosethisdiscussion byfindingtheworkdonewhenthemag­ netization changesby8MinafieldBowhichisduetoothersources, i.e.Bodoesnotincludethefieldduetothemagnetic material itself. Asimpleexample wouldbeapieceofmagnetizable material insidea solenoid; Boisthenjustequaltothefield11'0Howhichthesolenoid would produceintheabsenceofthemagnetizable substance, andcanbecal­ culated fromthestandard formulae. Westartbyconsidering two circuits, forwhich,fromequation (6.39),inaninfinitesimal change, 8U=8Up+28(Ml11z). ButUp=-mZ.B1=-lzfB1.dSz,andthefluxthroughthesecond circuitisM11=fB1.dSz.Hence 8U=+812fB1·dS2=8(B1·m2), wherem2istheequivalent magnetic moment ofthesecondcoil,and weassumethatB1isconstant overtheareaofthiscoil.IfB1isfixed, andm2increases by8m,then'8U=B1•'8m.Clearlyitdoesnotmatter whatthesourceofB1is,sothatwecanwriteingeneral '8U=Bo·8m=f(Bo·'8M)dT, (6.46) whereBoexcludes thefieldcontribution fromthemagnetized substance itself. PROBLEMS 6.1.Twoinfinitecoaxialcylinders carryequalandopposite currentsI.Calculate theself-inductance Lperunitlength(equation (6.18»byequating thestored energyperunitlengthtoILI2. 6.2.Provethattheinductance perunitlengthoftwoinfiniteparallel wiresof radiusaseparated byadistance 2d(2d>a)is L=(1-'I-'olTT)loge(2dja). VARYING CURRENTS 177 6.3.Provethattheinductance ofalongsolenoid oflength 1,radiusa,withmturns perunitlengthisapproximately L=mnBaB/L/Lo{(l2+a2)!-a} (assumethatthefieldisuniform overanycross-section: if1>lOa,thisformula isaccurate within2percent). 6.4.Showthatthemutualinductance between twocoplanar coaxialcoilsof radiiaandb(b~a),withturnsn1andn2respectively, isapproximately (invacuo) M=/LoTTaBn1 n2/2b. Usethisresulttoshowthatthefluxthroughthelargercoilduetoasmallmagnetm placedatitscentreandpointing alongitsaxisis N=/Lomn2/2b. 6.5.Ifthemagnetinthelastproblem iswithdrawn alongtheaxisatauniform velocityv,showthatthee.m.f.inducedinthecoilwhenthemagnetisatadistance zfromthecentreofthecoilis V=3/Lomn2zvb2/2(b2+z2)f. Iftheresistance ofthecoilisR,showthatthetotalchargewhichflowswhenthe magnetisremoved fromthecentretoinfinityis Q=/Lomn2/2bR. 6.6.Aplanecirculardiskofradiusarotatesataspeedof]revolutions persecond aboutanaxisthroughitscentrenormaltoitsplane.Auniform induction Bexists paralleltothisaxis.Showfromfirstprinciples thatthereisane.m.f.between the centreofthediskanditsrimofmagnitude V=]BTTa2•(Lorenz's methodof determining theunitofresistance depends onthisresult;see§7.4.) 6.7.Acircularcoilofnturnsofradiusa,totalresistance R,andnoself-inductance isrotatedwithuniform angularvelocity waboutaverticaldiameter inahorizontal induction B.Provethatthemeanpowerrequired tomaintain thecoilinmotionis W=tn~2a4B2w2/R, andthatthisisequaltothepowerdissipated intheresistance ofthecoil. Asmallmagnetic needle,whichisfreetoturnslowlyinahorizontal plane,is placedatthecentreofthecoil.ShowthatitwillsetatananglerPtoBwhere cotrP=4R/(7Tn2/Lowa). (Rayleigh's methodofdetermining theunitofresistance isbasedonanexperiment ofthistype.) 6.8.Atorsional pendulum consists oftwospheresofIemdiameter ateither endofathinrod10emlongsuspended atitsmid-point. Itswingsinahorizontal planesothatitsinstantaneous angulardeflexion isTTcostnt.Findthemagnitude anddirection ofthecurrentflowingintherodatanyinstant, assuming that thecapacity ofeachsphereisthesameasifeachwereisolatedinspace,andthe rodandsuspension havenegligible resistance. Showthattherewillbedamping oftheswingifthesuspension hasafinite resistance, butnototherwise. (Vertical component ofearth'smagnetic field=6x10-5weber/m2.) (Amwer: 3·2x10-19cos!TTtamp.) ~nw N 178 ELECTROMAGNETIC INDUCTION 6.9.Anaeroplane isinlevelflightatagroundspeedof300km/hr.Itsmetal propeller measures 3 mfromtiptotipandrotatesat3000r.p.m.Findanex­ pression forthep.d.between theendsofthepropeller whentheaeroplane isflying alongthemagnetic meridian. (Vertical component ofearth'smagnetic field=6X10-5weber/m2.) (Answer: 0'015cos 1007rtvolts.) 6.10.Showthatforauniformly magnetized spherical permanent magnet, SiB.Hd-r,integrated overthevolume insidethesphere, isjustequaland opposite tothevalueoftheintegral overthevolume outsidethesphere.Thus SiB.Hd-roverthewholeofspaceiszero.(Thisfollowsbecause thereareno currents-see thederivation ofU=StB.Hd-rin§6.5.) 6.11.Themagnetostatic energyofapermanent magnet isf~fLoH2d-rtaken overallspace.Asolidspherical permanent magnet ofradiusbisuniformly magnetized withamagnetization Mperunitvolume. Showthatthetotal energystoredinthefieldoutsidethesphereisP-oVM2/9,whereVisthevolume ofthesphere.Showalsothattheenergy(thusdefined) storedinthefieldinside thesphereisfLoVM2jl8. ThusthetotalenergyisfLoVM2j6,andthisisthe magnetostatic enegyofthesphere. (Hint:UsetheresultofProblem 5.9.Thequestion ofthemagnetostatic energyofapermanent magnet isdiscussed byV.Heine(1956),Proc.CamlJ. Phil.Soc.52,546.) 7 DIRECT CURRENT MEASUREMENTS 7.1.Galvanometers, ammeters, andvoltmeters; thewattmeter AGALVANOMETER isaninstrument fordetecting andmeasuring electriccurrent;ifitisprovided withascalealreadycalibrated, itis calledanammeter. Theonlytypeofgalvanometer nowincommon use isthemoving-coil type,wherethecurrentispassedthrough acoil, usuallyrectangular inshape,whichissuspended inauniform constant magnetic induction, asinFig.7.1.Thesuspension isadjusted sothat theplaneofthecoilisparalleltothelinesofmagnetic fieldwhenno currentispassing. Flowofcurrentthroughthecoilgivesitamagnetic moment, andtheactionofthemagnetic fieldonthismoment produces acouplewhichtendstoturnthecoil;thisisbalanced bytherestoring torqueduetothesuspension, andanequilibrium position isreached wherethetwoareequal.Ifthecoilhasnturns,eachofareaA,and thecurrentisI,themagnetic momentmwillbenAI.Whenthecoil isdeflected throughanangle8,thecoupleonitduetothesuspension isc8,andonequating thistothecoupleexertedbythemagnetic field, wehave 8c=m1\B=InABcos8. (7.1) InpracticethelinesofBareshapedsothattheyarealwaysparallelto theplaneofthecoil,andthetermcos8willthenbeunity,sothatthe equilibrium deflexion 8willbelinearlyproportional tothecurrent. Themagnetic induction, usually ~0·4weberjmetre2(4,000gauss), isproduced byasmallpermanent magnet, withpolefacesshapedas inFig.7.2.Thecentreofthegapisfilledwithiron,sothatthelines offorceinthegapareasnearlyradialaspossible. Thecoiliswound onaformerwhichfitsinthegapbetween thispieceofironandthepole faceswithout touching either.Inthelesssensitive butmore rugged typeofinstrument thecoilissuspended bytwospiralmetalsprings whichalsoserveasleadsforthecurrent,andthedeflexion ofthecoil isindicated byapointermovingoverascale.Inacommon typeof instrument full-scale deflexion (corresponding torotation ofthecoil through about60°ofarc)isobtained foracurrentof1rnA,though moresensitive instruments canbeobtained. Thishighersensitivity is 180 DIRECT CURRENT MEASUREMENTS [7.1 achieved bywindingmoreturnsonthecoil;thisincreases theresistance, whichrisestomorethanahundred ohms.Suchahighresistance isun­ desirable inmanyuses,andingalvanometers forcurrents oftheorder ofmicroamperes alightsuspension ofphosphor bronzewireisused / -----.~B / / / FIG.7.1.Coilinafluxofmagnetic induction B.Areaofcoil=A,numberofturnsn. :.~00·· /.~,/ /'./ FIG.7.2.Polepiecesofapermanent magnetinamoving-coil galvanometer. insteadofsprings.. Thisgivesasmallrestoring torqueandagreater sensitivity. Thistypeofsuspension isnotstrongenoughtosupporta pointer,andthedefiexion isobserved bymeansofalampandscale, asmallmirrorbeingattached tothesuspension justabovethecoil. Withthescaleat1metredistance, acommon typeofgalvanometer givesadefiexion ofabout100mmfp,A, withacoilresistance ofabout 10ohms.Thustheincreased sensitivity isobtained partlyfromuseof asuspension withasmallerrestoring torque,andpartlyfromthegreater readingaccuracy ofthelampandscalemethod(thedeflexion justquoted corresponds toarotation ofthegalvanometer coilthrough about3°). Theperiodofswingofsuchaninstrument isabout2sec.Highersensi­ tivitycanbeobtained byuseofaweakersuspension (givingalsoalonger period)andmoreturnsonthecoil(givingahigherresistance, andalso (Courtesy ojMessrs.H.Tinsley'" Co.) FIG.7.3.Amodern galvanometer suspension andmagnet. e-coil Z-zeroadjustment M-mirror L-lens $-suspension strip X-clamping mechanism P-magnet polepieces Y-magnet yoke T-suspension tuhe Whenreadyforusethecoilassembly islowered sothatthemirror isbehindthelensandthepolepiecesinlinewiththeyoke. 7.1] DIRECT CURRENT MEASUREMENTS 181 increasing theperiodthroughthegreatermoment ofinertiaofthecoil unlesscorrespondingly thinnerwireisused).Atypicalgalvanometer construction isshowninFig.7.3. Thepointertypeofinstrument described aboveiscommonly known asamilliammeter. Itmayreadilybeadapted tomakethefull-scale readingcorrespond toanygivengreatercurrentbyuseofashunt.If thecoilresistance isR,andthecurrentrequired forfull-scale reading 1r R(1o{R/r}) FIG.7.4.Shunting amilliammeter. is10,thenbyconnecting ashuntofresistance racrossthecoilasin Fig.7.4,full-scale readingwillbegivenbyacurrentI=Io{l+R/r). HenceifRisknown,thesensitivity maybereduced byanygiven amountbythecorrectchoiceofr.Thenetresistance oftheinstrument becomes rR/(r+R), andsobecomes verylowifthecoilisshunted to readhighcurrents. Thisisadesirable feature,sincetheidealammeter shouldhavethelowestpossibleresistance inordertominimize itseffect onthecircuitinwhichitisintroduced tomeasure thecurrent. Thee.m.f.required toproduce full-scale deflexion intheinstrument justconsidered isRIo,andhencethemetercanbeusedtomeasure voltageifRisknown. SinceRisabout50ohmsforaninstrument with afull-scale readingof1mA,thevoltagerequired isabout50mV.To reducethesensitivity, aresistance risaddedinserieswiththecoil,since thismakesthevoltagerequired forfull-scale readingIo(r+R). Byad­ justment ofr,anygivenfull-scale readingcanbeobtained, andthe instrument isthencalledavoltmeter .Useofaseriesresistance increases thenetresistance oftheinstrument; sinceavoltmeter isconnected across thecircuitbetweenthepointswherethevoltagedropistobemeasured, itisdesirable thatitsresistance shallbeashighaspossibleinorderto changethecurrentflowbytheminimum amount.Itisobviousthat, whatever thereduction insensitivity byaddingseriesresistance, the currentdrawnbythemeteratfull-scale deflexion is10;thusameter withthesmallest possiblevalueof10shouldbeusedtoconstruct avolt­ meter.Similarly, ifthemeterisshunted tomakeanammeter, the 182 DIRECT CURRENT MEASUREMENTS [7.1 voltageacrossitforfull-scale deflexion isconstant andequaltoRIo; thusameterwiththelowestvalueofRIoshouldbeusedtomakean ammeter. Themostaccurate measurements ofvoltagearemadewithapotentio­ meter,asdescribed in§3.6;sincenocurrentisdrawnfromthesource ofvoltagetobemeasured whenthepotentiometer isbalanced,itbehaves asavoltmeter withinfiniteresistance. Currents canbemeasured by determining thevoltagedropacrossaknownresistance, asinthecali­ brationofanammeter described in§3.6. Insomeapplications itismoreimportant toobtainthemaximum galvanometer readingthantohaveaninstrument withverylowresis­ tance,andweshallnowdiscussthefactorsinvolved inthecorrectchoice ofgalvanometer forthispurpose. Obviously thesensitivity ofamoving­ coilinstrument isincreased byusingahighervalueofB,andweshall assumethatBisalreadymadeashighaspossiblebythecorrectdesign ofthepermanent magnet. Thismeansthatthemagnetgapinwhich thecoilmovesisfixed,andthedimensions ofthecoilitselfarefixed. Thisdetermines alsothesuspension, sincethisdepends mostlyonthe massofthecoil,andtherestoring torqueistherefore alsofixed.Thus theonlyvariables leftarethenumberofturnsnonthecoil,andthe cross-section ofwireused.Iftheavailable cross-section ofthecoilisex, thecross-section ofthewiremustbeex/n;iftheperimeter ofthecoilist, thetotallengthofwireisnt.Hencethecoilresistance willbe R=ntp/(ex/n)=n2tp/ex, wherepisthespecificresistance ofthewire.IfthesourceofvoltageV hasinternal resistance r,asin.Fig.3.2,anddrivesacurrentIthrough thegalvanometer coilresistance R,thenI=V/(R+r), andfromequa­ tion(7.1)thecoildeflexion willbe ()_nABI_ABV( n) (72)--c----c-r+(n 2tp/ex). . Differentiation withrespecttonshowsthat()isamaximum when n2tp/ex=r,orR=r.Henceforoptimum sensitivity thenumberof turnsonthecoilshouldbeadjusted sothatthecoilresistance isjust equaltotheinternal resistance ofthesource(cf.§3.5).Inpracticeit isonlynecessary tochooseagalvanometer whosecoilresistance is approximately equaltothatoftheload,sinceafactor2intheratio oftheresistances onlycausesthedeflexion tofallby6percentbelow theoptimum. Theanalysis showsthatalowresistance instrument shouldbeusedtomeasure, forexample, thee.m.f.ofathermocouple, 7.1] DIRECT CURRENT MEASUREMENTS 183 whichhasalowinternalresistance. Ontheotherhand,ahighresistance instrument shouldbeusedtomeasureanionization current(suchasis causedbythepassageofX-raysthrough agas)sincethisbehaves as averyhighresistance source. Inthemoving-coil instruments considered sofarthemagnetic induc­ tionBisconstant andprovided byapermanent magnet. Inanother classofmoving-coil instruments, knownasdynamometers, theinduc­ tionBisprovided byasecondsetoffixedcoilscarrying acurrentII AIIrB I 121 RLoad V 10 FIG.7.5.Awattmeter. surrounding themovingcoilwhichcarriesacurrent12,SinceBispro­ portional toII'itisobviousthatthetorqueonthemovingcoilwillbe proportional to1112;moreexactly,ifMisthemutualinductance be­ tweenthefixedandmoving coils,byequation (6.41)thetorqueis I1I2(dM/dO), andhencetheequilibrium deflexion willbegivenbythe equation cO=I1I2(dMldO), wherecOistherestoring torqueduetothesuspension (usually aspiral spring). Thefactor(dMldO)lc isfoundbycalibration withknown currents. IfacurrentItobemeasured isallowedtoflowthroughthefixedand movingcoilsinseries,sothatI=II=12,theinstrument willactasa milliammeter withasquarelawdeflexion; withapointertypedynamo­ meterthefull-scale readingisabout10mAoAmoreimportant applica­ tionisitsuseasawattmeter; thecoilsarethenconnected asfollows. Thefixedcoilrisconnected inserieswiththeload,sothatIIisnearly equaltoI,thecurrentintheload;themovingcoilisconnected inseries withalargeresistance Racrosstheload,asinFig.7.5.Thenthecurrent 12=VIR,whereVisthevoltageacrosstheload.Thedeflexion ispro­ portional toII12=IVIR=WIR,whereWisthepowerbeingdissipated intheload.Thustheinstrument maybecalibrated tomeasure power 184 DIRECT CURRENT MEASUREMENTS [7.1 directly. Notethatwiththecoilsconnected asinFig.7.5,thecurrent throughthefixedcoilisreally1+12'sothatthepowermeasured isthe sumofthatintheloadandinR(seeProblem 7.6). 7.2.Galvanometer dampin~ Sofarwehaveconsidered onlytheequilibrium deflexion ofagalvano­ meterwhenasteadycurrentflows;thebehaviour duringthetimein­ tervalbetween switching onthecurrentandtheattainment ofthefinal deflexion mustalsobediscussed. Whenthecurrentstartstoflow throughthegalvanometer coil,acoupleisexertedonitwhichgivesit anangular acceleration. Thiscouplediminishes asthecoilnearsits equilibrium position, owingtothereversecoupleexertedbythesus­ pension,butifthedamping issmall,thecoilmayovershoot andoscillate aboutitsfinalposition.Ifthedamping islargenoovershoot mayoccur, andthecoilapproaches thefinalposition veryslowly.Ineithercase sometimemustelapsebeforetheequilibrium deflexion canbeobserved, andtheeffectofthedamping isthusofconsiderable importance. Thoughairresistance andothersourcesofenergylosscontribute tothe damping, themostimportant sourceiselectromagnetic damping, due tothemotionofthecoilinthemagnetic fieldofthepermanent magnet. Withthenotation ofFig.7.1,thefluxthroughthecoilN=nABsin0, andtherewilltherefore beane.m.f.inducedinthecoilequalto -dNjdt =-nABcosO(dOjdt). If0issmall,orifthelinesofmagnetic induction arealwaysradial(as inFig.7.2),wecanputcos0=1.IfVistheexternal sourceofe.m.f. andRisthetotalresistance inthecircuit(including thegalvanometer coilresistance), thenthecurrentIwillbegivenbytheequation V_dN=R1dt atanyinstant. Theequation ofmotionofthecoilwilltherefore be ~d20+bdO+cO=nAB1=nAB(V_d~=nABV_(nAB)2 dO dt2dt RdtJR R dt' (7.3) where ~isthemoment ofinertiaofthecoil,b(dOjdt)thetorquedueto airdamping, etc.,andcOtherestoring torqueduetothesuspension. Rearranging theequation gives ~d20+{b+(nAB)2} dO+cO=nABV (7.4) dt2 RdtR' 7.2] DIRECT CURRENT MEASUREMENTS 185 fromwhichitisseenthattheeffectofthee.m.f.inducedbythemotion ofthecoilistoincreasethedamping torquebyanamount (nAB)2(d(Jjdt)jR. Thedifferential equation issimilartothatfortheoscillatory circuitof §6.3,andthesameanalysismaybeapplied. Therearethreetypesof solution, according towhetherthedamping islargeorsmall: (a)Smalldamping, b'<2(~c)l,whereb'=b+(nAB)2j R.Themotion isadamped harmonic motion,ofperiodnearlyequalto27T(~jc)l. Thecoilovershoots itsequilibrium deflexion (J=(nAB)V/cR, and thenoscillates withdiminishing amplitude aboutthisposition. (b)Oriticaldamping, b'=2(~c)l.Thecoilapproaches itsequilibrium without overshoot. (c)Highdamping, b'>2(~c)l.Thecoilapproaches itsfinalposition without overshoot, butmoreslowlythanincase(b). Ingeneraltheelectromagnetic damping ismuchgreaterthanother sourcesofdamping, unlessthetotalresistance Rofthegalvanometer circuitisverylarge.AsRdecreases theelectromagnetic damping in­ creases,andcriticaldamping willbeobtained withacertainvalueof resistance Regivenbytherelation (7.5) assuming thatbisnegligible undertheseconditions. Itisimportant to operateagalvanometer atornearcriticaldamping sinceitthentakes upitsequilibrium deflexion (or,moreprecisely, comestoadeflexion withinacertainfractionofitsequilibrium value)intheshortestpossible time;Ifthegalvanometer iseitherhighlyover-damped orverymuch under-damped itwilltakemuchlongertosettledown,andthisreduces therateatwhichreadings canbetaken.Ifthesourceofvoltagevaries withinthetimerequired forthegalvanometer tosettle,thereadingwill alwayslagbehindthetruevalue;indetecting thebalancepointofa bridge,quitemisleading indications canbeobtained ifthebridgearms arealteredtoorapidly. Sincethecriticaldamping resistance isofsuchimportance, itisalways specified foragalvanometer, together withthecoilresistance, thesensi­ tivity,andtheperiod.Asaroughrule,thecriticaldamping resistance is10to20timesthecoilresistance inamodernreflecting galvanometer. Inapointerinstrument thecoilmaycarryashort-circuited turn,orthe coilformermaybemadeofmetal,inordertogiveadequate damping. 186 DIRECT CURRENT MEASUREMENTS [7.3 7.3.Theballistic galvanometer andfluxmeter Theballisticgalvanometer isasuspended moving-coil instrument with verylightdamping, whichcanbeusedforthemeasurement ofcharge byobserving themaximum deflexion initsoscillatory motion. Forthis purposethechargemustpassthroughthegalvanometer inatimeshort compared withitsperiodofswing,sothatthecoildoesnotdeflect appreciably duringthepassageofthecharge.Ifthecurrentatany instantduringthispassageisI,thecoupleexertedonthecoilisnABI, andtheimpulseofangularmomentum giventothecoilwillbe fnABIdt=nABQ, whereQisthetotalchargeflowingthroughthecoil.Theeffectofthis impulseistogivethecoilaninitialangularvelocity (d()jdt)t=o =nABQj~, where ~isthemomentofinertiaofthecoil.Weanalysethesubsequent motionofthecoilassuming ittoremainconnected toacircuitoftotal resistance R(including thecoilresistance) butwithnoexternal e.m.f. Thedifferential equation ofthemotionwillbethesameas(7.4)butwith V=0,sothatwehave d2() Id() ~dt2+bdt+c()=0, (7.6) whereb'=b+(nAB)2jR isthetotaleffective damping, whichweshall assumetobesmall.Thenthesolution is ()=De-b't/2:Jsinwt, wherew=(cj~)ltoagoodapproximation. Theconstant Disfound bydifferentiation, andsettingtheinitialangular velocity equalto nABQ/~, giving nABQj~ =(d()jdt)t=o =Dw, whence ()=(nABQj~w)e-b't/23sinwt. (7.7) Iftherewerenodamping thedeflexion wouldoscillate between maxi­ mumandminimum values±()ooccurringattheinstantswhensinwt =1, andthechargeQwouldbegivenby Q=~w()oj(nAB) =c()oj(wnAB) =()O(2:)(n~B)' (7.8) wherewehaveusedinsuccession therelations (~jC)w2 =1,andw=27TfT, where Tistheperiodofswingofthegalvanometer. Thiscanbemeasured experimentally, andtheconstant (cjnAB) canbefoundfromthede­ flexionproduced bypassingasteadycurrentthroughthegalvanometer. 7.3] DIRECT CURRENT MEASUREMENTS 187 Theeffectofthedamping istomakesuccessive defl.exions smalleras inFig.7.6,andalsotomakethefirstthrowB1(att=IT)smallerthan Bo,sincefromequation (7.7) B1=Boe-b'T/8:J. () ............()e-(b'/2~)' ........:: 0-t--+----1~-+-~:---\--~__\r___I_-_\___-__I_------ T 2T FIG.7.6.Thedeflexion ofaballistic galvanometer plottedasafunction ofthetime 8=80e-(b'/2:Jll sinwt. Thesizeofthecorrection canbefoundbymeasuring thelogarithmic decrement .\ofsuccessive swingsonthesameside,since .\=10g(B1/B2)=10g(B2/Ba),etc. =b'T/2<;J. Hence, Bo=BIelA (7.9) or,since.\issmall, Bo=B1(1+1.\). Foraccurate usethedamping oftheballisticgalvanometer shouldbe smallandtheresistance oftheexternal circuitmusttherefore behigh. !fusedto'measure thechargeonacapacitor byconnecting thecapacitor tothegalvanometer andallowing ittodischarge throughthegalvano­ meter,thiscondition isfulfilledbecausethecapacitor iseffectively an 188 TCURRENT MEASUREMENTS [7.3 opencircuitforthelowlyvarying induced e.m.f.produced bythe galvanome~er swing.Aconvenient methodofcalibrating theballistic galvanomet~r directlisbydischarging throughitaknowncapacitance chargedtoaknown oltage.Analternative methodisbymeansofa standard mutualindutance,usingthecircuitofFig.7.7.Thesecondary I M pl- FIG.7.7.Aparatusforcalibrating aballistic galvanometer. GGalvanom ter. MStandard utualinductance. PPotentio terconnected acrossstandard 1ohmresistance. ceisconnected tothegalvanometer, andaknown theprimary coil.IfListheinductance ofthe dRthetotalresistance inthesecondary circuit, Iflowingatanyinstantisgivenbytheequation LdI+RI= _dN, dt dt ateatwhichthefluxischanging through the atingovertheduration oftimeoccupied bythe thatthetimeconstant RjLisveryshortcom­ ofthegalvanometer) gives 2=Lf~;dt+RfIdt=L[IJ~+RQ =RQ, (7.10)wheredNjdtisthe secondary coil.Inte fluxchange(assumi paredwiththeperio f_dNdt=N1­ dtofthemutualinducta currentisreversed i secondary windings a thesecondary curren 7.3] DIRECT CURRENT MEASUREMENTS 189 (7.11)showingthatthechargemeasured bythegalvanometer isjustequalto thetotalfluxchangeinthesecondary dividedbythetotalresistance. If10isthecurrentreversed intheprimary, N1-N2=2Mlo'andso Q=2M101R.Thecurrent10mayeitherbemeasured byasubstandard ammeter, orbymeansofthepotential dropitproduces acrossaknown resistance. Toreducethedamping oftheballistic galvanometer, either Rmustbelarge,whichingeneralwillmeanaddingconsiderable resist­ ancetothecircuitandthusreducing thesensitivity ofthegalvanometer, orthesecondary circuitmustbebrokenimmediately aftertheprimary currentisreversedandbeforethegalvanometer hasdeflected appreciably. Oncetheballistic galvanometer hasbeencalibrated, itmaybeused tomeasure afluxchangeoranunknown mutualinductance ifthe resistance Risknown.Analternative method istouseafluxmeter, whichgivesadirectreadingofthefluxchange. Thisinstrument con­ sistsofamovingcoil,suspended insuchawayastogivealmostzero restoring torque,inastrongmagnetic induction ofapermanent magnet. Theelectromagnetic damping isverystrong,andifotherformsofdamp­ ingcanbeneglected, theequation ofmotionis ~d20+(nAB)2dO=nABl=nAB(_dN-Ld£\ dt2Rdt Rdtde)' where1istheinstantaneous currentproduced bythee.m.f.-dNldt inducedbythefluxchangetobemeasured, andListheselfinductance. Theeffectofthiscurrentistoimpartanangular momentum tothe fluxmeter coil,whichisthenbrought torestundertheactionofthe electromagnetic damping. Integration ofequation (7.11)overtheentire timeoccupied bythemotionofthecoilgives ~[dO]O (nAB)2[O]OO =nAB[N,-R]-nABL[I]0dt+R 0R12 R 0o and,sincetheangularvelocity andcurrentarezerobothinitiallyand finally,nABOO=N1-N2, (7.12) where00istheultimate deflexion ofthefluxmeter. Thefluxchange through thefluxmeter coilisjustnABOo,andsothisequation shows thatthetotalfluxthreading thesystemisthesameatthebeginning andtheend.Thefluxmeter isgenerally calibrated directlyintermsof flux;sincethereisnorestoring torqueonthecoilithasnostableposi­ tionofequilibrium butcanrestanywhere inneutralequilibrium. Itis therefore necessary toleveltheinstrument carefully topreventthecoil driftingduringthemeasurement. Asaroughrule,afluxmeter isaccurate 190 DIRE TCURRENT MEASUREMENTS [7.3 toabout1percent,hileaballisticgalvanometer, properly calibrated, isaccurate toabout·1percent. 7.4.Absolute mearements Relative measure entsofelectrical quantities canbemadeusing bridgesorpotentio terstoanaccuracy ofsomepartsin105•For absolute measureme sitisnecessary tohaveultimate standards (for example, ofresistan andcurrent) withwhichanunknown quantity canbecompared. T eresistance ofacoilwasdetermined byRayleigh intermsoftheconstnt1-'0andthestandard oflength, byrotating the coilinamagnetic fild(seeProblem 6.7).Hereweshalldescribe in somedetailthemethdusedbyLorenz,whichhasthetwoadvantages ofbeinganullmethd,andofnotneedingtheresistance tobeinthe formofacoil.Ifask-shaped conductor isrotatedinamagnetic field Bwithafrequency frevolutions persecond,andBisparalleltothe axisofrotation, a vltageisinducedinthediskbetween theaxisand therimofmagnitud V=kfB,wherekisaconstant (withthedimen­ sionsofanarea:seeProblem 6.6)determined bythegeometry ofthe apparatus. InLoren's experiment (Fig.7.8)thefieldBisproduced by coilscarrying acurrntIwhichalsoflowsthroughtheresistance Rto bestandardized. TenB=I-'rl/k',wherek'isaquantity withthe dimensions ofalenh,againfixedbythegeometry oftheapparatus. ThevoltageVisbancedagainstthepotential dropRIacrossthe resistance, sothat =(k/k')l-'ofI, andatthebalancepoint R=(k/k')l-'of. (7.13) Thefrequencyfcanemeasured accurately, 1-'0isdefinedinthem.k.s. systemas4'7T10-7he/metre,and(k/k')hasthedimensions ofalength. InfactI-'o(k/k')isthmutualinductance between thefieldcoilandthe rotating conductor, hichcanbecalculated fromthedimensions and thegeometry. Fundmentally thecomparison isbetween theresistance andtheproduct inctanceXfrequency, therotating conductor being adevicebywhichasteadyvoltageisdeveloped acrosstheinductance forbalanceagainst hevoltagedropintheresistance. Thearrangement ftheapparatus isshowninFig.7.8.Therotating conductor isformedbythedisksP,P'mounted onashaftdrivenby amotor.ThegalvaometerGreadszerowhenthevoltageinducedin thecircuitabedisexctlyequaltothepotential acrosstheresistance R, which is inseriesththetwocoils00'andDD'.Thecurrentflows through thesecoilsntheopposite sensesothatthevoltages induced inthetwodisksadroundthecircuitabed.Contact withthedisksat ------------- '1.4] DIRECT CURRENT MEASUREMENTS 191 aanddismadebybrushes,andthecoilsaredesigned sothatthemag­ neticinduction atthesecontacts iszero;thisminimizes anyerrorin determining theactualradiusofthedisk;thatis,thedistance from thecentreatwhichcontactismade.Sometroublearisesfromthermal e.m.f.s.atthebrushcontacts, whichwouldgiveafinitegalvanometer 1 o p p' 0' 1D'--.------ FIG.7.8.Lorenz's apparatus fordetermining aresistance Rintermsof11-0andthemetre. 00',DD'arecoilsthrough which1flows(inopposite senses)producing themagnetio fieldsinwhiohtheplatesPP'arerotated. readingatthetruebalancepoint.Thisdifficulty isovercome byrevers­ ingthecurrentthroughout thewholesystem,whichreversesanytrue unbalance currentbutnotthatduetothethermal e.m.f.Hencethe correctbalancepointiswhenthegalvanometer readingremains un­ changed onreversing thecurrent,andthisprocedure alsoeliminates theeffectofanyvoltages inducedbystraymagnetic fields.Withade­ quateprecautions itispossibletoobtainanaccuracy ofafewpartsin 105;otherresistances canthenbedetermined relativetothestandard bymeansofaWheatstone's bridge. Fromthefundamental equations of§5.1,itcanbeseenthatacurrent maybedetermined inabsolute unitsbypassingitthroughtwoconduc­ torsandmeasuring theforcebetween them.Themethodofdoing 192 DIRE TCURRENT MEASUREMENTS [7.4 this,bymeansofa'centbalance', isillustrated inFig.7.9.A,B,0, D,E,Faresixsingle-layer coilswoundonmarbleformersandcon­ nectedinseries.CoilsA,0,D,andFarefixed,whileBandEare carriedonabalancearm;thecurrentflowsthroughthevariouscoilsin suchadirection thattheforceonEisupwards whilethatonBisdown­ wards.Thebalancearmisbroughtbacktoitsequilibrium positionby movingastandard massalongacalibrated scaleonthearm.Fromthe distance ofthemassfromthefulcrum, thetorqueandhencetheforce .....-..A 11-_...0 FIG.7.9.Current balance. Rstandard resistance. Ggalvanometer. Sstandard cell. between thecoilscanbeevaluated.IfMisthemutualinductance be­ tweenBorEandeitherofthecoilsA,°orD,F,thenthetotaltorque duetothecoilswhencarrying acurrentIis412(dMjd8), where8isthe angledefiningtherotation ofthebalancearm.SincedMjd8mustbe calculated fronithegeometry, theremustbenoironneartheapparatus. Effectsduetothemoreremotecoils(thatis,forexample, theforce between coilAandcoilE)areeliminated byrepeating thereadings with thecurrentthrough allthecoilsononesidereversed. Anaccuracy of onepartin105canbeachieved, butthemeasurements areverylaborious. Onesuchdetermination iscarriedoutatastandardizing laboratory, andatthesametimethevoltageacrossastandard resistance, placed 7.4] DIRECT CURRENT MEASUREMENTS 193 inserieswiththecoilssothatthestandardized currentflowsthroughit, iscompared withthee.m.f.ofastandard Westoncell.Bypotentiometer methods, otherWestoncellscanbecalibrated againstthestandard cell foruseassubstandards; subsequently, acurrentismeasured bycom­ paringthepotential dropitproduces inastandard resistance withthe e.m.f.ofthesubstandard cells.Ammeters canbecalibrated inthisway, asoutlined in§3.6. FIG.7.10.Determination ofacapacitance 0intermsofastandard resistance R. Thevalueofaresistance isfoundinabsolute unitsbycomparing it withaninductance whosevalueisfLotimesafactorwiththedimensions ofalength.Thecapacitance 0ofacapacitor is£0l,wherelisafactor withthedimensions ofalengthwhichisdetermined bythegeometry ofthecapacitor. Thusifliscalculated, andthecapacitance 0isde­ termined bycomparison withastandard resistance R,thevalueofthe constant £0canbefound.Anaccurate experiment ofthistypewas carriedoutbyRosaandDorseyin1907;theapparatus consistsessen­ tiallyofthebridgecircuitshowninFig.7.10.ThecontactXvibrates between PandQataknownfrequency, andthecapacitor 0isalter­ natelychargedanddischargedftimesasecond. ThevoltageVacross itisq/O,andthecharging ofthecondenserftimesasecondsendsa currentfqthroughthearmAB,sothattheeffective resistance ofthis armisV/(/q)=1/(/0).Henceatthebalance point,R2/R1=fOR. RosaandDorseyusedbothspherical andcylindrical capacitors, and tookmanyreadings tocorrectforalargevarietyofpossible errors. 851110 0 194 DIRECT CURRENT MEASUREMENTS [7.4 Theirfinalresult;expressed inourunits,was EO=(8'8547)10-12 Fjmetre withanaccuracy ofabout4partsin106• PROBLEMS 7.1.Abridgeconsistsofaself-inductance Landresistance ofabout10ohmsin oneaI'IIlandthreenon-inductive resistances intheremaining arms,sothatan accurate steadybalance isobtained. If,withthegalvanometer incircuit,the batterykeyisdepressed, aballistic deflexion of10emofthegalvanometer is recorded. Ifinasecondexperiment aresistance of0·02ohmisconnected inseries withL,asteadydeflexion of12emisobtained withthekeydepressed. The galvanometer isamoving-coil instrument withaperiodof9sec.Aswitchin thegalvanometer circuitisopenedimmediately aftertheflowofchargethrough itwhenitisusedballistically. ShowthatLisapproximately 24mHo 7.2.Aballisticgalvanometer iscalibrated byputtingitinserieswitha2-Vbattery andaresistance of106ohm.Asteadydeflexion of17emisobserved. Thetime ofswingis3'8sec.Acapacitor ischargedbya4-VbatteryamIwhendischarged through thegalvanometer givesathrowof24·2em.Finditscapacitance. (Answer:0=0·43(kF.) 7.3.Asmallsearchcoilwith8turnsofmeanarea1·5cm2isplacedbetween the polesofanelectromagnet withitsplanenOI'IIlaltothemagnetic field.Itis connected toaballistic galvanometer andthetotalresistance ofthecircuit is1000ohms.Whenthecoilissuddenly removed toaplacewherethefieldis negligible, thethrowofthegalvanometer is23divisions. Whenacapacitance of1(kFcharged to1 Visdischarged through thegalvanometer, thethrowis 25divisions. Calculate Bbetween thepolesofthemagnet. (Answer: B=0·77weberfmetre2.) 7.4.Aballistic galvanometer givesathrowof10emwhenachargeof3·5X10-7 coulombs ispassed.Itsperiodis2·2sec.Calculate thedeflexion whenasteady currentof3(kAispassed. (Answer: 30em.) 7.5.Acapacitance isconnected acrossamoving-coil ballistic galvanometer of negligible resistance; showthattheperiodofthegalvanometer isincreased. Ifthe capacitance is10(kF,thecurrentsensitivity ofthegalvanometer per(kAis10emat ametre,themoment ofinertiaofthesuspended system10gem2,andtheperiod onopencircuit10sec,showthatthefractional changeinperiodisabout8X10-3• 7.6.InthecircwtofFig.7.5,themoving coil(inserieswithE)maybeconnected eitherasshownorbetween thepointsAandO.Showthatthefractional error inthereadingofthepowerintheload(resistance Z)islesswiththelattermethod ofconnexion ifZ>(rE)l,andviceversa,whereristheresistance ofthefixedcoil. 8 MAGNETIC MATERIALS ANDMAGNETIC MEASUREMENTS 8.1.Origins ofmagnetism THEfactthatasubstance placedinamagnetic fieldacquires amagnetic moment wasintroduced inthetheoryofmagnetostatics in§5.3.The ratioofthemagnetic moment perunitvolumetothemagnetic fieldHis knownasthesusceptibility X'andsubstances areclassedasdiamagnetic, paramagnetic, orferromagnetic according tothenatureoftheirsuscepti­ bility.Inthefirsttwooftheseclassestheinduced magnetization is proportional totheappliedfieldunderordinary conditions, sothatthe susceptibility isindependent ofthefieldstrength. Indiamagnetic substances themagnetization isintheopposite direction totheapplied field,sothatXisnegative, whileinparamagnetic substances itisinthe samedirection, givingapositive valueofX.Ferromagnetic substances aredistinguished byverylarge(positive) valuesofX,whicharenotinde­ pendent ofthefieldstrength; inaddition theymaypossessamagnetic moment evenintheabsenceofanappliedfield,asinapermanent magnet. Inthischapterweshallgivefirstabriefdescription oftheoriginsofthe magnetization, anddesoribe methods ofmeasuring magnetic properties. Afulleracoountofthetheoryofparamagnetism andferromagnetism is giveninChapters 20"and21. Mterthediscovery thatasmallcoilcarrying acurrentbehaves like amagnet, Ampere suggested thattheoriginofallmagnetism layin smalloiroulating ourrents associated witheachatom.These'amperean currents' wouldeachpossessamagnetic dipolemoment, andthetotal magnetic moment ofanysubstance wouldbejustthevectorsumof themagnetic dipolemoments ofitsconstituent atoms.Thisgavea naturalexplanation ofthefactthatnoisolated magnetic polehadever beenobserved, sinceevenontheatomicscaleonlydipolesexisted,and thesewereduetoelectriccurrents anddidnotconsistoftwoactual magnetic polesofopposite signseparated byasmalldistance. Ampere's theoryisessentially thesameasthatofmodernatomicphysics,theorigin ofhiselementary currentcircuitsbeingthemotionofthenegatively­ charged electrons inclosedorbitsroundthepositively-charged atomic nucleus. 196 MAGNETIC MATERIALS AND [8.1 (8.2) ThequantityWhenaparticle ismovinginaclosedorbitinasystemwhereno external forceisacting,itsangular momentum isconstant.Ifthe particle ischarged, amagnetic moment willbeassociated withits motion,andthereisalinearrelation between theangularmomentum andthemagnetic moment (seeProblem 5.10forthesimplecaseofa circularorbit).Ageneralexpression forthemagnetic dipolemoment associated withadistributed currenthasalreadybeenfoundin§5.5. Inequation (5.52)wecanreplaceJdTbyvdqtofindtheequivalent expression for·amovingcharge,giving m=f!(rAv)dq. (8.1) Intheabsenceofanexternal force,thequantity (r1\v)isaconstant, sincem(rAv)isequaltoG,theangularmomentum foraparticleof massm.Hence m=Gf(dqj2m)=(qj2m)G=yG, whereq=fdqisthetotalchargecirculating intheorbit. y=qj2miscalledthemagnetogyric ratio. Thiscloserelation between magnetic moment andangularmomen­ tumisofgreatimportance inatomictheoriesofmagnetism, becauseon quantum theorytheangular momentum ofanelectron inanatom, whichisaconstant ofthemotion,canonlyhavediscretevalues.The electron possesses angularmomentum notonlyinrespectofitsorbital motionroundthenucleus, butalsoinrespectofitsintrinsic rotation (spin)aboutitsownaxis.Theresultant angularmomentum ofanatom isthevectorsumoftheindividual angularmomenta ofitselectrons, andtheresultant magnetic moment isasimilarvectorsumoftheindi­ vidualmagnetic moments oftheelectrons. Becauseofthelinearrelation between thetwo,anatom,ion,ormolecule willhavenoresultant per­ manentmagnetic moment ifthetotalangularmomentum iszero.If thetotalangularmomentum isnotzero,theatom,ion,ormolecule will haveapermanent magnetic dipolemoment. Mostfreeatomspossess apermanent magnetic dipolemoment becausetheyhavearesultant electronic angularmomentum. Suchmagnetic dipolemoments arefundamentally different from electricdipolemoments, whereithasbeenshown(§2.3)thatifparity isconserved inanatomornucleus, noelectricdipolemoment canexist. Thedifference isclearifwewriteoutthecomponents ofthemagnetic dipolemoment givenbyequation (8.1);e.g. mz=Il(x:~-y::)dq. 8.1J MAGNETIC MEASUREMENTS 197 Ifparityisconserved, wehaveinversion symmetry; thatis,thevalue ofdqisthesameatthepoint(-x,-y,-z)asatthepoint(x,y,z).So alsoisthequantity {x(oyjfJt)-y(oxjot)}, sothattheintegrand retainsthe samesignundertheinversion operation, andtheintegral canhavea finitevalue.Thedifference fromtheelectricdipolemoment isthatthe latterinvolves thefirstpowerofthecoordinates (seeequation (2.25)), whilethemagnetic dipolemoment involves thesecondpower.The difference alsoshowsclearlyinanoperation suchasreflection inaplane, asshowninFig.8.1.Thepresence ofmagnetic dipolemoments inatoms (a)o c:=:>---++ +t (b)++-- +t FIG.8.1.(a)Reflection ofcirculating currents inaplane. (b)Reflection ofdipolesinaplane. Notethatthecirculating currents behavedifferently fromtheequivalent dipoleson reflection. andnucleiconfirms thattheyarisefromcirculating currents associated withmovingelectriccharges, sinceadipolemoment duetoapairofreal magnetic chargeswouldfollowthesamerulesasadipoleduetoapair ofelectriccharges. Ingeneral, however, atomsdonotexistinthefreestatebutare combined intomolecules, anditsohappensthattheforcesresponsible forchemical bindingstrongly affectthearrangement oftheindividual magnetic moments ofthevariouselectrons inthemolecule. Asaresult thestablestateofthemolecule isnearlyalwaysoneinwhichthevector sumoftheseindividual moments isjustzero,sothatthemolecule as awholehasnoresultant permanent magnetic moment. Similarly, in themajority ofsolidsandliquids,theatomicconstituents areionswith nopermanent magnetic dipolemoments. Whensuchasubstance is placedinafluxofmagnetic induction B,eachindividual electron, being amoving charge,experiences aforce.Itsorbitalmotionroundthe nucleusisalteredinsuchawaythatitacquires anangularmomen­ tumandhenceamagnetic dipolemoment whichisproportional to theappliedfield,butintheopposite direction. Thisgivesanegative 19lJ MAGNETIC MATERIALS AND [8.1 susceptibility, whosevaluewillnowbecalculated. Itshouldbenotedthat thisdiamagnetic effectispresentinallsubstances, butinparamagnetic substances thereisamuchlargerpositive contribution tothesuscepti­ bilitywhichgenerally faroutweighs thediamagnetic contribution. The positive contribution arisesfrommolecules (orions)wherethevector sumoftheindividual electronmoments isnotzero,sothatthemolecule orionpossesses apermanent magnetic moment. Thusallsubstances showadiamagnetic effect,givingasusceptibility oftheorderof10-5 (usingm.k.s.units,or10-6withe.m.u.),butinparamagnetic substances thereisapositive susceptibility contribution whichismuchgreater. 8.2.Diama~netism Thefactthatanatomplacedinamagnetic fieldBacquires amag­ neticmoment paralleltoBcanbeshownbyfindingtheeffectonthe atomofestablishing thefieldBfromzero.In§6.1itwasshownthat thischangesthemomentum ofaparticleofchargeqfrommvotomv, wheremv=mvo-qA andAisthevectorpotential associated withthefieldB.Thecorre­ sponding currentdensityisgivenby J=pv=pvo-p(qjm)A. (8.3) IfAisincreased bySA,theincreaseinpotential energyis,byequation (5.45), SUp= -f(J.SA)dT = -fp(vo·SA)dT+fp(qjm)(A.SA)dT. (8.4) Forsimplicity wetakeBtobeauniform field(whichitwillbeover atomicdimensions), sothat A=t(B/\r),and(A.SA)=1(BSB)r2sin28, where0istheanglebetweenBandr.Hence SUp= -ftpvo·(SB/\r)dT+(qj4m) fpBSBr2sin20dT =-SB·ftp(r/\vo)dT+(qj4m)BSBfpr2sin20dT =-SB.mo-SB.mi=-SB.m. (8.5) Thisresultshowsthatthemagnetic moment mconsistsoftwoparts, ofwhichthefirst,mo,isclearlythepermanent magnetic dipolemoment (seeequation 8.1),whilethesecondmiisaninduced dipolemoment whichisproportional tothefieldstrength B.Thevalueofmiis mi=-B(qj4m)fpr2sin20dT=-B(q2j4m) (r2sin20), 8.2] MAGNETIC MEASUREMENTS 199 whereq=IpdTisthetotalcharge.Ifpisindependent of(J, 1T (r2sin2B)=(r2)4~fsin2B.21Tsin(JdB=i(r2), o where(r2)isthemeansquareradiusoftheorbit.Onsumming over allelectrons intheatomorion,momustbereplaced bytheresultant (ifany)ofthepermanent dipolemoments, whiletheinduced moIhent peratombecomes m..=-(e2f6m)B!(r2), (8.6) wheretheelectronio charge-ehasbeensubstituted forq.Hencethe diamagnetic susceptibility ofasamplecontaining natoms(orions)per unitvolumewillbe X=nm..fH=-nJLo( e2f6m)!(r2). (8.7) ThisresultcanbederivediIianotherwayusingLarmor's theorem (Appendix A.II).Theresultofestablishing afieldBistosetup aprecessional motionoftheelectronic orbitswithangular velocity w=-(qf2m)B, asaresultofwhicheachelectronacquiresanadditional angularmomentum G..=m(a2)w=-m(a2)(qf2m)B, where(a2)=(r2sin2{)isthemeansquaredistanceoftheelectronfroman axisparalleltoBthroughthecentreofgravityoftheatom(thenucleus). Associated withG..isanadditional magnetic momentofmagnitude (from equation (8.2» m..=(qf2m)G ..=-(q2f4m)(r2sin2{)B. Onevaluating theaverageandsumming overallelectrons intheatom, thisgivesequation (8.7)above. Thisequation showsthatthediamagnetic susceptibility isinherently negative insign,anddoesnotdepend, forexample, onthesignofthe electronic charge.Fundamentally, thenegative signfollowsfromLenz's law.Whentheexternal magnetic induction Bisswitched on,thereis achangeinfluxthroughtheelectronorbitswhichinducesamomentary e.m.f.Thechangeintheorbitswhichthiscausesgivesaninducedmag­ neticmoment totheatomwhichisinsuchadirection astoopposethe changeinfluxthroughtheorbit,i.e.themagnetic moment isduetoan inducedcurrentwhoseownlinesofmagnetic fieldthroughtheatomare intheopposite direction tothoseoftheexternal field.Wehaveassumed that(r2)isunaltered bythepresence ofthemagnetic field;thisisjusti­ fiedbecausethemagnetic forcesatordinary fieldstrengths arenegligible compared withtheinternal atomicforces(seeAppendix A.II). (8.9)XM=-3'55x 1092(r2). (8.8) Inmostreference tablesthesusceptibility isgiveninelectromagnetic units,andforsolidsandliquidsthediamagnetic volumesusceptibility (percm3)isoftheorder-10-6e.m.u.(forgasesitismuchsmallerowing tothelowernumberofatomsperunitvolume). Inourm.k.s.unitsthe volumesusceptibility (permetre3)isgreaterbyafactor 47Tandsois oftheorder-10-5•Asaroughrule,thesusceptibility (m.k.s.)per gramme atomisoforder-1O-11Z,whereZistheatomicnumber, equal tothenumberofelectrons intheatom.Thisindicates thatthemean valueof<r2>isabout10-21metres2,asexpected fromatomictheory. Thesusceptibility ofadiamagnetic substance issubstantially indepen­ dentoftemperature, since2<r2>ispractically unaltered bytemperature. Inthefirstapproximation 2<r2>isconstant foraparticular typeof atomorion,andisnotgreatlyalteredbyitssurroundings. Thusaqueous solutions ofalkaliandalkalineearthhalidesobeyquiteaccurately (and mostsubstances approximately) anadditivity ruleknownasWiede­ mann'slaw.According tothisrulethemasssusceptibility Xmofasolu­ tioncontaining amassm1ofasaltofmasssusceptibility Xlinamass m2ofsolventofmasssusceptibility X2is m1Xl+m2X2Xm=m+m.1 2 Similaradditivity rulesareapproximately validforchemical com­ pounds. Thus,forcompounds whichionizeinsolution, themolarsus­ ceptibility ofthecompound isgenerally closetothesumoftheionic susceptibilities ofitsconstituent ionsinsolution. Byassuming atheo­ reticalvalueforthesusceptibility ofonetypeofion,approximate values forthesusceptibilities ofotherionscanbeobtained frommeasurements200 MAGNETIC MATERIALS AND [8.2 AsdefinedinChapter 5,thesusceptibility referstounitvolumeof substance, andnisthenthenumberofatomsinunitvolume. SinceX islinearlyproportional ton,itispermissible totakesamplesofdifferent size,andrefertothe'susceptibility perunitmass',or'susceptibility per gra;mme atom'.Thesearesimplyequaltothesusceptibility perunit volume(or'volume susceptibility' forshort)multiplied bythevolume ofthesample. Thusthe'masssusceptibility' Xm=xlp,wherepisthe density, sinceunitmass(1kg)occupies avolumeof(lip)metre3• Similarly, thesusceptibility pergramme atomorgramme molecule will beXM=10-3MXm'whereMistheatomicormolecular weightin grammes. Herethefactor10-3occursbecauseourXmreferstoakilo­ gramme ofsubstance. Foragramme moleequation (8.7)givesthe numerical value 8.2] MAGNETIC MEASUREMENTS 2~1 (8.10)oncompounds, andthevalidityoftheadditivity rulestested.(Fora reviewofthediamagnetism ofions,seeMyers(1952).) 8.3.Parama~netism Asalreadypointed out,paramagnetism occursinthosesubstances wheretheindividual atoms,ions,ormolecules possessapermanent magnetic dipolemoment. Intheabsenceofanexternal magnetic field, theatomicdipolespointinrandomdirections andthereisnoresultant magnetization ofthesubstance asawholeinanydirection. Thisrandom orientation istheresultofthermalagitation withinthesubstance. When anexternal fieldisapplied,theatomicdipolestendtoorientthemselves paralleltothefield,sincethisisastateoflower energythantheanti­ parallelposition. Thisgivesanetmagnetization paralleltothefield, andapositive contribution tothesusceptibility. Sincethethermal agitation, whichtendstogivearandomorientation, islessatlowtem­ peratures, abiggerproportion ofthedipolesareabletoalignthemselves paralleltothefield,andthemagnetization isgreaterforagivenfield. Itwasdiscovered byCuriethat,forordinary fieldsandtemperatures, thesusceptibility ofmanysubstances followstheequation M0x=H=p' where0isaconstant, andTistheabsolute temperature; thisisknown asCurie'slaw.Forlargefieldsatlowtemperatures themagnetization produced isnolongerproportional totheappliedfield,andtendstoa constant value.Thissaturation effectisproduced whenalltheatomic dipolesarealignedparalleltothefield,sothatthemagnetization reaches alimiting maximum value. Thetheoretical explanation ofCurie'slawwasgivenbyLangevin, usingtheclassical statistics ofBoltzmann. Heassumed thateachatom hadapermanent magnetic moment m,andthattheonlyforceacting onitwasthatduetotheexternal fieldB.Then,ifagivenatomicdipole ispointing inadirection makinganangle()withB,itsmagnetic poten­ tialenergyisW=-mBcos0.Now,onclassical statistics, thenumber ofatomsmakingananglebetween ()and()+d()is dn=ce-W/kTsin0dO, (8.11) wherekisBoltzmann's constant andTistheabsolute temperature. cisaconstant definedbythefactthatintegration of(8.11)overthe wholepossible rangeofenergies mustgivejustn,thetotalnumberof 202 MAGNETIC MATERIALS AND [8.3 atomsinthesystem. Henceforourcase dn=cexp(mBcos0jkT).mBsin0dO andnisequaltothisintegrated overallanglesfrom0to7T.Thecom­ ponentofeachdipolemoment paralleltoBismcosB,andhencethe averagecomponent peratomism,where nm=mfcosOdn. 0·2 o 2 4 6 y 8 HenceFIG.8.2.TheLangevin function L(y)=cothy--Ijy. '7rfcosB.cexp(mBcosBjkT). mBsinB dBm()-::........-=--------------~m '7rfcexp(mBcosBjkT).mBsinO dB () OnwritingmBjkT =y,cosB=x,thistakestheform +1fxexp(xy)dxm 1 --~1 =cothy-- =L(y), mfexp(xy)dx y -1(8.12) whereL(y)isknownastheLangevin function.Itisplottedasafunc­ tionofyinFig.8.2.Forlargevaluesofythefunction tendstounity, saturation beingreachedwhenalltheatomicdipolesareparalleltoB. (8.14)8.3] MAGNETIC MEASUREMENTS 203 Fors~allvaluesofythecurveislinear,andL(y)=y/3=mB/3kT. Thenthes'USceptibility is nmfLonm2 X=H=3kT' (8.13) wherenisthenumberofatomsperunitvolume. Thisisthesameas Curie'slaw,equation (8.10),ifweidentifytheCurieconstant 0with fLonm2/3k. Theonlyunknown quantity in(8.13)istheatomicdipole moment mso,thatbymeasuring thesusceptibility asafunction ofthe absolute temperature inaregionwheremB/kTissmall,themagnitude oftheatomicdipolemoments maybefound.Ingeneraltheseareof theorderof10-23ampere-metre 2(10-20e.m.u.),orslightlygreater,and thevolumesusceptibility atroomtemperature ofsolidparamagnetic substances whichobeyCurie'slaw~+10-3.Thustheparamagnetism considerably outweighs thediamagnetism whichisalsopresent. Langevin's theoryappliesstrictlyonlytogases,wherethemolecules aresufficiently farapartfortheirDlutualinteractions tobenegligible. Inliquidsandsolidssuchinteractions maybelarge,andmanysub­ stancesobeythemodified Curie-Weiss law o X=T-()' ()iscalledthe'Weissconstant' andischaracteristic ofthesubstance; itmaybeeitherpositiveornegative (seeChapters 21and22).Equa­ tion(8.14)holdsonlyattemperatures whereT>18/,andformany substances nosingleequation represents thesusceptibility variation adequately overawidetempe!ature range. Asalreadynoted,thetendency inchemical combination istowards zeroresultant angularmomentum oftheelectrons andhencetozero permanent magnetic moment. Ofthecommon gases,onlyoxygenO2 andnitricoxideNOareparamagnetic. Inthesolidstateparamagnetism occursinsaltsofthe'transition group'ions(seeChapter 20),andthe magnetic moment isassociated withthemetallic ionitself.Thusin compounds suchasCrK(S04)2,12H 20('chrome alum')orCuS04,5H 20 (coppersulphate) onlytheCr+++ionandCu++ionrespectively have permanent magnetic moments, theotherions(K+,SO"4-) andwater molecules givingonlyadiamagnetic contribution tothesusceptibility. Inmostmetalstheouterelectrons aredetached fromtheindividual atoms,whicharethusleftasdiamagnetic ions.Thedetached electrons arefreetomovethroughthemetalandformtheconduction electrons; thesegiverisetoadiamagnetic andaparamagnetic effect,bothofthe 204 MAGNETIC MATERIALS AND [8.3 sameorderofmagnitude andbothindependent oftemperature (see §18.7).Theoutstanding exceptions aretheferromagnetic metals,iron, cobalt,nickel,andafewothers. 8.4.Ferromagnetism Ferromagnetic substances areallsolids,andeachischaracterized by acertaintemperature knownastheCuriepointatwhichitsproperties changeabruptly. AbovetheCuriepointthesusceptibility isindependent S H(ampere/metre)/R /r·OI I I0·5 I -I'e -27 H(oel'sted) IA,I 2 :1 _---'--------,-_---L_-,-------L-+-_....':C..j""'---_ ---'---.-----"--------r- ---"- -200 -100, I100 200 I II-0,5 , I II/1-1,0_I ". "./ ,-_.....' .....""""S--...,;;;;;;------- -1-5--B(inunitsofweber!metre2or104gauss) 1·5 (oersted) -3 (ampere/metre) FIG.8.3.Magnetization curve(fulllineOABS)andhysteresis loop (broken line)foriron. offieldstrength, andfollowsapproximately aCurie-Weiss law(equation (8.14))withaWeissconstantBwhosevalueisclosetothatoftheCurie point.BelowtheCurietemperature thebehaviour isquitedifferent; verylargevaluesofmagnetization areproduced byquitesmallfields, andthemagnetization variesquitenon-linearly withthefieldstrength. Thisisshownbyacharacteristic plotofthemagnetic induction Basa function ofthefieldHinasampleofiron,Fig.8.3.Iftheironisinitially unmagnetized, andafieldofslowlyincreasing magnitude isapplied, BfollowsthefulllineinFig.8.3,knownasthe'magnetization curve'. Inafieldofafewhundred amperefmetrethevalueofBbecomes practi­ callyconstantatabout1·5weberfmetre2•Ifthemagnetic fieldHisnow reduced, theinduction Bdoesnotreturnalongthemagnetization curve butfollowsthebrokenline,andevenatH=0,corresponding tothe pointRinthefigure,Bisstillnearthesaturation value.Thevalueof Batthispointisknownasthe'residual induction', andtheretention 8.4] MAGNETIC MEASUREMENTS 205 ofmagnetization inzerofieldisknownas'remanence'. Onapplying areversefieldthevalueofBfaIlsandfinallybecomes zero(point0in Fig.8.3);thevalueofthefieldatthispointiscalledthe'coercive force'. Asthemagnitude ofthereversefieldisfurtherincreased, areversein­ ductionissetupwhichquicklyreachesthesaturation value.Finally, ifthereversefieldisgradually removed andapositive fieldapplied, 10,000 2·5 2 0·5 1'0 1·5 B(weber/metre2) FIG.8.4.Curveofpermeability floagainstinduction Bforiron.4,000 theinduction tracesoutthebrokencurveinthedirection SfBS.This brokencurveiscalledthe'hysteresis curve'.Itshowsthatthechange inthemagnetic induction alwayslagsbehindthechangeintheapplied magnetic field. Themagnetization curvemayalsoberepresented intheformofa permeability curve,showing thevariation ofI-'=B/(l-'oH) asafunc­ tionofeitherBorH.Suchacurve(I-'againstB)isshowninFig.8.4. Whenafieldisapplied I-'goesthroughamaximum andthenfallsrapidly asthematerial becomes saturated. Thevaluesof1-',oftheorderof104, areenormously greaterthaninparamagnetism (1'001orsoatordinary temperatures). Toexplainthisbehaviour, Weisssuggested thataferromagnetic substance contains atomswithpermanent magnetic moments, asina paramagnetic substance, butthattherearelargeforcesactingbetween 206 MAGNETIC MATERIALS AND [8.4 neighbouring atomicdipoleswhichcausegroupsofthemalltopoint inthesamedirection. Thesubstance wouldthenbepermanently mag­ netizedwithineachgroup;suchgroupsarecalled'domains' andtheir sizeisnowknowntovaryfromabout10-6emSto10-2emS,orgreater insinglecrystals. Inanunmagnetized polycrystalline specimen the domains areorientedatrandom, sothatthereisnoresultant magnetic moment inanydirection. Whenafieldisapplied, domains wherethe (a) (b)(e) FIG.8.5.Schematic representation ofdomains inaferromagnetic substance: (a)un. magnetized; (b)magnetization through movement ofdomainboundary wall;domains oriented paralleltoHgrowattheexpense ofanti-parallel domains; (e)magnetization byrotation ofthemagnetization ofwholedomains. Thedomains remainmagnetized alongapreferred direction ineachcrystallite; verylargefieldsarerequired toswingthe magnetization awayfromsuchadirection towards theappliedfield. magnetization isparalleloratasmallanglewiththefieldgrowatthe expenseofthosewherethemagnetization isanti-parallel ornearlyso, sothattheboundary between domains isdisplaced. Initially (OAin thefullcurveofFig:8.3)themagnetization ofthesubstance asawhole proceeds bysmall(reversible) boundary displacements, butthesteeper part(AB)ofthemagnetization curveisduetolarger(irreversible) dis­ placements. Above thekneeofthecurve,magnetization proceeds by rotation ofthedirection ofmagnetization ofwholedomains; sucha processisratherdifficultandtheincreaseinmagnetization isrelatively slow.Theseprocesses areshownschematically inFig.8.15.Whenthe appliedfieldisreduced, thereislittlechangeinthedomainstructure sothatthemagnetization remains quitehighuntilreversefieldsare applied, thusgivingrisetothehysteresis described above. Ferromagnetic substances maybebroadly dividedintotwoclasses: (a)magnetically softmaterials, whichhavehighpermeability, andare easilymagnetized anddemagnetized, and(b)magnetically hardmaterials, whichhavearelatively lowpermeability andaredifficulttomagnetize 8.4] MAGNETIC MEASUREMENTS 207 ordemagnetize (highcoercive force).Thechiefusesoftheformerarein electromagnetic machinery andtransformers, andofthelatterinperma­ nentmagnets. Toobtainasoftmagnetic material thedomainwalls mustbeabletomoveeasilyandreversibly, sothatthemagnetization changesbylargeamounts forsmallchangesinthemagnetizing field. Thisrequires amaterial asfreeaspossibleofirregularities inthecrystal structure duetostrainsorsmallparticles ofimpurities. Themaintreat­ mentofsuchmaterials consiststherefore ofheatingtoatemperature wheresufficient movement oftheatomsispossible forthemtosettle intoanorderedlattice,followed byaslowcooling(annealing) soasnot todisturbit.Ontheotherhand,ahardmagnetic material isonein whichdomainwallmovement isdifficultowingtolatticeimperfections. Theseareproduced byheatingthematerialandthenplungingitsuddenly intocoldoil(quenching), whichsetsupinternal stresses. Somealloys arethenreheated toalowertemperature tocauseoneoftheconstituents partially toseparate outinsmallparticles dispersed throughthealloy. Inanalternative processmagnets areconstructed fromcompressed powders ofveryfineparticles.Iftheparticles arebelowacertainsize, eachformsasingledomainandtherearenodomainwallswithinthe particle. Magnetization anddemagnetization canonlybeaccomplished byrotation ofthedirection ofmagnetization ofeachparticle, which requires ahigherfieldthanwallmovements, andsogivesahigher coercive force. 8.5.Production ofmagnetic fields Formanypurposes itisnecessary tomaintain alargemagnetic field whichisconstant overacertainvolume. By'large'ismeantvaluesof Brangingfrom0·1to10weberjmetre2(loato105gauss),andthevolume mayvaryfrom100cm3uptomanycubicmetresinamodemaccelerator foratomicparticles. According totheparticular application, permanent magnets orelectromagnets (withorwithoutiron)maybeemployed, and theprinciples oftheirconstruction areoutlined below. Amodernpermanent magnetoftypicalshapeisshowninFig.8.6. Theimportant quantities arethevaluesofBintheairgap,andthe volumeoftheairgap;sincetheenergydensityatanypointislB.H (equation (6.44)),thetotalenergystoredintheairgapis(!BH)Ya, where Yaisthevolumeofthegap,andthevectorial representation ofBand Hcanbedropped sincetheyareparalleltooneanother. Theenergy storedinthegapincludes boththeimportant parameters (thevalueof Bandthevolume), andissimplyrelatedtothequantities involved in 208 MAGNETIC MATERIALS AND [8.5 B.Aa=BmAm> (8.15) whereAa,Amarethecross-sections oftheairgapandthemagnetrespec­ tively,andBmisthevalueofthemagnetic induction inthemagnet.themagnetdesign,asfollows.Ifweassumethatthereisnoleakage, so thatallthelinesofmagnetic induction withinthemagnetpassthrough thegap,wehave ~----- ------~ / ' / 'I \ I \ I 'I I I I I I I I I I I \ I , /, / ~--------------------~ FIG.8.6.Permanent magnet. Now,ifweconsider acircuitthrough thegapandthemagnetasindi­ catedbythebrokenlineinFig.8.6,thetotalmagnetomotive force(see §5.2)iszero,sincenoelectriccurrents areinvolved. Hence fH.ds=Hda+Hmd m=0, (8.16) whereda,dmarethepathlengthsintheairgapandthemagnetrespec­ tively,andHmisthemagnetic fieldinthemagnet. Oncombining equations (8.15)and(8.16)wehave (!BH)Va=(lBH)Aad a =-(!BmHm)Amd m=-(!BmHm)Vm' (8.17) Thisimportant relation showsthatforamagnetofgivenvolumeVm' thegreatestamountofenergystoredinthegapisobtained iftheproduct (BmHm)hasitsmaximum value. Thevariation oftheproduct (BmHm)withBmforatypicalmaterial isshowninFig.8.7,together withthe'demagnetization curve';thatis, thepartofthehysteresis loopcorresponding totheapplication ofa 8.5] MAGNETIC MEASUREMENTS 209 reversefield.Inapermanent magnetthefieldinsideisthe'demagnetiz­ ingfield'duetothe'freemagnetic poles'neartheendsofthemagnet; thisfieldisintheopposite direction toBm>andthenegative signin equation (8.17)arisesfromthis.Thesizeofthedemagnetizing fieldis 1·5 (weber/metre!} (oersted} 400 2000·5 5 X10' 5 X10' (amp/metre} (joule/metre3) Ilm4 •(BmHm) FIG.8.7.Demagnetization curve(totheleft)andplotof(BmHm)againstEm (totheright),forAlcomax III. determined bytheshapeofthemagnet, andthismustbedesigned so thatthematerial isatthepointwhere(BmHm)isamaximum. Agood working ruleisthatatthispointtheratioofBmtoHmisequaltothe ratioBr/Hc'whereBristheresidualinduction andHcthecoercive force. Inpractice thereisalwayssomeleakageoflinesofmagnetic fieldso thattheenergystoredinthegapislessthanthetheoretical valuegiven byequation (8.17).ThebestshapeofthemagnetisoneinwhichBmHm ateverypointinthematerial isclosesttoitsmaximum value.Sincethe productBmHmisacharacteristic ofthematerial thevolumeofmaterial required increases linearlywiththeenergystoredinthegap.Many specialalloys,suchasAIcomax (Alnico), havebeendeveloped withhigh valuesoftheenergyproduct. AIcomax IIIhasthecomposition 50% iron,25%cobalt,13·5%nickel,8%aluminium, 3%copper,andt% niobium. Itscoercive forceHc~5X104A/metre, theresidualinduction Br~1·3weber/metre2,andthemaximum valueof(BmHm)~4x104 851110 P 210 MAGNETIC MATERIALS AND [8.5 c yy p pcjoules/metre3•Thedisadvantage ofmostofthesespecialalloysisthat theycannotbemachined, andthemagnets mustbecastorsintered frompowder. Theyareveryusefulwhenconstant fieldsuptoIweber/ metre2arerequired. Whenfieldsofgreatermagnitude, oradjustable fields,arerequired, anelectromagnet isused.Forfieldsuptoabout2weber/metre2 the normaltypeofconstruction isasshown inFig.8.8.ItconsistsofayokeYof softironor,morecommonly, mildsteel, whichhasareasonably highmagnetiza­ tionforsmallvaluesoftheappliedfield. 0,0arecoilsofcopperwire,orcopper tubethrough whichcoolingwatermay flow,carrying currentfromad.c.gen­ erator.Ifthecoilshaveatotalofn turnseachcarrying acurrentI,the ofelectro- magnetomotive force(m.m.f.) is nI=fH.ds=Hda+Hmd m(8.18) roundacircuitthrough theairgapand yokesimilartothatshowninFig.8.6,withthesamenomenclature as before.Sinceequation (8.15)stillholds,wehaveFIG.8.8.Weisstype magnet. 0,0coilscarrying electriccurrent; P,Ppoletips;Yyoke. (8.19) wherethequantity inbrackets isknownasthemagnetic 'reluctance' ofthesystem.Inthisanalogy between them.m.f.andthee.m.f.ina circuit,theflux(BAa)isanalogous totheelectriccurrent,andtheflux passesthroughthetwocomponents-the airgapandtheyoke-in series. Thetotalreluctance ofthe'magnetic circuit'isthesumofthetwoparts duetothegapandtheiron.SincefLforironisverylarge,thegreater partofthereluctance isgenerally associated withtheairgap. Inmostiron-cored magnets thepolefacesmayberemoved and replaced byothersofdifferent shapeforspecialinvestigations, andthe widthoftheairgapmaybealtered. ToobtainhighervaluesofB,coned polepiecesmaybeused.InthedesignofFig.8.9,ifitisassumedthat themagnetization iseverywhere paralleltotheaxisandhasthesatura­ tionvalue~, thefieldatthecentreofthegapduetotheconicalportions indicated byshadingmaybeshowntobe(seeProblem 8.7) B=fLoH=fLo~sin2ep coseploge(b/a). (8.20) 8.5] MAGNETIC MEASUREMENTS 211 Hx=m3(2CosOcosc/>+sinOsinc/».47rrThishasamaximum valueatc/>=54'70 ;inpracticethepolepiecesare noteverywhere completely saturated, andavalueofabout600gives thebestresults. Thepoletipsmaybemadeofaspecialcobaltsteel whichhasahighersaturation induction thanordinary mildsteel.Aplot ofthefieldinthegapagainstexciting currentisusuallyfairlylinear FIG.8.9.Conedpoletipsformagnet. (apartfromasmallinitialfieldduetotheremanence) untilthesteel becomes saturated; therateofincrease thenbecomes muchlower,any extrafieldbeingthatduetothecurrentinthecoilsthemselves. ItwaspointedoutbyBitter(1936)thatthisarrangement, inwhich theironismagnetized everywhere inthesamedirection, doesnotmake thebestuseoftheiron.InFig.8.10thefieldattheorigin0duetothe dipoleatA,oriented asshown,hasacomponent paralleltothex-axis equalto Iftheorientation ofthedipoleisvariable, thenHxhasamaximum valuewhendHx/dO=0;i.e.tan0=}tanc/>,andthisvalueisthen (8.21) If,ontheotherhand,thedipolepointsparalleltothex-axis,sothat 0=-c/>,thevalueofHxisonly mHx=47rr3(3cos2c/>-1). (8.22) Boththeangularfunctions in(8.21)and(8.22)havethevalue2at 212 MAGNETIC MATERIALS AND [8.5 ep-O,butthelatterfallstozeroatep=54.70andthenchanges sign (thereasonwhytheconedpolepiecesdiscussed abovehavethisastheir optimum angleisthatdipolesatalargerangleoriented paralleltothe axiswouldgiveareversefieldandthusreducethefieldinthegap). Ontheotherhand,atthisangle(1+3cos2ep)1hasfallenonlyto";2,and itssmallest valueis1atep=900 • o ~----~--.-x FIG.8.10.Illustrating thecalculation ofthefieldattheorigin0duetoadipolematA. Atfirstsightitwouldseemimpracticable tosetupamagnetization intheironwiththeangulardistribution ofthedirection ofmagnetiza­ tionrequired bytheequation tan()=!tanep.Theorientation ofthe magnetization atanypointis,however, justthesameasthedirection of thelinesofforcesetupbyapointdipoleattheoriginpointing along thex-axis,anditfollowsthatsuchadipolewouldmagnetize theironin justtherightdirection. Bittertherefore designed amagnetofthetype showninFig.8.11,wherethemagnetizing coil,whosefieldisapproxi­ matelythatofapointdipole,issurrounded bysoftiron.Thisgavean appreciably betterperformance thantheoldertypeofdesign.Bitter considered alsothequestion ofthecurrentdistribution andshapeofthe magnetizing coil,andconcluded thathigherefficiency couldbeobtained ifthecurrentdensitywerenotuniform, asitisinacoilwoundinthe ordinary way,butfelloffinversely withtheradius.Inthedesignof Fig.8.11thisisachieved bywinding aspiraloutofcopperstripof gradually increasing width;thecurrententersalongacentralbrasstube, flowsroundthespiralwithdecreasing densityasthewidthincreases, andleavesattheouteredge.InsuchamagnetBitterobtained afluxof 8.5] MAGNETIC MEASUREMENTS 213 30000gauss(3weberjmetre2),usingapowerof21000W;them.m.f. was5X104ampere-turns, andthecoilresistance about0·03ohm. Sinceironandothermagnetic materials saturate atabout1to2 weberjmetre2,thecontribution theymakeinmagnets forstillhigher fieldsisnotsufficient tojustifytheexpense. Bitterhasdesigned air­ coredsolenoid magnets ontheprinciple outlined above.Thenon-uniform Iron --------------------b~ A A ~con~ /------~ B B Iron FIG.8.Il.Bittermagnet, withcentralcoilsurrounded withiron. Thecoilconsists ofaspiralstrip,whosewidthincreases withthe radius;currententersatAA,flowsroundthecoilproducing anaxial field,andleavesatBB. currentdistribution isobtained byusingflatconductors oftheshape showninFig.8.12;thecurrentisledinalongtheedgeAAandoutalong BB,sothatthecurrentdensityvariesinversely withtheradiusbecause theresistance alongapathofradiusrincreases withr.Thesolenoid isconstructed ofanumberofsuchdisksmounted oneabovetheother, withholesdrilledinthemsothatcoolingwatercanbeforcedthrough. Thelowresistance ofsuchadesign(,-..,0·01ohm)reducesthechance ofanybreakdown intheinsulation, andcorrosion through electrolysis ofthecoolingwater.InonesuchcoilBitter(1940)obtained afieldof 10weberjmetre2,uniformto1percentoveravolumeof25ems,with asupplyof10000Aat170V.Itcanbeshownthatthefieldobtained fromanair-cored coilcanbeexpressed as(WAjpr)1 timesafactor depending onlyontheshapeofthecoilandthecurrentdistribution; hereWisthepowerdissipation, Athefractionofthecoilvolumeoccu­ piedbyconductor ofresistivity p(theremainder beinginsulation and coolant), andralineardimension suchastheinnerradius.Thusto doublethefieldoverthesamevolumerequires fourtimesthepower. 214 MAGNETIC MATERIALS AND [8.5 Thediscovery ofsuperconducting wireswhichremaininthesuper­ conducting stateuptofieldsof105gauss(10weber/metre2)ormorehas madeitpossibletoconstruct solenoids inwhichasteadyfieldcanbe maintained withoutanypowerconsumption, sincetheresistance inthe superconducting stateiszero.Themaindrawback isthatthesuper­ conducting stateisonlyattained belowsomecriticaltemperature 1;, A B FIG.8.12.Shapeofcopperdiskknownasa'Bitterpancake'. whichliesbelow20°Kforvirtually allsubstances, sothatthesolenoid mustbemaintained atliquidheliumtemperatures. Acomparison ofthe powerrequirements ofvarioustypesofmagnets isgiveninFig.8.13; thepowershownforthesuperconducting magnet(10k\V)isarbitrary, beingthatrequired toruntheliquidheliumrefrigerator! 8.6.Measurement ofmagnetic fields Thecommonest methodofmeasuring thevalueofBatanypoint inairisbymeansofa'flipcoil'andaninstrument formeasuring flux. Theflipcoilconsistsofanumber nofturnsofwirewoundonasmall formerofknownareaA;thisismounted onahandleandtheleadsto thecoilaretwistedandbroughtoutthroughthehandle.Whenthecoil isplacedwithitsaxisparalleltoafieldB,thefluxthroughitisnAB; ifthecoilisthenquicklyremoved toapointwhereB=0,theflux changeisjustnABandthiscanbemeasured eitherbyaballistic gal­ vanometer orafluxmeter (see§7.3).Foraccurate resultstheballistic galvanometer shouldbestandardized asdescribed in§7.3usingamutual 8.6] MAGNETIC MEASUREMENTS 215 inductance, withthesecondary windingandtheflipcoilinserieswith thegalvanometer throughout allthemeasurements sothatthetotal resistance ofthecircuitremains constant. Flipcoils·canbemadewith different valuesoftheproductnA(turnsXarea),sothatbychoiceof therightcoilforthefieldtobemeasured asuitable deflexion canbe 200 OJ OJ ::>100- ~-70-;§. ;g ~40- ":z ~120 10I I I I ISuperconducting magnet I Alnicopermanent magnet 10,000----L-~ ~ ~_.~__~._~ --------' o10 100 1000 TotalPower(kilowatts) FIG.8.13.Comparison offieldproduced byvarioustypesofmagnet, andthepower requirements (forthesuperconducting magnetthisisjusttherefrigerator powercon­ sumption). Volumeoffield=500cm3•(Courtesy Dr.J.Hulm,Westinghouse Research Laboratories. ) obtained ontheballisticgalvanometer orfluxmeter. Asnotedin§7.3, theaccuracy obtainable withanaveragefluxmeter isabout1percent, andwithaballisticgalvanometer abouttentimesgreater. Ifthecoilisrotatedrapidlyinthefield,analternating voltageisset upwhichisproportional tonABtimestheangularvelocity; measure­ mentofthisvoltagerequires alesssensitive instrument thantheflux­ meterbecausetheenergyavailable ismuchgreaterthanwithasingle throw.Inatypicalinstrument, acoilof3mmouterdiameter isrotated at30cis,andgivesfullscaledeflexion inafieldofabout500gauss; withlargerandsmallercoils,fieldsrangingfromtheearth'sfieldto105 gausscanbemeasured quicklywithanaccuracy ofabout1percent. Anabsolute methodwhichiscapableofgivinghigheraccuracy isthe electromagnetic balanceofCotton,inwhichtheforceduetothefieldB onalengthofwirecarrying aknowncurrentismeasured directly. This methodcanonlybeusedforratherstrongfieldswhichareuniform over 216 MAGNETIC MATERIALS AND [8.6 w=-yB,afairvolume. Alongrectangular coilissuspended fromananalytical balancewiththelowerendofthecoilinthefieldtobemeasured, this fieldbeingdirected horizontally. Thelongsidesofthecoilarevertical, sothatnoforceisexertedonthemintheverticaldirection; theyact asleadsforthecurrentinthehorizontal loweredgeofthecoil,andthe forcemeasured isjustthatonthisloweredge,assuming thatthevalue ofBalongtheupperedgeisnegligible.Iftheloweredgeisdirected perpendicular tothelinesofB,thenetverticalforceisF=1IBdx, wheretheintegralismeasured alongtheloweredge.Thustheintegrated valueofthefieldalongthisedgeisdetermined, andtofindthevalue atanypointthefielddistribution mustbeknown. Thecurrentis measured withastandard resistance andpotentiometer, andtheforce bythechangeinthebalance reading whenthecurrentthrough the rectangular coilisreversed indirection. Withabalanceofthistype Thomas, Driscoll, andHipple(1950)wereabletomeasure afieldof about0·5weberjmetre2withanaccuracy ofafewpartsin105•The purposeoftheirexperiment wastomakeanaccurate absolute measure .. mentofthemagnetogyric ratioofthenuclearmagnetic moment ofthe proton,bydetermining theprecession frequency (Wj27T)ofthenuclear moments inafieldB.Theprinciple ofthismethod('nuclear magnetic resonance') willbediscussed inChapter 23,butitisbasedonthe equation whereyisthemagnetogyric ratio(seeequation (8.2)).Sincethefrequency (Wj27T)islinearlyproportional toB,andthefrequency ofaradio-oscilla­ tionmayreadilybedetermined withhighaccuracy (seeChapter 15), thisgivesanaccurate measure ofBifthevalueofyisknown. 8.7.Measurement ofsusceptibility -=-Mostmethods ofdetermining thesusceptibility ofweaklymagnetic substances dependonmeasuring theforceonthesubstance inanin­ homogeneous magnetic field~Byanalogywithequation (1.15),theforce onamagnetic dipolemhasanx-component .F=m(8Bx)+m(8BY)+m(8Bz).xx8xy8x zax(8.23) Now,ifinsteadofapermanent dipolewehaveaparticleofmagnetizable matterofsusceptibility Xandvolumev,itsmoment willbe m=XvH 8.7] MAGNETIC MEASUREMENTS 217 FIG.8.14.Curie'smethod forthe susceptibility ofasmallspecimen S. P,Parethemagnet poletips.andthex-component oftheforceonitis F (HoHxHoHyHoHz)I(0H2)x=XJLoV xox+yox+Zax=2XJLOVox• Iftheparticlehasasusceptibility Xlandisimmersed inamedium (such astheatmosphere) withsusceptibility X2'thentheforceonitis oH2Fx=t(XI-X2)JL OV-, (8.24)ox Thiscanbeseenfromthefactthatanydisplacement oftheparticle inthex-direction requiresanopposite displacement ofanequalvolume ofthesurrounding medium. Thisequation mayalsobederivedbyconsidering thestoredenergy. Theeffectofthepresence oftheparticleofvolume vistoincreasethe storedenergyby U=v(tBI.H-tB2.H) =tv[(I+Xl)-(I+X2)}JLoH2 =tV(XI-X2)JLOH2. Fromequation (6.40)theforcecomponent isgivenbyFx=+(oU/ox), givingthesameformulaasabove(equation (8.24)).Inboththesede­ rivations wehaveassumed thatthemagnetic fieldinsidethespecimen isthesameasthevaluemeasured before thespecimen wasintroduced. Thesetwo quantities differonlybythedemagnet­ izingfieldinthespecimen, whoseorder ofmagnitude isM=XH,whichis negligible forthevaluesofX(~10-3 orless)ordinarily encountered. ,._Theforceequation (8.24~isthebasis ofmethods formeasuring thesuscepti­ bilityofsmallspecimens, andwasused byCurie.Thespecimen issuspended fromonearmofasensitive torsion Jbalance, andhangsbetween thepole \Jjpsofanelectro!I!~~t..as inFig.8.14. -Thearrangement ofthepoletipsgives alargevalueofoH~/ox,whilealongtheaxisthequantities oH~/oxand om/oxareverysmall.Valuesoftheorderof1011A2/metre3canbe obtained foroH:/oxwithHyoftheorderof106A/metre (,-...,104oersted). Themaindifficulty arisesfromthefactthatthevalueof8H~/8xis 218 MAGNETIC MATERIALS AND [8.7 usuallyconstant overonlyarathersmallvolume, anddifferent speci­ mensmustbeplacedratheraccurately inthesameposition inthe magnettoobtaincorrectresults. Thisdifficulty isreducedbytheuse ofspecialshapesofpoletipsdesigned tomake8HU8xuniform overa largervolume. Whenlargerquantities ofasubstance areavailable, abettermethod ofmeasuring thesusceptibility isthatduetoGouy.Thespecimen is (a) (0)~/...."./....•/...~•.•."./~ //~./" //.// (c) FIG.8.15.Gouy'smethod formeasuring susceptibility. madeintoalongcylinder ofuniform cross-section, andissuspended fromonearmofasensitive balancesothatitslowerendhangsbetween thepolesofanelectromagnet, asinFig.8.15(a).Ifthecross-section ofthespecimen isA,theverticalforceinthex-direction isdFa;onan element Adx,andhencethetotalforceisfound,byintegrating over thelengthofthespecimen, tobe Fa;=tA/LO(XI-X2)J(8H2/8x)dx =tA/LO(XI-X2)(H~-H~), (8.25) whereHIisthefieldatthelowerendofthespecimen andH2thatat theupperend;generally H~willbenegligible compared withH~.Itwill,... beseenthat,although theforcearisesbecausethespecimen isinanin­ homogeneous field,onlythevalueofthehomogeneous fieldatthecentre ofthemagnetgapisrequired. Withfieldstrengths oftheorderof105 to106A/metre (103to104oersteds), forcesoftheorderofmilligrammes areobtained whichcanbemeasured onanordinary laboratory balance. 8.7] MAGNETIC MEASUREMENTS 219 Forliquidsandpowders, acylindrical container ofuniform cross­ sectionmaybeused,half-filled withthesubstance ordividedintotwo compartments (seeFigs.8.15(b)and(e)).Themid-point ofthecon- Itainerisinthecentreofthefield,andtheforcesonthetwohalvesofthe container areequalandopposite. Theytherefore cancel,leavingonly theforceonthespecimen. Forliquids,avariantoftheGouymethod duetoQuincke maybeused,inwhichtheforceduetothemagnetic fieldisbalanced byhydrostatic pressure. I!l-.thesi1!!pl(lst forIrl_gfJhis .. methodtheliquidiscontained inaU-tube~d ~he~ell!~~~US inonearm"" isplacedintheuniform fieldbetween themagnetpoletips.,Whenthe fieldisswitched on,themeniscus risesorfalls,according towhetherthe liquidismoreorlessparamagnetic thantheairinthetubeabovei;y Ifthedensityoftheliquidisknown,thehYd.r:ostatic pressure caused bythemagnetic forcesmaybefoundfromthechangeinheiglltofthe meniscus ineitherlimbofthetUbe~orea(Jcurate-method, which eliminates errorsduetotheliquidsticking onthewallsofthetube,gs 1]9restorethemeniscus toitsoriginalheightbychanging therelative levelsofthetwolimbs(e.g.bytilting). ThismethodwasusedbyAuer indetermining thediamagnetic susceptibility ofwaterwithanaccuracy ofabout0'1percent'] Forcemethods ofmeasuring thesusceptibility havethedisadvantage thattheforcearisesfromaninhomogeneous magnetic field.Inmodern measurements, themagnetic moment ofasmallsinglecrystalisoften required asafunction offieldstrength andofthedirection ofthefield relativetothecrystalaxes(aswellasoftemperature)., Forthispurpose itispreferable touseauniform magnetic field,.whosevaluecanbe determined accurately muchmoreeasilythanaquantity suchas8H2J8x. whichisneededinCurie'smethodK Anumberofmethods havebeen developed wherethesampleismoveninandoutofacoil(orfromone coiltoanother) inthefield,andthechangeinfluxthroughthecoilis measured. Movement ofthesampleissuperiortomovement ofthecoil, becausethelattergivesafluxchangeevenintheabsenceofasample duetoresidualinhomogeneity inthefield.Anullmethodisoftenused; onewayofachieving thiswithacylindrical sampleistowindasmallcoil roundthesampleandadjustthecurrentthroughituntilthemagnetic moment ofthecoiljustcancelsthatduetothesample,asshownbythe zerofluxchangeinapick-up coilfromwhichthe(sample+coil) are suddenly removed. Thisisagoodexample oftheequivalence ofa currentcircuitandamagnetic shell,asdiscussed in§5.2. Asensitive magnetometer, duetoFoner(1959),inwhichthesampleI. 220 MAGNETIC MATERIALS AND [8.7 isvibrated at90c/s,isshowninFig.8.16.ThesampleS,attheend ofalongsupport whichreachesdownintoadewarvesselforworkat lowtemperatures, isvibrated vertically byaloud-speaker transducer T fedbyalternating currentat90c/s.Thesampleisatthemid-point between twocoilsA,Bwhoseaxesarevertical; theverticalcomponent ofthefluxfromthehorizontal magnetic moment ofthesamplethreads Voltage DividerPhaseshifter Amplifier and NullDetector90cIsvoltagefromA,BDOcIsvoltagefrom0,D sI'1-.._­I\\ I,"//,......_,.,."'"---yDirection ofDOcIsvibration!T/ '"V G~M~D ~ ~ I I I I I A BI Magnet ~~~IMagnet poleIpole face faceI FIG.8.16.Foner'sVibrating Magnetometer. ThepairofcoilsA,B(andsimilarly 0,D) areinseriesbutopposing, thusreducing spurious voltages duetomagnetic fieldinstability orunwanted mechanical vibration. Ssample, producing 90cIsvoltageincoilsA,D. Mpermanent magnet producing comparison voltageincoils0,D. Tloudspeaker transducer driving sampleSandmagnetMinvertical vibration at90cIs. thetopandbottomhalvesofeachcoilinopposite sense,sothatthe netfluxiszeroifthesampleisexactlyopposite themid-point. When thesampleisdisplaced vertically duringvibration, thenetfluxthrough eachcoilbecomes finiteandinfirstapproximation islinearlypropor­ tionaltothedisplacement ofthesample. Thusane.m.f.alternating at 90c/sisinducedinthecoils,whichcanbebalanced againstasimilar e.m.f.inducedbyasmallpermanent magnetMinthecoils0,D.This 8.7] MAGNETIC MEASUREMENTS 221 givesanullmethodwherethemagnetic moment ofthesampleisread offonthecalibrated voltagedivider(thephaseshifterisrequired because thetwoalternating voltages mayhaveasmallphasedifference--see Chapter 9).Theaccuracy isabout1percent,andchangesinmagnetic moment ofabout10-4e.m.U.(10-7amperemetre2)canbedetected, corresponding toavolumesusceptibility inthesampleofabout10-8 e.m.u.(10-7m.k.s.)inafieldof104gauss(1weberfmetre2).Rotation ofthewholeassembly aboutaverticalaxisenablesthemagnetic moment tobemeasured throughout ahorizontal plane;permanent moments can bemeasured aswellasinduced moments. 8.8.Experimental investigation ofthehysteresis curve Inordertodetermine thehysteresis curveofasubstance itisnecessary toknowthevaluesofBandHinsidethesubstance. Forthispurpose themostsatisfactory shapeofthesubstance isintheformofananchor ring,ortoroid,sincethenthereisnodemagnetizing fieldduetothe 'freepoles'attheendsofthespecimen.Iftheradiusoftheringis largecompared withthedimensions ofitscross-section, acoilwound uniformly roundtheringwillproduce auniform fieldHeverywhere withinthering.IfnIisthenumberofturnsperunitlengthinthis(the 'primary' coil),and1thecurrentflowing,thenH=nIl.Tomeasure B,asmallsecondary coiliswoundovertheprimaryatsomepointon thering;ifthiscoilhasn2turns,andthecross-section oftheringisA,the fluxthroughthecoilisn2AB.Changesinthefluxthroughthesecondary coilaremeasured byconnecting ittoaballistic galvanometer; thegal­ vanometer iscalibrated atthesametimeasdescribed in§7.3. Acircuitdiagram oftheapparatus isshowninFig.8.17.Inorder todetermine theinitialmagnetization curve(OABSinFig.8.3),the ringmustpreviously havebeendemagnetized; thisisusuallyaccom­ plishedbypassinganalternating currentthroughtheprimary coiland slowlydiminishing itsamplitude tozero.Thentheprimary coilissup­ pliedwithdirectcurrentmeasured ontheammeter A:thiscurrentcan beadjusted invaluebymeansofthevariable resistance RI.Byadjust­ ingRIinsteps,thecurrentintheprimary coilisincreased stepwise andtheballisticthrowofthegalvanometer measured ateachstep.This givestheincrement inBateachstep,andfinallythesaturation valueS (Fig.8.3)isreached. Beforestarting toplotoutthehysteresis curve,thefullcurrentin theprimary shouldbereversed anumberoftimesuntilthefluxchange ateachreversal reachesaconstant value.Thematerial isthenina 222 MAGNETIC MATERIALS AND [8.8 'cyclicstate'andreproducible resultsforthehysteresis curvecanbe obtained. Suppose thematerial isatthepointSinFig.8.3,withthe fullcurrentflowingthroughtheclosedswitchS2inFig.8.17.Onopen­ ingtheswitchS2withasuitablevalueoftheresistance H2,theprimary currentisdiminished andapointbetweenSandRonthehysteresis curveisattained. Onthrowing thereversing switchintheprimary FIG.8.17.Apparatus forB-Hcurvemeasurement. Misastandard mutualinductance forcalibrating theballistic galvanometer B. Pprimary winding , . . Ssecondary windingJon torOIdal speCImen. circuitthecurrentisrestoredtoitsfullvaluebutflowingintheopposite sensethroughtheprimary coil,sothatthematerial isnowatS',andcan bereturned toSbyreturning thereversing switchtoitsoriginalposition afterclosing82,Byrepeating thisprocedure withvariousvaluesofR2, andthenrearranging thecircuittothatR2isincircuitwhenthecurrent flowsinthenegative sense,pointsallroundthehysteresis curvecanbe obtained.Itisimportant thatthecycleisalwaysfollowed inthesame sense,andcompleted byreaching thepoints8andS'everytime,in ordertoretainthecyclicstate.Ifthedirection ofmovement roundthe hysteresis curveisreversedatsomepointbetweenSandS',aninter­ mediate curvewillbetracedout,andthecyclicstatemustberestored byanumberofreversals ofthefullcurrent. Inmanycasesthematerial tobetestedisintheformofalongbar 8.8] MAGNETIC MEASUREMENTS 223 ratherthanatoroidalring.Specialmethods (see,forexample, Vigoureux andWebb,1946)mustthenbeused,butthegeneralprinciple issimilar tothatgivenabove. Theareaenclosed bythehysteresis curveisofimportance becauseit represents theworkdoneintakingthematerial onceroundthehysteresis curve.Thusinatransformer foralternating currentoffrequencyf,the hysteresis curveistraversedftimes persecondandpowerisdissipated whichappearsasheatinthemagnetic material ofthecore.Fromequa­ tion(6.45)theworkdoneperunitvolumeinmovingfromonepointon thehysteresis curvetoanotheristheintegral, takenalongthecurve~ w=fHdB. (8.26) Itisreadilyseenthatinacomplete cyclethevalueoftheintegralisjust givenbytheareaenclosed bythehysteresis curve.Sincethiscurve istraversed oncepercycle,theenergydissipated riseslinearlywiththe frequency ofthealternating current. 8.9.Terrestrial magnetism Ithasbeenknownsincethesixteenth centurythatthereisasmall permanent magnetic fieldatthesurfaceoftheearth.Thegeneralnature ofthisfieldissimilartothatofauniformly magnetized spherewhose magnetization isslightlyinclinedtotheaxisofrotation. Attwopoints thelinesofforcearenormaltotheearth'ssurface. Theseareknownas the'magnetic poles';thenorthmagnetic poleattractsthe'north'poleofa suspended magnetorcompass needle,andthelatterismoreaccurately termedthe'north-seeking pole',sinceitisapoleofopposite signtothe earth'smagnetic pole.Ingeneral,themagnetic fieldatanypointonthe earth'ssurfacemakesananglewiththehorizontal, knownastheangle ofdip.Thedirection ofthehorizontal component iscalledthemagnetic meridian, andtheanglebetween thisandthegeographical meridian is theangleofdeclination. InEngland thesizeofthehorizontal component isabout0·18oersted (~14A/metre), andtheangleofdipis58°. Although itisaconvenient firstapproximation tothinkoftheearth asauniformly magnetized sphere,itmustberemembered thatthisim­ pliesthatthefieldoutsideitisjustthesameasthatofasmalldipole atthecentre,andnoimmediate deductions canbedrawnfromthe natureofthisfieldabouttheactualdistribution ofmagnetization within theearth.Themagnetic potential associated withtheearth'sfieldcan beanalysed inaseriesofspherical harmonics. Apartfromsmalllocalized distortions duetoiron-bearing minerals intheearth'scrust,thereisa 224 MAGNETIC MATERIALS AND [8.9 dipoletermwhichhasdecreased inmagnitude byabout5percentin thelasthundred years,whilethequadrupole andhighertermshave strongandfairlyrapidsecularvariations withlifetimes lessthana hundred years.Theselattertermshavenoconstant components and itisbelievedthatallthenon-dipole fieldcomponents wouldaverageto zerooverasufficiently longperiodoftime.Thevariation withtimeof thefieldatanyone pointalsocontains diurnalvariations whichare irregular andunpredictable. Thesearecausedbycurrents intheiono­ sphereduetosolarandlunarperturbations, anddaysofgreatmagnetic disturbance canoftenberelatedtoepochsofmaximum sunspots, the intensity showing asimilarll-yearcycle. Theoriginofthemainfieldismoredifficulttoaccountfor.Aplausible guessofthecomposition oftheinterioroftheearthmaybemadeby studying thecomposition ofmeteorites, thesun,stars,andotherplanets, andusingthedataonthedensityobtained fromthevelocityofseismic wavesthroughtheearth.Thelattershowthatthereisacentralcore, witharadiusof3473±4 km,whichisassumed tobeliquidsinceno transverse seismicwavesaretransmitted through it.Although this contains muchiron,thetemperature andpressure aretoohighforit tobeferromagnetic; itisassumed toconsistmostlyofliquidsilicates ofiron,magnesium, andcalcium, whichhaveanappreciable electrical conductivity athightemperatures. Thepresentviewisthatthemain partoftheearth'sfieldisduetoelectriccurrentsinthiscore,associated withconvective currents causedbyradioactive orchemical sources. Themathematics oftheprocess (energy source-+kineticenergyof fluid-+electrical energy)hasbeenstudiedbyElsasser, Bullard, and others,anditseemsprobable thatelectrical currents canbemaintained inthisway.Fordetailedaccounts reference shouldbemadetoChapman andBartels(1940),andElsasser (1950,1955-6). REFERENCES BrrrER, F.,1936,Rev.Sci.lnstrum. 7,479;7,482. --1937, ibid.8,318.--1940,ibid.11,373. CH.U'MAN, S.,andBARTELS, J.,1940,Geomagnetism (G.U.P.). ELSASSER, W.M.,1950,Rev.Mod.Phys.22,l.--1955-6,Am.J.Phys.23,590;24,85. FONER, S.,1959,Rev.Mod.Phys.30,548. MYERS, W.R.,1952,ibid.24,15. THOMAS, H.A.,DRISCOLL, R.L.,andHIPPLE,J.A.,1950,Phys.Rev.78,787. VIGOUREUX, P.,andWEBB,C.E.,1946,Electric andMagnetic Measurements (Blackie). MAGNETIC MEASUREMENTS 225 PROBLEMS 8.1.Thesusceptibility ofagramme moleofheliumgasis-2·4X10-11(m.k.s. units).Showthatthiscorresponds toavalueforthemeansquareradiusofeach electronic orbitintheheliumatomofl'22a~,whereao=0'528X10-10metreisthe radiusofthe:firstBohrorbitinthehydrogen atom. 8.2.Ifthemutualrepulsion ofthetwoelectrons intheheliumatomisneglected, thewavefunction ofeachelectron inthegroundstateis if1=(ZS/1TWo)i exp(-Zr/ao), where-eif12isthedensityofelectronic chargeatadistance rfromthenucleus, andZeistheeffective chargeoftheheliumnucleus. Showthatthiswavefunction leadstoavalueforthemeansquareradiusofeachelectronic orbitof T2=3a~/Z2. Verifythatagreement withthevalueof1·22a~intheprevious problem isobtained ifwetakeZasabout1'6(weshouldexpectittobelessthan2becauseeachelectron partially shieldstheotherfromthefieldofthenucleus). 8.3.When23·15gofNiCl 2aredissolved in100gwater,thedensityofthesolution is1255kg/m3.Showthatthemaximum heightofacolumnofthesolution which canbesupported bymagnetic forcewhenonesurfaceofthecolumnisinauniform induction of1weber/metre 2is3·0mm. (Susceptibilities oflkgofNiCl 2andwaterare+0·438X10-6and-0·0090X10-6 respectively; volumesusceptibility ofair=+0'4X10-6(m.k.s.units).) 8.4.Thesusceptibility ofagramme moleofNiK2(S04)2,6H20 is foundtobe 1.610-5T-1.Assuming thatthediamagnetic contribution isnegligible, andthat theonlyparamagnetic contribution comesfromtheNi++ion,calculate thesize ofthepermanent dipolemoment oneachNi++ion. (Answer: 3·0X10-23ampere-metre 2.) 8.5.AnironanchorringoflargemeanradiusRanduniform cross-section hasa gapofthickness dcutinit.Itiswoundwithasinglelayercoil.Overtherange underconsideration thepermeability oftheironis(I+a/H),whereHisthefield intheiron.Showthat,ifd<:R,fourtimesasmuchpowerisrequired tomaintain afield3ainthegapasisrequired forafield2a. 8.6.ItwasshownbyRayleigh thatatlowvaluesofthemagnetic induction the hysteresis loopwithtipsatBo, Hoand-Bo'-Hoisdescribed bytheequations B=JLJLoH+ta(~-H2) (upperhalfofloop) and B=JLJLoH-ta(~-H2) (lowerhalfofloop), where JL=Bo/JLoHo.Showthattheenergylosspercyclerepresented bythearea oftheloopis W=~aHg ineachunitvolumeofthesubstance. ThisrelationisvalidonlyforlowvaluesofBo(inironbelowabout0·05weber/ metre2).AthighvaluesWvariesapproximately asB~·6,anempirical lawdueto Steinmetz. 851110 Q 226 MAGNETIC MATERIALS 8.7.Deduceequation (8.20),bythellileofequivalent magnetic shellsandintegra­ tionofequation (5.55),orfromintegration ofthemagnetic induction duetothe 'freepoles'(polarization charges) onthesurfaces ofthecones. [Wtn]!b{l+(l2+ a2)!} H=87Tl2ploge(bJa) logea{l+(l2+b 2)!}" Notethatthisisoftheform(W,\Jpr)! timesafactordepending ontheshape ofthesolenoid (as givenin§8.5),sinceA=ntJ2l.8.8.Anair-cored solenoid oflength2lisconstructed fromnequally-spaced Bitterpancakes eachofthickness t,innerradiusa,outerradiusb,andresistivity p.Ifnislarge,andWisthepowersupplied, showthatthefieldHatthecentre isapproximately 8.9.ItwasshownbyMaxwell thatstressesarepresentinthemagnetic fieldwhich canberepresented byastresstensorsimilartothatintheelectrostatic case(§1.7) ifE,Darereplaced byH,Brespectively. Showthattheforceequation (8.25)can beobtained byconsidering thestressesontheendofthespecimen. 9 ALTERNATING CURRENT THEORY 9.1.Forcedoscillations IN§6.3weconsidered thetransient currents whichflowwhenacapaci­ tor,initially charged, isallowedtodischarge through acircuitcon­ tainingbothinductance andresistance, andfoundthatanoscillatory currentofdecaying amplitude flowedthroughthecircuitprovided that theresistance inthecircuitwasnottoohigh.Thetheoryofsuch transients isduetoLordKelvin,anditscorrectness wasverifiedbyearly experimenters. Withtheinvention ofthedynamo and,later,the electronic vacuum tube,itbecame possible toproduce continuous alternating currents whosefrequency ofoscillation maybeanything uptoabout1011cis.Inthesimplest casetheformofthecurrentis thatofasimplesinewave,andmaybewrittenas 1=10coswt, where1isthevalueofthecurrentattimet.Themaximum valueof1 is10,knownasthe'amplitude' ofthecurrent,andthefrequency of alternation isf=(wI27T)cis.Thecurrentgenerated byadynamo or otherdevicemayormaynothaveasimplesinusoidal waveform,but whatever theactualwaveformitmayberesolved byFourieranalysis intoasumofsineandcosinetermswhosefrequencies areintegral multiples ofthefundamental frequency. Thisfrequency isgivenby theinverseoftheperiodbetween instantsatwhichthewholewave formisrepeated. Sincethebehaviour ofacircuitisingeneraldifferent atdifferent frequencies, itisnecessary toconsider eachcomponent of suchaFourierseriesseparately, andinthetheorythatfollowsweshall assumethatthewaveformissinusoidal, varyingatonefrequency only. Exceptinnon-linear circuitswherethebehaviour ofacircuitelement depends onthesizeofthecurrentorvoltageappliedtoit(Le.elements inwhichtheamplitudes ofcurrentandvoltagearenotlinearlypropor­ tionaltooneanother) anynon-sinusoidal fluctuations mayberesolved intotheirFourier components, andtherequired solution issimplya sumofsuchcomponents. InthecircuitofFig.9.1,avoltage Yocoswtisappliedtoaninductance, aresistance, andacapacitance inseries.If1isthecurrentflowingat ------- -------- 228 ALTERNATING CURRENT THEORY [9.1 (9.1)anyinstant,thee.m.f.setupintheinductance is-L(o,1jo,t), andthe voltagedropacrossthecapacitance isqjO,whereqisthechargeonthe capacitor. Wehavetherefore forthecircuit Yocoswt-L(o,1jo,t)-qjO =R1 0,1Ldt+R1+qjO =Yocoswt. or L R I f'\...,V=Vocoswt c+q -q FIG.9.1.Forcedoscillations inacircuitcontaining L,C,R. Nowtherateofincreaseofthechargeonthecapacitor o,qjdt=1,the currentflowing,andhencebydifferentiation wehave d210,11 .Ldt2+Rdt"+O=dVjo,t=-wYosmwt. (9.2) Thisisadifferential equation whosesolution consists oftwoparts. Thefirstofthese,knownastheComplementary Function, isfoundby solvingtheequation obtained bysettingtheright-hand sideequalto zero;thatis,itisasolutionofequation (6.27),andhenceisoftheform givenbyequation (6.28).Thissolution represents atransient flowof currentproduced bytheactofapplying thee.m.f.Vocoswt,itbeing assumed thatthisstartstoactattheinstantt=O.Inallpractical applications thistransient currentdecaysrapidlyinamplitude, owing totheexponential termexp(-tRj2L) andbecomes negligible withina fewsecondsorlessofthecircuitbeingclosed.Ifconditions aresuch thatthetransient currentisoscillatory, itsfrequency isthenatural frequency determined bythevaluesofL,0,andR,andnotthatofthe appliede.m.f. Thesecondpartofthesolution isknownastheParticular Integral, andforequation (9.2)itmaybewrittenas 1=(YojZ)cos(wt-ef», (9.3) 9.1] whereALTERNATING CURRENT THEORY 229 (9.4) andthephaseangle1>isgivenby (9.5) Thisisalsoknownasthe'steadystate'solution, sinceitgivesthe currentflowatanytimeafterthetransient currenthasbecomenegli­ gible.Thefrequency ofthecurrentisthesameasthatoftheapplied e.m.f.,sothatthecircuitisin'forced'oscillation. Ingeneralthephase ofthecurrentisdifferent fromthatoftheappliedvoltage, exceptwhen wL-1/wC =O.Thisoccurswhen .w=Wo=l/,J(LC), (9.6) andhenceisthesameastheangular frequency ofnaturaloscillation ofthecircuitintheabsenceofanydamping resistance. Thecircuitis thensaidtobein'resonance', andtheamplitude ofthecurrentis,by equations (9.3)and(9.4),amaximum. Thequantity Zinequation (9.4)iscalledtheimpedance ofthecircuit, andatresonance thevalueofZisjustequaltoR,thetotalresistance in thecircuit.Atotherfrequencies thevalueofZisrelatedtothequanti­ tiesR,L,andC,butisnotgivenbythesimpleadditive relationthat holds(§3.4)forresistances inseries.Thereasonforthisisthatthe voltages acrossthedifferent elements arenotinphase,andthetotal voltageamplitude istherefore notjustthesumoftheindividual ampli­ tudes.Inthefollowing sections weshallseehowthisdifficulty canbe overcome bytheintroduction ofcomplex numbers torepresent theim­ pedances. Theuseofsuchcomplex impedances enablesustoapply Kirchhoff's laws(§3.4)toalternating current(a.c.)networks, andwe canfindthesteady-statevaluesofthecurrentandvoltageinanybranch without havingtosolveadifferential equation. Animportant consideration ina.c.circuitsistherateofdoingwork. AtanyinstanttherateWatwhichworkisdonebythegenerator in Fig.9.1is W=VI=(Yocoswt)xCfo/Z)cos(wt-:p) =(VUZ){cos2wtcos4>+coswtsinwtsin4>}. TofindthemeanrateWofdoingwork,thisexpression mustbeaveraged overoneormoreperiodsofoscillation. Nowthemeanvalueofcos2wt averaged inthiswayisjust!,whilethatofcoswtsinwtisO.Hencethe 230 ALTERNATING CURRENT THEORY [9.1 meanpowerdrawnfromthegenerator is W=t(v8/Z)cos¢> =!VoIocos¢> =F8Zcos1>, (9.7) where10=Vo/Zistheamplitude ofthecurrentgivenbyequation (9.3). Fromequation (9.5)itisreadilyshownthatcos¢>=RIZ,andtheex­ pression forthemeanpowercantherefore bewritten W=F8R. (9.8) Nowthemeanrateatwhichpowerisdissipated intheresistance ofthe circuitistheaverage valueofRIgcos2(wt-¢»takenoveracomplete period,andthisisjustequalto!RIg.Henceallthepowerdelivered by thegenerator, averaged overaperiod,isdissipated intheresistance of thecircuit.Thetermincoswtsinwtintheexpression forVIrepresents workdonebythegenerator inincreasing theenergystoredinthein­ ductance andcapacitance; sincetheproduct coswtsinwt isasoften negative aspositive, thisworkisreturned tothegenerator inother partsofthecycleandnomeanpowerisdrawnfromthegenerator for thispurposeinthesteadystate.Ofcourse,powerWasdrawnfromthe generator initiallytoprovidethestoredenergy,andthisisrepresented bythetransient current; whenthishasdecayed, themeanstoredenergy remains constant andnofurtherworkisdonebythegenerator onthe averageexcepttosupplythatdissipated intheresistance. Sincetherateatwhichpowerisdissipated isproportional tothe squareofthecurrent,itisconvenient tospecifytherootmeansquare valueIofthecurrent, definedbythefactthat (1)2=<12), wheretheaverageofthesquareofthecurrentistakenoverawhole period.TherootmeansquarevalueofthevoltageVmaybedefined inasimilarway,(V)2=<V2).Ifthewaveformissinusoidal, theroot meansquarevalueisjust(1/-v2)timestheamplitude, andwecanwrite equation (9.7)as W=(V2/Z)COS ¢>=vIcos¢>=I2zcos1> (9.9) andtheratioW/VI=cos¢>iscalledthe'powerfactor'ofthecircuit. Itrepresents thefraction oftheproductvIwhichisdissipated as Jouleheat.Ifthecircuitbehaves asapureresistance, asoccurswhen theresonance condition (9.6)isfulfilledinthecircuitofFig.9.1,the powerfactorisunity,whileifthecircuitcontains noresistance thepower factoriszero. Thegeneralpracticeina.c.circuitsistospecifytherootmeansquare valuesofthecurrentandvoltage, anditshouldbeunderstood that 9.1] ALTERNATING CURRENT THEORY 231 anyvaluesquotedarerootmeansquarevaluesunlessthecontrary is specifically stated. 9.2.Useofvectorsandcomplex numbers Thevaluesofthevoltageacrosstheindividual components ofthe circuitinFig.9.1willnowbeconsidered inmoredetail.Forthispurpose weassumethatacurrentI=10coswtflowsthroughthemallinseries, asinFig.9.2.Thevoltages acrossthethreecircuitelements are: acrossR:V=1R=R10coswt acrossL:V=L(d1jdt)=-wL10sinwt =wL10cos(wt+i 7T) across0:V=qjO=(ljwO)losinwt =(ljwO)locos(wt-!7T) whereinthelastcasewehaveusedtherelationq=JIdt.Itwillbe seenthatthevoltageacrosstheresistance isinphasewiththecurrent, l=locoswt R L FIG.9.2.CurrentIflowingthroughR,L,0inseries. thevoltageacrosstheinductance leadsthecurrentinphaseby90°, whilethatacrossthecapacitance lagsbehindby90°.Wemayrepresent thesevoltagesbyvectorssuchthat,ifthevoltageacrosstheresistance isrepresented byavectordrawnparalleltothex-axis,thatacrossthe inductance isrepresented byavectorparalleltothey-axis,andthat acrossthecapacitance byavectorparalleltothelatterbutinthe opposite sense.Thelengthsofthevectorsareproportional toR,wL, andl/wO,respectively. Ifwerequirethevoltageacrosstwooftheelements, saytheresistance andinductance, itmaybefoundbyaddingthetwoindividual voltage vectorstogether, asinFig.9.3(a).Forthetotalvoltagewillbe V=1R+L(d1/dt) =R10coswt-wL10sinwt =10(R2+w2L2)lcos(wt+ef» =10Zcos(wt+ef», (9.10) wheretanef>=wL/R.FromFig.9.3(a)itwillbeseenthat(R2+w2L2)i isjustthelengthofthehypotenuse ofthetriangle, whilethephase angle ef>isjusttheanglebetween thevectorsRand(R2+ W2L2)l.Thus themagnitude ofthetotalvoltageisrepresented inamplitude bythe hypotenuse, anditsphaserelativetothecurrentisgivenbytheangle 232 ALTERNATING CURRENT THEORY [9.2 through whichthisvectorisrotatedwithrespecttoR.Similarly itcan beshownthatforaresistance andcapacitance inseriesthetotalvoltage maybefoundbyaddingthevectorsRand-(l/wO) asinFig.9.3(b), whilethecaseofaresistance R,inductance L,andcapacitance 0allin seriesisrepresented byFig.9.3(c).Heretheamplitude oftheresultant vectoris{R2+(wL-1/wO)2}l', whichisjustthevalueofZ(equation (9.4))andthevoltageleadsthecurrentbythephaseanglecp,where tancp=(wL-1/wO)/R, asinequation (9.5).Theterm'impedance' hasalreadybeenintroduced forZ,whichrepresents theratiooftheamplitude ofvoltagetocurrent R (a)(b) 1 roOI -roO 11()lC)2) 1\2-\-\(~1- 'b""''';' cP R (e)roL eiwt=coswt+jsinwt,FIG.9.3.Vectordiagram forimpedance. (a)R,Linseries;(b)R,0in series;(e)R,L,0inseries.Thevoltagevectorsarethesarnoastheim­ pedance vectorsifthecurrentisrepresented byaunitvectorparalleltoR. forthewholecircuit.Thequantities wLandl/wOassociated with inductance andcapacitance respectively areknownasreactances, and areusuallydenotedbythesymbolX.Thusthetotalreactance ofthe circuitofFig.9.2isX=(wL-1/wO), andtheimpedance isgivenbythe vectorsumofRandX,represented bymutually perpendicular vectors. ThusZ=(R2+X2)t, andthephaseangleisgivenbytancp=X/R. Thevectorrepresentation oftheimpedance issimilartotherepre­ sentation ofacomplex number ontheArgand diagram. Usingthe relation wherep =-1,wemayreplaceourcosineandsinefunctions bycom­ plexexponentials ontheunderstanding thatweareinterested onlyin 9.2] ALTERNATING CURRENT THEORY 233 therealorimaginary partsrespectively. Thenwemaywritethecurrent 10coswtas10fJl(eiwt),andquantities suchasdI/dtorq=JIdtbecome dI/dt=Io(d/dt)coswt =IofJl(ieiwt) =IofJl(jweiwt) and q=10Icoswtdt=10fJl(Ieiwtdt)=10fJlC~eiwt) respectively. HencethetotalvoltageacrossthecircuitofFig.9.2is V=RI+L(dI/dt)+(I/C)IIdt =(Reiwt+jwLeiwt+~eiwt)IoJWC =(R+jwL+j~dIoeiwt =(R+jwL+j~c)I. (9.11) Ifthecurrentis10coswt,thentofindthevoltagewetaketherealpart of(9.11).Thus V=10fJl(Rcoswt+jRsinwt+jwL coswt-wLsinwt-·1)-~Ccoswt+ wCsinwt =10(Rcoswt-wLsinwt+ wIcsinwt), wherethethreecomponent termsarejustthevoltages acrossthethree circuitelements derivedatthebeginning ofthissection. Similarly, if thecurrenthadbeenIosinwt =10J(eiwt),thevoltagewouldbefound bytakingtheimaginary partof(9.11),giving V=Io(Rsinwt+wLcoswt- wICcoswt). Theimportance ofequation (9.11)liesinthefactthatitshowswe mayrepresent theinductance andcapacitance byimpedance operators jwLandI/(jwC)respectively, andtheseoperators maybeaddedtoone anotherwhentheelements areinseriesinasimilarwaytoresistances. Thephaseofthevoltage,whichleadsthecurrentbyi1Tintheinductance andlagsby~inthecapacitance, istakencareofbythepresence ofj. If,asisusualintheArganddiagram, realquantities arerepresented byvectorsdrawnparalleltothex-axis,andimaginary quantities by vectorsparalleltothey-axis,thenthecomplex impedance operator isas 234 ALTERNATING CURRENT THEORY [9.2 showninourvectordiagram Fig.9.3(c).Thecircuitimpedance isgiven bythemodulus ofthecomplex impedance operator, andthephase anglebyitsargument.Ifwewritetheimpedance operator as Z=R+jwL+~ =R+jX=Zeiq,JWO thenZ=(R2+X2)1, andtanep=XIR,asbefore.Then V=ZI=Zeiq,Ioeiwt=ZIoei(wt+<p>. If1=Iocoswt, therealpartofthisgivesV=ZIocos(wt+ep), asin equation (9.10).If,ontheotherhand,thevoltageisgivenasYocoswt, asin§9.1,thecurrentisfoundbytakingtherealpartof I=VIZ=Yoeiwtl(Zeiq,) =(YoIZ)ei(wt-q,>, whichgives I=(YoIZ)cos(wt-ep), asinequation (9.3). Fromanextension ofthetreatment givenaboveitmayreadilybe shownthattheimpedance operator foracircuitconsisting ofanumber ofimpedances Zl>Z2'"''Zninseriesis (9.12) (9.13)Thecorresponding formula foranumberofimpedances inparallelis foundbynotingthatthevoltageacrosseachisthesame.Thecurrent throughtheimpedance ZkisVIZk'andthetotalcurrentisthesumof anumberofsimilarterms.Hencethenetimpedance isgivenby I I I I IfV=Z=Z+Z+'''+z'1 2 n Whenanumberofelements areinparallelitisgenerally convenient to workintermsofthereciprocal oftheimpedance, knownastheadmit­ tanceY.ThusI=VIZ=YV,andequation (9.13)maybewrittenas (9.14) IngeneralYiscomplex; thusY=G+jS,whereGiscalledthecon­ ductance andSthesusceptance. Forthethreesimplecircuitelements wehaveconductance G=IIR;susceptance ofaninductance is S=(jwL)-1=-jI(wL);susceptance ofacapacitance isS=jw0;but notethatforacircuitcontaining bothresistance andreactance these simplereciprocal relations donothold.For Y=liZ=I/(R+jX)=(R-jX)/(R2+X2), 9.2] ALTERNATING CURRENT THEORY 235 sothatY=IYj=jZ-l/=Z-l,but 0=R/(R2+X2) =Ycos4>, 8=-X/(R2+X2)=-YSin4>} andsimilarly . R=0/(02+82)=Zcos4>,X=-8/(02+82)=Zsin4> (9.15) Inordertocalculate thepowerconsumed inacircuitwemustuse therelationW=9t'{V}X9t'{I};notethatthisisnotthesameas9t'{VI}, (:,"' OJ ~ '/- ISwC't II -\ R=l/G (a) (0)C FIG.9.4.(a)Imperfect capacitor withconductance G,repre­ sentedbycapacitance Cshunted byresistance R=I/G; (b)corresponding admittance diagram. for9t'{Voexpjwt}x9t'{I oexpj(wt-4>H =Vocoswtxlocos(wt-4», which doesnotequal9t'{Vol oexpj(2wt-4>H =Volocos(2wt-4». However, the meanenergyWdissipated isgivenbyanyoftherelations (where17 and[aretherootmeansquarevalues) W="V29t'{Y}=1720=V2R/(R2+X2) =[29t'{Z}=[2R=[20/(02+82), (9.16) ascanbeverifiedbycomparison withequation (9.9). Theunitofreactance andimpedance isthesameasthatofresistance, theohm;whenwisexpressed inradians/second, Linhenries,and0in farads,thecorresponding reactances areinohms.Similarly, theunit ofadmittance andsusceptance isthesameasthatofconductance; the reciprocal ohm,ormho. Wemayillustrate theuse.ofadmittance byconsidering thecaseof alossycondenser, represented byapurecapacitance 0shunted bya resistance R,asinFig.9.4(a).Theadmittance operator is Y=(I/R)+jwO =O+jwO. 236 ALTERNATING CURRENT THEORY [9.2 Thismayberepresented bytheadmittance diagram showninFig.9.4(b). Thephaseangleoftheadmittance isgivenbytanif;=wOIG=wOR; byuseofequation (9.15)orbyplotting thecorresponding impedance diagram itisreadilyshownthattanif;=-tanr/J, sothatif;=-r/J. Hencethepowerfactorcosr/J=cosif;=(1+w202R2)-t.Foraperfect capacitance, asforaperfectinductance, thepowerfactoriszero;in practice thisisnotquitetrue,sinceatradiofrequencies powerislost through eddycurrents intheplatesandimperfections inthedielectric. Thelatterarecausedbyhysteresis effectswhichcausethevectorDto lagbehindthevectorEinthedielectric asinthecorresponding case ofBandHforaferromagnetic substance. Thequalityofadielectric isexpressed intermsofits'losstangent', tan8,where 8=iTT-if;=iTT+r/J; hence tan8=cotif;=-cotr/J=GI(wO)=(wORt1. Inanoscillating electricfieldthehysteresis loopistraversed onceper cycle,sothatthelossconductance Gisproportional tothefrequency. Thevalueoftan8shouldtherefore beindependent offrequency, and thisisgenerally trueexceptforstrongly polardielectrics (seeChapter 17);foragooddielectric, suchasquartz,thevalueoftan8isabout10-4• (9.17) Z=R+j(wL-1/wO),9.3.Tunedcircuits ThecircuitofFig.9.1consisting ofaninductance, resistance, and capacitance inseriesisknownasa'seriesresonant' circuit.Theim­ pedance operator is and,aspointedoutin§9.1,themodulus ofthishasaminimum value ifthefrequency isadjusted tomakewL=llwO.Thisistheresonant frequency ofthecircuit,givenbyequation (9.6) wo=1/../(LO). Atthisfrequency thecurrentthroughthecircuitisamaximum andis inphasewiththeappliedvoltage, sinceZisreal;themagnitude ofthe currentisVIR.Thevoltageacrosstheresistance Risthusequaltothe voltageacrossthewholecircuit,andthevoltages acrosstheinductance andcapacitance aretherefore equalandexactly1800outofphasewith oneanother, makingthevoltageacrossthetwozero.Thevoltageacross thecapacitance aloneatresonance is 1}.3] ALTERNATING CURRENT THEORY 237 andhence,using(9.6), 1JL wLIVoJVI=(woGR)-l= R(j=~=Q, (9.18) whereQisthe'quality factor'ofthecircuitasdefinedin§6.3.Itis alsoknownasthe'circuitmagnification factor',sincefromequation (9.18)weseethatitequalstheratioofthevoltageacrossthecapacitor tothevoltageacrossthewholecircuit.Thusthetunedcircuitactsas atransformer; sincethecurrentthrough thecapacitor isamaximum atresonance, theratioofthevoltagesVo/Visalsoamaximum atthis point.Thevoltageacrosstheinductance Lisequaltothatacrossthe inductance atresonance, butingeneralRisassociated withLand onlythevoltageacrossthecombination (R+jwL) canbemeasured in practice. Thebehaviour ofthecircuitatfrequencies nearresonance isof particular interest. Thereactance ofthecircuitchangesratherrapidly inthisregion,passingthrough zeroattheresonant frequency because tworatherlargequantities, wLandl/(wG),cancelatthispoint.For frequencies nearresonance wemaywritew=wo+Swandthenthe impedance operator becomes Z=R+j(woL+8wL- w~G+:td=R+2j8wL, (9.19) since1j(wgG)=L.Onintroducing Q,thismaybewritten Z=R{1+2j(Sw/w o)Q} (9.20) andtheimpedance is (9.21) showingthatifagivencurrentflowsthrough thecircuit,thevoltage acrossthecircuitrisesbyafactor"';2whenthefrequency deviates from theresonant frequency byafraction Sw/wo=±1/(2Q). Similarly, if agivenvoltageisappliedtotheci~cuit,thecurrentthroughitfallsto 1j..J2ofitsmaximum valuewhenthefrequency deviatesbythisamount; thepowerdissipated inthecircuitfallstoone-halfofthemaximum, and thesepointsaretherefore oftenreferredtoasthe'half-power' points. Theformofequation (9.21)showsthatthevariation ofimpedance with frequency givesacurveofuniversal shapebutwhosespreadinfrequency isdetermined bythevalueofQ.SinceingeneraltheratioofIJOItoIVI is(wGZ)-l, thevoltagestep-upobtained fallssharplyawayoneither sideoftheresonance point.Thusthecircuitisselective initsresponse 238 ALTERNATING CURRENT THEORY [9.3 tosignalsofdifferent frequency; its'selectivity' isdetermined bythe valueofQsinceQistheratiooftheresonant frequency tothedijierence offrequency between thetwohalf-power points: Q=wo/(2ow)=fo/(2of). Aplotoftheamplitude ofthecurrentinaseriesresonant circuitwhen asignalofgivenvoltagebutvaryingfrequency isappliedtoitisshown inFig.9.5.Sincethevoltageacrossthecapacitance Va=I/(wO),and '~---T-' ._.-jf------ 2 3 ---~ (Qbf/fol-0,4(111maxl r I ---~-'---T-L-3 -2 -1 0 FIG.9.5.Variation ofcurrentinseriestunedcircuitnearresonance. nearresonance thevariation inIisverymuchmorerapid(ifQislarge) thanthatofw,thevoltagestep-upobtained isalsogiventoagood approximation byacurveofthesameshapeasinFig.9.5. Theapproximation usedinthededuction ofequation (9.20)isthat ow/wo~1.Atfrequencies wherethisapproximation isnotvalid,the variation ofZstillfollowsauniversal curve,sincewemaywrite (9.22) wherex=(w/wo)=(f/fo).Sincethereactivepartoftheimpedance variesas2Q(ow/w o)nearresonance, thecurrentfallstoquitesmall values-beforetheapproximation ow/wo~1becomes invalid,provided 9.3] ALTERNATING CURRENT THEORY 239 thatQisfairlyhigh.Inanordinary tunedcircuittheresistance isthat ofthewireusedinwindingtheinductance coil;ifweusealargerin­ ductance inordertoincreasethevalueofQatagivenfrequency, the resistance goesupandsodoestheself-capacitance betweenthedifferent partsofthecoil.Thelattersetsupanupperlimittothesizeofthecoil sinceatsomepointthecoilwillresonateatthedesiredfrequency with­ outanyexternal capacitance, becauseofitsself-capacitance. Asarough guideonemaytaketheQofacoildesigned toresonateatanaudio­ frequency asabout20;atfrequencies oftheorderof1Mc/saQof100­ 200maybeobtained; atfrequencies of109and1010ciswheretuned I L R G r FIG.9.6.Seriesresonant circuitwithlossycapacitor. transmission linesandwaveguide cavities(seeChapters 11,14,and15) areusedinsteadoflumpedcircuits,valuesof1000to10000areobtained forQ.Thusinmostradioworkwithtunedcircuitsequation (9.20)is agoodapproximation. Hitherto wehaveassumed nolossinthecapacitor;ifweincludesome lossthecircuitbecomesthatshowninFig.9.6.Theimpedance operator isnow Z=R+jwL+.c1I=R+jwL+(r-jwCr2)J(I+w2C2r2).JW+1r Ifthepowerfactorofthecapacitor issmall,wCr~1,andtheim­ pedance operator isapproximately (cf.Problem 9.3) Z=R+jwL+(w 2C2r)-1-jl(wC). (9.23) Thustheresonance frequency isunaltered inthefirstapproximation, buttheresistance ofthecircuitisincreased. ThevalueofQisnow givenby R1IJQ={R+(w~ C2r)-1}/(woL)=-L+-C =IJQL+tanS, WoWor whereQL=woLJRistheQofthecircuitduetothelossintheinduc­ tancealone.Sinceforagoodcapacitor tanS<10-3,whileforlumped circuitswithinductive coils(asdistinctfromtransmission linecircuits) l/QL~10-2,lossinthecapa.citor cangenerally beneglected. 240 ALTERNATING CURRENT THEORY [9.3 Parallelresonant circuits Aparallelresonant circuitconsistsofacapacitance shunted across aninductance+resistance, asinFig.9.7.Theadmittance operator for thecircuitis Y= .0__1__.0R-jwL JW+R+jwL-Jw+R2+W2L2' I rvV G FIG.9.7.Parallel resonant circuit.(9.24) L Wewilldefinetheresonance pointforthiscircuitasthepointatwhich theadmittance isreal.Then,equating theimaginary termstozero gives R2+wgL2=LjO (9.25a) andhence Wo=(10)l(1- R~O)i=(L1di(1-~2)1, whiletheadmittance atthispointis(9.25b) (9.26) Weseethattheresonance frequency isnotquitethesameasintheseries tunedcircuit,butthedifference issmallandcanusuallybeneglected if Qislarge(ifQ=100,thefractional difference inWois110-4).Because ofthisdifference, thealternative formulae woLjRand1j(woOR)forQ arenotexactlyequivalent to(LjO)ljR,buttheyareagoodapproxima­ tionifQislarge.Thereciprocal oftheadmittance atresonance iscalled the'parallel resistance' ofthecircuit,andisequaltoQ2Rwithoutany approximation. Thedefinition ofresonance asthepointatwhichtheadmittance is realistoacertainextentarbitrary, butisveryconvenient touse;ithas thefurtheradvantage ofbeing unique. Theadmittance ofthecircuit atthispointisverysmall,butisnotnecessarily aminimum.Ifresonance isdefinedasthepointatwhichYisaminimum, thentheresonance con- 9.3] ALTERNATING CURRENT THEORY 241 ditiondepends onwhatisbeingadjusted tomakeYaminimum.Ifthe capacitance isaltered,andotherquantities arekeptfixed,then,by differentiating Y=IYIwithrespectto0,itcanreadilybeshownthat Yisaminimum whenthevalueofthecapacitance satisfies equation (9.25a).Ontheotherhand,ifLortheappliedfrequency isaltered, slightlydifferent resonance conditions areobtained; thefractional dif­ ferenceis,however, onlyoforderI/Q2,andsocanbeneglected ifQis large.Thecalculation involved infindingtheseresonance conditions by differentiation isverytediousincomparison withthesimpledeviceof settingtheimaginary partofYequaltozero,andthesimpledefinition ofresonance asthepointatwhichthecircuithasunitypowerfactor (i.e.theadmittance isreal)hasmanyadvantages. Itisusefultohaveanapproximate expression forYnearresonance similartoequations (9.19-21) fortheseriesresonance circuit.Writing w=wo+8w, wehavefromequation (9.24) Y=R2+~~L2{1- ~~~~~2}+j{8wO- R2::~L2(1- R;+~~~2)} =R~{1-28w(woOL)}+2j8w02Lw~, wherewehaveusedequation (9.25a).Inthetermoforder8wwemay putw~LO=I,giving Y=RO{I_28w}+2j8wO.L Wo Ingeneralthesmallchangeintheconductance canbeneglected, sothat Y=(RO/L)+2j8wO =(RO/L){1+2jQ(8w/wo)), (9.27) showing thatthebehaviour oftheadmittance nearresonance forthe parallelresonant circuitissimilartothatoftheimpedance oftheseries tunedcircuit.Theselectivity isthesameinthatYrisesbyafactor"';2 whenthefractional deviation ofthefrequency is (8f/fo)=(8w/wo)=±1/(2Q). Thusthecurrentdrawnfromthevoltagegenerator isaminimum at resonance (apartfromthesmallcorrections whichcanbeneglected when Qislarge).Thecurrentineachofthearmsislargerthanthatdrawn fromthesourcebyafactornearlyequaltoQ.Thustheratioofthe currentthroughthecapacitance tothatdrawnfromthegenerator is Jl0/ll=(woOV)/(ROV/L) =woL/R=Q (9.28) inourapproximation; thisrelationiscomplementary tothatgivenby 851110 R 242 ALTERNATING CURRENT THEORY [9.3 equation (9.18).Thatequation showedthattheseriesresonant circuit canbeusedasatunedtransformer wheretheoutputvoltageacrossth condenser islargerbyafactorQthanthatinjectedinserieswiththein ductance. Theimpedance measured attheoutput(e.g.byfindingth currentdrawnfromagenerator appliedacrossthecapacitance asi Fig.9.7)isQ2Ratresonance (byequation (9.26))whereastheimpedanc oftheseriestunedcircuitisR.Thustheratiooftheimpedances isQ2,th squareofthevoltageratio.This,whichisacharacteristic ofalltrans formers, canbeseenasfollows:ifweapplyagenerator ofvoltage totheseriestunedcircuit,thecurrentdrawnfromitisI;tohaveth samecurrentflowingthrough thecapacitance intheparallelresonan circuitformedfromthesameelements, wemustapplyagenerator 0 voltageQVacrossthecapacitance, andthecurrentdrawnfromitw' beIIQ.Thepowerdissipated inthecircuitisVI,thesameineachcase, Theimportance ofthequalityfactorQindetermining thepropertie ofaresonant circuitcanbereadilyappreciated fromthefollowing sum mary.Forasimpleresonant circuitasconsidered hitherto: (a)Q=(LjO)ijR ~woLjR ~(woOR)-l; (b)Qisequaltothevoltagestep-upobtained byusingthecircuita atunedtransformer; (c)theparallelresistance isQ2timestheseriesresistance; (d)thefractional frequency difference (281jlo)between thepointsa whichtheimpedance oradmittance changesbyafactor"';2 is1jQ (e)Q=17j>',where>'isthelogarithmic decrement offreeoscillation inthecircuit(see§6.3); (1)inforcedresonance, WoX(storedenergy) Q=rateofenergydissipation ThislastrelationcanbeusedtodefineQinmorecomplicated resonan circuits,andinotherresonant systems suchaswaveguide cavities(se Chapterll)wherethevaluesofL,0,Rcannotbespecified. Forth seriesresonant circuitofFig.9.2itisreadilyshownthatthisdefinitio agreeswiththatin(a)above.Forthestoredenergyatanyinstant' tLI2+lqHO =ILI~cos2wot+l{I~j(w~ 0)}sin2wot =lLIgatresonance, whilethemeanrateofenergydissipation istRI~.Hence Q=wo(tLI~)j(tRI~) =woLjR. 9.4] ALTERNATING CURRENT THEORY 243 9.4.Coupled resonant circuits Twocircuitsaresaidtobecoupledtogetheriftheyhaveacommon impedance. Theimpedance maybearesistance, inductance, orcapaci­ tance,andmaybeapartofeachcircuit,asintheexample shownin Fig.9.8(a),orconnected between thetwocircuitsasinFig.9.8(b). (a) (b) FIG.9.8.Typesofcoupledcircuits: (a)withcommon impedance (Oa);(b)withimpedance (03)connected between them. Twocircuitsmayalsobecoupledtogetherifoneofthemisinanelectric ormagnetic fieldsetupbytheother;forexample, theoscillating mag­ neticfluxduetoacoilinonecircuitmayinduceavoltageinacoilin thesecondcircuit,sothatthereisamutualinductance betweenthetwo circuits. InFig.9.9tworesonant circuits1and2arecoupledtogether bya mutualinductance Mbetween thecoilsL1andL2•Thevoltage ~ ALTERNATING CURRENT THEORY 244 appliedtothefirstcircuitproduces acurrent 11>andthisinducesavoltag M(dIljdt) inthesecondary circuit.Forasimplesinusoidal wavefor thisvoltagemaybewrittenasjwM11'wherejwMistheimpedanc operator forthemutualinductance. Similarly, if12isthecurrentint Fro.9.9.Tworesonant circuitscoupledbymutualinductance M. secondary, avoltagejwMI2willbeinduced intheprimary. thetwocireuitswehave Ti=IlZl+~wMI2}, 0=12Z2+JwMll whereZl'Z2aretheimpedance operators fortheprimaryandseconda circuitsintheabsenceofthemutualinductance. Oneliminating 12w have (w2M2)Ti=IIZl+--z;. Thequantity (w2M2jZ2) iscalledtheimpedance 'reflected intoth primary circuit'. Forthesecondary circuit,elimination of11gives jwM (w2M2)---Ti =12Z2+-- ,Zl Zl showingthatthecurrentflowisthatproduced byanapparent voltag -(jWMjZl)Ti working intoZ2plustheimpedance 'reflected intoth secondary circuit'(w2M2jZl)' Toinvestigate thebehaviour oftheprimary circuitimpedance i moredetailwewriteZl=Rl+jXl,Z2=R2+jX2.Then TijIl=Rl+jXl+w2M2j(R2+jX2) =Rl+R2W2M2j(R~+X~)+j{Xl-X2W2M2/(R~+Xm =Rp+jXp- (9.3) 9.4] ALTERNATING CURRENT THEORY 245 Thecurrentandvoltageintheprimary areinphaseifXpiszero.This requirement issatisfiedifXl=X2=0,andinthefollowing analysis weshallassumethattheelements oftheprimaryandsecondary circuits arethesame,sothatRI=R2,Xl=X2atallfrequencies (thelatter impliesL1=L2,01=O2),ThenXpiszerowheneither X=°orX2=w2M2_R2. (9.33) Thesecondoftheseconditions canbefulfilled onlyifwM>R.If wM=R,thethreerootsareidentical, andifwM<Ronlyonereal rootexists. Thesignificance ofthesethreerootsbecomes apparent whenwe examine thebehaviour ofthesecondary current. When Xl=X2=X=0, fromequation (9.31)wehave -jwoMJ;. 12=R2+w~M2' (9.34) where Wo=(LO)-iistheresonance frequency ofeithercircuitbyitself. Ifwecanvarythemutualinductance M,thenthevalueof12atthis frequency risesasMisincreased fromzero,passesthrough amaximum valueof-jJ;./2R whenwoM=R,andthenfallsagain.Atthemaxi­ mumtheimpedance reflected intotheprimary fromthesecondary is justequaltoR,sothatthesecondary circuitis'matched' totheprimary circuit,andthepowerdissipated inthesecondary circuitisamaximum atthispoint.Fromequations (9.30)and(9.34)itcanbeseenthatthe currents intheprimaryandsecondary circuitsarebothequaltoY:t./2R atthispoint,buttheydifferinphasebyt1T. Atthesecondtwopointsgivenbyequation (9.33)wheretheeffective primary impedance isreal,thesecondary currentis 1 _-jwMJ;. _ -jwMJ;. 2 -Z2+w2M2-R2_X2+ w2M2+2jRX -jwMVr -jwMJ;. =2R2+2jRX =2R(R+jX)' wherewehaveusedthecondition givenbyequation (9.33).Hence 11IwMJ;. Y:t. 2=2R(R2+X2)i =2R' Theseresultsshowthatwhenthecircuitsare'over-coupled' (wM>R) thesecondary currentrisesto(J;./2R)atthe.secondtwopointsgiven byequation (9.33),whileithasfallenbelowthisvalueatthepoint 246 ALTERNATING CURRENT THEORY [9. x=O.Theprimary circuitimpedance ispurelyresistiveatallthre points,butonlyatthesecondtwopointsisitequalto2R,ascanb seenbysubstituting inequation (9.32).Thusthesecondary currenti amaximum atthesetwopointsbecausethesecondary circuitisagai 'matched' totheprimary circuit. Thegeneralbehaviour ofthesecondary currentisshowninFig.9.10 wherethedegreeofcoupling isspecified intermsofthe'coefficient 0 coupling' definedbyequation (6.14)as k=Mj(L1L2)!=MjL whenthetwocircuitsareidentical.(9.35 -0.04° +0·04 tJf/fo- FIG.9.10.Secondary currentinthecircuitofFig.9.9 fortwoidentical circuitseachwithQ=100.Critical coupling occursatk=0·01. Thecriticalcondition WoM=Rthenoccurswhenthecoefficient 0 coupling hasthevalue ko=Rj(woL)=1jQ. (9.36 ThecurvesinFig.9.10aredrawnforQ=100,andcoefficients 0 coupling ofk=tko,ko,and2korespectively. Thefirstcurveissimila toanordinary resonance curve,butthecurrentisalwayslessthanth maximum possible value ~j2R.Thesecondcurvehasasinglepeaki thecentre,butthepeakisdecidedly flattened because ofthethre coincident roots.Assoonask>kowehavetwosidemaxima inth currentwithacentralminimum. Askjkoincreases thesepeaksmov outwards andthetroughinthemiddledeepens. WhenQislarg andthefrequency isclosetoresonance wemaywriteX=28wL,an 9.4} ALTERNATING CURRENT THEORY 247 equation (9.33)givesfortheseparation 2S/ofthepeaks (281//0)=(28w/wo) ::::1{~:-Wf~2}t=(k2-:.I/Q2)j-=(k2-kB)!· (9.37) Although identical circuitshavebeenassumedinthisanalysis, the behaviour ofanypairofcoupled circuitswhosenaturalresonant fre­ quencies arethesameissimilar. ThusfortwocircuitswiththesameX butdifferent R1,R2,optimum coupling occurs(forX=0)when woM=(R1R2)!{k=1/(QIQ2)!} andthesecondary currentisthenJi/2(R1R2)l.Sidepeaksoccurwhen thecoupling isgreaterthanthisvalue,andthesecondary currentat thesepeaksisJi/(R1+R2),whichislessthantheoptimum. Thevalues oftheeffective primary impedance are2R1whenthecoupling isopti­ mumandX=0,and(R1+R2)atthesidepeakswhenthecircuitsare over-coupled. Coupled circuitshaveanimportant application inthereception of radiosignalswhenitisdesiredtoacceptanarrowbandoffrequencies andrejectfrequencies outsidethisband.Thusinbroadcast reception ofsoundauniform response overabandofabout9kc/swidthisre­ quiredwithtotalrejection outside, sothattheidealresponse curve wouldberectangular inshape.Bytheuseoftunedcircuitswithrather morethancriticalcoupling aresponse curvewithsteepsidesisobtained, andtheslightdipinthemiddlecanbecompensated bytheuseofsome­ whatunder-coupled circuits (withacentralpeak)elsewhere inthe receiver. Inatunable receiver ofthiskindaconstant bandwidth is required independent ofthecentralfrequency. Thiscannotbeachieved bymutualinductance coupling alone,forequation (9.37)showsthat thebandwidth isthenproportional tothecentralfrequency. Acombina­ tionofmutualinductance coupling together withcapacitance coupling as inFig.9.8(a)maybeusedtogiveamoreorlessconstant bandwidth, for thecoupling throughthecapacitance decreases withfrequency, because ofthefallintheimpedance common tothetwocircuits. 9.5.Low-frequency transformers Thecoupledresonant circuitsdiscussed inthelastsectionmaybe regarded asatunedtransformer; theyareusedassuchatradio frequencies wheretheeffectsofstraycapacitance canbereducedby makingitpartofthetuningcapacitance. Atlowfrequencies (suchas thoseofpowersupplies) effectsofstraycapacitance aresmall,and 248 ALTERNATING CURRENT THEORY [9. transferofpowercanbemadeveryefficiently byuseofthetransforme whoseprinciple wasmentioned in§6.2.Itconsistsoftwocoils,th primary orinputcoilofn1turns,andthesecondary oroutputcoilofn turns,whicharecloselywoundtogether onanironcoresothatallth fluxduetoonecoilpassesthroughtheother.Inpracticethereisalway asmallleakageofflux,sothatnotallthefluxofonecircuitpasse through theother.Thenthetransformer mayberepresented byth circuitofFig.9.11,whereLvLzaretheself-inductances oftheprima. FIG.9.11.Circuitdiagram oftransformer withloadZ•. Ji=(Zl+~wL1)I1+~wMlz}. 0=(Zz+JwL z)Iz+JwMl1 Theseequations differfrom(9.29)inthattheself-inductances Lv arenotincluded intheimpedances ZvZzbecausetheirreactances ar normally verymuchlargerthananyotherimpedances inthecircuit. Alternate elimination ofIzand11between theseequations givesth following formulae fortheprimary andsecondary circuitsandsecondary windings, andMisthemutualinductance betwee them.Analternating voltageJiofangularfrequency wisappliedt theprimary inserieswithanimpedance Zvandthesecondary iscon nectedtoaloadimpedance Zz.Then,iftheprimary andsecondar ourrents areIvIz'theequations fortheprimaryandsecondary circuit are Ji/I;=Zl+jwL1+wzMz/(Zz+jwL z) and -(M/L1)Ji/lz=Zz+Zl(L z/L1)+jw(Lz-M2/L 1), whereinthesecondequation atermZlZz/(jwL1)hasbeenomitte sinceitisanorderofmagnitude smallerthanZz. 9.5] ALTERNATING CURRENT THEORY 249 Foraperfecttransformer, thereiscomplete coupling between the twocoils,sothatM2=L1L2.Sincetheself-inductances arepropor­ tionaltothesquaresofthenumberofturns,theturnsration=(L21L1)i, andwithcomplete coupling wehavealson=MILl=L2IM.Hence, assuming Z2~wL2,fromequation (9.39)theprimary impedance Zp maybewritten ~/I1=Zp=Zl+jwL1+w2M2(Z2-jwL2)/(Z~+w2L~) ~Zl+Z2(M2IL~)+jw(L1-M2jL2) =Zl+Z2jn2. (9.41) Thecurrentinthesecondary circuitisthesameasthatduetoane.m.f. -(MjL1)~ =-n~working intoanimpedance Zs'where Zs=Z2+Z1n2• (9.42) Theseequations showthatforaperfecttransformer onload(Z2~wL2) theinductive termssuchasjw(L1-M2jL2)areexactlyzerowhenthere iscomplete coupling betweenthetwocoils.Theeffectofthesecondary circuitonthecurrentintheprimary isrepresented bytheadditional impedance Z2jn2,knownasthe'reflected impedance'. Inthesecondary circuittheeffective e.m.f.is-n~,withanapparent internalimpedance Zln2•Hencetheimpedances aretransformed byn2whilethevoltages aretransformed byn;thecurrenttransformation ratioislIn,sothat thepoweronthetwosidesisthesame (~I1=~12),asweshouldexpect foraperfecttransformer withnolosses. Inpractice M2isslightlylessthanL1L2becausenotallthefluxfrom onecircuitpassesthroughtheother,andwewriteM=k(L1L2)!,where kisthecoupling coefficient definedbyequation (6.14).Itisusefulto deriveanequivalent circuitforthetransformer, andinordertoinclude thecasewhereZ2isofthesameorderofmagnitude aswL2,werewrite theexpression fortheprimary impedance (equation (9.39»)inthefollow­ ingway Z=Z+.L(l-k)+w2k2L1L2+jwL1k(Z2+jwL 2) p 1JW1 Z+'L .2JW2 Thelasttermmaybewrittenintheform jwkL1{Z2+jw(1-k)L 2} {Z2+jw(1-k)L 2}+jwkL1n2' sinceinthedenominator jwkL1n2-jwkL2=O.Thelasttermisnow {I n2}-1 jwkL1+Z2+jw(1-k)L 2' whichisequivalent totwoimpedances inparallel. Hencetheprimary 250 ALTERNATING CURRENT THEORY [9. circuitmayberepresented byFig.9.12.Here(l-k)Llisthe'leakag inductance' duetotheimperfect coupling, andtheimpedance onth extreme rightisthereflected impedance ofthesecondary, whichisi parallelwiththeremainder oftheprimary inductance. Z.+jw(l-k)L. n' I,,,,. FIG.9.12.Equivalent circuitofprimary ofimperfect transformer. n'=L./L1;k=MI(L1L.)l. I. FIG.9.13.Equivalent circuitforsecondary ofimperfect transformer. Forthesecondary circuit,sinceMILl=k(L2ILl)1 =kn,wehave, fromequation (9.40), -kn~/I2=Z2+n2Zl+jwL2(I-k)+jw(kL2-J1[2/Ll) =Z2+n2{Zl+jwL l(l-k)}+jwL 2k(l-k), whichisrepresented byFig.9.13.Thereflected impedance isthe transformed valueofZlplvstheprimary leakageinductance Ll(l-k), whilethesecondary leakageinductance is,apartfromafactork,just thatwhichappearsinserieswithZ2intheimpedance reflected inthe primary.Ifkisclosetounity,thisapproximation isagoodone,forthe 9.5] ALTERNATING CURRENT THEORY 251 leakageinductance willbesmallcompared withZzexceptforverysmall valuesofZz.Theequivalent circuitofthetransformer canbedrawn asinFig.9.14,wherethecentreportionenclosedinthedottedrectangle isregarded asaperfecttransformer. Theresistances r1andrzarethe resistances ofthewindings, whichpreviously wehaveregarded aspart ofZlandZz.Theresistance RinparallelwithkLlallowsforthedissi­ pationofenergythrough hysteresis andeddycurrentsintheironcore. r Il:nIL__--l FIG.9.14.Approximate equivalent circuitofimperfect transformer. rt>r••resistances ofprimary andsecondary windings. R,resistance equivalent tohysteresis andeddycurrent lossesinironcore. Theportionwithinthedottedrectangle isregarded asa.perfecttransformer ofturnsration. Theprimerequirements inatransformer aretherefore ahighprimary inductance, tokeepthe'magnetizing current' throughkLlinFig.9.14 small,andthesmallest possible leakageoffluxbetween primary and secondary. Theserequirements arefulfilledbywindingtheprimaryand secondary roundanironcorewhosemagnetic circuitiscompleted by ayoke,asinFig.9.15.Ifthetwocoilsareinterwound theleakageis reduced toaminimum, butifgoodinsulation between primary and secondary isrequired theymaybewoundsidebyside. Intransformers themagnetic material issubjecttoanalternating magnetic fieldsothatthehysteresis loopistraversed onceeveryperiod ofthealternation. Fromequation (8.26)thisrequirestheexpenditure ofenergy,theamountpercyclebeingjustequaltotheareaenclosedby thehysteresis loop.Thisloopshouldtherefore beasthinaspossible(or, roughlyspeaking, thecoercive forcemustbesmall).Asoftmagnetic material istherefore required (see§8.4),andideallythepermeability ALTERNATING CURRENT THEORY FIG.9.15.Construction ofiron-cored transformer. Ilaminated ironcoreandyoke. Pprimary winding. Ssecondary winding.252 shouldbehighandconstant, withtheareaofthehysteresis loopzer, sothattheB-Hcurveisastraight lineofhighslopethroughth origin.Alloysofironwithafewpercentofsiliconapproach thiside morecloselythanpureiron,andar usedinpowertransformers forsuppl frequencies. Forsmalltransformers an otherusesathigherfrequencies, mo expensive alloysrequiring lengthyhe treatment giveimproved performanc Forexample, analloyofabout78p centnickeland22percentiron(per­ alloy),ifslowlycooledfrom9000Can thenrapidlycooledfrom6000C,givs aninitialpermeability ofnearly104an amaximum permeability ofnearly10. Another alloy(supermalloy) witht e composition 80percentnickel,15p r centiron,5percentmolybdenum, aftr heatinginverypurehydrogen at12000to13000C,givesinitialan maximum permeabilities sometentimeshigher. Thesehighperm­ abilities areaccompanied byverylowvaluesofthecoercive fore, ""4A/metre forpermalloy and0·3A/metre forsupermalloy. Another effectofthealternating magnetic fluxinthecoreistos t upinduced voltages andproduce powerlossthrough eddycurrent. Thecoreistherefore constructed ofverythinlaminations, insulatd fromeachother,andoriented sothattheinsulation liesacrossthepa oftheeddycurrent. Thisistherefore constrained toflowwithint e lamination, andthelossesarereduced (seeProblem 9.14).Int s respectthespecialmagnetic alloysmentioned abovehavethefurthr advantage ofahighelectrical resistivity. Atradio-frequencies (105-18 cis)theuseofmagnetic coresalsoreducesthesizeofinductors andi ­ provesthecoupling between thecoilsoftransformers; suchcoresa e madeeitherfromfinemetallic powders mixedwithinsulating bindes orfrommagnetic oxides('ferrites') whicharethemselves electricI insulators. ALTERNATING CURRENT THEORY 253 PROBLEMS 9.1.Aresistance R,inductance L,andcapacitance 0areconnected allinparallel. Showthattheadmittance ofthecircuitatfrequencies nearresonance is Y=I/R+2j8wO. IfR=3X105ohms,L=10-3henries, 0=100p,p:F,calculate thecurrentin eacharmwhenavoltageof10Vr.m.s.atafrequency of0'5Mc/sisapplied, and thephaseofthetotalcurrentdrawnfromthegenerator. (Answers: 0·033,3·18, and3·14mA; 51°.) 9.2.Acircuitisrequired toacceptasignaloffrequency 1·1Mc/sandtoreject asignaloffrequency 1·2Mc/s.Acoilofself-inductance 200p,Handresistance 10ohmsistunedtoparallelresonance at1·2Mc/sbyacapacitance 01"Acapaci­ tanceOaisthenplacedinserieswiththecombination, sothatthewholeisin seriesresonance at1·1Mc/s.Findthevaluesof01and0a. (Answers: 88and16p,p,F.) 9.3.Showthatacapacitance 0shuntedbyaresistance risequivalent toacapaci­ tance0'inserieswitharesistance Ratanygivenfrequency. IfwOr~I,show thatapproximately R=(W302r)-1,and0'=O. 9.4.Fourimpedances Z1'Za,Za,Z4'inthatorder,areplacedinthearmsofa 'generalized' Wheatstone's bridgeusingalternating current. Showthatthe balance condition isZ1/Za=Z4/Z3' Notethatthisisreallyadoublebalance condition, sincetherealandimaginary partsofthisequation mustbeseparately satisfied. Thisisbecause anullreading isobtained onthedetector onlyifthevoltagesateachofitsterminals areequal bothinamplitude andphase. 9.5.ThefourarmsofaWheatstone's bridge,takenincyclicorderroundthe bridge,area,b,c,d.aandbareequalresistances R;cisaresistance Rinseries withacapacitance 0;disaresistance Rshuntedbyacapacitance O.Showthat suchabridgewillnotbebalanced atanyfrequency, butthatiftheresistance in armbisdoubled, abalancewillbeobtained atafrequency f=(2mRO)-1. 9.6.Acapacitor ofcapacitance 0,aresistance r,andacoilwhoseinductance is Landresistance isR,areconnected allthreeinparallel. Ane.m.f.ofvariable frequency isapplied acrossthecapacitor. Showthatthefrequency ofparallel resonance isindependent ofthevalueofr,andprovethattheparallel resistance ofthecombination isrL/(L+RrO). IftheQofthecircuitishigh,showthatitisgivenapproximately by ~=~J~+RJ~. 9.7.Aseriesresonant circuitisconnected acrossaconstant voltagegenerator operating at6·50Mc/s.Asthecapacitance isvaried,thecurrent isobserved to fallto1/".12ofitsmaximum valuewhenthecapacitance is12·47p,p,F,andagain whenthecapacitance is12·64p,p:F.FindthevaluesofQ,L,Rforthecircuit. (Answer: ~150;48pH;13ohms.) 254 ALTERNATING CURRENT THEORY 9.8.Analternating voltageisapplied.totheterminals A,Bofthenetwork sho inFig.9.16.Showthat,asRisvaried,theamplitude ofthepotential differenc betweentheterminals X,Yremainsconstant, butitsphaseisshiftedby7Tradia. Explain yourresultsbymeansofavectordiagram. (Thisnetwork iscommonl usedforproducing avariable phaseshiftwithout changeoftheoutputamplitude. 9.9.Ahighfrequency transformer hasprimary inductance 100fLHandprima resistance 5ohms,secondary inductance 2·5X103fLH,andsecondary resistanc 100ohms.Ifthecoefficient ofcoupling M/(L1L2)iis0·9,showthatathig frequencies thevoltage acrossthesecondary isapproximately 4'5timestha appliedtotheprimary. Ifprimary andsecondary areeachseparately tunedbycapacitors toresonance a 50kc/s,showthatthehighfrequency powerrequired toproduceanr.m.s.voltag of2000Vacrossthecapacitance inthesecondary circuitisapproximately 650W A Bx rG_~_I FIG.9.16.Phaseshiftnetwork (seeProblem 9.8). 9.10.Awire-wound resistance hasasmallinductance andself-capacitance whic mayberepresented byplacinganinductance Linserieswiththeresistance R, andshunting thecombination byacapacitance O.Showthatthereactance a lowfrequencies iszerointhefirstapproximation ifthewireiswoundsotha L/O=H2.Showalsothatundertheseconditions theapparent resistance is(to thesecondapproximation) R(I+w 2LO). 9.11.Aninductance Lwithsmallresistance rhasself-capacitance whichcanbe represented approximately byacapacitance 0shunted acrosstheseriescom­ bination ofLandr.Show thatasthefrequency increases theapparent self­ inductance ofthecoilisincreased bythefactor(l+w2LO)whiletheapparent seriesresistance increases bythefactor(l+2w2LO)"intheregionwherew2LO~1. 9.12.Aparallelplatecapacitor isfilledwithamedium whichhasadielectric constant Eandaconductivity a.Showthatatafrequencyf=W/27Tthepower factorofthecapacitor.is cose/>=sinS,wherethelosstangent ofthedielectric medium isgivenbytherelationtanS=a/WEEo' Notethatthisisindependent oftheshapeofthecapacitor, aswouldbeexpected fromequation (3.11). ALTERNATING CURRENT THEORY 255 9.13.A100Vdynamo isconnected toamagnet whoseresistance is10ohmand self-inductance 0·1henry.Showthatthepercentage increase intheheatingof themagnetcausedbythepresence ofa100cisripplevoltageof5 Vamplitude intheoutputofthedynamo is0·0031percent. 9.14.Theironcoreofatransformer haslaminations oftljickness aandresistivity p;itissubjecttoasinusoidally varying induction withamaximum valueBand frequencyf.Showthatthepowerlossperunitvolume(neglecting skineffects, seeProblem 10.13)duetoeddycurrents isapproximatelY1TsBza~z/6p. Ifa=0·1 rom,p=4X10-7ohm-metre, B=0'5weber/metreS, andf=50cis,showthat thepowerlossisapproximately 2·6X10-5W/cm3•Compare thiswiththepower lossthrough hysteresis, iftheenergydissipated percycleforpermalloy atthiS induction is200ergs/cm3• (Answer: Hysteresis loss=10-3W/cm3.) 10 ELECTROMAGNETIC WAVES 10.1.Maxwell's equations oftheelectroma~netic field Sofarwehaveconsidered thepropagation ofelectrical currents i material conductors. Thepossibility ofthepropagation ofanelectr­ magnetic wavethrough spacewasfirstsuggested byFaraday, andt s suggestion wasconfirmed bytheworkofMaxwell. Maxwell wasabe toshowthatthelawsofelectromagnetism couldbeexpressed int e formofsomefundamental equations which,withanimportant mo'­ fication, leadtoadifferential equation whosesolutions represent tran­ versewavestravelling through freespacewiththevelocity oflig. Further workshowedthattheproperties ofthesewaves-reflectio , refraction, diffraction-are thesameasthoseestablished experimenta y forlightwaves,andwearetherefore justifiedinassuming thattheya e identical, andthatlightwavesareaformofelectromagnetic radiatio . ThetheoryofMaxwell dealsentirelywithmacroscopic phenomen , makingtheassumption thatmatteriscontinuous andhasnoatomisic structure. Thisassumption placescertainlimitations onthetheor; thusitoffersnoexplanation ofthephenomenon ofdispersion-t e changeofrefractive indexwithfrequency. Thisphenomenon wille discussed inChapter 17;itarisesfromthechangeinthedielectic constant andmagnetic permeability ofamediumwithfrequency. Thse changescanberelatedtotheeffectofelectromagnetic wavesonthein'­ vidualelectrons inanatom,butforthepresentweshallregardt e dielectric constant andmagnetic permeability asmacroscopic quantitas whosevaluesareobtained byexperiment. Thefundamental lawsofelectromagnetism whichhavealreadyben derivedmaybesummarized asfollows: (a)thetheorem ofGaussappliedtoelectrostatics (equation (1.20»: divD=p; (101) (b)thecorresponding resultformagnetic fields(equation (5.24»: divB=0; (102) (c)Faraday's andLenz'slawofelectromagnetic induction (equatin (6.3»: curlE=8B. (103)-at' 10.1] ELECTROMAGNETIC WAVES 257 (10.5) (10.6)divJ=-~(divD) =dive-aD/at),at diV(J+a~)=o. J'=J+aD,ator HenceifwedefineJ'as(d)Ampere's lawformagnetomotive force(equation (5.21)): curlH=J'. (10.4) ThereasonforwritingJ'ratherthantheordinary currentdensityJ inthislastequation isasfollows. Since(divcurl) ofanyvectoris identically zero,itfollowsthatequation (lOA)impliesthatdivJ'iszero. IfwehadwrittenJinsteadofJ',weshouldhavehaddivJ=0,and thisconflicts withtheequation ofcontinuity (3.3)whichgives divJ=_ap. at Thisequation represents thelawofconservation ofcharge,andiscon­ firmedbyallexperiments. Maxwell realizedthatthedifficulty arose fromanincomplete definition ofthetotalcurrentdensityinequation (lOA),whichisnotentirelygivenbythecurrentflowduetothemotion ofelectriccharges. Byusingequation (10.1)wemaywrite(10.5)inthe form thendivJ'=0,andAmpere's lawtakestheform aD aDcurlH=J+-=uE+-.at at(10.7) ThetermJisgenerally calledthe'conduction current'andthesecond term(aD/at)the'displacement current', sinceitariseswhentheelectric displacement Dischanging withtime.WemayobtainsomephySIcal pictureofwhatisimpliedbythedisplacement currentbyconsidering asimplecircuitsuchasinFig.10.1,whereacurrentIisflowingfrom abatterytochargeacapacitor O.IfweapplyAmpere's lawtoaclosed circuitsuchasLMNLwhichencircles thewirewefindthatJH.ds roundthiscircuitisjustequaltoI.Indefining thecurrentwhich threadsthecircuitwemusttakesomesurfacebounded bythecircuit, andintegrate thenormalcomponent ofthecurrentdensitycrossing thissurface.Ifwetakeasurfaceintersecting thewire,thisclearlygives justI,thecurrentinthewire.Butifwetakeasurfacewhichpasses betweenthetwoplatesofthecapacitor, thenfJ.dS=0,sincenocon­ ductioncurrentflowsthrough thesurface. Thevalueofthisintegral shouldbeindependent ofwhatsurfacewechoose,sincefH.dsdepends 851110 S 258 ELECTROMAGNETIC WAVES [10.1 (10.) (10.) (10.1) (10.1)onlyonthecircuitbounding thissurface;thusitisclearthatwehay omitted somecontribution. Ifforsimplicity wetakeaparallelplat capacitor withplatesofareaA,surrounded byaguardring,thefiel inbetweentheplatesisuniform, andthedisplacement Dhasthevalu FIG.10.1.APflication ofAmpere's lawtocalculate themagneto­ motiveforceH.dsroundthecircuitLMNL fortheeaseofa currentIcharging acapacitor O. D=q/A,whereqisthetotalchargeonthepositive plate. totaldisplacement currentbetween theplatesis A(oD/ot) =oq/ot=I andourdifficulties withthesurfaceintegral ofthecurrentdensit disappear ifweincludethedisplacement current. Forthecaseofaninfinitehomogeneous mediumofdielectric constat €andmagnetic permeability /-',containing nofreecharges (p=0)an havingzeroconductivity (a=0),ourequations become divD=div(€€oE) =EEOdivE =0, divB=div(,u,uoH) =,u,uodivH=0, curlE=-oB/ot =-,u,uo(oH/ot), curlH=oD/ot=E€o(oE/ot). 10.1] ELECTROMAGNETIC WAVES 259 Theseformasetofsimultaneous partialdifferentia.! equations whose solutions canbefoundbyeliminating oneofthedependent variables, EorH.Thiscanbedonebymeansofthevectoridentity curl(curIE) =grad(divE)-V 2E=-,-V2E(usingequation (10.8». Then V2E=-curl(curIE) =curl(fLfLooH/ot) a=P-JLo-(curlH)=fLfLoEEo(o2E/at2). (10.12)at Similarly itmaybeshownthat V2H=fLfLoEEO(02H/ot2). (10.13) Thesetwoequations areeachofthegeneralformforawavemotion inthreedimensions; ifthevelocitywithwhichthewavesarepropagated isv,thegeneralwaveequation is V2X=.!.-o2X v2ot2' whereXissomescalarorvectorquantity. Comparison withourcase showsthatthewavevelocity mustbe v=(fLfLoEEo)-1 (10.14) withtheparticular valueinfreespace(fL=1,E=1)of C=(fLoEo)-l. (10.15) NowthevalueoffLohasbeendefinedas4rr10-7,whilethebestvalue ofEOisthatofRosaandDorsey(see§7.4):thisleadstoawavevelocity ofourelectromagnetic wavesinfreespace(seeBirge,1941)of 299784±10 km/sec. Withintheexperimental errorthisisthesameasthevelocityoflight determined bydirectmeasurement, anditisnowaccepted thatlightis aformofelectromagnetic radiation, ofthesameformasradiowaves, heatwaves,X-rays,andy-rays,whichdifferfromlightwavesandfrom eachotheronlyinfrequency andwavelength. Thefrequency rangeis from104to1011cisforradiowavesto1020cisandoverfory-rays. Maxwell's theorydoesnotgiveanadequate accountoftheinteraction ofelectromagnetic radiation withatoms,andithasbeenfoundnecessary (forexample, inthephotoelectric effect)toconsidertheelectromagnetic energyastravelling aboutin'packets' or'quanta', whichareindivisible. Thesizeofaquantum forradiation offrequency vishv,wherehisa universal constant knownasPlanck's constant. Forradiowavessuch quantum effectsaretoosmalltoaffecttheinteraction withmatter seriously, andweshallnotconsider themfurtheratpresent. 260 ELECTROMAGNETIC WAVES [10.1 Ifweaccepttheidentification ofourelectromagnetic waveswithlight andradiowaves,thenwemayuseoneoftheaccurate methods ofdeter­ miningtheirvelocity (see§15.6)toinferthevalueoftheconstant EO' sincethisgivesamoreaccurate valuethanthedirectcomparison 0 acapacitance andresistance. Thevaluethusobtained isgivenin Appendix B. Equation (10.14)showsthatthevelocity ofthewavesinamateria medium islessthaninfreespace,since v=C/(fL€)t. Foralightwavethevelocity inamedium isv=c/n,wherenisth refractive indexofthemedium. Hencewehave n=(fLE)t. (10.16 Inusingthisequation wemustremember thatitappliesonlyifw determine thevaluesofn,fL,andEatthesamefrequency, andnon sensicalresultsmaybeobtained ifweusevaluesdetermined atdifferen frequencies. Thus,forwatertheopticalrefractive indexis1,33,bu thestaticvaluesofEandfLare81and1,leadingtoarefractive inde (forverylowfrequencies only)of9!Thisisoneofthemoreglarin examples ofthemisuseofequation (10.16)tocompare valuesdeter minedatdifferent frequencies, towhichweshallreturninChapter 17 10.2.Planewavesinisotropic dielectrics Inordertoexamine thebehaviour oftheelectricandmagnetic field inmoredetailweshallconsider thecaseofaplanewave.Forsimplicit weassumethatthisismovinginthedirection ofthe:r-axisofaset0 right-handed Cartesian axes,x,y,z.Thenthedefinition ofaplanewav isoneinwhichthequantities havethesamevalueoveranyplan normaltothedirection ofpropagation; mathematically thisisexpresse bysettingallpartialdifferentials withrespecttothey-andz-coordinate (Le.O/oyand%z)equaltozero.Iftheseconditions areputintoth equations (10.8)and(10.9)wefindthat oEx/ox=oHx/ox=0, andfromthex-components ofthecurlequations ((10.10)and(10.11) wehave oEx/ot=oHx/ot=O. Thisshowsthatapartfromauniform steadyfieldinthex-directio , whichisnotpartofanywavemotion,bothExandH"mustbezer Thewaveistherefore purelytransverse inthatnocomponents ofth electricormagnetic fieldsexistinthedirection ofpropagation. Th 10.2J ELECTROMAGNETIC WAVES 261 (10.19)remaining components ofthecurlequations are,takingthex-com­ ponentsofEandHtobezero,and%y=%z=0, -oEz/ox =-flP.ooHy/ot, -oHz/ox =€€ooEy/ot, oEy/ax=-fLfLoa~/at, aHy/ax=€€oaEz/at. Theseequations showthatthez-component ofEisassociated withthe y-component ofH,whilethey-component ofEisassociated withthez­ component ofH.Thetwocomponents ofEorHcorrespond towaves whichareplanepolarized indirections normaltooneanotherandnormal tothedirection ofpropagation. Thuswehavetwolinearlyindependent solutions, ineachofwhichthemagnetic fieldisnormaltotheelectric field.Theydifferonlyintheplaneofpolarization, whichweshaJItaketo bethatoftheelectricvector;thisdiffersfromtheconvention accepted inlightbeforetheelectromagnetic natureoftheradiation wasunder­ stood,whichadopted theplanenowknowntobethatofthemagnetic vector.Theelectricvectorismoreimportant inthetheoryofdispersion (Chapter 17)asitdetermines theforceonanelectroninanatom. Sinceweneedconsider onlyonestateofpolarization, weshalltakethe solution wheretheelectricvectorisparalleltothey-axis;itthenfollows thatthemagnetic vectorhasonlyacomponent paralleltothez-axis. Thewaveequation forEy(equation (10.12»becomes a2E, a2E, ox:-fLfLo€€o a/=0, (10.17) whichhasageneralsolutionoftheform Ey=Fl(x-vt)+F2(x+vt), (10.18) whereFl,F2maybefunctions ofanyform.Thetwofunctions represent wavestravelling withvelocity v,whosevalueisgivenbyequation (10.14). ~represents awavetravelling inthedirection ofxincreasing, sinceanygivenpointinthewavemovesaccording totheequation x=vt+constant, while.F;isawavetravelling intheopposite direction, agivenpoint movingasx=-vt+constant. WemayfindthevalueofHzbyusingoneoftheremaining components ofourcurlequations. For a~/8x=-€€o8Ey/8t=€€ov{F~(x-vt)-F~(x+vt)}, whereF'isthedifferential ofF.Hence,sincev=(€€ofLfLo)-t, wefind onintegration 262 ELECTROMAGNETIC WAVES [10. IE.ds=V, IH.ds=I,Thisshowsthatthevalueof~bearsaconstant ratiotoEyineach0 thetravelling waves,buthastheopposite signforawavetravelling t theleft.TheratioofEytoHzinaplanewaveisoftenuseful,anditha beencalledthe'intrinsic impedance' Zoofthemedium. Inourcase0 anon-conducting medium, wherethewavevelocity isv, Zo=Ey/Hz=(I-'I-'o/££o)t =I-'I-'ov. (10.20 ForaplanewaveinfreespaceZo=(1-'0/£0));=I-'oc=(EOC)-I,andit valueisapproximately 376·7ohms.Thefactthatthedimensions ofZ arethesameasthoseoftheimpedance discussed inearlierchapters . readilyseenfromtherelations whereVisthepotential dropinaconductor; itappearsalsofromth factthatEismeasured involts/metre andHinamperes/metre. Th factthattheintrinsic impedance isarealquantity independent ofx0 tshowsthatthewaveformof~iseverywhere thesameasthat0 Ey,withoutanyphasedifference, inatravelling wave.Thisistrueonl ofanon-conducting medium. Another important restriction isthatth wavevelocitymustbeindependent offrequency ifthewaveformisto remainthesameasthewaveprogresses. Ifitisnot(i.e.ifwehav adispersive medium) wemustperform aFourieranalysis oftheinitia waveformandtreateachcomponent ofagivenfrequency separately. Intheremainder ofthischapterweshallassumethatwearedealin withwavesofonefrequency only,andourfieldcomponents arethereal orimaginary partsoftheexponential functions oftheform expUw(t±x/v)}. Thenthegeneralsolution forwavestravelling bothtotherightandto theleftwillbe Ey=Aexp{jw(t-x/v)}+A' exp{jw(t+x/v)} }(10.21) ZoHz=Aexp{jw(t-x/v)}-A' exp{jw(t+x/v)} . Theexponentials aresometimes writtenintheformsexp{jw(t±nx/c)} orexpj(wt±f3x); f3iscalledthe'phaseconstant', andisequalto27T/A, whereAisthewavelength inthemedium. Thephasevelocity visgiven bytherelation v=w/{3. (10.22) Itmightbethoughtthataphaseconstant 8shouldbeincludedinone ofthewavesinequation (10.21),sinceotherwise someparticular choice ofzeraisimpliedforeitherxortwhichmakes8=o.Weshall,however, 10.2J ELECTROMAGNETIC WAVES 263 FIG.10.2.Directions oftheelectricand magnetic fieldsEandHandthePoyn­ tingvectorNforaplanewavepropa- gatedinthex-direction.allowA'(orA)tobecomplex, andoftheformA"exp(j8),andthus includethephaseconstant inA'ratherthaninthecomplex exponential. 10.3.ThePoynting vectorofenergyflow Sinceanelectromagnetic waveconsistsofelectricandmagnetic fields, wemayexpectthatstoredenergyisassociated withthesefields.Ifthe energydensityforwhichwenowusethesymbolUisgivenbythe equations derivedforstaticfields,thenforaplanewaveinanisotropic dielectric itwillbe U=!(D.E+B.H) =!(EEOE2+JLJLoH2). Forawavepropagated inthepositive x-direction, theenergycrossing unitareapersecondwilljustbethevelocity timestheenergydensity. Thisis vU=!{J(;~J~+Jt::)H:} =!(E;/Zo+Zom) =ElIHz=E;JZo=ZoH:. Thesevariousrelations showthattheenergystoredinthemagnetic fieldisjustequaltothatinthe electricfield.Ifwedenotetherate E. ofenergyflowbyavectorNwhose direction isthatoftheenergyflow, thenforthiscasewemaywrite Nx=ElI~' orinvectorform,sinceElI,~,andNx formaright-handed triadofaxes (x-axis) (Fig.10.2), N N=EI\H. (10.23) ThevectorNisknownasthePoyn­ tingvector,andweseethatthe direction ofenergyflowisreversed H, forawavetravelling intheopposite direction becausethephaseofHrela­ tivetoEisreversed (seeequations (10.21)). ThevalueofNgivesthe instantaneous rateofenergyflow.Inaperiodic wavethevaluesof EandHatanypointareoscillating functions ofthetime,andthe meanrateofenergyflowisfoundbyaveraging Noveracomplete period.IfEandHaregivenasrootmeansquarevalues,thenNas computed from(10.23)givesthemeanenergyflow. I(D.E+B.H)264 ELECTROMAGNETIC WAVES [10. ThePoynting vectorisofgreatersignificance thanmightbethough fromthewayinwhichithasbeenintroduced above,andweshallno deriveitforthegeneralcase.Suppose thereisaregionofspacewher anelectricfieldEcausesacurrentflowofdensityJ.Thenthepowe dissipated perunitvolumeisE.J,andthetotalpowerdissipated willb IE.Jd'T=I(E.curlH) d'T-IE.(oDjot)d'T, wherewehavesubstituted forJtheexpression givenbyequation (10.7) Now,usingthevectoridentity div(E/\H) =H.curlE-E.curIH andtheexpression forcurlEgivenbyequation (10.3),wefindthatth powerdissipated is -IE.(oDjot)d'T-IH.(oBjot)d'T-Idiv(E/\H)d'T. Normally thepermeability fLanddielectric constant €donotvarywit time,andthefirsttwotermsmaybewrittenas 10f---(D.E+B.H)d'T2ot whilethelasttermmaybetransformed intothesurfaceintegral -I(E/\H).dS takenoverthesurfacebounding thevolumeunderconsideration. Henc therateatwhichthefielddoesworkmaybeequatedtothesumoftw terms,thefirstofwhichweinterpret astherateatwhichtheenerg storedintheelectromagnetic fielddiminishes, andthesecondasth rateatwhichenergyflowsintothevolumeunderconsideration. Thu wetake asthedensityoftheelectromagnetic energy,inagreement with0 earlierresults,andthevector N=E/\H astherateatwhichenergyflowsacrossunitareaoftheboundary Strictlyspeaking, onlytheintegralofNoveraclosedsurfacehasbee showntorepresent theenergyflow,butinmostcasesNdoesrepresen theflowofenergyperunitareaateachpoint.Anobvious exceptio tothiswouldbethecaseofaregionwithanelectrostatic andastati magnetic fieldarisingfromdifferent sources,butProblem 10.12show thatthePoynting vectorcanbeusedinthecaseofasteadycurren -------- -- 10.3] ELECTROMAGNETIC WAVES 265 flowinginacircularwire,wherebothelectricandmagnetic fieldsare associated withthecurrentflow. (10.24)10.4.Planewavesinconductin~ media Inamedium withafiniteconductivity uthefieldequations arethe sameas(10.8-11) exceptfortheadditional term+uEontheright-hand sideofequation (10.11).Theseequations maybesolvedquitegeneraJIy inthesamewayasin§10.1,andelimination ofHgivesthedifferential equation (10.25)aEylax=-J-tJ-to(aHzlat) } -aHzlax=EEo(aElIlat)+uE y•Anexactlysimilarequation holdsforH,anditmaybeshownthateach oftheseequations represents adamped wavemotionwheretheampli­ tudedecaysasthewaveprogresses owingtotheextraterminaElator aHjat.Weshallnotpursuethisgeneralsolution, butspecialize tothe caseofaplanewavetravelling paralleltothex-axis.Thenallpartial derivatives (ajay)and(ajaz)vanish,andthesamemethods asusedin §10.2showthatbothExandHxarezero,sothatweagainhaveapurely transverse wave.Ifweassumethatwehaveaplanepolarized wavewith asinglecomponent ofEparalleltothey-axis,thefieldequations reduce to (10.25a)Theseequations showthat,asbefore,Eyisassociated withllz.(Thereis anindependent solution, polarized intheperpendicular direction, where Ezisassociated withHy.)Elimination ofHzwouldgiveawaveequation forEyofthesametypeasthegeneralequation above,butasimpler methodofsolution isavailable onassuming thatwearedealingwith wavesofasinglefrequency.Itwillappearthatthevelocityofpropaga­ tionisnowdependent onthefrequency, sothatitisnecessary tocon­ sidereachfrequency separately inanycase;wetherefore introduce this restriction atthebeginning, andtakeEyandHztovaryintimeas exp(jwt). Wewillalsotryasolution wherethevariation withxtakes theformofacomplex exponential, sothatbothEyandHzvaryas exp{jw(t-nxjc)}, wherenisacomplex refractive index.Thenourequations become jw(njc)E y=jwJ-tJ-to~ } jw(njc)llz =(jWEEO+u)EII• 266 ELECTROMAGNETIC WAVES [10. Elimination ofEyorHzshowsthattheseequations aresatisfied provide that 2'( j)n=p.€-;ap.WEO,. wherewehavesubstituted (p.o€o)forIjc2• Toseparate therealandimaginary partsofnwewriteitas n=n-jk, (10.27 n2-k2=P.E}giving (10.28nk=(ap.j2w€0) . Tounderstand thesignificance ofnandkwenotethatthewavei propagated as exp{jw(t-nxjc)} =exp(-wkxjc)exp{jw(t-n:cjc)}, showingthatthevalueofkdetermines therateatwhichtheamplitud ofthewavedecays(kappearsintheargument oftherealexponential whilendetermines thewavevelocityinthemedium. Thusacomple refractive indexnmeansthatthewaveisbeingabsorbed asitprocee becausethefiniteconductivity ofthemedium givesrisetoapowerlos through theJouleheating. Thesolutions wehavegivenaboveappl toawaveproceeding inthepositive x-direction; forawaveinthere versedirection wereversethesignofn,andhenceofbothnandk. Theintrinsic impedance ofourmedium isdefinedasEyjHz,sothat Zo=cp.p.ojn= {p.p..o}!. (10.29E€o-;ajw Thustheimpedance isacomplex quantity, butisindependent ofxor• Thismeansthattheratiooftheamplitudes ofEyandHziseverywher thesame,butthereisaphasedifference between them.Sincealso Z _cp.p.o(n+jk) 0-(n2+k2) weseethatthephasedifference 4>isgivenbytherelationtan4>=kj Thefullsolution ofequations (10.28)tofindnandkgivesrathe complicated expressions, butthesesimplifygreatlyforthecaseofagoo conductor. Forametaltheconduction currentisenormously greate thanthedisplacement currentatallfrequencies uptothoseofultr­ violetlight,aswillbeseenbycomparing thevaluesofaand(WEE) whichoccurinthesecondofequations (10.25a).Formostmetalsai 107(ohm-metre)-l orgreater, whileforlightofwavelength 5000 (WEEO)~(4X1015X8'85X10-12XE)~3'5X104€.Wedonotknowwhat isforametal,butprovided itisnotverymuchgreaterthantheordina 10.4] ELECTROMAGNETIC WAVES 267 valuesfoundindielectric substances, thedisplacement currentevenfor visiblelightwillbemuchsmallerthantheconduction current. For lowerfrequencie$ theinequality increases, andagoodapproximation isobtained byomitting thedisplacement current. Thisisequivalent to setting E=0,andequations (10.28)thengive n=k=(uftj2wEo)1. Ifweintroduce aquantity 8suchthatn=k=(cjw8),thefieldcom­ ponentsinthemetalarepropagated as exp(-xj8)expj(wt-xj8), (10.30) fromwhichitcanbeseenthat8hasthedimensions ofalength.8is knownasthe'skindepth',andtheamplitude ofthewavefallstoIjeof itsinitialvalueinadistance 8,whiletheapparent wavelength inthe metalis2778.Theequation for8is(writing w=277J) 8=(!uwft/Lo)-l =(77uJft/Lo)-l (10.31) andtheintrinsic impedance ofourmetalcanbewrittenas Zo=(l+j)j(u8) =(1+j)(pj8), (10.32) wherepistheresistivity. Themagnitude oftheskindepthdecreases withtheinversehalf-power ofthefrequency, andofthepermeability andconductivity ofthemetal.Anideaofitsorderofmagnitude is obtained fromthefactthatforcopperithastheapproximate values: 6'6X10-3cmatafrequency of1Mc/s,6·6x10-5cmat104Mc/s(a wavelength of3cm),and2·7X10-7cmat6x108Mcjs(greenlight). Thusthewavelength oftheradiation inthemetalisverysmallcom­ paredwiththewavelength infreespace;andZoisalsomuchsmaller thanthevalueforfreespace(forcopperitisabout0'026(1+j) ohmsat 104Mcjs). 10.5.Theskineffect Sinceanelectromagnetic wavevariesinphaseveryrapidlyinsidea metal,onemustaskwhether thisaffectsthedistribution ofalternating currentinsideaconductor. Anelectromagnetic fieldisassociated with suchacurrent,andProblem 10.1showsthatthereisaflowofenergy intothesurfaceofaconductor whichisjustequaltotheenergydissi­ patedasJouIeheat.Thisimpliesthatanelectromagnetic waveispassing throughthesurface,andfromwhathasbeenfoundaboutthebehaviour ofsuchawaveweshouldexpectittobeveryrapidlyattenuated inside. Thecurrentassociated withitwouldtherefore flowonlyinathinskin 268 ELECTROMAGNETIC WAVES [10. ofthickness oftheorderofS,aphenomenon knownasthe'skineffect' Forsteadycurrents w=0andS=00,sothatthecurrentdistributio wouldbeuniform, butasthefrequency risesSdecreases andthecurren isincreasingly confined tothesurface. Forthisreasonthintubesar justasgoodconductors ofhighfrequency currents assolidrods,bu theresistance isofcoursehigherthanforasteadycurrentinasoli rodowingtothesmallereffective cross-section. Wewillnowcomput thehighfrequency resistance ofawire.Neglecting displacement cur rents,andwritingJforaEinequation (10.24),wehave V2J=ftftoa(oJjot)=jwftftoaJ (10.33 FIG.10.3.Current flowatsurfaceof cylindrical stripopenedouttoflatstrip ofwidthb=27Ta. Jy=Joexp(-xjS)expj(wt-xjS),xz AAforacurrentoscillating withfrequency (Wj27T).Foracylindrical wir weshouldexpresstheoperator V2incylindrical coordinates andth general solution involves Besse functions withacomplex argument ThevalueswehavefoundforSi coppershowthatathighfrequencie b=27Taitwillordinarily besmallcompare withtheradiusaofthewire.I thiscasewemayobtainanapprox imatesolution byneglecting th curvature ofthesurfaceofthewir andregarding itasathintubewhic canbesplitandunrolled toforma infiniteflatstripofwidthb=27Ta inwhichacurrentflowsparallelt thesurfaceofthestripintheIon direction.If(asinFig.10.3)w takethisdirection asthey-axisan thenormaltothesurfaceofth stripasthex-axis,thenthecurren densityhasacomponent aEyinth y-direction whichvarieswithxandtasgivenbyequation (10.30),s that whereJoisthecurrentdensityatthesurfaceofthewire(x=0).Thi equation showsthatthecurrentchangesphaseaswemoveintothewir aswellasdecreasing rapidlyinamplitude. Tofindtheeffective resistance Rperunitlengthofthewirewemus calculate thetotalcurrentIflowinginthestripatanyinstantandth 10.5J ELECTROMAGNETIC WAVES 269 powerdissipation perunitlengthW,andsetW=t/12/R. Now 00 00 1=bJJydx=bJoexp(jwt)Jexp{-(l+j)xjS}dx o 0 =(M.lo.)exP(jwt) =.~bSJoexpj(wt-l7T)·l+J v2 Thisexpression showsthatthetotalcurrenthasadifferent phasefrom .lo;werequiretoknowonlytheamplitude ofthetotalcurrentwhose squareis 1121=i(M.lo)2. Theinstantaneous powerdissipation perunitvolumeofthemetalispJ;, andhencethetotalaverage dissipation perunitlengthofthewireis 00 00 W=bJiplJ~1dx=ibpJ~Jexp(-2xjS)dx=!bpSJ~, o 0 wherepistheresistivity ofthewire.Hencetheeffective resistance per unitlengthofthewireis R=2Wj112/=pj(b8)=pj(2-rraS). (10.34) Thisequation showsthattheresistance isthesameasiftherewere acurrent1ofuniform densityflowinginathintubeofradiusaand thickness equaltoS,andthereasonforthename'skindepth'isthus apparent. Thehighfrequency resistance ofsuchawireisthusgreater thanthed.c.resistance byafactor(aj2S),whenS~a. Thistypeofcalculation maybeappliedtothemoregeneralproblem ofthepowerdissipated perunitareaofaplanemetallic surfacewhen thereisatangential magnetic fieldll:;=Hocoswtatthesurface. At anypointinthemetalthecurrentdensityis Jy=aEy=aZoll:;=(1+j)ll:;jS (fromequation (10.32», andhence.lo =-v'2Ho/S.Fromtheaboveitisreadilyseenthatthemean powerdissipation perunitareais W=IpSJ~=ipH~/8. (10.3480) ThisisjustequaltotheaveragevalueofPoynting's vectoratthesur­ face(seeProblem 10.1). 10.6.Reflection andrefraction ofplanewavesattheboundary oftwodielectrics Thereflection andrefraction oflightwavesatthesurfaceseparating twomediaofdifferent refractive indicesisafamiliarphenomenon and wemustnowinquirewhether electromagnetic theoryoffersasimple explanation ofit.Weassumetwonon-conducting dielectric media, (10.35270 ELECTROMAGNETIC WAVES [106 separated byaplaneboundary whichwetaketobethexy-plane (t planez=0).Wealsochoosethedirection ofthex-axistobeint planeofincidence, i.e.theincident rayliesintheplaney=0,asi Fig.10.4,makinganangle8withthenormaltotheboundary. (J' knownasthe'angleofincidence'. Thenallfieldcomponents ofth incident wavevarywiththespaceandtimecoordinates as exp[jw{t-n1(xsin (}+zcos(})/c}], wheree/n1isthevelocityofthewaveinthefirstmedium, andW/21Ti itsfrequency ofoscillation. Whentheincident wavefallsontheboundary therewillingener bebothareflected waveandatransmitted wave.Weknownothin aboutthesewaves,eitherastotheirfrequency ortheirdirection. W have,however, boundary conditions forthecomponents ofEand atthesurface,andweassumethatthebehaviour ofEandHmustb thesameasderivedearlierinChapters 1and5.Hencethetangenti components ofEandHmustbecontinuous onthetwosidesofth boundary atalltimesandforallvaluesofxandy.Thefirstofthes conditions showsthatthereflected andtransmitted wavesmusthav thesamefrequency astheincident wave.Secondly, atallpointsinth boundary planealongalineforwhichxisconstant thefieldcomponent oftheincident waveareconstant inamplitude andphase.Ourboundar conditions canonlybesatisfied everywhere ifthisistruealsoofth reflected andtransmitted wavesaswell,andtheymusttherefore als travelintheplaneofincidence. Thecorresponding raysareshown. Fig.10.4,makingangles (Jfand(}"withthenormalrespectively, kno asthe'angleofreflection' andthe'angleofrefraction'. Thefieldcom ponentsofthetwowavesmustbepropagated as (reflected wave) exp[jw{t-n1(x sin8'-zcose')/e}], (transmitted wave)exp[jw{t-n2(xsin8"+zcoself)Ie}], wheree/n2isthevelocityinthesecondmedium. Attheboundary plan (z=0),theboundary conditions canonlybesatisfied everywhere i thearguments oftheexponentials fortheincident, reflected, andtrans mittedwavesareallidentical. Hence n1xsin8=n1xsin(}'=n2xsin(}", whichgives 8=(J', sothattheanglesofincidence andreflection areequal,and n1sin8=n2sin8", whichisSnell'slawforrefraction. 10.6] ELECTROMAGNETIC WAVES 271 Theselawsofreflection andrefraction aretruenotonlyforelectro­ magnetic wavesbutforallkindsofwavemotion, sincetheydepend onlyontheassumption thatthecharacteristic quantities involved in thewavemotion(inthiscasetheelectricandmagnetic fields)shallbe continuous attheboundary. FIG.10.4.Reflected waveOBandrefracted wave00attheboundary (z=0)between twomedia.AOistheincident wave,andtheelectricvectorsareintheplaneofincidence asshown.Themagnetic vectorisparalleltothey-axis,andhencenormaltotheplane ofthepaper.Ifn2>nl'andthepermeability ofbothmediaisunity,thenthedirection ofE'isasshownwhentheangleofincidence isgreaterthantheBrewster angle tan-l(n2/nl),butintheopposite direction whenitislessthantheBrewster angle. Intensity relations Inordertoobtaintheamplitudes ofthereflected andrefracted waves wemustexamine theprobleminmoredetail,andmatchtheamplitudes ofthetangential components oftheelectricandmagnetic fieldsonthe twosidesoftheboundary. Todothiswemustconsiderthetwoprincipal directions ofpolarization oftheincident waveseparately. Thesedirec­ tionsarewiththeelectricfieldintheplaneofincidence andnormalto 272 ELECTROMAGNETIC WAVES L106 (10.3)theplaneofincidence respectively. Fortheformerofthesetherea components ofEparalleltothex-andz-axes,butE!I=0;theonl component ofHisparalleltothey-axis.Thesestatements holdforaI thethreewaves,sothatwehave(seeFig.10.4): Incident wave: Ex=AcosOexp[jw{t-n1(xsinO+zcosO)je}] } Ez=-Asin0exp[jw{t-n1(x sinO+zcosO)jc}] Hy=(AfZl)exp[jw{t-n1(xsinO+zcosO)je}] Reflected wave: (10.3) (A-A')fZl =A"jZ2'AcosO+A'cos0=A"cos0" andE~=A'cosoexp[jw{t-n1(x sinO-zcosO)jc}] } E~=A'sinOexp[jw{t-n1(x sinO-zcosO)jc}] H~--(A'fZl)exp[jW{t-nl(xsinO-zcosO)jc}] Refracted wave: E;=A"cos0"exp[jw{t-n 2(xsinO"+zcosO")!e}]} E:=-A"sin0"exp[jw{t-n 2(xsin0"+zcos8")Ie}] HZ=(A"jZ2)exp[jw{t-n 2(xsinO"+zcosO")!c}] whereZl>Z2aretheintrinsic impedances ofthetwomediarespectivel . Onmakingthetangential components ofEandH(i.e.ExandH) continuous attheplanez=0,weobtaintherelations whichgiveA'Z2cos0"-Zlcos0 A-Z2COSO"+ZlCOSO(10.41 (10.43(10.42 A'sm20"-sin 20 A=sin20"+sin20'andA"_2Z2cos0 A-Z2COSO"+ZlCOSO' IfthemediaaresuchthatJ1-1=J1-2=1,asisthecaseforlightofvisibl wavelengths, Zl!Z2=n2/n1=sinO/sinO", andtheseformulae canb writtenintheform A"4sin0"cos0 A=sin20"+sin20· Thecondition thatthereshallbenoreflection (A'=0)isthat sin20=sin20",(10.44 10.6] ELECTROMAGNETIC WAVES 273 whichissatisfiedif28and28"aresupplementary angles;thatis 0+0"=17T, sothatthereflected andrefracted raysarenormaltooneanother. UsingSnell'slaw,wefindthatthisoccurswhen tanO=n2(nl• (10.45) TheanglewhichsatisfiesthisrelationisknownastheBrewster angle. FIG.10.5.Incident waveAO, reflected waveOB,andrefracted waveOCattheboundary between twomedia.Theelectricvectorisnormaltotheplaneofincidence andhence normaltotheplaneofthepaper;themagnetic vectorisintheplaneofincidence. The theoryshowsthatifn.>n1,andthepermeability ofbothmediaisunity,theactual direction ofH'isopposite tothatgiveninthefigureforallvaluesof6. Thefieldcomponents ofthewavewhichisplanepolarized withits electricvectornormaltotheplaneofincidence areasfollows(see Fig.10.5). Incident wave: E 1I=BPI } H~=-(B(Z~)cos8F;. (10.46) Hz=(B/Z1)smOP 1 851110 T 274 ELECTROMAGNETIC WAVES [10. Reflected wave: E~=B'F2} H~=(B:jZl)C~S()~ (10.47 Hz=(B/Zl)sm()F2 Refracted wave: E;=B"Fa )H:=-~,B"/Z~)co:(}" Fa (10.48 Hz=(B/Z2)sm()Fa wherefor simplicity wehavewrittenFl,F2,Faforthethreecomple exponentials representing thepropagation ofthewaves,whichareth sameasinequations (10.36), (10.37), and(10.38)respectively. Theboundary conditions forElIandHxnowgive B+B'=B", (B-B')coS(}jZl =B"cos(}"jZ2' thesolutions ofwhichare (10.51(10.50(10.49) andB'Z2cos(}-Zlcos()" B=Z2cos8+Zlcos0" B" 2Z2cos()Ii=Z2COSO+Zlcos8'" Theseequations aresimilartothoseobtained fortheotherdirectio ofpolarization butnotidentical, sothatthemagnitudes ofthereflecte andrefracted wavesaredifferent. Thisisstilltruewhenwespecializ tothecaseoffJ-l=fJ-2=1,whentheformulae become B'sin(O"-(}) B=sin(O"+O) , (10.52B"2sin0"cos() B=sin(O"+()). Sinceingeneral ()"isnotequalto0,equation (10.51)showsthatther isnoangleatwhichthereflected waveforthisdirection ofpolarizatio haszeroamplitude. Iftherefore westartwithunpolarized light,whic consistsoflightcontaining asuperposition ofmanycomponents wit theirelectricvectorsinrandom orientations, andreflectitfrom dielectric suchasglassattheBrewster angle,thereflected wavew' onlycontainonedirection ofpolarization, thatwiththeelectricvecto normaltotheplaneofincidence. Thisphenomenon cantherefore b usedtoproduceplanepolarized light,andoriginally itwasusedtodefin theplaneofpolarization ofthereflected lightattheBrewster anglea 10.6] ELECTROMAGNETIC WAVES 275 beingtheplaneofthereflected ray.Ourtreatment showsthatthisis infacttheplaneofthemagnetic vectorinthereflected light,notofthe electricvector. Thefactthattheexpressions for(A'fA)and(A"fA)arerealshows thatthephasechangesattheboundary between twoperfectdielectrics arealways0or77.Inspection oftheequations showsthatforawave withtheelectricvectorintheplaneofincidence, passingfromamedium oflowerrefractive indextooneofhigherrefractive index,thereisa changeofsigninthetangential component Ex(butnotinEz'By)ofthe reflected wavewhentheangleofincidence islessthantheBrewster angle,whereasthereverseisthecasewhenitisgreaterthanthisangle. Forthewavewiththeelectricvectornormaltotheplaneofincidence, thereisachangeofphaseinEyandllzinthereflected wave(assuming n2>nt),butnotinHx'whatever theangleofincidence. Equations (10.43), (10.44), (10.51), and(10.52)areknownasFresnel's formulae, aftertheirdiscoverer. Fornormalincidence theygiveindeter­ minateresults,butitisreadilyseenfromequations (10.41)and(10.42) (or(10.49)and(10.50»thatwehavethen (10.54)(10.53) andA'Z2-Z1 n1-n2 A=Z2+Z1=n1+n2 A"2Z22n1 -==--,AZ2+Z1 n1+n2 wheretheexpressions ontheextreme rightapplyonlywhen iL2=iLl=l. The'reflecting power'ofthesurfaceisequalto(A'fA)2,sincethepower intheincidentandreflected wavesisproportional tothesquareofthe amplitude. IfZ2>Zt,AII>Asothattheamplitude oftheelectric vectorisgreaterinthetransmitted wavethantheincident wave.Since thepowervariesas(amplitude ofelectricvector)2fimpedance, thetrans­ mittedpowerislessthantheincident power,anditisreadilyverified thatthedifference isequaltothereflected power. Totalinternalreflection Whentherefractive indexn2ofthesecondmedium islessthannt, application ofSnell'slaw(equation (10.35» sinO"=(ntfn2)sinO tofindtheangleofrefraction leadstovaluesofsin0"greaterthanunity whensinOisgreaterthann2fnt.Sincethereisnorealangleforwhich 276 ELECTROMAGNETIC WAVES [10. thesineisgreaterthanone,weconclude thatthereisnorefracted wav andthatalltheenergyisreflected. Thisisconfirmed byinspection 0 equations (10.41)and(10.49),forwhensinO">1,cosO"=(1-sin20") isapurelyimaginary quantity, andtheexpressions for(A'/A)an (B'/B)areeachoftheform(a-jb)/(a+jb) whosemodulus isunity. Thusthereflection istotal,butthereisachangeofphaseonreflectio whichisdifferent forthetwodirections ofpolarization. Iftheinciden waveisplanepolarized inadirection whichisnotinornormaltoth planeofincidence, thetwocomponents intheseplanesofthereflecte wavewillnotbeinphaseandthewavewilltherefore beellipticall polarized. Thefactthatthereisno'refracted wave'doesnotmeanthatthere isnodisturbance inthesecondmedium, forequations (10.42)and(10.50) showthat(A"/A)and(B"/B)arefinite.Inordertofindwhatkind0 waveispropagated inthesecondmedium, wewritesin0"=coshy (sincecoshyisalwaysgreaterthanone),andthen cosO"=(1-sin20")t=j(cosh2y-1)t=±jsinhy. Hencethefieldcomponents inthesecondmedium arepropagated as exp[jw{t-n2(xcoshy-jzsinhy)/c}] =exp{(-wn2Zsinhy)/c}exp[jw{t- (n2xcoshy)/c}] =exp{(-21TzsinhY)/A2}exp[jw{t-(nlxsinO)fc}], (10.55) where>"2isthewavelength oftheradiation inthesecondmedium. Equation (10.55)showsthatthewaveisrapidlyattenuated onthefar sideoftheboundary, sinceatadistance z=A2itsamplitude fallsby exp(-21TSinhy). Foragivenvalueofx,thereisnophasechangeasweproceedinthe z-direction, butthereisaphasechangeaswemovealongtheboundary inthex-direction. Thisisbecause awavefrontobliquely incident on theboundary arrivesatpointsofgreaterxatalatertime. Ifalltheincident energyisreflected, noenergycanbetransmitted in thesecondmedium.Ifwecompute thevalueofPoynting's vectorfora direction normaltotheboundary (i.ethevalueofN"z=ExNyor-EyHx' according tothedirection ofpolarization) inthesecondmedium, we findthatithasapurelyimaginary value,becauseHyisc~7Toutofphase withEx.ThismeansthattherealpartofNziszero,andnoenergyis transported awayfromtheboundary ontheaverage. Energy does flowintothesecondmedium, sincethestoredenergymustbefiniteif thefieldcomponents arefinite,buttheflowisintheopposite direction 10.6] ELECTROMAGNETIC WAVES 277 duringalaterpartofthecycleandthestoredenergyisreturned tothe firstmedium. 10.7.Reflection fromthesurfaceofa.metal Whenanelectromagnetic waveisincident ontheplanesurfaceofa conducting medium, theamplitudes ofthereflected andtransmitted wavescanbecalculated inamanneressentially similartothatused fordielectrics. Therefractive indexandintrinsic impedance ofacon­ ductingmedium arenowcomplex numbers, andtheformulae deduced fordielectrics maybetakenoverastheystandbyuseofacomplex valueforn2andZ2'Fromequations (10.41)and(10.49)itisthen apparent thatthephasechangeonreflection isnotingeneral0or77, sothataplanepolarized wavewillbecomeelliptically polarized after reflection. Hereweshallonlycalculate therefl.ecting powerofametal forawavefallingonitfromfreespaceatnormalincidence, usingequa­ tion(10.53).Foragoodconductor Z2=(l+j)/(ooS) fromequation (10.32), whileforfreespace Zl=P,oc, whereSistheskindepthinthemetalandcisthevelocityoflightin freespace.Thusthereflection coefficient is A'Z2-Zl (l+j)-P,ooocS A=Z2+Zl=(l+j)+P,ooocS' wherethecomplex valueshowsthatthereisaphasechangeevenat normalincidence, andthereflecting poweris IA'12=!(1-t)+jI2 =2+t2-2t A(l+t)+j 2+t2+2t' wherewehavewrittentforthequantity (P,ooocS).Forametal,where 00:::::::107(ohm-metre)-l, t:::::::(4X109S),andisthusmuchgreaterthan unityatallfrequencies uptothatofvisiblelight(f:::::::1015c/s).Hence approximately I(A'/A)12=1-4/t=1-2'1X1O-5(p,f/oo)!. (10.56) Thisformula showsthatmetalsshouldbealmostperfectreflectors of electromagnetic radiation forallfrequencies uptothoseofvisiblelight. Forcopperatafrequency of1010c/s(awavelength of3em),thereflect­ ingpowerdiffersfromunitybyabout2·7X10-4,andthisformulawould leadustoexpectthatitshouldonlyfallbelowunitybyabout0·09at afrequency of1015c/s(awavelength of3000A).Thefactthatcopper 278 ELECTROMAGNETIC WAVES [10. appearsstrongly coloured showsthatthisformula failsforopticalfre quencies, whenelectrons intheatomotherthanthoseresponsible fo theconductivity begintoplayarole.Inaddition, theeffective con ductivity ofametalathighfrequencies islessthanthelowfrequenc value(seeProblems 18.2and18.5). Thehighreflection coefficient ofametalarisesfromthefactthatit intrinsic impedance isverymuchsmallerthanthatoffreespace.Fo copperat1010cjs,Z2is0·026(1+j) ohms,whileforfreespace Zl=376·7ohms. Thustosatisfytheboundary conditions thewavemustbealmosttotall reflected withaphasechangeintheelectricvectorbutnotinthemag neticvector,makingEalmostzeroatthemetalsurface,andHalmos twicetheamplitude duetotheincident wavealone. 10.8.Thepressure duetoradiation Whenaplaneelectromagnetic wavetravelsthrough aconductin medium inthex-direction, aconduction current flowsofdensit Jy=uEy(assuming thewavetobelinearly polarized paralleltoth y-axis). Associated withthewaveisafluxofmagnetic induction Be whichwillexertaforceonthecurrentwhosemagnitude onavolum element d-risdFx=(J/\B)xd'T=JyBed'T inthepositive x-direction. Thisforceisinthedirection inwhichth energyistravelling, andgivesrisetoa'radiation pressure'. Fro equation (10.7),Jy=-oHejox-oDyjot, sinceallothercomponent vanishinaplanewave,andhence dFxjd-r=(-oHejox-oDyjot)B e =-(o~jox)Be-o(DyBe)jot+(oBe/ot)DIf· Butfromequation (10.3),BBe/ot=-BEy/OX, sothat dFjd'T= -(noEy+BoHz)_~(nB )x yox eoxBtyz =-(oU/ox)-€€ot-tt-to(oNxlot) inamedium whereD,Barelinearly proportional toE,H.Inth steadystatethesecondterm(oNxlot)vanishes whenaveraged over cycle,andthesignificance oftheminussignbefore(0U/ox)isthatth forceisintheforwarddirection provided thatUdiminishes asthewav travelsonwa.rds, asitnecessarily willinaconducting medium. Weca 10.8] ELECTROMAGNETIC WAVES 279 interpret thevolumeforceasduetothegradient ofapressureP,and, fromthegeometry showninFig.10.6, dFa:=-(oPjox)dxdydz =-(oPj8x)dT. U=vG=N/v. (10.58) Thisresultisconsistent withquantum andrelativity theorybywltich radiation consistsofphotonsofenergyhv,whosemomentum invacuoHence P=U(normalincidence). (10.57) Thepressure isexertedinthedirection ofPoynting's vectorN,andcan beattributed toamomentum Gperunitvolumewhichflowsacrossunit areaattherate Pdydz---(P+~dX) dydz 4 FIG.10.6.Interpretation ofvolumeforceasgradient ofapressure. ishv/c.Fordiffuseradiation, consisting ofwavestravelling inalldirec­ tions,onlyone-third ofthetotalenergydensitywillonaverage be associated withwavestravelling normaltothesurface,sothatwehave p=lU(diffuseradiation). (10.59) Whenawaveistotallyreflected, itsmomentum isreversed sothat thepressure isdoubled; however theenergydensityisalsodoubled so thattheequationP=Ufornormalincidence andequation (10.59)for diffuseradiation stillhold.Theseresultsareeasilyobtained fromthe conceptofmomentum flowintheelectromagnetic wave;othermethods aremuchmorecomplex, asisillustrated byconsidering theradiation pressure onagoodconductor (suchasametal)onwhichaplanewave fallsatnormalincidence. Letthesurfaceofthemetalbetheplane x=0,withmetalatx>0,andvacuumatx<0,asinFig.10.7.Within 280 ELECTROMAGNETIC WAVES [10.8 themetaltherewillbeaforceontheconduction current, which,from above,isJyl.tl.to~dx perunitareainanythickness dx.Nowfromthe equation curlH=J,wherewehaveneglected thedisplacement current becauseitisverysmallincomparison withtheconduction currentin Vacuum Stressonouter surface =t.uoH~x=o HoH{;;Metal Stressoninner FIG.10.7.Pressure onatotallyreflecting magnetic metal,arising fromstressatthesurfaceandthevolumeforceontheconduction current. ametal,wehave ~/=-oHz/ox. HencetheoverallpresHure duetoth forceontheconduction currentis 00 00 I.tl.tofJyHzdx=-I.tl.tofHz(oHz/ox)dx =tl.tl.toH~, o 0 whereHoistheinstantaneous valueofHzatthesurfaceandwehav usedthefactthatHz=0atx=00. Inaddition tothisvolumeforce,wemustincludethepossibility 0 stressesattheboundary duetotheelectricandmagnetic fields.I generalsuchstressestakeatensorform(the'Maxwell stresstensor' see§1.7andProblem 8.9),butinthepresentcasethefieldsareparalie totheboundary, andgiverisetoapressure t(EoE~+f-loH~) fromth vacuum side,andt(EEOE~+,u,uoH~) onthemetalside.Poragoodcon ductorthereflection coefficient ispractically unity,andsincethefield aretangential andcontinuous acrosstheboundary, Ey=O, Hz=Ho=2H, 10.8J ELECTROMAGNETIC WAVES 281 (10.3)(10.1) (10.2)whereHistheamplitude intheincident wavealone.Hencethenet pressure onthemetalis (stressonvacuum side)-(stress onmetalside)+ +(force onconduction current) =!JLoHg-!fLfLoHg+!fLfLoH~ =!fLoH~ =2fLoH2. Sincetheenergydensityintheincident waveis U=!(€OE2+ fLoH2)=fLOH2, thenetpressure isjusttwicetheenergydensityintheincident wave alone;thatis,itequalsthesumoftheenergydensitiesintheincident andreflected waves. Thisexample showsthatingeneralwemustincludeboththestresses ataboundary andthevolumeforceontheconduction current. Inthe derivation ofequation (10.57)weconsidered acasewithnochangeof medium sothattheboundary stresseswereabsent,andweobtained the correctanswerfromthevolumeforcealone.Ontheotherhand,when anelectromagnetic waveispartially reflectedattheboundary between twonon-conducting dielectrics thereisnoforceonanyconduction currentandthepressure attheboundary arisesentirely fromthe difference intheMaxwell stresstensoroneithersideoftheboundary. It canagainbecalculated muchmoreeasilyfromthemomentum balance byconsidering theenergydensitiesintheincident, reflected, andtrans­ mittedwaves;thatthetwoapproaches givethesameanswerinthecase ofnormalincidence isverifiedinProblem 10.14. 10.9.Radiation fromanoscillating dipole InChapter 5theconceptofavectorpotential A,suchthat B=curIA, (10.60) wasintroduced. Thevectorpotential isnotofgreatuseinelementary problems, butisofconsiderable assistance incalculating theradiation fromanaerial.Thefundamental equations oftheelectromagnetic field aredivD=p, divB=0, 8BcurlE=--,8t 8DcurlH=J+-.8t(10.7) 282 ELECTROMAGNETIC WAVES [10. Onsubstituting fromequation (10.60)into(10.3)wehave,asin§6.1 curl(E+a:)=0, thesolutionofwhichis aAE=---grad V,Bt(10.61 wheregradVisthe'constant ofintegration', Vbeingsomescalarfunc tion.Sincecurlgradofascalarfunction iszero,anysuchfunctio satisfies equation (10.3).Inastaticproblem, whereAisconstan , equation (10.61)reducestoE=-gradV,whereVistheordinary scala electrostatic potential, asdefinedinequation (1.6). Equation (10.60)doesnotdefinethevectorpotential Acompletel , sinceAisthesolutionofadifferential equation andwecanaddtothi solutionanyvectorwhosecurliszero.Inequation (5.43)weaddedth condition divA=0,butforourpresentproblemitismoreconvenien togeneralize thiscondition intheform . BV 1BVdivA=-fLfLoEEO- =---,atv2Bt wherev=(fLfLoEEO)-iisthevelocityofelectromagnetic wavesinth medium. Thisdefinition reducestothatusedpreviously whenV . constant, andhastheadvantage thatitenablesustoseparate t variables VandA.Fromequation (10.61)wehave(usingequatio (10.1)) -V2V=-divgradV=diV(E+0:) =divE+~divA=L_.!..B2V,ot €€ov2at2 sothat -V2V+.!..82V=L. (10.6)v28t2EEO Again, curlB=curl(curlA) =graddivA-V2A =_12gradBV_V2A,vBt whilefromequations (10.7)and(10.61) (BD) DEcurlB=fLfLocurlH=fLfLoJ+-at=fLfLoJ+fLfLoE€0at 1{B2A BV}=fLfLoJ---n-+grad-.v2ot2at Identifying thesetwoequations forcurlBgives 1a2A-V2A+-_ =fLfLoJ. (10.6)v28t2 10.9]-------- ELECTROMAGNETIC WAVES 283 Forstaticsystems, orsystems whichvarywithtimeonlyslowly, equations (10.63)and(10.64)reducetotheequations (2.1)and(5.46) obtained earlier.Thegeneralsolutions ofournewequations arealso similartothoseoftheearlierequations, andmaybewrittenas V=_1_f[p]dT, (10.65) 4?TEEO r A=fLfLaJ[J]dT, (10.66) 4?Tr wherethesquarebrackets roundpandJhavethefollowing significance. ThevaluesofVandAatatimetandatapointdistance rfromthe elementofvolumecontaining pandJarerelated,nottothevaluesof pandJattheoriginatthesametime,buttothosevalueswhichobtained atatime(t-rjv).Inotherwords,thedisturbance setupbythevalues ofpandJattheoriginispropagated withthevelocity vandreachesa pointdistance rawayatatimelaterbyrjv.Thusthedisturbance at thispointisrelatedtowhathappened attheoriginatimerjvearlier, justasthelightreaching theearthfromastartellsuswhatwashappen­ ingonthestar,notatthesameinstant,butatthetimewhenthelight leftthestar.ThevaluesofVandAgivenbyequations (10.65)and (10.66)areknownas'retarded potentials'. Theseequations willnowbeappliedtothecaseofashortlengthof wiresattheoriginofcoordinates, carrying acurrent I=10coswt. Then,sinceJdT=Ids,theretarded value[J]dT=[I]ds,andifsis veryshortcompared withr,sothatrdoesnotchangesignificantly duringtheintegration, wemaywrite A=:rf[1]ds=:r[I]s, (10.67) wherewehaveassumed thatthewireisinvacuoandhavesetfL=1. Thenwecanalsowritev=c,thevelocityoflightinvacuo.Equation (10.67)showsthatthevectorpotential Aiseverywhere paralleltos, asshowninFig.10.8. Themagnetic fieldHcannowbefoundatanypoint,for H=BjfLo=(ljfLo)curIA =(lj41T)curl([ l]sjr) =(lj4?T){[~]curlS-S!\grad[~]} =-(lj4?T){S!\ grad([l]jr)} 284 ELECTROMAGNETIC WAVES [10. sincecurls=O.Nowgrad([IJjr) =r1o([IJjr)jor, wherer1isauni vectorinthedirection ofr,since[IJvariesinspaceonlywithr.Henc H=-(lj41T){s 1\r1}:r{[1Jjr}, showingthatHisnormaltosandtorl'andithasthusonlyoneco­ ponent, H.pinspherical polarcoordinates. z A FIG.10.8.Radiation fromashortdipoles;atthepoint(I',0,.p)the magnetic vectorpotential Aisparalleltos;Poynting's vuetorNis alongtheradiusvectorr,andthefieldcomponents areEo(alonga lineoflongitude) andHef>(alongalineoflatitude). Theretarded value[IJ=10cosw(t-rjc), andhence o L wIo.-{[IJjr} =_.J!cosw(t-rjc)+-slllw(t-rjc)or r2 cr L 21TIo.=-~cosw(t-rjc) +~slllw(t-rjc),r r/\ whereAisthewavelength oftheradiation. Themagnetic fieldisth givenbythesolecomponent 8LsinO 8LsinO.H.p=:rrr2cosw(t-rjc) -~rA-slllw(t-rjc). (10.6) Thefirsttermpredominates atshortdistances (r~A)fromtheorigi, andwillberecognized asjustthefieldgivenbythelawofBiota d Savart(equation (5.37».Itisknownasthe'induction field'.Atlare distances ofmanywavelengths fromtheoriginonlythesecondterm's 10.9] ELECTROMAGNETIC WAVES 285 (10.72)(10.69) (10.70)significant; itfallsoffas('\r)-linsteadofr-2andisknownasthe'radia­ tionfield'. Theelectricfieldmaybefoundbyusingequation (10.3),oritmay befoundfromequation (10.61)ifthescalarpotential Visfirstcomputed bymeansofequation (10.62). Weareinterested onlyinitsvalueat alargedistance fromtheorigin,whenourspherical wavefronthasvery smallcurvature andoverasmallregionmaybetakenasaplanewave. Aswewouldexpect,EisthennormalbothtoHandtothedirection of propagation, anditsonlycomponent is Eo=ZoH,p=(Jl-o/€o)!H,p. Theenergycrossing unitareapersecondisgivenbyPoynting's vector N=EoH,p=ZoH~, showingthattheenergyflowisradiallyoutwards. Themeanvalueof Naveraged overacycleofoscillation is }{=lZo(los/r,\)2sin20 andthetotalenergyradiated inalldirections persecondis W-I7T 22}{'OdB_7TZO-PoS2 -7Trsm-3,\2. o Thepowerradiated maybeexpressed intermsofaresistance Rr,called the'radiation resistance', obtained bywritingW=tRr1~,sincethis isjustthemeanpowerwhichwouldbedissipated inarealresistance Rr• Forourcurrentelement equation (10.70)gives Rr=2~Zo(Xr =789(s/,\)2 ohms, (10.71) showingthattheradiation resistance depends onlyonthesquareofthe ratioofthelengthofthewiretothewavelength. Wemayimagine ouroscillating currentelementtobeduetotwo oscillating charges±qosinwt,separated byashortdistance s.Thisis anoscillating electricdipoleofmoment Posinwt=sqosinwt,andthe currentis I /=dqdt=wqocoswt, gIvmg sl=wPocoswtorsIo=wPo- Thetotalradiated powermaythenbewrittenas Z2 2 A_32Z4 2 4 2JV=7T0wPo=~=0wPo=Jl-owPo 3,\2 3€0,\4 127Tc2127Tc (usingZo=(€OC)-l=Jl-oc),showingthatWisproportional tothesquare oftheoscillating dipolemoment andthefourthpowerofthefrequency. 286 ELECTROMAGNETIC WAVES [10. Sincetheoscillating currentelementcanberegarded asanoscillatin electricdipole,itisinteresting tocalculate thescalarpotential an compare itwithourearlierresultsforastaticdipole.SinceAhas component onlyparalleltothepolaraxis(whichwewillcallthez-axis), divA=oAzjoz.Inthedifferentiation x,yareconstant and (ojoz)x,y =(zjr)(ojor) =cosfJ(ojor) (cf.§2.2). Hence,usingequations (10.62)and(10.67), • c2JLL8cosfJaoVjot=-c2divA= _ 0 0_{r-1cosw(t-rjc)}. 41Tor Sincerandtareindependent variables wemayintegrate withrespec totinsidethedifferential, giving VC2JLo108cosfJa{-1'(tj)1.= - -rslnW-rcf477'wor =_PocosfJ ~{r-1sinw(t-rjc)} 41T1'oor =__I_[po.grad{r-1sinw(t-rjc))]. 477'£0 Comparison withequation (1.11a)showsthatthisisofthesamefor exceptthatallowance mustbemadefortheretardation intakingth gradient. Itmustbeemphasized thattheequations wehavederivedapplyonl whenthelengthoftheoscillating dipoleissmallcompared withthewav­ length.Mostoftheradiation emittedbyatomsisduetooscillations electrons, andistherefore 'electricdipole'radiation. Thesizeoftheele­ tricdipolewillbeoftheorderoftheelectronic chargetimesanatomi dimension, andtheeffective lengthofthedipoleisthusabout10-8c ; sincethewavelength ofvisiblelightisabout10-5cm,ourtheorycoul beappliedtothiscaseiftherestrictions duetoquantum theorycouldb neglected. Thusclassical theorypredictsthatanelectron movingina orbitroundthenucleuswouldbehaveasatwo-dimensional oscillat andloseenergycontinuously byradiation untilitfinallyspiralsintoth nucleus. Thisdifficulty couldnotbeovercome inattempts tofindasati­ factorymodeloftheatomuntilBohrintroduced theconceptofstationa orbitsinwhichtheelectron didnotradiate. Thusthedifficulty aros fromtheuseofclassica.ltheory,whichisagoodapproximation formacr­ scopicoscillators, inanatomicproblem whereitwasnotapplicabl . Nevertheless, theclassicaltheorygivesa.goodexplanation ofthephen­ menaofdispersion andscattering (seeChapter 17),andmostoft 10.9] ELECTROMAGNETIC WAVES 287 (10.74)qualitative resultswhichitprovides, suchasthepolarization ofthe radiation andtheabsence ofradiation inthedirection inwhichthe dipoleispointing, arevalidforatomicsystems. Atradiofrequencies theaerialorantenna usedforthereception or transmission ofbroadcast signalsisnotingeneralshortcompared with thewavelength. Itsradiative properties maybecalculated bymethods similartothoseusedabove,thevalueofAbeingobtained bytheuse ofequation (10.66).Forasinglestraightwirethisgives A=PP'of[1]ds, 4-n-r wheretheintegration isalongthewireand1isthecurrentintheelement ds.Twopointsmustbenotedinperforming theintegration: (a)the currentisnotingeneralconstant alongthewire,sinceitmustfallto zeroattheends;(b)inevaluating theretarded potential, allowance must bemadeforthephasedifference insignalscomingfromdifferent parts ofthewireowingtothechangeinthedistance. Theproblemissimilar tothatofthediffraction patternofasingleslitwheretheamplitude variesovertheaperture. Theinterference concepts usedforlightwaves maybeappliedin·findingtheradiation patternproduced byanaerial arrayconsisting ofmanyelements; theproblem isessentially similarto thatofadiffraction grating. REFERENCE BmGE,R.T.,1941,Ann.Rep.Progr.PhY8.(Physical Society, London), 8,90. PROBLEMS 10.1.Ametallic sheetisbounded bytheplanex=0andatangential oscillating magnetic fieldHII=Hocoswtexistsatthesurface. ShowthatthetotalcurrentI perunitwidthofthesurfaceflowinginthemetalhasaninstantaneous valueequal toHIIandthatitsdirection isnormaltoHII•Verifythatthisconforms with equation (5.22),ifthefH.dsistakenroundasuitable rectangular circuitwith twosidesparalleltothex-axisandtwoparalleltothey-axis,oneofthelatter beingjustoutsidethemetalandtheotherinsideata.depthmuchgreaterthan theskindepth. Compute themagnitude ofPoynting's vectorjustinsidethesurfaceofthe metalandshowthatitsmeanvalue(averaged overaperiodofoscillation) is justequaltothepowerperunitareadissipated inheatingthemetalasgiven byequation (10.3411.). 288 ELECTROMAGNETIC WAVES Showalsothat10.2.Showthatthesuperposition oftwowaveBofequalalllplitude, onewit1 angular frequency w+dwandphasec!")Ilstant [3+d[3,theothol'withw-dwa [3~d[3,givesawaveoffrequency wandphaseconstant [3,whoseamplitude varis as(sin)(tdw-xd[3). Henceapointofmaximum amplitude movesascos (tdw-xd[3)=constant, orwithvelocity u=(dw/d[3). Sincetheenergyisproportional tothesquare theamplitude, ugivestherateatwhichenergyispropagated, knownast e 'groupvelocity'. Showthatinagoodconductor, suchasametal,wherethedisplacement curre canbeneglected, thephasevelocity v(=w/[3)iswo,andthegroupvelocity (=dw/d[3)is2wo,where°istheskindepth. 10.3.Showfromequation (10.41)thatthecondition fortheretobenoreflect waveattheboundary between twoinsulators (El,JLl; E2,JL2)whentheelectrc vectorisintheplaneofincidence requiresthat Eltan8=E2tan8". Thisequation showsthatatthisparticular anglethelinesofelectric fielda e refracted asintheelectrostatic case(cf.Fig.1.13)sothatthoboundary con'. tionsaresatisfied without anyreflected wave.Similarly, whenthemagnet c vectorisintheplaneofincidence, thecondition fornoreflection is JLltan8=JL2tan8/1 sothatthelinesofmagnetic fieldarerefracted asinthemagnetostatic case. Verifythatingeneraltheboundary conditions forthenormalcomponents f DandBareautomatically satisfiedinthetheoryof§10.6whentheconditio forthetangential components ofEandHaresatisfied. 10.4.Therateatwhichsolarenergyfallsontheearth'ssurfaeoisapproximate y 2cal/cm2/minute. Calculate ther.m.s.valuesoftheelectricandmagnetic fiels attheearth'ssurface, andthepressure exertedonit,assuming ittobehave s aperfectabsorber. (Answer: E=730V/metre; H=1·9A/metre; pressure, 4·7>10-6newton/m .) 10.5.Showthatbytheintroduction ofacomplex dielectric COnstant E=E'_j/I, where E/I=(a/wEo),theequations (10.25a) maybewritteninthesameforms foranon-conducting dielectric. Showthatthisleadstotherelation n2=E, andthatE/I/E'=tano,wheretanoisthelosstangent ofthcdielectric (8 Problem 9.12). 10.6.Aslightly imperfect dielectric hasasmalllosstangent. Showthatint e firstapproximation thevelocity ofelectromagnetic wavesisthosameasifta werezero,butthepowerfallsasexp(-aZox) intravelling adistance x,where istheintrinsic impedance ofthemedium neglecting theconductivity a.Tis resulthasthesimpleinterpretation: inathickness dxthepowordissipated p r unitcross-section isaE2dx,whiletheincident powerW=E2/Z0•Hence -dW/dx =aZoW. aZo=217(losstangent ofthedielectric)/'\, where,\isthewavelength oftheradiation inthedielectric. ELECTROMAGNETIC WAVES 289 10.7.Whenaplanewaveisincident onaconducting wiretheelectricfieldsetup inthewireisequaltothetangential component oftheelectricfieldstrength in thewave.Showthatinashortstraight wirethepowerpickedupisproportional tosin2(1,where(Iistheanglebetween thewireandthedirection oftravelofthe wave.Thisshowsthatthedirectional properties ofthewirearethesamefor receiving asfortransmitting (cf.equation (10.69». 10.8.Whenaplanewavefallsonasmallplaneloopofwire,thee.m.f.induced intheloopisdetermined bytherateofchangeofmagnetic fluxthrough it.Show thatthedirectional properties ofthelooparethesameasthoseoftheshortwire, if(Iismeasured fromthenormaltotheplaneoftheloop,buttheplanesofthe electricandmagnetic vectorsinthewavemustbeinterchanged. Thisisconsistent withthefactthataloopcarrying analternating currentbehaves asanoscillating magnetic dipole. Showthattheratioofthee.m.f.setupinashortwireoflength 8tothatina smallloopofareaAis(A8/21TA), whereAisthewavelength oftheradiation.Ifthe lineardimensions oftheloopareroughlythesameasthoseofthewire(A::::J82), thisshowsthattheloopisamuchpooreraerialwhen 8<A. 10.9.Atransmitter radiates apowerWfromashortdipoleaerial.Showthatthe r.m.s.electricfieldatadistance Dintheequatorial planeofthedipoleaerialis E=(3ZoW/81TD2)!, whereZoistheintrinsic impedance offreespace. IfW=1kW,showthatthefieldstrength atadistance of10kmisabout 0·021V/metre. 10.10.Atransmitter radiates apowerWfromashorthorizontal dipoleaerial locatedataheightHabovethesea.Showthatthesignalreceived atatarget whosedistance isDandheightabovetheseaish(D~H,h)isamaximum when h=DAj4H,whereAisthewavelength ofthetransmitter. Theseamaybeassumed toactasaflat,perfectly reflecting (conducting) surface. Showthatiftheheightofthetargetisverymuchsmallerthanthisvalue,the powerincident onunitareaofit(assuming ittolieintheequatorial planeofthe dipole)is67TWH2h2/D4A2.Thisequation showsthattheeffectoftheseaistomake thepowerfalloffwiththeinversefourthpowerofthedistance insteadofthe inversesquare;itshowsalsotheimprovement gainedbyusingshortwavelengths. IfW=1kW,D=10km,H=h=A=10metres, showthatther.m.s. electricfieldstrength atthetargetisabout2·7X10-4V/metre. 10.11.Obtainequation (10.56)bytheuseofequation (1O.34a) andtheLawof Conservation ofEnergy. 10.12.Acylindrical conductor offiniteresistance carriesacurrentI.Calculate thevalueofPoynting's vectoratthesurfaceofthewire,andshowthattheenergy flowingintothewireisjustequaltothatdissipated inheatingthewire. 10.13.Athighfrequencies themagnetic fieldfallsinsideeachlamination ofa transformer owingtotheskineffect;theeffective permeability athighfrequen­ ciesisfoundbycalculating thetotalfluxinalamination inphasewiththatat eachsurfaceofthelamination. Showthattheeffective r.f.permeability isless thanthestaticvaluebyafactorS/awhentheskindepthSisverymuchsmaller thanthethickness aofthelamination. MUro UJ I I I 290 ELECTROMAGNETIC WAVES Whatthickness oflamination isrequired tomakea=aforamaterial 0 permeability 105andresistivity 5X10-7ohm-metre atafrequency of50c/s? (AnBWer: 0'16mm.) 10.14.Aplanewavefallsatnormalincidence ontheboundary (z=0)between twomediawithconstants (lOl'1-'1)and(lO2'1-'2)respectively. Ifthefieldcomponents attheboundary areEa;,H'II'showthatthestressis !{(lOl-lO2)lOO E~+(p.l-1-'2)l-'om} andthatthisisequaltothesumofthemomentum flowsintheincident and reflected waveslessthatinthetransmitted wave;i.e.to E2+E'2 EN2 ZlVlZ2V2' whereE,E',ENaretheelectricintensities intheincident, reflected, andtrans­ mittedwavesrespectively, andZl'VI;Z2'v2aretheimpedance ofandphase velocity inthetwomedia. Hint:NotethatEa;=EN,HlI=EN/Z2andshowthatbothexpressions can bereducedto {I Z2~!EN2__+_1 _ ZlVIZ~VlZ2V' 10.15.Inaregionofspacecontaining nparticles perunitvolumeofchargeq andmassm,wherethepressure issolowthatcollisions maybeneglected, equations (10.3)and(10.7)maybewritten curlE=-/-'o(oH/ot); curlH=nqv+lOo(oE/ot). Showthatbyusingtheequation ofmotionqE=m(ov/ot), andeliminating v between theseequations (takinggraddivlOo E=grad(nq) =0),onecanobtain theequation o2Enq2-+- E=e2V2E.ot2lOom r=(nq2/47r2lOom)+(e 2/.\2). Forverylongwavelengths thisreducestotheequation forplasmaoscillations, equation (4.47).Thisistheequation ofTonksandLangmuir forpropagation ofwavesinanionized medium. Show,byassuming Etovarysinusoidally withfrequencyfandwave­ length.\,that 11 FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 11.1.Elements offiltertheory AFREQUENT requirement inradioandtelephony istheseparation of twosignalsofdifferent frequencies. Anycircuitwhoseimpedance varies withfrequency canbeusedforthispurpose, asimpleexample already considered beingthetunedcircuit,whichcanbeemployed toaccept orrejectanarrowbandoffrequencies centredonitsnaturalresonant frequency. Thisactsasa'bandpass'or'bandstop'filter.Another typeoffiltermayberequired topassallfrequencies uptoacertain LA_e-01~-a1-ea B-e-I--- T eD FIG.11.1.Simplelow-pass filtersection. value,andstopallhigherfrequencies; thisisa'low-pass' filter.The reversecaseisa'high-pass' filter,whichrejectsalllowfrequencies up toacertainvalue,andpassesallhigherfrequencies. Theactionofafiltercanbeunderstood byconsidering asimple example, thelow-pass filtershowninFig.Il.l.Acommon useofsuch afilteristoremovetheripplevoltagefromtheoutputofarectifierunit whichisconverting ana.c.voltagetoasteadyvoltage. Theoutputis appliedtotheterminals ABofthefilter,whichisrequired topassthe steadyvoltagecomponent ontotheterminals OD,butnotthealter· natingcomponent. Ifthelatterhasafrequency of,say,100cis,and thecapacitance 0ischosentohavealowimpedance at100ciswhilethe inductance Lhasahighimpedance, onlyasmallfractionoftheinput voltageatthisfrequency willappearattheterminals OD,becausethe inductance andsecondcapacitance actasavoltagedivider. Thefraction -,' ..,iL/292FILTERS, TRANSMISSION LINES, ANDWAVEG UIDES [1I.l isapproximately XC/XL=(l/wO)/wL =1/(w2LO),andif0=10fLF, L=25henries,thefraction isabout1/100.Ontheotherhand,ifthere isnoleakageinthecapacitor, thefullsteadyvoltagecomponent will bepassedonto0D.Thusthefilteracceptsthesignalofzerofrequency, FIG.II.2.Chainoflow-pass filtersections. andpartially rejectsthesignalat100CiSfrequency. Betterrejection isobtained byaddingmoresections ofthiskind,asin:Fig.11.2.This· arrangement iscalledastep-orladder-type filter.Itisclearthat evaluation ofthecurrents andvoltages inthedifferent elements by IZlI IZlI IZlI I I II I I r-- .....- .....- r-..... Z2~Z2r:;Z2(---.. Z2 \Tnil ,~ '--r- -..... .....- -..... FIG.II.3.Generalized typeofuniform ladderfilter. application ofKirchhoff's lawswouldbeverylaborious, anditisprefer­ abletoproceedinadifferent way,makinguseoftherecurrent nature oftheelements. Weshallbeginbyconsidering auniform filter,consisting ofachain ofsimilarsections, asin:Fig.11.3,whicharerepeated indefinitely, forminganinfinitechain.Ifagenerator isappliedatsomepointearlier inthechain,currents willflowinthevarioussections; letthecurrents insuccessive sectionsbeIn-vIn'In+1'Application ofKirchhoff's lawto 11.1JFILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 293 thecentralsectiongives /' II Z2(In-In-l)+ZlIn+Z2(In-In+1) =0 or -Z2In_l+(Zl+2Z2)In-Z2In+1 =O. '>(t1J.l) WemaywriteIn=aIn_l'whereaisarealorcomplex number; thenin aninfinitechainwherewecannotdistinguish between sectionswemust slsohaveIn+1=aIn"Equation (11.1)thengives t(a+1/a) =1+Z1/2Z2. (11.2) Thisequation determines theattenuation constant a.Weshallcon­ fineourattention tothecasewhereZl/Z2isreal,whichcorresponds to ZlandZ2beingbothpureresistances orpurereactances. Thenacan beeitherrealorcomplex, butnotapurelyimaginary quantity. The realrootsarisewhentheright-hand sideofequation (11.2)liesoutside therange+1to-I,i.e.Zl/4Z2liesoutsidetherange0to-1.We consider separately thethreecaseswhereitisgreaterthan0,between oand-1,andlessthan-1. (i)(Zl/4Z2)>0 aisrealandpositive. Sincethenetwork is'passive' (i.e.itcontains nopower-generating elements), thecurrents mustdecrease aswemove awayfromthegenerator attached tooneendofthefilter.Thuswetake a<1forawavetravelling towardstheright(i.e.generator attached to theleft-hand endofthefilter)anda>1forawavetravelling towards theleft.Thisinterpretation isconsistent withthefactthattheroots ofequation (11.2)arereciprocal; thusthewaveisattenuated bythe sameamountpersectioninthedirection inwhichitprogresses, which­ everwayitisgoing.Thesignificance ofthepositive signofaisthat thewaveisattenuated without changeofphase.Ifwewritea=e-<X, where existheattenuation constant persection,equation (11.2)becomes coshex=I+Z1/2Z2. (11.3) (ii)0>(Zl/4Z2)>-I aiscomplex, withmodulus unity,sothatwemaywritea=e-jfJ. Thewaveisnotattenuated atall,butsuffersaphasechangebyan anglef3ineachsection,where cosf3=I+Z1/2Z2. (11.4) (iii)(Zl/4Z2)<-1 aisthenrealandnegative, sothatthewaveisattenuated witha phasechangeof7Tinsuccessive sections.Ifwewritea=-e-<X,equation (11.2)becomes -cosh ex=1+Z1/2Z2. (11.5) 294FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.1 Herewehavealreadyspokenofthecurrentaspartofa'wave'and thisterminology needssomejustification. Ifthecurrentinonesection alternates atagivenfrequency, sothatwecanwriteIn=I~exp(jwt) , thenthecurrentinalatersection(n+m)willbe In+m=aml~exp(jwt). Ifwewritea=exp(-h), thecurrentinthesectionn+mwillbe In+m=I~exp(-hm)exp(jwt) =I~exp(jwt-hm), whichissimilartotheexpression forawavemotioninacontinuous medium exceptthatm,thenumberofsections, whichcanonlybean integer,replacesx,thedistance travelled inthemedium.Ifhisreal, wewriteitasex,andwehaveanattenuated wave,butifhisapure imaginary quantity jf3wehaveanunattenuated wavewithaphase change f3persection. Thevaluesofexandf3aregivenbyequations (11.3-5). Thereciprocal rootsareobtained bychanging thesignofh, corresponding toawavetravelling intheopposite direction. Thethird case(iii)above,wheretheattenuated wavechanges signinsuccessive sections, couldnotariseinacontinuous medium. Thebehaviour of thefilterisdetermined immediately fromthevalueof(Zl/4Z2).Ifit lieswithintherange0to-1,wehavea'passband'withnoattenua~ tion;outsidethisrangewehavea'stopband'. Theword'section' hasbeenusedsofarwithoutanyprecisedefinition ofitsmeaning. Theuniform chaincanberegarded asmadeupof similarsectionsjoinedtogether, buttwotypesofsectioncanbeobtained bycuttingthechainindifferent ways.Thesetwotypesareshownin Figs.11.4and11.5andforobvious reasonsareknownasT-sections and'1T-sections respectively. Itisclearthatasuccession ofeithertype ofsectionjoinedtogether givesthesamechainfilter(seeFig.11.6),so thatthetransmission characteristics derivedearlierapplytoeithertype ofsection. Ifagenerator ofvoltageVisconnected toasemi-infinite chainand thecurrentIdrawnfromitismeasured, adefinitevalueoftheratio (VII)isobtained, knownasthe'characteristic' or'iterative' impedance ofthechain.Itisobviousthatthesamevaluewouldbeobtained ifa numberofsectionswereremoved andtheimpedance ofthesemi-infinite chainmeasured atalaterpoint.Again,thechainmaybeseveredat somepointandtheinfinitetailreplaced byanimpedance equaltothe iterative impedance without alteringtheimpedance measured atthe inputterminals. Thisgivesusamethodofcalculating theiterative im­ pedance. Fig.11.4showsaT-section terminated byanimpedance ZT; 11.1]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 295 theimpedance measured attheinputterminals ABcanbecalculated bystandard methods, andonequating thistoZTwehave ZT=tZl+(~2 +IZl~ZJ-l. Solution ofthisequation gives ZT=(ZlZ2+1Zi)i =(ZlZ2)i(I+Z 1/4Z2)i. (11.6) IZlI 0' I I ...-- r-~ r-'- 2Zz 2Zz z. '----','-r- '-r- D'A'_-------.----1 B'_------- .........-------....l---- ...... --1 FIG.11.5.'IT-section A'B'O'D' tenninated byitsiterative impedance. Thisisthevalueoftheiterative impedance ZTforaT-section. By applying thesamemethodtoa'IT-section, asshowninFig.11.5,wefind theiterative impedance ofa'IT-section tobe Z7T=(ZlZ2)i(I+Z1/4Z2)-i. (11.7) Thus ZTZ7T=ZlZ2. (11.8) Theimportance oftheiterative impedance liesinthefactthatsections ofdifferent kindsmaybejoinedtogether toformanon-uniform filter, withoutdisturbing awavetravelling downthefilter,provided thatthey havethesameiterative impedance. Ifsectionswithdifferent iterative 296FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [ILl iInpedances areuseda'reflection' ofthewaveoccursatthejunction andthesimplefiltertheoryisnolongerapplicable. Similarly, thefilter mustbeterminated byaloadequaltoitsiterative impedance inorder toavoidareflection attheoutputterminals. Anysuchreflection will diminish thepowerdissipated intheload,sincepartoftheincident energywillbereflected. H-~~~- T-section 2Z2 ,'t-----t...--- ----- T-section 1T-section-~~~""'" FIG.11.6.FilterofT-sections suchasABODterminated bytheiterative impedance ZT­ Itmayalsobedividedinto7T-sections terminated bythenetwork totherightof0'D'_ 1 \ !.':'y ; \. C,-, \'" Zw2Z2~ 2Z2 Zw i \. (a) (b) FIG.11.7.Half-sections usedastransformers (a)converting ZTtoZ,,;(b)Z17toZT- Afiltercorrectly terminated bytheiterative impedance ZTisshown inFig.11.6,whereitisregarded asmadeupofT-sections (ABGD). Alternatively, wemayregarditasmadeupof7T-sections (A'B'G'D'), whichmuststillbecorrectly terminated. Thustheimpedance ofthe portiontotherightofG'D',shownseparately inFig.11.7(a),must beZ1T'Hencethehalf-section actsasatransformer fromtheimpedance ZTtoZ1T'asmayalsobeverifiedbydirectcalculation. Similarly, the ll.I]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 297 otherhalf-section showninFig.11.7(b)transforms theimpedance Z.. toZT'Toalimitedextentsuchhalf-sections canbeusedtomatcha­ loadtoafilter,buttheirespecial importance comesinthedesignof composite filters(seeProblem 11.4). '.'-}!(I11.2.Somesimpletypesoffilter Low-pass filter Thesimplest typeoflow-pass filteristhatshowninFig.11.2with capacitances intheshuntarmsandinductances intheseriesarms.Then Zl=jwLandZ2=1/(jwO), giving Zl/4Z2=-w2LOj4. _1_L Thecut-offfrequency Woisobtained bysettingthisequalto-I,giving Wo=2(LO)-i. Frequencies belowthisarepassedwithoutattenuation, i o 000_ 00p i 000---+00 000------'00~.,~t-It:. FIG.11.8.Variation of0:,{3andtherealpartofZTforasimplelow-pass filter. '1\j. whilehigherfrequencies areattenuated. Inthepassbandthephase changeisgivenby cos{3=l-w2LO/2=1-2{w/W O)2, showingthat{3changesfrom0to7Tasthefrequency increases fromzero tothecut-offvalue.Thecharacteristic impedance ofaT-section is ZT=(~_w:L2)t=(~)t(l_:Vl, whichisresistive inthepassband,butvariesfrom{LjO)ltozeroat thecut-offfrequency. InthestopbandZTisapurereactance andthe attenuation isgivenby -cosho:=l-w2LOj2, whichriseswithfrequency fromzeroatw=wo0Thenegative sign showsthatcurrents insuccessive sections arereversed indirection. Thegeneralbehaviour of0:,(3,andZTisillustrated inFig.11.8.The variation ofZ7Tiseasilyfoundfromtherelation Z7T=ZlZ2/ZT =L/{OZT)'.'-! 298FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.2 Asanexample, wemaytaketheproblem ofsmoothing theoutputof arectifierconsidered atthebeginning ofthischapter. Then0=10p,F, L=25henries,andtheripplefrequency is100cIs.Theattenuation persectionatthisfrequency isgivenby -cosha= HenceeCX~100,ortheattenuation inpowerisapproximately 104per section.Ifwehavensuchinductances andcapacitances, itisvery v!(LjO) 0 OJo_w 0 OJo_OJ fJ -rr0 Wo~(J) FIG.11.9.P-section ofsimplehigh-pass filter,withvariation ofrealpartofZT'ex,andfJ. muchbettertojointhemasaladderfilterofnsectionsthantolump themallintoonesection.Thelatterwouldreducetheripplevoltageby afactorecx'~(w2n2LO)=100n2approximately, whiletheladdertype filterwillreduceitbyencx~(100)n.Thusthreesections wouldreduce theripplevoltageby106,whilethesinglesectionwiththreeinductances inseriesandthreecapacitances inparallelgivesonlyafactor900. Thehigh-pas8 filter Aswouldbeexpected, thesimplest high-passfilterisformedbyputting capacitances intheseriesarmsandinductances intheshuntarms,and atypicalT-section isshowninFig.11.9. 11.2]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 299 ThenZl=I/jwG,Z2=jwL,and Zl/4Z2=-1/4w2LG. Thisliesbetween zeroand-Iforw~l(LG)-i, thelatterbeingthe criticalfrequency wooThephasechangepersectioninthepassbandis givenby ()2cosfl=I-IJ(2w2LG)=1-2:0, andthemodulus ofthisexpression givesalsothevalueofcoshexinthe stopband.~Z'l" l!Q•-r0s· •0 l<ll l<ls------.l<l FIG.11.10.Simpletypeofband.p8B8 filter,showing T.section, andrealpartofZT,ex, andfJ88functions offrequency. Theiterative impedance is Zp=(~-4w~Gi)t =(~)!(I_ :~t. whichisimaginary inthestopband,andrisesfromzeroatthecritical frequency toalimitingvalueof(LIG)!inthepassband.Thebehaviour ofex,fl,andZpisillustrated inFig.11.9. Band-pass filters Asimpletypeofband-pass filterisshowninFig.11.10.Qualitatively itsbehaviour canbeseenasfollows. Atverylowfrequencies theim­ pedanceintheseriesarmwillbedominated bythecapacitance, sothat 300FILTERS" TRANSMISSION LINES, ANDWAVEGUIDES [11.2 thesectionwillactasasimplecapacitance-type attenuator (seeProblem 11.1).Atfrequencies higherthantheresonant frequency ofthecom­ binationLand01'wL>(wOl)-1andtheimpedance oftheseriesarm isinductive; thenthesectionbehaves asalow-pass filter,thehighest frequencies againbeingstopped. Quantitatively, theanalysis is ZI=jwL+I/(j WOl)' Z2=I/(jw02), ZI/4Z2=~[~:-~2L02J. whichispositiveatzerofrequency, buttendsto-00asu)-';>-00.Whenit liesbetween 0and-1,wehaveapassbandwhoselowestfrequency is WI=(Wl)~!' whereZI/4Z2=0, andhighestfrequency (ic.S+40)1 (~>:z+f!::;: - 2 1 2 'h 11W2-LqO 2'were4Z 2= - . Thebehaviour ofex,{J,andZTisshowninFig.11.10. Disadvantages ofthesimplefilter Asimplefilter,consisting ofachainofsimilarsections, suffersfrom twoprincipal disadvantages: (a)theiterative impedance varieswithfrequency inthepassband, makingitimpossible toterminate thefiltercorrectly throughout thepassband; (b)theattenuation inthestopbandvarieswithfrequency, beinglow nearthecut-offfrequencies. Thesedrawbacks canbereducedbyusingacomposite filter,contain­ ingsectionsofdifferent types,insteadofauniform filter.Forexample, theattenuation justabovethecut-ofTfrequency ofalow-pass filtercan bemadehighbyusingasectionwithaparalleltunedcircuitintheseries arm.,oronewithaseriestunedcircuitintheshuntarm(asinFig.11.11), adjusted toresonate atafrequency justabovethecut-offfrequency. Iftherewerenoresistive lossinthecomponents, thiswouldgiveinfinite attenuation attheresonant frequency. Suchasectionwillhavelowat­ tenuation atthehighfrequencies, butitcanbecombined withasimple low-pass sectionsothatthecomposite filterhasasufficiently highat­ tenuation throughout thestopband.Insuchacomposite filtereach sectionmusthavethesameiterative impedance atallfrequencies, or l1.2JFILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 301 reflections willoccuratthejunctions between sections. Thiscanbe achieved bytheuseof'm-derived' filters,anexampleofwhichisgiven inProblem 11.3.Half-sections mayalsobeusedtomaketheimpedance G (b)••w_."I... D•..TG • (a) FIG.11.11.T-section andtenninal half-section ofanm-derived filter.Thevaluesmust obeytheequations L1=mL,G.=mG,L.=L(1-m')/4m. FIG.11.12.T-section ofconstant kband.pass filter(L./G1=L1/G.=P). moreconstant inthepassband(seeProblem 11.4).Thesimplelowe passandhigh-pass filtersconsidered earlierbelongtotheclassof'k­ derived'or 'constant k'filters,sincetheirimpedances obeytherelation Z1Z2=k~, wherekisaconstant independent offrequency. Theband-pass filter considered earlierisnotofthistype,buttheT-section showninFig. 11.12doesobeythisrelation provided thatL2/01=L1/02=k2(see Problem 11.2). 302FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.3 11.3.Travelling wavesontransmission lines Intheelectrical circuitsconsidered hitherto wehavebeenableto identifythecircuitelements asinductances, resistances, capacitances, orcombinations thereof,knownas'lumpedimpedances'. Theseelements areconnected together andtogenerators anddetectors bylengthsof wirewhoseeffeotisassumed tobenegligible. Thisistrueonlywhenthe lengthsofwireinvolved areverysmalloompared withthewavelength II+dI I I I I I I I I I 1ill1-(I+dI) x+dx1 -dI=V(Ydx) xII I I I I I I I I I 14 FIG.11.13.Infinitesimal sectionoftransmission line,showing currentandvoltage(the lowerconductor isassumed tobeearthed, sothatVisthevoltagebetween thetwo conductors) . oftheradiation flowingalongthem.Whenthiscondition isnotfulfilled, thesignalchangesphaseasitflowsalongthewires,inmuchthesame wayasinthepassbandofafilter.Infaotwemayregardthewiresas alimitingcaseofalow-pass filterwheretheelements aremadeinfinitesi­ mallysmall,butthereareaninfinitenumberofseotionsperunitlength sothatthe'distributed impedance' perunitlengthremains finite. Normally twowiresarerequired tocomplete thecircuitbetween any twopiecesofapparatus, andweshallassumethatthesetaketheform eitheroftwoparallelwires,ortwocoaxialoylinders. Inanelement dx oftwosuohconduotors, asshowninFig.11.13,thevoltageandourrent willbelinearlyrelatedtooneanother, sothatthevoltagedropina lengthdxwillbe v-(V+dV)=-dV=I(Zdx), whereZistheseriesimpedanoe assooiated withunitlengthofthetwo 11.3]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 303 wires.Similarly, thecurrentflowingacrossbetween thetwowiresin theelementdxmaybewritten -dI=V(Ydx), whereYistheshuntadmittance between thetwoconductors perunit length.Sincechargeisconserved, (+dI)isthechangeinthecurrent flowinginthewireswhenwemovethedistancedx.IngeneralZandY willbecomplex; boththeseriesresistance andtheinductance ofthe wirescontribute toZ,andtheleakageconductance andcapacitance between theconductors toY.Thus,ifweassignaresistance Band inductance L,aconductance Gandcapacitance 0,allperunitlength, tothetwoconductors, ourbasicequations become -CoY/ox) =IB+L(oI/ot), -(aI/ox) =GV+O(oV/ot). (11.9) Theseformapairofsimultaneous differential equations whichmaybe solvedbyeliminating eitherIorV,whenasecond-order differential equation similartoequation (10.24)foraone-dimensional wavemotion isobtained. Wemayanticipate thisresultbyassuming thatequations (11.9)possessasolutioninwhichbothvoltageandcurrentvarywith tandxexponentially, sothattheychangeasexp(jwt-hx). Thenthe partialdifferentials (O/ot)and(a/ox)maybereplaced bymultiplication byjwand-Itrespectively, sothattheequations become ltV=I(R+jwL), hI=V(G+jwO). (11.10) Elimination ofIandVbetween thesetwoequations gives -h2=LOw2-jw(BO+GL)-RG. (11.11) Sincethisequation isquadratic inh,italwayshastwosolutions of opposite sign.Thesecorrespond towavespropagated inopposite direc­ tions,justasinthefilterthereweretwosolutions, aandl/a,correspond­ ingtoexp(-hx)andexp(+hx)inourpresentnomenclature. Thegeneralsolutionofequation (11.11)givesacomplex valueofh, indicating thatthewaveispropagated withattenuation, asweshould expectwhenthereisresistance andconductance present. Toobtaina clearerpictureofthewavemotionweshallfirstconsider thecasewhen bothBandGarezero;thatis,aloss-free line.Thisisgenerally agood approximation forshort,non-resonant lines(see§11.5).Equation (11.11) thenreducesto. -h2=LOw2. (11.12) Hencehisapurelyimaginary quantity, whichmaybewrittenh=jfJ. ThenfJ=w,.j(LO), andthewavevelocity isw/fJ=1/,.j(LO). 304FILTERS, TRANSMISSION LINES, ANDWAVBGUIDES [11.3 Forapairofcoaxialcylinders, radiiaandb(b>a),wehave°=27TEEo/loge(b/a) (Problem 1.8), L=(27T)-lfLfLologe(b/a) (equation (6.18», whileforapairofparallelcylinders radiusa,separation 2d(2d~a) 0=7TEEo/loge(2d/ay (equation (2.51», L=(7T)-lfLfLologA2d/a) (Problem 6.2), wheretheunitsarefarad/metre andhenry/metre respectively, andE,fL refertothemedium between thecylinders.Itisreadilyseenthatthe product(LO)isindependent ofthegeometry ofthecylinders, andthe wavevelocity is v=(LO)-l=(EEofLfLo)-l =c/(EfL)l. Thisisthesameasforanelectromagnetic wavetravelling intheun­ bounded medium, anditcanbeshownthatthisisthecasewhatever theshap~orsizeoftheparallelconductors. Thereasonforthisonly becomes apparent iftheproblem issolvedrigorously bystarting from Maxwell's equations, whenitisfoundthatapurelytransverse wave (thatis,withtheelectricandmagnetic fieldsbothnormaltothedirec­ tionofpropagation) ispossibleiftheconductivity ofthecylinders is infinite. Thistreatment showsalsothattheassumptions ofourmethod, expressed inequations (11.9),arevalidforperfectconductors butnot forimperfect ones.Theapproximation isquiteagoodone,however, so longasR~wLandG~wO,conditions thataregenerally fulfilledin practice (see§11.5). Sincethewavevelocityisindependent offrequency onaloss-free line, 'weneednotconfineourselves tosinusoidal waves,andthegeneralsolu­ tionofequations (11.9)(withR=G=0)is V=Fl(X-vt)+~(x+vt), (11.13) (11.14) (11.15)ZoI=F1(x-vt)-F2(x+vt), Zo=(L/O)l. wherewithv=(LO)-l.F1andF2represent wavesofarbitrary waveform travelling inopposite directions withvelocity±v.Further application ofequations (11.9)showsthatthecorresponding expression forthe currentis Thederivation oftheseequations issimilartothatofequations (10.18­ 19);andZoisknownasthe'characteristic impedance' oftheline.It playsasimilarroletotheintrinsic impedance ofamedium forelectro­ magnetic waves,ortheiterative impedance ofafilter(itisequaltothe (11.16)- -~~--- 1.3]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 305 limiting valuesofZTandZ7Tforalow-pass filter,iftheinductance and capacitance persectionareallowedtogotozerokeeping theirratio constant). Inawavetravelling towards positive valuesofx,theratio ofthevoltagetothecurrentisZo,whileforawaveintheopposite direction theratiois-Zoo FromthevaluesofLand0forourtwospecialtypesofline,wehave: coaxialline:Zo=21(ftfto)iloge(bja), 7TEEO parallelwireline:Zo='!'(ftfto)!loge(2dja). 7TEEO Typical valuesofZoare:foranair-spaced coaxiallinewithb::::::!3a, Zo::::::!70ohms;foraparallelwireline,2dja::::::!20,andZo::::::!400ohms. Inacoaxialcableacontinuous dielectric isusedtosupporttheinner conductor, andtheratiobjaisadjusted tomakeZosomestandard value, usuallyeither70or50ohms. Animportant quantity isthepowerflowingalongtheline.Ifwe confineourselves toawavetravelling alongtowards positive x,the energyflowingpastanyplanenormaltothex-axis-is, atanyinstant, W=v(iLI2+tOV2) =(tLI2+t0V2)j-J(LO) =iI2Zo+!V2jZo =I2Z0=V2jZo=IV. Herewehavegivenalltheequivalent expressions forW,andwesee thatZobehaves likeapureresistance, exceptthatWrepresents the energyflowingalongthelinepersecondratherthantheenergydissi­ patedasJouleheatingofarealresistance. Inasensetheenergystored inasectionofthelineisbeingdissipated, sinceitflowsawayfromthat section,andthestoredenergywouldtherefore diminish unlessitwere continually replaced bytheenergyflowingintoitfromtheprevious section.Wrepresents theenergycrossing agivenpointinthelineat anyinstant;theaverageflowofenergyisfoundbyusingtherootmean squarevalues1andPinequation (11.16). ThefactthatZoisrealshows thatthephasedifference between VandIis0or7T,according towhether thewaveistravelling towards positive ornegative x.Fromequation (11.16),WhasthesamesignasZo,showing thattheenergyflowsin oppositt directions inthetwowaves,asweshouldexpect. Thisresult followsalsofromthedirection ofPoynting's vector(seeProblem 11.5), sinceVandIarelinearlyrelatedtotheelectricandmagnetic fieldsE andH. 851110 x 306FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.4 11.4.Terminated loss-free lines Hitherto wehaveregarded thetwowavestravelling inopposite direc­ tionsalongalineastwoquiteindependent solutions ofthewaveequa­ tion.Usually, however, thereisonlyonegenerator attached totheline, producing awavetravelling, say,towards positivex.Ifthelineisin­ finiteinthisdirection, therewillbenoreturnwave.Ifthelineistermi­ natedinsomeway,areflection mayoccuratthetermination andthis A_--------------, x=OB_-------------..A X=-l--------.x FIG.11.14.Lineterminated byimpedance Zatx=o. willgenerate areturnwave,whichwillnotbeindependent ofthe incident wave.Ifthelatterhasagivenfrequency, thereturnwave musthavethesamefrequency inorderthattheratioofcurrentto voltageatthetermination (assumed toconsistofsomeconstant im­ pedance) shallbeindependent ofthetime.IfZistheterminating impedance (asinFig.11.14),thefactthatVjImustequalZatthis pointforallvaluesofthetimeconstitutes theboundary condition, from whichonecancalculate themagnitude andphaseofthereflected wave relativetotheincident wave. Iftheterminating impedance isnotapureresistance, itsvalue dependsonthefrequency, andsoalsowillthereflection coefficient. We musttherefore assumeawaveofagivenangularfrequency w.(Ifthe waveisnotpurelysinusoidal, wemustperform aFourieranalysisand treateachharmonic separately.) Theequations forcurrentandvoltage maythenbewrittenascomplex exponentials, where,asalways,thereal orimaginary partmustbeextracted attheend,according towhether theinputvoltageisacosineorsinefunction. Equations (11.13)and (11.14)become V=Aexp{jw(t-xjv)}+A' eXP{jw(t+x/v)}} ZoI=Aexp{jw(t-xjv)}-A'exp{jw(t+xjv)} .(11.17) Forsimplicity weshalltakethetermination tobeattheoriginof coordinates x=0,notingthatallpointsonthelinewillthenhave 11.4]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 307 negative valuesofx.Inserting theboundary condition, wehave Aeiwl+A'eicutI I(V/Zol),r;=o =Z/Zo=A"wiA'"cut=(A+A)/(A-A ).e'-e' Hence (11.18) whichdefinesthereflection coefficient A'/A.IfZiscomplex, AIwillbe complex (assuming Areal,whichcanalwaysbemadetmebychoosing thezeroofthetimescalecorrectly), showingthatthereisaphasechange inthereflected wave.IfZisapureresistance, A'isrealandthereis nophasechange(wemayexcludeaphasechangeof'1Tbyallowing negative valuesofA'). Ifthelineisopen-circuited, thevoltageattheendis2A,sinceAI=A. IfZisfinite,thevoltageacrossitisA+A'=2AZ/(Z+Zo), showing thatthelinebehaves asagenerator ofvoltage2A,withaninternal resistance ZOoThepowertransferred totheload,ifZisapureresistance R,isW=V2/R=2A2R/(R+Z O)2.IfZ=R=Zo,thepowertrans­ ferredtotheloadisamaximum; thatis,theloadismatchedtothe generator. Reference toequation (1l.18)aboveshowsthatunderthis condition AI=o.Thusmaximum powertransfertotheloadCorre­ spondstonoreflected wave;hencethematching condition hasasimple physical meaning, sinceanyreflected wavewouldcarryenergyaway fromtheloadandreducethepowerdissipated init.Thepowerinthe incident waveistA2/Zo,andthatinthereflected waveislA'2/Z0•The difference willbefoundtoequalthepowerdissipated intheload,as calculated above. Examination oftheformulaforA'/A,whenZisapureresistance R, showsthatasRgoesfromzerotoinfinity,A'/Achangescontinuously from-1to+1.WhenR<Zo,thereflected voltagewavehasopposite signfromtheincident wave,whilethereflected currentwavehasthe samesign.ThevoltageacrossRisthenlessthanA,andthecurrent throughitgreaterthanA/Zo.ThereverseistmeifR>ZOo WhenZisapurereactance jX,A'/Aiscomplex anditsmodulus is unity.Thisistobeexpected, sincenoenergyisdissipated inapure reactance, andtheamplitude ofthereflected wavemusttherefore equal thatoftheincident wave.Wemaywritethereflection coefficient A'/A inthiscaseaseiS,andanalgebraic reduction showsthat ei8=(X2-Z~+2jXZo)/(X2+Z~). Hencetan8=2XZO/(X2_Z~), whichmaybewritteninthemorecon­ venientformtant8=Zo/X. 308FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.40 (11.19) andAlj8_Z~-Z~+2jZI Zosinep Ae-Z2Z2 -1..'1+o+2Z1Zocos'I' Al_J(Z~+Z~-2ZIZ0COSep)} A- Z~+Z~+2ZIZ0COSep t"2Z1Zosinepana=Z2_Z2 I 0 Themaximum voltageonthelineis(A+A1),andoccursatavoltage antinode wheretheincident andreflected voltages areinphase;the minimum is(A-AI)atanodewheretheyareinanti-phase. Theratio (A+A1)j(A-A 1)iscalledthevoltagestanding waveratio(v.s.w.r.), andmeasurement ofittogether withthepositionofthenode(whichis relatedto8)formthebasisofamethodofmeasuring anunknown im­ pedance Zatshortwavelengths (see§15.3).givingInthegeneral case,whenZiscomplex, theformulae arerather cumbersome, butmaybereduced somewhat bywritingZ=ZleN, A'=Alej8•Then,onclearingimaginary termsfromthedenominator, onefinds ~=A"expjw(t-xjv 2), Z212=A"expjw(t-xjv 2), where VI'V2arethewavevelocities onthefirstandsecondlinesrespec­ tively.Ifthejunction isatx=0,thevoltageandcurrentatthispoint mustbethesameonthetwolines,sothat A+A'=A",li=Aexpjw(t-xjvl)+A' expjw(t+xjv l), Zl11=Aexpjw(t-xjvl)-A'expjw(t+xjv l); secondline:Transmission lineterminated byanotherlineofdifferent impedance Aspecialcaseofaterminated lineisonewithimpedance Zljoined ontoanotherlineofdifferent characteristic impedance Z2"Inthiscase therewillbeanincident andareflected waveonthefirstline,anda transmitted waveonthesecondline.Ourequations arethen: firstline: (A-A')jZI =A"jZ2' thesolutions ofwhichare A'jA=(Z2-ZI)j(Z2+ZI)' A"jA=2Z2j(Z2+ZI)' (11.20) Theformeroftheseisthesameasforalineterminated byaresistance Z2'asweshouldexpect.Thereflected poweristhesameasinthecase ofarealresistance Z2'andthetransmitted powerthesameaswouldbe 11.4]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 309 dissipated inarealresistance. NotethatifZ2>Zl'thevoltageonthe secondlineisgreaterthanthatintheincident wave.Thepowerisless, however, sincethecharacteristic impedance ishigher. Equations (11.20)areidentical withtheformulae forreflection and transmission ofaplaneelectromagnetic waveatnormalincidence atthe boundary oftwomedia(equations (10.41), (10.42), (10.49),and(10.50». Thisshowsthatthereisacloseanalogy between thecharacteristic im­ pedance ofatransmission lineandtheintrinsic impedance ofamedium transmitting anelectromagnetic wave.Thesimilarity appears alsoin theexpressions forthepowertransmitted: inaplanewavewehave (powertransmitted acrossunitarea)=N=E;lZo=ZoH~(see§10.3), whileforatransmission lineW=V21Zo=ZOI2(equation (11.16». The analogous behaviour makesitpossibletoadaptmanyoftheformulae derivedbelowtothecaseofplanewaves. Inputimpedance ofterminated lines Whenthe'reflection coefficient duetotheloadZusedtoterminate . alineisknown,itisasimplemattertocalculate thecurrentandvoltage, andhencetheeffective impedance, atanypointintheline.Thiscan bedoneforanarbitrary loadZ,butweshalllimitourselves toafewof thesimplerandmoreinteresting cases. Forashort-circuited line,Z= 0andA'=-Ainequations (11.18), sothatatapointx=-lontheline(i.e.attheterminals A,BinFig. 11.14)v=A[exp{jw(t+llv)}-exp{jw(t-llv)}] =j2Aexpjwtsinwllv =j2Aexpjwtsin27Tl/A and ZoI=2Aexpjwtcos27Tl/A. Theimpedance atthispointisthen VII=jZotan27TlIA. (11.21) Thisformula showsthatasectionofshort-circuited linebehaves as apurereactance. If1islessthanaquarterofawavelength, then tan27Tl/Aispositive andthelinebehaves likeaninductance. If1lies between aquarter- andahalf-wavelength, thetangentisnegative and thelinebehaves likeacapacitance. Thesestatements holdalsoifwe increase1byanintegralnumberofhalf-wavelengths. Ifthelineisopen-circuited, A'=+A,andtheequations forVand ZoIarejustinterchanged. Theimpedance atx=-listherefore VII=-jZocot27TlIA. (11.22) 310FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.4 Anopen-circuited line,lessthanaquarter-wavelength long,therefore behaves likeacapacitance; ifitslengthliesbetween aquarter- anda half-wavelength, itbehaves likeaninductance. Ifitslengthisexactly aquarter-wavelength itsimpedance iszero.Thusifwehaveanopen­ circuited linewecancutoffaquarter-wavelength andreplaceitbya shortcircuitwithout affecting theconditions earlierontheline.For wethenhaveashort-circuited lineoflength(l-lA),sotheimpedance attheinputterminals becomes +jZotan(27TlIA-tn") =-jZocot(27Tll>"), inagreement withthevaluefounddirectlyfromequation (11.22). Theseresultsshowthatalumpedreactance Xattheendofaline canbereplaced byasuitable additional lengthofline,eitheropen-or short-circuited. Wefoundearlierthatthewavereflected byareactance hasthesameamplitude astheincident wave,butaphasechangeS, wheretanis=Zo/X.Ifthereactance isreplaced byanopen-orshort­ circuited lengthofline,thenthewavereflected fromthefarendwill havethesameamplitude astheincident wave,butthephasechange arisesfromthetimetakenbythewavetotraveltheextradistance to theendofthelineandback.Atmetrewavelengths, suitable lengths ofeithercoaxialorparallelwirelinesarecommonly usedasinductances becausetheyhavealowerresistance thanacoilofwireofthesame ~nominal' r.f.resistance. By'nominal' r.f.resistance ismeantthevalue whichwouldbecalculated fromtheskindepthforastraight wire.In acloselywoundcoilthereisanadditional energylossbecauseofeddy currents inducedbytheoscillatory currents inneighbouring turns;this isknownasthe'proximity effect',andincreases theeffective r.f.re­ sistance. Itseffectisminimized byusingstraightwires,asinasection ofatransmission line. Thetransmission lineasatransformer Sincethevoltageandcurrentareindifferent ratioattheinput terminals ofalinefromtheratiotheybearattheoutputterminals, it followsthatalinecanbeusedasanimpedance transformer. Thecase ofgreatest interestisthatofalineone-quarter wavelength long.Ifthe terminating impedance Zisatx=0,thenatthepointx=-lA,the voltageandcurrentare V=Aexpj(wt+!7T)+A' expj(wt-!7T) =j(A-A')expjwt, ZoI=j(A+A')expjwt, and VII=Zo(A-A')/(A+A') =Z~/Z, (11.23) showingthattheterminating impedance hasbeentransformed toZ~/Z. 11.4]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 311 Inthisrespectaquarter-wave linebehaves likeatunedcircuit,which transforms aseriesresistance Rintoaparallelresistance LIOR.Since Z~=LIO,theformulae aresimilarinthetwocases. Thequarter-wave transformer maybeusedtomatchaloadZtoa transmission lineofimpedance Zobyinserting immediately beforethe loada1;\sectionoflinewhoseimpedance ZlissuchthatZVZ=ZOo IfZandZlarelinesofthesamedimensions butindifferent dielectric media,thesituation isexactlyanalogous tothe'blooming' ofoptical lenses.Thefractionoftheincident intensity reflected fromanair-glass surfaceisabout4percent,andthelossoflightinanopticalsystem withtenortwentysurfacesisserious.Thereflection maybereducedby depositing onthesurfaceaquarter-wave thicklayerofmaterial oflow refractive index,ideallyequaltothesquarerootoftherefractive index oftheglass.Thethickness isadjusted tobecorrectforthemiddleof theopticalregion,andisthusnotquitecorrectfortheendsofthis region.'Bloomed' surfaces appearslightlypurple,therefore, owingto reflection oftheredandbluerays.Quarter-wave filmshavealsobeen usedtoproduce highlyreflecting layers.Ifthefilmisofcharacteristic impedance Zl'andtheinitialandfinalmediaarethesame(impedance Zo),thefilmactsasamediumofimpedance ZlIZo,orrefraotive index n-2,wherenistheactualindexofthefilm,assumed tobeinair.Afilm ofglass(n=1'5)willthenreflect38percentoftheincident intensity. Further detailsoftheseopticalapplications aregivenbyKuhn(1951). Thehalf-wave transformer isalsoofinterest.Itmayberegarded as twoconsecutive quarter-wave transformers, givinganimpedance ~/(ZMZ) =Z. Alternatively, thisresultmaybeobtained directly, sinceifwemoveone half-wave alongaline,allvoltagesandcurrents arethesameexceptfor theirreversed sign.Thehalf-wave lineistherefore a1:1transformer A typicaluseisthatofaconnecting linkbetween twopiecesofapparatus, whichmakestheimpedance ofeitherappearunchanged. Thisisoften usefulatveryshortwavelengths whereconnecting wiressufficiently shorttogivenoimpedance transformation arenotpracticable. 11.5.Attenuation onlossylines,andresonant lines Whenthereislosspresentonatransmission line(R,Gnotzero),the velocityofawaveisaltered,anditisattenuated. Writing hasOI.+jfJ, equation (lI.ll)canbeseparated intorealandimaginary parts,giving 0I.2_fJ2=RG-LOw 2,201.fJ=(GL+RO)w. 312FILTERS, TRANSMISSION LINES, ANDWAVEG VIDES [11.5 Theseequations maybesolvedexactlyfor0:andf3,butitismore instructive tosolvethemapproximately, assuming R,Gtobesmall. Then f3=w,,(LO){1+ 8:2(g-~)l 0:=MRJi+GJ~}.(11.24) Thevelocity isapproximately w/f3="(~0){1-8:2(g- ~)} (11.25) showingthatitisalteredonlyinthesecondorder.Thepowerflowing alongthelinedecaysasittravelsalongas(usingZo=,,(L/O)) exp(-2a:x) =exp{-(R/Zo+GZo)x}. Thetwotermsintheexponential represent justthefraction ofthe storedenergywhichisdissipated perunitlengthintheresistance and conductance respectively. WeseethatifR/ZoandGZoarebothsmall, thelineisdistortionless inthefirstapproximation, sineeneitherthe velocity ofthewavenoritsattenuation dependonthefrequency to thisorder.Inthenextapproximation, distortion arisesfromthechange invelocity withfrequency andthisismostseriousatlowfrequencies. Athighfrequencies (owingtoskineffect)theresistance rises,increas­ ing0:.IfGisnegligible, distortion maybereducedbyincreasing L.On telephone landlinesthisisaccomplished byintroducing inductances in serieswiththelineatregularintervals. Thisalsoreducestheattenua­ tion,sinceitincreases Zo,butgivesthelineaperiodicstructure sothatit behaves likealow-pass filter.Thecut-offfrequency mustbekeptabove theaudio-frequency range,andtodothisthespacingoftheinductances mustbesmallcompared withtheshortest wavelength whichitisre­ quiredtotransmit. Thedistortion canalsobegreatlyreducedbytrans· mittingavoice-modulated signalof,say,100000c/sfrequency, instead oftheactualvoicefrequency rangeof100to10000c/s.Although the band-width required isthesame,thefractional changeinfrequency involved isverymuchsmaller. Inthelaboratory, thelengthsoflineusedaresoshortthatattenuation isnegligible exceptatthehighestfrequencies, whereRrisesowingto theskineffect.Toobtainanumerical value,letustakeanair-spaced coaxiallineatafrequency of3X109c/s(.\=10em).Thenforcopper theskindepth8isabout1·2X10-4cm.Iftheconductor dimensions are a=2·5mm,b=8mm(givinga70-ohmline), R=p(_I_+_I_) =1.2ohm/metre. 21Ta821Tb8 11.5]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 313 Thena:=8X10-3permetre,andthepowertransmitted alongtheline willfallbyafactoroflIeinadistance of60metres. Ifthespacebetween theconductors isfilledwithadielectric, the attenuation duetodielectric losswillbeimportant unlessthedielectric isofthehighestquality.Hitslosstangentistan8,thenG=wCtano, and 20:=GZo=w(LC)ltan8 =(wlv)tan8 =(27T/A1)tan8, whereAlisthewavelength inthedielectric.Ifitsdielectric constant is 2,2,andtan8=2x10-4,a:=9x10-3permetreatA=10em,which isaslargeasthatduetotheresistance. Forthisreasonhigh-frequency cablesareoftenmadewithsomedevicesuchasanopenspiralofpoly­ thenestringsupporting thecentreconductor inordertoreducethe amountofdielectric inthepositionofmaximum electricfield. Incalculating thelossonthelinewehavetakennoaccountofenergy lostbyradiation, thoughwemightexpecteachelementoftheconductors toradiatesinceitcarriesanalternating current.Atransmission line hastwoconductors carrying equalandopposite currents, however, and incomputing theradiation wemustallowforthedestructive inter­ ferencebetween theirtworadiation patterns. Foraparallelwireline thisdoesnotgiveanexactnull,butthemaximum phasedifference between thesignalsfromthetwowiresinanydirection willdifferfrom 7Tatmostby27T(2d)/A, where2distheseparation between thetwocon­ ductors. Hencetheradiated energyislessthanthatfromasinglewire byafactoroftheorder(dIA)2,andissmallifd~A.Thecoaxialline givesanexactnullbecausetheonecurrententirely enclosestheother, andthemagnetic fieldatanyexternal pointiszero.Forthisreason coaxiallinesaretobepreferred atwavelengths lessthanaboutametre. Transmission linesastunedcircuits Ifwehaveaquarter-wave sectionofloss-free line,short-circuited at oneend,thentheimpedance measured attheotherendisinfinite. Similarly,ifitisopen-circuited atoneend,thentheinputimpedance attheotherendiszero.Wehavealreadyseenthatifaresistance Ris connected acrossoneend,theimpedance attheotherendis Z~/R=LICR. Thusinallrespectsthesectionbehaves likeatunedcircuit.IfRiszero orinfinity,theinputimpedance willonlybeinfinityorzerosolongas thelineiscompletely loss-free. This,ofcourse,willneveroccurin practice, andtoassesstheproperties ofthesectionasatunedcircuit (11.26)314FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.5 wemustincludetheeffectofdistributed losses.Wecandothisby bringing intheattenuation constant ex. Wewillassumethatthesectionisopen-circuited atthefarend,so thatA'=A.Thenthevoltageandcurrentatapointx=-lare (sinceAistheincident voltageamplitude atx=0) v=A[exp{j(wt+2;l)+exz}+exp{j(wt- 2;Z)-exl}], ZoI=A[exp{j(wt+ 2;l)+exl}_ex p{j(wt_2;Z)_exl}]. Theimpedance Zatthispointisthengivenby Z 1+exp{-j 417l/'A-2exl} Zo=l-exp{-j 417l/'A-2exl}' Wenowassumethatthelengthoftheline1isclosetoanoddmultiple ofaquarter-wavelength, andexamine howtheimpedance changesin theneighbourhood ofthispoint.Inthecomplex exponentials ofequa­ tion(11.26)theimaginary partoftheargument givesarapidvariation andtherealpart(whichisassumed tobesmall)aslowvariation. We therefore treatthemseparately, andwrite1={(2n+l)A/4}+~l inthe imaginary partonly.Thentheexponential becomes exp{-j(2n+l)l7-j 417~l/>'-2exl} =-exp(-j417~l/>'-2exl) ~-(I-j417~Z/A-2exl), wherewehaveassumed thatboth ~l/>'andexlaresmall.Thustheim­ pedance becomes Z=Z1-(I-j417~l/>'-2exl) ,...,Z(l+'2 ~l/>')=ZexZ(I+ .217~!\ 01+(I-j 417~l/'A-2exl)"'" 0exJ17 0Jex>'fJ' (11.27) whereonlysmallquantities ofthefirstorderhavebeenretained. This equation isofthesameformasequation (9.20)foraseriestunedcircuit nearresonance, whichis Z=r(l+j 2Q~w/wo) since ~l/l=-~>'/>'=~w/wo(heretheminussignisintroduced inre­ lating ~l/lto~/>'becauseincreasing thelengthofthelinehasthesame effectasshortening thewavelength oftheappliedradiation). Hence atresonance thequarter-wavelength line(oralineanoddmultiple of thislength)behavesasaseriestunedcircuitwitharesistance Zoexland aqualityfactorQ=l7/ex>'. (11.28)11.5]FILTERS, TRANSMISSION LINES. ANDWAVEGUIDES 315 Inmakingmeasurements atshortwavelengths itisalwaysadvisable tokeepthegenerator frequency constant, ifpossible, andvarythe elementundertest,sincethisavoidserrorsduetodetectors, connecting lines,etc.,beingfrequency sensitive. Equation (1l.27)showsthatwe mayconveniently measure Qbyfindingthefractional changeinlength oftheline(tJ.l/l)required tomovebetween thepointsatwhichtheim­ pedance risesto-./2ofitsminimum value. Tofindanumerical valueforQwetakethevalue0:=8 X10-3per metrefoundearlierforanair-spaced coaxiallineat10emwavelength. ThisgivesQ=4000,whichisverymuchhigherthancanbeobtained normally withalumpedcircuitatmedium radio-frequencies. Qisinde­ pendent ofthenumberofquarter-wavelengths inthesection,butthe seriesresistance r=Zo0:1=1·4X1O-2(2n+1)ohms.Thusrincreases whenwemakenlarger,butQdoesnotchange. ThisisbecauseQ depends ontheratioofthestoredenergytotheenergydissipated, and bothoftheseincrease asthelengthofthelineincreases. Theimpedance ofashort-circuited quarter-wave linecanreadilybe calculated fromtheabove,sinceinthiscaseA'=-A,andtheformulae arethesame,ifweinterchange VandZoI.Thisgives ZoI ZYl(.21TtJ.~V=0=0:1+Jcx,\Tf' showingthatatresonance theimpedance isZo/al=ZUr=L/Cr,the sameexpression asforaparalleltunedcircuit.Forasinglequarter­ wavelength ofthecoaxiallineconsidered previously, theparallelim­ pedance is350000ohms,showing thatthelinemakesagoodanode loadforanoscillator oramplifier. Inpracticethelinewouldberather shorterthanaquarter-wavelength, whenitbehaves asaninductance whichcanbeadjusted toresonate withtheanodecapacitance ofthe vacuum tube(seeChapter 14). 11.6.Guided waves-propa~ation between twoparallel con­ ductin~ planes Whenanelectromagnetic waveislaunched fromanaerialintofree space(oranon-conducting medium) itsamplitude fallsoffinversely withthedistance owingtothespreading outofthewaveinaspherical wavefront.Thereisnodissipation ofenergy,butthepowerflowing through unitareanormaltothewavefrontfallsoffaccording tothe inversesquarelaw.Ontheotherhand,awavesentalongacoaxialline suffersnodiminution inamplitude, apartfromthatduetoresistive 316FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.6 losses,becauseitisconfined tothespacebetween thecouductors and doesnotspreadout.Suchawaveisaguidedwave,andamorerigorous approach thanthatadopted inthepreceding sections wouldbeto solveMaxwell's equations withtheboundary conditions thatthetan­ gentialcomponents oftheelectricfieldmustbezeroattheconductors (assuming thesetobeperfectconductors). Suchanapproach would z=c z=orc xz y FIG.11.15.Coordinate systemforpropagation between twoparallel planes. haveshownthatwithtwoparallelconductors asolution canbefound givingapurelytransverse wave(nocomponents ofEorHinthedirec­ tionofpropagation) whichisfreelypropagated atallfrequencies with thesamevelocity asawaveintheunbounded medium. Withasingle hollowconductor thisisnolongerthecase,thoughthefactthatitis possible toseedownametaltubeshowsthatsomeformofelectro­ magnetic wavecanbepropagated through it.Suchawaveisagain aguidedwave,sinceitmustmoveinthedirection ofthetube.Asa preliminary tostudying propagation through suchatube(knownas a'waveguide'), weshallinvestigate theproblem ofpropagation between twoparallelinfiniteperfectly conducting planes,separated byadistance c,asshowninFig.11.15. ACartesian coordinate systemmaybedefinedbytakingthecon· ductorstobetheplanesz=0andz=c,andassuming thatthewave ispropagated paralleltothex-axis,whichisnormaltotheplaneofthe paper.Theboundary conditions nowdemandthatanycomponents of theelectricfield(ExorEy)tangential totheplanesmustvanishatthe planesz=0andz=c.Wetrytofindthesimplest possible solution ofMaxwell's equations consistent withthesedemands, andbeginby assuming apurelytransverse planewaveinwhichbothExandHxare zero,suchaswewouldhaveintheabsenceoftheconductors. Ifthis waveispolarized withitselectricvectornormaltotheplanes(i.e. (11.29) (11.30) (11.31)11.6]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 317 Ey=0)itisreadilyseenthattheboundary conditions aresatisfied automatically, andawaveofthispolarization ispossible. Thesolutions areofthesameformasforawaveintheunbounded medium, andthe velocity isalsothesame.Ontheotherhand,aplanewaveinwhich thefieldcomponents donotvarywithzisobviously impossible ifthe electricvectorisparalleltotheplanes,sinceEymustbezeroatz=0 andz=candwillbezeroeverywhere unlessweallowittovarywithz. Wemusttherefore examine whether itispossibletohaveawavein whichEyisfinite,butEzandExarebothzero,sothattheelectricfield ispurelytransverse. Thefieldcomponents ofHcanthenbecomputed fromMaxwell's equations. Weshallassumethatthewaveispropa­ gatedasexp(jwt-hx), sothatwecanreplacedifferentiation withrespect totandxbymultiplication byjwand-hrespectively. Then,ifthe medium between theplaneshasdielectric constant E,magnetic permea­ bilityft,andzeroconductivity, thecurlequations (10.3)and(10.7)give thefollowing components: -~wftftoHx =-OEY/OZj -JwftftoHy =0 , -jwJLJLoH z=-hEy o=OHz/OY-OHY/OZj jW€EOEy=oHx/oz+hH z, o=-hHy-oHxjoy wherewehavealreadyassumed Ex=Ez=O.Equations (11.29)show immediately thatHy=0,butHxcannotbezerounlessoEyjoziszero, andthisisnotallowedbytheboundary conditions. Hencethewave willnotbepurelytransverse, butwillhaveacomponent ofHinthe direction ofpropagation. OnputtingHy=0inequations (11.30),we seethatoHzjoyandoHxjoyarebothzero,sothatthereisnovariation inthey-direction, andexamination ofthecomponents ofdivE=0 showsthistobetruealsoofEy•Theremaining components ofequations (11.29)and(11.30)therefore reduceto jWftftoHx=oEyjoz1 jWftftoHz=hEy . jW€EOEy=oHx/oz+hH z Elimination ofHxandHzbetween thesethreeequations gives -W2ftfto€€oEy=o2EyjoZ2+h2Ey or o2Eyjoz2=-(h2+w2jv2)Ey, (11.32) wherev=(ftfto€EO)-~isthevelocity ofanelectromagnetic waveinthe • (11.33) andhence318FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.6 unbounded medium (intheabsenceoftheconducting planeswecould put8/8z=0andobtainthisresultdirectlyfromequation (11.32),since hmustthenbeanimaginary quantityjfJ,andv=w/f3).Thesolution ofequation (11.32)canbewrittenintheform Ey=Asin(?Tnz/c)+Bcos(l7nz/c) andtheboundary conditions Ey=0atz=0andz=crequirethat B=0andnmustbeaninteger. Tosatisfyequation (11.32)wemust have Hencehiseitherrealorpurelyimaginary according towhether the quantity insidethesquarerootispositive ornegative.Itisclearthat atlowfrequencies hwillbereal,andthewavewillthenbeattenuated. Atsufficiently highfrequencies hwillbeimaginary, andwaveswillbe freelytransmitted withoutattenuation; thusthesystemactsasahigh­ passfilter.Thecondition forfreetransmission ofwavesisthat w/v>nl7/c. Sincew/v=217/Ao,whereAoisthewavelength oftheradiation inthe unbounded medium, thiscondition maybewrittenintheform Ao<2c/n. Hence2c/nisthecut-offwavelength Ac'andonlyradiation ofshorter wavelength isfreelytransmitted. Inthepassbandwemaywrite h=jfJ=j(217/A g),whereAgistheapparent wavelength oftheradiation intheguide;thatis,itisthedistance between pointsalongthex-axis wherethephasediffersby217.Equation (11.33)thenreducesto f21 1 1 1)2=Xi=A2+A2' (11.34)o gc Thisisknownasthe'waveguide equation', anditisfoundtoholdfor anyshapeofwaveguide, although ithasherebeendeduced onlyfora simplespecialcase.Thecut-offwavelength Acdepends ontheshape anddimensions ofthewaveguide, andonthemodeofpropagation (i.e. inthepresentcase,onthevaluesofcandnrespectively). Equation (11.34)showsthatthewavelength intheguideisalways greaterthanthatintheunbounded medium Ao'When"0~Ac'Agap­ proaches Ao,whilewhenAo~Ac'Agtendstoinfinity. Thephasevelocity intheguidebehavesinthesamewayasAg,since vg=w/fJ=fAg, (1l.35) 11.6]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 319 showingthatVoisalwaysgreaterthanthevelocityintheunbounded. medium.Iftheguidecontains nomaterial medium, thephasevelocity willbegreaterthanthevelocityoflight(fAo)infreespace,sinceAo>'An. Thisdoesnotmeanthatenergyistransmitted withavelocity greater thanthatoflight,sincewehavedispersion: thephasevelocity depends v, I I I I V.I / / / /./-"--- L...- ------''---__ "..1•----.Ao FIG.11.16.Variation ofthephasevelocity "/1(brokenline) 8lldgroupvelocity u/1(fullline)inawaveguide. onthefrequency, anddoesnotequalthegroupvelocityug=dwjdfJ. Fromequation (11.33) - h2=fJ2=(;r-(:7Tr. andhence 2fJ(dfJjdw) =2wjv2, gIvmg (wjfJ)(dwjdfJ) =vgUo=v2• (11.36) Sincevgisalwaysgreaterthanv,itfollowsthatugisalwayslessthanv, andisthusalwaysles8thanthevelocityoflight.Thebehaviour ofug andVoisillustrated inFig.11.16;theserelations holdforallwaveguides, sincetheydependonlyonthewaveguide equation (11.34). Thepropagation ofwavesbetween two-parallel conducting planes maybeconsidered inanotherwaywhichisilluminating, particularly inrespectofthegroupandphasevelocity. Thewavemotionmaybe regarded asconsisting ofanordinary planewave,withthesamepro­ pertiesasawaveintheunbounded medium, whichismultiply reflected 320FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.6 fromthetwoplanes.Thenormaltothewavefrontisa:-;:mmed tomake anangle0withthenormaltotheconducting planes, a:-;inFig.U.l7. FromFigs.10.4, 10.5, ifwereplace0by17-0,thecomponents ofsuch incident andreflected wavesaregivenbyequations (10.36)and(10.37), iftheelectricvectorisintheplaneofincidence, orequations (10.46) and(10.47)ifitisnormaltotheplaneofincidence. Ineithercasewe mustsatisfytheboundary conditions, thatthetangential components ofEmustbezeroallovertheplanesz=0andz=c.Fortheformer z=c z xF GHL" \.", FIG.11.17.Reflection ofordinary planewavesbetween twoparallel planes. FG,GH normals towavefrontsincident andreflected atplanez=O.LL',MM'incident wave fronts;LL",MM"reflected wavefronts,differing inphaseby217.LJIgivestheguide wavelength Ag• case(electric vectorinplaneofincidence) thefirstoftheseconditions givesA'=-Ainordertomakethex-component ofEzeroatz=0; thesamecondition isobtained atz=cifwetakecos(J=0(0=!17). Thismakesthex-component ofEzeroeverywhere, butEzandHyare finite,sothatwehaveasimpleplanewavemovinginthex-direction; thiswaveispurelytransverse, andmoveswiththesamevelocity asin theunbounded medium. Whentheelectricvectorisnormaltotheplaneofincidence, tomake they-component ofEzeroatz=0,wemusttakeB'=-Binequa­ tions(10.46)and(10.47).Atotherpointstheamplitude ofthey-com­ ponentofEisthen(replacing 0by17-0) Ey=B(F;.-F2) =Bexp{jw(t-xsin Ojv)}{exp(jwz cosOjv)-exp( -jwzcos Ojv)} =2jBsin(wzcosOjv)exp{jw(t-xsinOjv)}, (11.37) wherewehavewrittenvforthevelocityintheunbounded medium. At theplanez=cthefieldcomponent givenbyequation (11.37)iszero provided that wCcosOjv=217CcosOjAo=n17 or cosO=Ao(n/2c)=Ao/Ac' (11.38) 11.6]FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 321 whereAcisthecut-offwavelength aspreviously defined,andAoisthe wavelength intheunbounded medium. Thisequation showsclearly thatnowaveispossibleforAu>Ac'forthentpereisnorealvalueof() whichsatisfiesit. Thewavelength Aoisdefinedasthenormaldistancebetween twowave frontssuchasLL'andMM'intheplanewaveswherethephasediffers by21T;thesewavefrontshaveanintercept LMontheplanez=0,and thelengthofthisintercept, whichgivestheapparent wavelength Allof awavepropagated inthex-direction, isAu/sin(). Renee sinO=Ao/AIl, (11.39) andoncombining thiswithequation (II.38)wehave 1/A~=(sin28+cos28)/~ =1/A~+1/A~, whichisthe'waveguide equation' alreadyderived(equation (11.34)). Weseethatitfollowsfromthefactthatonlyoneangle0ispossiblefor thedirection ofourmultiply reflected planewaveinordertosatisfy boththeboundary conditions. Theenergyflowtravelswithvelocity v intheplanewaveinadirection normaltotheplanewavefront(that is,alongFGorGH);thecomponent ofthisvelocityinthex-direction isvsin8,andthisisthespeed 'Ullatwhichtheenergyflowsintheguided wave.Ontheotherhand,thephasevelocity VIIoftheguidedwave,from equation (11.37),isvlsin8;hence 'UIlVIl=v2,asshownearlier(equation (11.36)).(Itshouldbenotedthatthepossibility ofVIIbeinggreaterthan visnotpeculiartoelectromagnetic waves;theeffectcanbeobserved bywatching themovement paralleltoareflecting boundary ofacrest inanywavemotion,as,forexample, inwaterwavesbeingreflected at ananglefromabreakwater.) Thebehaviour ofourguidedwavewhen Auisequaltothecut-offwavelength Accanbeunderstood ifweremember thatthisrequires 8=0;thatis,thewavemotionisanordinary plane wavebeingreflectedatnormalincidence betweenthetwoplanes.Then noenergyispropagated inthex-direction, sothat'Ull=0;butthephase atagivenvalueofzisindependent ofx,sothattheapparent phase velocity VIIinthex-direction isinfinite. (11.40) 85111011.7. Wave~uides Thetypeofwavewehavebeenconsidering, propagated between two parallelplanes,hasthefollowing components: Ey=Asin(21TzjAc)sin(wt-21TxjA g) } Hx=-A(AuIZlAc)cos(21TZIAc)COS(wt-21TXIAg) . Hz=A(AoIZl\)sin(21TzIAc)sin(wt-21TxIAg) y 322FILTERS,TRANSMISSION LINES, ANDWAVEGUIDES [11.7 Thesecomponents mayeitherbeobtained fromequations (10.46)and (10.47)(e.g.ElIisfoundbytakingtherealpartofequation (11.37)and writing2B=-A),orbytakingElIastheappropriate solutionofequa­ tion(11.32)andusingequations (11.31);Zlistheintrinsic impedance ofthemedium between theconducting planes.Theseequations show thatforagivenvalueof'\c'thecomponent Hxinthedirection ofpropa­ gationdiminishes inamplitude as'\0isdecreased, sothatthewave i4--b----. E. c y (a) xrK--!=.- ~-·I--~f.--I-~)~\/.IK.-;.--+...:::X\ +1f__/\illi/\~__,(\i. IJI·( ~.IIIII1( \IIIITI,IIIIII ~I~ '~\-_./~J "t~.\\.._.....~II\"'-4-- J\\--t-...)/ ...-.--_---.,;........_- --...",_- --."", (b) FIG.11.18.Rectangular waveguide withTEolmode. (a)Linesofelectricfield--. (/I)Linesofmagnetic field----andcurrentBow---c>-. approaches apurelytransverse wavetravelling alongthex-axis.As "0--?"c'Hz--?0since"o/,\g--?0;givinginthelimitatransverse wave travelling alongthez-axis. Sincetheonlyelectricfieldcomponent isinthey-direction, itispos­ sibletoinsertconducting planesnormaltothey-axis withoutintroduc­ inganynewboundary conditions. Wehavethenaclosedrectangular waveguide, asshowninFig.11.18(a),bounded bytheperfectly con­ ductingplanesz=0,z=c;y=0,y=b.Thefieldcomponents within theguidearegivenbyequations (11.40),andarezerooutside. This typeofwaveisdesignated TEon(orHon);TEmeans'transverse elec­ tric',indicating thatthereisnoelectricfieldcomponent inthedirection ofpropagation; thesubscripts 0,nmeanthatthereisnovariation in they-direction ofanyfieldcomponent, whileinthez-direction theyvary assinorcos(-7Tnz/c). Thesimplest wave(n=1)isshowninFig.11.18, andthisisalsothemodewiththelargestcut-offwavelength ,\C=2c. Itistherefore usedasthestandard modeforwaveguide propagation, andtheguidedimensions arechosensothat2c>'\0>cforthewave- 11.7JFILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 323 lengthitisdesiredtopropagate. Nohighermode(withn=2ormore) canthenbepropagated; thishastheadvantage thatwavesofhigher modes,setupbyalocaldisturbance ofthefield(duetodiscontinuities orchanges intheguidedimensions), decayexponentially alongthe guide.Bymakingthedimension blessthan>.0/2,noTEmodecanbe propagated withtheelectricvectorpolarized inthez-direction; andit canbeshownthatallothermodeshavestillsmallercut-offwavelengths, andsocannotbepropagated. Thefieldcomponents intheTEolmodeareshowninFig.11.18. Eyisamaximum inthecentreoftheguide,andzeroattheplanes z=0andz=c;itvariessinusoidally withz,withjustonehalf-period ofvariation (modeswithhighervaluesofnmakenhalf-periods of variation, andsorequireacorrespondingly largervalueofcforagiven Ao).Thecomponents ofthemagnetic fieldareeverywhere tangential to theboundaries, andthelinesofmagnetic fieldareshowninFig.11.18(b), wheretheguideisviewedlookingdownonthebroadface.Thelinesof magnetic fieldencirclethepointsatwhich8Ey/8tisgreatest; thatis,the pointswherethedisplacement currentisgreatest. Thiscorresponds to Maxwell's equation curlH=8D/8t,whichimpliesthatadisplacement cUJ.'l"Emt(achanging electricdisplacement) isencircled bylinesofmag­ neticfield.Thelinesofdisplacement currentflowarecompleted by conduction currentflowingintheconducting walls;suchcurrentflow is always normaltothemagnetic fieldatthesurfaceofthewall.The direction ofconduction currentflowisalsoshowninFig.11.18(b).The linesofcurrentflow(displacement plusconduction) encircletheregions ofchanging magnetic flux,corresponding thustocurlE=-8B/at. Wemayformaphysical pictureofthepropagation ofthewavein thisway,sinceanoscillating electricfieldsetsupandisencircled byan oscillating magnetic field,whichinturnsetsupandisencircled byan oscillating electricfield;thishasacomponent furtheron,ahalf-wave­ lengthfromtheoriginaldisturbance. ThelinesofEandaB/atarein perpendicular planes(similarly HandaD/at),andmaybecrudelyrepre­ sentedbythelinksofachain.Inanordinary planewaveinfreespace thelinesofforcegotoinfinity,andmaybeconsidered tojoinupthere. Studyofthepropagation characteristics ofwavesinguidesofother thanrectangular shapeinvolves theuseofmorecomplex mathematics, andwemention onlythecylindrical waveguide. Thisinvolves the solution ofthewaveequation orMaxwell's equations incylindrical coordinates, andrequires theuseofBesselfunctions. Wavesmaybe designated asTMmnorTEmn,according asthereisnomagnetic orno 324FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.7 c Sections through c-d(a) (b)o d c (c) d --Linesofelectricfield •towards observer---Linesofmagnetic fieldoawayfromobserver FIG.11.19.Approximate configurations ofelectricandmagnetic fieldsinacylindrical waveguide. Propagation isdirected awayfromtheobserver ortotheright. (a)TEnorHnmode;(b)TMo1orE01mode;(c)TEo1orHOImode(afterSouthworth, 1936,BellSystemTechnical Journal, 15,287(bycourtesy ofBellTelephone Laboratories) ; orseeProc.I.R.E.1937,p.237). electricfieldinthedirection ofpropagation; thefirstsubscript indicates thatthefieldcomponents varyascosorsinmep,whereepistheazimuthal angle,whilethesecondgivesthenumberofvaluesoftheradiusatwhich theelectricfieldcomponents otherthantheradialcomponent E,are zero.Thesimplest modesareTEll'whichisrathersimilartotheTEO! modeinrectangular guide;theelectricfieldispurelytransverse, and distributed asshowninFig.11.19(a);thecut-offwavelength is1·707d, 11.7JFILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 325 wheredisthediameter oftheguide.TheTMOlmodehasatransverse magnetic fieldwhoselinesofforcearecircular; thecut-offwavelength is1·30Sd,andthemodeissimilartothatinacoaxiallineexceptthat theconduction currentinthecentreconductor isreplaced bydisplace­ mentcurrent, withlinesofErunning downthecentreandturning outwards toterminate onthewallasinFig.11.19(b).TheTEoImode israthersimilar,butwiththelinesofelectricfieldandmagnetic field interchanged; theelectricfieldhasclosedcircularlinesofforceandis purelytransverse, whilethemagnetic fieldisgreatest downtheaxis (seeFig.11.19(c»;thecut-offwavelength isO·S20d. Cavityresonators Ifalengthofwaveguide isclosedbyconducting wallsateachend, itwillresonateatwavelengths suchthatthedistance between theend wallsisamultiple ofhalfaguidewavelength. Inthecaseofarectan­ gularguideclosedbyconducting wallsnormaltothex-axisadistancea apart,thedistance amustbe!l,\gwhereIisaninteger,inorderthatthe electricfield(whichistangential totheendwalls)maybezeroatthe twoends.Fromequation (11.34)thewavelength Aointheunbounded mediumatwhichtherectangular cavitywillresonate inaTEonmode isthusgivenby 1 (I)2(n)2 ~=2a+2c• (11.41) ThisisaspecialcaseofthemoregeneralformulaforaTEmnmode,for which ~=(:aY+(;Y+(~Y (11.42) forarectangular cavityofdimensions a,b,c;thisformulamayberecog­ nizedasthatusedinthetheoryofheatradiation incomputing the resonant modesofahollowrectangular cavity(seeProblem 11.12). Themostimportant quantity foranyresonant systemisitsquality factor,Q;thismaybefoundforawaveguide cavitybyusingtherelation (see§9.3) Q=c.o(energy stored)/(energy dissipated persec). Thetotalstoredenergymaybecomputed byintegrating theenergy densityinthecavity,whilethetotalenergydissipated inthemetallic walls(owingtotheirfiuiteresistivity) canbefoundbyusingequation (10.34a). Theorderofmagnitude ofQcanreadilybefoundwithout carrying throughthedetailsoftheintegration inthefollowing way.If theamplitude oftheoscillating magnetic fieldinthecavityisRo,the 326FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES [11.7 storedenergy ~!fLoH~V,whereVisthevolumeofthecavity(assumed tobeevacuated), whiletheenergylostatthewalls~tpH~AIS, where Aisthetotalwallarea,pandStheresistivity andskindepthinthe wall.Hence Q~W(!fLoH~ V)/(lpH~AIS) =(VIA)(wSfLolp) ~V/(AS) (11.43) assuming themagnetic permeability ofthewalltobeunity,andusing equation (10.31). Thisresultshowsthatatagivenwavelength the valueofQincreases withthelineardimensions oftheresonator; while atdifferent wavelengths, ifthelineardimensions arescaledinpropor­ tiontothewavelength, Qvariesas~~,sinceSvariesas,\g,andVIAas ~o.Foragivenwavelength andsizeofcavityQdoesnotvarygreatly withthemodeofresonance, withoneexception. TheTEO!modein acylindrical cavityhasratherahighQ,andthereisnoradialflowof currentontheendwalls;forthisreasonitisusedinwavemeters (see §15.4)whereoneendisamovable plunger. Agoodcontactbetween thisandthecylindrical wallisnotessential toahighQ,sincethereis nocurrentflowacrossthecontact. Avalueof10000maybeobtained forQatcentimetre wavelengths, andthesharpness ofresonance isthus ratherhigherthanforaresonant coaxialline,mainlybecausethereis nocentreconductor withitsratherhighcurrentdensitytocontribute tothedissipation ofenergy. REFERENCE KUHN,R.,1951,Ann.Rep.Progr.Phys.(Physical Society, London), 14,64. PROBLEMS 11.1.ShowthatifZ1andZ2inasimplefilterarebothpureresistances orpure capacitances thefilteractsasanattenuator atallfrequencies. Calculate theattenua­ tionpersectionwhenbothZ1andZ2arepureresistances of100ohms,andfind theiterative impedance ofaT-section. (Answers: Powerfallsbyfactor6·8persection;ZT=112ohms.) 11.2.AfilterwhereZ1Z2=k2,aconstant independent offrequency, iscalled a.constant-k filter.Showthatthesimplelow-pass andhigh-pass filtersof§11.2 a.reofthistype,buttheband-pass filterofFig.11.10isnot. ShowthatthefiltersectionofFig.11.12isaband-pass sectionoftheconstant-k typeprovided thatL101=L202,whenk2=L2/01=L1/02•Iffl,f2arethelower andupperfrequency limitsofthepassband,showthattheysatisfytherelations FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES 327 IL3.The valuesofthecomponents inthem-derived T-section shownin Fig.11.11(a)obeytherelations (m<1) L1=mL, O2=mO, L2=L(I-m2)/4m. Showthatthesectionbehaves asalow-pass filterwiththefollowing properties: (a)thecut-offfrequency faisindependent ofm;(b)theiterative impedance is ZT=(L/0-w 2L2/4)!,andisthusthesameasthatofasimplelow-pass filter sectionin§11.2;(0)theattenuation inthestopbandisinfiniteatafrequency f=fo/(I-m2)i. 11.4.AchainoftheT-sections ofFig.11.11(a)istenninated bythehalf-section showninFig.11.11(b),wherethevaluesofthecomponents obeythesamerelations asinthepreceding problem. Provethattheimpedance atthetenninals ODis Z=(!:.)i{I-(I-m2)Plf~} 0·'(I-Pin)! . Ifm=0'6,showthatthisdoesnotdepartbymorethan4percentfromthe value(LIO)!forfrequencies upto85percentofthecut-offfrequency fo.Thus theuseofahalf-section asatransformer givesamoreuniform impedance inthe passband. 11.5.Findexpressions fortheelectricandmagnetic fieldsinacoaxialtransmission linecarrying acurrentIandavoltageV,andshowbyintegrating Poynting's vector overthecross-section between thetwoconductors thatthepowerflowingalong thelineisIV. Iftheconductors haveafiniteresistivity, compute thepowerflowingintothem perunitlengthbymeansofequation (10.34a),andshowthatthisgivesthesame attenuation ascalculated in§11.5. 11.6.Inaninfinitetransmission linealeakdevelops atonepointwhoseresistance .isjustequaltothecharacteristic impedance oftheline.Showthatofthepower in-theincident waveone-ninth isreflected, four-ninths istransmitted, andfour­ ninthsisdissipated intheleak. 11.7.Alengthofloss-less transmission lineisfirstshort-circuited atoneendand thenopen-circuited; theimpedance measured attheotherendisZlinthefirst caseandZ3inthesecond. ShowthatZlZ2=~,whereZoisthecharacteristic impedance oftheline.Thisisaconvenient wayofmeasuring Zoforacableof unknown electrical length. 11.8.Afilmofcryolite (refractive index1·35)one-quarter wavelength thick isdeposited onaglasssurface(n=1'50).Showthatthereflected intensity is reduced toabout1percent. Areflecting filmismadeupofalayerofcryolite (n2=1'35)placedbetween twolayersofTiOz(n1=2'45).Eachlayerisone-quarter ofawavelength thick. Showthattheratioofthereflected totheincident amplitude is(n~-ntll(n~+nt), andthatthereflected intensity isabout81percentoftheincident intensity (assume normalincidence). 11.9.Aquarter-wavelength, air-spaced, parallelwiretransmission lineisfoundto beinresonance withanoscillator whenitslengthis25em.Whenacapacitance 328FILTERS, TRANSMISSION LINES, ANDWAVEGUIDES of1p,p,Fisconnected acrosstheopenend,itisfoundthatthelengthoftheline mustbereduced to12'5emtoobtainresonance. Showthatthecharacteristic impedance ofthelineisapproximately 530ohms. 11.10.Show,eitherbytheuseofequations similarto(11.29)and(11.30)but withtheassumptions H",=II.=0,orbytheuseofequations (10.36)and(10.37), thatawavecanbepropagated between twoparallelconducting planeswiththe following fieldcomponents: HII=(A/Zl)cos(21TZ/Ac)COS(wt-27rx/Ag), E",=(AAo/Ac)sin(27TZ/Ac)sin(wt- 27rx/Ag), E.=-(AAo/Ag)cos(21TZ/Ac)cos(wt-21Tx/A g), whereAcandAghavethesamevaluesasforthetransverse electricwavederived in§11.6.Thiswaveisatransverse magnetic wave,andmaybedesignated as TMon;notethatitcannotexistinaclosedrectangular guidebecausethetangential components ofEmustthenvanishatthewallsy=0andy=b.Thelowest transverse magnetic wavethenpossible wouldbeT~Hwithcomponents HII,H., E""E.eachvarying sinusoidally inboththey-andz-directions. 11.11.Ahollowcubicalboxofsidearesonates intheTElOlmode(thatisl=n=1 inequation (11.41». Calculate theenergystoredandenergydissipated persecond, andshowthatthevalueofQ=a/2S,:whereSistheskindepthinthemetalwalls attheresonant frequency. 11.12.Ahollowrectangular boxisbounded byperfectly conducting planesat x=0,x=a;y=0,y=b;z=0,z=c.Showthatthestanding wavesystem E",=A",coscxxsinf3ysinyzexp(jwt), Ell=Allsincxxcosf3ysinyzexp(jwt), E.=A.sincxxsinf3ycosyzexp(jwt) satisfiestheboundary conditions provided thatcxa=l1T,f3b=rn1T,yc=n7T,and thatthewaveequation issatisfiedif 1 (l)2(m)2(n)lI(w)21-(CX2+f32+y2)= -+ - + - = -=-47T2 2a2b2c 27TV A~• Showalsothat,tosatisfydivD=0, cxA",+f3AII+yA.=O. (Ingeneraltherecanbeonlytwoindependent amplitudes, corresponding tothe twopossible polarizations ofanelectromagnetic wave.) 12 THERMIONIC VACUUM TUBES IFatungsten wireisheatedinvacuotoatemperature ofabout2500°K, itisfoundthatelectrons areemitted fromthesurfaceofthemetal. Othermetalsandsomemetallic oxidesshowthesameeffect,knownas thermionio emission (§4.4).Ifaseoondeleotrode isplaoedinthesame evacuated envelope, andheldatapositivepotential withrespecttothe first,thentheemittedelectrons willbeattracted tothesecondelectrode, andacurrentwillflow.Thisphenomenon isthebasisoftheradiotube, andthedevicejustdescribed isknownasadiode.Thesurfaceemitting electrons isoalledthecathode, andthatreceiving themtheanode.Ifthe anodeiscold,andemitsnoelectrons, thennocurrentwillflowifitis madenegative withrespecttothecathode; thedeviceactsasavalve, permitting onlyaunidirectional flowofcurrent. Thediodemaythusbe usedto'rectify'; thatis,toconvertanalternating currentintoadirect current.Ifathirdelectrode intheformofagridisinserted betweenthe oathodeandtheanode,a'triode'tubeisformed,whichmaybeusedto amplifyanalternating voltage, ortosustainanalternating currentin atunedcircuit;thatis,toactasagenerator ofoscillations. Inthese twooperations thetubeisconverting energyfromad.c.sourceintoa.c. energywhile,inrectification, a.c.energyistransformed intod.c.energy. Thegenericnameforthediode,triode,andsimilardevicesutilizing the flowofelectrons fromahotsurfaceisthethermionic vacuumtube,and itisoneofthefundamental toolsofmodernphysicsandofmodern technology. Inthischapteranoutlineisgivenofthemodeofaction ofthethermionic vacuumtubeanditschiefuses. J=AT2e-q,lkT, wheretheconstants Aandepdependonthematerial, afewvaluesbeing giveninTable4.1.Thetemperature atwhichadequate emission is12.1.Construction ofthethermionic vacuum tube Thenumberofmaterials available foruseascathodes isseverely limitedbytherequirement ofhighelectron emission attemperatures wherethematerial doesnotdisintegrate. Theemission currentper unitareaofacathode surfaceatabsolute temperature Tisgivenby equation (4.21) 330 THERMIONIC VACUUM TUBES [12.1 obtained isdetermined primarily bythevalueoftheworkfunction, cPo Ofthematerials listedinTable4.1thoseinmostgeneralusearetungsten, thoriated tungsten, andabariumoxide-strontium oxidemixture. The temperatures required areapproximately 2500°,1900°,andHOO°K respectively forcurrentdensities oftheorderof1Ajcm2• Tungsten andthoriated tungsten (oftenknownas'brightemitter' and'dullemitter' respectively) areusedintheformoffinefilaments heatedbythepassageofelectriccurrentthrough them,thisbeingthe onlypractical methodofmaintaining thehightemperatures required. Puretungsten is"\Teryresistant to'poisoning' byresidual gasandwill givelonglifeintransmitting tubeswheretheanodepotential maybe 10000Vormore.Thoriated tungsten islessgoodintheserespects, but theloweroperating temperature isaconsiderable advantage. Onetotwo percentofthorium oxideisaddedtothetungsten duringmanufacture, andafterthetubehasbeenevacuated thefilament is'activated' by temporarily runningitataveryhightemperature. Someofthethorium oxideistherebydecomposed, thethorium atomsmigrating tothesur­ facewheretheyformamonatomic layerwithalowerworkfunction. Theoxide-coated cathode, owingtoitslowworking temperature, has thegreatadvantage thatitcanbeheatedindirectly, thusmakingitpos­ sibletohaveanequipotential cathode. Insmallreceiving triodesthe voltagedropalongadirectlyheatedfilament iscomparable withthevol­ tagedifference between gridandcathode, sothatthisvoltagedifference cannoteverywhere betheoptimum. Inaddition, thefilament cannot beheatedwitha.c.,sincethealternating voltagedifference between variouspartsofthefilamentandthegridwouldbeamplified andcause anintolerable hum.Theindirectly-heated cathode isgenerally made ofanickeltube,ofcircularorrectangular cross-section, withaninternal heateroftungsten wirecoatedwitharefractory insulator suchas alumina. Forcathoderayandothertubeswhereaflatcathode isre­ quired,ahollowdiskisusedwiththeheaterintheformofaflatspiral. Sincethebarium-strontium oxidemixtureisunstable inair,thematerial isdeposited intheformofcarbonate, usuallybyspraying onasuspension ofitinavolatileorganicsolvent. Onheatinginvacuocarbondioxide isevolvedandpumped away,andinsomecasesthecathodesurfacehas tobeactivated bydrawing currentfromitatanelevated temperature. Itisgenerally believedthattheemission takesplacefromparticles of freemetal(barium) atthesurfaceoftheoxidecoating. Thefreemetal gradually evaporates andhastobereplaced byreduction oftheoxide; thisiscausedpartlybypositiveionbombardment, partlybyelectrolysis (12.1)12.1] THERMIONIC VACUUM TUBES 331 through thepotential gradient inthecoating, andpartlybyreaction withthesurfaceonwhichtheoxideisdeposited. Asinthecaseofthethoriated tungsten filament, theoxide-coated cathode iseasily'poisoned' bythepresence ofgas,especially oxygen. Itistherefore essential tomaintain ahighvacuum forthewholelife ofthetube.Mostmetallic surfaces containoccluded gas,whichisvery gradually evolvedifthesurfaces aremaintained invacuoatroomtem­ perature, butisrapidlyevolvedathightemperatures. Nickeliscom~ monlyusedfortheanodeandotherelectrodes, andis'out-gassed' by heatingtoabout1300°Kinvacuoorinhydrogen beforethetubeis assembled. Gridsaregenerally woundoftungsten wire,owingtoits stiffness andhighmelting-point. Mterthetubehasbeenassembled, it isevacuated and,whilestillonthepump,isheatedtojustbelowthe softening pointoftheglassenvelope toremoveoccluded gasfromthe glass.Themetallic electrodes arethenoutgassed atredheatbyinducing eddycurrents inthemwithahighfrequency oscillator. Theadvantage ofthismethodisthattheglassisnotheateddirectly, andtheelectrodes cantherefore beraisedtoatemperature wellabovethemelting-point ofglass.Thecathodeisthenactivated, andimmediately beforethetube issealedoffafilmofanactivemetalsuchasbariumisdeposited by evaporation overpartoftheinsideoftheglassenvelope. Thepurposeof this'getter'istoabsorbresidualoxygenandnitrogenbychemical action. 12.2.Thediode Thesimplest typeofthermionic vacuum tubeisthediode,andwe shalldiscussfirstthewayinwhichthecurrentflowtotheanodedepends ontheanodevoltage. Iftheelectrons wereemittedfromthecathodewithzerovelocity, and therewerenocontactpotential difference between anodeandcathode, weshouldexpectthecurrentflowtobezerowhentheanodevoltageis negative, andtoriseimmediately toaconstant value,equaltothetotal emission fromthecathode, assoonastheanodeismadepositive. In facttheelectrons areemittedwithfinitevelocities, corresponding toa Maxwellian distribution. Thenumberwithanenergybetween Wand (W+dW)isthen0exp(-WjkT)dW,and,ifanegative potential Vis appliedtotheanode,onlythoseelectrons withenergyWgreaterthan (-e)Vwillreachtheanode.Hencethecurrentis W=oofexp(-WjkT)dW =OkTexp(eVjkT) =loexp(eVjkT) W=-eV 332 THERMIONIC VACUUM TUBES [12.2 andtherateofchangeofIwithV,knownastheslopeofthecharac­ teristic,ortubeconductance, is (dI/dV)=(e/kT)Ioexp(eV/kT) =(e/kT)I. (12.2) Foradiodewithanoxide-coated cathodewhosetemperature isabout 11000K,thevalueof(dI/dV)/1 isapproximately lOY-I.Inpractice thisslopeisnotattained becausetheflowofcurrenttotheanodeis limited,notbythevelocityofemission, butbythemutualrepulsion of theelectrons inthespacebetween cathodeandanode.Theseelectrons areknownasthe'spacecharge',andthecurrentflowunderthesecon­ ditionsiscalled'space-charge limited'. Onlyatverylowcurrentdensity, whenthespacechargeissmall,isthecurrent'temperature limited'. Theoriginofthelattertermarisesfromthefactthatboththemaximum current, andtheshapeofthecharacteristic, aredetermined bythe temperature ofthecathode. Ingeneraltubesareoperated under'space­ chargelimited'conditions anditispossibleformostpurposes toneglect thefinitevelocity ofemission anditsspread. Similarly, anycontact potential difference between anodeandcathode, whichhastheeffectof shiftingthecharacteristic upordownbyafewvolts,willbeneglected. 12.3.Thethree-halves powerlaw Toexamine theeffectofspacechargeontheflowofcurrent,weshall consider thecaseofadiodewherethecathodeandanodeformparts ofparallelplanesdenotedrespectively bytheequations x=0andx=d. Weshallfurtherassumethatthepotential ofthecathodeiszero,while thatoftheanodeis"Va.Thepotential between theelectrodes canbe determined bysolvingPoisson's equation (12.3) where-eistheelectronic charge,andnthenumberofelectrons per cubicmetre.Ifthemassofanelectron ism,anditsvelocityuatthe pointxwherethepotential isV,thentheenergyequation gives tmu2=eV, (12.4) whilethecurrentdensityis J=neu. (12.5) (Heretheflowisunidirectional anditisnotnecessary totreatJasa vectorquantity; wehaveomittedthenegative signwhichdenotesthat thedirection ofpositive currentflowisfromanodetocathode.) The 12.3] THERMIONIC VACUUM TUBES 333 velocity canbeeliminated between theseequations, giving ne=JJ(2:V)' Substitution ofthisinPoisson's equation gives d2Vjdx2=aV-t, wherea=(Jj€o)..j(mj2e). Thisequation maybeintegrated ifbothsides aremultiplied by2(dVjdx), giving (~~2-(m: =4aVt. (dVjdx)o istheelectricfieldatthecathode, whereVandxarezero. Sincetheconstant aisproportional toJ,itisevidentthatthemaximum currentdensityisattained when(dVjdx)o =O.Thenwemaywrite dVjdx=2atV*, integration ofwhichgives (12.6) wheretheconstant ofintegration iszerobecauseV=0atx=O. Sinceattheanodex=dandV="fa,wehave V~=9ad2j4=(9j4€o).j(mj2e)d2J, (12.7) showingthatthecurrentdensityJisproportional tothethree-halves poweroftheanodevoltage, andinversely proportional tothesquare oftheseparation between cathodeandanode.Thisrelation wasfirst derivedbyChild,andissometimes knownasChild'slaw. Sincethecurrentdensityisindependent ofx,itfollowsfromequation (12.5)thatthedensityofelectrons isgreatest wheretheirvelocity is smallest; thatis,nearthecathode.Itisthisconcentration ofelectrons whichreducestheelectricfieldatthecathode, sincetheirelectricfield isoppositely directedatthispointtothatduetothepositivepotential ontheanode.Theelectron concentration cannotrisetoagreatervalue thanthatrequired tomakedVjdxzeroatthecathode, sincenoelec­ tronscouldthenleavethecathode, andthespacechargewouldfallas electrons moveawaytotheanode,withouttheirbeingreplenished from thecathode. Neartheanodetheelectricfieldisgreaterthanthatdue totheanodepotential alone,becausethefieldishereincreased bythe repulsive forceduetothenegative spacechargenearthecathode. The potential variation isshowninFig.12.1.CurveAshowsthelinearpo­ tentialgradient whichwouldexistintheabsenceofspacecharge,while curveBisthatcalculated above,ontheassumption thattheelectrons 334 THERMIONIC VACUUM TUBES [12.3 areemittedfromthecathodewithzerovelocity.Itiseasilyseenfrom equation (12.6)thattheequation ofcurveBmaybewrittenintheform VjT:=(xjd)!.Owingtothefinitevelocity ofemission, electrons can leavethecathodeevenwhenthereisasmallreverseelectricfield,and thespacechargecanthenincreasetotheextentofsettingupapotential v r o~=;;;;;-.~ _ x=o FIG.12.1.Potential distribution inadiodewithplaneparallel electrodes. CurveAnospacecharge. CurveBspacechargelimited, electrons emitted withzerovelocity. CurveGspacechargelimited, electrons emitted withfinitevelocity.v. x=a minimum, asshownbycurveC.Thedepthofthisminimum isofthe orderWje,whereWis theaverageenergyoftheemittedelectrons, since onlythoseelectrons withsufficient energytopenetrate thepotential minimum willeventually reachtheanode. Oninserting numerical values,theequation forthecurrentdensity maybewritten J=2.34X1O-6V~jd2, (12.8) whereJisinAjcm2,T:involts,anddinem.Obviously thisequation cannotholdindefinitely asT:isincreased, sincethecurrentdensitywill eventually belimitedbytheemission fromthecathode, andwillthen reachaconstant value.Thecurrentwilldepartfromthethree-halves powerlawassoonasthespacechargeisnolongersufficiently denseto nullifytheelectricfieldatthecathode. Theformofthecurrent-anode potential curvewilltherefore beasinFig.12.2.Atlowanodevoltages, thecurrentislimitedbythespacecharge,anditsmagnitude isinde- 12.3] THERMIONIC VACUUM TUBES 335 pendent ofthecathodetemperature. Athighpotentials thesizeofthe currentislimitedonlybythecathode emission, andthus,inthefirst approximation, depends onlyonthecathodetemperature. Inpractice itwillbefoundthatthesaturation currentdoesincrease slightlywith ~,owingtofieldemission (§4.4).Thisincrease ismorenoticeable with oxide-coated cathodes thanwithpuretungsten cathodes. 0·8 J (amp/eml) 0·6- 0·4 0·2- ____--.L ~___L __LI _ 2,000 4,000 6,000V.(Volts) FIG.12.2.Current-anode potential eurveforadiodewithplaneparallel electrodes, 1cm apart,fortwodifferent cathode temperatures T1andTI(TI>T1). Although thederivation ofthethree-halves powerlawhasbeengiven hereonlyforthecaseofthediodewithplane-parallel electrodes, ithas beenshowntoholdalsoforelectrodes intheshapeofcoaxialcircular cylinders andofconcentric spheres. Byadimensional argument itmay beshowntoholdforanyelectrode geometry, assuming alwaysthatthe electrons areemittedwithzerovelocity. Mostvacuum tubesforlow frequency applications areconstructed withelectrodes intheformof coaxialcylinders, sometimes, butbynomeansalways,withcircular cross-section. Atveryhighfrequencies, wheretheclearance between theelectrodes mustbemadeverysmall,theplane-parallel arrangement isusedforspecialtubes. 12.4.Usesofthediode Theprimary useofthediodeisasarectifier, converting analternating voltageintoasteadyvoltage. Thebasiccircuitforthispurposeisshown 336 THERMIONIC VACUUM TUBES [12.4 inFig.12.3.Thediodeisconnected inserieswithaloadresistance R toasourceofalternating voltagesuchasatransformer, andacapacitor oisplacedinparallelwithR.Tounderstand themodeofaction,con­ siderfirstthecasewhereRisinfinite. Onapplying thealternating voltage,currentwillflowroundthecircuitthroughthediodeonlywhen theanodeofthediodeisatapositivevoltagewithrespecttothecathode. Whilethecapacitor isuncharged thisoccurseveryotherhalf-cycle. Since thecathode loseselectrons totheanodeduringthishalf-cycle, and Supply voltage VocoswtA+ '0 B FIG.12.3.Thediodeasahalf-wave rectifier. cannotregainthemduringthereversehalf-cycle, thecathodeandthe plateofthecapacitor connected toitwillbecomepositively charged. Theflowofelectrons fromcathodetoanodewillcontinue solongasthe anodereachesapositive voltagewithrespecttothecathodeatany pointduringthecycle.Thechargeonthecapacitor willthuscontinue torise,thelimitbeingreachedwhenthevoltageacrossthecapacitor is equaltothepeakvalueYoofthealternating voltage Yocoswt.Atthis pointthevoltageacrossthediodeis-Yo+Yocoswt, showingthatthe anodeneverbecomes morepositivethanthecathode, andhenceno currentflows.Atonepointinthecycletheanode-cathode voltage difference is-2Yo.Thisisknownasthe'inversepeakvoltage', and thediodemustbeconstructed sothatitcanwithstand theinversepeak voltagewithoutfailure. Ifavoltmeter isplacedacrossthecapacitor, itwillregisteravoltage equaltothepeakvalueYoofthealternating voltage. Thisisthebasic circuitfortheuseofthediodeasa'peak'vacuumtubevoltmeter.If thecapacitor hasnoleakage,itwillremainchargedtothegreatestpeak voltageeverappliedtothesystem,butifthevoltmeter hasafinite 12.4] THERMIONIC VACUUM TUBES 337 resistance R,itwillbeabletofollowchangesinthepeakvoltageso longastheydonotoccurwithinatimeoforderRG,thetimeconstant oftheR-Gcombination. IngeneralthecircuitofFig.12.3isusedtodeliverdirectcurrentinto aloadR.Undertheseconditions thecapacitor willdischarge slightly through theresistance duringthatpartofthecyclewhenthediodeis notconducting, beingrecharged tothepeakvoltagewhenthediode conducts. ThevoltageacrossRistherefore notconstant, butcontains Half-wave rectification ---.Time ....,..,..,..,..,..,'-,..,",, ..','..,....,,,,, "......,,==~Ripplevoltage Full-wave rectification --+Time FIG.12.4.Half-wave andfull-wave rectification. V=voltageacrossR. acomponent fluctuating atthefrequency oftheappliedalternating voltage, asinFig.12.4.Thiscomponent isnotsinusoidal, owingtothe asymmetrical natureofthecapacitor charge-discharge system. The 'ripplevoltage',asthefluctuating component isgenerally called,becomes largerifRisreduced, sincethetimeconstant oftheR-Gcombination isthensmaller,andthecapacitor discharges toalowervoltagebefore beingrecharged. Theripplevoltageistherefore moreseriouswhenthe diodeisonload.Theripplevoltagemustbeeliminated orverycon­ siderably reducedifthesystemisusedtosupplyd.c.powerforan amplifier orotherelectronic device,sinceanyalternating voltageap­ pliedtotheearlystageswillbegreatlymagnified attheoutput. Reduc­ tionoftherippleiseffectedeitherbyusingaverylargecapacitance G, sothatthetimeconstant RGisverylongcompared withtheperiodof thealternating supply,orbytheuseofasimplelow-pass filtercircuit. Eitherofthesemaybeusedwithafull-wave rectifier, whosecircuit, withafiltersection,isshowninFig.12.5. Theadvantage offull-wave rectification overhalf-wave rectification 851110 Z 338 THERMIONIC VACUUM TUBES [12.4 canbeseenfromacomparison ofthetwowithoutthefiltersection,that iswithacircuitconsisting ofthecapacitor 0inparallelwithaload resistance Rconnected acrosstheterminals ABineachcase.Withthe full-wave rectification thecapacitor ischargedtothepeakvoltageby thepassageofcurrentthrough onediodeduringthefirsthalfofthe cycle,andisthenagainchargedduringthesecondhalf-cycle bycurrent FIG.12.5.Afull-wave rectifierwithfiltercircuit. flowingthrough theseconddiode.Ifthetimeconstant oftheO-R combination islongcompared withtheperiodofthesupplyvoltage, so thatthevoltagedropduringthedischarge periodisonlyasmallfraction oftheinitialpeakvoltage,thenthevoltageon0willfallnearlylinearly withtimeduringthedischarge interval. Withfull-wave rectification, thisinterval isonlyhalfacycle,andthesizeoftherippleisthusonly halfasgreataswithhalf-wave rectification (seeFig.12.4).Inaddition itsfundamental frequency isnowtwicethesupplyfrequency, making thefiltering problem easier.Thesimplest typeoffilter,asshownin Fig.12.5,consistsofthecapacitance 0withaseriesinductance Land asecondshuntcapacita:q.ce 01,Thesemustbeofsuchamagnitude that theimpedance of01attheripplefrequency isverysmallcompared withR,whiletheimpedance ofLisveryhighcompared withthatof01, Thisarrangement actsasapotentiometer whichdeliversthefullsteady voltageoutputacrossR,butreducestheripplevoltageroughlyinthe ratio(l/WOl)/(wL) =1/W2W1,wherewistheripplefrequency. Fora full-wave systemdelivering about100mAat300V(Le.R~3000ohms) fromasupplyfrequency of50cis,theconditions laiddownaboveare amplyfulfilledformostpurposes ifLisabout25henries,and0and01 12.4] THERMIONIC VACUUM TUBES 339 about10p.F.Ifthereduction ofrippleisinsufficient, furtherfiltersec­ tionsmaybeadded(seeChapter 11). Thesystemdescribed inthelastparagraph isknownas'capacitor input',sinceacapacitor isconnected immediately acrosstheterminals AB.Iftheimpedance ofthiscapacitor atthemainsfrequency issmall, thenthewholeofthetransformer outputvoltageisappliedacrossthe diodeand,onnoload,thecapacitor chargesuptothepeakvoltage developed between thecentretapandeitherendofthetransformer secondary winding. Onload,thecurrentthroughthediodeconsistsof shortpulsescentredontheinstantwhenthesecondary voltagereaches itspeakvalue.Thepeakcurrentistherefore muchhigherthanthe steadycurrentdrawnbytheload,andifthelatterismanyamperes, thepeakdiodecurrentmaybesohighastodamagethetube.Thisis avoidedbyomitting thefirstcapacitor 0,sothatan'inductive input' systemisused.Sincetheactionoftheinductance istoopposeany changeinthecurrentflowingthroughit,thecurrentthrougheachdiode issubstantially constant duringthehalf-cycle whenitconducts, the currentbeingswitched totheotherdiodeinthesecondhalf-cycle. The voltageoutputislowerthanwithcapacitor input,andthepercentage rippleishigher.Ontheotherhand,the'regulation' (thechangeofout­ putvoltagewithoutputcurrent) isimproved. Wheretheloadcurrent isfairlyconstant, andnotmorethanafewhundred milliamperes (e.g. inaradioreceiver), capacitor inputisgenerally usedforthepower supplyunit. 12.5.Thetriode Inthetriodetubeathirdelectrode, knownasthegrid,isinterposed between thecathodeandtheanode.Iftheelectrodes areplanar,the gridtakestheformofacoarsewiremesh;iftheelectrodes arecylindrical thegridiswoundintheformofahelix.Thetotalcurrentdrawnfrom thecathode willnowdependonthepotentials ofbothgridandanode, sincebothcontrolthefieldatthecathode. Intheabsence ofspace charge,thechargeonthecathode, assumed tobeatzeropotential, is equalto -(OeqYg+OeaYc,)=-Ocg(Yg+ Oe~o0=-Oeq(Yg+~), whereOeq,Oeaarethecoefficients ofcapacitance between cathodeandgrid (atpotential Yg)andanode(atpotential ~)respectively. Thesecapaci­ tancesareinthesameratioasthedivision oflinesofforcefromthe cathode between thegridandtheanodewhenYc,=Yg.Theratio 340 THERMIONIC VACUUM TUBES [12.5 p,=0eg/Oeaisknownasthe'amplification factor'ofthetube(however, thevaluesof0eg,0eagivenintubemanuals donotnormally fitthis relation because theyincludethecapacitances oftheleads).Ifthe currentfromthecathode islimitedbyspacecharge,sothatthefield atthecathode (asintheassumption madeinderiving Child'slaw)is zero,thennolinesofforcewillreachthecathode, butallwillterminate onthespacecharge.Sincethelatterismainlylocatedveryclosetothe cathode, itisaffected equallybylinesofforcefromthegridandfrom theanode,anditfollowsthatthecurrentleavingthecathodedepends ontheequivalent voltage V~=(Vy+"fa/p,). Experimentally itisfound thatthecurrentdepends verynearlyonthethree-halves powerofthis equivalent voltageintheregionofcomplete spacechargelimitation, corresponding tothefactthatnearthecathodethepotential distribu­ tionisthesameasinan'equivalent diode'.Wemaytherefore write J=b(Vy+"fa/p,)i/d2, (12.9) wheredisknownasthe'equivalent diodespacing'; fortubeswithfairly highvaluesofp,(i.e.whennearlyallthelinesofforceterminating on thespacechargecomefromthegrid)disnearlyequaltothecathode­ gridspacinginaplanetriode.Theconstant bhasthesamenumerical valueasforthediode.Ingeneralthetriodeisusedwiththegridvoltage negative, sothatnocurrentflowstoit,andthetotalcurrentleaving thecathode (towhichequation (12.9)applies)isalsotheanodecurrent. Itisobviousthatcurrentwillleavethecathodesolongastheequivalent voltage (Vy+~/p,) ispositive, andasmallnegative gridvoltagemustbe combined withalargepositive anodevoltage. Thishastheadvantage thatasourceofvoltageappliedtothegridwillinfluence theanode currentwithoutanycurrentbeingdrawnfromthesourcebythegrid. 12.6.Characteristics ofthetriode Characteristic curvesmaybedrawnforthenegative gridtriodeshow­ ingtheanodecurrentasafunction ofthegridvoltageforvariousvalues oftheanodevoltage. AtypicalsetofcurvesisgiveninPig.12.6.Since theanodecurrentdepends ontheexpression (Vy+"fa/p,),itisobviousthat allthecurveswillbesimilar,butwillbedisplaced tomorenegative valuesofVyas"faisincreased, theshiftinVybeing-(1/p,)ofthatin"fa. Asecondsetofcharacteristic curvesmaybeformedbyplottingthe anodecurrentasafunction oftheanodevoltageforvariousvaluesof thegridvoltage, asshowninFig.12.7.Athirdsetofcharacteristics maybeobtained byplottingthegridvoltageagainsttheanodevoltage atvariousconstant valuesoftheanodecurrent. 12.6] la (rnA) 10 5THERMIONIC VACUUM TUBES 200150 100341 oL-_L~~L....l---L~-L..._-----L_- __-L_ -8 -4 0 +4V.(volts) FIG.12.6.la-VIIcurvesforasmalltriodetube. Urn-4mA/V,fL-30. la (rnA) 15 10 o 100 200 300Va(volts) FIG.12.7.la-Vacurvesforasmalltriodetube. Urn-4mA/V,fL-30. Inordertocarryoutcalculations ontheperformance ofatriodetube invariousapplications, itisconvenient tohavesomesimplemethodof specifying thetubecharacteristics. Thethree-halves powerlawofequa­ tion(12.9)isclumsytohandleandnotamenable tocalculation, aswell asbeingonlyapproximately true.Formostapplications, wearein­ terestedinsmallchangesinfaresulting fromtheapplication ofanalter­ natinggridvoltage. Forthispurposetheanodecurrentmaybeexpanded 342 THERMIONIC VACUUM TUBES [12.6 (12.10)intheform.ofaTaylor's series,forsmallchangesaboutitsmeanvalue. Wehavethen,forsmallchanges vg,vainthegridandanodevoltages respectively, I=10+Vg(:~) +Va(:~) + gv" av, {~2(821a) (821a) 12(821a)}+2Vg8V2+VgVa8E:8J!: +2Va8V2+.... gVa gaVa,V, aV, Thepresence ofthesecond-ordertermsresultsinthechangeofanode currentnotbeinglinearwiththechangeintheappliedvoltages. This isgenerally undesirable, sinceitcauses'distortion' oftheappliedsignal. Itcanbeavoidedbykeepingthemagnitude ofvgandvasmall,whenthe second-order termsbecomerelatively lessimportant. Ifweadoptthe convention ofusinglower-case symbols forsmallchanges (e.g.wewrite iafor1-10),wemaysimplifythenotation and,onomitting thesecond­ orderterms,equation (12.10)becomes ia=Umvg+(l/p)va. (12.11) HereUm=(81a/8Vy)vaisknownasthemutualconductance ofthetube, andp=(8~/81a),;" iscalledtheanodesloperesistance (oroftenjustthe anoderesistance ofthetube.Ifiaiszero,then Ik= -(:~t.= -(::t..=o=Urnp, (12.12) whichgivesasimplerelation between thethreeconstants ofthetube. These'constants' willvarywiththeworking conditions ofthetube,and areconstants onlyinsofarastheapproximations wehavemadeare justified. Theseapproximations areequivalent toconsidering thecharac­ teristicsofthetubeasstraightlinesintheneighbourhood oftheworking point. Whenthethree-halves law(equation (12.9))isagoodapproximation, thevalueofUmis Urn=J.(b2J/d4)1=en, showingthatUmincreases withtheone-third poweroftheanodecurrent SinceIk=Oeg/Oea'itissubstantially independent oftheworking condi­ tions,beingdetermined bythegeometry ofthetube.Itfollowsfrom equation (12.12)thatPwilldecrease astheinverseone-third powerof theanodecurrent. Typicalvaluesofthetubeconstants forsmalltriodesareasfollows. Urnvariesfrom1to10mA/V,whichissometimes expressed as1000to 10000micromhos. Ikvariesfrom20to100,andPfrom5000to100000 ohms.Theworking voltages areabout-3Vonthegrid,and+100to 12.6] THERMIONIC VACUUM TUBES 343 +250Vontheanode.ThevalueofiLiscontrolled bythecloseness of thewinding ofthegrid,andtherelative distances ofgridandanode fromtheoathode. Ifthenumberoflinesofforcereaching thecathode fromtheanodeisverysmallcompared withthenumberfromthegrid (asisthecasewithacloselywoundgrid),themagnification factoriLis high.Ifthegridisfairlyopenlywound, iLislow. R v.+ HoT. FIG.12080Triodetubewithresistance inanodecircuit 12.7.Equivalent circuitofthetriode Themostimportant useofthetriodeisasanamplifier.Ifthevoltage appliedtothegridischanged byasmallamount, therewillbeacorre­ sponding changeintheanodecurrent.Iftheanodeisconnected toits hightensionsourcethrough aresistance R,asinFig.12.8,thechange inanodecurrentwillcauseachangeinthepotential dropacrossR.The ratioofthechangeinthisvoltagetothechangeinthegridvoltageis knownasthevoltageamplification. Thechangeinthepotential drop acrosstheloadresistance Rwillcauseacorresponding changeinthe anodevoltage, whichwillfallifthegridvoltagerises.Thisfallreduces theanodecurrent,andhencetocalculate thevoltageamplification we proceedasfollows. Ifia,Varepresent thechanges inanodecurrentandvoltagedueto achangevginthegridvoltage, ia=(1mvg+va/p· ButVa=-iaR,andhencethevoltageamplification is A_Va_iaR_(lmpR_ iLR (1213) - vg-----:v;- -(R+p)- -(R+pf . Thisresultisthesameaswouldbeobtained fromagenerator ofvoltage -iLVgwithaninternal resistance p,ascanbeseenfromthecircuitof Fig.12.9(a).Thisisknownastheequivalent circuitofthetriode,and 344 THERMIONIC VACUUM TUBES [12.7 itsusegreatlyfacilitates calculation. Forexample, theanodeloadis oftennotapureresistance Rbutacomplex impedance Z=R+jX. Thevoltageamplification isthen A= _iaZ= _p,Z= _p,(R+jX) . (12.14) vgZ+p (p+R)+jX Inthisexpression thepresence ofacomplex number showsthatthe anodevoltagechangeisnotinphasewiththegridvoltagechange,where­ aswithapureresistive loadtheyareexactlyinanti-phase. Usuallythe p (a)p (b)z FIG.12.9.(a)Equivalent circuitofatriodetube,considered asavoltagegenerator. (b)Equivalent circuitofatriodetube,considered asacurrentgenerator. phaseshiftbetween anodeandgridvoltageisofnosignificance, and theusefulamplification isgivenbytakingthemodulus ofequation (12.14).Ifthereactance Xisafunction offrequency, asisusuallythe case,thevoltageamplification willalsovarywithfrequency unless Z>poverthewholerangeoffrequencies whichitisdesiredtoamplify. Forthisreason,resistive loadsaregenerally usedforwidebandamplifiers Analternative equivalent circuitforthetriodewhichisoftenuseful isshowninFig.12.9(b).Thetubeisreplaced byacurrentgenerator ofmagnitude -gmvg'whichhasinfiniteinternalimpedance, acrosswhich istheanoderesistance pofthetube.Theexternal loadZisconnected inparallelwithp,andafractionpl(Z+p) ofthecurrentfrom thegenera­ torflowsthrough Z.ThevoltageacrossZisthus-grnVgZpl(Z+p), givingthesameresultasinequation (12.14). 12.8.Inputimpedance ofthetriode Atradiofrequencies (105cisandupwards) theeffectsofthefinite capacitances betweenthevariouselectrodes ofatubebecomeimportant. Thefulltreatment ofthisproblem iscomplicated, sincethegrid-cathode, 12.8] THERMIONIC VACUUM TUBES 345 grld"anode, andcathode-anode capacitances formanetwork together withtheanoderesistance andloadasinFig.12.10.Asimplified treat­ mentisgivenbelow,whereCcaisdeemedtobepartoftheloadZ,and thecurrents through theotherelectrode capacitances areassumed to besmallcompared withtheanodecurrentthroughthetube. + H.T.-------1 I I ~o,. I I --------~ rv-/-IV. Cathode FIG.12.10.Thetriodewithitselectrode capacitances (above)anditsfullequivalent circuit(below). Theprincipal effectofthecapacitances fromthegridtotheother electrodes istodrawafinitecurrentfromthesourceofthevoltage appliedtothegrid.Ifthisvoltageisvrelativetothecathode(takenas thezeroofvoltage) thenthecurrentflowingfromgridtocathode is jwCgCv.Inaddition, thereisthecurrentflowingthroughthegrid-anode capacitance, whichalsocompletes itsreturnpaththrough thesignal source.Tocompute thiswenotethattheanodevoltageisAv,where A=-p,Z/(Z+p) istheamplification produced bythetubeworking (12.18)346 THERMIONIC VACUUM TUBES [12.8 intoananodeloadZ.Thusthetotalvoltageacrossthegrid-anode capacitance is (1A)V-Va=V- , andtheresulting currentflowisjwOga(I-A)v. Thusthetotaladmit­ tanceassociated withthegridis ~=I/Zg=jwOge+jwOga[I+JLZ/(Z+p)], (12.15) IfZisapureresistance R,thentheinputimpedance ofthegridis thatofapurecapacitance Ogofmagnitude Og=Oge+Oga[1+JLR/(R+p)]. (12.16) IfZiscomplex =R+jX,then ~_.[{JL(Rp+R2+X2)}] p'wXpOgaZg-JWOge+Oga 1+(p+R)2+X2 (p+R)2+X2' (12.17) ThusI/Zgcontains aresistive component whosevalueisnegativeifX ispositive, Le.iftheanodeloadisinductive. Thisnegative resistance meansthatpowerisflowingbacktothegridthrough thegrid-anode capacitance, andifthispowerismorethanisrequired tosupplythepower dissipated inanypositive resistance inthesourceofthegridvoltage, oscillations willresult(seeChapter 13).Thisisoneofthechiefdiffi­ cultiesintheuseoftriodesasamplifiers atradiofrequencies. Onthe otherhand,iftheanodeloadiscapacitative, theresistive component of thegridinputimpedance isalwayspositive. Inthistreatment wehaveneglected thefactthatthecurrentthrough thegrid-anode capacitance flowsalsothroughtheanodeload,thereby alteringtheanodevoltageslightly. Amoreaccurate treatment shows thatOgainequation (12.15)shouldbereplaced by {(Zp) .}-lOga1+Z+pJwCga. Thisintroduces aresistive component tothegridinputimpedance evenwhentheloadZisapureresistance R.Itsvalueisthenapproxi­ mately anditisinparallelwiththegridinputcapacitance. Whentheanode loadisinductive, thisreducesthenegative conductanee effectatthe grid. Toestimate themagnitude oftheseeffectswetakep,=30,p=104 ohms,R=2·5X104ohms,w=106,Cge=Cga=5JLp,F.Then Rg~2·5X105ohms, Cg~120JLp,F. 12.8] THERMIONIC VAeUUM TUBES 347 Atthisfrequency Gucorresponds toareactance ofonly8000ohms,so thattheinputimpedance ofthetriodeisverylow,andalmostwholly capacitative. Thebulkofthiscapacitance isduetothegrid-anode capacitance, whichismagnified becauseithastheamplified signal voltageacrossit.Theinputimpedance willbegreatlyincreasedifGua canbediminished, andforthispurposethescreen-grid tetrodetubewas introduced. 12.9.Thescreen-grid tetrode Inthescreen-gridtetrodeafourthelectrode isinsertedintheformof anextragridbetween thecontrolgridandtheanodeofthetube.This extraelectrode, thescreengrid,ismaintained atafixedpositivepoten­ tialwithrespecttothecathode, sothatthevoltageonitdoesnotchange whenasignalisappliedtothecontrolgrid.Thescreengridiswound sothatmostofthelinesofforcefromthecontrolgridterminate on thescreengrid,andcomparatively fewreachtheanode.Inthiswaythe grid-anode capacitance isreduced toabout10-9F,andalthough the capacitance betweenthecontrolgridandscreengridisofthesameorder asthatbetween gridandanodeinthetriode,thealternating voltage acrossthiscapacitance isthesameasthatacrossthegrid-cathode capacitance andisnotmagnified bytheactionofthetube.Thus ~hecapacitance between thetwogridsisjustaddeddirectlytothe grid-cathode capacitance. Sincethescreengridisatapositive potential withrespecttothe cathode, itwillcollectelectrons whichwouldotherwise havegoneto theanode.Mostofthelinesofforcewhichpenetrate throughthecontrol gridtothespacechargeroundthecathodewillcomefromthescreengrid ratherthantheanode,andtheformerwillexercisemuchmorecontrol overthecurrentthantheanode.Thustheanoderesistance ofthetube willbehighsince (alala~)v. issmall,andsowillbetheamplification factor,whichdepends ontheratioofthenumberoflinesofforcereach­ ingthespacechargefromthegridtothenumberfromtheanode. Thescreencurrentandanodecurrentinatetrodeforgivenvoltages onthescreenandcontrolgridsareshownasfunctions oftheanode voltageinFig.12.11.Astheanodevoltageisincreased fromzero,the anodecurrentshowsinitiallyasteeprise,followedbyafallandasecond risewhentheanodevoltagebecomes ofthesameorderasthescreen voltage. Thescreencurrentshowstheinversebehaviour, andthesum ofscreenandanodecurrents issubstantially constant sincethetotal currentleavingthecathode depends practically onlyonthecontrol 348 THERMIONIC VACUUM TUBES (12.9 andscreen-grid voltages, whicharebothconstant. Thefallintheanode currentandriseinthescreencurrentareduetosecondary emission of electrons bytheanode.Thisisnegligible atverylowanodevoltages, butappreciable whentheanodevoltagerisesabove10Vorso.The ratioofthenumberofsecondary electrons tothenumber ofincident primary electrons maybehighwhenacomposite surface(see§4.4)is formedontheanodebybariumevaporated fromthecathode. Inthe 300Va(volts) 20024(rnA) 10 68 FIG.12.11.Curvesofscreencurrentandanodecurrentagainstanodepotential, with zerocontrolgridvoltage, foratetrodetube.(Screenvoltage ~60V.) InitialriseofLainregionABoccurswhenanodevoltageistoolowtogiveappreciable secondary emission. ThelattersetsinatVa"'"10Vandincreases overrangeBO, causingnetanodecurrenttofall.Itrisesagaininregiononwhen ~becomes greater thanscreenvoltagev.,sincesecondary electrons arethenattracted backtotheanode. triode,secondary electrons areemittedbytheanodewithlowvelocities, buttheyfindthemselves inastrongpotential gradient whichreturns themtotheanode,sothatthereisnoneteffectontheanodecurrent. Inthetetrode, however, thefieldattheanodeisreversed whenthe screenisatahigherpotential thantheanode,andsecondary electrons emittedbythelatterwilltherefore traveltothescreen.Thiscausesa reduction inthenetcurrentflowingtotheanode,andanincreaseinthat flowingtothescreen.Theresulting kinkintheanodecurrentcharac­ teristicisaconsiderable drawback, sincestrongdistortion oftheampli­ fiedsignalwilloccurunlessthetubeisworkedsothattheanodevoltage isalwaysgreaterthanthescreen-grid voltage. Thisispossible with smallsignals,butnotwithlargesignals. 12.10] THERMIONIC VACUUM TUBES 349 12.10.Thepentode Theundesirable kinkinthecharacteristic ofthetetrodeiseliminated inthepentode, whereanextraelectrode isinserted betweenthescreen gridandtheanode.Thiselectrode, agridofcoarsemeshoranopen helix,isknownasthesuppressor grid,andismaintained atcathode potential (inmanytubesitisinternally connected tothecathode). Its f.(mA) 8__----1. 6 4 2------f. Ol...-.-----::~---~---~___;~_____o:_:300V.(volts) FIG.12.12.Curvesofscreencurrentandanodecurrentagainstanode potential, atzerogridvoltage, forapentode. function istomaintain thefieldattheanodealwaysinadirection such thattheforceonanysecondary electrons emittedbytheanodewill returnthemtotheanode,andtheywillnotreachthescreen.Aswould beexpected, theanodecurrentispractically independent oftheanode potential, sothattheanodesloperesistance isveryhigh,andsoalsois theamplification factorft,sincehardlyanylinesofforcefromtheanode penetrate tothespacechargenearthecathode. Thevoltage amplifica~ tionobtained fromapentodewitharesistance Rastheanodeloadmay, fromequation (12.13),bewrittenintheform A=-gmR/(l+R/p), whichisapproximately -YmRifP~R.Itisobviousthattheampli­ ficationisgreaterthanthatobtained fromatriodewhichhasthesame mutualconductance Ym,butalowervalueofp. Theanodecurrentandscreencurrentofatypicalpentode areshown 350 THERMIONIC VACUUM TUBES [12.10 asfunctions oftheanodevoltageinFig.12.12.Thecurrentdrawnfrom thecathodeisvirtually independent oftheanodevoltage, whichaffects onlythedivisionofcurrentbetween screenandanode.Thecapacitance between thecontrolgridandtheanodeinthepentode isusuallyofthe orderofafewthousandths ofamicromicrofarad, anditscontribution totheinputcapacitance isnegligible. Thekinkless characteristic ofthepentode mayalsobeachievedin specialtetradesknownas'beam-power' tetrodes. Inthesetubesthe spacechargeformedbytheelectronsintheregionbetween screengrid andanodeproduces apotential minimum infrontoftheanodewhich, likethatduetothesuppressor gridinthepentode, returnstotheanode anysecondary electrons emittedbyit.Theeffectofthespacecharge isenhanced byusingaratherlargespacing between screengridand anode,andbymakingthescreengridofthesamepitchasthecontrol grid(ordinarily itismuchcoarser). Thescreen-grid wiresareinthe 'shadow' ofthecontrolgridwires,sothattheelectrons flowinbeams between themandthescreencurrentissmallerthaninanordinary tetrode. Thebeamaction(enhanced bytheuseofsideplatesatcathode potential whichlimittheareaoverwhichcurrentflows)increases the electron densityandthespacechargeeffectneartheanode. GENERAL REFERENCE ROLLIN, B.V.,1964,AnIntroduction toElectronics (O.U.P.). PROBLEMS 12.1.Calculate theanodevoltage V~oftheequivalent diodeforatriodewith planeparallel electrodes inwhichthegrid-cathode spacing is0·03emandthe current density is0·02A/cm2,assuming thatthe'equivalent diodespacing' is thesameastheactualgrid-cathode spacing. IfVg= -3V,Va=+150Vonthetriode,whatmustbethevalueofJL? (Answer: V~=3·9V;JL=22.) 12.2.Provethatinaplane-parallel diodethetransittimeofanelectron is3/2times aslongunderspacechargelimitedconditions asitwouldbeintheabsence of spacecharge. Whatwillbethetransittimebetween cathode andgridinthetriodeof Problem 12.1? (Answer: 7·7X10-10sec.) 12.3.Inthearrangement ofFig.12.3,assumethatthecapacitor chargesinstan­ taneously toYowhenthediodeconducts justatthepeakoftheappliedvoltage Yocoswt, andthatthetimeconstant RO=Tofthecapacitor-resistor combina­ tionissolongthatthevoltageonthecapacitor fallslinearlyduringthedischarge period. ShowbyFourier analysisthattheamplitude ofthecomponent sin(nwt) ofthevoltageonthecapacitor is(2Yo/nw'T). 13 APPLICATIONS OFTHERMIONIC VACUUM TUBES INtheprevious chapterthechiefcharacteristics ofthebasictypesof thermionic vacuumtubeswereoutlined. Inthischaptertheirmainuses asamplifiers, oscillators, anddetectors willbeconsidered inmoredetail. 13.1.Audio-frequency volta~eamplifiers Amplifiers maybeclassedundervariousheadings, andweshalldeal firstwithsmall-signal amplifiers, wherethemagnitude ofthealternating voltageappliedtothegridofthetubeissuchthattheresulting changes intheanodecurrentareonlyasmallfractionofthemeananodecurrent. Thefundamental circuitfortheuseofthetriodeasanamplifier was discussed in§12.7.Ifmoreamplification isrequiredthancanbeobtained fromasingletube,severalstagesmaybeusedincascade. Someform ofcoupling isthenrequired totransfertheamplified voltageappearing attheanodeofonetubetothegridofthenexttube,whilepreserving thecorrectsteadypotentials ontheseelectrodes. Themostcommon methodemploys RO-coupling. Acapacitor 0isconnected fromthe anodeoftheprevious stagetothegridofthenext,asinFig.13.1,and thegridisconnected toearth(ortoitsbiasbattery) through alarge resistance RI.Itisimportant thatthecapacitor 0(knownasthe blocking capacitor) haveaverysmallleakagecurrentunderthesteady voltagewhichithastosustain,sinceotherwise thisleakagecurrentwill flowalsothrough thegridresistance RIandchangethegridvoltage fromitsoptimum. Thesizeofthecapacitor mustbesuchthatitsim­ pedance issmallcompared withRIatthesignalfrequency, sincethen alltheamplified voltageattheanodewillbeimpressed onthegridof thenexttube. Theequivalent circuitofanRO-coupled amplifier isshowninFig. 13.2.Sincethehightensionsupplymustformalowimpedance forthe signalfrequency, bothterminals areatearthpotential asfarassignal voltages areconcerned. Theresistances RandRIhavetherefore a common terminal, asalsohasOg,whichrepresents theinputcapacitance ofthefollowing tube.Atlowaudio-frequencies theimpedance ofOgis largecompared withRI,andmaybeneglected.Iftheimpedance ofoissmallcompared withRI,asshouldbethecase,thenitwillbeseen 352 APPLICATIONS OF [13.1 thatthegridresistorR1iseffectively inparallelwithR,andthecom­ bination formstheloadforthefirsttube.ItisusualtomakeR1large compared withR,sothattheeffective loadisnotmaterially smaller thanR. + H.T. FIG.13.1.RO-coupled amplifier. ""\.."-p,v. FIG.13.2.Equivalent circuitforFig.13.1. Theeffectofthecapacitances 0andOgcanreadilybeseenfromcon­ sideration ofacommon requirement, anaudio-frequency amplifier to covertherangeof50to10000c/swithconstant amplification. Weshall assumethatthetriodeconstants arep,=30,P=10000ohms;then, withR=25000ohms,theinputcapacitance ofthetriodeOgisapproxi­ mately 120p,p,F(see§12.8).R1maybemade1megohm, sothatits shunting effectonRisnegligible. Atthelowfrequency limittheeffect 13.1] THERMIONIC VACUUM TUBES 353 ofOgisnegligible andtheratioofthevoltageacrossRItothatacross Ris IRI+~ijwol =(1+w2~2R0-!· Thustherequired condition is(1/wO) ~RI>whichisamplysatisfiedby making0=0·1ftF,whentheratiodiffersfromunitybyjustover 1percentat10cis.Atthehighfrequency end,theshunting effectof theinputcapacity ofthenexttriodemustbeconsidered. (Reference to§12.8showsthattheinputresistance ofthetriodewillbeabout60 megohms atafrequency of10kc/s,anditsshunting effectmaythere­ forebeneglected.) Thecapacitance OgisinparallelwithbothRandRI, anditspresence becomes noticeable onlywhenitsimpedance becomes comparable withthelowerofthese,R.Theeffective loadforthetube isthenOgandRinparallel, andtheamplification becomes Il~/zl =Il+P/4jwogp!={(1+p/R/+'w2C:p2}1' (13.1) showing thattheamplification isaffected onlywhenwOgpbecomes comparable withl+p/R. Thevaluesofthesetwoquantities are~re­ spectively 0·075and1·4at10kc/s,sothattheeffectofOgisnegligible. At100kc/stheamplification wouldbereducedby13percent,andfalls rapidlyasthefrequency isincreased stillfurther. Phaseshiftsinthe amplifier mayalsobeimportant (seeProblem 13.7). Incertainapplications, suchaspulsedradar,theamplification of shortpulsesoftheorderofmicrosecond duration isrequired, andifthe outputistobeundistorted theamplifier musthaveauniformmagnifica­ tionfromverylowfrequencies uptoseveralmegacycles persecond.It isclearthattriodescannotbeusedinsuchanamplifier, owingtotheir largeinputcapacitance, andpentodes mustbeusedinstead. Thesecond pointisthattheanodeloadisshuntedbysmallcapacitances fromseveral sources: (a)acapacitance of5to10ftftFbetween theanodeandthe suppressor- andscreen-grids, (b)theinputcapacitance ofthenextstage, againfrom5to10ftfJoF,and(c)straycapacitance fromthewiring.The totalcapacitance maybeasmuchas15fJofJoFanditseffectmaybeana­ lysedasfollows. Sincetheanoderesistance ofthepentode isveryhigh, itisconvenient tousetheconstant currentgenerator fortheequivalent circuit,asshowninFig.13.3.HereRistheloadresistance and0the totalcapacitance shunted acrossit.Thevoltageacrosstheloadis -gmvglZI =-gmvgR/(1+w2C2R2)1, showingthatthemagnification perstagewillfallbyafactor../2when 851110 Aa 354 APPLICATIONS OF [13.1 wOR=1.If0=15p,p,FandR=10000ohms,thispointisreached atafrequency of1·1Mc/s.Iftheamplifier hasaltogether nstages,each withmagnification A,thentheoverallmagnification isAn,andat1·1 Mc/stheoverallamplification willbedownby2n/2•Ifnis5or6,this isfartoomuchdistortion. Amoreuniform magnification perstagecan besecuredbyreducing R,whichreducesthestagegaininproportion. Constant currentgenerator G FIG.13.3.Current-generator circuitforapentode. Theanodoresis­ tanceofthetubewouldbeshunted acrossR,butitissohighitcan beneglected. R=loadresistance. G=totalcapacitance shunted acrossload. Morestagesmustbeadded,buttheoverallmagnification willbemore uniformbecausethedistortion dependsontheR2terminthedenomina­ tor.Asecondmethodofmaintaining thestagegainatthehighfrequency limitistointroduce asmallinductance inserieswiththeresistance, which,shuntedbythecapacitance, formsalowQresonant circuit.The valuesofRandLshouldbechosensothattheirimpedance isaboutthe sameasthatofthecapacitance 0atthehighestfrequency itisdesired toamplify. 13.2.Negative feed-back amplifiers Inanegative feed-back amplifier, afractionoftheoutputvoltageis fedbacktotheinputinsuchphaseastoreducethenetinputvoltage. Aschematic diagram isshowninFig.13.4.Thegainoftheamplifier in theabsenceoffeed-back isA,and,8isthefractionoftheoutputreturned totheinput.Theoutputvoltage Voisthen Vo=A(vi+,8vO), glvmg Vo=AVi/(I-A,8). (13.2) ThegainisnowA/(I-A,8), anditistherefore reduced if,8isnegative, i.e.ifthefeed-back voltageopposestheinputvoltage. Thisdrawback isoffsetbyseveraladvantages, inparticular thereduction ofdistortion. 13.2] THERMIONIC VACUUM TUBES 355 Anydistortion voltagewhichwouldappearattheoutputisfedback totheinputandreducedinthesameratioastheamplification. Ifnow theinputvoltageisincreased bymeansofapreceding amplifier untilthe overalloutputisrestored toitsformerlevel,thedistortion voltagewill remainatthereduced levelprovided thepreceding amplifier doesnot introduce distortion. Thiscondition isgenerally fulfilled becausethe signalisatalowlevelinthepreceding amplifier, anddistortion arises onlywhenthesignalishigh,asinthelaststagesofanamplifier. IIv; Iv. Amplifier FIG.13.4.Amplifier withfeed-back. If(-AfJ)ismadelargecompared withunity,thenthevoltage ampli~ ficationbecomes simply1JfJ.Thusitdepends onlyonthefeed-back ratio andnotatallontheactualamplification factorofthereceiver.Itis therefore independent ofanyvariations inAduetofluctuations inthe h.t.voltage, etc.Ifthefeed-back ratioisindependent offrequency, a wide~band amplifier withaveryuniformfrequency response isobtained. Fortheseconditions tobesatisfied inareceiverwithconsiderable net gain,{JmustbesmallandA,theamplification intheabsenceoffeed­ back,large.Greatcareisthenrequired inthedesign,forifthefeed-back {Jbecomes positiveatanyfrequency whereAissufficiently largetomake (I-AfJ)zeroornegative, oscillation willsetin.Toavoidthis,thefeed~ backmustbenegative overawiderrangeoffrequency than_the receiver willamplify. 13.3.Audio-frequency poweramplifiers Thediscussion sofarhasbeenconcerned with'voltage amplifiers', whereanamplified voltageoutputisrequired working intosuchahigh impedance thatnopowerisdrawn.Thisistruefortheintermediate stagesoflow-frequency amplifiers, butthelaststageisgenerally re­ quiredtosupplypowertoafiniteload.Ifthisloadisapureresistance R,thenuseoftheequivalent circuitofthetriode(Fig.12.9(a))shows thatthemeanpowerdeveloped intheloadwillbe ~R=fJ-2~RJ(p+R)2. (13.3) 356 APPLICATIONS OF [13.3 IfRcanbevaried,thenmaximum powerwillbedeveloped inRifit ismadeequaltop,ascanbeshowneitherbyuseofthemaximum power theorem (§3.3)orbydirectdifferentiation ofequation (13.3)withregard toR.Formosttriodesthismeansthattheoptimum valueofRisin theregionof10000ohmsormore.Sincethepoweroutputunderthe optimum condition R=pcanbewrittenasp,2v~/4p,itisobviousthat forafixedvalueofp,vg,morepowercanbeobtained byreducing p.For thisreasonlowimpedance triodes,withpoftheorderofafewthousand ohms,areusedforoutputstages.Inpracticetheallowable valueofp,vg isfixedbythesizeoftheh.t.voltage, forthevoltageswingontheload willbe!p,vg,andthiscannotapproach theh.t.voltagetoocloselywith­ outcausingconsiderable distortion (seebelow).Lowimpedance triodes havelowvaluesofp"sincegmisfixedbythecathodeemission, sothat toobtainthedesiredpoweroutputlargervaluesofvaarerequired. Typicalvaluesforanoutputtriodearegm=2·5mA/V,p=1500ohms, p,=3·75. Push-pull amplifiers Inordertoreducedistortion intheoutput,amethodofworking using twoidentical tubesin'push-pull' iscommonly employed. Thecircuitis showninFig.13.5,thegridsbeingexcitedinantiphase bymeansofa transformer withcentretappedsecondary winding. Theanodesofthe twotubesareconnected totheh.t.supplythrough thetwohalvesof thecentre-tapped primaryofatransformer whosesecondary winding is connected totheload.Sincethechangeinthegridvoltageofonetube is+vgwhilethatontheotheris-vg,theanodecurrents ofthetwo tubescanbeexpressed asseriesexpansions bo+b1Vg+b2V~+bsv~+b4 v~+} (13.4) and bo-b1va+b2v~-bs V~+b4 v~-. Theseflowinopposite directions throughthetwohalvesoftheoutput transformer, sothatitistheirdifference 2(b1vg+bsV~+b5 v~+...) (13.5) whichformsthemagnetizing currentforthetransformer, andwhich inducesavoltageinthesecondary winding. Thus,ifthetwohalvesof thecircuitareequallymatched, alltheevenharmonics disappear from theoutput. Soalsodothesteadycomponents booftheanodecurrents sincetheyflowinopposite directions through thetwohalvesofthe primary. Thishastheadvantage thatsaturation ofthetransformer corebythesteadycomponents oftheanodecurrents isavoided. 13.3] THERMIONIC VACUUM TUBES 357 Useofatransformer hastwofurtheradvantages: (a)iftheturnsratio isn:1,theloadseenbythetubeisn2timesgreaterthantheactualload (seeequation 9.41),andncanbechosentomatchtheloadtothetube; (b)thed.c.resistance ofitsprimary winding islowsothatthemean voltageontheanodeisalmostequaltotheh.t.voltage. Input_1....__ FIG.13.5.Push-pull amplifier. Efficiency ojpoweramplifiers Weconsider nowthesourceofthea.c.powerwhichanamplifier delivers intoaload.Itclearlycannotcomefromthesignalsource appliedtothegrid,forthepowerdrawnfromthissourceispractically zerobecause ofthehighinputimpedance ofthevalve.Theultimate sourceoftheoutputpoweristheh.t.supply,thoughthisisnotimme­ diatelyobvious, forthecurrentdrawnfromtheh.t.supplydoesnot changewhenasignalisbeingamplified ifthereisnodistortion. For simplicity, weshallanalysethecaseofatriodewhoseanodeiscon­ nectedthrough aresistance Rtoah.t.supplyofVuvolts.Iftheanode currentisla'thepowerdrawnfromtheh.t.islaVo,ofwhichapart l~R=la(Vo-v..),wherev..istheanodevoltage,isdissipated intheload resistance. Theremainder, lav..,isdissipated inthetubeandappears asheatattheanode.Theelectrons formingthecurrent fathroughthe tubegainkineticenergylav..astheymovethroughthepotential differ­ encev..between cathodeandanode,andthiskineticenergyisdestroyed whentheycollidewiththeanode,beingturnedintoheat.Whenasignal isappliedtothegrid,analternating component isaddedtotheanode current, whichbecomes la+iasinwt, whiletheanodevoltagebecomes 358 APPLICATIONS OF [13.3 Yo-R(1a+i asinwt).Themeanpowerdissipated ontheanodeisnowthe meanvalueoftheproductofthesetwoexpressions. Onmultiplying out, itisseenthattheproduct contains termsinsinwtwhoseaveragevalue iszero,andtheremainder is (Yo-R1a)1a-Ri~sin2wt. Thefirsttermisthesameastheanodeheatingintheabsenceofasignal, butthepresence ofthesecondtermshowsthatthereisareduction in themeanpowerdissipated ontheanodeoftRi~.Thisisjustequaltothe a.c.powerdissipated intheloadR.Thephysical reasonforthereduc­ tionintheanodeheatingarisesfromthefactthattheanodepotential fallsastheanodecurrentrises.Thusmorecurrentreachestheanode whileitspotential islowerthantheaverage value,andlesscurrent whileitishigherthantheaverage, withaconsequent reduction inthe meanpowerdissipated attheanode. Sincethesourceofthea.c.poweristheh.t.supply,wedefinethe efficiency ofthepoweramplifier astheratioofthea.c.poweroutputto thepowerdrawnfromtheh.t.supply. Thusintheaboveexample, the efficiency is lR'2/1TT"2~aaYo' Tofindthetheoretical efficiency wetakeanidealized casewherethe characteristics ofthetubearestraight linespassingthroughthepoint 1a=0atYc"=o.Weassumetheloadresistance Rtobeconnected through a1:1transformer withzeroresistance initsprimary winding sothatnovoltagedropacrosstheprimary occursintheabsenceofa signal.ThenthemeananodevoltageisYoanditsinstantaneous value isYo-vasinwt, whereVaistheamplitude ofthealternating voltage developed acrosstheload.Ifthecharacteristics arestraight downto "fa=0,wecanincrease vawithout introducing distortion uptothe valueYo,whentheinstantaneous anodevoltagebecomes zeroatone pointinthecycle.Thea.c.poweristhentV~/R,whilethepowerdrawn fromtheh.t.is1aYo=(V5!p),wherepistheanodesloperesistance of thetube.Aswasshownearlier,Rshouldbemadeequaltopforopti­ mumoutput,andthetheoretical efficiency isthen50percent. Practical valuesoftheefficiency aremuchlessbecausethecurvature ofthecharacteristics prevents largevoltageswingsbeingemployed. In thepush-pull amplifier, withitscancellation oftheevenharmonics, biggerswingscanbetolerated.Itisthenpossibletodepartfromthe typeofworking (knownasClassA)wehaveconsidered hitherto, and touseClassBworking, wherethetubesarebiasedtothecut-offpoint 13.3] THERMIONIC VACUUM TUBES 359 onthegrid.Currentflowsineachvalveofthepush-pull paironlyduring thehalf-cycle whenthealternating voltageappliedtoitsgridispositive. Duringthishalf-cycle theanodevoltageislow,andnocurrentflowsin theotherhalf-cycle whentheanodevoltageishigh.Thismakesthe efficiency high,thetheoretical valuebeing78percent(seeProblem 13.5).Inpractice, valuesof50to60percentarerealized. G + H.T. FIG.13.6.Amplifier withtunedcircuitasload.Forradio frequencies ascreengridorpentode tubewouldbeused. 13.4.Radio-frequency amplifiers Atradiofrequencies (bywhichismeantfrequencies oftheorderof 1Mcjsandhigher)itisusualtoemployatunedcircuitfortheanode load.Resistive loadsareunsatisfactory becausetheyareshuntedbylow reactances formedbytheinputcapacitance ofthenexttube,theanodeto earthcapacitance, andstraycapacitance inthewiring.Aparalleltuned circuitisemployed, asinFig.13.6,toobtainahighimpedance forthe anodeload;thesevariouscapacitances arethenshuntedacrossthetuned circuit,andformpartofthetotalcapacitance 0required totunethe coiltothedesiredresonant frequency. Sincetheimpedance ofaparallel tunedcircuitishighonlyneartheresonant frequency, suchanamplifier isselective, themagnification fallingrapidlyoneithersideofresonance. Nearresonance theimpedance oftheparalleltunedcircuitmaybe writtenapproximately as(seeProblem 9.1) ~=~+2jdwO, whereR=L/(Or),anddwisthedeparture ofwfromtheresonant 360 APPLICATIONS OF [13.4 (13.6)valueWo=l/(LO)!. Hencetheamplified voltage acrossthetuned circuitis -fLVg -fLvg -gmvg l+p/Z=1+p/R+2j,1.wOp =1/p+1/R+2j,1.wO· Attheresonant frequency themagnification isfL/(l+p/R)anditfallsby afactor";2atfrequencies deviating fromtheresonant valuesuchthat ±2,1.wO =(1/p+1/R). Iftheselectivity isdefinedasf/(2,1.f) =w/(2,1.w), sothatitisanalogous totheQofaresonant circuit,weseethattheselectivity isthesameas thatofourtunedcircuitshuntedbytheanoderesistance pofthetube. Reference totheequivalent currentgenerator circuitofFig.12.9(b) showsthatpiseffectively inparallelwiththeloadZ. Thevaluesofinductance andcapacitance forthetunedcircuitare determined asfollows. Thedesiredresonant frequency isusuallyfixed, sothatLO=l/wg.Forhighvoltageamplification, Rshouldbeashigh aspossible, sayabout105ohms.NowR=Q.j(L/O), andagoodworking ruleisthatQisoftheorderof100atfrequencies ofafewmegacycles persecond. Thusatafrequency of1·6Mc/s(wo=107),wehave .j(LO)=10-7,.j(L/O)=R/Q=103,givingL=100fLH,0=100fLfLF. Transformer coupling isoftenemployed inr.f.amplifiers, thesecondary windingbeingtunedasinFig.13.7.Theinputcapacitance ofthefollow­ ingstagethenformspartofthetuningcapacitance. Themagnification atresonance (theratioofthevoltageacrossthetunedcircuittothe voltageappliedtothegrid)isthen(seeProblem 13.2) A woMQ (7) =gm1+(woM)2/rp' 13. whichisamaximum whenthecoupling isadjusted sothat(woM)2=rp. HereQisthemagnification factorofthetunedcircuitintheabsence ofanycoupling. Inmanyapplications theprimary ofthetransformer istunedbyaparallelcapacitance aswell,andthecoupling isadjusted to givethe'band-pass' tuningobtainable withcoupledcircuits(see§9.4). Triodesareseldomusedforr.f.voltageamplifiers becauseofthefeed­ backthroughthegrid-anode capacitance. Itwasshownin§12.8that thisfeed-back givesafinitevaluefortheinputadmittance ofthetube. Thisadmittance consistsoftwoparts,oneprimarily capacitative which canbetunedoutifitisnottoogreat(intheexample of§12.8theinput capacitance wasfoundtobe120fLfLF,whichisofthesameorderasthe tuningcapacitance required aboveat1·6Mc/s.)Thesecondtermisre­ sistive,butmayhaveeithersign,beingnegative iftheanodeloadis 13.4] THERMIONIC VACUUM TUBES 361 inductive. Sinceaparalleltunedcircuitwillbeinductive atfrequencies belowitsresonant frequency, theamplifier willbreakintooscillation if thenegative conductance resulting attheinputisgreaterthananyposi­ tiveconductance inthesourcewithwhichitisinparallel. Toavoidsuch instability intheamplifier, screen-grid orpentade tubesaregenerally used,sincetheirlowgrid-anode capacitance makesthefeed-back very Voltage dropping .resistorforscreen o Screenby-passcondenser FIG.13.7.R.F.amplifier withtunedtransformer coupling andpentocle tubes.Thecapacitor 01isusedifcoupledtunedcircuitsareneededtoobtain bandpass tuning. small.Withhighgainamplifiers usingseveralstageseachstagemust bescreened byenclosure inanearthedmetalboxtopreventfeed-back fromonestagetoanotherthrough straycapacitances orinductances. Suchaboxisaneffective screenprovided thatitsthickness isgreater thantheskindepthforr.f.currents inducedontheinsideofthewalls, sincesuchcurrents arethenhighlyattenuated beforetheyreachthe outside. Iftriodetubesareused,asisgenerally thecaseinpoweramplifiers, thefeed-back throughthegrid-anode capacitance mustbe'neutralized' bytheprovision ofasecondfeed-back pathofopposite phase.Thiscan bedonebyanumberofmethods, oneofwhich,the'neutrodyne circuit', isshowninFig.13.8.Theanodecoilissplitintotwohalves,theh.t. supplybeingconnected tothecentrepoint.Thetwoendsofthecoil arethenatequalandopposite potentials withrespecttoearthasfaras theamplified signalisconcerned. Thefeed-back tothegridthroughthe 362 APPLICATIONS OF [13.4 grid-anode capacitance fromoneendofthecoilisthenbalanced outby thatthrough theneutralizing capacitor Onfromtheotherendofthe coil.Thearrangement iseffectively abridgecircuitasshowninFig. 13.9.Atbalance, forwhichthecondition isL20n=L10IJa,noneofthe outputvoltageappearsacrosstheinputterminals. Solongasinductance o Neutralizing capacitor On Grid-anode capacitance + H.T. L- --1~----- .....- FIG.13.8.Theneutrodyne circuit. oOutput voltage 1 FIG.13.9.Equivalent circuitoftheneutrodyne. intheleadsandotherstrayreactances arenegligible, thebalanceand hencetheneutralization isindependent offrequency. Forhighefficiency, r.f.poweramplifiers mayberununderClassC conditions. Thegridofthetubeisthenbiasedbackwellbeyondcut-off, sothatcurrentflowsthroughthetubeonlyforasmallfractionofacycle nearthepositivepeakofthealternating potential appliedtothegrid. Theamplitude ofthegridswingmustbeofthesameorderasthenega­ tivebiasonthegridinordertocarrythetubeintotheconducting 13.4] THERMIONIC VACUUM TUBES 363 region.Therelations between gridvoltageandanodecurrentareillus­ tratedinFig.13.10.Thecharacteristic plottedhereisa'dynamic characteristic', thevariation oftheanodecurrentwithgridbias being shownnotatconstant anodevoltage, asina'staticcharacteristic', but underworking conditions witharesistive loadintheanodecircuit.The anodecurrentishighlydistorted, consisting ofshortpulses,butatuned circuitisusedastheanodeloadsothatahighimpedance ispresented Dynamic characteristic I. V!---- ---...;..- /J I ~ VI c::;::! ':::J c:::::I J:::, c::::;;: I ::;:::I. e:: Ir t I FIG.13.10.Therelation between gridvoltageandanodecurrentinaClassCamplifier. Thegridhasalargenegative d.c.bias,andtheappliedalternating voltagehasalarge amplitude. Anodecurrentonlyflowsforafraction ofthepositive halfofthecycle. totheanodecurrentonlyatthefundamental frequency. Thiseliminates theharmonics fromtheoutputvoltageacrossthetunedcircuit. The theoretical efficiency ofClassCoperation is100percent,sinceunder idealized conditions theanodecurrentflowsonlyinpulsesofinfinitesi­ malduration whichcoincide withthepointinthecycleatwhichthe anodevoltageiszero(weassumethattheanodevoltageswingisequal inamplitude totheh.t.voltage). Inpractice efficiencies of60to80per centareobtained. ClassCoperation mayalsobeusedforthepurpose offrequency multiplication. Theformoftheanodecurrentmakesitveryrichin harmonics. Ifatunedcircuittunedtooneoftheharmonics isusedas theanodeload,theoscillatory voltageacrosstheloadwillhaveafre­ quencywhichisanexactmultiple ofthatappliedtothegrid.Frequency 364 APPLICATIONS OF [13.4 multiplication ofthistypeisusedinfrequency measuring equipment whereanunknown frequency istobedetermined bycomparison with astandard ofmuchlowerfrequency (see§15.4). 13.5.Tunedanodeandtunedgridoscillators Inthediscussion ofpoweramplifiers itwasshownthatthetubeacts asaconverter whichtransforms d.c.powerfromtheh.t.supplyinto a.c.powerin'theload.Forthispurposethesignalappliedtothegrid ofthetubeactsmerelyasatrigger,littleornopowerbeingdrawnfrom thesignalsourcesolongasthegriddoesnotgopositiveinanypart ofthecycle.Ifasmallfractionofthea.c.powerintheloadisusedas asourceofthesignalappliedtothegrid,theconversion ofd.c.power intoa.c.powermaybemadeautomatic, andnoexternal 'trigger' is required. Thetubethenactsasaself-sustained oscillator. Forthisto occur,certainconditions mustbefulfilledbythesizeandphaseofthe voltagefeed-back tothegrid.Reference toequation (13.2)showsthat theoutputvoltage Vois Vo=AviJ(l-Af3), whereAisthegainoftheamplifier without feed-back, andf3isthe fractionoftheoutputvoltagefedbacktotheinput.IfVoistobefinite whentheinputvoltage Vifromanindependent sourceismadezero, thenthedenominator mustbezero.Inotherwords,theproductAf3 mustbepositiveandequaltounity.Thisimposes conditions onboth thephaseandthemagnitude ofthefeed-back, whosenaturecanbemore clearlyunderstood byreference toasimplecase. Thetunedanodeoscillator Acircuitdiagram fora'tunedanode'oscillator isshowninFig.13.11. Aparalleltunedcircuitformstheanodeloadofatriodetube,anda voltageisfedbacktothegridbymeansofamutualinductance. We shallanalysethecircuitstartingfromfirstprinciples. IfV=Vo-~is thevoltageacrossthetunedcircuitandIthecurrentthroughthein­ ductance L,wehavetherelations 1a=1o+flm ¥g+~/p=10+flm¥Y+CVo- V)/p, ¥g=Md1/dt, V=r1+LdIJdt, Ia=1+CdVJdt. Fromthissetofsimultaneous equations, allvariables butonemaybe 13.5] THERMIONIC VACUUM TUBES 365 eliminated. ItissimplesttoretainIasthedependent variable, andthe resulting equation is LOd2Ijdt2+(Or+Ljp-gmM)dljdt+(1+rjp)I =Io+Yojp. (13.8) Theright-hand sideisindependent ofthetime,andthedifferential equation isthatofadamped harmonic oscillation. Thedamping will bezeroifconditions arechosensothatthecoefficient ofdIjdtismade zero,i.e.(13.9) c +VO H.T. o FIG.13.11.Thetuned-anode oscillator. (13.11)Thefrequency ofnaturaloscillation ofthecircuitisthengivenbythe relationLOw2=(l+rjp), (13.10) andany oscillation ofthisfrequency whichexistswillcontinue withthe sameamplitude. Equation (13.9)isthecondition forthemaintenance ofoscillation, anditcanbeshownthatitcorresponds tothecondition Af3=1(seeProblem 13.3). Inpraoticethefeed-baok isnotadjusted soastomaketheooeffioient ofdlldtinequation (13.8)exaotlyzer01sincethiswouldnotgivestable oscillations (asmallohangeintheconditions leadingtoareduotion in Ym'forexample, wouldmakethecoefficient ofdlldtpositive, andthe oscillations woulddieaway).Thefeed-back istherefore madesolarge thatthecoefficient ofdlldtisnegative, andthesolutionofequation (13.8)isthenoftheform 1=(Io+Yojp)+e-bt(Ae(b'-c·lt+Be-(b.---c.)i t), 366 APPLICATIONS OF [13.5 whereb=(Cr+LjP-(JmM)/2LC, andc2=(1+rjp)/LC. Ifc2>b2, theoscillatory partofthecurrentmaybewritten 1=e-bt(A'coswt+B' sinwt), wherew=.J(c2-b2).Thisrepresents anoscillation whichdecaysaway ifbispositive, isjustmaintained ifbiszero,andincreases inampli­ tudeifbisnegative. Thecondition forthelatteris gmM>(Cr+Ljp) orM>(L+pCr) , fL showingthatthereisaminimum valueofMrequired togiveoscilla­ tions.Whenbisnegative, anytransient oscillation intheanodecircuit (suchaswouldbecausedbyswitching ontheh.t.voltage, orbynoise (seeChapter 16)buildsupinamplitude insteadofdyingaway.Our equations suggestthattheamplitude wouldincrease indefinitely, but thisisnotso,becausethe'constants' gm'pofthetubearetrulyconstant onlyforsmallamplitudes ofoscillation, limitedtothestraightportion ofthetubecharacteristic. Whentheamplitude issogreatthatthepeaks oftheoscillation carrythetubeontotheflatportions ofthecharac­ teristicatsaturation andcut-off,theeffective valueofgmfalls,andthe amplitude willreachasteadypointwhereitseffective valueissuchas tomakeb=O.Ingeneralthispointisreached whentheamplitude ofthevoltageswingacrossthetunedcircuitisofthesameorderasthe h.t.voltage Vo. Thecoefficient bmaybewrittenintheform r1gmM b=2L+2pC-2LC . (13.12) Herethefirsttermgivestherateatwhichoscillations woulddecayin theanodecircuitifthetubewerenotconnected, ornotswitched on;the secondtermrepresents theextradamping causedbytheanoderesistance ofthetubep,whichiseffectively shunted acrossthetunedcircuitwhen thetubeisrunning; andthelasttermshowstheeffectofthetubeand thefeed-back inreducing thedamping ofthetunedcircuitevenwhen Misnotlarge enoughtomaintain oscillations. Thiseffectisknownas 'regneration'. Itisimportant toremember thatthesignofMcanbe negative iftheconnexions tothemutualinductance arereversed. The damping ofthetunedcircuitisthenincreased bytheactionofthetube, aneffectknownas'degeneration'. TheQandselectivity ofthecircuit aretherebydecreased, whereastheyareincreased byregeneration. The latterhastheeffectofcreating a'negative resistance' inthetuned 13.5] THERMIONIC VACUUM TUBES 367 circuit,andspontaneous oscillation occurswhenthenegative resistance islargeenoughtooutweigh thepositiveresistance. Apositiveresistance isoneinwhichpowerisdissipated, anegative resistance oneinwhich powerisgenerated. Thetunedgridoscillator Asecondimportant typeofoscillator isobtained byattaching the tunedcircuittothegridofthetube,andfeedingbackavoltageintothis circuitbymutualinductance coupling fromacoilintheanodelead; Thisisknownasthe'tunedgrid'oscillator andisshowninFig.13.12. +VO H.T. o FIG.13.12.Thetuned.grid oscillator. Theanalysisofthiscircuitissimilartothatusedforthetunedanode oscillator. Letfbethecirculating currentinthegridcircuit,Vbethe voltagedeveloped between gridandcathode, ~theanodevoltage,and Iatheanodecurrent. Thentheequations forthegridcircuitare V=M(dfajdt)+rI+L(dljdt),1=-O(dVjdt), whilefortheanodecircuitwehave fa=fo+YmV+~/P. Yo=~+(Lld~a+M~0· Asimplesolution oftheseequations ispossibleifweassumethat theeffectofthetermsinthelastbracketissmall,sothateffectively ~isconstant andequaltoYo.Thisisusuallytrueinpractice. Then 368 APPLICATIONS OF [13.5 elimination offandfabetween thefirstthreeequations leadstothe expression d2V dV LOdt2+(rO-ymM)dt"+V=O. (13.13) Thisrepresents anoscillatory motion,andissimilartoequation (13.8). Oscillations willbemaintained orwillbuildupifM;?:rOjYm' Whentheoscillator isrunning steadily, thecoefficient oftheterm dVjdtintheoscillatory equation (13.13)iszero,andthefrequency of oscillation isgivenbytherelation w=(LO)-l,showingthatitisdeter­ minedbythenaturalresonance frequency ofthetunedcircuit.Ifwe returntothecorresponding equation (13.8)forthetunedanodeoscilla­ tor,theangularfrequency isfoundtobew={(1+rjp)jLG}l showing thatitdepends slightlyontheanodeimpedance pofthetube.Since thelattermaychangewiththerunning conditions, thefrequency will alsovary,andwhengoodfrequency stability isdesired,thetunedgrid oscillator isgenerally preferred, sinceherethetubeconstants donot enterdirectlyintotheequation forthefrequency. Inpracticethefre­ quencywilldependtosomeextentonthetube,fortheinputcapacitance ofthelatterisshunted acrossthetuningcapacitor ofthegridcircuit ofthetunedgridoscillator, andtheinputcapacitance varieswiththe running conditions (see§12.8).Another causeoffrequency driftis changeinthetemperature ofthecomponents, withconsequent changes intheirelectrical constants. Asaroughguideitmayhesaidthatthe frequency ofanordinary smalloscillator, following theinitialwarming upperiodafterswitching on,isstabletotheorderofapartin1000. lfhigherstability isrequired quartzcrystaloscillators areused(see §15.4).Thesearelowpoweroscillators (afewwattsatmost),which arethenfollowed byr.f.poweramplifiers tosupplytherequired output. Forthehighestefficiency, suchamplifiers arerunas'ClassC'(see§13.4). 13.6.Poweroscillators Whenalargepoweroutputisrequired, butitisnotessential tohave thehighestfrequency stability, anoscillator rununder'ClassC'condi­ tionsisused.ThisissimilartotheClassCamplifier (see§13.4),and giveshighefficiency; itmayberegarded asaClassCamplifier with regeneration tosupplythegridexcitation voltage. Themeanpotential ofthegridiswellbeyondthecut-offvalueforthetube,andtheexcita­ tionvoltagemusttherefore havesufficient amplitude tocarrythetube intotheconducting regionatthepositive peaks.Aconvenient circuit givingalargegridexcitation isthatduetoHartley, whereatapped inductance isusedasanauto-transformer tosupplytherequired feed- 13.6] THERMIONIC VACUUM TUBES 369 back.ThebasiccircuitisshowninFig.13.13.Thecathodeisconnected tothemid-point oftheinductance, andthegridandanodethroughtheir respective voltagesuppliestotheopposite endsoftheinductance, where G.B. + Tunedcircuit·+ H.T. FIG.13.13.BasiccircuitofHartley oscillator. thealternating potentials areinopposite phasewithrespecttothe cathode, thusgivingtherightsigninthefeed-back foroscillation. Analternative formoftheHartley circuitisshowninFig.13.14. Thisisknownasthe'shunt-feed' typeofcircuit,theh.t.voltagebeing connected totheanodeinparallelwiththetunedcircuit,insteadofin Tuned circuit oChoke + H.T. FIG.13.14.Shunt-feed Hartley oscillator withautomatic grid-bias. Appropriate values ofthecircuitelements forafrequency of1Mc/s:L=125p.H,L1=0·1H;R=10000 ohms;0=200p.p.F; 01=O·OIp.F, 03=0·001p.F. serieswithit,asinFig.13.13.Thisrequires achokeL1intheh.t.lead topreventoscillatory currents flowingtotheh.t.,andablocking capaci­ tor01toisolatethetunedcircuit,whosemeanpotential isthatofthe cathode, fromtheh.t.ThevaluesofL1andqmustbesufficiently large atthefrequency ofoscillation thattheirimpedances arerespectively largeandsmallcompared withthatofthetunedcircuit. ForClassCoperation ofanoscillator, aspecialtypeofgridbiascircuit 861110 Bb 370 APPLICATIONS OF [13.6 isrequired, forthefollowing reason.Ifalargesteadynegative bias,such asthatprovided byabattery, isappliedtothegrid,oscillations cannot startbecausenocurrentcanflowthroughthetubewhentheamplitude ofoscillation issmall,thoughoscillations canbemaintained atahigh levelsufficient toswingthegridintotheconducting region.Toovercome thisdifficulty, anautomatic formofgridbiasisrequired whichisinitially zero,andincreases withthelevelofoscillation. Thisisprovided bythe ROscombination showninFig.13.14.Astheoscillations increase in amplitude, thegridisswungpositive forpartofthecycle,andcollects electrons. Thisgivesagridcurrentwhich,flowingthroughRonits returnpath,makesthemeanpotential ofthegridnegative provided thatthesizeofthecapacitor Osissuchastomakethetimeconstant oftheROscombination longcompared withtheperioclofoscillation. Thentheshortpulseofelectron currenttothegridwhenitswingsposi­ tivechargesupOs,andtheslowdischarge ofOsthroughRcreates themeannegative potential required forthegridbias.Inpracticethe optimum valueofRisusuallyaround10000ohms;lowervaluesgive insufficient bias,andmuchhighervaluesaredangerous. For,ifthe voltageswingbecomes toolarge,andtheanodepotential fallstoolow whilethegridpotential ispositive, thegridmaystarttoemitmore secondary electrons thanitreceives primaries. Thisreverses themean gridcurrent, andthebiasbecomes positive insteadofnegative; the excessive currentwhichresultsmaydestroythetube. Thecapacitor OsshouldbechosentomaketheROstimeconstant about10periodsofoscillation. Ifthetimeconstant ismadetoolong, intermittent operation knownas'squegging' maybecaused,forthe gridbiascannotadjustitselfquicklyenoughtofollowrandomchanges intheamplitude ofoscillation. Ifthelatterstartstofall,butthebias isnotreduced, currentceasestoflowthroughthetube,andtheoscilla­ tionswilldieaway;theycannotrestartuntilOshasdischarged through Rsothatanodecurrentcanflowagain.Thusoscillations maybeinter­ ruptedperiodically atafrequency determined bytheRCscombination. Typical valuesofthecircuitconstants foraHartley oscillator ata frequency ofabout1McjsaregivenaboveinFig.13.14.Theresistance oftheinductance isomitted fromthediagram, butitsvaluecanbe foundiftheQofthecoilisknown. 13.7.TheKipprelayandthemultivibrator Theoscillators whichhavebeenconsidered sofarproduce sinusoidal oscillations whosefrequency iscontrolled almostentirelybythecon- 13.7] THERMIONIC VACUUM TUBES 371 stantsof atunedcircuit.ThisistrueevenoftheClassCtype,where theanodecurrentisveryfarfromsinusoidal, forthetunedcircuitoffers anappreciable impedance onlytothefundamental frequency, andthe voltagedeveloped acrossitisalmostsinusoidal (inthisrespectthe oscillator issimilartotheClassCamplifier). Thequestion arises,what willhappenifwetakeanuntuned amplifier, andintroduce feed-back oftherightsigntoproduce instability? Suchadeviceisshownin Fig.13.15,wherethecircuitconsistsofatwo-stage aperiodic amplifier, r---.......--------i~-- ......---....+H.T. ......-------- ......--------...f~ -H.T. FIG.13.15.TheKipprelay.B!,Baarebatteries tosupplygrid-bias voltage. withfeed-back fromtheanodeofthesecondtubetothegridofthefirst tube.ThepurposeofthebatteriesBl'B2istoprovidedirectcoupling fromanodetogridwhilepreserving thecorrectsteadyvoltages onthese electrodes. Ifthevoltageamplification ofeachstageisA,whereAis negative toallowforthechangeofphasebetween gridandanode voltages, thentheoverallamplification isA2.Thefeed-back factorf3is practically unity,sothatifA2>I,thedeviceshouldbeunstable. Intheanalysisofthissystemwemusttakeaccountoftheelectrode capacitance oftheanodeandotherstraycapacitance between anodeand earth;thisisrepresented bythesmallcapacitance Cwhichshuntsthe anodeloadrofeachtube.Theanodecurrentofthefirsttubeis il=gmvI+V2!P, wherelower-case symbols areused,sinceweshalldealonlywiththe fluctuating components. HereVIisthevoltagechangeappliedtothe gridoftubeI,andV2isitsanodevoltagechange,whichisthesameas 372 APPLICATIONS OF [13.7 thegridvoltagechangeofthesecondtube.Sincetheanodecurrentil flowsthroughrandCinparallel, wehavealso -il=C(dvz/dt)+vz/r. Elimination ofilgivesthefollowing relationbetween VIandvz,together withanexactlysimilarrelationwithVIandVzinterchanged, obtained byapplying thesameanalysistothesecondtube: -gmVI=C(dVz/dt)+Vz(r+p)/r p}. (13.14) -gmVz=C(dvl/dt)+vl(r+p)/rp Since-gmpr/(r+p) =A,theamplification ofeachtube,wemaywrite theseequations intheform AV1--r(dvz/dt)+V z},(13.15) Avz=-r(dvl/dt)+vI where -r=Crp/(r+p) isthetimeconstant ofthecapacitance Cin parallelwithrandp.Thesolution oftheseequations is VI=-Vz=voexp{-(A+l)t/-r}. (13.16) Ifthevalueof-Aforeachtubeisgreaterthanunity,thissolution showsthatasituation witheachtubeconducting willnotbestable,since anydisturbance ofthegridpotential ofonetubeduetonoise,etc.,will increase exponentially. Thegridofonetubewillriseinpotential while theothergoesnegative atthesamerate.Thefirsttubewilltherefore conductatanincreasing rateuntilitsaturates, whilethesecondwillcon­ ductatadecreasing rateuntilitiscutoff;orviceversa.Thissituation willremainuntilashortpulseappliedtothegridofthetubewhichiscut offbringsitintotheconducting region;thentheexponential increase ofitsgridvoltagewillcarryittosaturation whiletheothertubewill changefromsaturation tocut-off. Thetimerequired forthisvoltage 'landslide' isveryshort.If-A~1,theeffective timeconstant ofthe exponential isapproximately -r/(-A)=C/Ym>andtypicalvaluesare C=50JLJLF,Ym=5X10-3A/V,givingC/gm=10-8sec.Hencethe timerequired foraninitialdisturbance (whichmightheoftheorder ofamicrovolt) toincreaseto100Vis t=10-8{2·31ogI0(10z/1O-6)} =0·2X10-6sec. Thisresultoflessthanamicrosecond givesaratheroptimistic value fortheduration ofthevoltagelandslide, however, fortworeasons: (a)whensaturation setsinthevalueofgmislowerthanthatassumed, and(b)whenthegridofthesecondtubeisswungnegative beyondthe cut-offpoint,thedischarge ofitsanodecapacitance isincomplete and 13.7) THERMIONIC VACUUM TUBES 373 cancontinue onlythroughtheanodeloadresistance r,whichisusually muchgreaterinvaluethanl/gm-Thismakesthelandslide oflonger duration forthetubewhichisbeingcutoffthanfortheothertube. Thedevicewhichhasjustbeenconsidered isknownastheKipprelay. Itwillrespondto·averyshortvoltagepulse,andonceswitched overwill remainsountilapulseoftheopposite polarity isapplied. Theanode currentofthetubewhichiscausedtoconductbythepulsemaybe .....-----------..-------+H.T. "----e.------ ......----.....------(:"---H.T. FIG.13.16.Themultivibrator. usedtooperateamechanical relay.Inpractice thebiasbatteries Bll B2maybeeliminated byasuitableautomatic biasingarrangement. Ifthedirectcoupling between stagesoftheKipprelayprovided by thebatteries B1,B2isreplaced byROcoupling,asinFig.13.16,asystem isproduced inwhichaperiodicchangeoverfrom(tube1saturated, tube2 cutoff)to(tube1cutoff,tube2saturated), andviceversa,isproduced automatically. Thisisknownasthemultivibrator. Itsactioncanbe understood asfollows. Supposeatsomeinstanttube1issaturated, and tube2iscutoff.Thenthevoltageongrid2isnegative, butthecharge oncapacitor 01whichisholdingitnegative isgradually returning to itsequilibrium valueandthevoltageacrossthegridresistance R1is returning tozero.Whenitbecomes sufficiently smalltoallowanode currenttostartflowinginthesecondtube,theexponential voltageland­ slidewilltakeplace.Theanodevoltageofthistubewilldropsuddenly, andthisdropwillbetransferred through O2R2tothegridoftube1, whichwilltherefore becutoff.ThevoltageacrossR2willthendecay APPLICATIONS OF [13.7 asO2recharges toitsequilibrium value,andthereverselandslide will occurwhenithasfallensufficiently forconduction tobeginintube1. Thecycleisnowcomplete, andthewaveformsofgridandanodevoltage fortube1,andgridvoltageoftube2areshowninFig.13.17.The positivekicksofgridvoltagewhichoccuratthechange-over pointsare cutoffatasmallpositive voltagebytheflowofelectrons tothegrid. Theanodeofeachtubeisalternately attheh.t.voltage(duringcut-off) L o.r----......-- -ve FIG.13.17.Voltage changes inthemultivibrator onthegridandanodeoftube1, andthegridoftube2. andatalowvoltagedetermined byrandthesaturation currentofthe tube.Ithastherefore arectangular waveform,thesteepness ofthesides depending ontherapidity ofthevoltagelandslides, whicharecontrolled mainlybythestrayanodecapacitance. AsalreadynotedfortheKipp relay,thechangeoverfromconduction tocut-offtakesratherlonger thanthereversechange,sothatthetwosidesofthesquarewaveare notequallysteep. Theperiodofacomplete cycleismainlydetermined bythecharging ofqthroughRvandO2throughR2•Atsaturation thevoltagedrop acrossthetubeissmall,sothatthesuddenchangeofanodevoltage whenthetubeconducts ispractically equalinamplitude totheh.t. voltage; soisthenegative voltagekickappliedtothegridofthenext tube.Thisvoltagemustdecaytothecut-offpointofthegridcharac­ teristic, whichisnearlyequaltotheh.t.voltage dividedbyfL,the 13.7] THERMIONIC VACUUM TUBES 375 amplification factorofthetube.Thetimeofdecayforthecircuitof Fig.13.16willtherefore benearlyR1qlogp.foronetube,andR2021ogp. fortheother,thetimerequired foracomplete cyclebeing (R101+R2°2)logp.. Amoreaccurate analysis showsthateachRshouldbereplaced by Rrp +(r+p) ineachcase,since°reallychargesthroughRinserieswith(randpin parallel). Themultivibrator, withitsrectangular waveform,isofgreatusein generating squarevoltagepulses,andharmonics ofastandard frequency. Itisreadilysynchronized withaninjected sinusoidal signal,appliedto thegridofonetube,ifitsnaturalperiodisclosetothatofthesignal. TheeffectofsuchasignalistodelaythereturnofthegridvoltageW theconducting pointifitwouldbeearly,andtospeeditupifitsnatural periodissuchthatitwouldreturntoolate.-Thisproperty ofsynchroniz­ ingwithanappliedsignalisofuseinfrequency measurement, since theharmonics generated bythemultivibrator arethenexactmultiples ofthestandard frequency, andanunknown frequency maybecompared withthenearestharmonic. Themultivibrator mayalsobeusedforfre­ quencydivision, foritwillsynchronize withasignalwhoseperiodis closetoanexactfractionofitsown,i.e.afrequency upto5or10times itsown. 13.8.Amplitude modulation anddetection .InChapter 12theuseofvacuumtubesforrectification wasoutlined; thatis,theconversion ofanalternating voltageintoasteadyvoltage. Aprocesssimilartothisisemployed inthereception oframosignals,and isgenerally knownasdetection. Thedifference liesinthefactthatthe radiosignalismodulated insomewayinordertoconveyinformation, suchasspeechormusic,whosecharacteristic frequencies lieintheaudio range,whilethesignalitselfisatamuchhigherfrequency, knownas thecarrierfrequency. Onesystemusedforthispurposeiscalledampli­ tudemodulation, sincetheamplitude ofthecarriersignalismadeto varywiththeperiodoftheaudiofrequency, andbyanamountwhich isproportional tothestrength oftheaudio-frequency information. For simplicity weshallconsider onlyasingleaudiofrequency ofconstant strength. Theamplitude-modulated radiosignalmaythenbewritten intheform V=A(I+mcospt)ooswt. (13.17) 376 APPLICATIONS OF [13.8 HereAistheamplitude ofthecarriersignalintheabsenceofmodula­ tion,andW/21Titsfrequency. Theconstant misknownasthedepthof modulation, andcannotbegreaterthanunity,andp/21Tistheaudio frequency. Thenatureofanamplitude-modulated signalcanbeseenfromFig. 13.18(a),whichshowsthevariation ofthevoltage Vwithtime.Its ~--------,.27Tlp-------~ FIG.13.18.(a)Amplitude-modulated signal,beforedetection. (b)Amplitude-modulated signal,afterdetection. Normally wismuchgreaterthanp. amplitude fluctuates slowlybetween amaximum valueofA(1+m)and aminimum ofA(I-m), theperiodofacomplete cycleofthisfluctua­ tionbeing21T/p.Manipulation ofequation (13.17)showsthatitmay berewritten as V=Acoswt+!mA cos(w+p)t+!mA cos(w-p)t. (13.18) Thisindicates thatthemodulated signalmayalsoberegarded ascom­ posedofthecarriersignalAcoswt,together withtwootherfrequencies, higherandlowerbyp/21T,whichareknownasside-bands, andwhose amplitude isproportional totheproductofthecarrierstrength andthe depthofmodulation. Thepresence oftheseside-bands showsthatany receiver withr.f.circuitsmustbedesigned tohaveapassbandwhich willacceptthefrequencies (W±p)/21T aswellasthecarrierfrequency W/21T,asotherwise theaudio-frequency modulation willbecutout.The presence oftheside-bands maybedemonstrated byapplying the 13.8] THERMIONIC VACUUM TUBES 377 modulated signaltoasharplytunedfrequency-meter, whichwillshow responses atthethreefrequencies (W-p)/217, (/)/217,and(W+p)/217. Ingeneralthemodulation willnotconsistofasingleaudiofrequency, butofawholerangeoffrequencies. Forspeechormusicthesecover therangefromabout50c/stoseveralkc/s,whilefortelevision aband ofseveralMc/sisrequired. Thisisbecause thepictureconsists, for example, of400X400separate dots,scanned 25timesasecond,sothat 400X400X25=4X108piecesofinformation mustbetransmitted per second.Thepulsecorresponding toaspotmusttherefore lastlessthan amicrosecond, andareceivertoamplifysuchpulsesrequires aband­ widthoftheorderof4X108c/s.ByFourieranalysis anymodulation canalwaysberesolved intoasetofsinusoidal oscillations, andour analysiscantherefore proceedintermsofonesuchfrequency, bearingin mindthatthevariouspartsofareceivermustthenhavethebandwidth required toaccommodate allmodulation frequencies uptothehighest. ThemeanvalueofthesignalvoltageViszerooveranyperiodlong compared withthatofthecarrierfrequency, anditwilltherefore produce noeffectinareceiverdesignedtoacceptonlyaudiofrequencies. Ifthe signalispassedthrough arectifierstagesothattheportions whereVis negative arewipedout,asinFig.13.18(b), themeanvalueoftheresultant isnotzeroandfluctuates attheaudio-frequency ratecorresponding to themodulation. Thisprocessisknownasdetection, sincetheinforma­ tionwhichisconveyed bythemodulation cannowbedetected bythe earifthesignalfromtherectifier, aftersuitableamplification, isapplied toheadphones oraloudspeaker. Anobviousrequirement ofadetector isthatitsoutputsignalshallbeasnearlyaspossibleatruereproduction oftheoriginalmodulation, i.e.theoutputvoltageshouldbelinearly proportional tothedepthofmodulation m,andtheconstant ofpropor­ tionality shouldbethesameforallmodulation frequencies. Acircuitusingadiodeforthedetection ofamplitude modulated waves isshowninFig.13.19.Itwillbeseenthatitisessentially thesameas thatofFig.12.3,butcertainlimitations mustbeplacedonthevalues ofRand0toobtainefficientanddistortionless detection. Thesemay besummarized asfollows: (I)Theloadresistance Rshouldbelargecompared withtheeffective outputresistance ofthediode,p.Thelatterisapproximately equalto thereciprocal oftheslopeofthediodecharacteristic, andformsavoltage dividerwithRjustasinthecaseofthetriodetube.Sincepvarieswith thesizeoftheappliedsignal,thecondition R~pnotonlymakesthe fractionofthepossibleoutputvoltageappearing acrossRnearlyunity 378 APPLICATIONS OF [13.8 (highefficiency) butalsomakesthisfraction nearlyindependent ofp andhenceofthemagnitude oftheappliedsignal(lowdistortion). (2)Thetimeconstant oftheBOcircuitshouldbelongcompared withtheperiodofthecarriervoltage,toavoidvoltages ofthisfrequency appearing intheoutput(i.e.1/wO~B). a.f.amplifier II I I I I I I I I I I I I I IG Diodedetector stageII I II Circuittuned tor.r. ~-.----.Inputfrom r.f.amplifier FIG.13.19.Diodedetection circuit. V1=modulated inputvoltage. V2=outputvoltage. (3)AnupperlimittoBOissetbytherequirement thatthevoltage across0shallchangesufficiently rapidlytofollowthemodulation. This requires(l/pO)>B,ormorestrictly,(l/pO)?Bm/(1--m2)t(forproof ofthisrelation, seeE.Williams, 1952;thepresence ofrnarisesbecause therateofchangeofthecarrieramplitude depends onthedepthof modulation). (4)0shouldbeseveraltimesaslargeasthecathode-anode capaci­ tanceOcaofthediode,since0andOcaformavoltagedividerforther.f. voltageappliedtothediode. ThecircuitofFig.13.19showsthemodulated inputvoltagebeing supplied fromatunedr.f.transformer. Thecondition B?>p(see(1) above)makesitnecessary forthesizeoftheinputvoltagetobeofthe orderofavoltorso,inordertoworkonaportionofthediodecharac­ teristicwheretheslopeisfairlyhigh.Inthereception ofbroadcast signalsrangingfrommillivolts downtomicrovolts, itistherefore neces­ sarytoamplifythesignalbeforedetection. AttherightofFig.13.19 --- ---~ -~ 13.8] THERMIONIC VACUUM TUBES 379 theoutputfromthedetector isshownappliedtothefirststageofari a.f.amplifier. Theblocking capacitor 01isinsertedtopreventthesteady component oftherectified voltageacrossRbeingappliedtothegridof the:firsttubeandsochanging itsbias.Thesizeofqshouldbesuchthat (lJpOl) ~R1forthelowestfrequency (pJ27T)presentinthemodulation, andR1shouldbeofthesameorderorlargerthanR,since,inparallel withR,itformspartoftheloadresistance forthediodedetector. Theuseofthediodedescribed above,wheretheappliedsignalislarge enoughtooperatethediodeonthestraightpartofitscharacteristic, is knownas'lineardetection' ,sincetheoutputvoltageislinearlypropor­ tionaltotheamplitude oftheinputvoltage.Iftheinputvoltageisvery small,aswouldbethecaseifabroadcast signalwereappliedtothe diodedirectlywithoutprevious amplification, thediodeisoperated only overaverytinyportionofitscharacteristic, anddetection orrectifica­ tionresultsorilyfromthecurvature ofthecharacteristic ofthisregion. Theoutputcurrentorvoltageisproportional tothesquareoftheinput voltage,andtheprocessisknownas'squarelawdetection'. Anapproxi­ mateanalysis maybemadebyassuming theloadresistance issmall compared withthemeanoutputresistance ofthediode;thelatteris veryhighwhentheappliedsignalissmall.Thenthecurrentthrough thetubewhenasmallsignalvoltagevisappliedmaybewrittenas I=Io+(:~)v+~(:i2)V2+ ...=Io+av+bv2+...,(13.19) whereIoisthecurrentflow(ifany)whenv=0,andaandbaredeter­ minedbytheslopeandcurvature ofthecharacteristic nearthepoint I=Io.Ifv=VIcoswt,then (13.20) showingthatthereisachangetbviinthemeancurrent,whichispropor­ tionaltothesquareoftheappliedsignal.Ifthelatterismodulated, sothatVI=B(l+mcospt), thenthelowfrequency currentchangeis !bB2(1+2mcospt+im2+!m2cos 2pt),showingthatthedetected signal willhaveharmonic distortion owingtothepresence ofthetermincos2pt. Forthisreason,andbecauseoftheverylowefficiency, squarelawde­ tectionisnotusedinradioreception.Itisusedinsomevacuum-tube voltmeters, butusuallywithatriodetuberatherthanadiode.The triodeisworkedonacurvedportionofitsanodecurrent-grid voltage characteristic, .andtheanalysis givenabovemaybeappliedifvisthe changeingridvoltageandithechangeintheanodecurrent. Thechange 380 APPLICATIONS OF [13.8 inanodecurrentmaybeobserved onamilliammeter inserted inthe anodeload.Thissystemisknownas'anodebend'detection, sinceit depends onthecurvature oftheanodecurrentcharacteristic. Thead­ vantageofusingatriodeinsteadofadiodeisthatcomparatively large changesintheoutputcurrentmaybeobtained, whileahighinputim­ pedance isofferedtothesource. Itshouldbenotedthatthetriodecanbeusedforlineardetection if thegridisbiasedtocut-off,andthesizeoftheinputsignalissufficient toswingthegridontothelinearportionoftheanodecurrent-grid voltagecharacteristic duringthepositivepeaks.Themeananodecurrent willthenchangelinearlywithanychangeintheamplitude oftheapplied signal.Thiscanbeusedeitherfordetection ofamplitude modulated signals,orinfrequency changing, discussed inthenextsection. 13.9.Frequency changing Sincesquarelaw·detection isveryinefficient compared withlinear detection (seeProblem 13.4),itisalwaysdesirable thatasignalbe amplified sufficiently, beforebeingappliedtothedetector, toworkthe latterinitslinearregion.Oftenitisundesirable, andsometimes im­ possible, toprovidesufficient amplification forthispurposeatthecarrier frequency. Adeviceknownasfrequency changing isthenused,inwhich, asthenamesuggests, thecarrierfrequency isalteredtoanothermore convenient frequency, themodulation beingpreserved intact.Inthe formulae (13.17)and(13.18)aboveforamodulated signal,thismeans thatwischangedtoanothervalue,butthatthetermsinmremainthe same. Thischangeoffrequency isaccomplished byaddingtotheoriginal signalanalternating voltageofanother frequency (w1/2rr)generated locally,andpassingthetwointoarectifying stageknownasthemixer. Theoutputfromthemixerthencontains voltagecomponents which fluctuate at,apartfromthemodulation frequencies, (w1-w)/2rrand (wl+w)/2rr. Ifthedifference frequency (wl-w)/27T liesintheaudible range,itmaybeamplified andmadetoworkheadphones oraloud­ speaker. Thissystemisknownasheterodyne reception andisusedin telegraphy wherethecarriersignalismodulated onlybybeingswitched onandoffinaccordance withsomeprearranged codesuchasthedot­ dashsystemoftheMorsecode.Thedotsanddashesarethenheardas audiblenotes(usuallyabout1000cis). Inasuperheterodyne system,thesumanddifference frequencies are outsidetheaudiblerange,andoneofthemisselected andamplified. 13.9J THERMIONIC VACUUM TUBES 381 Thefrequency selectedisknownastheintermediate frequency (i.f.)and thei.f.amplifier magnifies thesignal,withitsoriginalmodulation, toa levelatwhichitcanbedetectedbyadiodeoperating inthelinearregion. Sincethemixingstagemustincorporate anon-linear device,itisoften calledthe'firstdetector', whilethatfollowing thei.f.amplifier iscalled the'seconddetector' . Theoperation offrequency changing (or'frequency conversion') canbereadilyunderstood asfollows. Suppose analternating voltage 111cosW1tissupplied bya'localoscillator' ,andtothisisaddedasmall signalvoltagevcoswt,wherev~V1andW1isclosetow.Thenthetotal amplitude ofthealternating voltagewillfluctuate between (v1+v)when thetwocomponents areinphase,and(v1-v)whentheyareoutof phase.Thetimeintervalbetweeninstantsatwhichthetwoareinphase is27T/(W1-W); theamplitude therefore fluctuates sinusoidally atafre­ quencyequaltothedifference ofthetwooriginalcomponents, andthe sizeofthefluctuation isthesameasthatofthesignalvoltagev.This constitutes anamplitude modulated voltagewhichcanbedetected as described inthelastsection,thedifference beingthatthe'modulation' frequency isdetermined bythedifference between thesignalandlocal oscillator frequencies. Theamplitude ofthelocaloscillator voltagev1 maybeadjusted sothatthedetector isworkedonthelinearportionof itscharacteristic, andthe'modulation' ofthelocaloscillator voltage produced bythesignalappearsintheoutputasacomponent atthei.f. frequency whoseamplitude isproportional tothatoftheoriginalsignal. Anyslowfluctuation ofthelatter,suchasthatduetoanaudio-frequency amplitude modulation, ispreserved, andthesignalatthei.f.amplifier differsfromtheoriginalonlyinthefrequency ofthecarriervoltage. Forthepurposeofmathematical analysis, theactionofthelocaloscil­ latorvoltageonthedetector maybeassumed toproduce aperiodic variation ofitsslopeconductance dI/dV(ortransconductance inthecase ofanodebenddetection). Thisfluctuating conductance maybeanalysed asaFourierseriesoftheform (J=(Jo+!hcosw1t+(J2 cos2w1t+.... (13.21) Theeffectofaddingasmallvoltagevcoswtistochangethedetector currentbyanamount (Jvcoswt =(Jovcoswt+(J1 vcosw1tcoswt+ ... =(Jovcoswt+tfh v{cos(W+Wl)t+cos(w-w 1)t}+...,(13.22) showingthatthereareFouriercomponents atboththesumanddiffer­ enceofthesignalandlocaloscillator frequencies. Thei.f.amplifier may 382 APPLICATIONS OF [13.9 betunedtoaccepteitherofthese;components at(nw1±w) alsoexist, buttheyareusuallysmallbecausethecoefficients gndecreaseinmagni~ tudeasnincreases. Thisanalysis showsthateitherthesumorthe difference frequency maybeused,although intheprevious discussion onlythedifference termwasconsidered. Inworkatveryhighfrequen­ ciesthedifference isgenerally used,sincethisismoreconvenient in buildinganLf.amplifier. Inaddition, whereselectivity isrequired, itis easiertogetanarrowpassbandfromcircuitsatthelowerfrequency. Forexample, aQof100wouldgiveapassbandofabout10kc/sinthe ~Circuittuned . tosignalfrequency Signalfrequency ._~,........ r----1I--~.., Loosecoupling FIG.13.20.Diodefrequency changer circuit.Fromloeal]oscillatorCircuittuned tointermediate frequency (i.f)~ :Toi.f.amplifier circuitsofani.f.amplifierat1Mc/s,whereastoobtainthesamelimited bandatar.f.ofsay100Mc/s,wouldrequireaQof104•Inasuper­ heterodyne receiverusingsuchfrequencies, thelocaloscillator mightbe at99or101Mc/s,andther.f.circuitswouldhavetobesufficiently sharplytunedtorejectanysignalat98or102Mc/srespectively, which wouldproduce thesamebeatfrequency. Thisrequirement isknown as'secondchannelsuppression'. Ingeneralitmeansthatthetunedr.f. circuitsforthesignalfrequency w/2Tfmustbesufficiently selective to rejectanyunwanted signalatthe'imagefrequency' (2w1-w)/2Tf, which isseparated fromthelocaloscillator frequency wl/2Tfbythesame amount, andhencewouldalsobeaccepted bytheLf.amplifier. Atfrequencies oftheorderofafewmegacycles persecondorless, anodebenddetection isgenerally usedinthemixingstage,andspecial tubessuchasthehexodeandpentagrid (orheptode) areemployed. The formerisascreen-grid tubewithtwocontrolgrids,oneforthesignal voltageandtheotherforthelocaloscillator voltage, separated byan 13.9J THERMIONIC VACUUM TUBES 383 extrascreengridwhichprevents eitherofthetwovoltages beingfed backintothecircuitsoftheothersectionthrough theinter-electrode capacitance. Aseparate localoscillator tubeisrequired, thoughthismay beenclosed inthesameenvelope, asinthetriode-hexode. Inthepenta­ gridtube,thefirsttwogridsformthecontrolgridandanodeofatriode whichisusedaslocaloscillator. Theelectronstreamemerging fromthe secondgridisthusmodulated atthelocaloscillator frequency beforetra­ versingthesecondpartofthetube,againeffectively ascreen-grid tube. Athighfrequencies diodefrequency changers areused,theessential circuitbeingshowninFig.13.20.Themaindifference fromthesimple detection circuitofFig.13.19istheaddition ofaloosecoupling tothe localoscillator, andtheuseofatunedtransformer coupling tothei.f. amplifier, insteadofanROcircuitcoupledtoana.f.amplifier. 13.10.Frequency modulation Thetransmission ofintelligence byaradiowaverequires someform ofmodulation, andamplitude modulation, whereacarrierwaveofa fixedhighfrequency ismodulated inamplitude atalowfrequency, has beenoutlinedin§ 13.8.Analternative systemis'frequency modulation' , inwhichthesignalwavehasaconstant amplitude, butitsfrequency isvariedperiodically inaccordance withthemodulating signal.The amountoffrequency variation isproportional totheamplitude ofthe modulating voltage, andtherateofvariation isproportional tothe modulating frequency. Theunmodulated carrierwave,forwhichwis constant, maybewrittenas V=Acosep(t)=Acoswt, wherethefunction ep(t)=Iwdt.Ifthefrequency ofthiscarrierwave ismodulated byasingleaudio-frequency (pI27T)ofconstant amplitude, thentheinstantaneous angularfrequency becomes Wi=w+dwcospt, wheredwisthemaximum deviation ofWifromw,thefrequency ofthe unmodulated carrier.Thenforthefrequency modulated wave V=Acos(JWidt)=ACOs(wt+ ~wsinPt)=Acos(wt+m1sinpt). (13.23) Thequantity m,=(dwlp)iscalledthemodulation index.Forexample, iftheunmodulated carrierwavehasafrequency (wI27T)=108cis,and themodulation isatafrequency (pI27T)=500cis,andthemodulation 384 APPLICATIONS OF [13.10 indexismf=0,04,thenthemodulated carrierwavewillvaryinfre­ quencyfrom(108+20)cisto(108-20) cisandbackagain500timesa second.Ontheotherhand,ifthemodulation indexis20,thefrequency ofthecarriervariesfrom(108+104)cisto(108-104)cisandbackagain 500timesasecond. Sinceafrequency modulated waveisnotasimplesinewave,itcontains side-bands, whicharemorecomplicated thanthoseforanamplitude -A mf=0·5 mf=5-A I I I I I I FIG.13.21.Sidebandsinafrequency modulated wavewithmodulation index0·5 and5respectively. OA=amplitude ofunmodulated carrier. modulated wave.ByFourieranalysisitmaybeshownthatthevoltage waveformofequation (13.23)canbewrittenas V=A.fo(mf)coswt+A~(mf){cos(w-tp)t-cOS(w-p)t}+ -tAJ2(mf){cos(w-t2p)t-tcos(w-2p)t}-t ... co =A[.fo(mf)coswt+ !In(mf){cos(w-tnp)t+( -1)ncos(w-np)t}]. n=l (13.24) Herethenumerical coefficients In(mf)canbefoundfromtablesofBessel functions, forInisaBesselfunction ofordern.Although theside-band frequencies stretchtoinfinity,themoredistantside-bands havesmall intensity.Ifthemodulation indexmf=0,5,thefirstorderside-bands (w±p)haveamplitude 0,24,andthesecondorderside-bands (w±2p) haveamplitude 0·03relativetotheunmodulated carrier;higherorder side-bands arenegligible.Ifmf=5,theamplitudes oftheside-bands arelarger,as showninFig.13.21,theamplitude ofthecarrierismarkedly reduced, andmostoftheenergyisintheside-bands (thetotalenergyis 13.10] THERMIONIC VACUUM TUBES 385 independent ofmf).Thisrepresents aneconomy intransmitter power overamplitude modulation, wherethecarrierwaveisfixedinamplitude andcarrieshalftheenergyevenwith100percentdepthofmodulation. Asaroughrulethewidthofthefrequency bandoverwhichtheside­ bandshaveappreciable amplitude isapproximately Inatypicalsystemfortransmitting speechandmusic,themaximum frequency deviation (D.Wj21T) is±75kc/s,andthemaximum audio­ modulation frequency 15kc/s,sothebandwidth requiredis 2(75+15) =180kc/s. Thoughthebandwidth required isthusconsiderably greaterthanfor transmission ofanamplitude modulated wavewiththesamemaximum audiofrequency, afrequency modulation systemhasthegreatadvantage inthatitcutsoutallamplitude modulated disturbances causedby interference andnoise,andsogivesmuchimproved reception. The carrierfrequencies usedforfrequency modulation transmission arehigh (::::::100Mc/s),partlybecausethefractional frequency deviation (D.w/w) isthensmallandeasiertorealizeintransmission, andpartlybecause onlythedirectrayfromthetransmitter isthenreceived. Anyrayre­ ceivedindirectly (e.g.byreflection fromtheionosphere) wouldbemore seriously distorted byselective fading(unequal transmission ofdifferent frequencies) thaninanamplitude modulated system, becauseofthe greaterbandwidth required. Inthereception ofanf.m.transmission itisnecessary toconvertthe frequency modulation intoanamplitude modulation, andthisisaccom­ plishedbya'discriminator'. Severaltypesofdiscriminator areinuse, theessential ingredient beingacircuitwhoseimpedance depends on frequency. Asimpleexample isatunedcircuitadjusted sothatthe meansignalfrequency (W/21T)liesonthesideoftheresonance curve, atthepointofinfl.exion wherethechangeincurrent(seeforexample Fig.9.5)varieslinearlyforsmallchangesinfrequency. Twosuchcir­ cuits,onewithitsnaturalresonance frequency tunedabovethesignal frequency, andtheotherbelow,canbeusedwithapush-pull circuit tobalanceoutdistortion, aswellasunwanted amplitude modulation. Thelatterismainlysuppressed, however, bypassingthefrequency modulated signalfirstthrough a'limiter', suchasapentoderunatan abnormally lowanodevoltagesothatitcanbeswungfromcuteoffto saturation byachangeofafewvoltsinthegridpotential. Thereceived 851110 Cc 386 APPLICATIONS OF [13.10 signalisamplified tosuchalevelbeforebeingappliedtothelimiterthat thegridswingonthepentode iswellintothecut-offregioninonedirec­ tionandintothesaturation regionintheother.Thentheamplitude of thesignalvoltageinthetunedcircuitusedasanodeloadisdetermined entirelybythetubecharacteristics andispractically independent ofthe amplitude ofthesignalappliedtothegrid.Twosuchpentodes, one following theother,aregenerally usedtomaketheremoval ofany amplitude modulation morecomplete. Local oscillator Frequency changer (mixer) Loudspeaker ordisplay .systemLF.'Lm.,~. 'tI'fi - - 1m1er LP1ersignal a.m.signal -AUdi~--:~",r[I~S~~ond I~--Discriminatorvideo-amplifier detecto~~ Fla.13.22.Blockdiagram ofaradioreceiver. Thetwostagesontheextreme right arerequired onlyforthereception ofafrequency modulated transmission. 13.11.Radioreceivers Wearenowinapositiontooutlinebrieflythecomponent partsofa typicalreceiver, asexemplified intheblockdiagram inFig.13.22.The signalfromtheaerialisfedintoanr.f.amplifier whichmustbetunedto thesignal frequency~ Ifthelatterisvariable thenallthetunedcircuits intheamplifier mustbeadjusted eachtimeasignalofdifferent frequency isreceived. Thisiscumbersome andexpensive andthestagesofr.f. amplification aretherefore kepttoaminimum, orevenomitted. Inthe lattercasetheonlytuningrequired isthatofthelocaloscillator resonant circuittogether withthecircuitintowhichtheaerialsignalisfed.The latteristunednotonlytoachieveavoltagestep-upbutalsotosuppress thesecondchannelattheimagefrequency whichwouldotherwise be passedintothemixer.Thissuppression isofcourseimproved bythe useofanr.f.amplifier, andsoalsoisthesensitivity, sincetheamplifier canbedesigned togivelownoise(seeChapter 16). Thelocaloscillator isgenerally asimpletunedanodeoscillator witha 13.11] THERMIONIC VACUUM TUBES 387 triodetube,andapoweroutputofafewwattsissufficient todrivethe mixingstageinthelineardetection regionwithout usingtightcoupling fromthelocaloscillator. Tightcoupling makesitdifficulttotunethe signalcircuitsandlocaloscillator circuitsindependently, andmayalso resultinlossofsignalintothelocaloscillator circuits. Thepresence ofside-bands inamodulated signalmeansthatallcircuits inthereceivermusthavesufficient bandwidth topasstheside-bands if themodulation istobepreserved. Inther.f.stages,simpletunedcircuits willgenerally suffice,butinthei.f.amplifier someformofband-pass tuning,suchascanbeobtained bytheuse'ofcoupledresonant circuits (see§9.4),mayberequired.Itisthenconvenient toplaceonetuned circuitintheanodeleadoftheamplifier tube,andcoupleitbyamutual inductance orcapacitance (orboth)toanotherresonant circuitconnected tothegridofthenexttube,asinFig.13.7. Inareceiver foramplitude modulated signalsthepurposeofthei.f. amplifier istomagnify thesignaluntilitislargeenoughtoworka detector (theseconddetector) inthelinearregion.Inareceiver for frequency modulation thei.f.amplifier magnifies thesignalsbeforethey areappliedtothelimiteranddiscriminator detector.Itisreadilyseen thatitismoreconvenient toperform theseoperations ataconstant frequency thanatavariable one,sothatthesuperheterodyne system isaconsiderable advantage inareceiver designed tocoverarangeof frequencies. Inallreceivers thefinalamplification isbyanaperiodic amplifier designed topassallfrequencies uptoafewkilocycles per secondforsoundorafewmegacycles persecondforvision.Thusthe onlysubstantial difference between areceiver fora.m.andoneforf.m. isthatthelatterrequires twoextrastages,alimiterandadiscriminator. REFERENCES ROLLIN, B.V.,1964,AnIntroduction toElectronics (O.U.P.). WILLIAMS, E.,1952,Thermionic ValveCircuits (Pitman). 388 APPLICATIONS OF PROBLEMS A_:S:_ gm -Vg-gm+(I/p)+(I/Z) andthattheequivalent circuitconsists ofaconstant current generator gmvg, shunted byaconductance gm'working intoaloadconsisting ofimpedances p,Z inparallel. 13.2.InthecircuitofFig.13.7,theamplification maybedefinedastheratio (voltage acrosscapacitor C)/(input voltageatgridoffirsttube).Showthat A_ Mgm -C{w2M2/p+Z2(l+jwL1/p)}' whereZ2=seriesimpedance ofthetunedcircuitL2,C,rbyitself.Ifthecircuit istunedtoanangularfrequency Wowhichmakesthedenominator ofthisequation purelyresistive, showthatAcanbewrittenas A_ woMQ -gm{l+w~M2/(rp)+w~LVp2}' whereQ=(woCr)-I.Ingeneral (WOL1/P)2~ 1,typical value:,;beingWo=107 sec-I,L1=10-4henry,p=105ohms,andtheexpression forAthenreducesto thatgivenbyequation (13.7).13.1.A'cathode-follower' circuitisshowninFig.13.23.Showthattheamplifica­ tionis + H.T. FIG.13.23.The'cathode-follower' circuit. Af3=gmM/(Cr+L/p).13.3.Inthetunedanodeoscillator circuitofFig.13.11,showthatthefraction ofthevoltageoutputwhichisfedbacktotheinputis f3=-jwM/(r+jwL). Bymeansofequation (12.14)calculate theamplification Awhichthetriodewith itstunedcircuitwouldgiveattheoscillation frequency givenbyequation (13.10), andshowthat Hencethecondition Af3?1givesthesamecondition foroscillation asequation (13.9). 13.4.Anamplitude-modulated voltagesignalv=B(I+mcospt)coswt isapplied totwodifferent receivers: (1)adiodedetector withthecharacteristic givenby equation (13.19),working inthesquarelawregion,followed byanaudio-frequency amplifier withoverallamplification A;(2)asignalfrequency amplifier with overallamplification A,followed bythesamediodeworking inthelinearregion. THERMIONIC VACUUM TUBES 389 Showthattheoutputvoltages (assuming thatthediodeworksintoaresistance R ineachcase)fromthetwosystems areintheratiobB:a,andshowthatif B=10-5V,a=10-3AfV,b=10-5A/vatheratiois10-7•Thisillustrates the inefficiency ofsquarelawdetection. 13.5.Inapush-pull ClassBamplifier theanodecurrentwaveformineachtube consistsofahalf-period ofasinewave,thecurrentbeingzerointheotherhalf. period. Assuming thattheamplitude oftheanodevoltage swingcannotbe greaterthantheh.t.voltage, showthatthegreatest efficiency istrr. 13.6.The'flip-flop' circuitisahybridoftheKipprelayandthemultivibrator, inwhichthebatteryB1ofFig.13.15isretained butthebattery Baisreplaced bycapacitor andresistance (e.g.0aandRaofFig.13.16).Showthatthisarrange­ menthasastableposition withtube1conducting andtube2cutoff,butifa shortpositive pulseisappliedtogrid2(orashortnegative pulsetogrid1)the circuitexecutes onecycleofoscillation (similartothemultivibrator), returning toitsstableposition. 13.7.InthecircuitofFig.13.2thevoltage vaisnotexactlyinphasewith(-vo) showingthattheamplification A=valvoiscomplex. Writing A=-IAlexp(i8), showthatthephaseangle8isgivenbytheexpression (1+p/R)(wOR1)-1- pwOo tan8=p/R1+(l+p/R)(1+00/0)" Thisshowsthatthephasedelayvarieswithfrequency, andifitisappreciable it willcausedistortion. Thusasquarewavewillnotappearsquareafteramplifica­ tionbecausethephaseofthehigherfrequency components isalteredrelativeto thelowerfrequency components; theearis,however, insensitive todistortion of thiskind. 14 THERMIONIC VACUUM TUBES AT VERYHIGHFREQUENCIES ATfrequencies aboveabout50Mc/stheperformance ofthermionic vacuumtubesbeginstofalloffforanumberofreasons. 'Thesemaybe brieflyclassedasfollows: (a)Effectsofelectrode impedance, whichmakethevoltageappearing attheactualelectrode differfromthatappliedtotheleadoutside thetube. (b)Effectofthefinitetimetakenbytheelectrons intravelling from oneelectrode toanother, causingthecurrentflownottobeexactly inphasewiththeappliedvoltageatthevariouselectrodes. (c)Increased powerlossintheexternal circuits, duetoskineffectin conductors (a~dproximity effectincoils),dielectric lossinim­ perfectdielectrics suchastubebases,andradiation. Itisconvenient todiscusstheseeffectsseparately, andthenshowhow thedesignoftubeandcircuitismodified inordertoimprove their performance. 14.1.Effectsofelectrode impedance Atverylowfrequencies theeffectsofstrayinductance andcapacitances associated withthevariouselectrodes ofathermionic vacuumtubemay beneglected. Asthefrequency israised,theinterelectrode andother capacitances becomeimportant, asdiscussed in§12.8,whereitwas shownthattheeffectofthegrid-anode capacitance inatriodeisto reducetheinputimpedance. Thisdifficulty iseliminated inthepentode tube,whichistherefore generally usedforamplification atfrequencies between about100kc/sand100Mc/s.Atthehighfrequency endof thisrangetheinductance ofthecathode leadbecomes important, for thisinductance iscommon tothegridandanodecircuits,anditthere­ foreintroduces feed-back, asinthecathodefollower circuitofProblem 13.1.Inparticular theperformance isadversely affected becausethe flowofcurrentthrough thegrid-cathode capacitance andthecathode leadinductance resultsinalowinputresistance. Thisresistance is shunted acrossthetunedcircuitwhichisnormally usedfortheinput 14.11 VERYHIGHFREQUENCIES 391 athighfrequencies, andmayseriously reducethevoltagemagnification whichthiscircuitwouldotherwise give.Thesizeoftheinputresistance maybeestimated asfollows,usingthecircuitofFig.14.1. Letvbetheexternal voltageapplied,andvgtheactualvoltageexisting between gridandcathode. Thesedifferbecauseofthevoltagedeveloped acrosstheinductance Lthroughtheflowofanodecurrentthroughit. v R, G•• ~ ~ FIG.14.1.Effectofcathode leadinductance athighfrequencies. Rg=(gmwSW(IC)-l. Iftheanodeloadissmallcompared withtheanodeimpedance ofthe tube,asisusuallytrueinthepentodes usedinr.f.amplifiers, theanode currentisapproximately equaltogmVg,andwehave v=vg+gmvg{jwL). Nowthepresence ofthegrid-cathode capacitance willcausegridcurrent toflow,ofmagnitude ig=vg{jwOgc). Hencethegridadmittance Y willbe Y=ig/v=jwOgc/(I+gmiwL) ';:::jjwOgc(l-gmjwL) (sincegmwLissmall) =jwOgc+gmw2Wgc' (14.1) Fromthisequation itisseenthattheinputcapacitance ofthetube isshuntedbyaconductance whosevalueisproportional tothecathode leadinductance, thecathode-grid capacitance, andthesquareofthe frequency. Toestimate themagnitude oftheeffect,weshalltake gm=5mA/V, OgC=5fLpJ!, L=5X10-8henries, wherethevalueoftheinductance isoftherightorderforastraight wire5emlongand1mmindiameter. Thenatafrequency ftheinput conductance isapproximately 5X10-20/2mhos;at50Mc/s,thiscorre­ spondstoaresistance Rgof8000ohms,andat500Mc/s,ofonly80ohms. 392 THERMIONIC VACUUM TUBES AT [14.1 Thisresistance isshunted acrossanyparalleltunedcircuitwhichmay beattached totheinput,andwilltherefore loweritsQ,withresulting lossofmagnification ofthesignalvoltageintheinputcircuit. The figuresgivenaboveshowthatthiseffectwillbeseriousat50Mc/s, whilethepowerdrawnfromasignalsourceappliedbetween gridand cathodeat500Mc/swouldbeintolerable. -A--0L- -8 o--0 (b)-A A 0 0 00 1----"---"'---0 ~----8L-------O------0---------A(a) 1P=====p FIG.14.2.(a)Tubewithleadsbrought outofglass,through bakelite base(showndetached below)topinsP. (b)Tubewithpressed glassbasehasmuchshorterleads, asthepinsaresealedintotheglass. Aanode,frontportionremoved toshowinside.ogrid. Ppins. Ccathode. Ssealing-off point. Reduction oftheinputconductance ofatubeathighfrequency can beachieved byadesigninwhichboththegrid-cathode capacitance andthecathode leadinductance arekeptassmallaspossible. Since thegrid-cathode separation cannotbeincreased, owingtotransittime limitations, theelectrodes mustbemadewiththesmallest possiblearea, andtheleadstogridandcathodemustbekeptwellapart.Thecathode leadmustbekeptasshortaspossible, sinceitsinductance increases with itslength.Forthisreason,apressedglassbaseisusedasinFig.14.2, sinceaseparate baseentailsgreaterleadlength,butthelengthoflead 14.1]------- VERYHIGHFREQUENCIES 393 insidethetubeisfixedbythenecessity ofproviding adequate heatin­ sulation between thehotcathodeandthepointwherethecathodelead issealedintotheglassenvelope. Onemethodofreducing theinductance istobringoutseveralleadsfromthecathode; theinductance ofeach leadisinparallelwiththatoftheothers,andthenetinductance is therefore reducedbyafactorequaltothenumberofleads. Areduction intheinputconductance byafactorofabout10belowthevaluesgiven abovecanbeachieved bymodifications ofthissortinthedesign.An additional advantage ofusingatuberequiring noseparate baseisthat dielectric lossesinthematerial ofthebaseareavoided. Toavoidsuch lossesinthematerial ofthetubeholderitmustbeagooddielectric, and specialinsulating materials withlowpowerfactorhavebeendeveloped forthispurpose. 14.2.Effectoftransittimeoninputconductance Whileanelectron isleavingoneelectrode ofatubeandapproaching anotheritinducesachargeoneachoftheseelectrodes. Asitmoves,the induced chargeontheelectrode whichithasleftdiminishes, whilethat ontheelectrode whichitisapproaching increases. Thiscanbeseenquite simplybyconsidering twoplaneparallelelectrodes whicharemaintained atvoltages 0and~respectively bymeansofabattery. Suppose acharge -qisemittedfromtheplaneofzerovoltage.Itwillbeaccelerated to­ wardstheotherplane,andwhenithasmovedthrough apotential Vthe workdoneontheelectron willbeqV.Thisworkmustbesupplied by thebattery, whenceitfollowsthatacharge q(V/~)musthaveflowed through thebattery. Thedirection offlowissuchthattheplaneat potentialv,.willhaveacquired acharge +q(V/~), whilethechargeon theotherplane,whichwas+qatthemoment aftertheelectron was emitted, isreducedto+q(l- V/~).Thusthemovement ofthecharge isaccompanied bychanges intheinduced chargesonthetwoplanes, corresponding tothechangeinthenumber oflinesoffieldfromthe electron whichterminate oneitherplane(seeFig.14.3).Theduration ofthesechanges isequaltothetransittimeofthechargebetween the twoplanes,andacurrentpulseflowsforthislengthoftime. Similararguments holdifoneplaneisreplaced byagrid,andthe passageofachargethrough agridtherefore causesamomentary flowof chargetothegridwhichreverses insignasthechargepassesthrough. Itisnotnecessary forthechargetohitthegridtocreateaninduced charge,andthecurrentflowaccompanying thepassageofthechargeis showninFig.14.4.Theareaunderthecurveuptoanypointrepresents 394 THERMIONIC VACUUM TUBES AT [14.2 thechargeinducedatthatmoment. Thetotalareaiszeroifthegrid potential isconstant, sincethepositive andnegative sections annul oneanother provided thatallthechargeflowsthrough thegridand FIG.14.3.Induced chargesonelectrodes. ~--1"l----.I<lf-­ I I I I I I I I 1 --~ Time FIG.14.4.Current flowtogridduringtransitofanelectron. tisinstant atwhichelectron passesthrough grid.Transittimeis1"1+1"2' noneisintercepted. Thecancellation isonlycomplete ifthegridpoten­ tialisconstant overatimegreaterthanorequaltothetransittime. Thisisabout10-9secforelectrons inanormaltube,andatfrequencies upto10Mcjsthecancellation isvirtually complete. Athigherfrequen­ cies,wherethetransittimeisanappreciable fraction ofanr.f.cycle, theeffectofthepassageoftheelectrons ininducing allr.f.currentto 14.2] VERY HIGHFREQUENCIES 395 flowtothegridisappreciable. Fullanalysisoftheeffectiscomplicated, butanestimate ofitsorderofmagnitude canbeobtained bythefollow­ ingmethod. Atatimet,letthevqltageappliedtothegridbeV=Yosinwt.Let thetransittimefromcathodetogridbeTl'andthatfromgridtoanode beT2'Thenthecurrentinducedinthegridbytheelectrons approach­ ingitwillbeapproximately II=gmYosinW{t-Tl) sincethesizeofthecurrentisdetermined bythevalueofthegridvoltage atthetime(t-Tl)whentheelectrons leftthecathode(or,morestrictly, thespacechargeregion). Similarly thecurrentinducedinthegridby theelectrons leavingfortheanodemaybewritten 12=-gmYosinw{t-T1-T2), theminussignarisingfromthereversalofthecurrentfordeparting electrons. Thenetcurrentistherefore 11+12=2gmYocosw{t-(Tl+V2)}sintWT2 =2gml'osintWT2{COS wtcosW(T1+tr2}+sinwtsinW(Tl+tr2}} -2 . ~{dVCOSW(Tl+fr2}+V' (+~_}}-gmSmyWT2dt W smwTlY'2• Thiscontains bothacapacitative andaresistive component. The latterismoreimportant sinceitcausesaloadingoftheinputcircuit. IfbothWTlandWT2~1theinputconductance maybewritten G=gmw2(Tl T2+tr~). (14.2) AfullanalysisbyNorthshowsthatfortubesofcommon sizetheinput conductance Gisapproximately equaltogmw2T~/10. Ifwetakeourstandard valueof5rnA/Vforgm>andtX10-9secforT1, thevalueofGisfoundtobeabout5X1O-21j2mhosatafrequencyf. Itistherefore ofthesameorderastheinputconductance duetocathode leadinductance intheimproved vacuum tubesmentioned in§14.1. 14.3.Modified circuitsandtubesformetreanddecimetre wave­ lengths Athirdcauseofloweredefficiency ofoperation ofvacuum tubesat veryhighfrequencies isincreased powerlossintheexternal circuits. Atafrequency of100Mc/stheskindepthincopper(cf.§lOA)isonly ~0·007mm,andthecurrentflowistherefore confined toaverysmall partofthecross-section ofanyconductor, withconsequent increasein theeffective resistance. Inacloselywoundcoilthereisafurtherloss 396 THERMIONIC VACUUM TUBES AT' [14.3 ofpowerandincreaseofresistance duetoeddycurrents inducedbythe alternating currents inneighbouring partsofthecoil(principally inthe nearbyturns).Thisproximity effectcanbereducedtoaminimum, and soalsoistheself-capacitance, byusingstraight conductors ratherthan coils.Itwasshown.in §11.4thatashort-circuited lengthofatrans­ missionlinebehaves asareactance, andin§11.5thatlengthswhichare r.f.choke H.T.+ H.T.- FIG.14.5.Lecher-wire oscillator. TheRO-combination enclosed bybrokenlinesisan automatic biascircuitforClassCoperation. oddmultiples ofquarter-wavelengths behaveasparalleltunedcircuits ofhighimpedance. Atmetrewavelengths (frequencies""" 30-300Mc/s) shortlengthsofparallelwirelinesmaybeusedforthetunedcircuits, atypicalcircuitforatriodeoscillator beingshowninFig.14.5.Thisis theequivalent oftheHartleyoscillator discussed in§13.6.Theblocking capacitor 0'servesonlytoseparate thesteadyvoltages onanodeand grid,anditsimpedance shouldbelowsothatitiseffectively ashort circuitforther.f.currents. Thenthisformstheclosedendofthetrans­ missionline,andthetwowiresattheopenend,wherethevoltages aregreatest andofopposite phase,areconnected toanodeandgrid respectively. Thisgivesfeed-back ofthecorrectsignforoscillation, as intheHartleycircuit.Theactuallinelengthrequired willberatherless thanone-quarter ofawavelength, sincetheelectrode capacitances must betunedtoresonance byaninductive lengthofline.Sincetheanode andgridelectrodes havesomewhat different capacitances toearth,the currents flowingintheLecherwireswillnotbequiteequalandopposite. Thisincreases thelossofenergybyradiation, whichissmallifthe currents areexactlybalanced andthedistance apartofthewiresis madesmallcompared withaquarter-wavelength. Thisdifficulty maybeavoidedbyusingapairoftubesworking in push-pull. ThecircuitshowninFig.14.6isofthistype,beingatuned 14.3] VERY HIGHFREQUENCIES 397 grid-tuned anodeosciIlator withfeed-back throughthegrid-anode capa­ citance. Thelattergivesanegative inputresistance atthegrid(see §12.8),provided thattheanodecircuitistunedtoheinductive atthe H.T.+------, A H.T.-_.L----4-=::=_'G FIG.14.6.Push-pull Lecher-wire oscillator. Aanodeline. Ggridline. A FIG.14.7.CV273triodewithgrounded grid. A,Ganodeandgrid,oncopperdiskssealedthrough glassenvelope. Gcathode. Hheaterconnexion. H'(cathode andheater)connexion. Grid-cathode separation 0·07rom. Grid-anode separation 0·25rom. p.=30.gm=7rnA/V.Maximum frequency, 3700Mc/s. frequency ofoscillation. Toavoidmagnetic coupling betweentheanode andgridlines,theyareusuallybroughtoutatrightanglestooneanother. InboththecircuitsofFigs.14.5and14.6itmaybenecessary touser.f. chokesinthesupplyleadstopreventtheflowofunwanted r.f.currents. TheRO-combination shownprovides automatic gridbiasforClassC operation. 398 THERMIONIC VACUUM TUBES AT [14.3 Atdecimetre wavelengths (frequencies between 300and3000Mc/s) considerable modifications inthedesignofvacuum tubesarerequired. Toreducetransittimeeffects,triodesareusedwithsmallclearances be­ tweentheelectrodes. Leadinductance iscuttoaminimum byavoiding thinwireleadsandbringing largediameter metaldisksthroughtheglass envelope (alogicaldevelopment fromthepracticeofputtinginseveral leadsofthinwiretothecathodetoreduceinductance). Suchdisksgive ------·_---1 I IC,.== I I_______ .-l-f------- -f------~ _r.~-----I FIG.14.8.Thegrounded-grid triodeconnexion. goodelectrical connexion totheexternal circuits,whichareintheform ofcoaxiallinestoavoidlossofenergybyradiation; thediameter of theconductors isusuallyfromIto5cmtoreduceresistive losses.A common formoftubeconstruction usesacopperdisksealtocarrythe grid,asinFig.14.7,andanodeandcathodearealsoplanestructures. Thedisksealreducestheanode-cathode capacitance toaverysmall value,animportant pointsincethetubeisnormally usedinthe'grounded grid'connexion, whoseequivalent circuitisshowninFig.14.8.Ithas theimportant advantage thatfeed-back tothegridcircuitthroughthe grid-anode capacitance isavoided, sincethecurrentthrough thiscapa­ citancedoesnothavetoflowthroughthesourceofsignalvoltageapplied between gridandcathode, asisthecaseintheordinary 'grounded cathode' connexion (see§12.8).Thisgreatlyincreases thestability of thesystemwhenusedasanamplifier, andthisisfurtherincreased bythe presence ofnegative feed-back duetotheflowofanodecurrentthrough theinputcircuit.Inthelatterrespectthecircuitissimilartothe 'cathode follower' or'grounded anode'connexion (seeProblem 13.1). TheanalysisofthecircuitofFig.14.8,neglecting thecathode-anode capacitance 0ea'isasfollows. Theusualequation fortheanodecurrent takestheform .ia=gmvl+(-i aZ2+v1)/P, or 14.3] VERYHIGHFREQUENCIES 399 Thevoltagemagnification is A=V2/Vl=iaZ2/V1=(l+fL)Z2/(P+Z2)' whiletheinputimpedance is(14.3) (14.4) Theseformulae showthatthecircuitisequivalent toavoltagegenerator ofmagnitude (1+fL)V1,withinternalimpedance pworkingintoaloadZ2' I \ Al:ro \G/ 7" 0I FIG.14.9.Disk-seal triodewithquarter-wave coaxial-line circuits. Aanode;acathode; Ggrid. I,0inputandoutputcoaxiallineswithloopcoupling. Thegrid-anode capacitance mustbeincluded inZ2andwillbetuned outbytheinductance oftheattached coaxiallineatresonance. The cathode-anode capacitance, ontheotherhand,actsasabypassforr.f. currentandmustbekeptsmall.Sincetheanodecurrentflowsthrough thesignalsource,theinputimpedance isfinite.Thisisnotseriousas theeffectsdiscussed in§§14.1and14.2wouldlimittheinputimpedance inanycase.Inspection oftheequations givenaboveshowsthatthe voltagemagnification isjustequaltoZ2/Z1,aresultwhichcouldhave beenobtained directlysincetheanodecurrentflowsinseriesthrough bothofthem. Aschematic diagramofatriodewithitscoaxiallinecircuitsisshown inFig.14.9.Asanamplifier itisusefuldowntoabout20-omwave­ length,onesuchstagebeingusedbeforeasuperheterodyne mixingstage. Oscillation atausableefficiency (afewpercent)isobtained inlowpower tubes(suitable a810caloscillators forasuperheterodyne receiver) down to10-cmwavelength, butmuchhigherefficiencies areobtained atlonger Witvelengths. 400 THERMIONIC VACUUM TUBES AT t14·4 14.4.Theklystron Wehaveseenalreadythatthefinitetimewhichanelectron takesto passfromcathodetoanodecausesdifficulty intheoperation ofconven­ tionaltubesatmetreanddecimetre wavelengths. Atcentimetre wave­ lengthstheproblem ofreducing thecathode-grid clearance soastokeep thetransittimedowntoasmallfractionofacyclebecomes practically insuperable. Itistherefore necessary tolookforsomeothermeansof reducing thetransittime.Oneobvioussolution istoshoottheelectrons atalargevelocitythroughagridwhichthenactsasaneffective cathode. Forexample, ifelectrons areaccelerated byapotential of2500V,their velocityisabout3X109cm/sec,andtheywilltraverse adistanceofImm in3X10-11sec,whichisonlyone-tenth ofaperiodatawavelength of 10cm.Thus,ifthecathode-grid systemisreplaced bytwogrids,be­ tweenwhichthehigh-frequency voltageisimposed, andtheelectrons areshotthrough thesegridsatahighvelocity, thetransittimecanbe keptshort. Suchanarrangement mustdependonsomedifferent principle forits working fromthatofaconventional tube.Inthelattercasethegrid voltageinfluences thespacechargeinthepotential minimum justin frontofthecathode, andthuscausesachangeinthenumberofelectrons flowingtotheanode.Whenanalternating voltageisappliedbetween gridandcathode aperiodic 'density modulation' issetupintheelec­ tronstreamflowingtotheanode.If,now,thecathode, whichemits electrons withanaverage energy corresponding toaboutone-tenth eV, isreplaced byagridthroughwhichelectrons areinjectedathighvoltage, therewillbepractically nospacechargebetween thispseudo-cathode andsecondgrid.Itisobviousthatapplication ofasmallr.f.voltage between thetwogridswillnotthencauseanychangeinthedensity ofelectrons leavingthisspace.Itwillcausea'velocity modulation', forsomeoftheelectrons willbeaccelerated byther.f.field,whileothers whichgothrough thisfieldhalfaperiodlater,whenitisreversed in sign,willberetarded. Theprinciple of'velocity modulation' ratherthan'spacecharge modulation' isfundamental intheworking oftheklystron andother centimetre wavetubes.Velocity modulation isnotofitselfsufficient to produce amplification oroscillation, sinceforthiswerequireadensity modulation oftheelectron beam.However, ifavelocity modulated beamis allowed to'drift'alonginafield-free space,adensitymodula­ tionwillbesetupinthefollowing way.Theelectrons whichwereac­ celerated byther.f.fieldwillgradually overtake theslowerelectrons in 14.4] VERYHIGHFREQUENCIES 401 _1800Distance frombuncher L-------------l>1i3600 8 ~ II2700 ""." 01«: <:>1800 'r::: ~., 01 .....; 's., ~ ~frontofthem,whichwereretarded bythefield.Inthisway'bunching' oftheelectrons willoccur,asillustrated inFig.14.10.Herethedistance coveredbyanumberofelectrons, initiallyuniformly spacedinthebeam, isplottedagainsttime.Linescorre· sponding tofastelectrons overtak· ingslowelectrons converge, while atpointsappropriate tohalfa periodearlierorlaterinther.f. field,thelinesdiverge. Theformer givesa'bunch' sincetheconver- genceofthelinesmeansthatmore electrons occupyagivenvolume, ., whilethelattergivesa'rarefac- ~ali tion'.Ifnowthebeamtraversesi;...01 asecondpairofgrids,between '"whichanr.f.fieldofthesame8..,frequency isapplied insucha <:> phasethatabunchi,retMded by:"11 thefield,whileararefaction is.: accelerated, energywillbetrans- ~ ferredfromthebeamtothefield]<:> becausemoreelectrons areslowed :<=, downthanarespeeded up.This ~~ constitutes aconversion ofenergy ~ fromtheh.t.supplyusedforthe initialacceleration ofthebeaminto energyinthealternating electro­ magnetic field,inasimilarmanner tothatinaconventional radioFIG.14.10.Bunching ofelectrons after velocity modulation. tube.There,inanordinary ampli- fieroroscillator, thedensercurrentHowtotheanodecoincides withthe moments atwhichtheanodepotential islow,sothattheseelectrons are slowedupbythealternating component oftheelectricfieldinfrontof theanode,thusdoingworkagainstthisfield. Aschematic diagram ofaklystron isshowninFig.14.11.Electrons accelerated andformedintoabeambyasuitable gunpassthrough aresonator Bwhereavelocity modulation isimposed onthembythe r.f.field.Theythentravelthroughthefield-free 'drift-space' inwhich bunching occurs,andenterasecondresonator 0calledthe'catcher', tunedtothesamefrequency asB.Finally,theelectrons arecollected 851110 Dd 402 THERMIONIC VACUUM TUBES AT [14.4 onanelectrode ath.t.potential, whichplaysnoessential roleinthe actionofthetube,butprevents thebeamfromstrikingtheglassen­ velope.Ifsomeofther.f.signalinthecatcherisfedbackthrough a coaxiallinetothebuncher, oscillations willbesetupifthephaseiscor­ rectlyadjusted, andifmoreenergyisextracted fromthebeambythe catcherthanisdissipated inthecombined resistances ofthecatcherand B c FIG.14.11.Schematic diagram ofklystron oscillator. Gelectrongun. Bbuncher. Ccatcher. Aanodetocollectelectrons. I,0inputandoutputcoupling loops. A,B,Cath.t.positive, Gath.t.negative potential. buncher. Inthisconnexion itshouldbenotedthatthevelocitymodula­ tionofthebeambyther.f.fieldinthebuncher requires nonetpower ifthetransittimeisshort,forasmanyelectrons aresloweddownas arespeededup.Forhighefficiency, itisnecessary todevelopthegreatest r.f.electricfieldintheresonator atthepointwherethebeamtraverses it,withthesmallest dissipation ofpowerintheresistive walls.The cavityresonator givesthebesttypeofcircuitinthisrespect,andits shapeisdetermined primarily bytherequirement ofashorttransittime fortheelectrons.Ifthelattertravelwithone-tenth ofthevelocityof light,andtheirtransittimeistobenotmorethanone-tenth ofthe periodofoscillation, thegapwhich th~,traverse mustbeaboutone­ hundredth ofawavelength. Thisrequires' anindented cavityofthe shapeshowninFig.14.11.Itmayberegarded eitherasashortsection ofcoaxialline,slightlylessthanaquarter-wavelength longsothatits inductance resonates withthecapacitance acrossthegap,orasanin­ dentedwaveguide resonant cavity. Aswithmostoscillators, thefullmathematical theoryisrather 14.4] VERYHIGHFREQUENCIES 403 complex, butitispossibletoderivethestartingcondition foroscillations fromelementary considerations asfollows. Weassumethattheelec­ tronsleavethegunwithpotential Vo,andthepaththeytraverseinthe resonant cavityBhasapotential difference VcoswtoacroB.8itatthe instantto'ThenifV~Vo,aswillbethecaseforsmallamplitudes of oscillation, wemayusethedifferential relation 3v/v=i(3V/V)tofind thefractional changeintheirvelocityaftertraversing thecavity;their finalvelocitymaythenbewrittenas v=vo(l+Vcoswto/2Vo), (14.5) where Vo=(2e'Vo/m)1 isthevelocity withwhichanelectron leavesthe gun.AnelectronwhichleavesAattimetoreachesthesecondresonator B,atadistancexaway,atatime t=to+x/v=to+(x/vo)(l+Vcoswto/2Vo)-1 ~to+(x/vo)(l-Vcoswto/2Vo), (14.6) wherewehaveagainusedtheapproximation V/'Vo~1. Nowthecurrentatanypointisequaltodq/dt,therateatwhichcharge passesthatpoint.Tofindthecurrent,weshallconsider asmallsection ofthebeamcontaining chargedq,andfollowitalongthebeam.Owing tothevelocity modulation, thefrontandrearportions ofthissection travelatdifferent speeds,andthetimedtwhichthesectiontakesto passagivenpointtherefore changeswiththedistanceithastravelled. Thebeamcurrentatthetimetistherefore 1=dq/dt=(dq/dto)(dto/dt) =10(1+wxVsinwto/2voYo)-l ~10{1-(wxV/2v oYo)sinw(t-x/v o)},(14.7) wherewehaveusedtherelations (dq/dto)=10,theinitialbeamcurrent, and (dto/dt)=(dt/dto)-l =(1+wxVsinwto/2voVo)-l obtained bydifferentiation ofequation (14.6). Equation (14.7)showsthatwehavenowadensitymodulated current, whoseamplitude ofmodulation increases linearlywithx,thedistance travelled, solongaswerestrictourselves tosmallvelocitymodulation. Ifthiscurrentnowpassesthrough asecondresonator, withapotentia] difference J;coswt,themeanpowerextracted fromthebeamwillbe 27T/W -1~coswt=-;:rf10J;coswt{1-(wxV/2vo'Vo)sinw(t-:J}dt o 27T/W 10VJ;wxwft' (tX)dt10V~wx•wX =-coswSlnw--=- Sln-,2voYo21T Va 4voYo Voo 404~----_ ....._----------- THERMIONIC VACUUM TUBES AT [14.4 sinceonlytermsincos2wtocontribute tothemeanpower.Thepower extracted fromthebeamwillbegreatest when-sinwx/v o=+1,i.e. wx/vo=21T(n+!), wherenisaninteger.Inotherwords,ifoscillations inBandaareinphase,thebeammusttake(n+!)periodstotravel fromBtoO.Thetimeoftravelcanbeadjusted byalteringtheinitial accelerating voltageYo' Sofarwehaveconsidered thepowerextracted fromthebeamwhen r.f.signalsfromanexternal sourcearefedintoresonators BandO.It isclear,however, thatifalittleofthepowerthatisfedfromthebeam intoaisreturned toBtoactthereasthesourceofsignal,oscillations canbesustained solongasthepowerextracted fromthebeamisgreater thanthatdissipated inresistive heatingofresonators BandO.Wemay represent thisdissipation byaresistance Rforeachresonator. The oscillations willbesustainedif 10~wX>V2+V~. 4voYo2R 2R IfV=a:~,where a:issmall,thepowerdissipated inthebuncher may beneglected, andthisrelationmaybeexpressed intheform minimum startingcurrent10=2Vr,(vo/wx)/(exR) =Yo/{exR1T(n+i}}. (14.8) Thisshowsthatthesmallest beamcurrentrequired forsustained oscilla­ tionsisoftheorderofthebeamvoltagedividedbytheparallelimpedance oftheresonator. Thecurrentrequired isdiminished ifexisincreased, or ifthetimeofdriftbetween theresonators (x/vo)isincreased, sincethe densityofthebunches reaching thesecondresonator isenhanced in eithercase. Withlargeelectrode voltagesandcurrents, theklystron isanefficient andpowerful oscillator, andcanbeusedasatransmitter, butitsprincipal useisasalowpowerlocaloscillator, forwhichanoutputofafewmilli­ wattsissufficient, inasuperheterodyne receiver. Forthispurposethe klystron mustbetunable, andthisisnotfeasiblewhentwoseparate resonators ofhighQ(severalthousand) mustbeadjusted simultaneously. Asingleresonator orreflexklystron istherefore employed, asinFig. 14.12,wheretheelectronstreamafterpassagethroughtheresonator is confronted byanelectrode whosepotential isnegative withrespectto thecathode potential. Theelectrons aretherebyhaltedandreflected backthrough thesameresonator. Bunching occursbecausetheelec-_ tronswhicharespeededupinthefirstpassagethrough theresonator travelfurthertowardsthereflector electrode andsoreturnlater(asin FIG.14.12.Reflexklystron. Elec. tronsfromthegunGpassthrough thegapintheresonator R,andare returned backthroughthegapby thereflector Xwhichisatapoten­ tialnegative withrespecttothe cathode.14.4] VERYHIGHFREQUENCIES 405 thecaseofaballthrownintotheair),together withtheelectrons which passedthroughtheresonator halfaperiodlaterandwereretarded by ther.f.field.Thusthebunches occuratpointsinthereturning beam whichthefasterelectrons reachlaterratherthanearlier,butasthey areretarded ontheirreturnpassagethroughtheresonator whenther.f. fieldisdirected awayfromthecathodeinsteadoftowardsit(asinthe two-resonator klystron), therequired transittimeforoscillation isstill(n+i) periods.Theminimum startingcurrentis nowgivenbyequation (14.8)withex=1. InanearlytypeoftubeforlO-cmwave­ length, "Va=1200V,R=70000ohms, R I.....-~"(n+i)=Ii,givingastartingcurrentof about3mAoTherunningcurrentisabout 8rnA,andthepoweroutputofanaverage tubeis~300mWcorresponding toan efficiency of3percent.Latertypesrun atabeamvoltageof300V,andacurrent of20rnA,witharatherlowerpowerout­ put. Reflexklystrons ofthistypehavebeen madetooscillateatwavelengths down toabouticm,whichseemstobeabout thelimit.Themaindifficulties inmaking suchoscillators forshorterwavelengths arisefromthesmallersizeof theresonant cavity,withitscorrespondingly lowerparallelimpedance R,whichmeansthatahigherbeamcurrentisrequired tostartoscilla­ tions.Thishighercurrentneedstobesentthrough asmallerholein thecavity,butthecurrentdensitywhichcanbeobtained inthebeam islimitedbythemutualrepulsion oftheelectrons. 14.5.Themagnetron Oscillations ofhighpoweratcentimetre wavelengths areproduced bythemagnetron; anoutlinediagram ofatypicaltubeisshownin Fig.14.13.Electrons areemittedfromacentralcylindrical cathode, andareaccelerated towards acoaxialcylindrical anodeconsisting of asolidcopperblockwithanumberofresonant cavities. Thesemay havetheshapeshowninthefigure,butothershapesarealsopossible. Essentially theyformasetofquarter-wave resonant linesorcavities, theopenendofthelinebeingattheinnersurfaceoftheanodeblock. 406 THERMIONIC VACUUM TUBES AT [14.5 Thus,whenoscillations takeplace,astrongr.f.electricfieldissetup attheinnersurface,thefieldlinesrunning mainlyinthecircumferen­ tialdirection acrosstheopenendofthecavity,asshowninFig.14.14. Themagnetron operates withastrongaxialmagnetic fieldofsome fewkilogauss (afewtenthsofaweberfmetre2),whichisnormally pro- O-------III~II o----H-~t;::::::::===::=l 1----1-1$=$;1-- - F ---jTr+-tI-Hnl+ nmllIJUU T ~I_---- '\II'\11----L ----R 1----0 ---T (a) (b) FIG.14.13.Atypicalmagnetron: (a)fromside,(b)alongaxis(parallel toexternal magnetic field)(afterWillshaw etal.,1946,J.I.E.E. 93,Part3a,985). 0,outputsidearm;0,oxide·coated cathode; I,insulated heater; P,cooling fins; L,outputcoupling loop;R,resonator system; T,tungsten heaterandcathode leads (cathode connected toonesideofheater). videdbyapermanent magnet,andapotential of10to50kVonthe anode.Anelectrononitswayfromthecathodetotheanodeexperiences amagnetic forceperpendicular toitsdirection ofmotionandaradial electricforce.Itstrajectory undertheactionoftheseforcescanbest bepictured byreference toacasewithsimpleplanegeometry. Sup­ poseauniform electricfield-E(i.e.inthesensewhichaccelerates a negative electroninthepositivey-direction) existsbetween twoparallel 14.5] VERYHIGHFREQUENCIES 407 z=o. "_+eE+e B· y-- - x,m mconductingplanesy= Oandy=a,withauniformmagnetic fieldBinthe z-direction. Thentheequations ofmotionforanelectronofcharge-eare ..eB.x=--y,m Iftheelectron startsfromrestattheorigin,itmovesinacycloid whoseequations are x=vt-psinwt =p(wt-sinwt), Y=p(l-coswt), wherev=EjB,p=mEjeB2, andw=eBjm.Thecycloidisthesame asthepathfollowed byapointonthecircumference ofacylinderof radiusp,rollingalongtheplaney= 0withangularvelocityw;visthen thelinearvelocityofitscentreinthex-direction. Inthecaseofcylindrical geometry, theelectronorbitisapproximately anepicycloid generated byrollingacylinderonthecylindrical cathode, andsoisrathersimilartothecaseoftheparallelplanesifweimagine thelattertobegivenasmallcurvature. Theapproximation arises becauseweareneglecting theradialdecrease intheelectricfieldwhich occurswithcylindrical geometry aswegofromcathodetoanode.In addition weareneglecting themutualrepulsions ofthevariouselectrons ('spacecharge')ineithercase.Ifthedifference betweentheradiusaof thecathodeandtheradiusboftheanodeissmallcompared witheither, atapointmidwaybetweencathodeandanodewemaywritetheangular velocityoftheelectroncloudasapproximately v2E(V){2}2V!(a+b)=B(a+b)=b-aB(a+b)=B(b2-a2)' (14.9) whereVisthesteadyvoltageappliedbetweenanodeandcathode. This expression showsthattogivetheelectroncloudacertainangularvelocity ofrotation, wemustmaintain acertainlinearrelationbetweentheanode voltageandthemagnetic field.Theimportance ofthisangularrotation oftheelectroncloudarisesfromthenecessity ofsynchronizing themove­ mentsoftheelectrons withthealternation ofther.f.electricfieldsin theresonators, inordertopreserve therightphaserelationships. This isessential fortheefficienttransferofenergyfromtheelectroncloudto ther.f.field,thebasisofanyoscillator. Beforeconsidering themechanism ofthistransfer, itisnecessary to discusstheresonator system.Eachresonant cavitybehaveslikeatuned circuitbutthesystemismorecomplicated thanthatofasimpleoscilla­ torbecauseofthepresence ofNsuchcircuits,allcoupledtogether. This coupling ispartlyelectrostatic, linesofelectricfieldoriginating fromone 408 THERMIONIC VACUUM TUBES AT [14.5 cavityterminating inanother, butitispredominantly magnetic; the linesofr.f.magnetic fieldgoingdownthroughonecavityarecompleted byreturning upthrough another cavity.WithasystemofNcircuits alltunedtothesamefrequency, andstrongly coupled together, the .~-_I-I-L FIG.14.14.Electron trajectories (E)andlinesofr.f.electric field(L)(arrows show direction offorceonelectrons) inthe'17'mode'magnetron. Sspacechargecloud,enclosed bybrokenlines. naturalfrequencies ofoscillation aresplitapartinthesamewayaswith twocircuits(see§9.4),butanalysisofthesystemismuchmorecomplex. Thedifferent resonant frequencies correspond tooscillations wi:thvary­ ingchangeofphasebetween successive resonators; theycanbeanalysed intosystemsofstanding waves,oroftravelling waves,oramixtureofthe two.Thesimplest systemisastanding wavewherethephasedifference between successive cavitiesis7T(theso-called '7Tmode'),andthisisalso oneofthemostefficient modesofoperation ofthemagnetron. Atany 14.5] VERYHIGHFREQUENCIES 409 instantthedirection ofthelinesofforceinsuccessive cavitiesisexactly reversed, aswouldarisefromasimplepotential distribution where alternate segments arejustplusandminusinther.f.voltage(seeFig. 14.14). Theinteraction between theelectrons andther.f.fieldmustnowbe considered. Undernormalconditions ofoperation, butintheabsence ofoscillation, theelectrons wouldtravel(approximately) incirclessuch thattheirfarthestpointfromthecathodeisabouthalf-way acrossthe cathode-anode space.Atthemoment whentheyreturntothecathode, theirvelocity wouldbezero,sincethemagnetic fielddoesnoworkon them,andthatdonebytheelectrostatic fieldastheymoveinitially awayfromthecathodeisallregained onthereturnpath.Suppose now thatanelectron isjustmovingtangentially atthefarthestpointinits trajectory fromthecathode. Thentheforceexertedonitbythemag­ neticfieldistowardsthecathode, whilethatexertedbytheelectrostatic fieldistowardstheanode.Ifthetangential velocityoftheelectron is increased atthismoment throughbeingaccelerated bythefringingfield ofoneofthecavities,themagnetic forceonit(whichisproportional to itsvelocity) willbeincreased, whiletheelectrostatic forceisunchanged, sothattheeffectistoreturnittowardsthecathode. If,ontheother hand,theelectron isretarded byther.f.field,themagnetic forceis decreased andtheelectronwillmoveinapathwhichbringsitcloserto theanodethanitwouldhavegotintheabsenceofther.f.field.Ifnow itarrivesopposite anothercavityatthemoment whenitisagainre­ tarded,itagaingivesupenergytother.f.field,andmovesstillcloserto theanode.Notethatasitdoesso,itmovesintopositions wherether.f. fieldisstronger andsoagreaterproportion ofthekineticenergyofthe electron istransferred tother.f.field.Ontheotherhand,anelectron whichisspeededupreturnstowardsthecathodewhereitsinteraction withther.f.fieldissmaller. Thus,iftherightphaserelationship canbe maintained, someoftheelectrons willgiveupenergytoseveralcavities insuccession, andeventually reachtheanodewithkineticenergymuch lessthanthatcorresponding toeXV,whileotherswillbereturned to thecathode. Onthewhole,thelattertakemuchlessenergyfromthe r.f.fieldthantheformergivetoit,andthenettransferofenergywill maintain oscillation. Togettherightphaserelationship, theangularvelocityoftheelectron cloudmustcoincide withtheangularvelocityofrotationofoneofthe Fouriercomponents ofther.f.fieldsystem.Forthe'IT-mode,thissimply meansthattheelectron cloudmustrotatethrough theangle(2'ITfN), 410 THERMIONIC VACUUM TUBES AT [14.5 between successive cavities, inn+!cycles(wherenisaninteger). Its angularvelocitymusttherefore be(27T/N)/(n+!)T =27T!/N(n+!), where 1=l/Tisthefrequency oftheoscillations. Equating thisto(14.9)gives V=TTIb2B(1_a2), (14.10)k b2 wherek=N(n+!). Assuming theratioa:bisroughlyconstant, itwill beseenthattomaintain operation atagivenfrequencyIinagiven modek,atafixedfieldB,theanodevoltagemustbeincreased withthe squareoftheanodediameter.Ifitisdesiredtokeepthevoltagefixed andtoconstruct amagnetron ofhigherfrequency (shorterwavelength) butwithequivalent operating conditions, thentheresonator systemand anodediameter mustbescaledinproportion tothewavelength (boc1/f), andBmustbeincreased inproportion toI. Highpoweroutputfromthemagnetron canbeachieved onlyifhigh anodevoltages andhighanodecurrents areused.Byrunningthetube inshortpulsesroughlyofl!-'secduration, witharepetition rateofabout 1000/sec, thepowerinthepulsecanbemadeover1000timesasgreat ascanbeobtained undercontinuous operation. Thesehighpowersare mainlyduetothreefactors: (a)theelectronic conditions aresuchthathighefficiency isattained athighlevel; (b)oxide-coated cathodes cangiveveryhighcurrents perunitarea, 100timesgreaterunderpulsedconditions thanundercontinuous running; (c)themeanpowerdissipated ontheanodeisreduced, andiseasily removed byconduction throughthesolidcopperanode. Animportant factorunder(a)isfocusing actionbyther.f.field,which helpstoconcentrate thespacechargeintoanumberofnarrowspokes (seeFig.14.14).Eachspokethenpassesthrough theT.f.fieldatthe moment whenitisamaximum, givingtheequivalent ofClassCopera­ tioninordinary triodes. Themaintechnical difficulties havebeenthe construction ofruggedcathodesurfaces, whichcanwithstand theheavy bombardment bythereturning electrons accelerated byther.f.field, andavoiding 'modejumping', wherethefrequency changesasthetube jumpsfromonevalueofktoanother. Powerisextracted bymeansof aloopcoupling inoneoftheresonators, orthrough awaveguide slitin oneresonator. Typical operating conditions foramedium highpowermagnetron operating at10-cmwavelength are:magnetic field,B=0·28weber/ 14.5] VERYHIGHFREQUENCIES 411 metre2,anodevoltage31kV,anodecurrentduringpulse35A,output powerinpulse750kW.Inthistubethecathodediameter is6·0mm, andtheinsidediameter oftheanodeis16·1mm;thelengthoftheanode blockis2cm,andtheoveralllengthofthetubeis3·2cm.Thedimensions ofthetubearethuscomparatively small,andthehighpowerobtainable inthepulseisduetothehighefficiency (70percent),whichalsoreduces thedissipation ontheanodeblocktoonly30percentoftheinputpower. AtlO-cmwavelength, outputpulsepowersofafewmegawatts canbe achieved, butthepowerdecreases rapidlyasthewavelength isreduced, owingtoanumberoffactors. Experimental tubeshavebeenmadeto operateatwavelengths ofafewmillimetres, andtheshortwavelength limitisaboutthesameasoralittlelowerthanthatoftheklystron. Mostcavitymagnetrons arefixedfrequency tubes,butsomemagnetrons tunable overarangeof10-20percentinfrequency havebeen con~ structed, thevariation beingobtained byplungers movingintothe resonators fromoneend. 14.6.Crystaldiodes Atcentimetre wavelengths themostcommon typeofreceiver uses afrequency-changing system(§13.9),withareflexklystron asthelocal oscillator. Thethermionic vacuumtubediodeisunsatisfactory asade­ tectorormixer,because, tomakethetransittimesufficiently short, averysmallclearance between cathodeandanodeisrequired. Thisin­ creasestheinter-electrode capacitance, andsincetheoxidecoatingofthe cathodeactsasalossydielectric, thecapacitance iseffectively shunted byacomparatively lowresistance; thuswhenthediodeismadepart ofatunedcircuit,ther.f.voltageacrossitisratherlow.Forthisreason acrystaldiodeisusedinstead, consisting ofasmallpieceofsiliconon whichapointcontactismadebymeansofafinetungsten 'whisker'. Siliconisasemi-conductor, andelectrons canflowacrossthecontact withthetungsten verymuchmoreeasilyinonedirection (towards the silicon)thanintheother(seeChapter 19).Hencethecurrent-voltage characteristic isasymmetrical asshowninFig.14.15.Thecharacteristic israthersimilartothatofathermionic diode,withasomewhat higher slopeintheforward direction, butwithasmallcurrentflowinthere­ versedirection.Itisclearthatitwillactasadetector ormixerinthe samewayasanordinary diode.Thetransittimeoftheelectrons and thecapacitance acrossthepointcontactarebothverymuchsmallerthan inathermionic diode,andthesiliconcrystaldiodecanbeusedupto muchhigherfrequencies. Twotypicalmountings areshowninFig.14.16: 412 THERMIONIC VACUUM TUBES AT [14.6 acapsuletype,forwavelengths of10emandlonger(analternative coaxialconstruction ispreferred forwavelengths of1-10em),anda waveguide mounting formillimetre wavelengths. -2 -I 0 1 :l Voltage FIG.14.15.Current-voltage characteristic ofsilicon-tungsten crystaldiode. v.,t.'t¥tA--- C v..+-- O }--\,L+-.--- TV VA1~t4-~- s •,,-l-------B----B ~_--1~ __~W ~""T"~$i~~t ---s c' ~C '"'-........-f/k~~ ............---C__B (b) (a) FIG.14.16.Crystaldiodes(a)capsuletype,(b)waveguide mounting formillimetre wavelengths. Bbrass. Wtungsten whisker. Ssilicon Cinsulator. 14.7 Travellin~ wavetubes Animportant classofelectronic tubesforcentimetre wavelengths, whichweshallnotdiscussindetail,isthatofthe'travelling wave'tube. Thisisavelocity modulation deviceinwhichthebeaminteracts con­ tinuously withtheelectromagnetic wave,insteadofonlylocally,asin theklystron. Tomakethispossiblethewavevelocitymustbereduced tocoincidewiththebeamvelocity; thisisaccomplished bya'slow-wave structure', suchasawirehelixsurrounding thebeam,whichbehaves as 14.7] VERY HIGHFREQUENCIES 413 anartificial transmission linewithwavevelocity l/.J(LC) (see§11.3). Travelling wavetubescanbeusedasoscillators oramplifiers, anim­ portant application beingasamplifiers incommunication repeater stations, wherethelargebandwidth whichcanbeamplified makesthem ofconsiderable commercial importance. GENERAL REFERENCE ROLLIN, B.V.,1964,AnIntroduction toElectronics (O.D.P.). 15 ALTERNATING CURRENT MEASUREMENTS 15.1.Measurement ofvoltage, current, andpower IFanalternating voltageisappliedtotheterminals ofad.c.instrument suchasamoving-coil galvanometer, thereadingobserved isusuallyzero. Themovement ofthegalvanometer istoosluggishtofollowthealterna­ tionsoftheappliedvoltageiftheseoccuratmorethanafewcyclesper second. Theinstrument therefore recordsonlythemeanvalueofthe currentovermanycycles,whichiszerofor.asymmetrical waveform. Thusthemeasurement ofalternating currentsandvoltages requiresthe useofspecialinstruments whichmaybedividedintothreeclasses according totheprinciple involved intheirconstruction. Inthefirst classareinstruments withveryrapidresponses sothattheycanfollow thealternating waveform;second,'squarelaw'instruments, socalled becausetheyrespondtothesquareofthecurrentorvoltageapplied; andthird,rectifier instruments, wherethealternating voltageiscon­ vertedtoasteadyvoltagewhichcanbemeasured onad.c.instrument. Inpracticethemostwidelyusedinstruments arethosewiththegreatest frequency range,andFig.15.1showsthatthesearethethermoammeter, asquarelawinstrument, forcurrent; thevacuum tubevoltmeter, a rectifierinstrument, forvoltage;andthecathoderayoscillograph, ashort timeconstant instrument forthedisplayofwaveformandmeasurement ofvoltage. Ofmorerestricted usearemovingironinstruments andthe dynamometer. Thelatterisoneofthefewinstruments whichmeasures powerdirectly, butitsuseisconfined tosupplyfrequencies. Atradio frequencies powerisnormally determined fromthevoltagedeveloped acrossaknownresistance, orthecurrentflowthroughit.Thusingeneral currentandvoltagearetheprimary quantities measured. Thechief typesofinstrument aredescribed below. Thecathode-ray oscillograph Thecathoderayoscillograph isaninstrument whereby thewaveform ofanalternating voltagemaybedisplayed onascreen.Adiagramofthe instrument isshowninFig.15.2,thevariouspartsbeingcontained inan evacuated glassenvelope. Anelectron gunconsisting ofacathode C, agridG,andanodesAlandA2isusedtoformanarrowbeamofelectrons 15.1] ALTERNATING CURRENT MEASUREMENTS 415 Quantity to bemeasured Frequency(c{s)-->I02 10" 1()8 Current Voltage Power Impedance Frequency Wavelength.....~----_Thermoammeter'_-----+. Movingiron instruments .....I------Vacuum tubevoltmeter'----- ..... ....~-----C.R.O.-----__.~ +-Dynamometer-+ Unshielded bridges .....~----Shielded bridges • ~Q-meter---+ VSWRandresonance • 'd•inlinesandwavegm es, ...1---- Bridges • ....1------- Quartz-crystal andharmonicsl------ ...... ~Resollallt lines--' •Reso~~llt • cavItIes FrG.15.1.Frequency rangesofvarioustypesofmeasuring instruments . •f---yyxx S FIG.15.2.Thecathode-ray oscillograph (nottoscale). ocathode. YYy-deflecting plates. Ggrid. XXx.deflecting plates. Alfirstanode. Sscreen. A2secondanode. travelling paralleltotheaxisofthetube.XXandYYaretwopairsof platesorientedatrightanglestooneanother, andvariousvoltages may beappliedacrosstheplatesofeitherpairtodeflecttheelectron beam. Thesedeflexions inthex-andy-directions arenormaltotheaxisofthe tubeandproportional tothevoltages appliedtotheX-andY-plates. 416 ALTERNATING CURRENT MEASUREMENTS [15.1 ThebeamfinallystrikesascreenS,coatedontheinsidewithafluo­ rescentsubstance suchaszincsulphide sothattheposition ofarrival ofthebeamisshownbyasmallluminous spot.Thebrightness depends onthebeamcurrent,whichiscontrolled bythegridGandtheaccelerating voltageonanodeA2•Thisvoltagerangesfrom2000Vormoreonlarge tubes(6-in.diameter faceorgreater)to500Vonsmallertubes.The magnitude ofthedeflexion, andhencethesensitivity (defined asthe deflexion perunitvoltageappliedtotheX-orY-plates), isinversely proportional totheaccelerating voltage(seeProblem 15.1).Thesensi­ tivityisincreased byreducing theseparation between thetwomembers ofapairofdeflecting plates,andtheyaretherefore splayedasinFig.15.2 inorderthattheyshallnotintercept thebeamatlargedeflexions. Electrostatic deflexion, asthissystemiscalled,causesacertainamount ofdistortion, andmagnetic deflexion, usingfieldsgenerated bysmall coilsplacedoutsidethetube,ismorecommon fortelevision tubes,where verylargedeflexion anglesareemployed. Electrostatic deflexion isused formostlaboratory work,andthepatternobserved onthescreenisthen determined bythevoltages appliedtothetwosetsofdeflecting plates. ItisusualtoapplyaknownvoltagewaveformtotheX-plates (the 'time-base'), whiletheunknown voltageisappliedtotheY-plates. The mostusefultypeoftime-base isonewherethespotmovestotheright acrossthescreeninthex-direction atconstant velocity, followed bya rapid'fly-back' totheleft-hand side.Thisiscalledalineartime-base, andrequires asaw-tooth voltagewaveformasshowninFig.15.3.To obtainastationary picture,therepetition frequency ofthetime-base mustbeanexactsubmultiple ofthebasicfrequency ofthewaveform appliedtotheY-plates.Thusthetime-base frequency mustbeadjustable, andsynchronization isusuallyobtained byapplying alittleofthevoltage fromtheY-plates tothetime-base circuit,sothatthetime-base is 'lockedin'. Thebasicmethodofgenerating asaw-toothed waveformisalsoshown inFig.15.3.Acapacitor 0ischargedupthrough aresistance Rfrom anh.t.supply,andisthenperiodically discharged through another resistance rbyaswitchS.Ifr~R,thedischarge occupies averyshort periodcompared withthecharging, andsoprovides thefly-back, while theincreasing voltageacross0duringthecharging periodprovides the forward sweep. Thiswillnotbeexactly linearsincethecapacitor chargesexponentially, butiftheswitchSisarranged tooperatebefore thevoltageacross0hasrisentomorethanasmallfractionoftheh.t. voltage, thedeparture fromaconstant rateofcharging willbesmall. 15.1J ALTERNATING CURRENT MEASUREMENTS 417 Thelinearity isfurtherimproved bycharging thecapacitor notthrough aresistance, butthrough aconstant currentdevicesuchasapentode, wherethecurrentisalmostindependent oftheanodevoltageprovided thelatterdoesnotfalltoolow.Therateofcharging iscontrolled bythe (a) H.T.voltage orX Time-base voltage X R (b)8 FIG.15.3.(a)Saw-tooth voltagewaveform.ABgiveslinearforward sweep,BOgives rapidfly-back. (b)Basiccircuitforgenerating saw-tooth voltage. screenvoltageofthepentode, whichservesasafinefrequency control, coarsecontrolbeingprovided bychoiceofanumberofcapacitors 0of different values.Othertypesoftime-base includethesinglesweepfor observing transient phenomena (whichmustbetriggered bytheonset ofthetransient) andcircularorelliptical time-bases, obtained byapply­ ingsinusoidal voltages differing inphasebyi7TtotheX-andY-plates. Theseareusefulinthemeasurement offrequency (seebelow). Theoscillograph maybeusedtodetermine theamplitude ofan alternating voltagebymeasurement ofthedeflexion onthescreenfrom peaktopeak.Forthispurposeitmustbecalibrated usingaknownd.c. orlowfrequency a.c.voltage. Thesensitivity ofa6-in.diameter tube isusuallyoftheorderofafewtenthsofamillimetre deflexion pervolt. Therangemaybeextended bytheuseofanamplifier ofknowngain, andsignalsoftheorderofmicrovolts canbemadetogiveanobservable 851110 Ee 418 ALT.ERNATING CURRENT MEASUREMENTS [15.1 deflexion. Thistechnique mayalsobeusedforcurrentwaveform,by passingthecurrentthrough alowresistance andamplifying thevoltage developed acrossit. Otherapplications oftheC.R.O.arecomparison ofphaseandfre­ quency.Thephasedifference betweentwovoltagesofthesamefrequency maybefoundbyapplying onetotheX-plates andtheothertothe Y-plates.Iftheamplitudes areequal,andthephasedifference is90°, y x=v2sin(wt+",) FIG.15.4.Determination ofphaseanglefromthephaseellipse(seeProblem 15.2). theresultant ofthetwowavesisacircle,butforanyotherphasediffer­ ence,orunequal amplitudes, thepattern onthescreenisanellipse (asinFig.15.4),orastraight lineifthephasedifference iszeroor7T. Ifadouble-beam oscillograph isavailable, whereeachbeamhasalinear time-base ofthesamefrequency, thetwovoltages tobecompared may bedisplayed oneabovetheotheronthescreen,andthephasedifference ismeasured directly. Double-beam tubesarealsousefulforfrequency comparisons. The secondbeamisdeflected withastandard frequency, andifthetime­ baseissuchthatfiveorsixcomplete waveformsareshownonthescreen, asmalldifference between thefrequencies onthebeamsiseasilyseen. Onasingle-beam tube,thebestmethodistouseacirculartime-base, produced byapplying voltagesofequalamplitude, butdiffering inphase by90°,tothetwopairsofplates.Theunknown frequency isappliedto 15.1J ALTERNATING CURRENT MEASUREMENTS 419 theanode,modulating thesensitivity, andifitsfrequency isntimes thetime-base frequency, astationary picturewithnloopsisobtained (Fig.15.5a).Alternatively, theunknown frequency maybeappliedto thegridofthetube,thusmodulating theintensity. Thepatternonthe screenisbrokenup(asinFig.15.5b)intodotswhosenumbergivesthe frequency ratio.Gridmodulation ismoresensitive thananodemodula­ tion,anamplitude ofafewvoltsbeingsufficient fortheunknown frequency. (a) (b) FIG.15.5.Comparison offrequenoy withoiroular time-base. Time-base frequenoy =t(unknown frequenoy). (a)Anodemodulation. (b)Gridmodulation. Iftheratioofthefrequenoies isnotexaotlyaninteger,the patternisnotstationary butrotates. Thegreatadvantage ofthecathoderayoscillograph isitsabilityto portraythewaveformofanalternating voltageuptofrequencies of afewhundred megacycles persecond.Atthispointlimitations arise fromthedifficulty ofmakingsuitabletime-bases andamplifiers, aswell asfrominherent drawbacks inthetubeitself(seeProblem 15.3). Squarelawinatruments Anyd.c.instrument whosedeflexion depends onthesquareofthe currentorvoltagecanbeusedfora.c.measurements, andthereading obtained bycalibration withd.c.willgivetherootmeansquarevalue. Thustheelectrostatic voltmeter canbeusedforalternating voltages andcurrents; socanthedynamometer (§7.1),thoughitisconfined to audiofrequencies. Thethermoammeter hasagreaterfrequency range; thecurrentpassesthrough aresistive coilwhichheatsacopperdisk; thisissolderedtoathermojunction, thecurrentfromwhichisreadon amoving-coil galvanometer. Theinstrument canbecalibrated byd.c. orlowfrequency a.c.,andisusedforcurrentsoftheorderofmilliamps. 420 ALTERNATING CURRENT MEASUREMENTS [15.1 Thereadings areindependent offrequency uptoabout1Mc/s,butabove thatthereiscoupling between thecoilandthethermocouple dueto straycapacitance. Thesensitivity canbeincreased bymounting the thermojunction inanevacuated glassenvelope toimprove thethermal insulation; thecopperdiskissometimes joinedtothejunctionbyasmall glassbead,whichprovides thermalcontactbutinsulates ther.f.circuit fromthegalvanometer. Theheaterisverythin,toavoidanychange ofresistance withfrequency duetoskin-effect, andthedeflexion isvery nearlyproportional to(current)2 overalargerangeoffrequencies. The frequency rangemaybegreatlyextended byusingaseparate thermo­ junction whichmaybeinserted inthecircuitquiteapartfromthe instrument usedtomeasure itsd.c.outputvoltage. Leadstothelatter instrument mustbecarefully decoupled. Thedynamometer wattmeter canbeusedformeasuring power,by connecting itasinFig.7.5.Thescalereadingisthenproportional to theaverage valueofl{;10sinwtsin(wt+a:) overacycle,wherel{;sinwt isthevoltageacrosstheload,and10sin(wt+a:) isthecurrentthroughit. Thescalereadinggives!(l{;10)cosa:andthisisthepowerconsumed by theload,sothattheinstrument canbecalibrated toreadpowerdirectly, andnodetermination ofthephaseangleorpowerfactorisrequired. Thedynamometer issuitable onlyforaudiofrequencies uptoabout 1000cis.Atradiofrequencies powerisnormally measured bydeter­ miningthevoltageacross,orthecurrentthrough, aknownresistance. Thismethodcanbeusedatallfrequencies wherethecalibration ofthe voltmeter orammeter isreliable,butatcentimetre wavelengths itis replaced byadirectmeasurement ofpower.Forlowpowers(1Wdown toamicrowatt orso)a'bolometer' maybeused,consisting ofathin wiresuchastungsten of0·01mmdiameter andafewcentimetres long, enclosed inanevacuated envelope. Thethinwireisweldedtostout leads,whicharecollinear withthewire.Thebolometer canthenbemade thecentreconductor ofacoaxialline,whichistunedtoresonance (half awavelength long)asinFig.15.6.Theinputpowerisfedinfroma coaxialline,whichistappedontothecentreconductor atsuchapoint thattheresonant sectionismatched totheline.Thedissipation ofr.f. powerinthethinwireofthebolometer causesitstemperature andhence itsresistance torise,thechangeinthelatterbeingdetermined by including thebolometer asonearmofaWheatstone's bridge.Ifthe bridgeisbalanced withther.f.poweron,andthedirectcurrentthrough thebridgearmsisincreased soastoreturntothesamebalancepoint whenther.f.isswitched off,thenther.f.powercanbecalculated from 15.1] ALTERNATING CURRENT MEASUREMENTS 421 thechangeind.c.powerdissipated inthebolometer lamp.Analterna­ tivebolometer element isthethermistor, consisting ofatinybeadof various semi-conducting oxideswhoseresistance fallssteeplywith increasing temperature andhencewithpowerinput.Suchelements have theadvantage ofsmallsize,andtheirresistance canbeadjusted in manufacture tobeoftheorderofahundred ohms,whichisconvenient formatching toacoaxiallinewhosecharacteristic impedance isofthis order. .4----------)./2"------------.~ .,,- -...... n............--"1 I IIMovable plunger, insulated Evacuated glassenvelope fromcentreconductor Inputcoaxialline FIG.15.6.Resistance variation bolometer foruseatshortwavelengths. Theposition oftheinputtappingmustbeadjusted forcorrecttermination oftheinputline.(Wave­ guideinputcanalsobeused.) Powersofmorethanafewwattsaremeasured bydissipating the powerinwater,whosehighabsorption coefficient atcentimetre wave­ lengthsisconvenient forthispurpose (see§17.7).Thetemperature rise ismeasured, usuallywithacontinuous flowmethod. Rectifier instruments Sincerectification istheprocessofturningana.c.voltageintoad.c. voltage,itisobviousthatthismaybeusedasthebasisofamethodof measuring ana.c.voltage. Whenathermionic vacuum tubeisusedas therectifying devicetheinstrument isknownasavacuumtubeorvalve voltmeter. Thedetector circuitofFig.12.3maybeusedforthispurpose, thed.c.voltagebeingmeasured directlybyavoltmeter placedacrossthe loadresistance R.Whenmoresensitivity isrequired theoutputvoltage maybeappliedtothegridofatriode,asinFig.13.19,butwithoutthe blocking capacitor 01'Therectified voltagethencausesachangein theanodecurrentofthetriodewhichmaybereadonameterinthe anodecircuit. Theprimerequirement ofavoltmeter isaveryhighimpedance, and itistherefore morecommon touseatriodeemploying anodebend rectification thanadiodebecauseofitshigherinputimpedance. Italso 422 ALTERNATING CURRENT MEASUREMENTS [15.1 hastheadvantage ofproducing acertainamountofamplification of theinputvoltage. AtypicalcircuitisshowninFig.15.7,thepurpose ofthesecondtriodebeingtobalanceoutthemeterreadingduetothe steadyanodecurrentflowinthefirsttriodeintheabsenceofanapplied signal.Theinstrument maybeusedinvariousways.Ifthegridis biasedsoastoworkonacurvedportionofthecharacteristic, thenthe Gr Input FIG.15.7.Vacuum tubevoltmeter, usingdoubletriode. gridleaks. Amicroammeter. anodeloads. cathode biasresistance withvariable tapping toadjustmeter readingtozerointheabsenceofanyinputvoltagetothefirst triode.+ H.T. changeinanodecurrentwillbeproportional tothemeansquarevalue oftheinputvoltage(seeequation (13.20))provided theamplitude of thelatterisnottoohigh.Thisiscalledfull-wave squarelawaction. Half-wave actionisachieved ifthegridisbiasedjusttocut-off,sothat onlythepositive half-cycles oftheinputvoltagecausecurrenttoflow totheanode.Iftheappliedvoltageissmall,theanodecurrentwillbe proportional tothesquareoftheinputvoltage,butlargersignalswill swingthegridontothelinearportionofthecharacteristic, givinglinear rectification. Ifthegridisbiasedwellbackbeyondcut-offsothatonly thepositivepeaksoftheinputsignalwillcausecurrentflow,thedevice canbeusedasapeakvoltmeter. Thechiefadvantages ofthetriodevoltmeter areitshighinputim­ pedance (especially whenusedasahalf-wave orpeakinstrument), and 15.1]- - ~----- ALTERNATING CURRENT MEASUREMENTS 423 itslargefrequency range.Oncecalibrated atthesupplyfrequency, it willgivecorrectreadingsatfrequencies upto30Mcjsormore,thelimit beingsetbytheeffectsoftransittimeandcathode leadinductance discussed inChapter 14.Withcarefuldesignthesemaybereduced so thattheerrorissmalluptoabout200Mc/s.Inaddition themeterin theanodecircuitisprotected bysaturation oftheanodecurrentfrom theeffectsofaccidental overloads. Thesensitivity islimitedbythe stability ofthetubecharacteristics, sincetheseaffectthezerobalance ofthemeter.Itshouldbenotedalsothatthereadingmaybedependent onthewaveform,sincesharppositivepeaksaremoreeffective incausing anodecurrentflow,owingtothecurvature ofthecharacteristic. Itisoftenneededtomeasure avoltageofaparticular frequency separate fromotherfrequencies whichmaysimultaneously bepresent. Thiscanbedonebymeansofa'phase-sensitive detector', asimple designbeingamodification ofthecircuitofFig.15.7inwhichthesignal voltageisfedequallytothegridsofbothtriodes,insteadofjustone, whilealargeralternating voltageofthedesiredfrequency isimpressed acrossRabetween thetwocathodes. Intheabsence ofasignalthe cathodes arethusoscillating involtageinanti-phase. Ifasignalvoltage ofthesamefrequency isfedtobothgrids,thiswillbeinphasewiththe cathodeoscillation ononetube,andoutofphaseontheother;themean currentthroughthetwotubeswillalter,andtheammeter Awillregister acurrent. Sinceitisadirect-current instrument, itcanrespondonlyto cUrrents whichdonotfluctuate withinitsresponse time;thusthedevice issensitive onlytosignalswhichlieverycloseinfrequency tothevoltage impressed onRa•Inaddition, thesignofthecurrentthroughAdepends ontherelativephaseofthesignalandthevoltageacrossRa,making thedevice'phase-sensitive'. Manyothercircuitscanbeused,thebasic principle beingobservation ofthed.c.(zerofrequency) voltageobtained byheterodyning thesignalagainstalocaloscillation ofthesamefre­ quency;thedeviceissometimes calleda'homodyne'. Wheninterfering signalsornoisearelargecompared withthedesiredsignal,theymay overload thetriodesandinsuchcasesdiodes,whichhavealinear response uptolargervoltages, arepreferable; asuitable circuitisgiven byRollin(1964). 15.2.Measurement ofimpedance atlowfrequencies Themeasurement ofresistance usingdirectcurrentisusuallyaccom­ plishedmostprecisely bymeansofabridge,eitherWheatstone's bridge oroneofitsmodifications. Ataudiofrequencies themeasurement ofa 424 ALTERNATING CURRENT MEASUREMENTS [15.2 complex impedance isalsoreadilyachieved withhighprecision bymeans ofana.c.bridge.Todetermine acomplex impedance fully,twoquan­ titiesmustbemeasured-its realandimaginary parts.Atfirstsight thismightseemtorequiretwoseparate experiments, butinfactthe balancing ofana.c.bridgerequiresthattwoseparate conditions be A B Driving·voltage FIG.15.8.Generalized Wheatstone's bridge. satisfied simultaneously. Thesetwoconditions involvetherealand imaginary components oftheunknown impedance, andthusbothare determined when.bothconditions arefulfilled. Thereasonforthis extracomplexity inthebalancing ofana.c.bridgecanreadilybeseen fromconsideration ofasimplenetwork suchasthegeneralized Wheat­ stone'sbridgeshowninFig.15.8,withcomplex impedances ineachof thearms.Forabalancethevoltageappliedtothedetector mustbezero. Thisvoltageisequaltothedifference ofvoltagebetween thepointsA andB,whichistrulyzeroonlyifthevoltageatthesepointshasnot onlythesameamplitude butalsothesamephase.Inotherwords,the voltageateachofthesepointsmustberepresented byavectorwithtwo components, andforthevectorsatthetwopointstobeidentical, their components mustbeindividually thesame. Thepresence oftwobalanceconditions whichmustbefulfilledsimul­ taneously hasanimportant effectonthedesignofa.c.bridges.Inorder toavoiddisturbing onebalancecondition whenadjusting theother,it isessential thatthetwobalanceconditions shallbeindependent ofone another. Thiscanbeachieved bychoosing abridgewhereeachbalance condition canbemetbyadjusting avariable impedance whichdoesnot appearintheotherbalancecondition. Thefinalbalancecanbeobtained relatively quicklybyfirstadjusting onevariable untilaminimum detector readingisobtained, andthentheother.Onreturning tothe 15.2] ALTERNATING CURRENT MEASUREMENTS 425 firstafinerbalanceisobtained, andsoon.Asecondhighlydesirable qualityisthatthebalanceconditions shallbeindependent offrequency. Thereasonforthisisthatthesourceofpoweremployed forthebridge neverproduces apuresinewave,butcontains somedistortion whichcan berepresented inaFourieranalysisofthewaveformbyharmonics of thefundamental frequency. Thepresence ofquiteasmallharmonic contentwillbeimportant ifthebalanceconditions dependonfrequency, becausethesensitivity ofthebridgedepends ontheabilitytodetecta smallfractionoftheappliedvoltage,andthiswillbeobscured bythe harmonic contentunlessthisisbalanced outsimultaneously. Inpractice itisoftenfoundthattheharmonics donotvanisheveninabridgewhere thebalanceconditions areindependent offrequency, becausetheim­ pedances usedmayvarywithfrequency, usuallybecause ofstray reactances (seeProblems 9.10and9.11).Inthiscaseitisadvantageous touseeitheratuneddetector, suchasaphase-sensitive detector with phaseshiftsof0and!7Tsothatsignalscanbeobserved bothinphase andquadrature, ortoinsertafilterattheinputtothedetectortoelimi­ natetheharmonics. Thedrivingvoltageforthebridgeisusuallyprovided byasmallaudio­ frequency oscillator, afewvoltsbeingsufficient formostpurposes. The detector consistseitherofear-phones or,forgreatersensitivity, asmall audio-frequency amplifier followed byadetector orbyaC.R.O.The latterhastheadvantage thatitshowsthewaveformreaching the detector, andthepresence ofharmonics nearthebalancepointisreadily observed. Tosomeextentitispossibletoseparate visuallythefunda­ mentalandharmonics andtoreducetheformertoone-fifth orsoofthe harmonic. Theamplifier gainmustbevariableattheinputstage(a potentiometer beforethegridofthefirsttubeissufficient) inorderto avoidsaturation ofthelaterstageswhenthebridgeisfarfrombalance. Thegainisincreased asthebalanceisapproached andtheamplifier has theadvantage thatitisnotreadilydamaged byanoverload. Whenavacuum tubegenerator andamplifier arebothuseditmay happenthatoneterminal ofeachisearthed,orhasalargecapacitance to themainssupplywhichiscommon toboth.Thiswouldthroweithera shortcircuitoralargecapacitance acrossonearmofthebridge,andan isolating transformer (preferably onewithanelectrostatic screenbetween primary andsecondary) shouldbeusedbetween thebridgeandeither thegenerator orthedetector-amplifier. Variable impedances arerequired tobalancethebridge,andforthis purposeresistances andcapacitances aremuchpreferred toinductances. 426 ALTERNATING CURRENT MEASUREMENTS [15.2 (15.1) (15.2)Avariable self-inductance requires anadjustable contact, andhasan appreciable resistance; ataudiofrequencies theQisgenerally notbetter thanabout30,whilethelosstangent(=I/Q)ofagoodmicacapacitor isabout10-4•Theratioofreactance toresistance inastandard resistance isnormally muchlessthanthisatfrequencies atleastupto10kc/s. Agoodgeneralruleisthattheimpedances ofallarmsshouldbeofthe sameorderforoptimum operation ofthe])ridge. Driving voltage FIG.15.9.Schering bridgeforthemeasurement ofcapacitance. Thegeneralized Wheatstone's bridgeshowninFig.15.8isthebasis ofmostbridgecircuits,andthebalancecondition issimilartothatfor thed.c.bridge: Thetwobalanceconditions arecontained inthiscomplex equation, since therealandimaginary partsmustbesatisfied simultaneously. Thiswill beseeninthefollowing application totheSchering bridge,whichis commonly usedforthedetermination ofcapacitance. Thecircuitdiagram oftheSchering bridgeisshowninFig.15.9. Theunknown (lossy)capacitor isrepresented bytheseriescombination of0andR.qisagoodstandard capacitor, whosemagnitude should beofthesameorderasthatofthecapacitor undertest.R1isafixed resistance, andR2isavariableresistance shuntedbyavariablecapacitor O2,Thebalancecondition is jw01(R+j~O)=Rl(~2+j(02). Therealandimaginary partsofthisgive 0=Q(R2/R1)}. and R=R1(02/01) 15.2] ALTERNATING CURRENT MEASUREMENTS 427 Theseconditions fulfiltherequirements forana.c.bridgeoutlined earlier.Theyareindependent ofeachother,provided thatR2andO2 onlyarevaried,andtheyareindependent offrequency. Inaddition the capacitance oftheunknown capacitor isobtained intermsofaknown standard capacitor andtheratiooftworesistances, andthevariable capacitance O2entersonlyintotheequation fortheapparent series resistance oftheunknown capacitor. Withagoodcapacitor thiswillbe small,andhighaccuracy inthedetermination ofRisseldomrequired. Asmallvariable aircapacitor usuallysufficesforO2,anditsleakage resistance undernormalconditions willbesohighthatitdoesnotaffect R2,withwhichitisinparallel. Owingtothedifficulty ofconstructing astandard variable inductance, itisgenerally preferable todetermine anunknown self-inductance in termsofstandard capacitances andresistances. InMaxwell's Lj0bridge (seeProblem 15.5)anetworkoftheWheatstone bridgetypeisused,but tomakethetwobalanceconditions independent ofeachotherastandard variable capacitance isrequired. Amodification ofthisbridge,whichis commonly used,isduetoAnderson andhastheadvantage thatonly variable resistances arerequired, together withastandard fixedcapaci­ tance.ThecircuitisshowninFig.15.10.Theunknown self-inductance isL,withresistance r,whichformsonearmofthebridgewhenplaced inserieswithavariable resistance S.Thefixedcapacitor 0isinseries withavariable resistance T,thecombination beingshunted bya resistance P.Thedetector isconnected fromBtothejunction of0and T,insteadoftothepointA.Thebalance condition ismostreadily foundbycalculating thevoltages acrossFEandFBasfractions ofthe drivingvoltageV.ThevoltageacrossFEisafractionl/(l+jwOT) of thatacrossFA,whilethatacrossFA=VZj(Z+Q) =Vj(l+QjZ), whereZisthetotalimpedance betweenFandA.Since IjZ=IjP+jwOj(l+jwOT), thevoltageacrossFEis Vj{(l+jwOT)(l+QjZ)} =Vj{(l+jwOT)(l+QjP)+jwOQ}, whilethevoltageacrossFBisVRj(R+jwL+r+S). Equating these voltages gives 1+(jwL+r+S)j R=(1+jwOT)(l +QjP)+jwOQ. Therealandimaginary partsofthisequation giveseparately (r+S)jR =QjP and LjR=OT(l+QjP)+OQ. 428 ALTERNATING CURRENT MEASUREMENTS [15.2 (15.3)Itisgenerally convenient tomakeQ=P,inwhichcasethebalance conditions reducetor=R-S } L=OR(2T+P) . Itisobviousfromthisthatnobalanceispossible unlessORP<L;if thiscondition isbeingviolateditwillbeindicated bythefactthatthe nearestapproach tobalanceisobtained whenTiszero.Inspection ofthe balanceconditions showsalsothattheyareindependent offrequency. andofeachotherifSandTaremadethevariables. A B 1..- Driving_--------------' voltage FIG.15.10.Anderson's bridgeformeasurement ofself-inductance. Thesimplest methodofmeasuring mutualinductance isbymeansof adirectcomparison withavariable standard mutualinductance. The primary windings oftheinductance undertestandthestandard are connected inseriestoagenerator. andthesecondary windings arecon­ nectedinseriestoadetector.Ifthesecondary connexions aremadeso thattheinducedvoltages opposeoneanother. anullreadingisobtained inthedetector whenthevariable mutualinductance isequaltothat undertest.Inpracticeitisgenerally impossible togetagoodbalance becausethevoltageinducedinthesecondary ofeachinductance contains acomponent inphasewiththeprimary current.arisingfromeffectssuch asselfandmutualcapacitance inthecoils.Thismayberepresented bywritingthesecondary voltageas~=(p+jwM)ip- Hartshorn has shownthatthisdifficulty maybeovercome bytheinclusion ofavariable resistance whichiscommon toboththeprimaryandsecondary circuits, .(15.4) M1=M2 } r=±(Pl-P2)15.2] ALTERNATING CURRENT MEASUREMENTS 429 asshowninFig.I5.II.Theequation forabalanceatthedetectorD isnow rip±(pl+jwM1)ip=j=(P2+jwM 2)ip=0 whichseparates to Thesecondoftheseequations canonlybesatisfiediftheconnexions aremadesoastogivetherightsigns;wehavealreadyassumedthatthe secondaries areconnected inantiphase withrespecttooneanother. D p.+jroM. FIG.15.11.Hartshorn's mutualinductance bridge. 15.3.Measurement ofimpedance atradiofrequencies Asthefrequency isincreased thedifficulties associated withtheuse ofbridgesforthedetermination ofimpedance riserapidly. Eacharm ofthebridgemustbeenclosed initsownshield,andeachconnexion mustbeshielded. Thegenerator anddetector shouldalsobeshielded andcaremustbetakentoavoidanydirectpick-upfromgenerator to detector whichwouldgiveafalsezero-setting forthebridge.Thecapaci­ tancebetween eacharmanditsshieldmustbeincluded intheanalysis ofthenetwork.Itisusualtomaketwoarmsidentical, withequalresis­ tanceandequalcapacitance betweentheresistance anditsshield.This givesanequalratiointwoarmsinspiteoftheshielding capacitances. It isalsocommontousea'substitution' method,thebridgebeingbalanced firstwiththeunknown impedance inparallelwiththevariablestandard impedance (usuallyresistance pluscapacitance) andthenwithoutit.No generalaccountofr.f.bridgescanbegivenhere,butitmaybesaidthat 430 ALTERNATING CURRENT MEASUREMENTS [15.3 theirconstruction requiresexpertandspecialized knowledge ifaccurate resultsaretobeobtained. Ageneralpurposeinstrument whichiscommonly usedatradiofre­ quencies isthe'Q-meter'. Thebasiccircuitofthisinstrument isshown inFig.15.12.AsmallcurrentIfromanoscillator isreadonamilli­ ammeter Aandthenflowstoaknownlowresistance r.Aseriestuned circuitisconnected inparallelwithr,andthevoltageVdeveloped across thecapacitor Gisreadonavacuumtubevoltmeter whenthecapacitor c vv FIG.15.12.Q-meter circuit. Athermoammeter. Linductance undertest.VVvacuum tubevoltmeter. isadjusted forresonance. Thelatterisindicated byamaximum reading ofthevoltmeter. TheQofthecircuitundertestisthenequaltothe ratioV/lr,sincelristhevoltageintroduced inserieswiththetuned circuit.Itisnecessary thattheresistance rshallbesmallcompared with theseriesresistance Rofthetunedcircuit,inorderthatsubstantially allthecurrentregistered bythemilliammeter shallflowthrough r.This requirement maybestatedinanotherway--rmustbesmallcompared withRinordernottoloadthecircuitundertest.(Itiseasytoshowthat themeasured Qisthatforacircuitwhosetotalseriesresistance isthe sumofrandR.)Inacommercial instrument aninternal oscillator of calibrated variable frequency isprovided, andthecurrentfromitmay beadjusted tobringthemilliammeter readingalwaystoafixedmark. Aninternalvacuumtubevoltmeter maythenbecalibrated directlyto readtheQofanunknown coil.Thevariable capacitor Gisincluded in theinstrument, andiscalibrated sothattheinductance oftheunknown coilmaybecalculated fromtheknownoscillator frequency andthe tUningcapacitance. Anunknown capacitance mayalsobemeasured by thesubstitution method: asuitable coilLisinserted andthereading ofGrequired totuneittoresonance withandwithouttheunknown capacitor inparallelwithGisfound.Anunknown resistance canbe 15.3] ALTERNATING CURRENT MEASUREMENTS 431 measured byfindingtheeffectontheQofacircuitwhenitisplaced inserieswiththecircuit. Atfrequencies aboveabout100Mcjs(wavelengths of3metresandless) theleadstotheimpedance undertestarenotnegligible inlengthcom­ paredwiththewavelength anderrorsmaybeintroduced becausethe currentandvoltageatthemeasuring instrument arenotthesameas thoseattheunknown impedance. Theseerrorsmaybeeliminated by makingtheleadspartofatransmission lineofknownandconstant impedance, theunknown impedance beingplacedattheendofthisline andactingasitstermination. Theimpedance Zoofthetransmission line maybecalculated fromitsdimensions (see§11.3)andtheunknown impedance isdetermined asaratiotoZOoThismaybecarriedouteither bydetermining thevoltagestanding waveratio(v.s.w.r.) ontheline, orbyaresonance method. Thefirstofthesemethods hastheadvantage thattheresultsdonot dependonthegenerator impedance, andthegenerator maytherefore beconnected directlytotheline.Fromthetheoryoftransmission lines (Chapter 11)itfollowsthatthevoltageatanypointonthelinemaybe regarded asduetoanincidentwaveofamplitude Aandawavereflected fromtheterminating impedance ofamplitude A1.Theresultant voltage amplitude isamaximum (A+ A1)atanantinode wheretheincidentand reflected wavesareinphase,andaminimum(A-A1)wheretheyare 1800outofphase,thesepointsbeingaquarterofawavelength apart. Fromameasurement ofthevoltagestanding waveratio (A+A1)j(A-A 1), andthepositionofthenodesorantinodes, theratiooftheterminating impedance Ztothecharacteristic impedance Zomaybefoundusing equations (11.19): A1=J(Z!+Z~-2Z1Z0COSep)} A Zl+Zo+2Z1Z0cosc/>(11.19) t~_2Z1Zosinc/> 'ana-Z2 Z21-0 whereZ=Zlei4>,and3isthedifference inphasebetween thereflected andincidentwavesatthepointofreflection (thetermination oftheline). Thisphaseconstant canbefoundfromtheposition ofavoltagenode, thisbeinggenerally moreaccurate thanthelocation ofanantinode (especially ifthev.s.w.r.ishigh)becausethesensitivity ofthedetector canbeincreased asthenodeisapproached. Iftheendofthelineisat x=0,andthevoltagenodeatapointx=-l,thenthephaseofthe 432 ALTERNATING CURRENT MEASUREME~TS [15.3 incident waveatthispointis-w(-l/v)=27Tl/>",andthatoftherefl.ected waveis8+w(-l/v)=8-27Tl/>... Foranodethesemustdifferby7T, whence8=(4;l+7T).Todetermine laccurately, itisbesttoreplace theunknown impedance Zbyashortcircuitandfindthedistance between theprevious nodeandthenewone;thelatteris(electrically) exactlyanintegralnumberofhalf-wavelengths fromtheendoftheline. Leadstogalvanometer By-pass condenser forr.f. Detector unitonmovable carriage Scale Co-axial line(impedance Zo)Crystalrectifierorothertypeofdetector +---Choke providing d.c.returnpath Probepick-up FIG.15.13.Standing wavedetector oncoaxialline. Thevoltagestanding waveratiocanbemeasured bymovingany looselycoupledvoltagedetector alongtheline.Sinceonlyaratioofthe maximum andminimum readings isrequired, theabsolute calibration oftheindicator isunnecessary, andaknowledge oftherectifying charac­ teristic(d.c.currentorvoltageoutputagainstr.f.voltageinput)is sufficient. Withacoaxialline,asectionofair-spaced lineismadeup withknowndimensions, andanarrowslotiscutlengthwise alongthe outerconductor. Sincethisslotisparalleltothedirection ofcurrent flowintheline,itdoesnotdisturbconditions onthelinematerially. In thisslot(seeFig.15.13)isinserted asmallradialprobe,whichisparallel tothelinesofelectricfieldinsidethecoaxialline;itpicksupasmall fractionofthevoltageonthelineandfeedsittoadetector. Theintrusion oftheprobeiswadeassmallaspossibletominimize disturbance onthe line,andforthispurpose asensitive detector isrequired toobtain adequate sensitivity. Theprobeismounted onamovable carriage, carefully machined sothattheintrusion oftheprobedoesnotchange asitmovesalong.Thismaybecheckedbyobserving theconstancy of thedetector readingwhenthelineisterminated byitscharacteristic impedance, whenthev.s.w.r.shouldbeunity.Withaparallelwireline 15.3] ALTERNATING CURRENT MEASUREMENTS 433 asimilararrangement maybeusedwithaprobenearthewires,butthe indicator anditsleadsmustbeshieldedandkeptwellawayfromtheline sincetheelectricandmagnetic fieldsaroundthelinearenotnowrigor­ ouslyconfined astheyareinthecoaxialline. Fromthemeasurement ofthev.s.w.r.andofS,theratioofthereal andimaginary partsofZtoZocanbefoundbyusingequations (11.19), butthesearealgebraically soclumsytohandlethatgraphical methods arenormally employed. 'Impedance diagrams' canbeobtained from whichtherealandimaginary partsofZ/Zocanbereadoffatonce. Whentheimpedance tobemeasured hasonlyasmalldissipative component thestanding waveratiobecomes verylargeandisdifficult tomeasure accurately, principally becausethedetector lawmustbe knownoverawiderange.Itisoftenthenmoreconvenient tousea resonance method. Theunknown impedance isconnected acrosstheend ofalineasinFig.15.14,andanoscillator anddetector arelooselycoupled toit.Amovable bridge,whichshouldmakesuchgoodcontactasto beessentially ashortcircuit,isadjusted untilresonance isindicated bymaximum deflexion ofthedetector.IftheloadZisrepresented by aresistance Rinparallelwithareactance jX,thenresonance occurs whenthisreactance isequalandopposite tothelinereactance.Ifthe lengthofthelineatthispointisl,thenthelinereactance is jX'=-jX=jZotan27Tl/~, whichmaybepositiveornegative according tothevalueofl/~.Thus thevalueofXisdetermined fromtheresonant length.ThevalueofR maybefoundbymeasuring thesharpness ofresonance. Thisismost conveniently donebyvaryingthelengthofthelineuntilthedetector readingshowsthatthevol~age(orcurrent) onthelinehasfallento1/-./2 ofthemaximum. Atthispointthesusceptance formedbyX-Iinparallel with Zolcot27T(l±Sl)/~ hasrisenfromzerotobejustequaltol/R (cf.thetheoryoftheparalleltunedcircuitin§9.3).Ifthechangein lengthisSl,thenthevalueofthesusceptance is I-X-1+Z 01cot27T(l±ol)/AI =Zol(27TSl/~)cosec2(27Tl/~) =l/R, whenceRcanbedetermined. Iflossinthelinecannotbeneglected, ashasbeenassumed above,thenitcanbefoundbyaseparate measure­ mentwiththelineshort-circuited atbothends(oropen-circuited atone end)andacorrection applied. Thecalculation israthercomplicated, butfromequation (11.28)itcanbeseenthatthelengthoflinecanbe represented byacomplex admittance Y'whichisinparallelwithl/Z. IfY'isseparated intoitsrealandimaginary partsG'andS',thenthe 851110 Ff 434 ALTERNATING CURRENT MEASUREMENTS [15.3 calculation proceeds asbefore.Astheresonant lengthsoflineunloaded andterminated byZwillbedifferent, the1088onthelinemustbeex­ pressedintermsoftheattenuation coefficient ex(equation (11.24)). Theadvantage oftheresonance methodoverthes.w.r.methodisthat thedetector lawneedonlybeknownoverasmallrange,theother Movable short circuit-- H--------- ~--l--R -~ n Lo""ooopliog to,oIlb,,"'d "otooto< FIG.15.14.Measurement ofimpedance usingresonance method ontransmission line. measurements beingthoseoflengths. Bothtypesofmeasurements may alsobeusedwithwaveguides atcentimetre wavelengths, thoughhere theconceptofalumpedimpedance losesmostofitsmeaning. Afew examples ofsuchmeasurements willbegivenlaterinthisbook,butfor afulldiscussion reference shouldbemadetoBarlowandCullen,Micro­ waveMeasurements (Constable). 15.4.Measurement offrequency andwavelength Themeasurement ofthefrequency ofanaudiooscillation canbemade intermsofknownimpedances bytheuseofabridgewhosebalanceis dependent onfrequency. Itisobviousthattheoscillation tobemeasured mustbeconstant infrequency andfreefromharmonics, sincethelatter wouldbeoutofbalanceinthebridge(twousesofafrequency bridgeare thesuppression ofagiven frequency suchasatroublesome harmonic andtheanalysisofharmonic content). Alargenumberofbridgeshave 15.4] ALTERNATING CURRENT MEASUREMENTS 435 beendevisedwhichsatisfythedesiderata thatthebalance conditions shouldbemutually independent andonlyoneofthemshoulddepend onthefrequency. Asimplebridgeusingonlyresistances andcapaci­ tancesisduetoWienandisshowninFig.15.15.Thebalanceconditions are (02_1 )-RS0102,(15.5) O2QR 0 1=p-S Drivingvoltage FIG.15.15.Wien'sbridgeforfrequency measurement. whicharemutually independent ifthevariables RandSor01andO2 aregangedsothattheirratioiskeptconstant. Thenthesecondcondi­ tionwillremainsatisfied onceithasbeensetupandmeasurement of awiderangeoffrequencies isobtained byasingleadjustment. Atwavelengths lessthan1or2metresthemeasurement ofwavelength directlybecomes quiteconvenient, andhighaccuracy maybeattained becauseofthehighQofresonant transmission linesandwaveguide cavities. Parallelwirelinesmaybeusedforthelongerwavelengths, but coaxiallinesarebetteratdecimetre wavelengths, andcavityresonators atcentimetre wavelengths. Asimpletypeofcoaxiallinewave-meter isshowninFig.15.16.Thecentreconductor isvariable inlength,and movesthrough aspringcontactwhichformstheclosedendoftheline. Powerisintroduced bymeansofasmallloopwhichintersects someof themagnetic linesofforceattheclosedend,andasecondlooptakes powertoadetector (usually acrystalrectifier). Theseloopsmustbe keptsmalltogiveloosecoupling, andtoavoidpullingtheoscillator whosewavelength istobemeasured through coupled circuiteffects. Theequivalent circuitofthewave-meter isshowninFig.15.16,from whichitcanbeseenthatthedetector readingisamaximum whenthe lineisresonant. WithahighQitisusuallyundetectably smallaway 436 ALTERNATING CURRENT MEASUREMJ~~TS [15.4 fromresonance. Sincetheloopsintroduce smallimpedances whichalter theelectrical lengthoftheline,itispreferable tomeasure successive pointsofresonance, whichareexactlyhalf-wavelength apart.Thewave­ meterthenneedsnocalibration, thewavelength beingfounddirectly SourceofinputpowerIPowerfromsource Coaxialline wavemeter ITodetector Equivalent circuit r FIG.15.16.Coaxia11ine wave-meter andequivalent circuit. fromascaleandvernierattached tothemovingpart.Theaccuracy is usuallyabout1partin103,themaindifficulty beinginmakingagood contactbetween themovingconductor andthestationary end. Abasictypeofcavitywave-meter isshowninFig.15.17.Asection ofcircularwaveguide isclosedatoneend,theotherendbeingformed byaplungerdrivenbyamicrometer head.Powerisfedintothecavity fromawaveguide through asmallhole,andresonance isdetected by coupling alittlepoweroutthrough asecondholetoadetector. The holesshouldbekeptassmallaspossible, subjecttogettingafinite detector reading,inordertoavoidlowering thenaturalQoftheresonator, whichmaybeoftheorderof10000.Thewavelength inthecavityis foundfromthedistance(\/2)between successive resonance points.The wavelength infreespacemaythenbefoundfromthediameter ofthe cavity,andthemodeofresonance. Toavoidthedifficulty ofmaking agoodcontactbetween themovingplungerandthewalls,aparticular waveguide mode(TEO!orHOl)isoftenused,wherethereisnocurrent flowacrossthiscontact(see§11.7).Othermodesofresonance maythen alsobepresent,sincetheTEO!modeneedsratheralargecavitydiameter 15.4] ALTERNATING CURRENT MEASUREMENTS 437 (thecut-offwavelength isequalto0·82timesthediameter). Thesemay beavoidedbyusingspecialarrangements ofthecoupling holes(see Bleaney, Loubser, andPenrose, 1947).Anaccuracy ofoneortwoparts in104maybeattained, butacorrection isthenneededforthedielectric constant (1'0006)oftheairinthecavity. I-----Micrometer screwdriveforplunger Movable plunger---~:::=~=JI ----t-Resonant cavity Powerfromsource--I ...I Waveguide inputI------.Todetector i Waveguide output FIG.15.17.Resonant cavitywave-meter. Thequartzcrystaloscillator Wherehighaccuracy offrequency controlormeasurement isrequired, useismadeoftheproperties ofapiezo-electric crystal,quartzbeingthe mostsatisfactory forthispurpose. Aquartzcrystalgrowsintheform ofahexagonal prismwithpointedends,thecross-section oftheprism beingasshowninFig.15.18(a).Ifanelectricfieldisappliedtothecrystal y (a)L R (b)G FIG.15.18.(a)Quartzcrystal(x,yareoneofthethreepairsofX,Yaxes). (b)Equivalent circuitofcrystalanditselectrodes. intheX-direction, thecrystalcontracts orelongates intheY-direction according tothesignoftheelectricfield.Similarly,ifamechanical stress isappliedintheY-direction, anelectricpolarization issetupinthe X-direction andchargesappearonthefacesofthecrystal. Theseeffects arereversible andverynearlylinearlyrelated,andtheirimportance lies 438 ALTERNATING CURRENT MEASUREMENT,,, [15,4 inthefactthattheycoupletogether anelectrical andamechanical system.Ifanalternating voltageisappliedintheX-direction, an alternating stressappearsintheY-direction andtheamplitude ofthe resulting mechanical vibrations islargeifthefrequency ofalternation coincides withanaturalmechanical vibration ofthecrystal. Anumber ofdifferent modesofoscillation exist,butthosemostcommonly used arelongitudinal andshearvibrations. Thedamping ofthemechanical vibrations isverylow,andthesharpness oftheresonance makesthem verysuitable foruseasfrequency standards. Thedesirable properties forthispurpose are: (a)Zerotemperature coefficient offrequency ofoscillation. (b)Highpiezo-electric effect. (c)Asinglemodeofmechanical resonance wellseparated infrequency fromothermodes,sothatthereisnotendency tojumpfromone modetoanother. Thegreatest piezo-electric effectisobtained whentheelectrical andmechanical stressesareappliedalongtheelectrical orX-axisand mechanical orY-axisrespectively, butoscillations canbeexcitedbyany stresswhichhasacomponent paralleltotheseaxes.(Notethat,owing tothehighsymmetry ofthecrystal,therearethreesetsofX-andY­ axes,relatedtooneanotherbyrotations of1200and2400aboutthe Z-axis,theopticaxisofthecrystal.) AnX-cutcrystalconsistsofathin slabwithfacesparalleltotheYZ-plane, andthetemperature coefficient ofthefrequency ofoscillation isnegative, about-22X10-6per°e. AY-cutcrystalisathinslabwithitsfacesparalleltotheXZ-plane, andthetemperature coefficient ispositive withanumberofdiscon­ tinuities duetocouplings between different modesofoscillation. In generalitismoreimportant toobtainzerotemperature coefficient than highpiezo-electric activity, andintermediate cutsareusedsuchasthe AT-cut, athinplatewhosefacescontaintheX-axisandalineinthe YZ-plane makinganangleofabout35.50withtheZ-axis.Whena voltageisappliedbetween thelargefaces,ashearvibration issetup whosefrequency inmegacycles persecondis0·1675/(thickness incenti­ metres). Thisissuitable forfrequencies fromroughlytto10Mc/s. Lowerfrequencies maybeobtained frommodeswherethefrequency isdetermined byoneofthelongdimensions oftheslab,thefullrange ofquartzcrystalsbeingroughlyfrom25kc/sto15Mc/s.Toapplythe alternating voltagetheslabismounted between theplatesofacapaci­ tor;thesearegenerally formedbysputtering ametallic filmonto 15.4] ALTERNATING CURRENT MEASUREMENTS 439 thelargefaces.Thisreducestheloadingonthemechanical vibrations, whichisfurtherreducedbymounting thecrystalinvacuobetween light supports touching thecrystalatamechanical node.Forthehighest frequency stability thecrystaliskeptinanoventhermostatically con­ trolledto0.10orbetter,becausethetemperature coefficient iszeroonly overanarrowrangeoftemperature. Themechanical systemofaquartzcrystalmayberepresented by theequivalent electrical circuitshowninFig.15.18(b).Themechanical resonance isequivalent toaseriestunedcircuitandthisisshuntedby thecapacitance 01oftheelectrodes. Typicalvaluesare: X-cutquartz(lengthwise vibration) Dimensions: rectangular bar, X=1·4mm,Y=30·7mm,Z=4'1mm R=15000ohms q=3·54pF L=137henries Q=5150 o=0·0228pF io=89'87kc/s AT-cutquartz Dimensions: disk,25mmdiameter, thickness 1·10mm. R=24·2ohms 01=17'9pF L=0·119henries Q=46500 o=0·0945pF io=1500kc/s (FromW.G.Cady,Piezoelectricity (McGraw-Hill, 1946).) Thepresence ofqcausesthecircuittobehaveasaparalleltunedcircuit atafrequency justabovethatoftheseriesresonance (seeProblem 15.8). Thedifference between thesetwofrequencies isverysmallsothatthe phaseangleofthecircuitvariesveryrapidly.Asimpleone-tube circuit formaintaining thecrystalinoscillation isshowninFig.15.19.Feed­ backofenergytothegridcircuittakesplacethrough thegrid-anode capacitance Oga'andtoobtaintherightphasetheanodecircuitmustbe tunedtoafrequency higherthantheparallelresonance frequency ofthe crystalinitsmount,sothattheimpedance oftheanodecircuitisinduc­ tiveatthisfrequency (see§12.8).Thecrystaloscillation willbedamped iftheamplitude ofoscillation issohighthatgridcurrentflowsinthe tube,andvariousarrangements forcontrolling thefeed-back areused, suchasabridgesystemwhereonearmisalamporthermistor whose resistance varieswiththeamplitude ofoscillation. Thefrequency of 440 ALTERNATING CURRENT MEASUREME,NTS [15.4 oscillation canbeadjusted overaverynarrowrangebyasmallvariable capacitance inparallelwithqandthisisusedforfineadjustment. Oomparison ofunknown frequency withstandard frequency Thehighfrequency stability ofthequartzcrystaloscillator makesit extremely usefulasafrequency standard, andtheaccurate measurement ofanunknown frequency isinvariably madebymeansofacomparison Quartzcrystal+ H.T. FIG.15.19.Quartzcrystaloscillator. withsuchastandard. Theblockdiagram ofafrequency standard suitable forordinary laboratory purposes isshowninFig.15.20.The fundamental frequency generated is100kcls,usingaquartzcrystal contained inathermostat. Although long-term frequency stability of theorderofonepartin108ispossible, itisunnecessary tobuildthe complex systemthatthisrequires. Instead,thefrequency ofthestandard maybeadjusted immediately beforeuse,andcheckedduringoperation, againstoneoftheaccurate frequencies originated atastandardizing laboratory andradiated bystationMSF,Rugby,England, andstation WWV,Washington, U.S.A.Theworkingquartzcrystalstandards atthe national standards laboratories arecalibrated intermsofthefrequency ofanatomictransition ofthecaesium atom,andaninternational com­ mitteehasdecided(1964)thattheunitoftimeshouldbethusdefined, makingthecaesium frequency 9192631770 cis. Thisisamoreconvenient andamoreprecisestandard thanprevious onesbasedonthemeanlengthofthesolardayoryear,becausethe motionoftheearthisknowntobesubjecttofluctuations (see§23.6). 15.4] ALTERNATING CURRENT MEASUREMENTS 441 Inordertomeasure frequencies otherthanthoseclosetothe100kc/s fundamental itisnecessary togenerate higherandlowerfrequencies by multiplication anddivisionofthefundamental. Higherharmonics are generated byfeedingthefundamental intoaClassCamplifier stage, wheretheshortpulseofanodecurrenthasahighharmonic content. Thisexcitesacircuittunedtothedesiredharmonic whichactsasthe anodeload,andthisharmonic isthenamplified tothedesiredextent. Multipliers r--......---1 100kc/sStandard Harmonic generatorOutput: Markersat10kc/s intervals from 10kc/sto150Mc/s Dividers FIG.15.20.Frequency measuring equipment. Itisconvenient toworkwithharmonics risingbyfactorsof10(usually achieved bymultiplying firstbyfive,andthenbytwO).Byrepetition ofthisprocessfrequencies uptoafewhundred megacycles maybe generated withthesameaccuracy asthefundamental, andharmonics ofsuchfrequencies havebeengenerated upto""'1011cis(wavelengths of afewmillimetres). Frequency divisionmaybeachieved byanumber ofmethods, suchasuseofthemultivibrator (see§13.7).Abettersystem isillustrated bythefollowing methodofproducing 10kc/sfrom100kc/s: theoutputofanamplifier for10kc/sismultiplied to90kc/s,whichis heterodyned withthe100kc/stoproduce a10kc/ssignalwhichisfed backtotheinputofthe10kc/samplifier. Thiscausesittooscillateat afrequency precisely one-tenth ofthestandard 100kc/s,sinceonlythen isthefeed-back signalofthesamefrequency. Thisprocessmaybe repeated downto50cisifitisdesiredtorunaclockwhichcanbechecked againstradiotimesignalsinordertomonitorthelong-term stability Of thesystem. Comparison ofanunknown frequency withthestandard requiresthe useofanadjustable oscillator whichcanbeheterodyned againstboth theharmonics ofthestandard andtheunknown. Suppose thelatteris 442 ALTERNATING CURRENT MEASUREMENTS [15.4 known(byresonance withacalibrated tunedcircuit)tobeapproxi­ mately13Mc/s.Theadjustable oscillator isfirsttunedtozerobeat withthe10Mc/sstandard, andtheIMc/soutputisthenalsoswitched intothemixerstage,whichnowgenerates everyharmonic ofIMc/s. Thevariable oscillator isnowincreased infrequency, andthenumberof zerobeatnoteswiththeIMc/sharmonics passedbeforezerobeatwith theunknown isreachedarecounted. Suppose therearethree;thenthe unknown frequency liesbetween 13and14Mc/s.Thevariable oscillator isthenreturned to13Mc/s,andthe100kc/ssignalfromthestandard addedtothemixer.Thevariable oscillator isnowagainincreased in frequency, andthezerobeatsevery100kc/scounteduntiltheunknown isreached. Thisshowsthattheunknown lies,say,between 13·1and 13·2Mc/s,andtheprocessisrepeated withthe10kc/sstandard to establish thattheunknown liesbetween, say,13·16and13·17Mc/s.The ultimate heterodyne difference frequency betweentheunknown andthe nearestharmonic ofthe10kc/sstandard liesintheaudio-frequency rangeandmaybemeasured byafrequency bridge,orbycomparison withacalibrated audio-frequency oscillator, etc.,according tothe accuracy required. 15.5.Measurement ofdielectric constant Thedielectric constant ofasubstance affordssomevaluable informa­ tionastothestructure ofitsconstituent molecules (seeChapter 17), andaccurate measurement ofthedielectric constant istherefore ofsome importance. Sincethedielectric constant isdefinedbytheratioofthe capacitance ofacapacitor filledwiththesubstance undertesttothatof theemptycapacitor, itisobviousthatingeneraltwomeasurements of capacitance willsuffice.Forsolidsandliquidsthedielectric constant variesfromabout2to100,andanyofthebridgesdesigned tomeasure capacitance maybeusedtogiveaccurate results.Forthehigherdielectric constants caremustbetakentoavoidstraycapacitance whichmay seriously affectthereadingobtained withtheemptycapacitor, ifthis hasarathersmallcapacitance. Inthecaseofgasesthedielectric constant differsfromunityonlyby about0·001andspecialmethods mustbeused.Onesuchmethod(see, forexample, HectorandWoernley, 1946)makesuseofthehighaccuracy whichcanbeobtained inthemeasurement offrequency, byincorporating aspecially designed capacitor intheresonant circuitofatunedanode oscillator. Thefrequency ofthisoscillator isthencompared witha standard frequency fromaquartzcrystaloscillator, firstwiththe 15.5J ALTERNATING CURRENT MEASUREMENTS 443 capacitor evacuated, andthenfilledwithgas.Thechangeinfrequency mayeitherbemeasured directly, orthefrequency mayberestoredtoits originalvaluebyadjustment ofasmallstandard variable capacitor in parallelwiththecapacitor containing thegas.Theaccuracy ofthis lattermethodisusuallylimitedbythatofthevariable capacitor, and theformermethodistobepreferred. Standard oscillator Gas-tight box fortestcapacitor H.T.-.--------4.....-'A.F.frequency measurement Tuned-anode oscillator FIG.15.21.Measurement ofthedielectric constant ofagas. Ablockdiagram oftheapparatus isshowninFig.15.21.Toavoid dimensional changes whenthegasisintroduced, thecapacitor 0is surrounded, firstbyaperforated case,andthenbyaheavysteelgas-tight container. Thesurfaces ofthecapacitor aregold-plated tomaintain highconductivity andavoidtarnishing. Ifmeasurements aremadeover arangeoftemperature, inordertodetermine theelectricdipolemoment ofamolecule (see§17.3),acorrection mustbemadeforthermalexpan­ sion.Acorrection isalsorequired forstraycapacitance whichisnot alteredbytheintroduction ofthegas.Ifafrequency measuring equip­ mentisnotavailable, asmalltuningcapacitor 0'isadjusted when0is evacuated sothatazerobeatnoteisobtained between thetunedanode oscillator andastandard oscillator, preferably controlled byaquartz crystal.Ifthisfrequency f'isabout1Mc/s,thenonintroducing the gasanaudio-frequency beatnoteisproduced betweenthenewfrequency I"andthestandardf', whichmaybemeasured byaWien'sbridgeor bycomparison withatuningfork.Thensincef'=1/27T,J(LO), and I"=1/27T,J(LeO), wehavee=(f'/1")2. Lovering andWiltshire (1951)havecriticized theabovemethodon -~------------ 444 ALTERNATING CURRENT MEASUREMENTS [15.5 (15.6)(11.34)thegroundthatlong-term stability isnotattained, anditistherefore necessary tomeasure thefrequency changefairlyquicklyafterintro­ ductionorremovalofthegas.Thisintroduces errorsbecauseofadiabatic temperature changes. Theyusedasimplecapacitance bridgeat0·11Mc/s, anddetermined thecapacitance changebymeansofavariable cylindrical capacitor whoseinnerconductor wasadvanced byamicrometer screw. Themostaccurate measurements appeartobethoseofEssenand Froome (1951),usingacavityresonator andworkingatafrequency of 24000Mc/s.Thecavitywascylindrical, withadiameter ofabout5cm, andresonated intheTEOlmode.Thefrequency ofresonance wasdeter­ minedfirstwiththecavityevacuated, andthenfilledwithgas,by plotting outtheresonance curveusingaklystron oscillator whose frequency couldbedetermined to1partin108bycomparison withthe N.P.L.frequency standard. Thefrequency ofresonance isgivenby equation (11.34):f2p,Ef21 1 v2=C2=A2+A2'cg wherep,andEarethemagnetic permeability anddielectric constant of thegasfillingtheresonator andAc,Auarefixedbythediameter andlength ofthecavityrespectively. Thusiff'istheresonant frequency ofthe emptycavity,and!"thatofthegas-filled cavity,(f'If? =p,E.Hence theratioofthetwofrequencies determines n=,J(p,E),therefractive indexofthegas.Acorrection mustbeappliedforthepermeability, whichdiffersslightlyfromunityforairandoxygen, sincethelatteris paramagnetic. Acomparison ofthemeasurements ofEofanumberof workersatdifferent frequencies, together withthesquareoftheoptical refractive index,isgiveninTable17.4. Thecavityresonance methodmayalsobeusedformeasurement of thedielectric constant ofliquidsandsolids,provided thattheirloss tangentisfairlysmall(seeFaraday SocietyConference onDielectrics, 1946).Apartlyfilledcavitymustbeusedforsolidsorliquidsofhigh losstangent, butfornon-polar liquidsafilledcavitywasemployed by Bleaney, Loubser, andPenrose (1947).Atunablecavityresonant inthe TEolmodeofthesametypeasdescribed earlier(§15.4)wasadjusted toresonance withaklystron oscillator offixedfrequency, firstwiththe cavityempty,andthenfilledwithliquid.Bymeasuring anumberof successive resonant points,thewavelength intheguidewasfoundin eachcase,andthedielectric constant calculated fromtheequations Ea1 1Ell A2=A2+A2'A2=A2+A2'acdc I 15.5] ALTERNATING CURRENT MEASUREMENTS 445 where Eaisthedielectric constant ofairandEthatoftheliquid,Athe wavelength infreespace,andAa,Adthewavelengths intheair-and liquid-filled cavityrespectively. Thelosstangent oftheliquidwas foundfromthewidthoftheresonance curvedetermined bydetuning thecavity.Thusonlymeasurements oflength,depending onamicro­ meterthread,wereinvolved. Typicalresultsatatemperature of20°C aregiveninTable15.1.Whentwomeasurements aregivenat1'35-cm wavelength, theyweremadewithcavitiesofdifferent diameter. TABLE 15.1 L088tangent Dielectric constant € (tan8) .\=3-2om.\=1-35om,\=3·2om'\=1-350m Cyolo-hexane 2-0244 2-0246,2·0251 0·00005 0-00019 n.Heptane. 1·9220 1·9223 0-00037 0-00076 n-Hexane 1·9016 1-9016 0-00034 0-00076 CSs 2-6476 2-6477 0-00024 0-00072 CC14• 2-2386 2-2390 0·00031 0-00078 Allthesamples exceptthoseofn-hexane andCC14werespecially purified. Thelosstangent isconsiderably affectedbysmalltracesof polarimpurities, butitisnotcertainthatsuchimpurities wouldaccount forallthedielectric loss. 15.6.Measurement ofthevelocity ofradiowaves Thevelocityofelectromagnetic radiation haslongbeenregarded as oneofthefundamental constants ofphysics,andmuchefforthasbeen devotedtoitsaccurate determination. Apartfromonemeasurement ofthevelocityofradiowavesonatransmission linebyMercier(1924), mostoftheearlyworkhasusedlightwaves.Theresultsshowedagood dealofscatter,butinareviewbyBirge(1941)themeanvalueof 299776±4km/secwasadopted. From1945onwards anumberofnew determinations havebeenmade,ofgreateraccuracy, whichsuggestthat thetruevalueisnearly299793km/sec(seeTable15.2).Thesemethods havemadeuseofradiotechniques toimprove theaccuracy, andinsome casesthewavelength ofradiation usedhasbeenafewcentimetres. Abriefdescription isgivenbelow. In§15.4itwaspointedoutthatbothfrequency andwavelength can bemeasured atcentimetre wavelengths. Theproduct ofthesetwo quantities givesthewavevelocity, andthishasbeenthebasisofone typeofmeasurement attheNational Physical Laboratory. Itinvolves 446 ALTERNATING CURRENT MEASUREM}J~TS [15.6 theconstruction ofacavityresonator whoseresonant wavelength can becalculated fromtheinnerdimensions andwhoseresonant frequency canbedetermined bycomparison withafrequency standard. The dimensions weremeasured intheMetrology Department oftheN.P.L. IntheearlierworkofEssenandGordon-Smith(1948)acavityoffixed lengthwasemployed, consisting ofacoppercylinderofdiameter 7·4cm andlength8·5cm.Theresonant frequencies foranumberofdifferent modesweremeasured withtheevacuated resonator inatemperature controlled room,thefrequencies lyingbetween about3000and5000 Mc/s(wavelengths of10emand6cm).Thevelocity cmaybefound fromtheformula (15.7) wheref'istheobserved frequency ofresonance, LandDtheinternal lengthanddiameter, xisaconstant foraparticular mode(therootof aBesselfunction), nthenumberofhalf-wavelengths intheresonator, andQthequalityfactor.ThevalueofQwasabout15000andit appearsasasmallcorrection forthefiniteelectrical conductivity ofthe copperwalls.Theeffectofthismayberegarded asaneffective increase inthedimensions oftheorderoftheskindepthoftheradiation incopper. Theuseofseveralmodesofresonance isacheckon'end-effects', andthe changeintheresonant frequencycaused bytheintrusion ofthecoupling probesA,B(seeFig.15.22)wasdetermined. Thelengthoftheseprobes wasfinallyreduced beyondthepointatwhichanysuchchangecould beobserved. Themeasured valuesofthelengthLanddiameter Dwere accurate to3partsin106•Fourmeasurements ofclaybetween 299796 and299789km/sec,theaverage valuebeing299792±9 km/secwith aratherliberalestimate oftheerror. Inaseconddetermination Essen(1950)usedacavityresonator of variable lengthandmeasured thedistance required tomovebetween successive resonances. Thescatterinthesedistances (whichareeach halfaguidewavelength) wasabout±5X10-5cmwithatotaltravelof about12cm.Thisscatterispartlyduetovariations inthediameter (thoughnosystematic variation wasdetected) butalsoincludes errors arisingfromtemperature changes, frequency measurement, andsetting toresonance, givingaproportional errorincof3X10-6•Measurements weremadeat,..."6000,9000and11000Mc/s,andshowedasystematic decrease intheapparent valueofcwhentheresonant conditions were suchthatthediameter ofthecavityplayedagreaterpartindetermin­ ingtheguidewavelength. Sincethemeasured Qwaslowerthanthe 15.6] ALTERNATING CURRENT MEASUREMENTS 447 theoretical Q,itwasassumedthatasurfacefilmofpoorlyconducting tar­ nishedsilver(detectable byeye)causedtheeffective diameter tobegreater thanthemeasured diameter, sincether.f.currentrunsbeneaththisfilm. Themeasurements atdifferent frequencies madeitpossibletoapplyacor­ rectionforthis,andthefinalvalueofthevelocityinvacuowasfound tobe299792·5 km/sec,withamaximum errorof±3km/sec. R ~ Pumpo H.W.~~ F.S. FIG.15.22.Apparatus ofEssenandGordon-Smith formeasuring thevelocityof electromagnetic waves. A,B R Ta Lprobes. receiver. thermometer. cavityresonator. lagging.vo H.W. F.S.vacuum. klystron oscillator. heterodyne wavemeter. frequency standard. Thesedifficulties inthecavityresonator methodledFroome (1952) attheN.P.L.todeviseaninterferometer experiment at1·25emwave­ lengthwhichapproximates closelytoafreespacemethod. Thisusesa microwave analogue oftheMichelson interferometer, asshownin Fig.15.23.Powerfromastabilized klystron oscillator flowingalonga waveguide wasdividedintotwoportionsatahybridjunctionB(the analogue ofahalf-silvered plate).Onehalftraversed ashortlengthof waveguide andwasreflected fromashorting piston.Theotherwasfed toahornandlaunched asawaveinspace.Partofthisradiation was reflected backtothehornbya6-in.squaremetalplateMwhichcouldbe placedatpointsfrom6!to21!metresaway.Thisreflected waveon returning tothehybridjunction interferes withthatreflected fromthe shorting pistoninthesecondarm,andthevectorsumofthetwoampli­ tudesispassedalongthefourtharmtoadetector (asuperheterodyne 448 ALTERNATING CURRENT MEASUREMENTS [15.6 receiver). Thelatterisusedtodetectwhenthetworeflected wavesare exactlyinanti-phase andsogiveanullatthedetector. Themetalplate Misthenmovedthroughsuccessive nullpoints,whichoccureveryhalf­ wavelength. Thetotaldistance movedwas1·62metres,andthiscould bemeasured withanaccuracy of±O·003mm.Atthesametimethe frequency oftheklystron oscillator wasmeasured againstthequartz crystalstandard withanaccuracy of1partin108•Thusthewavelength Powerfromstabilized klystron oscillator. 1--+--1Quartzcrys!al frequency st>tlldard }lovable reflector drivenbyAlt-~-"'---­micrometerRadiator['-6!to21!metres-- ~ UrnMatching unit and B attenuatorWaveguide r------,~ L---_y HybridjunctionAdjustable shorting plunger FIG.15.23.Froome's microwave Michelson interferornpkr. inairandthefrequency weredetermined simultaneously. Intheformer casetwoimportant corrections mustbeappliedtofindthewavelength invacuo. (a)acorrection fortherefractive indexoftheair,basedonthe measurements ofEssenandFroome (see§15.5); (b)acorrection forthefactthatthewavefrontreaching themirror isnotaplane,buthasasmallcurvature, andsimilarly forthe reflected wave;thiscorrection wascalculated fromdiffraction theory,usingdatafromdifferent mirrordistances. Thefinalvalueobtained forthevelocityinvacuowas299792·6±O·7 km/sec.Inlaterexperiments (Froome, 1954,1958)hasusedafour-horn interferometer ofsymmetrical design,firstatawavelength of1·25em, thenat4mm.Thefinalresultsare 299792·75±0·3 km/sec 299792·5±O·1 km/sec(frequency 24000Mc/s), (frequency 72000Me/B). 15.6] ALTERNATING CURRENT MEASUREMENTS 449 Theseagreeverycloselywiththebestopticalmethodascanbeseen fromTable15.2.Adescription ofBergstrand's optical'geodimeter' andofFroome's laterinterferometer canbefoundinJ.H.Sanders, TheFundamental AtomicOonstants (OxfordUniversity Press,1961). TABLE15.2 Velocityofelectromagnetic waves Published r68ult Date Author (km/sec) Method 1941Birge 299776±4 Statistical surveyofearlierwork 1949Aslakson 299792·4±2·4 Radar,300Mc/s 1950ESBen 299792·5±3 Cavityresonator 1952Froome 299792·6±0·7 Microwave interferometer 1958Froome 299792'75±0'3 Ditto,24000Mc/s 299792·5±0·1 Ditto,72000Mc/s 1950Bergstrand 299792·9±0·25 Opticalgeodimeter 1957Bergstrand 299792·75±0·34 Ditto,averagewithearlierinstrument 299792·85±0·16 Ditto,averagewithlaterinstrument Selected values,basedonFroome (1952)andDumond (1959). REFERENCES ASLAKSON, C.r.,1949,Nature,Lond.164,711.--1951,ibid.168,505. BERGSTRAND, E.,1950,ArchivfurFysik,2,119. --1957, Ann.franc.Ohronom. 2,97. BmGE,R.T.,1941,Ann.Rep.Progr.Phys.,London, Physical Society, 8,90. BLEANEY, B.,LOUBSER, J.H.N.,andPENROSE, R.P.,1947,Proc.Phys.Soc. Lond.59,185. DUMOND, J.W.M.,1959,Ann.Phys.7,365. ESSEN,L.,1950,Proc.Roy.Soc.A,204,260.--andFROOME, K.D.,1951,Proc.Phys.Soc.B,64,862.--andGoRDON-SMITH, A.C.,1948,Proc.Roy.Soc.A,194,348. Faraday SocietyConference onDielectrics, 1946,Trans.Faraday Society, 42A. FROOME,K.D.,1952,Proc.Roy.Soc.A,213,123.--1954,ibid.223,195.--1958,ibid.247,109. HECTOR, L.G.,andWOERNLEY, D.L.,1946,Phys.Rev.69,101. LOVERING, W.F.,andWILTsHmE, L.,1951,Proc.I.E.E.98,PartII,557. MERCIER, J.,1924,J.Phys.Radium, 5,168. ROLLIN, B.V.,1964,AnIntroduction toElectronics (O.U.P.). 851110 Gg 450 ALTERNATING CURRENT MEASUREMENTS PROBLEMS 15.1.Acathode-ray tubehasplaneparalleldeflecting platesofseparation aand lengthbparalleltotheaxisofthetube;thedistance fromthecentreoftheplates tothescreenisL.Iftheelectrons areinitially accelerated byavoltageVo,show thattheirdeflexion onthescreenduetoavoltageVlonthedeflector platesis 8=!(LbVl/aVo), assuming thatL~b,thatthefieldisuniform between theplates,andthatedge effectscanbeneglected. Ifa=0'5em,b=4em,L=30em,andVo=1300V,showthatthedeflexion sensitivity is0·92mmjV. 15.2.Referring toFig.15.4,showthatthephaseangle e/>isgivenbytherelation sine/>=OP/OQ. 15.3.Ifthefrequency limitofthecathode-ray tubeofProblem 15.1weresetby thefinitetransittimeoftheelectrons throughthedeflector plates,showthatthe deflexion wouldfalltozeroatabout540Mc/s. 15.4.Ifinthebolometer ofFig.15.6alltheheatislostbyconduction tothe leads,whichremainatroomtemperature, showthatthefractional changein resistance (t:.R/R)whenad.c.powerWisdissipated inthethinwireisgivenby (t:.R/R)=(XWL/(12KA), where (Xisthetemperature coefficient ofresistivity, Lthelength,Kthethermal conductivity, andAthecross-section ofthewire. 15.5.Maxwell's bridgeforcomparing aninductance andacondenser hasthe circuitof:Fig.15.8,withthefollowing impedances: Zlaninductance Linserieswitharesistance Rr> Zsaresistance Rs' Zsaresistance Rs' Z4acapacitance 0inparallelwitharesistance R4• Showthatthebalance conditions are Rl/Rs=Rs/R4,L=RsRsO. Tomakethetwobalanceconditions independent, R4and0mnstbevaried. 15.6.Athigheraudiofrequencies resistances maypossessasmallinductive com­ ponent;inAnderson's bridgethismaybeallowedforbywritingthecomponents asP=P+jP',Q=Q+jQ', R=R+jR',T=T+jT'(weneglectanyinduc­ tivecomponent inSasthiswillbeaddedtoLatallfrequencies). IfPandQare identical impedances, showthatthebalance conditions are r+S=R-wO(2RT'+2R'T+RQ'+R'Q), L=O(2RT+QR-2R'T'-Q'R'HR'/w. Theseequations showthatitisimportant tomakeR'assmallaspossible.If R'=0,theerrorinthedetermination ofLiszero,whilethatintheresistance roftheinductance iswOR(2T'+Q'). ALTERNATING CURRENT MEASUREMENTS 451 15.7.Intheequivalent circuit(Fig.15.16)ofthecoaxiallinewavemeter, the sourceistakentobeagenerator ofvoltage VIwithinternal resistance R1,and thedetector hasaresistance R2•Iftheseriesimpedance ofthetunedcircuitby itselfisZ,showthattheratioofthevoltage V:;acrossthedetector totheinput voltageis ~=_ w2M1M2 VI Rl(Z+W2Ml/Rl+W2~/R2)" Thisequation showsthatV:;isamaximum whenZisaminimum, i.e.whenthe wave-meter isontuneandZisjusttheresistance r.Itshowsalsothatthecoupled impedances w2Ml/R1andW2~/R210wertheeffective Q;bywritingZ=r+2j'bwL nearresonance, showthatthe'loaded Q'=-vL/{-v0(r+w2Ml/Rl+w2~/R2)}' andthatitmaybemeasured byfindingthefractional changeinthefrequency required toreduceV:;to1/-v2ofitsmaximum value(neglect changesinthecoupled impedance whenvarying w). 15.8.Intheequivalent circuit(Fig.15.18b)ofaquartzcrystal,thecomponents foraparticular crystalareL=3.3henrys,0=0·042p.p.F,R=4500ohms, 01=5·8p.p.F.Showthatitbehaves asaparallelresonant circuitatafrequency approximately 8cyclesabovetheseriesresonance frequency (thenaturalmechani­ calresonance frequency). 16 FLUCTUATIONS ANDNOISE 16.1.Brownian motionandfluctuations THEirregular motionofsmallparticles suspended inafluidwasfirst observed byBrownin1828.This'Brownian motion'neverceasesand isaresultoftherandom motionofthemolecules bothoftheparticles themselves andofthefluid.Ifthemotionisobserved overalongtime, itisfoundthattheaveragecomponent ofthevelocity inanydirection iszero,sincepositiveandnegative valuesoccurwithequalprobability. Themeansquarevalueofthevelocity isnotzero,andfromclassical statistical mechanics itmaybeshownthattheaveragevalueofeachof theterms!mi:2,!m!j2,!mz2ofthetranslational kineticenergyis!kT, wherekisBoltzmann's constant (approximately 1·38X10-16ergsfdeg) andTistheabsolute temperature. Thisisaspecialcaseofthetheorem ofequipartition ofenergy:iftheenergyofasystemcanbewrittenasthe sumofanumberoftermseachcontaining onlythesquareofavariable, thentheaverage energyofeachofthesetermsis!kT.Thistheorem appliesjustasmuchtomacroscopic objectsastomicroscopic onesor molecules, butthemagnitude ofthefluctuations inthedynamical variable becomesmallerastheinertiaoftheobjectincreases, sincethe averageenergyisindependent ofsize.Givensufficient magnification, the motioncanalwaysbeobserved, anditsetsalimittothesensitivity of anymeasuring instrument, sincethefluctuations givearandom signal whichmasksanyappliedsignalofsmallermagnitude. Ifthistheorem isappliedtoasuspension galvanometer, thefollowing resultisobtained. Thesuspension hasonedegreeoffreedom, arotation measured bytheanglee.Thetotalenergymaybewrittenasthesum oftwoterms,thepotential energyofthesuspension duetoworkdone againsttherestoring torque,andthekineticenergy,sothat W=!ce2+!~B2, (16.1) wherecistherestoring torqueperunitangleoftwistand~isthemoment ofinertiaofthesystem. Toeachofthesetermswemustassignanaverage energy!kT,sothatfluctuations intheangleeandtheangularvelocityB willoccurwhosemeansquarevaluesaregivenby (16.2) 16.1] FLUCTUATIONS ANDNOISE 453 Asystemwhichismathematically similaristheelectrical tunedcir­ cuit,consisting ofaninductance, capacitance, andresistance connected together. Thetotalelectrical energyofsuchasystem,whereIisthe instantaneous currentandqtheinstantaneous chargeonthecapacitor, is W=!q2/0+!LI2. (16.3) Ifthetheorem ofequipartition ofenergyappliesalsotoelectrical sys­ tems,aswewouldexpectinviewofitsgeneralnature,thenthemean squarevaluesofthefluctuating chargeandcurrentwillbegivenby !q2/0=!LJ2=!kT. (16.4) Theserelations giveonlythemeansquarevaluesofthetotalfluctua­ tions,andtellusnothing aboutthefrequency distribution ofthe fluctuations. Ifweimagine thatweperform aFourier analysis of thefluctuations, andpostulate thattheyareduetosomerandomforce actingonthesystem,thenfortheelectrical tunedcircuitwewrite L(d2q/dt2)+R(dq/dt)+q/0 =Jjexp(jwt), (16.5) wherefJistheamplitude ofthecomponent oftherandome.m.f.causing thefluctuations atthefrequency1=W/27T.Wenowmakethefollowing assumptions aboutJj:itsmeansquarevalue11isindependent offre­ quency,butvoltagesofdifferent frequency areentirelyuncorrelated, so thattheaveragevalueoftheproductJjJj.iszero.Thejustification for theseassumptions willnotbediscussed here,butitisobviousthatthey areplausible inviewoftherandom natureofthefluctuations. On solvingequation (16.5)tofindthemeansquareamplitude qjofthe fluctuating chargeatthefrequencyI,wehave d(;:;2")- d(VJ) (166)qf-(Lw2-1/0)2+R2 w2' . Thefrequencies arecontinuously distributed, andthedifferentials are usedsincethisexpression givesthemeansquareamplitude ofthe fluctuations inthefrequency rangebetweenIandl+dl.Thetotalmean squarefluctuation mustbegivenbyequation (16.4),andhence,inte­ gratingoverallfrequencies, wemusthave co 1.kT=1.2/0-~fd(~)-_1_d(VJ)f dw • 22q-20qf-47T0dl(Lw2-1/0)2+R2w2 o Thisintegralmaybeevaluated asfollows. Onmakingthesubstitution w=x(LO)-!, itbecomes co co (03/L)!f-d(l/x) -(03/L)!f dx(x-1/x)2+R 20/L- (x-1/x)2+R2OfL' o 0 454 FLUCTUATIONS ANDNOISE [16.1 wherethesecond form isobtained byreplacingxby1Ix.Hencethe integralmaybewrittenas 00 00 i(C3jL)!f(X-~;:;;:~~2CjL =!(C3jL)!fZ2+~C(L =7TCj(2R). o -00 Hence or andl.kT=~2jC=~d(ry) 22q 8Rdf' d(lry)=4kTRdf, :::2 4kTRdf d(q,)=(Lw2-1jC)2+R2 w2'(16.7) (16.8) ThevalueoflLI2maybeshowntoequal!kT,asrequired byequation (16.4),fromtheseresults(seeProblem 16.1).Theequations leadto theinteresting resultthat,whereasthetotalmeansquarevaluesofthe fluctuations dependonlyonLandC,theexpression forthedistribution ofthevoltagefluctuations withfrequency involves onlyR.Theresult givenbyequation (16.7)maybeexpressed bysayingthatthemean squarevoltaged(VJ)ofthefluctuations inthefrequency rangedfis 4kTRdf,andisthusproportional tothebandwidth df.Theexistence ofsuchfluctuations wasfirstverifiedbyJohnson, andtheyareknown asresistance or'Johnson' noise.Theywillbeconsidered inmore detailin§16.3. 16.2.Fluctuations ingalvanometers Wereturnnowtothecaseofthegalvanometer, andconsider firsta moving-coil suspension galvanometer whenthecoilisonopencircuit. Thentheequation ofmotionis ':J(d2Bjdt2)+b(dBjdt)+cB =Pjexp(jwt), (16.9) where':Jisthemoment ofinertia,bthemechanical damping constant, andctherestoring torqueperunitangleoftwist.Weassumethatthe fluctuations arecausedbyarandomtorque,whoseFouriercomponent atthefrequency f=Wj27Thastheamplitude Pj.Ourfurtherpostulates aboutthenatureofFaresimilartothosemadeaboutVinthelast section. Thentheanalysis isexactlysimilartotheprevious caseofthe electrical tunedcircuit,sothatbycomparison weobtainatonce andd(Fj)=4kTbdf ii2 4kTbdf d(Bf)=(':Jw2-c)2+b2w2'(16.10) (16.1l) 16.2] FLUCTUATIONS ANDNOISE 455 (16.14)(16.13)Byintegration itmaybeshownthattheseexpressions satisfyequation (16.2). Ingeneralthegalvanometer willbeusedforobserving acurrentand willtherefore beconnected toacircuitwhosetotalresistance (including thegalvanometer coil)isR.Thenwehavetwoequations ::5(d20/dt2)+b(dO/dt)+cO =NI+Piexp(jwt)}(16.12)RI=-N(d(J/dt)+Vjexp(jwt+j3) , whereN=nAB,andIistheinstantaneous currentthroughthecircuit. Twosourcesoffluctuations havebeenincluded; arandomtorquedueto Brownian motionofthesuspended coil,andarandomvoltageassociated withtheelectrical circuit.Inequations (16.12)theFouriercomponents ofthesetwosourcesoffluctuations atthefrequencyJ=Wj27Thavebeen used,withaphasedifference 3between them.Sincethetwosourcesare independent, wedonotexpectanycorrelation inphase,andfordifferent frequencies thephasedifference 3willhaverandomvalues.Elimination ofthecurrentIbetween thetwoequations gives ::5(d20jdt2)+(b+N2jR)(d(J/dt)+c(J =(N/R)Vjexp(jwt+jS)+Piexp(jwt), andthesquareoftheamplitude ofthefluctuations atthefrequencyJ isfoundtobe (J2_(N/R)2V,+F'+2(NjR)V,Picos3 f-(::5w2_C)2+(b+N2/R)2 w2. Onsumming overarangeoffrequencies, 3takesallvaluesbetween 0 and27Tandthemeanvalueofcos3istherefore zero.Hencethemean squareangularamplitude inthefrequency rangeJtoJ+dJis d«(J2)={(NjR)2d(VJ)}+d(FJ) • f(::5w2-e)2+(b+N2jR)2 w2 Onsubstituting theexpressions ford("VJ)andd(FJ)givenbyequations (16.7)and(16.10),wefind (J2_4kT(b+N2/R)dJ d(f)-(::5w2-e)2+(b+N2jR)2 w2· Thisequation issimilartothatobtained forthegalvanometer onopen circuitexceptthatthetotaldamping constant (b+N2/R)appearsinstead ofjustthemechanical damping b.Integration ofequation (16.14)over allfrequencies willobviously givethesameresult,ieO!=IkT,asfor thegalvanometer onopencircuit,sincetheresultisindependent ofthe magnitude ofthedamping. Thus,although therearenowtwoinde­ pendent sourcesofrandom fluctuations, andtheseaddinthesquares 456 FLUCTUATIONS ANDNOISE [16.2 asshownbythenumerator ofequation (16.13),eachisassociated witha damping termsothatthetotalmeanenergy!c82storedinthesuspension remainsunaltered, provided thateachsourceisatthesametemperature. Thisargument couldbeextended byseparating themechanical damping bintotwoparts,oneduetoimperfect elasticity ofthesuspension and theothertodamping bytheviscosity oftheair.Thenitfollowsthatthe totalmeansquareangular fluctuations havethesamevaluewhether thegalvanometer isevacuated ornot;theadmission ofairprovides an extrasourceoffluctuations owingtothemolecular bombardment whose tendency toincreasethemeansquaredeflexion isjustcounterbalanced bytheviscousairdamping whichaccompanies it.Thefrequency distri­ butionofthefluctuations isofcoursechanged becauseoftheincrease inthedamping, butitisimportant torealizethattheBrownian motion isinherent inthesuspended coilandisnotcausedbythebombardment bythegasmolecules.Ifitwere,andthesuspension hadanimperfect elasticity, thenthemolecular bombardment wouldresultinthesus­ pensionbeingheated,throughthedissipation ofenergyinit,andthe gaswouldbecooled,eventhoughbothwereoriginally atthesametem­ perature. Thisiscontrary tothesecondlawofthermodynamics. Theprocesses whichweregardas'damping' inthegalvanometer represent adegradation ofmechanical energyintoheatenergy;in viscousdamping, intokineticenergyofthegasmolecules; inelectro­ magnetic damping, ultimately intothevibrational energyofthelattice oftheresistance intheexternal circuit(thecoilmovinginthemagnetic fieldactsasatransducer, converting mechanical motionintoelectrical voltage). Atthelevelofthemolecular fluctuations, thedamping pro­ cessesarejustthemechanisms bywhichthermal equilibrium isestab­ lished;withoutthem,anindividual component ofthesystem(galvano­ metersuspension, gasmolecules, latticeoftheresistor) wouldhaveno meansofknowing whatthetemperatures areoftheothercomponents. Intheelectrical case,resistance arisesfromtheconversion ofelectrical energyintoheatenergy,andatthefluctuation levelisthemechanism bywhichtheelectrical fluctuations reachthermal equilibrium withthe latticefluctuations. Thenatureofthecarriersoftheelectriccurrentis nomoreimportant inthisprocessthanthatofthemolecules ofthegas causingviscousdamping. Itisconvenient todefinetheminimum observable currentfora galvanometer asthatcurrentwhichwouldproduce adefiexion equal totherootmeansquarevalueofthetotalBrownian angularmotion. ForasteadycurrentIthedeflexion ()=I(nAB)jc =IN/c,andhence 16.2] FLUCTUATIONS ANDNOISE 457 theminimum observable current1mwouldbe 1m=(ckT)I/N. (16.15) Ingeneraltheelectromagnetic damping term(N2/R)ismuchlarger thanthemechanical damping termb,andthecriticaldamping resistance Rcisgivenbyequation (7.5), Rc=tN2j(:Jc)l, whiletheperiodT=27T(:JjC)t. Usingthesetworelations theminimum observable currentandvoltageareconveniently expressed intheform 1m=(7TkTjRcT)I,Vm=(7TkTRcIT)I, (16.16) sinceVm=Rc1mifthegalvanometer iscritically damped. Takingroom temperature as2900K,sothatkT=4X10-21joules,foragalvanometer ofperiod2secandcriticaldamping resistance 100ohms,wefindthat theminimum observable current andvoltage areapproximately 8X10-12Aand8X10-10V. Thecorrectness oftheexpressions derivedabovehasbeenverified experimentally byJonesandMcCombie (1952).Thedeflexions ofan ordinary galvanometer ofabout2secperiod(sensitivity 1mmdeflexion at1metredistance for10-8A)weremagnified byanopticallever.The beamoflightreflected fromthegalvanometer mirrorfellonasplit photocell, sothatrotation ofthemirrortransferred lightfromonecell totheother.Thedifference inthecurrents fromthetwophotocells was observed onasecondgalvanometer; adeflexion of15mmonthisinstru­ mentcorresponded toavoltageofabout10-9V(oracurrentof10-11A) appliedtothefirstgalvanometer. Tomakeuseofthisamplification, allexternal sourcesofdisturbance suchasvibration hadtobeeliminated. Typicaltracesobtained weresimilartothoseshowninFig.16.1.With thefirstgalvanometer onopencircuitthedamping issmall,andthe frequency distribution oftheangular deflexions islargeonlyinthe regionaroundthenormalfrequency ofthesuspension. Consequently thefluctuations resemble burstsofoscillation atthenaturalfrequency, thenumberofoscillations ineachbeingininverseratiotothedamping (androughlyequaltothe'Q'ofthesuspension). Whenthegalvanometer isjustcritically damped, (b+N2/R)2 =4:Jcandthedenominator of equation (16.14)canbewrittenas(:Jw2+C)2,showingthatthefrequency distribution ofthefluctuations nowhasitsmaximum valueatzero frequency. Theappearance ofthefluctuations isnowthatofarandom disturbance withoutanysinusoidal character (Fig.16.1(b)).Thevoltage sensitivity ofthesystemwasfoundbyapplying avoltageofabout10-8V, 458 FLUCTUATIONS ANDNOISE [16.2 obtained byattenuating aknownvoltage ~1 Vthrough aresistance chain,andathorough statistical analysisoftheresultsshowedthatthe magnitude ofthefluctuations agreedwiththetheoretical valuewithin 1percent. \~ FIG.16.1.Fluctuations ofagalvanometer (afterJonesandMcCombie, 1952). (a)Onopencircuit. (b)Nearlycritically damped. 16.3.Therelation between resistance noiseandthermal radia­ tion Inanevacuated enclosure containing thermalradiation atanabsolute temperature Ttheenergydensityinthefrequency rangeftof+dJis givenbyPlanck's law dU-87Thrdf (1617) - c3{exp(hf/kT)-1}' . wherehisPlanck's constant andkisBoltzmann's constant. Forall radiofrequencies hJ~kTatroomtemperature, since290kcorresponds toaquantum ofenergyforawavelength ofapproximately 21iOem.We maytherefore expandtheexponential, obtaining dU=87TJ2kTdf/c3, (16.18) whichissimplytheRayleigh-Jeans lawofclassical theory. Sincethe polarization oftheradiation israndom, ontheaverageonlyone-third ofthisenergycorresponds toradiation whoseelectricvectorisparallel toagivendirection (saythey-axis),andonlysuchradiation willinduce avoltageinashortdipoleaerialinserted intheenclosure parallelto they-axis.From§10.3themeansquareelectricfieldcomponent isthen givenbyE~=cZo(!U), whereZo=(fLo/Eo)!istheintrinsic impedance offreespace.Hencethemeansquarevoltageinduced inanaerialof lengthswillbe d(V~)=82d(E~) =87TJ2S2kTZodf/(3c2). (16.19) 16.3] FLUCTUATIONS ANDNOISE 459 Eveniftheaerialconsistsofaperfectly conducting wire,theresulting currentwhichflowswillbefinite,sinceenergywillbere-radiated by thisoscillatory current, andthisenergymustjustbeequaltothat pickedupbytheaerial.Theradiation musttherefore behaveasa generator ofopen-circuit voltagev,:withaninternal impedance R,.,as intheequivalent circuitofFig.16.2(a).ThisdrivesacurrentI,.when short-circuited, andthepowerdissipated isV~jR,.=I~R,.;thispower islostbyre-radiation, andfrom§10.9itfollowsthatR,.isjustthe radiation resistance givenbyequation (10.71)as R,.=27TZof282j(3c2). (16.20) Iftheaerialisnotaperfectly conducting wire,andhasarealohmic resistance R,theequivalent circuitwillbeasshowninFig.16.2(b),and (a) (b)R (c) FIG.16.2.Equivalent circuitofanaerial.Vr•voltageinducedbythermal radiation; Rr,radiation resistance ofaerial. (a)Aerialshort-circuited atcentre. (b)Aerialwithresistance Ratcentre. (c)As(b)butshowing noisevoltageduetoR. theenergydissipated intheloadRwillbeV~Rj(R,.+R)2. Thiswillheat theresistance R,whilelessenergyisre-radiated totheenclosure.IfR isinitiallyatthesametemperature Tastheradiation intheenclosure, theapparent resultwillbethatRisheatedandtheenclosure cooled, whichiscontrary tothesecondlawofthermodynamics. Inorderthat thenetexchange ofenergybetweenRandtheenclosure bezero,wemust postulate thatthereisafluctuation voltageassociated withR,asin Fig.16.2(c),ofmeansquarevoltageV2andinternalresistance R.This willsendapowerPR,.j(R,.+R)2 backintotheaerialwhichmustjust equalthatdrawnfromtheenclosure anddissipated inR.Thus V~R=V2R,.,andinthefrequency rangefromftof+df d(V2)jR =d(V2)jR_87Tj2s2kTZod/3c2=4kTdf,.,.- 3c2 X27TZof282 . (16.21) (16.22)460 FLUCTUATIONS ANDNOISE [16.3 Thisresultisidentical withthatobtained earlier(equation (16.7»by considering asimpletunedcircuit. Thevoltagefluctuations havea constant distribution withfrequency solongastheenergyquantum hf~kT;thislimitation corresponds toouruseoftheclassical expression (Rayleigh-Jeans law)fortheenergydensityintheenclosure. The fluctuations associated witharesistance Rcanberepresented byinsert­ ingavoltagegenerator V,whosemeansquareopen-circuit voltageis givenbyequation (16.21),forwhichRactsastheinternal impedance asinFig.16.2(c).Theequivalent currentgenerator willhaveamean squarecurrentd(12)=4kTdf/R anditwillbeshuntedbytheresistance R. Letussupposethatweareabletoconnecttoouraerialaloadof resistance Rwhichitselfproduces nonoise(e.g.aresistance keptata temperature verycloseto0°K).Thenthemaximum powerwhichcan bedrawnfromtheenclosure anddissipated inR,obtained bymaking RequaltoRr,isd(V~)/(4Rr) =kTdf;thisisthe'available noisepower'. IfRisinfactaradioreceiver, thispowerdrawnfromthethermal radiation incident ontheaerialformsasourceof'noise',andcanbe heardasahissfromaloudspeaker, orviewedonacathode-ray oscillo­ graph.Itwillobscureanysignalwhichitisdesiredtoreceiveunless thesignalpowerintheaerialislargerthanthatpickedupfromthe radiation background. Thisdifficulty cannotbeovercome byincreasing theoverallamplification ofthereceiver, sincebothnoiseandsignalwill beamplified together. Thustheradiation noisesetsalimittotheuseful sensitivity ofareceiver.Ifatheoretically perfectreceiver isdefinedas onewhichitselfintroduces nonoise,thentheamplified noiseoutputwill beAkTdf,whereAistheoverallamplification. Theamplified signal outputwillbeAP,wherePisthesignalpowerincident ontheaerial. Thentheminimum detectable inputsignalmaybedefinedasthatwhich givesasignaloutputequaltothenoiseoutput,fromwhich minimum detectable signalpowerPo=kTIi!(16.23) foraperfectreceiver.Itisclearthattheonlyvariableatourdisposal hereisthebandwidth df,andthereduction innoiseobtained onnarrow­ ingthebandwidth canbeseeninFig.16.3.Thisshowsthenoiseoutput fromareceiver covering abandfrom0to2Mc/s,beforeandafterthe insertion ofalow-pass filtercuttingoutfrequencies above0·1Mc/s. Thechangeincharacter ofthenoisewhenthehighfrequency components areabsentcanbeseenaswellasthereduction inamplitude. Ingeneral, however, thebandwidth cannotbereduced beyond acertainlimit 16.3] FLUCTUATIONS ANDNOISE 461 withoutimpairing thequalityofthereception, sincethehigher modula~ tionfrequencies willbecutout.Ifonlyaudio-frequency modulation is involved, thebandwidth willbeabout104cfsandtheminimum detect­ ablesignalpowerwillbe4X10-17W.Inatelevision receiveritis necessary tohaveabandwidth of~4Mcfstoincludealltheinformation necessary toformthepicture,andtheminimum signalpowertoequal noiseinaperfectreceiver is1·6X10-14W. (Photograph byL.J.Arundel.) FIG.16.3.Noiseoutputfromanaperiodic amplifier. (a)Covering thebandfrom0to2Me/s. (b)Afterinsertion ofalow-pass filterreducing thebandto0to0·1Mc/s. Inpractice, all'receivers generate acertainamountofinternal noise, withtheresultthatthenoiseoutputisgreaterthanforaperfectreceiver. Thesignalinput~required togiveasignaloutputequaltothenoise outputistherefore greater than~. Thequantity ~-Poisameasure oftheinternal noisegenerated inthereceiver, andbywriting ~-Po =kTedf itmaybeexpressed intermsofthe'excessnoisetemperature' Teofthe receiver. Inanidealreceiver Te=0,butinpractice littleisgainedby makingitsmallerthanaboutTJ10,whereTisthetemperature ofthe thermalradiation beingreceived intheapplication forwhichthereceiver 462 FLUCTUATIONS ANDNOISE [16.3 isdesigned. Inlaboratory applications thesourcetowhichthereceiver isconnected isgenerally atroomtemperature, anditisthenconvenient totakeavalueofT=2900KtodefinePo,makingitequalto4X10-21df (watts). Theratioof.lltothisvalueofPoisthendefinedasthe'noise figure'ofthereceiver, anddenotedbyF.SinceFisaratiooftwo powers,itisoftenexpressed indecibels. 16.4.Shotnoise Formostpurposes itissufficient toconsider theelectron currentin atubeasconsisting ofauniformflowofchargetotheanode.Sincethe currentconsistsinfactofthearrivalofafinitenumber ofelectrons persecond,thiscannotbetrue.Theflowofelectrons isarandom process,andwemayexpectthattherewillbeafluctuation inthenumber arriving inagiventimeinterval, ifwemeasure overanumberofsuch intervals.Ifthearrivaloftheelectrons consists ofasuccession of completely random events,thenthemeansquaredeviation fromthe averagenumberNpersecondisproportional toN.Thesefluctuations giverisetonoiseintheanodecircuitofthetube,knownasshotnoise fromtheobviousanalogywiththerandompatterofshotonatarget. Ingeneralweareinterested notinthetotaldeviation fromthemean, butinthefrequency distribution ofthefluctuations. Tofindthisitis necessary tocarryoutaFourieranalysisofthepulseofcurrentdueto thearrivalofasingleelectron ofchargee.Wewillassumethatthis pulse,occurring attimet=0,hassomeirregular shapebutisentirely confined withinthetimeinterval-T{2to+T{2.Sincethetotalcharge arriving ise,wehave +7{2 e=IIdt.. -7/2 Wedonotspecifyanything abouttheduration ofthepulse Texcept thatitisveryshort(-..thetransittime,see§14.2).TheFourierseries representing thefrequency distribution ofthecurrentduetothearrival ofeiswritten ~ 27Tnt~.27T1/t1=ao+L,ancos-p+L,bnsmT' n=1 n=1 Hereao,an'andbnarecoefficients tobedetermined, and']'isanundefined largeintervaloftime.Ineffectweregardallthefrequencies weare interested inasmultiples ofthefundamental frequency 1/T.Toobtain acontinuous frequency distribution weshouldmakeTinfinite,and replacethesummations intheseriesbyintegrations. Asthestudentis 16.4] FLUCTUATIONS ANDNOISE 463 likelytobemorefamiliar withaFourierseriesthanaFourierintegral weshallusetheformer,andbymakingTlargewecanobtainagood approximation toacontinuous frequency distribution. Fromtheordinary formulae ofFourieranalysis +T/2 +T/2 ao=~f1dt; an=~f1cos(2-rmt/T) dt; -T/2 -T/2+T/2 bn=~f1sin(2-rmt/T) dt. -T/2 Toevaluate thecoefficients werestrictourselves tofrequencies small compared withI/T.Then,sincethecurrentisfiniteonlyintherange -T/2to+T/2,andzerooutsidethisrange,wecanwritecos(2-rmt/T) =I andsin(2-rmt/T) =0overtherangeofintegration forwhichthecurrent isfinite.Hencebniszero,while Thuswehave d(12)=2e10df (16.24) forthemeansquarecurrentfluctuation inthefrequency rangeftof+df.+T/2 2ao=an=~f1dt=2e/T. -T/2 e~2e1=T+~Tcos(2-rmt/T) 110=1 andthemeansquarevalueofthenthcomponent is 1;=t(2e/T)2=2e2/T2. IfNelectrons arriveintimeT,theneachwillcontribute anequalamount tothevalueof1~.(Theelectrons arriveatrandom times,andtheir contributions totheFourierserieswillalldifferslightlyinphase.Thus wemustaddintensities, andnotamplitudes.) Then 1;=2e2NjT2=2e10/T, where10=Ne/Tisthemeanvalueofthecurrent. Nowthenumberof Fouriercomponents whosefrequencies liewithinarangebetweenf and f+dfisTdf,sincethecomponents areequallyspacedinfrequency by amounts(I/T).Addingtogether themeansquarevaluesofthesecom­ ponents gives 1nfluenceofspacecharge Inthisderivation oftheformula forshotnoisethearrivalofan electron isconsidered asarandomevent,completely independent ofthe arrivalofanyotherelectron. Weexpectthattheemission ofelectrons 464 FLUCTUATIONS ANDNOISE [16.4 fromthecathodehasthisproperty ofcomplete randomness, butthisis notnecessarily trueoftheirarrivalatanotherelectrode. Inpracticeit isfoundthatthevalueoftheshotnoiseismaterially lowerthanthat givenbytheaboveequation unlessthecurrenttotheanodeislimited onlybytherateofemission fromthecathode. Ingeneraltheanode currentisonlyafractionoftheemission currentbecauseoftheforma­ tionof'spacecharge'outsidethecathodewhichcausesalargenumber ofelectrons emittedfromthecathodetobeturnedbacktothecathode. Sincethisisduetothemutualinteraction oftheelectrons, wemay expectthattheflowofelectrons totheanodeisnotnowasuccession ofcompletely randomevents.Thevalueofthefluctuations isgreatest forrandom events,andfallsassoonastheybecomenotcompletely random. Physically, theactionofthespacechargemaybeenvisaged as follows. Supposethatatsomeinstantthenumberofelectrons emitted fromthecathoderisesmomentarily abovetheaverage. Thiswillcause atemporary increase inthespacecharge,andanumberofelectrons greaterthanaverage willleavethespacechargeregionfortheanode. Thisnumberissmallerthanthesurgefromthecathodebecausethespace chargeactsasareservoir; theeffectoftheincreased space charge isto turnsomeoftheexcesselectrons backtothecathode. Similarly, at instantswhenthecathodeemission fallsmomentarily belowtheaverage, thespacechargealsodropsandlesselectrons areturnedback.Toallow forthis'spacechargesmoothing', asitiscalled,afactorisinsertedin theequation fortheshotnoise.Thus (16.25) f1iscalledthespacecharge smoothing factor,andmaybeaslowas0·03, showingthatthesmoothing effectisveryconsiderable. Noiseinmulti-electrode tubes Thepresence ofgridsinatubedoesnotaffectthevalidityofthe equations givenaboveforshotnoisesolongastheydonotintercept any ofthecurrenton)tswaytotheanode.Thusequation (16.25)isstill validforanegative-grid triode,butthisisnotsoforascreen-grid tube orapentode, forthenthepositive screengridintercepts aconsiderable portionoftheanodecurrent. Sincethechanceofanelectron hitting thewireofthescreengridispurelyrandom, thescreencurrentwillhave thefullshotnoiseappropriate toitsmagnitude. Itisobviousthatsimilar fluctuations, thoughofopposite sign,mustbeimposed onthecurrent thatgoesthroughthescreengridtotheanode.Assuming thatlessthan 16.4] FLUCTUATIONS ANDNOISE 465 halfofthetotalcurrentgoestothescreen,wemaywriteapproximately fortheanodecurrent d(I2)=2Peladf+2e~df. (16.26) Sincepmaybelessthan0·1,whilethescreencurrentIsis0·2or0·3of la'thesecondtermisoftenmoreimportant thanthefirst.Hencescreen­ gridtubesandpentodes aregenerally morenoisythantriodes. The additional noiseiscalled'partition noise'.Insomehighfrequency pentodes anattempt ismadetoreducepartition noisebyincorporating anextragrid,carefully woundandplacedsothatitswiresareexactly infrontofthescreen-grid wires.Thisextragridiskeptatapotential negative withrespecttothecathode, sothatelectrons ontheirwayto theanodemustgothroughtheholesinthisgridanditcollectsnocurrent. Sincetheseholesareexactlyinfrontofthoseinthescreengrid,the electrons shootthroughthescreengridalso,andthescreencurrentis materially reduced, withacorresponding reduction inpartition noise. Itisoftenconvenient todefinetheamountofnoisebyreferring itto anequivalent resistance Rn(at2900K)inthegridcircuit.Thefluctuat­ ingvoltageatthegridduetoRnhasthemeansquarevalue d(V2)=4leTRndf sincethegridconsumes nopowerandtheequivalent noiseresistance is therefore onopencircuit.Thiscausesafluctuating anodecurrentwhose meansquarevalueis d(12)=U~d(V2)=U~4kTRndf, whereUmisthemutualconductance ofthetube.If12isduetotheshot noise,theequivalent noiseresistance maybecalculated bymeansof thisformula, Tbeingtakenasroomtemperature. Theadvantage ofthis methodofspecifying thenoiseisthatthevalueofRn,unlikethatof d(I2),isindependent ofthebandwidth, anditfacilitates comparison oftheshotnoisewiththeresistance noiseinthecircuitsattached to thegrid.Ifpartition noiseisincluded byreplacing equation (16.26)by (16.25)withaneffective valueP'insteadofp, Rn=p'elo/(2u~kT). (16.27) Anestimate oftherelativeimportance ofshotnoiseandresistance noisecanbeobtained fromtheformulafortheequivalent noiseresistance. Foratypicaltriode,Um=5rnA/V,P=0·03,10=lOrnA,e=1·6X10-19 coulombs; thisgivesRn=240ohms.Thevalueforapentode wouldbe somewhat higher,owingtopartition noise.Thesefiguresapplyat medium radiofrequencies (i.e.oftheorderofMc/s);athigherfrequencies 851110 Hh 466 FLUCTUATIONS ANDNOISE [16.4 (100Mc/sandup)Rnrisesowingtonoisevoltages inducedinthegrid whichhaveperiodofoscillation comparable withtheelectron transit time(cf.§14.2).Ataudiofrequencies theshotnoise(particularly from tubeswithoxide-coated cathodes) becomes abnormally large.Thisis knownastheflickereffect,andisthoughttobeassociated withchanges inthestateofthecathodesurfacewhichcauseabnormal fluctuations intheanodecurrent. 16.5. Desi~nofreceivers foroptimum performance (minimum noise fi~ure) Thecorrectdesignofareceiver isofgreatimportance. Ifitisbeing usedinanapplication wherethesignalstrength isfixed,suchasinr.f. astronomy orspectroscopy, thenthelimiting sensitivity attainable will dependentirelyonthedesignofthereceiver. Inradiocommunications animprovement ofafactorninsignal/noise ratiocanbeachieved by increasing thetransmitter powerbyafactorn,butaverymuchmore economical methodistoimprove thereceiverperfo;rmance bythesame factorinstead. Thefollowing remarksillustrate onl§thebasicprinciples, anddonotgointoanydetailofreceiverdesign. Ingeneralallthestagesofareceiverwillcontribute somenoise,but iftheamplification ofeachstageishighonlythefirststageortwois important. IfstagekgivesnoisepowerNk•andthestagegainism,then thesignal/noise ratioafternstagesis Smn/(N;.mn+N2mn-I+ ...+Nn)=S/(N1+N2m-I+Nam-2+...). (16.28) Withastagegainoftentoahundred eventhesecondstagewillcon­ tributelittletothenoiseoutput.Ifnot,thedesignofthesecondstage shouldfollowthesameprinciples asthatofthefirststage,andonlythe latterneedbeconsidered. InthecircuitofFig.16.4(a)Srepresents asignalsourceofvoltageS withoutputresistance R1•R1isassumed tobenoisy,attemperature T, anditsequivalent noisevoltageisrepresented byVnt>inserieswithS. ThesourceSmaybeasignalinducedinanaerial,inwhichcaseR1is theradiation resistance oftheaerialandTistheambient temperature whichwetaketobe2900K.Thesourceisconnected tothegridofthe tube,andR2isthegrid-bias resistance, orthefirsttunedcircuit,in whichcaseR2isitsparallelimpedance. IngeneralR2willalsogenerate resistance noise,whichisrepresented bytheinsertion ofavoltagesource Vn2inserieswithR2•Inthefirstinstance weshallassumethatthetube 16.5] FLUCTUATIONS ANDNOISE 467 contributes noshotnoise(Rn=0),andthatnoisefromsubsequent stagesisnegligible. Thenthesignalfnoise ratiowillbethesameatthe gridofthefirsttubeasatanylaterpointinthereceiver, andweneed onlycompute theratiooft.hemeansquaresignalvoltageonthegridto themeansquarenoisevoltage. Forsimplicity R1andR2aretakento havethesametemperature, whichinpracticewillnotbefarfromtrue. (a) (b) (16.30)FIG.16.4.(a)Equivalent inputcircuitofareceiver, showing noisevoltages. (b)Actualinputcircuit,showing aerialtappedontoinductance ofinput tunedcircuit. Since V~lfR1=V~2fR2=4kTdf,andR2actsastheloadforthenoise gene:rator ~l'andR1astheloadfor~2'themeansquarenoisevoltage onthegridis 4kTdf{R R~RRi}_kTdfR1R2 'J1(R1+R2)2+2(R1+R2)2-4'JXR1+R2' (16.29) SinceR1andR2arerandom noisesources,themeansquarevoltages havebeenadded;notethattheresultisthesameasthatforaresistance equaltoR1andR2inparallel, asweshouldexpect. Themeansquaresignalvoltageonthegridis 82R~f(R1+R2)2. Hencethesignaltonoiseratioatthegridis 82R21P R2 4kTdfR1+R2R1=kTdfR1+R2' whereP=82f(4R1)istheavailable signalpower.Ifequation (16.30)is putequaltounity,weobtainthesignalpower~required togivean 468 FLUCTUATIONS ANDNOISE [16.5 outputpowerequaltothenoiseoutputpowerofthereceiver. Thenoise figureFisdefinedastheratioofthissignalpowertothevaluekTdf foraperfectreceiver, andhencethenoisefigureis F=~/kTdf= (R1+R2)/R2• (16.31) Iftheaerialismatched tothefirstcircuit,R1=R2,andF=2.But ifR2>Rl>Fisreducedandtendstoitslimitingvalueofunityasthe ratioofR2toR1isincreased indefinitely. Hencetoobtainoptimum sensitivity itpaystomismatch theaerialtothereceiver, sincethe reduction innoiseatthegridwhenR2isshuntedbythelowerresistance R1isgreaterthanthelossofsignalatthegridduetothemismatch. Weseealsothat,intheabsenceoftubenoise,itispossibletoapproach verycloselytothetheoretical limitofsensitivity. Inapractical caseR1 (forahalf-wave dipoleaerial)wouldbe80ohms,whileR2couldbeof theorderof100000ohms,givingF=1·00l. Thisoptimum cannolongerbeattained iftubenoiseisappreciable. Inthiscaseitisnotsufficient tocompute thesignal/noise ratioatthe firstgrid,sincethereisalatersourceofnoise.Sincethissourceofnoise canberepresented asameansquarecurrentfluctuation inthetube, theanalysis needonlybecarriedonestepfurtherbytransforming any fluctuating voltageatthegridintoafluctuating anodecurrent. The effectoftheanodeloadontheanodecurrentneednotbeincluded since itaffectsallfluctuations inthiscurrentequally, whatever theirsource. Wehave: meansquaresignalcurrent =g~82R~/(Rl+R2)2, meansquarenoisecurrent=2[:3'eIdf+g~4kTdfRIR2/(RI+R2) =g24kTdf(R+_R1R2__) m nR1+R2' onsubstituting theequivalent noiseresistance ofthetube.Thesignal tonoiseratioatthisstageisnow P R 1R~[RR1R2]-1 kTdf(RI+R2)2n+R1+R2' wheretheavailable signalpowerhasbeenintroduced asbefore.Putting thesignaltonoiseratioequaltounity,wefindthenoisefigureFis (aftersomereduction) F=k;df= (~:+1){(~1+~jRn+l}. (16.32) IfR1isfixedandR2istheonlyvariable, thenthesmallest valueforF isobtained bymakingR2verylarge,whenF=1+Rn/RI•Foratypical 16.5] FLUCTUATIONS ANDNOISE 469 pentode, Rnisoftheorderof800ohms,andifR1is80ohms,wehave anoisefactorofII,whichisverypoorincomparison withthatobtained intheabsenceoftubenoise.Clearlythetroubleisduetothesmallvalue ofR1compared withRn,andthissuggeststhatweshoulduseatrans­ former between theaerialandthegridinordertostepupthevalue ofR1asseenfromthegrid.Thiswill,however, reducetheratioofR2 toR1,sothattherewillbesomeoptimum transformer ratio.Inpractice, thegridcircuitwillprobably beaparalleltunedcircuit,withtheaerial tappedintotheinductance asinFig.16.4(b).Ifthistappingpointis variable, thenatthegridtheequivalent circuitisasassumed, witha generator ofthesameavailable powerbutwithavariable internal impedance depending onthepositionofthetapping. Thismeansthat ourvariable isRvwhileR2isfixedastheparallelimpedance ofthetuned circuitwithouttheaerialbeingattached. Differentiating theexpression forFwithrespecttoR1wefindthattheoptimum valueoccurswhen R21 1 )R2=If+R• (16.33 1n2 IfR2ismuchlargerthanRn,thisreducestoR1="j(RnR2).Withthe valuesassumed previously (Rn=800ohms,R2=100000ohms),this givesR1=9000ohms,andtheoptimum valueofFisnow1·19.Though slightlyworsethaninthecaseofnotubenoise,itwillbeseenthatthis valueofFisverymuchbetterthanthatobtained previously bytapping theaerialrightacrossthetunedcircuit(R2).Ifthetappinghadbeen adjusted toobtainthemaximum signalvoltageonthegridbymatching theaerialtothetunedcircuit(R1=R2),thevalueofFwouldhave been2·03.Hencewehavegainedafactorofnearly2byover-coupling theaerial,justasinthecaseofnotubenoise.Thechiefdifference whentubenoiseispresentisthattheover-coupling mustnotbecarried sofarthatthenetimpedance ofaerial+tuned circuitbecomes lower thantheequivalent noiseresistance. Notethat,intheequivalent circuit, Rniseffectively inserieswith(R1inparallelwithR2):Sincenogrid currentflows,Rnmaybeinsertedintheleadimmediately attached to thegridasshowninFig.16.4(a),without affecting anyoftheother voltages imposed onthegrid. Thenoisefiguresderivedinthissectionapplytoreceivers atordinary radiofrequencies usingvacuum tubes;transistors (discussed in§19.8) alsoshowshotnoiseduetotherandommotionofthechargecarriers, andthenoiseproblems involved arebasically similar. Athigher frequencies thenoiseproperties ofvacuum tubesdeteriorate, though 470 FLUCTUATIONS ANDNOISE [16.5 travelling wavetubescangivenoisefiguresaslowas6dBatcentimetre wavelengths. Wavelengths ofthisorderareusedinradarandsatellite communication inordertoobtainhigWydirectional antennae; these pointattheopenskyandthebackground thermalradiation corresponds toatemperature ofafewdegreesabsolute. Thismakesitworthwhile tousereceivers ofverylownoise;thisisachieved inspecialdevices whereshotnoisehasbeeneliminated, andresistive elements areabsent orataverylowtemperature. Theparametric amplifier makesuseof anon-linear reactance, andthesolidstatemaserofaparamagnetic material inwhichanegative resistance isproduced atliquidhelium temperatures. Ineachcasesufficient amplification mustbeproduced tomakenoisefromthelater(conventional) stagesunimportant. 16.6.Measurement ofreceiver noise Although itisinprinciple possibletocalculate theconditions for optimum noisefigure,itisalwaysnecessary inpractice tohavesome methodofmeasuring thenoisefigureinordertobesureoftheper­ formance ofareceiver. Avacuumtubemaydeteriorate inuse,sothat itproduces excessive noise,oritmayloseitsgain,sothatnoisefrom thesecondstagebecomes important. Athighfrequencies theperfor­ manceofatubemaynotbesufficiently wellestablished, particularly intheexperimental ordevelopment stage,forthenecessary datatobe knownwithsufficient accuracy. Themoststraightforward methodofmeasurement ofnoisefigure istoreplacetheaerialbyacalibrated signalgenerator, andfindthe amountofsignalpowerwhichmustbeappliedtothereceiverinorder toproduce anoutputequaltothenoiseoutput. Bydefinition ofthe noisefigure,thissignalpower,dividedbykTdf,givesthenoisefigure Fdirectly. Thismethodneedscarefuldesignofthesignalgenerator. Themostobvious necessity isthatthesignalgenerator outputmust simulate theantenna; thatis,itmustbehaveasagenerator whoseoutput impedance isthesameasthatoftheantenna, sothatwhenthelatteris disconnected andreplaced bythesignalgenerator, conditions atthe inputofthereceiverareunaltered. Adjustment ofthegenerator output impedance maybeachieved bysomesimpletransformer circuit. Themostdifficulttechnical requirement inasignalgenerator isthat itmustproduce accurately knownoutputsoftheorderof10-14Wor less.Sincepowersofthisorderofmagnitude canonlybedetected by aradioreceiver, itisnotpracticable tomeasure theoutputdirectly. Instead,thepowerismeasured atahighlevel(e.g.10-3to10-6W)and 16.6] FLUCTUATIONS ANDNOISE 471 thenattenuated downbyknownamounts usingaresistance orcapaci­ tancenetwork. Aschematic diagram ofatypicalsignalgenerator is showninFig.16.5.' Powerisgenerated byasmalloscillator producing about1W.The oscillator istunable, andagivenfrequency maybeselectedbyadjust­ mentofacalibrated dial.Theaccuracy ofthefrequency calibration is Variable frequencyILevelindicatorIVariable-.oscillator IattenuatorOutput FIG.16.5.Blockdiagram ofsignalgenerator. usuallyoftheorderof1or2percent,whichissufficient formostpur­ poses.Asmallfractionofthepowerisfedtoaresistance, whichforms theinputtotheattenuator. Thevoltageacrossthisresistance isread onabuilt-invacuumtubevoltmeter; usuallytheamountofpoweris adjusted byanexternal controluntilthevoltmeter readssomestandard value,suchas1V.Thevariousstepsontheattenuator arecalibrated bythemakerandlabelledwiththevoltageoutputacrosstheoutput terminals eitheronopencircuitoracrossaloadequaltotheoutput impedance attheseterminals, whichisfixedatsomevalueindependent oftheattenuator setting. Theoutputimpedance isalwaysmarkedon thesignalgenerator. Sincetheoscillator generates about1 Wofpower,andthismustbe attenuated inaknownwaybyafactorof1014orso,allcomponents carrying radio-frequency currentsathighlevelmustbeverycarefully shielded. Thisisespecially trueatshortwavelengths, whereafewcenti­ metresofexposed wirewouldbeanefficientradiator. Atwavelengths belowafewmetres,thetypicallayoutofasignalgenerator isasfollows. Theoscillator isinitsownscreened box,andafractionofitsoutputis fedtoabolometer (cf.§15.1),alsoscreened, whosereadingshowswhen thepowerlevelisadjusted toitsstandard value.Theattenuator isa circulartubeforming awaveguide whichisbeyondcut-offforthefre­ quencyused.Thefieldcomponents ofanywavelaunched insuchatube areattenuated exponentially asexp(-hx),wherekisgivenbythe generalized formofequation (11.33), (16.34) where ~isthecut-offwavelength fortheparticular modelaunched in 472 FLUCTUATIONS ANDNOISE [16.6 thetube,andmaybecalculated fromthediameter. Atfrequencies con­ siderably belowcut-offthesecondterminequation (16.34)maybeneg­ lectedandtheattenuation isthenindependent offrequency. Ingeneral severalmodeswillbelaunched attheinputtothetube,whichshould bedesigned tokeepthenumberofmodestoaminimum; thehigher modes,withsmallervaluesofAc'areattenuated muchmorerapidlyand FIG.16.6.Signalgenerator outputwithpistonattenuator. Ainputfromoscillator. Bbolometer inscreened housing. Taccurately machined tubeofknowndiameter. Llooptopickupwaveintube. Ccoaxialline,drivenalongtubebymicrometer movement. Doutput. Atcentimetre wavelengths itissometimes preferable tolaunchthewaveinthetube fromtheendofawaveguide, insteadoffromabolometer lampactingasthecentre conductor ofacoaxialline. onlythelowestmodeneedbeconsidered exceptveryclosetotheinput. Adesignwherethebolometer lamplaunches aTEnmodeisshownin Fig.16.6;thishasamagnetic fieldcomponent, normaltotheplane.of thediagram, whichispickedupbyaloopconnected toacoaxialline whichslidesalongthetube.Sucha'pistonattenuator' givescomplete screening, andtheoutputcanbeadjusted overaverywiderange.Since theattenuator lawisnotknownaccurately overtheinitialrangewhere highermodesarepresent, thebestprocedure istomeasure thepower output(oftheorderof10-6W)byabolometer whentheattenuation is adjusted tothesmallest valuepossible consistent withitsfollowing the correctexponential law.Aknownsmalleroutputisthenobtained by theuseofequation (16.34). Themaintenance anduseofstandard signalgenerators forthemeasure­ mentofnoisefigurearerathercumbersome, astheinstruments require -------- ~----- 16.6] FLUCTUATIONS ANDNOISE 473 constant checking. Inaddition thebandwidth ofthereceivermustbe knowninordertodeducethenoisefigure.Forthesereasonsitis generally simplertouseasourceofnoiseofknownpowerratherthan asignalgenerator. Thistypeofsourceisalreadyroughlyatthelevel required, soobviating thenecessity ofcarefulscreening andattenuation ofsignalbylargeknownamounts requiredinasignalgenerator. Solong asthebandwidth ofthenoisesourceislargerthanthatofthereceiver, thebandwidth ofthelatterdropsoutofthecalculation, sincetheinput noisepowerisknownperunitbandwidth. Thusmeasurement ofthe receiver bandwidth isunnecessary. Asimpletypeofnoisesourceistheresistance noisefromaknown resistance whosetemperature maybevaried.Theavailable noisepower iskTaf,andforthistogiveasignaloutputequaltotheordinary noise outputofareceiverofnoisefigureFwemusthavekTaf=Fk(290)af, orF=T/290.ThusifFishigh,ahightemperature filament isrequired, sohighthatonlytungsten canbeused.Theprincipal difficulties ofthis methodaremeasurement ofthetemperature, andthechangeinresis­ tancewithtemperature ofthetungsten, whichaffectsthematching to thereceiver. Thecommonest typeofnoisesourceisadiodeoperated undertempera­ ture-limited conditions; thatis,atsaturation anodecurrent. Thisis achieved bymaintaining aconstant anodepotential of100to200V, theanodecurrentbeingcontrolled bythetemperature ofthefilament. Forthispurpose apuretungsten filament mustbeused,asanoxide­ coatedcathodewouldquicklydeteriorate whenunder-run intempera­ ture,aswellasgivingflickereffectandconsiderable driftintheanode current.Iftheanodeloadisaresistance Rwhosevalueissmallcompared withtheanodeimpedance ofthediode,theavailable noisepoweris 2elRaf,where1istheanodecurrent. Equating thistoFkTaf,we haveF=801R,where1isinamperes andRinohms.IfRismade equalto80ohmstosimulate ahalf-wave dipoleaerial,thenforanoise figureof10,adiodecurrentof1·6mAisrequired. Thisiseasytoproduce undertemperature-limited conditions. Todetermine whenthesignaloutputfromthereceiverisequaltothe noiseoutput,theyshouldbefedintoasquarelawdevicesuchasa thermo-junction milliammeter. Thenoiseoutputfromthereceiver aloneismeasured first,andthenthesignalornoisesourceinputis adjusted untilthemeansquarecurrentreadbythethermo-junction is doubled, when (signal+noise output)=2(noiseoutput). 474 FLUCTUATIONS ANDNOISE [16.6 Thethermo-junction methodismoresatisfactory thandisplayofthe outputonanoscilloscope, sincetheeyecandetectsignalswelldown intothenoise,andisnotagoodjudgeofthesignal/noise ratio. REFERENCES JONES,R.V.,andMCCOMBIE, C.W.,1952,Phil.Trans.244,205. LAWSON, J.L.,andUHLENBECK, G.E.,1949,ThresholdSigna18 (M.LT.Radiation Laboratory Series,McGraw-Hill BookCo.). ROBINSON, F.N.H.,1962,NoiseinElectrical Circuita (O.U.P.). PROBLEMS 16.1.Show,bydifferentiation ofequation (16.5)toobtainthedifferential equation forthecurrent I=dq/dt,andfollowing through ananalysis similartothatof §16.1,thattLP=tkT. 16.2.Asignalgenerator whoseoutputimpedance is500ohmsiscalibrated in termsofthepoweritwilldeliverintoamatched load(i.e.theavailable signal power).Itisconnected toareceiver whosebandwidth is10kc/s,andwhosefirst stageconsistsofatriodewhoseshotnoiseisnegligible, witha1000-ohmresistance connected between cathodeandgrid.Whatwillthesignalgenerator readingbe whenitisadjusted sothatthesignaloutputfromthereceiver isequaltothe noiseoutput? (Answer: 6X10-17W.) 16.3.Referring toProblem 10.10,assumethatthetargetislowoverthesea.and intercepts thepowerincident onanareaAi'Thispowerisscattered withthe sameangulardistribution asthatoftheradiation fromashorthorizontal dipole paralleltothetransmitter dipole.Someofthisscattered powerfallsonanaerial ofeffective areaA2locatedatthetransmitter, andisdetected byareceiverof noisefigureFandbandwidth df.Showthatthesignal/noise ratio,forthesignal returned fromthetarget,isunityforatargetdistance _(361T2WAiA2)i(Hh)!.D-FkTdf A (Thisformula showshowdifficultitistoincrease therangebyincreasing the transmitter powerW,andhowmuchbetteritistoreducethewavelength.) 16.4.Byfollowing thetreatment of§16.3usingPlanck's lawinsteadofthe Rayleigh-Jeans law,showthatthequantum-mechanical formula forresistance noiseis d(V2)4hfdf~=exp(hf/kT)-l• Verifyfromthisformulathatthetransition fromclassical regiontoquantum­ mechanical regionoccurswhenthenumberofquantaperunitbandwidth inthe noisepowerisoftheorderofunity. -~------------------- 17 THEORY OFTHEDIELECTRIC CONSTANT 17.1.Molecular structure andthedielectric constant FROMthestandpoint ofelectromagnetic theory,adielectric maybe regarded asacontinuous medium whichbecomes polarized underthe actionofanelectricfield.Theratioofthepolarization totheelectric fieldproducing itisproportional totheelectricsusceptibility, andis substantially independent ofthefieldstrength. Thevolumesuscepti­ bilityXisrelatedtothedielectric constant Ebytheformula E=l+X' Thedielectric constant variesnotonlyfromsubstance tosubstance, but alsowiththephysical stateofanyonesubstance. Hitherto ithasbeen takenasaconstant, experimentally determined, andnoinquirywas madeastotheoriginofthepolarization whichgivesrisetothesuscep­ tibility. Theconceptofacontinuous mediumisalientomodernatomictheory, bywhichanysubstance isregarded asanassembly ofatomsormolecules. Eachatomconsistsofaheavy,positively-charged nucleuswithnega­ tively-charged electrons surrounding it.Theatomiselectrically neutral, havingequalamounts ofpositiveandnegative charge.Thesameistrue ofamolecule, formedbyseveralatomsjoinedtogether, witheithera sharingoratransferofelectrons. Thedistribution ofelectronic charge inanatomissymmetrical aboutthenucleus, and,asdiscussed in§2.3, noatompossesses apermanent electricdipolemoment. Thisisnottrue ofmolecules, whichmaybedividedintotwoclasses-polar molecules, whichpossessapermanent electricdipolemoment, andnon-polar mole­ cules,whichdonot.Homonuclear diatomic molecules suchasH2,N2, O2haveasymmetrical chargedistribution andarenon-polar, butasym­ metrical molecules suchasKCIandHCIarepolar,sincethereisanet transferofelectronic chargefromoneatomtotheother.Asimplepicture oftheKCImolecule isthatoftwoionsK+andCI-,andonthisbasiswe shouldexpectthedipolemomenttobejustequaltotheproductofthe electronic chargeandtheinternuclear distance. Measured dipole moments aregenerally smallerthanbutofthesameorderofmagnitude assuggested bythiscrudemodel,andareexpressed intermsofthe 476 THEORY OFTHEDIELECTRIC CONSTA?\T [17.1 Debyeunit,definedas 1Debye=10-18e.s.u.=3·336X10-30coulomb-metre. Anumberofelectricdipolemoments andinternuclear distances for diatomic molecules aregiveninTable.17.1.Thealkalihalidescome neartohavingthemoments expected onthepictureoftwoions,butare somewhat smallerbecausethefieldofeachionpolarizes theotherion (seeFig.17.1),producing induced moments Piintheopposite senseto themainmoment. Theionicapproximation ismuchworseforthe TABLE 17.1 Internuclear distances andelectricdipolemoments ofsomediatomic molecules Electronic Observed Internuclear charge dipole distance r xr moment Molecule (A) (Debyeunits) (Debyeunits) CsF 2·345 11·2 7·88 CsCl 2·906 14·0 10·46 Cal 3·315 15·9 12·1 KF 2·55 12·2 7·33 KCl 2·667 12·8 10·48 KBr 2·821 13·5 10'41 KI 3·048 14·6 11·05 Hel 1·27 6·1 1·03 HBr 1·42 6·8 0·78 HI 1·62 7·8 0·38 hydrogen halides,HCI,HBr,HI,wherethedipolemoments actually decrease whiletheinternuclear distances increase inthisprogression. This,together withthefactthatthemoments aremuchsmallerthan theproduct oftheelectronic chargeandtheinternuclear distance, showsthatourpictureofthesemolecules astwoionsisanover-simpli­ fication. Infactmostoftheelectronic chargeresidesbetween thetwo nuclei.Thistendency increases aswegofromHCItoHI,andwespeak ofaprogressive changefromionicbindingtowards covalent binding, wherethevalenceelectrons aresharedbetween thetwoatoms. Thequestion ofwhether amorecomplicated molecule willhavea permanent dipolemomentornotdepends onitssymmetry; theproblem maybeillustrated byreference tothreetriatomic molecules. Water,H20, hasalargemoment, 1·84Debyes, andthisshowsthatitcannotbe linear;forthenitmusteitherbesymmetrical, likecarbondioxide, 0-0-0, whichhasnodipolemoment, orasymmetrical, likenitrous oxide, N~N-O, whichhasthesmalldipolemoment 0-17Debyes. The 17.1] THEORY OFTHEDIELECTRIC CONSTANT 477 latterpossibility isunlikely forvalencereasons.H20mustbetherefore abent,triangular molecule, withthenegatively-charged oxygenatthe apexandthepositively-charged hydrogens atthefootofthetriangle. Thustheabsenceofadipolemoment, oritsmagnitude, ifitispresent, isanimportant guidetothestructure ofamolecule.Itisalsointimately connected withthedielectric constant ofasubstance, thetheoryofwhich willnowbeoutlined. Sinceeachdipoleinteracts withtheneighbouring ----.P, p.--------- FIG.17.1.Theinduced dipolesPioneachionareintheopposite direction tothemaindipolePformedbythechargesonthetwo ions,sothatthetotaldipolemoment islessthanp. dipolesthroughthelocalelectricfieldwhichitpossesses, thetheoryfor densesubstances, wherethedipolesareclosetogether, ismorecompli­ catedthanthatforrarefiedsubstances. Weshalltherefore consider first thedielectric constants ofgases. 17.2.Dielectric constant ofnon-polar gases Inamolecule whichpossesses nopermanent electricdipolemoment, theelectron distribution issymmetrical aboutthecentre.Whena uniform electricfieldisapplied, notranslational forceactsonthe molecule asawhole,sinceitiselectrically neutral,andthecentreof massremains fixed(ormovingwithuniform velocity). Theelectrons andnucleiare,however, subjected toforcesofopposite sign,andthey willtherefore bedisplaced alittleinopposite directions untiltheinternal forcesbalancethoseduetotheexternal field.Themolecule thereby acquires aninduced moment whenthefieldisapplied. Theforces exertedonthecharged constituents ofthemolecule areparalleltothe fieldandproportional toit,andtheinduced moment isalsoparallelto thefield,andproportional toitatstaticfieldstrengths usedinthe laboratory. (Non-linear effectshavebeenobserved attheabnormal field 478 THEORY OFTHEDIELECTRIC CONSTANT [17.2 Pi=o:E, (17.1) wherePiistheinducedmoment, Ethefieldactingonthemolecule, and 0:isaconstant knownasthemolecular (oratomic,ifwearedealingwith atomsratherthanmolecules) polarizability. Thevalueof0:istypical ofeachdifferent typeofatomormolecule. ThefieldEisknownasthe localfield,sinceitistheactualfieldactingoneachmolecule. Thisis notnecessarily thesameastheexternal fieldEo,appliedforinstance bymaintaining avoltagedifference between twocapacitor platesand calculated therefrom, sinceeachmolecule issubjected alsototheelectric fieldsofneighbouring molecules which,likeit,haveacquired dipole moments undertheinfluence ofthefield.ThelocalfieldEisequalto thevectorsumofEoandthefieldsduetoneighbouring molecules; approximately (seebelow),Ecanbereplaced bythesumofEoandan average fieldduetotheneighbours whichisparalleltoEo.Thuswe shallfindanaveragelocalfieldEwhichisalsoparalleltoEo. Thegeneralrelationbetweentheelectricdisplacement D,theexternal fieldEo,thepolarization P,andthedielectric constant €isstrengths encountered whenthelightfromahigh-powered laseris broughttoafocus.)Sincetheelectrons aresomuchlighterthanthe nucleus, theirdisplacement iscorrespondingly greater, astheposition ofthecentreofmassisunaltered. Ingeneralweshallreferonlytothe electron displacement relativetothenucleus, sincethisdetermines the induced moment. Thestatements inthelastparagraph maybesummed upinthe mathematical equation (17.2) HerePistheinduced electricmomentp,;perunitvolume, anditis relatedtotheaveragelocalfieldEbytheequation P=noo:E, (17.3) wherenoisthenumberofmolecules perunitvolume(assumed tobe allofthesametype).Fromequation (17.2)wehavealso P=(£-1)£0Eo. (17.4) Inordertorelatethemacroscopic dielectric constant EOtothemolecular polarizability 0:,werequiretoknowtherelationbetween EandEo.The following approximate solutionofthisproblem isduetoI",orentz. Thesubstance isimagined tobedividedintotwoparts,andthe contribution ofeachisconsidered separately. Onepartconsistsofa spherewhosesizeissolargethatwhenconsidering thelocalfieldacting 17.2] THEORY OFTHEDIELEOTRIC OONSTANT 479 onamolecule atthecentreofthesphere,theeffectofthemolecules in theregionoutsidethespheremaybeevaluated byregarding theregion outsideasacontinuum. Thisisobviously asatisfactory approximation iftheradiusofthesphereislargecompared withtheintermolecular distance, sothatthespherecontains manymolecules. Thenthelocal fieldisE=Eo+E1+E2,whereE1isthefieldduetothemolecules outsidethesphere,andE2thatduetothemolecules inside.Thefield E1isthesameasthatduetothepolarization chargePnoverthesurface ofthesphere,wherePnistheoutward component ofthepolarization normaltothesurface. FromtheresultofProblem 2.1,wehave (17.5) J (17.6)HerePistheordinary polarization ofthemedium (wedonothaveto allowforanydistortion ofthefield,asinProblem 2.1,becausewehave notexcavated arealcavityinthedielectric). ThevalueofE2ismoredifficulttocalculate, sinceitdepends onhow themolecules arearranged withinthesphere.Lorentzshowedthatfor acubicalarrayofmolecules (asinasimpletypeofcrystal)E2=0,and thisisalsotrueofgasesandnon-associated liquidswherethemolecules aremovingatrandom, independently ofoneanother. Wehavetherefore E=Eo+E1,whencefromequation (17.3)P=noa:E=noa:{Eo+Pj3£0}' Elimination ofPusingequation (17.4)yieldstheequation, :firstderived byClausius andMossotti, andgenerally knownbytheirnames, £-1noa: £+2 3£0 Ifeachmolecule couldberegarded asaperfectly conducting sphereof radiusa,themoment acquired bysuchasphere(see§2.4)inafieldE is%£oa3E.Thissuggeststhatthevalueofa:willbecloseto%£oa3, whereaisthemolecular radius,.andtheright-hand sideofequation (17.6)isthenseentobeequaltotheactualvolumeoccupied byallthe molecules inunitvolume.Ifthevaluesofthemolecular volumecalcu­ latedinthiswayarecompared withthosederivedfromkinetictheory (e.g.frommeasurements ofviscosity), itisfoundthattheyareofthe sameorder,beinggenerally rathersmaller, asisillustrated bythe examples giveninProblem 17.1. Ifbothsidesofequation (17.6)aremultiplied byMjp,whereMis themolecular weightandpthedensity,itbecomes £-lM=Na:, (17.7) £+2P3£0 480 THEORY OFTHEDIELECTRIC CONSTANT [17.2 whereN=Mno{pisAvogadro's number. Thequantity NCi{3eOissome­ timescalledthemolarpolarizability. Thevaluesofthedielectric constants ofanumberofcommon gases atnormaltemperature andatmospheric pressure aregiveninTable17.2. Itwillbeseenthatthedifference between eandunityisoftheorderof 10-3,andfornon-polar gasesitincreases withthecomplexity andhence withthesizeofthemolecule. Thevaluesofe-lforgaseswhose molecules havepermanent dipolemoments aremarkedly higher,but atordinary pressures itisobviousthatitissufficient towritee+2as 3inthedenominator ofequation (17.7)above.Thisistantamount to ignoring thedifference between EoandE=(e+2)Eo{3,andthevalue ofe-lwiththisapproximation couldhavebeenobtained immediately fromequations (17.3)and(17.4).Athighpressures thisapproximation ceasestohold,andthevalidityoftheClausius-Mossotti relation(17.6) hasbeenverifiedbyvariousexperimenters usingpressures upto1000atm. Theconstancy ofthevalueofNCi{3eoascalculated usingequation (17.7) isshowninTable17.3,whilethevaluescalculated usingtheapproxima­ tione+2=3deviateappreciably athigherpressures. (17.8)np2 Pa=3kTE,17.3.Staticdielectric constant ofpolargases Thetheorydeveloped inthelastsectionholdsnotonlyfornon-polar gases,butforallgases,sincetheapplication ofanelectricfieldwillalways causeadistortion ofthemolecule andthusgiveaninduced dipole moment. Inthecaseofpolargases,however, thereisanadditional effectarisingfromthepresence ofthepermanent dipolemoments. In theabsenceofanappliedfieldthesepointinrandomdirections, andthere isnonetpolarization ofthegas.Whenafieldisapplied,thereisasmall excessinthenumberofdipolespointing withthefieldoverthosepointing againstthefield,andsothereisacontribution tothenetpolarization. Theexcessnumberisdetermined bytheBoltzmann distribution, since adipolepointing withthefieldhasaslightlylowerenergythanone pointing againstthefield,andsoisslightlymorefavoured inthedistri­ bution.Thisproblem hasalreadybeentreatedbyclassical methods for thecorresponding magnetic casein§8.3,andtheresultsobtained there maybeappliedimmediately totheelectrical caseifwewritepandE fortheelectricdipolemoment andfieldinsteadofthemagnetic quan­ titiesmandB.Thecontribution tothepolarization istherefore (cf.equation (8.13)) 17.3] THEORY OFTHEDIELECTRIC CONSTANT TABLE 17.2 Dielectric constants ofsomecommon gases atatmospheric pressure and0°0 Dipolemoment Gas(..-1)103DebyeunitB He 0·071 0 H. 0·270 0 O. 0·531 0 N. 0·588 0 CO. 0·988 0 CH. 0·948 0 C.H. 1·38 0 CO 0·692 0·10 N.O 1·08 0·17 NH38·34 1·45 SO. 9·93 1·59 TABLE 17.3 Dielectric constantofCO2andthe Olausius-M ossottirelation Dielectric ..-1M Experi- Pr68sure constant ..+2p menter8 (atm.) (at100°0) (em3) K.andK. 10 1·00753 7·49 30 1·0240 7·53 50 1·0431 7·57 70 1·0645 7·60 100 1·1041 7·69 151 H912 7·73 M.andM. 103·2 H086 7·71 194·5 1·2695 7·75 295·4 1·3895 7·70 365·0 1·4375 7·68 476·6 1·4900 7·67 588·3 1·5274 7·66 700·2 1·5570 7·66 812·3 1·5812 HI5 970·6 1·6097 7·62481 ..-1MThevalueof--2-atN.T.P.is7·33em"for1g-mole...+p Thedataindicate aslightriseinthemolarpolarizability withpressure, followed bya smalldecrease atthehighestpressures. References : K.andK.,F.G.KeyesandJ.G.Kirkwood, 1930,PhY8.Rev.36,754. M.andM.,A.MichelsandC.Michels, 1932,Phil.Trans.A,231,409. 851110 Ii 482 THEORY OFTHEDIELECTRIC CONSTA NT [17.3 wherekisBoltzmann's constant andTtheabsolute temperature. Here Eisagainthelocalfield,andtherelation between thelocalfieldand theexternal fieldEoismore complicated thanforinduced dipoles becausethepermanent dipolesarenotalloriented paralleltothefield (thisproblem willbeconsidered furtherin§17.6).Forgasesatsuchlow densities thatthedifference between EandEocanbeneglected, the staticdielectric constant lO8isgivenbytherelation nop2 lO8-lOi=PaJlOoEo=-kT' (17.9) 3lOo where lOiisthatpartofthedielectric constant duetotheinduceddipoles alone;inthelowdensitylimitlOi-1=noOl.JlOo.Itwillbenoticedthat wehaveusedtheformula appropriate tothecaseofpEJkT ~1,a condition whichiswellfulfilledatordinary fieldstrengths. Atroom temperature kTis4X10-21joules,sothatevenwithadipolemoment of4Debyes afieldof3X107VJmetre wouldberequired tomake pEJkT=0·1.Aslightdecrease inthedielectric constant hasbeen observed insomeliquidsatveryhighfieldstrengths, butonecannot approach saturation asinthecaseofmagnetic dipolesbygoingtovery lowtemperatures (see§20.5),sinceallpolargasestendtohavehigh liquefaction andfreezing-points owingtothelargeintermolecular forces between theirpermanent dipoles.Inthesolidstatethesearesolarge thattheelectricdipolescannotrotatewhenanelectricfieldisapplied, whereas magnetic dipolesinsuitable paramagnetic saltsarerelatively freetoorientthemselves inamagnetic field. Thederivation ofthecontribution tothepolarization fromtheper­ manentdipoleswhichwehavegivenisapurelyclassical one,andthe readermaywondertowhatextentitisconfirmed bywavemechanics. Theanswertothisisthatexactlythesameresultisobtained, butin asurprisingly different way.Thismaybeillustrated byreference toa diatomio moleoule. Therotational statesofsuchamolecule aredis­ tinguished byhavingquantized valuesoftheangularmomentum equal toJ(hJ27T),whereJiszeroorapositiveinteger,andhisPhmck's constant. Thecaloulation showsthatinsmallfieldsthestatesforwhiohJ=1=0 contribute nothingtothepolarization inrespeotofthepermanent dipole moment ofthemolecule. Thisisreasonable beoausewhenthemolecule isturningendoverend,theaverageprojection ofthedipolemoment on anydireotion inspaoeiszero.InthestateJ=0,however, themolecule isnotrotating, andthewholeofthecontribution comesfromthisstate. Athightemperatures alargenumberofrotational statesareoccupied, 17.3] THEORY OFTHEDIELECTRIC CONSTANT 483 andthefractionofmolecules whichareinthestateJ=0isproportional to1IT.Thisgivesthesametemperature variation astheclassicaltheory, anddetailed calculation showsthatthenumerical constant isalsothe same(seePaulingandWilson, 1935). Themolecular polarizability constant 0:isnotofgreattheoretical interestexceptinthecaseofaverysimpleatomsuchashelium,where awave-mechanical calculation ofitsmagnitude ispossible. Thesizeof thepermanent dipolemoment is,however, avaluable cluetothestruc­ tureofamolecule, aspointedoutin§17.1,andgivessomequantitative information aboutthenatureofthechemical binding.Itisobvious fromequation (17.9)thatthesizeofthedipolemomentmaybeobtained frommeasurements ofthedielectric constant ofthegas,experimental methods forwhichwerediscussed in§15.5.Inordertoseparate out thecontributions fromtheinduced polarization andthepermanent dipoles,measurements maybemadeoverawidetemperature range.If themolarpolarizability isthenplottedagainstliT,astraight lineis obtained fromtheslopeofwhichthedipolemoment canbecalculated usingequation (17.9).Theintercept atliT=0givesalsothevalue of0:,themolecular polarizability. 17.4.Dispersion ingases Thetheoryofelectromagnetic waves(Chapter 10)showsthatthe refractive indexofasubstance shouldbeequaltothesquarerootofits dielectric constant, ifthemagnetic permeability canbetakenasunity, asisusuallythecase.Acomparison ofthedielectric constants measured atlowfrequencies withtherefractive indicesmeasured intheoptical region(Le.atfrequencies oftheorderof1014)givesverypooragreement withthisrelationexceptinthecaseofsimplenon-polar gases.Values ofthedielectric constant ofafewsuchgasesmeasured overawiderange offrequencies aregiveninTable17.4together withthesquareofthe opticalrefractive index.Thelatterisextrapolated to'infinitewave­ lengths'tocorrectfordispersion intheopticalregion.Theagreement isseentobeexcellent inthecasesquoted. Intheopticalregion,variation oftherefractive indexwithwavelength hasbeenknownforaverylongtime,andiscalleddispersion. Ingeneral therefractive indexincreases asthewavelength decreases, andthisis knownas'normaldispersion'. Thereversecase,wheretherefractive indexdecreases withdecreasing wavelength, occursonlyinthevicinity ofanabsorption line,andisdifficulttoobservebecauseoftheabsorption. Thisisknownas'anomalous dispersion', butbothtypeshaveasimple (17.10)484 THEORY OFTHEDIELECTRIC CONSTANT [17.4 explanation intermsofclassical theory,basedontheassumption that anatomcontains electrons vibrating atcertainnatural frequencies characteristic ofthetypeofatom,andthattheapplication ofanalter­ natingelectricfieldsetssuchelectrons intoforcedvibration. TABLE 17.4 (£-1)10· atN.T.P. Gas 0·1Mcjs 1Mcjs 9000Mcjs24000Mcjs Optical Air 570 567·0 575·4 576·0 575·7 ±0·7 ±1·0 ±1·4 ±0·2 ±0·2 Nitrogen 578 579·6 586·9 588·3 581·3 ±0·7 ±1·0 ±2·9 ±0·2 Oxygen. 528 523·3 530·0 531·0 532·7 ±1 ±1 ±1'9 ±0·4 Argon 545 545·1 - 555·7 554·7 ±1 ±0·5 ±0·4 Carbondioxide 987 987·5 985·5 988 - ±1 ±2 ±3 ±2 Hydrogen 270 272 - - 272 ±1 A B C D E References : A.Lovering andWiltshire, 1951,Proc.I.E.E.98,PartII,557. B.HectorandWoernley, 1946,Phys.Rev.69,101. C.Birnbaum, Kryder, andLyons,1951,J.Appl.Phys.22,95. D.EssenandFroome, 1951,Proc.Phys.Soc.B,64,862. E.(nS-l)10·(various authors), extrapolated toiIrlinitewavelength. Letustakethesimplest possible caseofagasofdielectric constant E subjected toanoscillating electricfieldE=E'exp(jwt). Weshall assumethatthewavelength oftheincident radiation isverylarge compared withatomicdimensions (whichistrueuptotheregionofhard X-rays), sothatthefieldactingonanelectroninagivenatomisinde­ pendentofitspositionwithrespecttothenucleus, whichisassumed to bestationary. Eachelectron inthemolecule isdisplaced adistance s bythefield,andtherestoring forceiswrittenas-mw~s,where Wpj27T isthenaturalfrequency ofoscillation oftheelectron andmitsmass. Inaddition therewillbedamping duetocollisions, radiation ofenergy, etc.,whichmayberepresented byaterm-my(dsjdt). Hencewehave (d2SdS) .mdt2+ydt+w~s=-eE'e1",f. Thesolutionofthisis eE . S= - .+e-lyf{Acos[(w2_!y2)lt]+Bsin[(w2-!y2)lt]}. m{(w~-w2)+JYw} p p 17.4] THEORY OFTHEDIELECTRIC CONSTANT 485 (17.11) (17.12) (17.13)ThetermsinAandBaveragetozeroovermanyatomssinceAandB dependontheinitialconditions andareasoftenpositive asnegative. Theinstantaneous electricdipolemoment duetothedisplacement of theelectron isp=-es,and,iftherearenomolecules perunitvolume, thepolarization Pis noe2E 1P=nop=-- .m(w~-W2)+jyw. Forgasesathigherdensityacorrection forthedifference between the. localfieldandtheexternal fieldmaybeappliedinthesamewayasin §17.2,leadingtotheformula £-1n2-1noe2 1 £+2=n2+2=3mEo(w~-w2)+jyw' Thisformula showsthatboth £andnmustberegarded ascomplex. Writing £=E'-jE"=(n-jk)2, wherenistherealpartoftherefractive indexandkistheabsorption coefficient, wemayseparate therealand imaginary partsofequation (17.12). Theformula isclumsytohandle, however, andweshallassumethatwearedealingonlywithgasesat suchlowpressures thatwecanneglecttheLorentz correction. Since thevalueofkissmall,andnegligible exceptnearanabsorption line,we mayalsomaketheapproximation, ifthelineisnarrow,ofwriting (w~-W2) =(wp+w)(wp-w) ~2w(wp-w). Thenweobtaintheformulae E'=n2-k2~n2=1+_n_o_e2 _{.,--_w--"='P-:-::-w---,----,,}}2mwEo(wp-w)2+~w2, E"=2nk~2k=noe2{~w } 2mwEo (wp-w)2+~w2 wherethesymbol ~whasbeenusedfor1'/2,andwehaveassumed n~I,k~n. Thevariation ofnandkintheneighbourhood ofaweakabsorption lineisshowninFig.17.2.Theabsorption coefficient reachesamaximum attheresonant frequency wherew=wp,andfallstohalfitsmaximum valueatwp-w=±~w.Inopticalusage,thequantity 2~v=~W/7T iscalledthe'half-width' oftheline,meaning thefrequency difference between thepointsatwhichtheabsorption hasdropped tohalfthe maximum value.Microwave spectroscopists, however, prefertocall ~vthehalf-width. Ingeneral,eachatomormolecule possesses anumberofcharacteristic resonant frequencies, andtheexpressions givenabovefortherefractive 486 THEORY OFTHEDIELECTRIC CONSTANT [17.4 indexandabsorption coefficient shouldbereplaced byotherswith summations overthevariousvaluesofwp'Ifthenumberofelectrons permolecule whichhavearesonant frequency wpisdenotedbyfp,we maywriteequation (17.12)intheform £-1n2-1noe2"'" fp £+2=n2+2=3m£0~ (W~-w2)+jyw' (17.14) p Thevalueoffpisknownasthe'oscillator strength' ofanabsorption line,andonclassicaltheoryweshouldexpectittobeunity.Inpractice 0·250·751·0 0·5 o0·5- 0·25 -O'5I-;....J'---":-'---'-_-':----'-_~---'''''''"-J'-----'--...!--.L-J 0-2 0 +4 (01..-01)/1101(n-l) inthesamennitsask FIG.17.2.Variation ofnandknearanarrowabsorption line(fromequation (17.13». n-landkareinunitsofnoe·/4mw£o!lw. itgenerally hasvalueslessthanunity,andthequantum mechanical explanation showsthatthiscorresponds tothefactthateachelectron possesses anumberofpossible frequencies ofoscillation, anditstotal oscillator strength isdividedbetween them.Wehavethen!Up)=1 p foreachelectron. Atfrequencies farfromresonance theabsorption coefficient isnegli­ gible.Atverylowfrequencies, wherew<{anyvalueofwp,wehave £-1n2-1noe2"'"fpnoel: (17.15) £+2=n2+2=3m£0~w;=3£0 P bycomparison with(17.6).Thisshowsthatthemolecular polarizability el:isintimately connected withtheoscillator strengths andabsorption lines.Infact,asthefrequency israisedandwepassthroughanabsorp- 17.4] THEORY OFTHEDIELECTRIC CONSTANT 487 tionlineatWp/27T,therefractive indexgoesthrough theanomalous variation showninFig.17.2andapproaches asmallerlimiting value onthehighfrequency sidethanithadonthelowfrequency side.When thereareanumberofabsorption lines,thebehaviour isasshownin Fig.17.3,andfinally,when Wisgreaterthanallvaluesofwp,napproaches Atomicspectra Electronic transitions, Vibra.tion bandsMolecula.r spectra A I Rota.tion bands~] ~~ -J ~., ll:<11--------------"'-------\-f--+ __------ Micro-wa.ves I I 1010IOUFarInfra-red i I 1011 1013Nea.r Infra-red Visible Ultra-violet X-ra.y8' I I I I I 101&101610181017lQl8 Frequency (cIs) I 10'"I 10-'"I 10--I 10-3I 10-1I 10-1I 1I I 10-& 10--6 A(om). FIG.17.3.Schematic diagram showing variation ofrefractive indexwithfrequency. unity,butthevalueof(n-l)isslightlynegative. Thisisthewell-known anomaly intherefractive indexintheX-rayregion,andthevalueofn isthengenerally calculated byassuming thattheelectrons arefree,so thatequation (17.10)reducesto (17.16)d2sE'....• m-=-ee1<»<>dt2 • Thisisequivalent totheassumption thatw~(J)p,y. Ifamolecule hasapermanent electricdipolemoment, itsstatic dielectric constant contains anadditional terminvolving thedipole moment (seeequation (17.9».Fromthediscussion ofdispersion inthis section,weshouldexpectthatthistermwouldalsoberelatedtosome absorption lines.Thisisthecase,forsuchmolecules havea'pure rotational' spectrum inthefarinfra-red, duetotransitions between thedifferent rotational levelsofthemolecule. Suchtransitions canbe observed onlyifthemolecule hasapermanent dipolemoment, since thenanalternating electricfieldexertsacoupleonthemolecule which changesthestateofrotation. Intheoptical:fegionwearefaronthehigh frequency sideofsuchabsorption lines,sothattheygivenocontribution 488 THEORY OFTHEDIELECTRIC CONSTAW.r [17.4 totheopticalrefractive index.Ifthemolecule hasnootherabsorption linesintheinfra-red, thesquareoftheopticalrefractive indexwould(by equation (17.15))beequaltothatpartofthelowfrequency dielectric constant whicharisesfromthemolecular polarizability. Ingeneral, however, molecules showabsorption linesduetomolecular vibrations, sincethedistorted molecule mayhaveadipolemoment. Underthe actionofthevibration thisgivesthemolecule anoscillating dipole moment, whichcanemitorabsorbradiation. 17.5.Staticdielectric constants ofliquidsandsolids Thestaticdielectric constants ofliquidsandsolidsarerelatedto absorption lines(orbands)athigherfrequencies inasimilarwayto thatoutlined inthepreceding sectionforgases.Insimpleatomic substances, suchasthecondensed phasesoftheraregasesoftheatmo­ sphere,therearenoabsorption bandsintheinfra-red, andthelow­ frequency dielectric constant doesnotdiffergreatlyfromthatdeduced fromtheClausius-Mossotti formula, usingthemolecular polarizability measured forthegasphase.Forexample, atitsboiling-point liquid heliumhasadielectric constant of1,048,andadensityof0·125g/cm3• Thisgivesamolarpolarizability of(Nrx/3eo)=0'12,whilethatdeduced fromtheopticalrefractive indexofthegasat0°C(afterallowing for dispersion byextrapolating toinfinitewavelength), orfromthestatic dielectric constant ofthegas,is0·123. Onabroadclassification, asecondclassofsubstances contains those whichconsistofagglomerations ofmolecules heldtogether bythevan derWaalsforcesbetween themolecules; suchforcesarerelatively small (though largerthanthosebetween atoms),andthesesubstances have fairlylowmelting- andboiling-points. Mostorganicsubstances belong tothisclass.Thestaticdielectric constant corresponds toamolar polarizability considerably higherthanthatcalculated fromtheoptical refractive index,thedifference beingassociated withinternalvibrations withinthemolecules. Thesegiverisetoabsorption bandsintheinfra-red, provided thatthevibration setsupanoscillating electricdipolemoment. Fromequation (17.15)itfollowsthattheireffectislargestwhenthe oscillator strength fpishighandtheresonant frequency wpislow. Thesevibrations arecharacteristic ofthemolecule, andoccuratfre­ quencies whicharenotgreatlydifferent inthesolidorliquidfromthose inthegas.Thecharacteristic rotational frequencies areabsentinthe condensed phase,however, becauseoftheintermolecular forces.Inthe solidsuchrotations arecompletely inhibited inmostcases,butinliquids 17.5] THEORY OFTHEDIELECTRIC CONSTANT 489 wherethemolecules carrypermanent electricdipolemoments adisper­ sionbandisobserved atradio-frequencies (see§17.7). Thethirdclassofsubstances contains theionicsolids,consisting of latticesofpositively andnegatively chargedions;thesegiverisetostrong bindingforces,andthesubstances haveratherhighmelting-points. In ioniccrystalsanelectricfieldexertsaforceoneachion,causing a displacement ofthewholepositiveionlatticewithrespecttothenegative ionlattice.Thisgivesaratherlargepolarization, andahighdielectric constant. Inthelightofequation (17.15)thiscanbeinterpreted in termsoftheratherlow(farinfra-red) vibrational frequencies associated withdisplacements ofthepositive ionlatticerelativetothenegative ionlattice. Someofthesehavebeenmeasured spectroscopically, but thecalculation ofthedielectric constant is complicated because (a) oflocalfieldcorrections and(b)thechargecloudsoftheionspartly overlaponeanother, producing short-range forceswhicharenotade­ quatelyrepresented bytheLorentz localfield,whichisessentially a dipolarorlong-range force.Szigeti(1949)hasderivedtheformula E-nz=(nZ+2)Znoq:' (17.17) 3MrWt EO wherenZisthesquareoftheopticalrefractive index(extrapolated to infinitewavelength, qistheeffective chargeoneachion,Wtthecharac­ teristicfrequency fortransverse elasticwaves,andM,.thereducedmass, whichforasolidcontaining twotypesofionofmassMI,M2isgivenby 1 1 1 M=.M,+M.' (17.18) ......,. 1Z Theeffective chargeq=8(ze),wherezisthevalencyoftheion,ethe electronic charge,and8afactorclosetounitywhichisintroduced to allowforthechargeoverlapmentioned above.Sometypicalvaluesof8 aregiveninTable17.5.Incubiccrystalsthedielectric constant is isotropic, butthisisnotnecessarily trueofnon-cubic crystals. Inthe ioniccaseanisotropy ariseswhenthevibrational frequency Wtdepends onthedirection ofvibration, and,correspondingly, thedisplacement. ofanion(involving thesamerestoring forces)depends onthedirection oftheappliedfield. Ingeneralthedielectric constant ofasolidisnotgreatlydependent ontemperature, buttherearesomenotable exceptions. Inbarium titanate, BaTiOa,forexample, thedielectric constant variesathigh temperatures as(T-~)-l, andrisesashighas104justabove1200K. Belowthistemperature spontaneous polarization isobserved, which 490 THEORY OFTHEDIELECTRIC CONSTANT [17.5 canbereversed byanelectricfieldofsufficient strength, withhysteresis effects. Thisisaco-operative transition, showing manyresemblances toferromagnetism, andsubstances showing sucheffects(otherexamples TABLE 17.5 Dielectric constants ofsomeionicsolids SUb8tanu Lauicetype (! na(n2:~r8 Tiel Cubic(CaCl) 31·9 5·10 5·6 1-08 SrO. Cubic(Na.Cl) 13·2 3·31 3·14 0·6 TiOa(paralleltoaxis) Tetragonal 173 8·42 12·1 0·79.0·65 TiOa(perpendicular toaxis) 89 6·82 8·7 0·88.0·65 Twovaluesof8aregivenforTiOabecauseofanambiguity intheinterpretation ofthe infra-red absorption bands(afterSzigeti,1949). areRochelle salt,Pc=240C;andpotassium dihydrogen tartrate, Tc=-1500C)areknownas'ferro-electrics'. However, aferro-electric (unlikeaferromagnetic) doesnotcontainpermanent dipoleswhich becomespontaneously oriented belowthetransition temperature. The properties belowthistransition temperature areduetoaspontaneous latticedistortion inwhichionsofonetypeundergo asmalldisplacement relativetotherestofthelattice.Thisisaccompanied byachangein crystalsymmetry; inBaTiOsthesymmetry iscubicabovethetransition temperature of1200C,changing totetragonal symmetry belowthis temperature. Therearefurtherstructural changes toorthorhombic symmetry below00C,andtorhombohedral symmetry below_900C, thesechanges beingaccompanied bychangesinthedirection ofthe spontaneous polarization. Cochran (1960)hasshownthattheapparent Curie-Weiss law(cf.equation 8.14)abovethetransition temperature resultsfromatemperature dependence ofonevibrational mode,such thatinanequation oftheform(17.17)wrvariesas(T-1;,),Thelatter hasbeenverified experimentally byCowley (1962)forstrontium ·titanate. Thisfrequency fallstozeroatT=Pc,wherethelattice becomes unstable inrespectofthisonemodeandaspontaneous dis­ tortiontakesplace. Thepossibility ofanti-ferro-electrics, wherenonetpolarization exists belowthetransition temperature because equalnumbers ofionsare shiftedinopposite directions, waspointedoutbyKittel(1951),the firstsuchsubstance tobeidentified beingleadzirconate, PbZrOs (Tc=2300C). 17.6] THEORY OFTHEDIELECTRIC CONSTANT 491 17.6.Staticdielectric constants ofpolarliquids Inpolarliquidsthelocalfieldisverylarge,anditsrepresentation by theLorentz fieldleadstotheresultthatsuchliquidsshouldbecome ferro-electrics. Ifweconsider onlythatpartofthepolarization Pa arisingfromthepermanent dipolemoments, wehavefromequations (17.5),(17.8) PaPa nop2 nop2 Eo=E-(Pa/3£o)=3kT{I-(nop2/9kT£oH =3k(T-:Pc)' (17.19) where:Pc=nop2/9£o' Forwater:Pcwouldbeabout10000K,sothat watershouldbespontaneously electrified atordinary temperatures, as inthecorresponding ferromagnetic case(Chapter 21).Infacttheknown examples offerro-electrics arisefromspontaneous ionicdisplacements ratherthanspontaneous orientation ofdipoles(see§17.5).Thenonsensi­ calresultofequation (17.19)isduetothefactthattheLorentzmethod forthelocalfieldassumesthateachdipolehasamoment equaltothe averagemomentandparalleltotheappliedfield.Thisistrueforinduced dipoles,butelectricfieldsofordinary magnitudes causeonlyaslight departure fromrandom orientation ofthepermanent dipoles. The induceddipolesmustbetreatedseparately fromthepermanent dipoles, andOnsager (1936)hassuggested analternative methodoftreatingthe localfieldinwhicheachdipoleisregarded asbeingatthecentreofa spherical cavitywhosesizeisequaltotheaveragevolumeoccupied by eachmolecule. Intheabsenceofanypermanent dipolesitgivesthe sameresultastheLorentzmethod, ascanbeverifiedbyputtingp=0, £8=£iinequations (17.22)-(17.24) below;theequivalent localfieldE isthenEc/(I-OI.g) =(£i+2)Eo/3, whichisthesameasin§17.2. Eachpermanent dipoleppolarizes thedielectric outsidethespherical cavitycontaining it,andthisproduces areaction field(seeProblem 2.2) whichwillreactbackonthedipole.Thereaction fieldErisparalleltop, andproduces anextramoment OI.Erthroughpolarization ofthemolecule makingthenetmomentp'.Sincethereaction fieldisproportional to pi,thenetmoment, wehave pi=p+OI.Er=p+OI.gp/, wheregisthefactorrelating Ertopi,Hence IpP=l-cxg' (17.20) Asimilareffectoccurswiththeinducedmoment Pi'changing itto p'.----.RL (17.21) ~-l-cxg' 492 'fHEORY OFTHEDIELECTRIC CONSTANT [17.6 (17.23)Thesearethentheeffective moments, whichinteract withthefieldin thecavityEc.Fromequations (17.3)and(17.8)wehavethen(since Pi=G:Ei) , nop'2 {noG: noP2}P=nOPi+3kTEc=I-G:g+3kT(1-G:g)2 Ec•(17.22) Thisgivesustheresult,usingsomeformulae fromelectrostatics. IfEsis theactualdielectric constant ofthemedium, E=~E(fromequation (2.43)) c2Es+l0 (17.24)and g_2(Es-l)1 ---:3--"---'--- (fromProblem 2.2)-2Es+ 141TEOa3 2(Es-l)no= ,2Es+l3EO sincetheaverage volumeoccupied byonemolecule is47Ta3/3=Ilno• Thepolarizability G:,fromequation (17.6),isgivenby (17.25)Ei-1noG: Ei+2=3EO' whereEiisthatpartofthedielectric constant associated withthe induced dipolesonly.Onsubstituting theserelations intoequation (17.22),usingP=(Es-l)EoEofromequation (17.4),andcarrying out atediousalgebraic reduction, wefind (17.26)(ES-Ei)(2ES+Ei) nop2 Es(Ei+2)2 =9EOkT' Forwater,usingEi=4·9(see§17.7)andp=1·94Debyes,theformula gives Es~100,which,although higherthantheactualvalueof80,corre­ spondsmuchbettertorealitythantheLorentz prediction. Thedis­ crepancy ispartlyduetothefactthatwecanonlyexpectittohold forspherical molecules (sinceweassumed aspherical cavity), while H20istriangular, andpartlybecauseonlydipolar(longrange)forces havebeenincluded, short-range forceswhichactonlybetween neigh­ bouring molecules beingneglected. Inagasthemolecules aresofar apartthatonlythelong-range forcesneedbeconsidered, andOnsager's formula, equation (17.26),shouldbeusedathighdensities; atlow densitiesitreducestoequation (17.9). Inverydilutesolutions ofpolarmolecules innon-polar solvents, the dipolesaresufficiently farapartthattheirmutualinteractions canbe --------------- ------ 17.6] THEORY OFTHEDIELECTRIC CONSTANT 493 neglected. Thuswewouldexpecttobeabletoapplyequation (17.9),if wereplacepbytheeffective valueofthedipolemoment afterallowing forinteraction effectswiththesolvent. Forspherical molecules this canbedonebyanextension ofOnsager's theory,andthisgivesamethod offindingthemolecular dipolemoment. Thedielectric constant ofthe solution canbedetermined byoneofthestandard methods (see§15.5); thesolventsnormally employed arecarefully purifiedbenzeneandcarbon tetrachloride. Measurements overarangeofconcentrations areused, followed byextrapolation toinfinitedilution. Thedipolemoments measured inthiswayagreefairlywellwiththosefoundusingthegaseous method (§17.3),butdiscrepancies wouldbeexpected duetoshort-range forcesandnon-spherical molecules. Thegaseousmethodismoresatis­ factorywhenitcanbeused,butthesolventmethodisemployed for substances whosevapourpressure isverylow.Foranumberofsimple molecules (suchasthoseinTable17.1)accurate valuesofthedipole moments havebeenobtained frommicrowave spectroscopy (seeTownes andSchawlow, 1955)orelectricresonance inmolecular beams(see Ramsey, 1956),bymeasurements ofthesplitting oftherotational lines inanelectricfield. 17.7.Radio-frequency dispersion inpolarliquids Inthediscussion ofpolargasesitwaspointedoutthatthestatic dielectric constant ishigherthanthesquareoftheopticalrefractive index,thedifference beingmainlyduetodispersion intheinfra-red, associated withthepurerotational spectrum ofthemolecules. Inthe liquidstatethisdifference isevenmoremarked; thewell-known case beingliquidwater,whosestaticdielectric constant is80,whilethe refractive indexintheopticalregionis1·33.Sincethelargedielectric constant isduetoorientation ofthemolecular dipoleswhenafieldis applied,itwillclearlybemuchloweriforientation isinhibited forsome reason.Ifahighfrequency fieldisapplied,thedipolesmustbeableto re-orient themselves sufficiently quicklytofollowthereversal ofthe field,inordertomaketheirfullcontribution tothepolarization. Ifthis re-orientation takesafinitetimeT,thedipoleswillnotbeabletofollow afieldwhoseangularfrequency issuchthatWT~1.Intheregionwhere WT~1,thedielectric constantwillfall,andabsorption ofenergywill takeplacefromthealternating fieldintothedielectric. Toformanestimate ofT,wemustconsider themechanism inhibiting re-orientation. Inaliquidthisissimplythebombardment ofthemole­ culebyothermolecules; thatis,theBrownian motion.Ifaspherical 494 THEORY OFTHEDIELECTRIC CONSTANT [17.7 givingparticleofradiusaissuspended inaliquidofviscosity 1],thenthemean squarevalueoftherotational angle0inatimetis -kT()2=--t=t/T, (17.27)4'lr1]a3 where T=4'lr1]a3/kTisacharacteristic timefortheBrownian motion. Ifweapplythistothemolecules ofaliquidsuchaswater,taking a=2·3X10-8cm,thevaluefoundfromtheviscosity ofthevapour,and 1]=0·010c.g.s.units=0·001m.k.s.unitsat20°C,wefindT=3·7X10-11 sec.Sincethistimeislongerthananyofthecharacteristic periodictimes ofrotationofthefreewatermolecule, itfollowsthatthemolecule cannot rotateatanyofitsnaturalfrequencies intheliquidstate.Instead, the dispersion associated withthepermanent dipoleswilltakeplaceatfre­ quencies suchthatW~l/T,thatisatawavelength oftheorderof1em. Inordertointroduce Tintoourtreatment ofthedielectric constant, weconsider theeffectofmaintaining asteadyfieldonapolarliquid, andthensuddenly removing it.Undertheinfluence oftheBrownian motion,thepreferred orientations ofthedipoleswillgraduaIly disappear. Itisreasonable tosupposethattherateofdecayofthepolarization is proportional totheinstantaneous valueofthepolarization, andwewrite dP/dt=-P/T, P=f1exp(-t/T), where Tisacharacteristic 'relaxation time'whichwewouldexpectto beofthesameorderasthatfoundabove fortheBrownian motion.Here Pis,ofcourse,onlythatpartofthepolarization associated withthe permanent dipoles.Ifthefieldisnotswitched off,butchanged suddenly toavalueforwhichtheequilibrium polarization isPo,thentherateof changeofPisgivenbytheequation dPdP/dt=(PO-P)/T, orP+rdt=Po, Whenanalternating fieldE'exp(jwt) isapplied, wemaywriteour equation forthepolarization intheform(cf.equation (17.8» P+dPP.nop2E'(.t)T(fi=0=3kTexpJW, giving P=nop2E'exp(jwt). (17.28) 3kTl+jwT HereE'istheamplitude ofthelocalalternating field,andPisthat partofthepolarization dueonlytothepermanent dipoles. Allowing 17.7] THEORY OFTHEDIELECTRIC CONSTANT 495 (17.29)fortheseeffects,wefindforthedielectric constant theexpression E-Ei 1 Es-Ei=1+jWT where Eiisthatpartofthedielectric constant duetoinducedpolarization, andEsisthestaticdielectric constant. ThisresultholdsfortheOnsager treatment forthelocalfield,butitcanbeshownthattheformula is similariftheLorentz correction isused,exceptthatwemustusea ~.I----- o 0·1 1 10 «)T(logarithmic scale) FIG.17.4.Variation of~'andE"forapolarliquid. modified relaxation timeT'=T{Es+2)j{Ei+2). Thisdifference issignifi­ cantonlyinacomparison oftherelaxation timedetermined fromthe dispersion ofthedielectric constant withthatfromtheBrownian motion. Equation (17.28)aboveshowsthatP,andhence E,iscomplex, and wemayeitherwriteE=E'-jE",orE=(n-jk)2, wherenistherefrac­ tiveindexandktheabsorption coefficient. Then (17.30) 2k"{Es-Ei)WT (17.31)n=E=1+w2r 2• Thevariation ofthesequantities withfrequency iseasilyseenfrom Fig.17.4.E'fallsfromEsatlowfrequencies toEiathighfrequencies, the transition takingplacenearw=lIT.E"hasamaximum inthisregion atW=IjT,andfallstozeroatbothlowandhighfrequencies. Thetheoryofthedispersion ofthedielectric constant ofpolarliquids 496 THEORY OFTHEDIELECTRIC CONSTAN'l' [17.7 (17.32)HencewasfirstgivenbyDebye(1929),andthatgivenaboveisasimplified versionofhistreatment. Itwasfirstverifiedforglycerine andanumber ofalcohols, forwhich Tismuchgreaterthanforwaterowingtotheir higherviscosity andlargermolecular radius.Forexample, at22°Cthe valuesofe'ande"forglycerine atawavelength of9·5metresare42and 8·6respectively, showingthatwearealreadywellintotheregionof anomalous dispersion atthiswavelength. Vacuum tubeoscillators and detectors wereavailable forsuchwavelengths, butthedispersion in watercouldnotbemeasured accurately untilcentimetre wavetechnique wasestablished, owingtothesmallvalueofT.Weshallheredescribe themeasurements ofCollie,Hasted,andRitson(1948). Anaccurate methodofdetermining thedielectric constant oflowloss liquids,usingresonant cavities, wasdescribed in§15.5.Thismethod cannotbeusedwithwater,sincetheabsorption issogreatifthecavity isfilledwithwaterthatnoresonance canbeobserved. Thisdifficulty canbesurmounted byusingacavitypartlyfilledwithwater,thedegree offillingbeingadjusted togiveameasurable changeintheresonant frequency andQofthecavity. Analternative method, usedbythesameworkers, istodetermine the propagation constant inawaveguide filledwithwater.Thisconstant dependsonboththerealandimaginary partsofthedielectric constant, andonthelineardimensions oftheguide.Byusingtwoguidesofdifferent sizes,thevaluesofe'ande"canbefoundseparately, sincetheirrelative contributions tothepropagation constant dependonthesizeofthe guide.Fromequation (16.34)thefieldintheguidevariesas exp(-hx)=exp{-(ex+j,8)x}, where(Ao=wavelength infreespace) {If2}t{I£}i{Ie'_jeff}!-h=27TA~-V2=27T~-~=27T~---xr. .'-po~h'~~_~)). ex,8=27T2~ Thustwoseparate measurements ofex,withdifferent valuesofAc'suffice todetermine e'ande".Thevalueofexcanbefoundbymovingadetector throughtheliquid,andthemeasurement ofphase(i.e.,8),whichisvery difficultwhenlargeattenuation ispresent, isavoided. Essentially the method adopted wastousetwopistonattenuators (similartothat described in§16.6)inseries,onebeingfilledwithwaterandtheother 17.7] THEORY OFTHEDIELECTRIC CONSTANT 497 not.Thetwoattenuators areadjusted, onemovinginandtheotherout, soastokeepthepowerreaching areceiverconstant; thusnocalibration ofthereceiver isrequired. Theattenuation inthewateriscalculated fromtheknownlawoftheair-filled attenuator. Theresultsobtained maybefittedaccurately tothetheoryusinga valueof'T=1·01X10-11sec,asshowninFig.17.5,whereboththecalcu­ latedcurvesandtheexperimental pointsaregiven.Thegreatintensity oftheabsorption isillustrated bythefactthatatawavelength of 60 40 20 O':-:----:~----__::_'_:;---__=_L,:_--__::_'_::_----_='_;;;_--_:;;'0'1 0·2 0·51·02-0 5·0 10'0 11=v(cm.-1) FIG.17.5.Complex dielectric constant ofwaterat20°C. /'::,.Collie,C.H.,Hasted,J.B.,andRitson,D.M.,1948,Proc.Phys.Soc.60,145.oLane,J.A.,andSaxton,J.A.,1952,Proc.Roy.Soc.A,213,400. lll'To=1·90ii. 'To=1·01X10-11sec. 1·24em,thepowerinanincident wavewouldbediminished byafactor ofr2(21Tk)=raGorabout10-15-0inpassingthrough athickness of 1·24em.Thuswaterisquite'black'atsuchwavelengths. Itshould benotedthatinfittingtheseresults,thebestvalueof€i'thatpartof thedielectric constant duetoinduced polarization, isfoundtobe4·9. Thisisappreciably higherthanthesquareoftheopticalrefractive index (n=1'33),showingthattheremustbeotherstrongabsorption bands inliquidwaterintheinfra-red; theseareassociated withinternalvibra­ tionsoftheH20molecule. Rathersimilarresultshavebeenobtained byLaneandSaxton (1952)formethylandethylalcohols. Ithasbeenfoundthatthevaluesof€'and€"intheregionofdispersion varyquiterapidlywithtemperature, corresponding toavariation inthe 851110 Kk 498 THEORY OFTHEDIELECTRIC CONSTANT [17.7 relaxation timeT.Saxton(1952)hasshownthatAvariesfromabout 27X10-12secat_100C(insupercooled water)toonly4·7X10-12seoat +500C.Thisvariation isveryoloselyparalleltothatofthevisoosity, andindeedthevalueofTissurprisingly olosetothatwhichwouldbe obtained usingthesimpleformula T=47T7]a3/kT. Inthesolidstaterotation ofthepermanent dipolesisgenerally so restricted thattheymakepraotioally nooontribution tothedieleotrio oonstant. Thusthedieleotrio oonstant ofioeat3-omwavelength is about3.Attemperatures justbelowthemelting-point, however, ioe exhibitssomedispersion atlowfrequenoies (oftheorderof106o/s)andthe dielectrio oonstant fallsfromabout80atzerofrequenoy tothevalue quotedaboveathighfrequenoies. Thesechanges areconnected with residualrotation ofthedipolemoments similartothatinliquidwater, butwithaverymuchlongerrelaxation time,oorresponding tovery higheffective viscosity. 17.8.Scatterin~ Whenelectromagnetic radiation isincident onanysubstance, the intensity andangulardistribution oftheemergent radiation aredeter­ minedbytwodistinotphenomena whioharebothpresentinvarying degree. Thesetwophenomena arecollision damping andscattering, andbothresultinalossofenergyfromtheprimary wave,whiohthus suffersabsorption initspassagethroughthemedium. Whenaneleotron issetintovibration bytheeleotromagnetic fieldoftheradiation, itgains energywhichmaybelostifthemolecule containing itmakesaninelastio collision withanothermolecule. Theenergylostservestoinoreasethe kineticenergyofthemolecules, andsoappearsasheat.Scattering arises fromthefactthatwhenaneleotron issetintovibration, itradiates energyinalldirections. Theamplitude oftheradiation fromeach eleotron canbeoomputed usingtheformula foranoscillating dipole (§10.9).Ingeneralthelossofenergybyscattering issmallcompared withthatlostbyoollisions. Theangulardistribution ofthesoattered radiation depends onthe relative phasesoftheosoillating electrons, andtheirdistribution in spaoe.Thephaseoftheinoident waveisoonstant overanywavefront, andsoalsoisthedisplacement oftheelectron relativetothenucleus duetotheactionoftheincident wave.Thevibrating atomstherefore formanarrayofoscillating dipoles,whiohareallinphaseacrossawave frontoftheinoident wave.Thetotalamplitude oftheradiation from thesedipolesinanygivendirection isfoundbysumming theamplitudes 17.8] THEORY OFTHEDIELECTRIC CONSTANT 499 fromtheindividual dipoles,andinformingthissumwemustallowfor thephasedelayinthewavescomingfromthevariousdipoles. Thenew wavefrontmaybefoundbyusingHuyghens' principle.Itisobvious thatitwillbeparalleltotheoldwavefrontsothatthereisnobending ofthewave.Wemust,however, allowforthephasedifference between the'real'secondary wavelets radiated fromtheoscillating dipolesand the'virtual' wavelets fromintermediate points(thatis,those'virtual' wavelets usedtogenerate thenewwavefrontinvacuo).Theresultof thisistomodifythephaseatthewavefront,sothatthephasevelocity isdifferent fromthatinfreespace,i.e.themedium hasarefractive indexdifferent fromunity. Inordertocompute thescattering atanangletotheincident wave, itisnecessary tohavesomeinformation aboutthedistribution ofoscil­ latingdipolesoverthewavefront.Ifthisdistribution isuniform, asin acrystal,thenthescattered wavesmayreinforce strongly incertain directions, formingadiffraction pattern. Thisispossibleonlyatwave­ lengthsofthesameorderasthedistance betweenthedipoles;thatis,for acrystal,wherethespacingisoftheorderof10-8em,atX-raywave­ lengths. Formuchlonger(optical) wavelengths thereisnodirection of strongreinforcement excepttheforward direction, andthescattering (diffraction) ispractically zero.Scattering willoccur,however, when thecrystalcontains imperfections wheretheatomsarenotuniforntly spaced.Inagasthemolecules arerandomly spaced,andtherewillbe arandomphasedifference betweentheindividual scattered wavesinall directions exceptthatparalleltotheincident wave.Thetotalscattered amplitude inanyarbitrary direction willcontainasumoftheform 2acos(wt-8,) =2acoswtcos3,+2asinwtsin8" iii whereaistheamplitude duetoanindividual dipole.Intakingthesum thetermscos8,andsin8,willbenearlyasoftenpositive asnegative, andthesumwillbeverymuchsmallerthanaN,whereNisthetotal numberofdipoles.Infactweshallgetjustthestatistical deviation from zero,whichisa,yN.Thiscorresponds toaddingtheintensities ratherthan theamplitudes, forthetotalintensity is {2acos(wt-3,)}2 =2a2oos2wtoos28,+2a2sin2wtsin28,+ , , , +22a2coswtsinwtcos8,sin8,+ , +~!{a2cos2wtcos8,oos8j+a2sin2wtsin8,sin8j+ }~} +2a2coswtsinwtoos8,sin8j}. 500 THEORY OFTHEDIELECTRIC CONSTANT [17.8 Foraverylargenumberofdipoles,thesumsoverthevariousphases maybereplaced byintegrals, sincetherewillbeauniform distribution ofthephasesovertherange0to217.Theonlynon-zerotermswillbethe averages overcos28iandsin28iwhichareeacht.Thetotalintensity is thus 2 •!aNcos2wt+}a2Nsm2wt=N(ta2), wherela2isthemeansquarevalueofthescattered intensity fromeach dipole. Thefractionoftheincident intensity lostbyscattering maybefound asfollows. Fromequation (10.72)theenergyscattered byanoscillating dipoleofamplitude Pois(perunittime) Z4 2W=oW'PQ. 1217c2 Nowthemeanincident powerperunitareaisN=!E5/Zo,whereEo istheamplitude oftheelectricfieldandZoistheintrinsic impedance offreespace.Hence,writingPo=OI.Eo,wehave (17.33) (17.34)SinceWhasthedimensions ofpower,andNofpowerjunit area,(]'isan area,knownasthe'scattering cross-section'. ForX-rays, W~wP'y,andfromequation (17.11)wehavethen 01.=e2jmw2•Hencethescattering cross-section isindependent of frequency, andhastheclassical valuederivedbyThomson Z2e4 (]'=__0_=6.65x10-25cm2•o6wm2c2 Forvisiblewavelengths andsubstances suchasthemolecules oftheair wehavew<{wP'andhence 01.=e2jmw;ifweassumeonlyoneresonant frequency permolecule, giving (17.35) Thisisthewell-known formula, originally derivedbyRayleigh ina different way,whichshowsthatthescattering shouldvarywiththe inversefourthpowerofthewavelength. Henceshorterwavelengths are scattered toamuchgreaterextentthanlongerwavelengths. Theblue colouroftheskyisduetoscattered sunlight; thetransmitted lightis complementary incolour,andtherisingandsettingsuntherefore appear red.Macroscopic particles suchasraindrops, whosedimensions arelarge 17.8] THEORY OFTHEDIELECTRIC CONSTANT 501 compared withthewavelength, scatterallwavelengths equally, and hencecloudsappearwhite. Ifthefrequency oftheincident radiation coincides withoneofthe naturalfrequencies ofamolecule, theinduced dipolemoment isvery largeandthescattered radiation isabnormally intense. Thisisknown as'resonant scattering', andmayreadilybeobserved, forexample, if abulbcontaining sodiumvapourisilluminated withthesodiumD-lines. Inallsuchscattering phenomena, theinduced dipolesdonotradiate paralleltothedirection ofoscillation oftheelectriccharge.Henceifthe incidentradiation isplanepolarized, therewillbenoscattered radiation inthedirection oftheelectricvector.Thisfactisusedtodetermine the planeofpolarization ofX-raysandy-rays.Iftheincident radiation is unpolarized, thescattered radiation willbepartlypolarized, ascanbe observed bylookingattheblueoftheskythroughpolarizing sunglasses. REFERENCES COCHRAN, W.,1960,Adv.Phys.9,387. COLLIE, C.H.,HAsTED,J.B.,andRITSON, D.M.,1948,Proc.Phys.Soc.60,145. COWLEY, R.A.,1962,Phys.Rev.Letters,9,159. DEBYE,P.,1929,PolarMolecules (DoverPublications, NewYork). TOWNES, C.H.,andSCHAWLOW, A.L.,1955,Microwave Spectroscopy (McGraw- Hill,NewYork). KITTEL, C.,1951,Phys.Rev.82,729. LANE,J.A.,andSAXTON,J.A.,1952,Proc.Roy.Soc.A,213,400. ONSAGER, L.,1936,J.Amer.Ohem.Soc.58,1486. PAULING, L.,andWILSON, E.B.,1935,Introduction toQuantum Mechanics (McGraw-Hill, NewYork). RAMSEY, N.F.,1956,Molecular Beam8(O.U.P.). SAXTON,J.A.,1952,Proc.Roy.Soc.A,213,473. SZIGETl, 0.,1949,Trans.Faraday Soc.45,155. PROBLEMS 17.1.Thestaticdielectric constants ofCO2andNHsaremeasured at00Cand 1000Catapressure ofIatmandthevaluesof103(1:-1)arefoundtobe: CO2NHs 0·988 8·34 0·723 4·87 Calculate thepermanent electricdipolemoment foreachgas,andalsotheradius ofthemolecule, assuming thepolarizability tobethesameasthatofaconducting sphere. (Answer: p=0and1·45Debyes; radius=1·4and1·8A;fromviscosity data. theradiiare2·3and2·2Arespectively.) 502 THEORY OFTHEDIELECTRIC CONSTANT 17.2.Thedielectric constant ofliquidheliumatitsboiling-point is1,048,andits densityis0'12~g/cm3•Calculate therefractive indexofthegasatN.T.P.,and estimate theradiusoftheheliumatom,assuming ittobehavelikeaconducting sphere. Compare theradiuswith(a)thatgivenbytheBohrtheoryforanatom withanuclearchargeoftwounitsinitsgroundstate,(b)withthatcalculated fromthediamagnetic susceptibility (seeProblem S.I). (AnBWer: n=1'000034; radius=0'59A;Bohrtheoryradius=0·26A.) 17.3.Intheupperregionsoftheatmosphere (theionosphere) thegasmolecules areionized, mostlythrough theeffectsofultraviolet radiation fromthesun. Showthatinaregionwherethenumberoffreeelectrons perm3isno,therefrac­ tiveindexforwavesoffrequency] (cis)'is (1-noes)T=(1-W$)T~(1-Slno)T mws€o WS ]S ' wherewpl27T'istheplasma.frequency. Atthisfrequency, therefractive indexfallstozero,andtheionosphere istotally reflecting evenatnormalincidence. Thefrequency atwhichthisoccursiscalled thecriticalfrequency andatmidday isabout4Mc/sfortheE-Iayeratlatitude 400N.Estimate themaximum valueofnointheE-Iayerfromthisfigure. (AnBWer: no=2xIOll/m3•NeglecttheLorentz field.) 17.4.Showthatinanionizedregionsuchasthatintheprevious question the product ofthegroupvelocity andthephasevelocity isequaltothesquareof thevelocity infreespace. 17.5.Aparticleofcharge-eandmassmperforms a.simpleharmonic motion' 8=80coswtundertheactionofarestoring force.Showthatthrough theradia· tionofenergy(givenbyequation (10.72»thetotalenergyoftheparticle fallsas W=lVoexp(-1't},where l'=(Zoesw2/67T'mc2)=(27T'Zoe2j2/3mcS). HereZo=intrinsic impedance offreespace,and]=w/27T'.Onthequantum theory,thechancethatanatomspendsatimetinanexcitedstatehastheprob. abilityoftheorderexp(-yt);henceshowthatthemeanlifetime 7"=(II')')ofa sodiumatominanexcitedstatebeforeemitting oneofthesodium D-lines (A=5900A)isroughly 1·6X10-8sec. 17.6.Bytheuncertainty principle, thewidth!:l.Eoftheupperenergylevelofthe previous question isgivenbytherelation 'TilE=(hI27T'),wherehisPlanck's constant. UsetherelationilE=h(ilf)toshowthatthisgivesalinewidthof !:l.]=(1'/27T').Thisiscalledthenaturalorradiation breadthoftheline,andthe sameresultisobtained onclassical theorybyaFourieranalysis ofthespectrum ofanoscillator whoseenergyisdecaying exponentially asW=leVoexp(-1't). Estimate thelinewidthsduetotheDoppler effectandtocollisions inagasat 10000Kand10-3atmpressure, andshowthat(a)at]=1010cis(A=3'·0cm) collision broadening isdominant, (b)at!=1015cis(A=3000A)Doppler effect isdominant, (c)at!=1018cis(A=3·0A)naturallinebreadth isdominant. 17.7.Foravibrating molecule, thepolarizability a:variesasa:=a:o(1+bx2}, wherex=acosptisthechangeinthenormaldimensions ofthemolecule dueto thevibration. Showthatifincident lightoffrequency w/27T'fallsonthemolecule, thescattered radiation willcontainlightoffrequencies (w±2p)j27T'. (Thisisthe classical explanation oftheRamaneffect.) THEORY OFTHEDIELECTRIC CONSTANT 503 17.8.Showthatforanarrowabsorption linethemaximum andminimum ofthe refractive indexintheregionofanomalous dispersion occuratthefrequencies wheretheabsorption coefficient hasfallentohalfitsmaximum value. 17.9.Theconductivity ofsea-water at200Cisabout2(ohm-metre)-l. Showthat theabsorption atI-emwavelength duetothisconductivity issmallcompared withtheDebyeabsorption, butthetwoareroughly equalatawavelength of about10em.(UsethedatagivenforpurewaterinFig.17.5;infacttheDebye relaxation timeissomewhat alteredbythesaltsdissolved.) 17.10.Discussthepropagation inandreflection fromthesurfaceofthesea.of radiowavesinthelightofthedatagivenintheprevious question. 18 ELECTRONS INMETALS 18.1.Kinetics offreeelectrons inmetals INChapter 4anoutlineofDrude'stheoryofmetallic conduction was given,anditwasshownthatthisclassical modelgivesaplausible explanation ofthemechanism ofconductivity andisalsosuccessful in accounting fortheratioofthethermaltotheelectrical conductivity. Theclassical theorypredicts alargespecificheatof3R/2permolefor theconduction electrons, however, whichisnotobserved experimentally; thisdifficulty wasovercome onlywhenitwasrealizedthatquantum statistics mustbeusedratherthanclassical statisticswhen dealingwith electrons inmetals.ThisrequirestheuseoftheFermi-Dirac distribu­ tionfunction equation (4.17)insteadoftheMaxwell-Boltzmann function constant Xexp(-W/kT),towhichequation (4.17)approximates when (W-WF)/kT>1. Itisalsonecessary totakeaccount ofthewave-like properties ofthe electrons, andin§4.2anelementary accountofthiswasgivenusingthe deBroglierelationandtheanalogywithwavesinabox.Inarealsolid thewave-like natureisimportant foryetanotherreason:thewavelength iscomparable withtheinter-atomic distance, givingrisetodiffraction effects.Beforeconsidering these,weshalldiscussthesimplerproblem ofthewaveequation forfreeelectrons. Thewaveequation forafreeelectron oftotalenergyWinanun­ bounded regionwherethepotential isVis !!!..V2ifJ+(W -V)ifJ=O.2m(18.1) (18.2)",282ifJ--+(W-V)ifJ =0,2m8x2Forsimplicity weconsider firsttheone-dimensional case,forwhichthe waveequation is andforconvenience wefurtherassumeV=0everywhere. Thenthe solutions ofthisequation areoftheform (18.3) 18.1] ELECTRONS INMETALS 505 Themomentum oftheparticlePxisgivenby +00 +00 Px=-jlififJ*~=dx=-jli(jkaJfifJ*ifJdx=likx(18.4) -00 -co +00 sincethenormalization ofthewavefunction requiresfifJ*ifJdx=1. -00 Inordertosatisfyequation (18.2)wemusthave (li2/2m)k~ =W,or W=(li2/2m)k~ =p~/2m (18.5) sothatWcorresponds tothatpartofthekineticenergyoftheelectron associated withitsmomentum Pxinthex-direction. Thethree-dimensional equation (18.1)alsohasasimplesolution in Cartesian coordinates, corresponding totheproductofthreefunctions ofthetype(18.3).Thissolution is ifJ=Aexp(jkxx)exp(jkyy)exp(jkzz) =Aexpj(kxx+kyy+kzz) =Aexpj(k.r) (18.6) since(x,y,z)arethecomponents ofthevectorr,andwecansimilarly regard(kx'ky,kz)asthecomponents ofavectork,knownasthewave vector.Inordertosatisfyequation (18.1)wehave W=(li2/2m)(k~+k~+k~) =(p~+p~+p:)/2m =p2/2m, (18.7) wherepisthemomentum vector,withcomponents (Px'Py,Pz)'These components ofparejustIitimesthoseofk;thatis,wehavethe deBroglierelation k=pili (18.8) whichwasusedin§4.2.Thewavelength associated withtheelectron is21T/k=hlp. Inapplying thefreeelectron modeltoametal,weassumethatthe electrons moveinaregionofconstant potential, withasharprisein thepotential attheboundaries ofthemetal.Sincetheelectrons donot haveenoughenergytosurmount thisbarrier,theyareconfined within themetal(weneglectphenomena suchasthermionic emission, which areinsignificant, affecting onlyaminuteproportion oftheelectrons). Ourmodelthusassumes arectangular potential well,suchasisshown inFig.18.1(a)foronedimension. Takingthefloorofthewelltobeat V=0,thesolutions ofthewaveequation areoftheform(18.6)inside thewell.Outsidethewell,wherethepotential Vo~W,sothat(W-Vo) isnegative, thesolutions arerealexponentials, showingthatthechance offindinganelectron outside,whichisproportional to .1•••1.=[{8m("Vo-W)}l]..,....,..exp li2 x , 506 ELECTRONS INMETALS [18.1 fallsoffveryrapidlywithdistance. Apropersolution oftheproblem requiresthatthewavefunctions andtheirderivatives becontinuous at theboundary, butifwemaketheapproximation oftakingVotobe infinitethisreducestomaking.pvanishattheboundary. Thenthe problem issimilartothatofelectromagnetic wavesinaperfectly con­ ductingbox;ifthelatterisrectangular withdimensions (a,b,c),the FIG.18.1.(a)Rectangular potential wellassumedinfreeelectron modelofametal. (b)Actualpotential variation, showing sharpfallneareachpositively-charged ion. allowedsolutions (cf.Problem 11.12)areasetofstanding waveswhere .pisaproductoftermssuchas sin(TTlxja)sin(mnjb)sin(-7Tnzjc)cos cos ycos andthewavelength isgivenbyequation (11.42) Forelectrons atthetopoftheFermidistribution thewavelengths involved areoforder10-7cmorless,whichareverysmallcompared withthedimensions ofametalofordinary size.Thespacingofthe allowedwavelengths istherefore veryclose,andthenumberinagiven wavelength rangecanbecomputed usingtheapproximations adopted inthetheoryofheatradiation, asin§4.2. 18.2.Theener~ybandapproximation Atthisstagewearestillmakingthearbitrary assumption thatin ametalsomeoftheelectrons aredetached fromtheirparentatoms andaremerelyboundtothemetalasawholebyapotential wellinside whichtheymovequitefreely.However, manysolidsareverygood electrical insulators inwhichwemustassumetherearenosuchfree electrons. Thereisalsotheintermediate classofsolids,thesemi­ conductors, whicharemuchpoorerconductors thanmetalsandwhich generally possessanegative ratherthanapositive coefficient ofresis­ tivity.Tounderstand whythesedifferent typesofsolidsexistwemust 18.2J ELECTRONS INMETALS 507 consider theinteraction between theelectrons andnucleiwhenthey arecloselypackedinasolid,wheretheinteratomic distance isofthe sameorderastheatomicradius.Thepotential energyofanelectron thenvariesratherasshowninFig.18.1(b),fallingsteeplywhenthe electronapproaches apositively-charged nucleus. Obviously themotion oftheelectrons insuchapotential isaverycomplicated problem and cannotbesolvedexactly. Approximate methods mustbeused,whose natureisillustrated byapproaching theproblem fromtwodifferent standpoints......--------.""-....------<.............. ~~;"':"~...L-I'-"'''''''_____ --:c.::::-- ---~~~..;:..,;;,~~ Freeatom Solid FIG.18.2.Sharpenergylevelsinafreeatomandthecorrespond­ ingbandsinasolid(thetopbandisshownonlypartlyfull).The arrowsindicate allowed transitions givingX-rayemission bands whenanelectron hasbeenionizedoutofthelowestenergyband. Inanisolatedatomtheelectrons aretightlyboundandhavediscrete, sharpenergylevels.Whentwoidentical atomsarebrought together, theenergylevelsofeachatom,whichareinitiallythesame,aresplit intotwo,onehigherandonelowerthanthecorresponding levelsofthe separated atoms.Thesplitting onlybecomes appreciable whenthewave functions oftheelectrons ondifferent atomsbegintooverlapconsider­ ably;atagivendistance itistherefore greatest fortheoutermost electrons andleastfortheinnerelectrons.Ifmoreatomsarebrought together, morelevelsareformed,andforasolidofNatoms(whereN isaverylargenumber) thelevelsaresoclosetogether thattheyform analmostcontinuous band.Thewidthofthisbanddepends onthe degreeofoverlapofelectrons onadjacent atom.s,andisagainlargestfor theoutermost atomicelectrons. Fig.18.2isaroughdiagram showing howtheatomiclevelsdevelopintobandsastheatomsarebroughtcloser together. 508 ELECTRONS INMETALS [18.2 Theproblem maybeapproached fromtheopposite viewpoint bycon­ sidering howthemotionoftheelectrons, previously assumed tobe movingfreelyintheflat-bottomed potential wellof]j'ig.18.1(a),is modified whenweallowforthedropofthepotential neareachatomic nucleusshowninFig.18.1(b).Inacrystaltheatomsformaregulararray, andthepotential hastherefore aperiodicvariation inthreedimensions. v ,---TVo --a -bo (a-b) a whereV=0 whereV=Yo.FIG.18.3.Periodic rectangular potential wellassumedintheone-dimensional modelof KronigandPenney (1931). Theeffectofthisperiodicity inthepotential canbeunderstood bycon­ sidering asimpleone-dimensional case.Theelectricfieldsofother electrons willbeneglected, andthepotential assumed tohavetheform ofarectangular waveasinFig.18.3,where V=0when0<x<(a-b); V=Yowhen--b<x<O. Itmaybeshownthatthesolutions ofthewaveequation (18.2)inthis caseareoftheform .1.()('k)'f'=UxexpJxx , whereu(x)isaperiodic function ofxsuchthatu(x+a) =u(x),i.e.u repeatsitselfwiththesameperiodicity asthepotential. Wetaketwo different functions u1=[Aexp(jqx)+Bexp( -jqx)]exp( -jkxx) andu2=[Cexp(rx)+Dexp(-rx)]exp(-jkxx) Hereqandrmustsatisfytherelations q=(2mWjli2)tandr={2m(l'o- W)j1i2}t, andinaddition thesolutions forthetworegionsmustjoinsmoothly at theboundaries sothatwemusthaveU1=U2and(oul/ox) =(8u2j8x) bothatx=0andx=(a-b).Thisgivesfourequations fromwhichthe constants A,B,C,Dcanbeeliminated, yielding acomplicated relation 18.2] ELECTRONS INMETALS 509 betweenkandW.Considerable simplification isobtained byallowing b-+0andYo-+00insuchawaythattheproduct (bYo)remains finite. Then,ifcisthelimiting valueof(2m"Voabfli2),oneobtains sinqacoskxa=c--+cosqa. (18.9)qa Ther.h.s.ofthisequation isplottedasafunction of(qa)inFig.18.4for avalueofc=217:allowedvaluesofkxareobtained onlywhenthenmc­ tionliesbetween 1and-1,andhenceonlycertainrangesofvaluesofq ----..(qa) FIG.18.4.Plotofthefunction givenbyequation (18.9)whenc=217asafunction of (qa).Therangeswhichgiverealvaluesofkareshaded. areallowed. Sinceq=(2mWfli2)1,thismeansthattheenergyWis restricted toliewithin cert~inranges,whichformtheallowedenergy bands.Theallowedbandsarenarrowest forsmallvaluesofq(lowvalues oftheenergyW),andbecomebroaderasWincreases, theunallowed bandsgettingnarrower, justasinFig.18.2. Thistreatment ofasimplemodel,duetoKronigandPenney(1931), illustrates howallowedandforbidden energybandsariseinasolid.Their occurrence isassociated withtheperiodicstructure ofthecrystallattice, andtwoanalogies mayhelpinunderstanding thispoint:(a)X-rays whosewavelength satisfiesthecondition forinterference between suc­ cessiveBraggplanesinacrystalarestrongly diffracted, whileothersare transmitted. Theelectron wavelengths inasolidareofthesameorder 510 ELECTRONS INMETALS [18.2 asX-raywavelengths, andthestrongdiffraction corresponds tothe forbidden wavelengths (valuesofka;whicharenotallowed); (b)acon­ tinuoustransmission linetransmits allwavelengths freely,whereasthe periodic structure ofafilterrestricts freetransmission tocertainbands ofwavelength. InfacttheKronig-Penney modelcorresponds toa continuous transmission lineinwhichidentical lumpedimpedances have beeninsertedatregularintervals, adistance aapart.Wavelengths whicharelongcompared withaarefreelytransmitted, butasthe wavelength Aisreduceddispersion setsinasinafilter,andwhena=A/2 thereflections fromeachlumpedimpedance areinphaseandweenter astopband. (18.12) whereParticleaspects Theforegoing treatment ofelectrons inasolidconsiders themas wavesoccupying thewholevolumeofthesolid;thesewavesarethe stationary states,solutions ofthetime-independent waveequation. We needtoknowhowtheelectrons behaveundertheinfluence ofaforce (electric ormagnetic); thisisessentially aparticle description, which mustberelatedtothewaveaspectoftheelectron. Asshowninbooks onelementary quantum mechanics, afreeelectron mustbeconsidered asawavepacket,wherethegroupvelocity corresponds totheparticle velocity. Thex-component ofthegroupvelocityisgivenbytherelation Va;=a~J~=~:~ (18.10) whichisanalogous totheformula dw/dflusedforthegroupvelocity in§11.6;theenergyWcorresponds tonwandka;tothephaseconstant fl. Arigorous analysis showsthatequation (18.10)isvalidforanelectron movingintheperiodic potential ofacrystallattice,andalsothat dPa;/dt=Fa;,wherePa;-nka;andFa;isthecomponent ofanexternal force.Differentiation ofequation (18.10)thengives dVa;_d(lOa_lo2Wdka;_.1o2WdPa;_1F(1)dt-dthate-hok2fit-n2ok2dt-m*a;'·8.11a; a; a; 1lo2W m*=n2ak2•a; Thislastequation definesthequantity m*,whichinequation (18.11) clearlyhasthedimensions ofmass,andisknownasthe'effective mass'. Itfollowsalsofrom(18.11)thatPa;=m*va;' ThevalueofWforafreeparticleisgivenbyequation (18.5),andit caneasilybeverifiedthatequations (18.10)-(18.12) satisfy(18.5)with 18.2] ELECTRONS INMETALS 511 m*=m,sothattheeffective massisequaltothetruemassforafree particle. Foranelectroninaperiodicpotential theeffective massmay departmarkedly fromthetruemass.Theadvantage oftheconceptof effective massisthatthedynamic behaviour ofanelectroninaperiodic potential canbetreatedasifitwereaparticleofmassm*.ThedifferenQe between m*andthetruemassmrepresents theeffectonthemotionof theelectronwhichresultsfromtheelectricpotential oftheionsforming thecrystallattice;whenaforceisappliedtotheelectron, itschangein momentum isdifferent fromthatofafreeelectron, andp=1ikisoften referredtoasthe'crystalmomentum'. ThefactthatPz=m*vzand notmvzdoesnotrepresent abreakdown ofNewton's lawsofmotion, sincetheresidual momentum istakenupbythelattice.Experiments todetermine theratioofcurrenttomomentum, similartothoseof Kettering andScottdescribed in§3.1,havebeencarriedoutbyScott (1951)andBrownandBarnett(1951).Theyfindthateveninthecase ofsubstances wherethecurrentiscarriedby'positive holes'(seebelow), theratioofcurrenttonetmomentum hasthesamesignandisnumeri­ callythesameasforfreeelectrons. TherelationbetweentheenergyWandkz'asderivedfromtheKronig­ Penneymodelorotherwise, hastheformshowninFig.18.5.Itdoesnot differgreatlyfromthatforafreeelectronexceptneartheedgesofthe allowedband.Atthepointswherecoskza =±1,ofwhichthefirstis atkz=±1T/a,thereisadiscontinuity intherelationbetweenWandkz; differentiation ofequation (18.9)showsthatdq/dkzisproportional to sinkza,andhenceiszeroatsuchpoints.SinceWocq2,itfollowsthat 8W/8kzisthenalsozero,andfromequation (18.10)thismeansthatthe electronvelocityiszeroattheedgeofanallowedzone.Thisisthepoint atwhichthewavelength issuchthattheelectron wavesarestrongly diffracted (inonedimension thismeansreflected) andformasetof standing waves,notravelling wavesbeingallowed. ItcanbeseenthattheshapeofthecurveofWagainstkzinFig.18.5 meansthattheeffective massm*becomes negative nearthetopofan allowedband,because82W18k~becomes negative. Application ofaforce +Fzwillincrease kz'butaskzapproaches +1T/atheslope8Wj8kzdimin­ ishes,sothatbyequation (18.10)thevelocity Va:decreases, asweshould expectifthemasswerenegative. Justbelowffo,thetopoftheband,we haveapproximately, since8W/8kz=0atkz=1T/a,andm*isnegative, W=Jv.-dk8W+1.(dk )282W=Jv.+1i2(dkz)2=Jv.+(dpz)2.oz8k 2z8k2 02m* 02m*z a: 512 ELECTRONS INMETALS [18.2 Animportant property ofafullband(anallowedbandwhereallthe statesareoccupied) isthatitcancarrynoelectriccurrent,sinceforevery electronwithapositivevalueofkxthereisanotherwiththevalue-kx• Suppose wehaveabandwhichisfullexceptforonestateatthetop ofabandwhichhasanegative valueofkx•Ifanelectron occupied this state,itwouldhavenegative chargeandnegative mass;itsmomentum w.. k..(-vel. k.(+ve) FIG.18.5.PlotofW",against k",showing bandstructure duetoperiodic potential of lattice. - - -W'"against k",intheabsenceoftheperiodic structure. Px=likxwouldbenegative, butitsvelocity Vx=Px/m*wouldbe positivesothatitwouldcarryanegative current. However, thepresence ofsuchanelectron wouldfilltheband,andthenetmomentum and currentwouldbezero.Hencethemomentum andcurrentduetoall theotherelectrons mustbeequivalent tothatofoneparticlewith positivemomentum andgivingapositive current, andthesamevalue of[m*[.Suchaparticleiscalleda'positive hole',sinceitsbehaviour corresponds tothatofaparticlewithpositivecharge(andpositivemass); itisanalogous totheholeinthefilledbandsofelectrons inDirac'stheory ofthepositron. Theadvantage oftheconceptofpositive holesisthat themomentum andcurrentofanearly-filled bandwithnemptystates canbeattributed tothepresence ofanequivalent numbernofentities whichbehavelikeordinary particles withpositive chargeandeffective massm*;theenergythenbecomes W=JYo_(dpx)2/2m*. 18.2] ELECTRONS INMETALS 513 Threedimert8Wns Treatment ofathree-dimensional latticecorresponding toarealsolid, evenwithsimplified modelsofthepotential variation, isverycomplex andwillnotbediscussed here.ThewavefunQtion ofanelectron asso­ ciatedwithawavevectorkisoftheform I/J=u(r)expj(k.r), (18.13) where u(r+an)=u(r). Hereanisatranslation vectorrepresenting therepetitive property of thelattice;inparticular, thatthepotential energyisperiodic, obeying therulethatVatthepoint(r+an)isthesameasatr. Fromtheanalogy withX-raysitisclearthatthevaluesofkatwhichstrongdiffraction occurswilldependonthedirection ofk;thatis,thevalueofkatwhich thereisadiscontinuity intheenergy(theboundary ofazone)isa function ofdirection. Ifwedrawavectorkin'k-space' whoselength corresponds tothisvalue,andrepeatthisprocessforallpossibledirec­ tions,theendsofthevectorswillmapoutathree-dimensional figure, knownasa'Brillouin zone'.Itsconstruction involves onlygeometry, anditssymmetry isrelatedtothesymmetry ofthecrystallattice. Valuesofkwhosevectorsendonpointsinsidethezonecorrespond toallowedenergies; thosewhichterminate atthezoneboundary corre­ spondtodiscontinuities intheenergy. Higherzonescorresponding to higherallowedenergybandsexist,butitispossibletobringallwave vectorsintothefirstzone(intheone-dimensional casethisprocedure corresponds totakingvaluesofkzainequation (18.9)onlybetween -77 and+77).Valuesofkinthefirstzonethencorrespond tomorethan oneallowedenergy,butingeneralweareconcerned onlywiththeone bandwhichispartlyfilledwithelectrons. Attheabsolute zeroof temperature electrons fillthisuptoacertainenergy,theFermienergy, anditistherefore ofinteresttodrawplotsofconstant W,or'energy surfaces' ink-space. Calculation oftheenergysurfacesisverycomplex: theresultsobtained byonemethodareshowninFig.18.6forasimple cubiclattice.Energysurfaceswellwithinthezoneare'spherical inshape, butthisisbynomeanstruenearthezoneboundary; ingeneralatthe boundary BW/Bk=0andtheenergysurfacesmustendnormaltothe boundary. Theboundary ink-spacebetweenthefilledandemptystates at0°Kfollowstheenergysurfacecorresponding totheFermienergy, andisknownasthe'Fermisurface'. Onlyelectrons neartheFermi surface(see§18.3)cantakepartinconduction processes, andmany detailsoftheirbehaviour aredetermined bytheexactshapeoftheFermi 851110 L1 514 ELECTRONS INMETALS [18.2 surface. Animportant property istheeffective mass,whichingeneral isafunctionofdirection. Nearthetoporbottomofabandtheenergy maybeexpanded inapowerseriesink,thelowesttermsbeingquadratic; theycanbereducedbyasuitablechoiceofaxestotheform W=Jv.±ln2{k~+k;+k:}. (18.14)om* m* m*xyz Heretheuppersignmustbetakenforelectrons nearthebottomofa band,andJVoisthentheenergyatthebottom; whilethelowersign 1c~=-~a1c.=+~a k~=+!"a 1c.=-~a FIG.18.6.Section through theconstant energysurfaces forasimple cubiclatticeobtained byonemethodofcalculation (the'tight-binding' approximation). Theenergysurfaces endnormally tothezoneboundary exceptwheretwoendatthesamepoint. shouldbeusedforholesnearthetopofaband,andl¥oisthentheenergy atthetop.Thusforholestheenergyappearstobemeasured downwards fromthetopoftheband,apointwhichweshallreturntoinconsidering semiconductors (Chapter 19). Oorrelation energy Animportant effectwhichhasbeenneglected inourtreatment isthe electrostatic repulsion oftheconduction electrons. Thistendstokeep theelectrons apart,andthechanceoffindingtwoelectrons closetogether islessthanitwouldbeonourassumption thattheirmotioniscompletely independent ofeachother;inotherwords,thereisacorrelation between 18.2] ELECTRONS INMETALS 515 theirmotions. Thereisafurthereffectduetothefactthattheoverall wavefunction fortheassembly ofelectrons mustbeantisymmetric, whichissimilarinnaturetothe'exchange interaction' discussed in Chapter21.Thesetwoeffectscontributeto the'correlation energy'and areimportant incalculating thecohesive energyofametal.Theelectro­ staticrepulsion isalong-range interaction whichgivesriseto'plasma oscillations' (see§4.9),forwhichthecharacteristic frequency inametal isoforderIOU;cis.Atlowfrequencies thedynamical properties ofthe electrons arenotgreatlyaltered,afortunate circumstance whichmakes asimpletreatment neglecting thecorrelation energymoreaccuratethan mighthavebeenexpected. 18.3.Conductors andinsulators onthebandtheory Inoldertheoriesthefactthatsomesolidsareelectrical conductors whileothersareinsulators wasexplained byassuming thatinthe insulators alltheelectrons belonging toeachatomwerefirmlybound tothatatom,whileinconductors someoftheouterelectrons were detached fromtheirparentatomsandabletomovefreelythroughout thewholevolu.meofthesolid.Onthebandtheorythereisnosuch distinction between 'bound'and'free'electrons; theelectronic wave functions spreadoutthroughthewholevolumeofthesolid,thoughthe statesoflowerenergy(corresponding totheinnerelectrons ofasingle atom)havetheelectronic density(t/Jt/J*ofthewavefunction) greatest :neareachnucleus. Howthendoesthebandtheoryexplaintheoccurrence ofbothconductors andinsulators? Attheabsolute zerooftemperature theelectrons inasolidwillhave thelowestpossibleenergyconsistent withthePauliexclusion principle, andtheywillfilltheenergybandsfromthebottomupwards. Thelowest energybandswillbefll.llyoccupied, butthehighestoccupied levelmay occurinthemiddleofanallowedband.Thestateoflowestenergyis oneinwhichasmanyelectrons havepositivevaluesofkashavenegative values,sothatthenetcurrentiszero.Toestablish acurrentflowsome electrons mustbetransferred fromnegative valuesofktopositivevalues, butbecauseoftheexclusion principle thisispossibleonlyiftheymake transitions tounoccupied statesofhigherenergy,theenergybeinggained byacceleration throughtheapplication ofanelectricfield.Inweakfields thiscanoccuronlyifadjacent levelsareunoccupied; Le.ifthetopofthe Fermidistribution comesinthemiddleofaband.If,ontheotherhand, thehighestoccupied bandiscompletely full,anelectron mustgain sufficient energyfromamovement intheappliedelectricfieldtoraise 516 ELECTRONS INMETALS [18.3 itintothenexthigherband.Thisrequires enormous electricfields, andforordinary fieldstrengths thesubstance isaninsulator. Onthispictureitisreadilyseenthatthealkalimetalssuchaslithium, sodium,potassium, etc.,willbegoodconductors, fortheiratomspossess onlyonevalenceelectroninan8-state,whereastheenergybandinthe solidcorresponding tothisatomicstaterequirestwoelectrons peratom tofillit.Thealkalineearthelements, magnesium, calcium, etc.,have twosuchelectrons, whichwewouldexpecttofilltheband,makingthese substances insulators. Theyareinfactquitegoodconductors, andthe ,W ~"'----3d g(W)+-- FIG.18.7.Energybandsfornickel. Totherightisshownthebandwidth of48and3dstatesasafunction ofinteratomic distance a(aoisthevalueforsolidnickel). Totheleftisshowng(W),theshadedarea indicating thefilledpartsofthebands. reasonforthisisthattheenergybandscorresponding tothe8-andp­ statesintheatomaresobroadinthesolidthattheyoverlapappre­ ciably.Thestateoflowestenergyisthenonewheretheelectrons par­ tiallyfillboththe8-andp-bands,andconduction ispossible.Intransition elements thesituation ismorecomplex becauseoftheenergybands corresponding tod-electron states.Incopperthe3dbandiscompletely filledandthereisoneelectronperatominthe48band,makingitagood conductor. Iniron,cobalt,andnickelthe3dbandisnotcompletely filled;itisarathernarrowband,sinced-electron wavefunctions donot spreadasfaroutas8-electron wavefunctions, andinteracti9ns between d-electrons onadjacent atomsaresmallerthaninteractions between 8-electrons. Thenarrow 3dbandisoverlapped byabroad48band, 18.3] ELECTRONS INMETALS 517 andthe3dband(whichcancontain10electrons peratom)givesan abnormally highvalueofg(W)asshowninFig.18.7. Information aboutthewidthsofenergybandsinthesolidstatecan beobtained directlyfromsoftX-rayemission spectra(see,forexample, Skinner, 1938).Ifanelectron isexcitedoutofaninnershell,then electrons inoutershellsmaketransitions totheinnershell,emitting X-rays. ForafreeatomX-raysofdiscretewavelengths areobtained, sincewehavesharpenergylevels.Inasolidabandofwavelengths is obtained whosewidthisthesumofthewidthsofthebandswhichthe electron leavesandenters;ifthelatterbandcorresponds toaninner electron shell,itswidthis smallandtheobserved widthispractically entirelythatoftheinitialband(compare Fig.18.2).Sincetheelectrons comeonlyfromthefilledpartoftheband,theobserved widthisthat onlyofthefilledpart,notthewholewidth. 18.4.Specific heatoftheconduction electrons Thespecificheatoffreeelectrons canbefoundfromtheenergy distribution function equation (4.18)derivedfromFermi-Dirac statistics. Theenergyisgivenby 00 U=fWg(W)dW o andthespecificheat0=dUjdT. Sincetheintegration mustbecarried outbyapproximate methods, wequotetheresultfortheinternalenergy atatemperature T,whichis(fornelectrons) (18.15) Herethedifference betweenUandUoisthefirsttermofapowerseries inascending powersofT,butfurthertermsarenegligible atordinary temperatures. Thespecificheatoftheelectrons (perunitvolume) is (18.16) where{g(W)}FisthedensityofstatesatW=WF;onsubstituting from equation (4.13)weobtain 'TT2k2T0v=n2W F• (18.16a) Comparison withtheclassicalvalue,Ov=ink,showsthatthequantum statistical valueissmallerbyafactoroftheorder(kTIWF). Thereason IllS ELECTRONS INMETALS [IS.4 forthisisthatonlyasmallfraction ~(kTjWp.)oftheelectrons atthe topoftheenergydistribution curveareabletoincrease theirkinetic energy,asillustrated bythedistribution curveofFig.4.3.Theincrease inenergyoftheseelectrons is~kT,andsothetotalinternal energy increases byanamount ~n(kT)2jWp.. Electrons inthemiddleofthe bandcannotberaisedtohigherenergies unlesstheycanreachenergy levelsaboveWp.,sincealltheavailable energylevelsinthemiddleofthe bandarealreadyoccupied byelectrons. 0·06 A B o 10 20 TOK FIG.18.S.Specifioheatofoobaltatlowtemperatures (afterDuyckaerts, 1939). Aexperimental curve. Beleotronic contribution; 0",=12·0X1O-4T. olatticecontribution; 0",=465(T/443)3, where443istheDebyeeforcobalt. Theunitsarecal/gatom/deg. SincekT~m"atordinary temperatures, theelectronic specificheat willbeonlyasmallfractionofthatpredicted byclassical theory,and thedifficulty ofthelargeexcessspecificheatpredicted bythattheory formetalsisremoved. Anexperimental testofequation (18.16)is possible onlyatlowtemperatures, wherethespecificheatassociated withthelatticevibrations ofasolidfallsveryrapidly. According to thetheoryofDebye,thespecificheatfromthiscauseisproportional to T3atsufficiently lowtemperatures, andeventually thiswillbecome 18.4] ELECTRONS INMETALS 519 smallcompared withtheelectronic specificheat,whichfallsonlywithT. Thespecificheatsofanumberofmetalshavebeenmeasured, andbelow about200Ktheyarefoundtofollowalawoftheform Oy=aT3+bT. (18.17) Figure18.8showstherelativemagnitudes ofthetwocontributions to thespecificheatofcobaltattemperatures below200K.Thismetalis ferromagnetic, andlikeanumber ofothertransition groupmetals, ~ ~1·6 'i'1·2 ~ -3.:; ~-4 ~00 ~810 18 PI(Deg l) FIG.18.9.PlotofCIPagainstpIforcopper(Corak, Garftmkel, Satterthwaite, andWexler, 1955).Theintercept atpI=0givesthe coefficient boftheelectronic specificheat. showsanabnormally highelectronic specificheat.Inmanymetalsthe electronic specificheatispredominant onlybelowabout50K,and ratherprecisemeasurements arerequired todetermine itaccurately. Rearrangement ofequation (18.17)showsitmaybewrittenintheform OylT=aT2+b (18.17a) andbyplotting thequantity OylTagainstT2astraight-line graph shouldbeobtained whoseintercept givesthevalueofb.Figure18.9 showsatypicalgraphforcopper. Themeasured valuesoftheelectronic specificheatsofanumberof representative metalsareshowninTable18.1.Tocompare themwith valuescalculated fromequation (18.16)weneedtoknowthevalueofn, thenumberofconduction electrons perunitvolume,andof~,which byequation (4.11)isagaindependent onn.Itissimplesttodiscussnin termsofthenumberofconduction electrons peratom.Formetalssuch ascopperandsilver,wemayreasonably expectoneconduction electron peratom,andforberyllium andmagnesium two.Withthetransition metalsnickel,palladium, andplatinum, whichbelongtothe3d,4d, and5dtransition groupsrespectively, theenergybandsarenearlyfilled, andthenumberoffree'particles' isdetermined bythenumberof'holes' 520 ELECTRONS INMETALS [18.4 (18.20)(18.19)intheband;frommagnetic evidence theseamounttoabout0·6holes peratom.Usingthesevaluesofthenumberofcarriersperatom,we cancompute theelectronic specificheatfromequation (18.16),andin eachcasesomedeviation isfound.ThelastcolumnofTable18.1gives theratiooftheobserved tothecalculated electronic specificheatfor freeelectrons. TABLE18.1 Electronic specificheatsofsomemetals Number oJRatioojobserved conduction valueojelectronic CvlT(inunitsojelectrons spec.ht.tothatgiven Metal10-'calldegllg atom)peratombye,quation (18.16a) Cu 1·80 1 1'5 Ag 1·54 1 0·95 Be 0·54 2 0'46 Mg 3·25 2 1'33 Ni 17·4 0·6t 28 Pd 31 0·55t 27 Pt 16 0·6t 13 Theobserved valuesaregivenincolumn2;column 4givestheratioofthesevalues tothosecalculated fromequations (18.16a)and(4.11)assuming thenumberofconduc. tionelectrons (orholes)peratomgivenincolumn 3.Thisratioisinterpreted asthe ratiooftheeffective massm*tothefreeelectron massm. tNo.ofholesperatom,basedonmagnetic evidence. Thereasonforthesediscrepancies isthatweareusingformulae derivedforfreeelectrons, whereas weknowthatinasolidtheirmotion ismodified bytheperiodic potential.Itwaspointedoutin§18.2that thismodification canbeallowed forbyusingtheeffective massm* insteadofthetrueminmanyoftheformulae derivedforfreeelectrons. Thusequation (4.II)fortheFermienergybecomes 11,2WF=2m*(3172n)f, (18.18) whilethedensityofstatesg(W)becomes (W)=_I_(2m*)~ Wig 217211,2 =3nWi/2W}, thelastrelation beinginformthesameasforfreeelectrons. These relations showthat{g(W)}Irisproportional tom*,andinversely propor­ tionaltoWF•Thusforcopper(seeTable18.1)theobserved specificheat isabout1·5timeslargerthanthevaluecalculated onthebasisoffree 18.4] ELECTRONS INMETALS 521 electrons, fromwhichweconclude thatm*fm=1·5inthiscase,and thattheFermienergy JVp.shouldbeabout4·7eVinsteadofthevalue 7·0eVgiveninTable4.1. Thiseffectonthespecificheatcanbeseeninanotherwaywhichdoes notinvolvetheconceptofeffective mass.ThePauliexclusion principle restrictsthenumberofpointsinmomentum spacetotwo(including the electron spin)perelementary volume(h3fV),andsofixesthenumber ofstatesinagivenrangeofwavevectorktok+dk.Theeffectofband structure istoaltertherelation between Wandk,sothatthevalue ofg(W)ischanged. Fromequation (18.16)thespecificheatispropor­ tionaltog(W),sincedoubling thevalueofthedensityofstatesmeans thattwiceasmanyelectrons canincrease theirenergyforagiven temperature increase. Narrowbandshaveexceptionally largevalues ofg(W),asshowninFig.18.7,thusgiv.ingrisetoabnormally large valuesoftheelectronic specificheatintransition elements. 18.5.Electrical andthermal conductivity ofmetals Ontheclassical theoryoffreeelectrons, theelectrical conductivity ofametalisgivenbyequation (4.3).Forelectrons inaperiodicpotential thisformulaholdsprovided wereplacethetrueelectron massmbythe effective massm*,sothatwehave u=n(e2/m*)T=ne2l/m*v, (18.21) wherenisthenumberofelectrons perunitvolumeandTistherelaxation timedefinedin§4.1.listhemeanpathlengthbetween collisions, here takenasVT,wherevisthemeanelectron velocity. Sinceonlyelectrons attheFermisurfacecanbeaccelerated andgainenergy,thevalueof thevelocityrequired isthatcorresponding toJVp.;thisvelocity isabout 108cm/secformostmetals,andsince T-~10-14secatroomtempera­ ture,themeanfreepathisoftheorderof10-6cm,orabout100times theatomicspacinginasolid. Classical physicsgivesusnomethodofcalculating themeanpath length,nordoesitsuggestinwhatmanneritmightvarywithtempera­ ture.Sincethenumberandtheenergyoftheelectrons atthetopof theFermibandvariesinsignificantly withtemperature, equation (18.21) showsthatanychangeintheresistance mustbeassociated withachange inthemeanpathlength.Mostmetalsshowaresistance whichisroughly proportional totheabsolute temperature atroomtemperature and above,butatlowtemperatures theresistance fallsmarkedly belowthe valuegivenbythislaw.Anytheoretical approach tothisproblem must 522 ELECTRONS INMETALS [18.5 bemadethroughthewavetheory,andisextremely complicated. Here weshallattempttogiveonlyanoutlineoftheresults. ItwasfirstpointedoutbyHouston thatthemeanpat.hlengthofan electroninaperfectly regularlatticeofatomsshouldbeinfinite.Ifan electron isinanallowedenergystate,thenthatisastationary state, andintheabsenceofperturbations, theelectron willcontinue inthat stateoffixedenergy,andhencefixedvelocity, indefinitely. Realmetals donothaveperfectlatticesfortworeasons: (1)thelatticecontains foreign atoms(impurities) oratomsdisplaced fromtheirnormalposition (point defectsanddislocations), and(2)theatomsdeviatefromtheirmean positions becauseofthethermalvibrations. Eachoftheseimperfections causesscattering oftheelectron wavesinthesamewaythatadefective insulating crystalscattersalightwave,whereas aperfectcrystaldoes not.Analloyisanexample ofadisordered lattice,andwewouldthere­ foreexpectitsresistance tobehigherthanthatofapuremetal.The scattering insuchacase(orfromanyofthecauseslistedin(1)above) shouldbeindependent oftemperature, givingrisetotheconstant resistance whichis characteristic ofalloys. Thethermal vibrations oftheatomscanbeanalysed intonormal modesofvibration ofthecrystalasawhole.Inthelongwavelength limittheseareidentical withthestanding wavescomposed ofelastic waves(longitudinal andtransverse) propagated through acontinuous solid,butatshorterwavelengths comparable withtheinter-atomic spacingtheymustbetreatedbymethods (similartothoseusedfor electron waves)whichallowfortheperiodic structure ofthelattice. Eachmodehaswave-vector qandangularfrequency w.Theenergyis quantized, andattemperatures wherenw~kTquantum effectsmust beincluded incomputing themeanenergyofeachmode;thistreatment givesthewell-known Debyetheoryofthelatticespecificheat.Justas theelectrons havebothawaveandaparticleaspect,sodothelattice modes;theyareknownas'phonons', anamewhichemphasizes their resemblance tothephotons oftheelectromagnetic fieldandtosound wavesinasolid.Thefreepathsofthephonol1s arelimitedby'collisions' withotherphonons, andscattering bypointdefects,dislocations and ultimately bytheboundaries ofcrystallites. Inmetalst.hereisafurther scattering mechanism duetocollisions between thephonons andthe conduction electrons. Suchcollisions alsolimitthefreepathsofthe conduction electrons, andarethemaincauseoftheelectrical resistance atordinary temperatures. Atlowtemperatures, wherethelatticevibra­ tionsdieout,wewouldexpectthescattering tofallandtheconductivity 18.5] ELECTRONS INMETALS 523 toincrease rapidlyasthetemperature approaches theabsolute zero. Thisisfoundtobethecase,buttheconductivity reachesanupperlimit whichdepends ontheprevious historyofthespecimen. Thisisdueto thelatticedefects,whichcanbereducedbycarefulannealing. Then,in general,thepurerthespecimen thehigherthelimiting conductivity, showingthattheimpurities areresponsible fortheresidual scattering. Asemi-empirical formula, duetoGruneisen, whichrepresents theresis­ tancevariation ofmanypuremetalswell,is (18.22) wheretheconstants Aand0arechosentoobtainthebestfitwith experiment. Thisformulagivesavariation of(pIT)withtemperature whichisnotunlikethevariation ofthelatticespecificheatofasolid asgivenbyDebye's theory,andthevalueof0isclosetotheDebye characteristic temperature. Athightemperatures (pIT)approaches the constant valuel(AIO),butatlowtemperatures theresistivity variesas 125AT(TIO)4; thislatterrelationwasdeduced theoretically byBloch. Theelectrical resistance ofmanymetalshasbeenmeasured overa widetemperature range:theresistivity ofthreespecimens ofsodiumat lowtemperatures isshowninFig.18.10.Iftheconstant residualresis­ tanceobserved atlowtemperatures, whichisduetoimpurities, is subtracted fromeachcurve,anidentical remainder isobtained athigher temperatures whichwemaytaketobetheresistance ofideallypure sodium(thefactthattheresistance contributions duetoelectronscatter­ ingbyimpurities andbyphonons areadditive isknownas'Matthiesen's rule').Thecloseagreement withtheGruneisen formula isshownin Table18.2. Inasolid,heatcanbetransported bothbythephonons andbythe conduction electrons, thethermal conductivity ineachcaseincreasing withthemeanpathlengthofthecarriers. Inametalthephonons are scattered bycollisions withelectrons, andtheirmeanpathlengthis smallerthanitwouldbeinaninsulator wherethereisnosuchscattering process. Hencetheheattransport bythephonons shouldbesmaller inametalthaninaninsulator, whereas experimentally thethermal conductivity isfoundtobemuchlarger.Wetherefore conclude that thethermaltransport inametalisnearlyallduetotheelectrons, and infactthelatticeconductivity isnegligible incomparison atalltempera­ turesexceptinsuperconductors. 524 ELECTRONS INMETALS [18.5 Atroomtemperature thethermal conductivity Kofmostmetalsis practically independent oftemperature, butatlowtemperatures K increases, andfornearlyallpuremetals(seeRosenberg, 1955)its variation canbefittedtoaformula ofthetype I/K=OI.T2+fJ/T. (18.23) Thetwotermsinthethermalresistivity I/Karisefromscattering ofthe electrons bythephonons andbycrystalimperfections (orimpurities) o40... x J ~ o·Jn2---+--~+---l------'1;';;0:----'---~-_..L...-_---,-fn---'----- Temperature (OK) FIG.18.10.Lowtemperature resistance ofthreespecimens ofsodium(fromMacDonald andMendelssohn (1950). respectively; foranideallypuremetalwithaperfectlatticefJwouldbe zero.Thetemperature variation ofthethermalresistivity atlowtem­ peratures isdifferent fromthatoftheelectrical resistivity, causing departures fromtheWiedemann-Franz rule.Thisrule(see§4.1)states thatthequantity L=K/aTshouldbeauniversal constant forall metals;Lisknownasthe'Lorenznumber', andonthefreeelectron theory(seeProblem 18.1)itshouldhavethevalue 1T2k2 Lo=~~=2·45X10-8wattohmdeg-2, (18.24)3e2 18.5] ELECTRONS INMETALS 525 wherekisBoltzmann's constant andetheelectronic charge. The numerical constant isdifferent fromthatinequation (4.6)becausethe latterwasbasedonclassical statistics andtheaveragevelocity isthat ofalltheelectrons, whileinequation (18.24)wehaveusedthefactthat onlyelectrons attheFermisurfacewithasubstantially fixedvelocity areinvolved. Inthederivation oftheseformulae itisassumed that scattering oftheelectrons isequallyeffective asregardselectrical and TABLE 18.2 Ratiooftheresistance atTOKtothatat273'2°Kfor ideallypuresodiummetal Thecalculated valuesarefromtheGruneisen formula, equation (18.22). Theexperimental valuesarefromD.K.C.MacDonald andK.Mendelssohn (1950). Oalculated Observed TOK ratio ratio 273·2 1·0000 1·0000 90·0 0·2600 0·2420 20·4 0·00327 0·00326 15·95 0·00100 0·00098 1l·05 0·00015 0·00017 8·1 0·00004 0·00005 4·2 0·00000 0·00000 Thevalue8=202°Kisassumed inusingtheGruneisen formula tofindthecalculated ratio.Theresidual resistance duetoimpurity hasbeensubtracted fromthemeasured resistance beforefindingthe'observed ratio'.Forthepurestspecimen theratioofthe residual resistance totheresistance at273'2°Kwas0·0004. thermal transport, sothattheeffective meanfreepathisthesamefor bothprocesses. Thiswouldmakethequantities pandTJKvary together; thesinglepowerofToccursinthethermal casebecausethe quantity ofheatcarriedbytheelectrons isproportional totheelectronic specificheat,whichvarieslinearlywithT.Thereisnocorresponding termintheelectrical case,sothatboththeelectrical resistivity and T!Kareproportional tothereciprocal ofthemeanpathlength,i.e.to thescattering rate. Attemperatures approaching theDebyetemperature e,allphonons arefullyexcitedandwecanuseaclassical approximation. Themean freepathisinversely proportional tothemeansquareamplitude of thermallatticevibrations, whichisproportional totheabsolute tempera­ ture.HencepandTJKbothvaryasT(givingathermalconductivity independent oftemperature), andtheobserved valueofLisclosetoLo formostmetalsatroomtemperature. Attheopposite extremeofvery 526 ELECTRONS INMETALS [18.5 lowtemperatures wherethescattering ofelectrons isallduetoimpurities, themeanfreepathisindependent oftemperature, sothatpandTIK areconstant, andLagainapproaches Lo•However, inapuremetalat lowtemperatures wherescattering isduetophonons oflongwavelengths, themeanfreepathsforelectrical andthermal transfer aredifferent. Thenumberofphonons oftherightwavelengths toscatterelectrons is proportional toq2,andthescattering cross-section foreachvariesasq, whereqisthewavevectorforaphonon. Thisgivesusascattering rate aconstant (defectand impw;.ty scatteringr phononscattering asT_(J ---...T --_.T· FIG.IS.n.Variation ofelectrical conductivity uandthermal conductivity Kwith temperature inametal.8istheDebyetemperature. proportional toq3,andhencetow3,wherew=qfv,andwistheangular frequency ofthephononandvitsvelocity. Atanytemperature the preponderant numberofphonons arethoseforwhichnwisoforderkT, andhencewegetaT3dependence oftherateofscattering. Allsuch scattering collisions areeffective inenergytransfer, andTIKvaries asT3,sothatK-lvariesasT2,corresponding tothefirsttermin equation (18.23). Inconsidering theelectrical resistivity wemustallowforthefactthat thelongwavelength phonons carrylittlemomentum, andscatterelec­ tronsonlythrough smallangles.Theforward currentcarriedbyan electron scattered through anangleexisreduced onlybyanamount (I-cos ex),whichvariesasex2forsmallangles,andhencewithq2and withT2.ThisextrafactorofT2,together withthefactorofT3mentioned above,givesanoverallvariation ofpwithT5.Thebehaviour ofpand Kwithtemperature isillustrated inFig.18.11. 18.5] ELECTRONS INMETALS 527 Thedifference intheeffective meanfreepathforelectrical andthermal conduction inthisregionmakesLfallbelowLo'Atypicalplotofthe variation ofLisshowninFig.18.12,forcopper. Belowabout10°K LisclosetoLo,buthasaminimum atabout40°K.Forideallypure coppertheelectrical resistance canbefoundbysubtracting theresidual resistance observed atverylowtemperatures, andthethermalresistivity ~2·0x ---------------.........-------Q W ~ 00 ~ Temperature (OK) FIG.18.12.Lorenznumber forcopper(Berman andMacDonald, 1952). 1experimental curve; 2experimental curveforideallypurecopper,obtained bysubtracting contributions totheelectrical andthermal resisitivity fromimpurities. isgivenbythefirsttermofequation (18.23).Fromthesetwoquantities onefindstheLorenznumberLfortheideallypuremetal,andthisis shownbycurve2,whilethebrokenlinegivesthecurvecalculated by Sondheimer. Similardiscrepancies between theoryandexperiment have beenfoundforothermetals. Thereareanumberofeffectsinconnexion withtheelectrical resistance atlowtemperatures whichweshallnotdiscussindetail.Themost important oftheseis'superconductivity': foranumberofmetalsand compounds theresistance fallstozerobelowacertaintemperature characteristic ofeachsubstance. Atthesametimeallfluxofmagnetic induction Bthroughthesubstance isexpelled; thisisnotwhatweshould expectfromastraightforward application ofMaxwell's equations, since avanishing resistivity requires E=0,andhencefJBJfJt=0,sothat anyfluxofBinthemetalwhenitpassesintothesuperconducting state 528 ELECTRONS INMETALS [18.5 shouldremainfixed,notbereducedtozero.Thesubjectofsuper~ conductivity isfullydiscussed inmanybooksonlowtemperature physics. Athighfrequencies theskindepthinametal,ascalculated using theconductivity measured atlowfrequencies, becomes smallerthanthe meanfreepathofanelectron. Thehighfrequency resistivity isthen greaterthaniscalculated fromtheclassical formulafortheskindepth, fortheeffective relaxation timeT'isdetermined bythelengthoftime theelectron spendswithintheskindepth(i.e.thetimeitisactedonby theh.f.electricfield)ratherthantheactualtimeTbetween 'collisions'. Thisisknownasthe'anomalous skineffect'(see,forexample, Pippard (1949);alsoProblem 18.5). 18.6.TheHalleffect Whenablockofmetalcarrying acurrentofdensityjparalleltothe y-axisisplacedinafieldofmagnetic induction Bparalleltothez-axis, apotential difference appearsacrossthemetalinthedirection ofthe x-axis.Thiseffectwas discovered byHallin1879.Themagnetic induction Bexertsaforceonthechargedparticles carrying thecurrent, displacing theminthex-direction. Thissetsupanon-uniform charge densitywhichgivesrisetoanelectricfieldinthex-direction; inequili­ briumtheforceduetothisfieldmustjustbalancethatduetothe magnetic field,sothat F=eE+evi\B=O. (18.25) Ifwecanidentify vwiththedriftvelocity ofthecharged particles, thenj=nev,wherenisthenumberofparticles ofchargeeperunit volume. Thenwehave (18.26) RH=-1jniel,E=:-vi\B=-(ji\B)j(ne)=~RH(ji\B), whereRH,theratiooftheelectricfieldtotheproduct (current density Xmagnetic induction B),isknownastheHallcoefficient. Itsmagni­ tudeis wherewehaveintroduced thenegative signexplicitly toemphasize that wewouldexpectRHtobenegative forelectrons ofcharge-e.A rigorous analysis showsthatequation (18.26)iscorrectforametal whereonlytheelectrons attheFermisurfacetakepartintheconduction process, sothattheyallhavesubstantially thesamevelocity.Ifa velocity distribution oftheMaxwellian typeisused,anexpression for Risobtained largerbyafactor(37Tj8);thispointarisesinthetheory ofsemiconductors (Chapter 19). 18.6] ELECTRONS INMETALS 529 Acomparison oftheobserved valuesofRHforvariousmetalsand semiconductors withthosecalculated fromequation (18.26)isgivenin Table18.3.Theagreement isquitegoodforthemonovalent metals,but forothermetals,suchasthedivalent alkalineearthmetals,RHisfound tohaveapositive insteadofanegative sign.Thisunexpected result suggests thatthecurrentiscarriedbypositive insteadofnegative charges, forwhichtherewasnoexplanation untilthebandtheory TABLE 18.3 Observed andcalculated valuesoftheHalleffect Observed MetalRH(inunitsof10-6ems/coulomb) Calculated, a8suming qelectr01l8 peratom Lithium Sodium. Copper. Silver. Zinc Cadmium-17,0 -25-0 -0·5 -8·4+4·1+6-0-13-1(q=l) -24,4 (q=1) -7·4(q=I) -10-4 (q=I) -4·6(q=2) -6·5(q=2) showedthatanearlyfullbandofelectrons behaved inasimilarmanner toasetof'positive holes'(see§18.2).TheHalleffectisimportant in beingtheonlysimplewayinwhichwecantellwhether wehavetodeal withelectrons orpositive holes,anditsmagnitude givesthenumberof carriersnperunitvolume. Theseresultscannotbeobtained fromthe conductivity, butbycombining measurements oftheconductivity and theHalleffectwecanfindbothneandthemobility u(thedriftvelocity inunitelectricfield),sincea=neu.Thisisespecially important when dealingwithsemiconductors. 18.7.Dia-andparamagnetism ofconduction electrons Inmostmetalstheboundelectrons attached tothepositiveionshave closedelectron shellswithnopermanent magnetic dipolemoment and showonlyasmalldiamagnetism corresponding toequation (8.7).In amagnetic fieldtheconduction electrons areaffected intwoways: (1)theLorenzforce-e(v/\B)altersthetranslational motionandgives risetoadiamagnetic moment; (2)associated withtheelectron spinis amagnetic dipolemoment (seeChapter 21),whosecomponent isone Bohrmagneton f3=en/2mparallel oranti-parallel tothemagnetic field.Thesetwocomponents ofthedipolemoment areassociated with thetwoallowedcomponents oftheelectron spin(see§4.2),andwhena 851110 Mm 530 ELECTRONS INMETALS [18.7 magnetic fieldisappliedtheyhavedifferent energies, +f3Band-f3B respectively. Thelatterstate,whosedipolemoment isparalleltothe field,hasalowerenergythantheanti-parallel state,andwillhavethe largerprobability ofoccupation, givinga·net paramagnetism. Wecan­ notcalculate thisbythemethods usedin§8.3,however, fortheLangevin formuladerivedthereassumes aBoltzmann distribution function. The Fermi-Dirac distribution function mustbeusedforfreeelectrons, and, sincethisvariesverylittlewithtemperature, thesusceptibility turnsout tobepractically independent oftemperature. Weshallderiveanexpres­ sionfortheparamagnetic susceptibility attheabsolute zerooftempera­ ture,whichcanbedonerathersimply. Attheabsolute zero,twoelectrons withoppositely directed spins occupyeachtranslational energyleveluptoacertainenergyWF,thetop oftheFermidistribution. Whenamagnetic fieldisapplied,anelectron canonlyreverseitsspinmagnetic dipolefromananti-parallel toa parallel orientation ifthedecrease initsmagnetic energy(2f3B)is sufficient tosupplytheextrakineticenergyrequired toraiseittoan emptytranslational energylevel.ThisfollowsfromthePauliprinciple, whichshowsthattwoelectrons withparallelspinscannotoccupythe sameenergylevel.Theeffectonthedistribution ofelectrons inthe energybandisshowninFig.18.13.Thisdiffersfromtheearlierdiagram (Fig.4.3)inthatthebandisdrawnintwohalves,onecontaining the electrons whosespindipolesareparalleltothefieldB,theotherthose withtheirspindipolesanti-parallel. Thetwohalf-bands arethen separated inenergyby2f3B,thepotential energydifference inthe magnetic field.Forthetotalenergy,magnetic pluskinetic,ofthewhole systemtobeaminimum, theelectrons mustfillthetwodisplaced half­ bandsuptothesamelevel,asinFig.18.13.Anydeviation fromthis wouldrequireatransferofelectrons fromonehalf-band tohighervacant levelsintheotherhalf-band, andsoincreasetheenergy. Thetotalmagnetic momentofthesystemis2xf3,wherexisthenumber ofelectrons transferred fromtheanti-parallel totheparallelorientation, sincetheexcessinthelatteristhen2xandeachelectronhasaspindipole moment ofoneBohrmagnetonfl.Thevalueofxcanbefoundinthe following way.Weassumethattheenergydifference wbetween suo­ cessiveenergylevelsatthetopoftheFermidistribution isapproximately constant. Toturnroundthedipoleofoneelectronthenrequiresthatits kineticenergybeincreased byw,sincewemaytakeanelectronfromthe toPnlost filledlevelandputitinthenextlevel,whichisvacant.Toturn roundasecondelectronrequiresanadditional kineticenergyof3w,since 18.7] ELECTRONS INMETALS 531 thenexttwolevelswithparallelorientation arealreadyfilled.Thethird electronthenmustbegivenextraenergyequalto5w,andforthexth electrontheexcesskineticenergywillbe(2x-l)w.Ifxisverylarge compared withunity,thismaybetakenas2xw,andatequilibrium 2xw willjustequal2f3B,sothatthehalf-bands arefilledtothesamelevel, -band (spindipolesantiparallel tomagnetic field) I2l1(W)'--+bandC (spindipolesparallel tomagnetic field) I--. 2l1(W) FIG.18.13.Displacement of+and-bandsofconduction electrons byanappliedmagnetic field. Thedisplacement isequaltothedifference ofenergy2f3Bofaspin dipoleparallelandanti-parallel tothefieldB.Theresultant magneti­ zationisduetotheexcessofelectrons inthe+band. asinFig.18.13.Sincetwoelectrons withspinsanti-parallel canoccupy eachkineticenergylevel,thenumberofsuchlevelsintherangeW toW+dWisW(W),whereg(W) isthedensityofstatesinthisrange. Hencetheenergyseparation wbetween successive levelsatthetopof theFermidistribution is{ig(W)F}-l =2{g(W)}"F1•Hence 2x=2f3Bjw=f3B{g(W)}F, andthesusceptibility perunitvolumeis Xp=2;:=1-'0fJ2{g(W)}F' (18.27) Forfreeelectrons thevalueof{g(W)}Fmaybeobtained fromequation (4.13),andthen 3nfJ2 Xp=~~. (18.28) 532 ELECTRONS INMETALS [18.7 Thissimpleexpression wasfirstderivedbyPauli,andthephenomenon issometimes called'Pauliparamagnetism'. Sincethechangeinthe Fermidistribution withtemperature isverysmallsolongaskTf~is small,thesusceptibility ispractically independent oftemperature; Stonerhasshownthatthenextterminaseriesexpansion forthe susceptibility issmallerbyafactoroftheorder(kTfWF)2. Comparison ofequations (18.16)and(18.27) showsthatboththespecific heatandtheparamagnetism oftheconduction electrons aredetermined bythedensityofthestates{g(W)}FattheFermilevel,andthateachis smallerbyafactoroforderkT/Uj,thanthecorresponding quantity forasetofparticles obeying classical statistics (cf.equation (8.13)or (20.16)forthesusceptibility). Calculation ofthediamagnetic susceptibility arisingfromthetransla­ tionalmotionoftheconduction electrons inamagnetic fieldisconsider­ ablymorecomplicated, andwequoteonlytheresultforfreeelectrons, firstderivedbyLandau: Xa= _JLoe2(3n)1. 617m817 (18.29) Substitution oftheformula forWF(equation (4.11))inequation (18.28) showsthatXpisjustthreetimesasgreatasXaforfreeelectrons, sothe netsusceptibility ispositive. Thisexpression isvalidonlyinsmall fields;athighfieldsfurthertermsbecomeimportant whichgiveriseto anoscillatory variation ofXawithfundamental periodTlj"f2f3B. This isknownasthedeHaas-van Alpheneffect,andisobserved inmany metalsatlowtemperatures. Theformulae givenabovearevalidforfreeelectrons; forelectrons in aperiodicpotential theformulae aresimilar,provided wesubstitute the effective massm*form.Thusthediamagnetic susceptibility becomes Xa=_JLoe2 _(3n)!617m*817 (18.30) andsodecreases whenm*increases. Theparamagnetic susceptibility Xp isstillcorrectly givenbyequation (18.28),butasWE,!ccm*,XPincreases withm*.Thusanincrease intheeffective massmakesXvpredominate overXamorethanbyafactor3.Inaddition, interaction effectsbetween theelectrons causeafurtherincreaseinXP(forareview,seeVanVleck (1957)). Comparison oftheoretical resultswithexperiment iscomplicated by thefactthatastaticsusceptibility determination measures onlythe totalsusceptibility X=Xa+XP+Xc,whereXcisthediamagnetic suscepti­ bilityoftheelectrons boundtothepositiveioncores.However, thiscan 18.7] ELECTRONS INMETALS 533 beestimated fromvaluesforneighbouring non-metallic elements, or fromcalculated values of~(r2)(equation (8.7)).Thesusceptibility due totheelectronspinsalone,XP'canbemeasured byelectronspinresonance (seeChapter 23),andXdcanthenbefoundfromthevalueof(X-Xp-Xc)' Theexperimental andtheoretical resultsforlithiumandsodiumare summarized inTable18.4,whichisbasedonVanVleck(1957).Later measurements ofXPgiveslightly different values,butasatisfactory comparison withtheorymustawaitanexperimental determination of m*/m. TABLE 18.4 Experimental andtheoretical valuesojthevolumesusceptibility oJlithiumandsodium(afterVanVleck(1957)) (Inunitsof10-6e.m.u./cm3;toconverttom.k.s./m3multiply by47T) Lithium Sodium experiment thoory experiment thoory m*/m 1·46 0·985 Xp 2·08±0·1 l-l7t 0·95±O·1 0·64t 1'87:1: 0·85:1: X 1·89±0·05 0·70±0·03 Xc -0·05 -0'18 'Xd-0·14±0·15 -O·19§1-0.07±0.13 -0·22§ tFromequation (18.28),usingeffective mass. :f:Calculated byPines,including interaction effects. §Fromequation (18.30),usingeffective mass. REFERENCES BEATTIE, J.R.,1955,Phil.Mag.46,235. BERMAN, R.,andMAcDoNALD,.D. K.C.,1952,Proc.Roy.Soc.A,211,122. BROWN, S.,andBARNETT, S.J.,1951,Phys.Rev.81,657. CORAK,W.S.,GARFUNKEL, M.P.,SATTERTHWAITE, C.B.,andWEXLER, A.,1955, ibid.98,1699. DUYCKAERTS, G.,1939,Physica, 6,817. KRONIG, R.DEL.,andPENNEY, W.G.,1931,Proc.Roy.Soc.A,130,499. MACDONALD, D.K.C.,andMENDELSSOHN, K.,1950,ibid.202,103. PIPPARD, A.B.,1949,Physica, 15,45. ROSENBERG, H.M.,1955,Phil.Trans.A,247,441. SCOTT,G.G.,1951,Phys.Rev.83,656. SKINNER, W.B.,1938,Rep.Progr.Phys.5,257. VANVLECK,J.H.,1957,NuovoCim.6,857. GENERAL REFERENCES DEKKER, A.J.,1958,SolidStatePhysics (Macmillan). KITTEL, C.,1956,Introduction toSolidStatePhysics (Wiley). ROSENBERG, H.M.,1963,LowTemperature SolidStatePhysics (O.U.P.). 534 ELECTRONS INMETALS PROBLEMS 18.1.Usingthekinetictheoryexpression K=tlv(dUldT) forthethermal con­ ductivity Kofagas,showthatonthefreeelectron model 7T2nvlk2T K=6WF' wherenisthenumberofelectrons perunitvolumeofvelocity vand~eanfree pathl,kisBoltzmann's constant, andWFtheFermienergy. Thisexpression is validatverylowtemperatures wherelisdetermined bytheimpurity scattering, andcorresponds tothesecondterminequation (18.23). Verify,byusingequation (4.3),thatthisleadstotheexpression for Lo=KlaT giveninequation (18.24). 18.2.Theeffectofscattering ontheelectronic motionmayberepresented by adamping term,asinProblem 3.9.Ifanalternating electric fieldisapplied, theequation ofmotionbecomes m(dXldt)+mx!,r =eEoexp(jwt). Showthatthisleadstoaneffective conductivity a=ao/(l+jwT), whereaoisthe conductivity atlowfrequencies. Thustheconductivity atfrequencies where WT~1iscomplex; therealpartgivesacontribution totheconduction current, butwithareduced conductivity a'=ao/(1+w2.r 2),whiletheimaginary partis equivalent toadisplacement current(butofopposite signtothenormaldisplace­ mentcurrent), sothatthedielectric constant ofthemedium iseffectively reduced fromEtoE-a'T/Eo' 18.3.Usingthetreatment of§lOA,findanexpression forthecomplex refractive index(n-jk)ofametalintheregionwhererelaxation effectsintheconductivity areimportant, thatis,wheretheconductivity iscomplex asinProblem 18.2. Iftheordinary dielectric constant ofthemetalisneglected, showthat (n2-k2)/(2nk) =-WT. Themeasurements ofBeattie (1955)showthatforaluminium atroomtemperature atwavelengths between 6and12microns, thequantity(w-k2)/(2nk)isroughly equalto-lIlA,whereAisthewavelength inmicrons (1micron =10-6metre). ShowthatthisgivesavalueofTofabout0·6X10-14sec. 18.4.Theresistivity ofcopperat4°Kisapproximately 10-10ohm-metre (10-8 ohm-em). Assuming thatm*1m=1,5,andWF=4·7eV,showthatthemeanfree pathofelectrons incopperatthistemperature isabout7X10-4em,whilethe classical valueoftheskindepth(equation (10.31))atawavelength of3emisabout 5 X10-8em.Showalsothatatthiswavelength andtemperaturn thevalueofWT (seeProblem 18.2)isabout1.(Assume oneelectron peratomforn.) 18.5.Theanomalous skineffectmakestheeffective highfrequency conductivity a' lessthanthed.c.valueabyafactor ~(all),whereaistheeffective skindepth andlthemeanfreepathoftheelectrons. Assuming that a'ia=(3(all), where{3isanumerical factor(oftheorderofunity),andthattheeffective skin ELECTRONS INMETALS 535 depthisgivenbyequation (10.31)witha'insteadofa,showthattheeffective skindepthbecomes3=(21faflfLfLow}i. Fromequation (18.21)(lfa)isaconstant, andhence3becomes independent of temperature atlowtemperatures. (Thereflecting powerofpuremetalsinthe infrared atlowtemperatures isprincipally determined bytheanomalous skin effect,nottherelaxation effect.) 18.6.Inasimplecubiclatticetheenergysurfaces givenbythe'tight-binding' approximation areoftheform W="W;.-~(cosk",a+cosklla+cosk.a), whereaistheatomicseparation. Showthatthewidthoftheenergybandis6~, andthatnearthebottom(k",a-+0,etc.)theenergyisapproximately W=(T~-3~>+!~k2a2+ ..., whilenearthetop(k",a-+±1T,etc.)itis W=(Wl+3~)-l~k2a2+ ..., w:\1erek2=ki+k~+k~. Thisshowsthattheenergysurfaces arespheresabout thecentreofthezone,orthecornersofthezonerespectively (compare Fig.18.6). Notethattheeffective massm*=h2fa2TVz,andhenceisinversely proportional tothebandwidth. 19 SEMICONDUCTORS 19.1.Intrinsic andextrinsic conductivity ASUBSTANCE inwhichthenumberofelectrons isjustsufficient tofill thelowestenergybandsat00Kisaninsulator atverylowtemperatures. Atanon-zero temperature afewelectrons mayhavesufficient energy tobeexcitedintothelowestunoccupied band(the'conduction' band), leavingholesinthehighest'o,Ccupied' band(the'valence' band).This givesasmallelectrical conductivity whosemagnitude depends onthe temperature andonthewidthoftheenergygaplVgbetween thefulland emptybands.Weshallfindin§19.3thatthenumber ofelectrons excitedintotheconduction bandisproportional toexp(-UTy/2kT), and ifthegapisnotmorethanabout1eV,whichcorresponds toavalueof kTwithT;:::::120000K,therewillbeameasurable conductivity at roomtemperature. Thisphenomenon isknownas'intrinsic conduc­ tivity'andisacharacteristic ofpuresemiconductors; ithasbeenob­ servedinpuresilicon,germanium, indiumantimonide (lnSb)andsome othersubstances. Foreachelectronintheconduction band,therewillbeacorresponding 'hole'inthefilledband.Bothelectrons andholescontribute tothe conductivity a,sothat a=IeI(neue+n" Uh), (19.1) wherethesubscripts e,hrefertoelectrons andholesrespectively. As in§4.6themobilities ue'Uharetakenaspositive numbers andnosign isattached toIeI,thoughtheholesandelectrons driftinopposite directions undertheinfluence ofanelectricfield.Forintrinsic con­ ductivity ne=nh'sinceelectrons andholesoccuronlyinpairs.The equilibrium concentration risesrapidlyasthetemperature rises,but themobilities varymuchlessrapidlywithtemperature; hencethe increase innisthedominant factorandtheconductivity risesasthe temperature increases. Thisisthehall-mark ofasemiconductor, and oneofthefeatures (together withitsmuchsmallerconductivity) which distinguishes itfromametal.Another difference isthatelectrons inthe conduction bandareinexcitedstates,andhaveonlyafinitelifetime. Anelectron fromtheconduction bandcandropdownintothetopof thevalenceband,recombining withaholeandreleasing anenergyl¥g; 19.1] SEMICONDUCTORS 537 conversely anelectron-hole paircanbecreatedbyliftinganelectron fromthevalencebandtotheconduction band.Bothprocesses occur repeatedly, givingadynamic equilibrium concentration whichisa function oftemperature. Theproperties ofasemiconductor aregenerally profoundly modified bythepresence ofanimpurity, orsomeothercauseofirregularity in thelattice.Ifthesearepresentinnottoogreataconcentration, they produce discrete energylevels.Thereasonforthisisthatthelevels Emptyconduction band r EnergygapW.D-------donorimpurity level acceptor impurity levelA Filled valence band FIG.19.1.Energybandsinasemiconductor, showing thegapbetween thevalenceband andtheconduction band.At0°Kthevalencebandisfullandtheconduction bandis empty,sothatthesubstance behaves asaninsulator. Thediscrete levelsD,Aaredue tothepresence ofimpurities inlowconcentration, wheretheimpurity atomsaretoo farapartfortheirelectronic wavefunctions tooverlap. onlyspreadoutintobandswhentheimpurity atomsaresufficiently closefortheirwavefunctions tooverlap,andatlowconcentrations the impurity atomsaresofarapartthatanysuchoverlapisnegligible. Thesediscreteenergylevelsareimportant whentheylieintheforbidden band,andparticularly soiftheylieclosetotheconduction orthevalence band,asillustrated inFig.19.1.Intheformercaseelectrons may occupytheimpurity levelDatlowtemperatures, andarethenlocalized ontheimpurity atomandunabletopartakeinelectrical conduction. Asthetemperature rises,theseelectrons areexcitedintotheempty band.Theybehavethenasconduction electrons, withnegative charges; thematerial isknownasann-typesemiconductor, andtheimpurity levelsfromwhichtheelectrons comeareknownas'donor'levels.In thesecondcase,theimpurity levelsAwhichliejustabovethevalence band,willbeunoccupied at0°K,butasthetemperature riseselectrons areexcitedfromthevalencebandintotheselevels,whicharetherefore knownas'acceptor' levels.Thisprocessleavesholesinthevalenceband, whichbehaveaspositively charged carriers,andthematerial isknown asap-typesemiconductor. 538 SEMICONDUCTORS [19.1 Iftheimpurity levelliesclosetoaconduction orvalenceband,the temperature atwhichappreciable numbers ofelectrons orholesmaybe excitedisrelatively low,andthe'extrinsic conductivity' duetothis causemayoutweigh anyintrinsic conductivity, evenwithsmallcon­ centrations ofimpurities. Inpuregermanium atroomtemperature, for example, thenumberofintrinsic electrons intheconduction bandis onlyabout1013percm3,whereasthenumberofgermanium atomsper cm3is4·5X1022•Ifanimpurity atomwhich is easilyionizedatroom temperature ispresenttoaconcentration of1partperhundred million (4'5x1014impurity atomspercm3),itcangiverisetoanextrinsic con­ ductivity whichexceedstheintrinsic conductivity. Theextrinsic conductivity increases asthetemperature risesuntilall thedonorimpurity atomsarefullyionized, oralltheacceptor levels fullyoccupied. Thenumberofextrinsic conduction electrons, orholes, thenbecomes substantially constant; theconductivity becomes constant, ormayfallwithtemperature becauseofadecreaseinthemobility. This isknownasthe'exhaustion range'. Extrinsic andintrinsic conductivity mayofcoursebepresentsimul­ taneously, buttheformerwilldependontheimpurity contentwhilethe latterisaproperty ofthepurematerial. Ineachcaseconduction depends onexcitation intohigherlevels,andthechargecarriershaveafinite lifetime. Allsubstances wouldbeexpected toshowintrinsic conduc­ tivityatasufficiently hightemperature; 'insulators' aresubstances withsuchlargeenergygapsthatappreciable conductivity setsinonly attemperatures outsidethenormallaboratory range,andwhichmay beabovethemelting-point ofthesubstance. 19.2.Elementary andcompound semiconductors Anumberofelements areknowntobesemiconductors intheirnormal allotropic form;theprincipal onesaresilicon,germanium, boron, selenium, andtellurium. Ofthesethemostimportant aresiliconand germanium; theyareusedinmanysolid-state devicesandtheirproper­ tieshavebeenextensively investigated, sothatmuchmorereliable information isavailable abouttheirproperties thanforanyother semiconductor. Theelements carbon,silicon,germanium, tin,andleadbelongto groupIVoftheperiodic table.Silicon,germanium, andtheallotrope greytincrystallize inthediamond structure (seeFig.19.2)inwhich eachatomhasfourequidistant neighbours arranged intheformofa regulartetrahedron. Eachatomformsfourcovalent bondswiththese 19.2] SEMICONDUCTORS 539 neighbours, donating oneeleotrontoeaohbondwhosespinispairedoff withthatoftheoorresponding eleotrondonatedbytheneighbour. These eleotrons oanberegarded asbeinginafilledvalenoeband,abovewhioh isanenergygaptothenextbandwhiohisemptyandformsapossible oonduotion band.Thispioturerepresents thepositionat0°K,where thesubstanoes behaveasinsulators. Theenergygapsarelistedin Table19.1.Atanon-zero temperature someeleotrons maybeexcited FIG.19.2.Thediamond structure, consisting offouratomscentred onalternate cornersofasimplecubiclattice.bondedtooneatthe centreofthecube.Thestructure isrepeated sothateveryatom isinidentical surroundings. fromthevalenoebandintotheoonduction band,makingthesubstanoe anintrinsio semioonduotor. Onalocalized eleotron modelthiscorre­ spondstotakinganeleotronoutofabondtobehavelikea'freeelectron'. Thisleavesa'hole'inonebond;anelectroncanmigratefromanadjaoent bondtofillthishole,therebytransferring theholetoanotherbond.In thiswaytheholecanbepictured asmovinginarandomwaythrough thecrystal,andbeingmobilelikethe'free'electron, thoughnotneces­ sarilywiththesamemobility. Intrinsic conductivity isexhibited onlybyverypurespecimens, and theproperties ofsiliconandgermanium aredrastioally modified by smallamounts ofimpurity.Ifanimpurity fromgroupVoftheperiodic 540 SEMICONDUCTORS [19.2 table,suchasphosphorus, isintroduced, itenters'substitutionally', occupying theplaceofasiliconorgermanium atom.Liketheatomit replaces, itformsfourcovalent bondswithitsfourimmediate neighbours. Thisusesupfourofitsvalenceelectrons, leavingoneinexcess,which experiences anelectrostatic attraction tothephosphorus becauseofthe extrapositive chargeofitsparention.Thissystemofasinglycharged ionandexcesselectron resembles ahydrogen atom,butoneinwhich thepotential duetothepositively charged nucleusismodified bythe presence ofthesurrounding siliconorgermanium ions.Theseare electrically polarized bytheexcesspositive chargeofthephosphorus nucleus, andtheireffectontheelectrostatic potential canbecrudely approximated bytheintroduction ofadielectric constant, makingthe potential V=e/47T€€or. Obviously thisdeviceofusingadielectric constant isonlyrealisticatdistances largecompared withtheinter­ atomicdistance, wheretheelectron orbitissolargethatitembraces manyatoms,andwhoseeffectthenresembles thatofacontinuous medium. Foranelectronofeffective massm*inanorbitofprincipal quantum numberntheenergyisthenfoundtobe(seeProblem 19.1) m*R m*W=---- =---x 13·5eV, (19.2)m€2n2m€2n2 where13·5eVistheionization potential ofanelectron inthen=1 stateofafreehydrogen atom.Thebulkdielectric constant ofgermanium is16,andifweusethisvaluefor€,andavalueofm*/m=0·2(an averageofvaluesobtained fromotherevidence), wefindWI"-.J0·01eV fortheloweststaten=I.Thisisfairlyclosetothatactually observed forgroupVdonorsingermanium (forphosphorus theobserved value is0·012eV).Thecorresponding orbitradiusisover40angstrom units, whichisquitelargecompared withtheinter-atomic distance of2·45A. Since0·01eVisequivalent toatemperature ofonly1200K,itisevident thatsuchdonorimpurities willreleasenearlyalltheirelectrons intothe conduction bandatroomtemperature, aprocesssimilartothatof ionization offreehydrogen atomsattheveryhightemperatures inthe interiorofstars.ForgroupVdonorsinsiliconthecorresponding bindingenergyisabout0·04eV(seeProblem 19.1). IfagroupIIIelement isaddedasanimpurity wehaveadifferent situation. Theimpurity atomnowhasoneelectron toofewtofillthe fourbondswhichitshouldmakeonreplacing asiliconorgermanium atom.andwearetherefore leftwithaholeinonebond.Ifthishole movesawayfromitsparentimpurity, allfourbondstotheimpurity 19.2] SEMICONDUCTORS 541 atombecomefilledandithasonenetnegative charge. Sincethehole iseffectively apositive charge,ithasanelectrostatic attraction tothe negatively chargedimpurity ion,andwehavean'inside-out' hydrogen atomconsisting ofanegatively charged 'nucleus' withthepositive hole inorbitaroundit.Thisgivesan'acceptor' leveljustabovethevalence band,theheightabovethetopofthevalencebandbeingagainabout 0·01eV.At0°Kthevalencebandisfullofelectrons, andthehole occupies theimpurity level;itisthenlocalized ontheimpurity ion, forming aneutralatominaboundstate.Atafinitetemperature an electron maybeexcitedfromthevalencebandintotheimpurity level; thisleavesaholeinthevalencebandwhichisfreetomove,andcorre­ spondstoionization ofthe'insideout'impurity atom. Thisanalysis showsthatat0°Ktheholeisinthehighestlevel(the acceptor level)andasthetemperature risesmoreandmoreholesare excitedinthelowerlevels(thevalenceband).Thisbehaviour issimilar tothatofelectrons beingexcitedfromdonorlevelsintotheconduction band,exceptthatfortheholesenergymustbemeasured downwards insteadofupwards. In§19.3weshallfindthatholesobeysimilar equations toelectrons provided wemeasure energydownwards from thetopofthefilledband,anexample ofwhichhasalreadyoccurred inequation (18.14). Awiderangeofcompounds showsemiconducting properties ofwhich onlyafewwhichillustrate generalclasses,andforwhichsufficient information existstomaketheirproperties reasonably wellunderstood, canbementioned here.Following thegroupIVcompounds germanium andsilicon,itisnaturaltodiscussfirstthegroupIII-group Vcom­ pounds,takingasexample indiumantimonide, InSb,themoststudied ofsuchmaterials.- Thetwoelements, indiumandantimony, comein theperiodic tableimmediately beforeandaftertin.Theyformacom­ poundinwhicheachatomissurrounded byfourequidistant neighbours attheapicesofaregulartetrahedron, asinthediamond structure, but withthedifference thateachofthefournearestneighbours isofthe othertype(thisisknownasthezinc-blende structure, afteroneform ofthecompound ZnS).Thesefourbondsaremainlycovalent incharac­ ter,andlinklatticesiteswhichmayberegarded asoccupied byIn-and Sb+ionsinregularalternation. Eachoftheseionshasthesameelectron configuration astin,thegroupIVelement, andformscovalent bonds inasimilarfashion. However, thefactthatwenowhaveionsofalternate negative andpositive chargegivesrisetosomeionicbinding. Indium antimonide canbeprepared inasufficiently purestatetobehaveasan 542 SEMICONDUCTORS [19.2 Agintrinsic semiconductor, withanenergygapofabout0·24eVat0°K, decreasing to0·17eVatroomtemperature. Thenextbinarycompounds insequence aretheII-VIandI~VII compounds; thesegrowprogressively moreionicincharacter, withlarger FIG.19.3.Thezinc-blende structure (two-dimensional representation). Itissimilarto thediamond structure, exceptthatZnandSionsalternate. energygaps.Thisisillustrated inthesequence formedfromatomsin theseventhrowoftheperiodic table: 0·24eV /InSb"", CdInSnSbTeI '"greJtin/",o.o8ev / CdTe 1·6eV Silveriodideisagoodinsulator, andclearlyhasalarge energy gap. Thoughprobably notaspureapolarcompound asNaCl,wemayregard itasconsisting ofAg+andI-ions,withclosedshellsofelectrons. With aII-VIcompound suchasCdTewehavethedilemma ofwhetherto· regarditasapolarcompound, consisting ofCd++andTe--ions,or acovalent compound formedofCd--andTe++ions.However,the crystalstructure resembles thatofzinc-blende, suggesting acovalent 19.2] SEMICONDUCTORS 543 compound; ontheotherhand,thegroupofsaltsPbS,PbSe,PbTehave theNaCIstructure, suggesting apolarcompound. Thesedifficulties illustrate thereservewithwhichsuchextreme classification shouldbe regarded. Infactthegroupofleadsaltshavesmallerenergygapsthan CdTe,andtheirelectrical properties aremoretypicalofsemiconductors. Theenergygapsofanumberofsubstances aregiveninTable19.1. General(butnotinvariable) rulesarethattheenergygapdiminishes TABLE19.1 Valuesoftheenergygap(eV) GrcmpIVel6m6'nt8 Diamond ,...,5·3 Silicon 1·21 Germaniwn 0·78 Greytin--0·08llI-Vcompounds BN,...,10 AlP 3 GaAs 1·35 InSb 0·24ll-VIcompounds ZnSe ,...,5 CdTe 1·6 PbS ,...,0·4 theheaviertheatomsinvolved (thatis,reading downwards inthe table),butincreases onmovingfromcovalent topolarcompounds (that is,fromlefttoright).Muchlessinformation isavailable aboutthepolar semiconductors, owingtothedifficulty ofpreparing theminthepure state.Apartfromforeignatoms,whichactasacceptors ordonors according totheirgroupintheperiodic table,suchcrystalsmaybe non-stoichiometric. Forexample, leadsulphide mayhaveanexcessof lead,producing donorlevels,orofsulphur, producing acceptor levels. Theenergygapvariesconsiderably withtemperature, andmostof theabovearerounded values.Inthecaseofsilicon,germanium, and lnSbthevaluesgiveninthetablearethosefor0°K;atroomtempera­ turetheyareapproximately 1'12,0,66,and0·17eVrespectively. 19.3.Electron distribution andtheFermilevel Underconditions ofthermalequilibrium thenumberofelectrons with energybetween WandW+dWcanbecalculated bymeansofstatistical mechanics, usingofcoursetheFermi-Dirac statistics appropriate to particles ofhalf-integral spin.Thisnumberis(see§4.2) dn=f(W)g(W)dW, (19.3) wheref(W)istheFermi-Dirac function 1 f(W)=exp{(W-l¥z;.)jkT}+1 (19.4) andg(W)isthedensityofstates. l¥z;.istheFermilevel,definedasthe 544 SEMICONDUCTORS [19.3 energyatwhichthefunctionf(W) =t.Inametal,~~kTatordinary temperatures, andtheonlyelectrons whicharethermally excitedor cantakepartinconduction processes arethoseveryclosetotheFermi level. Inasemiconductor, itisnotsoobvious wheretheFermilevellies withrespecttotheconduction andvalencebands.Weshallconsider firstanintrinsic semiconductor, whereatlowtemperatures onlyafew electrons areexcitedintotheconduction band.Inthelimitofextremely fewelectrons thechanceofanelectron occupying agivenstateisvery low,andtherestrictions imposed bytheExclusion Principle playlittle role.Wearethusinasituation wheretheclassical Maxwell-Boltzmann statistics areagoodapproximation, sothatwecanwrite fc(W)=exp{-(W -WF)/kT} verynearly. (19.5) Thisisjusttheapproximation ofequation (19.4)inthelimitwhere (W-WF)~kT,sothatwecanneglectthesecondterminthedenomi­ nator;thisapproximation isappropriate forelectrons intheconduction band,whichisemptyofelectrons at0°K.Atthistemperature theval­ encebandisfull,andthuscorresponds toenergies wellbelowtheFermi level.Intheregionwhere (~-W)~kTthefirstterminthedenomi­ natorofequation (19.4)isnowverysmall,andwecanwrite 1-fv(W) =exp{-(~- W)/kT}, verynearly. (19.6) (19.7) (19.8)gAW)=C(m~)i(W-lJ;Y gv(W)=C(mt)!(Jv,;- W)!,andThequantity 1-fv(W)isrelevant tothenumberofholesinthevalence band. Wemustnowconsider thedensityofstates,g(W).Thisiszeroat theedgeofabandandweassumethatitvariesasthesquarerootof thedistance fromtheedgeoftheband.Thatis,wecanmodifyequation (4.16)andwrite whereJr,;,Jv,;aretheenergiesatthebottomoftheconduction bandand thetopofthevalencebandrespectively. Thetotalnumberofelectrons intheconduction bandisthus 00 ne=ffc(W)gc(W) dW w. 00 =C(m:)!f(W-Jr,;)!exp{-(W-~)/kT}dW. w. 19.3] SEMICONDUCTORS 545 Onwritingy=(W-We)jkT,thisintegralbecomes 00 ne=O(m:kT)fexp{-(We-~)jkT} Iy1e-lIdy o =(1Tlj2)O(mtkT)iexp{-(We-WF)jkT} =~exp{-(We-~)jkT}. (19.9) Thisresultisthesameasifwehadanumber ~ofstatesatenergyWe; thus~istheeffective densityofstatesatthebottomoftheconduction band,andonsubstituting for0fromequation (4.16)wefind ~=2(27Tm:kTjh2)!. (19.10) Similarly forthenumberofholesinthevalencebandwefind w. nh=I{l-fv(W)}gv(W) dW -00 (19.11) (19.12)=N;,exp{-(~-lYv)jkT}, N;,=2(27TmtkTjh2)!. Fortheproduct nenhwefind nenh=~N;,exp{-(We-lYv)jkT} =~N;,exp{-JYgjkT} (19.13) andsinceinanintrinsic conductor wemusthavene=nh=niweobtain ni=ne=nh=(~N;,)lexp{-lVyj2kT}, (19.14) whereJYg=We-lYvisthewidthoftheenergygapbetweenthevalence andconduction bands. TofindthepositionoftheFermilevel,~,wemustequatetheformulae forneandnh,whichyields N" 2~-J¥c-lYv N=exp kT 'c whence, since N;,j~=(mUm:)!, WF=!(J¥c:t-lYv)+ikTln(mt!m:). (19.15) Thisresultshowsthatformostintrinsic semiconductors, wheremt,m~ arenearlyequal,theFermilevelliesinthemiddleoftheenergygap, asshowninFig.19.4.Insomecases,suchasInSb,wheremUm: ~20, thelevelvariesmarkedly inposition withtemperature, andatroom temperature isshiftedwelltowardsthebottomoftheconduction band. Whenimpurities arepresent, andconduction ispartlyintrinsic and partlyextrinsic, theposition isagooddealmorecomplicated. Thereis, however, oneimportant generalresultwhichholdsprovided thenumbers ofelectrons intheconduction bandandholesinthevalencebandare 851110 Nnwhere 546 SEMICONDUCTORS [19.3 smallcompared withthedensityofstates.Inthatcasetherelations (19.9)-(19.13) arestillvalid,sincetheydonotdependonanysupposi­ tionabouttheposition oftheFermilevel.Thuswehaveinequation (19.13)animportant relation between thenumbers ofelectrons and holes,andinthelightofequation (19.14)wehavealsonenh=n~, whereniisthenumberofintrinsic electrons whichwouldexistatthe 1-+0 j-+1Conduction band / / I g{W/w..,l----------\-;,,------~ Valenceband FIG.19.4.TheFermi-Dirac distribution ofelectrons andholesin anintrinsic semiconductor; thefigureisdrawnforacasewheremt=m:.sothattheFermilevelWpisinthecentreoftheforbidden band.Theshadedareasindicate thenumbers ofelectrons inthecon- ductionband,andholesinthevalenceband. sametemperature. Theratioofnumbers ofelectrons andholesdepena.s onthepositionoftheFermilevel,forwhichweshallquotesomeresults onlyforextreme cases. Whendonorsoracoeptors (butnotbothtogether) arepresent, which produce discrete levelslyingclosetotheconduction orvalenceband respectively, theconductivity atlowtemperatures isdominated bythe 19.3] SEMICONDUCTORS 547 ionization oftheimpurity levels.TheFermilevelat0°Kthenlies between thedonorlevelandtheconduction band,orbetween the acceptor levelandthetopofthevalenceband(seeFig.19.5).Asthe temperature risestheFermilevelshiftsbecauseofatermsimilartothe secondterminequation (19.15),butinvolving In(Nd/N;;) orIn(Na/N,,), w.------------- Acceptors -------------w.Donorsw".-- FIG.19.5.Variation ofFermilevelwheneitherdonorsoracceptors arepresent. Atvery lowtemperatures theFermilevelliesmidway between theimpurity levelandthe conduction orvalenceband.ForsmaIlimpurity concentrations (Nt!.<NoorNa<Nwl thelevelmovestowards thecentreoftheforbidden bandwithrisingtemperature. At highertemperatures theelectron distribution isdominated bytheintrinsic contribution, andWpisatthecentreoftheforbidden gapifm1:=mt. whereNdandNaarethenumberofdonorandacceptor levelsperunit volumerespectively. Thusfordonorswehave ~=l(Jv.t+J¥c)+!kTln(Nd/N;;) (ne~Nd),(19.16) showingthatasthetemperature risestheFermilevelwillriseifNd>N;;, orfallifNd<N;;. Intheexhaustion rangethedonorlevelsarefullyionizedandne=Nd; inthiscasetheFermilevelisgivenapproximately by (19.17) IfNd<N;;theFermilevelliesbelowtheconduction band,thenumber ofelectrons excitedintotheconduction bandissmallcompared with thenumberofavailable statesandtheyobeytheclassical statistics. Thiscondition iscalled'non-degenerate'. Ontheotherhand,ifNd>N;; theFermilevelliesintheconduction band,andsincene=Nd,the nUIllperofconduction electrons isgreaterthanthenumberofavailable st,ates;thiscondition iscalled'degenerate'. Thissituation issimilarto 548 SEMICONDUCTORS [19.3 thatinametal,wheretheexclusion principle limitsthenumberof electrons inagivenenergyrange,andtheelectron distribution must betreatedbyFermi-Dirac statistics insteadofclassical statistics. Similarresultsareobtained foracceptor impurities, provided we countenergyasincreasing downwards fromthetopofthevalenceband ratherthanupwards fromthebottomoftheconduction band(cf. Fig.19.4). Theimportance oftheFermilevelliesinthefactthatitsvalueis equaltothatofthethermodynamic potential G=U-TS+eVofthe electrons.Iftheelectron distributions intwosubstances areinthermal equilibrium witheachother,thenthevaluesofthethermodynamic potential inthetwosubstances areequal,andhencesoalsoarethe Fermilevels.ThusthepositionoftheFermilevelplaysanimportant roleindiscussing theproperties ofjunctions. 19.4.Opticalproperties Semiconductors suchasgermanium andsiliconlookverymuchlike metals;theyareopaquetovisiblelightandhaveahighreflectivity. Thisisbecausethequantum carriedbyavisiblephoton,whichcorre­ spondstoanenergyroughly between 1·5and4eV,issufficient to exciteanelectron fromthevalencebandrightacrosstheforbidden energygapintotheconduction band.Iftheabsorption coefficient is measured atlongerwavelengths, asharpdropinabsorption wouldbe expected whenthephotonenergybecomes smallerthantheenergygap 1Yy;thatis,atwavelengths suchthathv<JVy.Thechangeinthe absorption coefficient canbequitedramatic, from104to105cm-1at wavelengths shorterthantheabsorption edge,downto10-1cm-1at wavelengths beyondtheedge,asillustrated inFig.19.6.Theabsorption beyondtheedgedependsonthepurityofthespecimen, Rinceimpurities produce levelsintheforbidden bandfromortowhichelectrons canstill beexcitedbyphotons oflowerenergy. Anopticaldetermination ofthepositionoftheabsorption edgegives, inprinciple, adirectmeasurement oftheenergygap,whoseaccuracy is limitedbythefactthatthedropinabsorption isspreadoutoverasmall butfiniterangeoffrequency. Carefulanalysis oftheexperimental resultsinthelightofadetailedtheoryofhowtheabsorption coefficient shouldvarywithfrequency inthevicinityoftheabsorption edgehas givenquiteaccurate measurements oftheenergygap,andshows directly howitvarieswithtemperature. However, careisneededin theinterpretation, through thepresence ofselection rulesconnected 19.4] SEMICONDUCTORS 549 withtheconservation oflinearmomentum. Aphotonofenergykv carriesmomentum kv/c,whichisnegligible compared withthemomen­ tumofaparticleofnon-zero restmass(suchasanelectron) ofthesame energy. Asaresultthemomentum ofanelectron excitedintothe ,conduction bandmustbethesameasitwasinthevalencebandbefore 1()4-1()I Lesspure ----.~l FIG.19.6.Theabsorption edgeinasemiconductor suchas germanium. Atshortwavelengths thesemiconductor isquite opaque, sincethelightintensity fallsasexp(-/XX);atlongwave­ lengthstheabsorption ishigherinimpuresamples becauseof electron excitation inandoutoftheimpurity levelsinthefor­ biddenband.Ingermanium theabsorption edgeliesinthe infra-red atabout1·4II-(14000A). theabsorption ofaphoton. Aquantum-mechanical analysisshowsthat thecrystalmomentum kmustbeconserved, sothatwehaveaselection rule~k=O.Ifthemaximum ofthevalencebandandtheminimum oftheconduction bandbothoccuratk=0,asinFig.19.7,nodifficulty arises.ThisappliestolnSb,butingermanium andsilicontheconduc­ tionbandhitsonlyasubsidiary minimum atk=0,thedeeperminimum occurring atafinitevalueofk,asshownalsoinFig.19.7.Thustransi­ tionsinthevicinityofk=0donotdetermine theminimum valueof ~,definedasthedifference ofenergybetween thetopofthevalence 550 SEMICONDUCTORS [19.4 bandandthebottomoftheconduction band.Itturnsout,however, thattransitions suchasthatmarked ~k=1=0inFig.19.7areallowed (thoughmuchweaker) provided thelatticecansupplyortakeupthe momentum requiredtomakethetotalmomentum oflattice pluselectron unchanged. Thisinvolves thecreation ordestruction ofaphonon "k=0 ~k kul='trIa alongIIIaxis ----=::~r___I_~~-----=~o;;;;:--_I_------W(k =0)-----r-----JVo -----1----w. FIG.19.7.Shapeofthebandedgesagainstcrystalmomentum kforgermanium. The momentum ofaphotonisnegligible, sothattherecanbenonettransferofmomentum onabsorption. Either11k=0fortheelectron, orthedifference inmomentum when 11k"#0mustbetakenupbythecreation ordestruction ofaphonon. Forgermanium Wg=0·75eVat0°K,butW(k=0)-We~0·14eV,sothatthet..koF0transitions giveafinestructure ontheabsorption edge. (processes involving morethanonephononhavenegligible probability). Transitions inwhich ~k=0areknownasdirecttransitions, andtransi­ tionsinwhich ~k=1=0arecalledindirecttransitions. (19.18) w=_mr~_, mE2n2Exciton8 Whenanelectronisexcitedfromthevalencebandintotheconduction bandbyadirecttransition, aholeiscreatedinthevalencebandwhose momentum mustbeequalandopposite tothatoftheelectron inthe conduction bandinordertomake~k=O.Theelectronandholethere­ foremoveapartinopposite directions. Inthevicinityofk=0,they moveapartratherslowly,andtheirmutualcoulomb attraction begins toplayarole;finallyatk=0itselftheelectronandholestaytogether. Undertheseconditions theirbehaviour resembles thatofanelectron andprotoninahydrogen atom;abettercomparison iswithanelectron anddonorimpurity atom,asdiscussed in§19.2.Electron andholemay moveindiscreteorbitsaboutthemutualcentreofmass,givingriseto aseriesofenergylevels 19.4] SEMICONDUCTORS 551 wheremristhereduced massgivenbytherelation 1 1 1 m=m*+m*'re 11,(19.19) ~j-71.=3--r--n =2 excitonlevels _+--71.=1 FIG.19.8.Exciton levelslyingjustbelow theconduction bandinthevicinity of k=O.Thequantwn ofenergyshownis thatrequired toexciteanelectron from thevalencebandtothe71.=2level,andis slightlysmallerthanthatcorresponding to theabsorption edge,whichrequires excita­ tiontothebottomoftheconduction band (71.=(0).W=0inequation (19.18)corresponds toseparation oftheelectron andholetosuchalargedistancethattheirmutualattraction isnegligible (i.e.to'ionization' oftheelectron­ hole'atom')andthuscorresponds to thebottomoftheconduction band atk=O.Alowerenergyisobtained whentheelectron andholeareto­ gether,sothatthelevelsofequation (19.18)liejustbelowtheconduction band(likethoseofadonorimpurity andboundelectron), asshownin Fig.19.8.Theenergyisalsoofthe sameorder;ifm:=mt=2mr,the energylevelsarejusthalfthose givenbyequation (19.2),andlie therefore veryclosetotheconduc­ tionband. Theelectron-hole boundpairis knownasan'exciton' andhasbeen identified throughitshydrogen-like spectrum insomesemiconductors withlargeenergygaps,together withCu20andGe.Ingermanium excitons areassociated bothwiththe directandindirect transitions. In theformercaseasharplinespectrum wouldbeexpected atfrequencies justshortoftheenergygapatk=0;intheindirect transitions the electron-hole paircanbeformedwithfinitemomentum andpossess kineticenergy,sothattheexcitonlevelsarenotsharpbutbroadened intobands. Photoconductivity Whenradiation whosewavelength issufficiently shortthattheenergy quantum hvislargerthantheenergygapisshoneonasemiconductor, electrons areliftedintotheconduction bandandholescreatedinthe valence band.Thepresence ofthisexcessofcarriersincreases the 552 SEMICONDUCTORS [19.4 conductivity, andthephenomenon isknownasphotoconductivity. Forsmallintensities ofillumination theincrease inconductivity is approximately proportional totheintensity oftheincident radiation, andtheconductivity changeisanimportant methodofdetecting infra­ redradiation. Suchadetector issensitive onlytowavelengths shorter thantheabsorption edge;PbScanbeusedforwavelengths uptoabout 4/J-,andInSbtoabout7J-t(theselimitsvarywithtemperature because theenergygapandhencetheabsorption edgearetemperature depen­ dent).Suchdetectors notonlyhaveahighsensitivity, butalsohavea Incident---f- ......'" radiation -~-f----" Rotating chopper wheelsemiconductor BL---_Tor---_ amplifier FIG.19.9.Useofphotoconductive effectinasemiconductor forthedetBction ofradiation. Theincident. radiation ismodulated inintensity byamechanical 'chopper', andproduces avariation intheresistance ofthesemiconductor atthemodulation frequency. The resulting alternating voltage acrosstheresistance Risamplified byanarrow-band amplifier tunedtothemodulation frequency. shortresponse time(varying from10-4to10-7sec)becausetheexcess carriersquicklydisappear through recombination, etc.Thismakesit possibletomodulate theincident radiation (e.g.byamechanical chop­ pingdevice)andobtainana.c.signalwhichcanreadilybeamplified anddetected, asillustrated inFig.19.9.Theuseofgermanium doped withsuitable impurities makesitpossibletoconstruct detectors which aresensitive tomuchlongerwavelengths, thephotoconductive effect thenbeingduetoelectrons excitedintotheconduction bandfromdonor impurity levels(orholescreatedbyelectrons beingliftedintoacceptor levelsfromthevalenceband).Sincethesmallest separation ofsuch discretelevelsfromtheadjacent bandsisabout0·01eV,suitably doped samplesaresensitive towavelengths uptoabout100J-t.Suchdetectors mustofcoursebecooledtoatemperature wherethermal ionization of theimpurity levelsisunimportant. 19.5] SEMICONDUCTORS 553 19.5.Transport properties Inaddition totheenergygaplVy,themostimportant quantities we requiretoknowaboutasemiconductor arethenumbers ofcharge carriersofeithersign,theireffective masses,andtheirmobilities. The bestmethodofmeasuring theeffective massisbymeansofcyclotron resonance, whichisdiscussed inChapter 23.Determination oflVyby opticalmethods ispossible onlyonverypurespecimens, andinmany casesitcanonlybededuced indirectly, usingequation (19.13).Tobring outthetemperature dependence explicitly werewritethisintheform nenh=(2'33X1031)(m: m~/m2)iT3exp( -Wg/kT), (19.20) wheretheunitsaremetre-6•Atfirstsightthesimplest methodof findingthenumberofcharged particles wouldbemeasurement ofthe Hallcoefficient RH,whichfromequation (18.26)isinversely proportional tothenumberofcarriers. Sincethisnumber issomuchsmallerina semiconductor thaninametal,theHalleffectismuchlarger (~105to 106cm3/coulomb forpureSiandGeatroomtemperature) andcorre­ spondingly easiertomeasure. However, weareimmediately facedwith thedifficulty thattheHallcoefficient changessignaccording towhether thecurrentiscarriedbyelectrons orpositive holes;inparticular, inan intrinsic semiconductor, withequalnumbers ofeach,wewouldexpect theHallcoefficient tovanish.Inpracticethisdoesnothappen, because theelectrons andholeshavedifferent mobilities, andsocarrydifferent fractions ofthecurrent. Thegeneralexpression fortheHallcoefficient is R= _B(neb2-nh), (19.21) Hlei(neb+nh)2 whereb=/Le//Lhistheratioofthemobilities ofelectrons andholes,and Bisacoefficient notfarfromunity.Thepresence ofBarisesfromthe factthatinafullanalysis, different averages overthedistributions of velocityandrelaxation timeareinvolved incalculating theconductivity andtheHalleffect.Forametaloradegenerate semiconductor, B=1; foranon-degenerate semiconductor withthermal (phonon) scattering, B=31T/8,butforionizedimpurity scattering B=1'93. Measurement oftheHallcoefficient RHandtheconductivity amakes itpossible todetermine bothneandnh'usingequations (19.1)and (19.21),provided thatthevalueofbisknown.Thedirectmeasurement ofmobility isdiscussed below,butthisispossible onlywithcertain substances suchassiliconandgermanium. Inothercaseswecan proceed onlybymakingsomeassumptions aboutb.Inanintrinsic 554 SEMICONDUCTORS [19.5 semiconductor, ne=nh=ni'andwehavetherelations (nipermetre3) ni=(4'8x1015)(m:mt/m2)!T!exp(-lVy/2kT) (19.22) B(b-l) and RH= -nilel(b+l)' (19.23) Although themobilities UeandUhbothvarywithtemperature, their ratiobdoesnotvaryrapidlyincomparison withni'whichisdominated bytheexponential factorinequation (19.22). Thusaplotofln(RHTi) againstI/Tshouldbeastraightline,andthisisfoundtoholdforsilicon andgermanium. TheslopeofthestraightlineyieldsavalueoflVy,but sincelVyisitselftemperature dependent, caremustbeexercised inthe interpretation (seeProblem 19.2). Another approximate methodoffindinglVyinvolves onlymeasure­ mentoftheconductivity. Foranintrinsic semiconductor (oranimpurity semiconductor athightemperatures wheretheconductivity isdomi­ natedbytheintrinsic electrons), wehave u=jelni(ue+uh) (19.24) andfrommeasurement oftheconductivity overarangeoftemperature wecanfindthevariation ofnandhencedetermine theenergygap provided weknowhowthemobilities varywithtemperature. Whenthe mobility isdetermined byscattering processes duetolatticevibrations (phonons) themobility (seep.557)shouldvaryasT-].Sincefrom equation (19.22)nivariesasT!exp(-W;/2kT), weshouldexpectthat theconductivity wouldfollowthelaw u=uoexp(-lVy/2kT) (19.25) sothattheenergygapcanbefoundfromaplotofInaagainstI/T. Thismethodwasusedinearlywork,butsuchplotsshowaslightcurva­ ture,indicating thatthemobility doesnotfollowaT-!lawexactly. Determination oftheenergygapfromtheopticalabsorption edge needscarefulexperimentation andinterpretation, butitisthemost satisfactory methodandtheonlyonewhichgivesdirectlytheenergy gapatagiventemperature. UseoftheHalleffectortheconductivity depends onassumptions aboutthemobility, whoseexperimental deter­ mination willnowbediscussed. Measurement ofmobility Themobility ofelectrons andholesinsemiconductors canbemeasured directly byamethod dueoriginally toShockley andHaynes. The specimen isintheformofanarrowrectangular bar,about0·05cm squareandafewcmlong.Asteadyvoltageisappliedbetweentheends 19.5] SEMICONDUCTORS 555 togiveafieldoforder10Vjcmalongthebar;twoelectrodes A,Bare appliedtothespecimen, asshowninFig.19.10.Ashortvoltagepulse ofduration about1microsecond isappliedtoelectrodeA;ifthesemi­ conductor isn-type,andAismadepositiveinthepulse,electrons are withdrawn fromthesemiconductor bytheelectrode; someofthesemay Microsecond injection pulse. Steady voltage-'"""'1--------..,.-- I I 14-j4---L ---~~I I I I IOscilloscope FIG.19.10.Apparatus fordirectmeasurement ofmobility. Thesteadyvoltageapplied atextreme leftproduces auniform fieldinthesemiconductor barunderwhoseinfluence amicrosecond pulseofminority carriersdriftfromAtoB.Themeandriftvelocityis LIt,wheretismeasured ontheoscilloscope. comefromthevalenceband,creating anexcessofholesinthesemi­ conductor. Topreserve electrical neutrality ofthespecimen, electrons enteratanyterminals whicharenegative withrespecttoA.Oneof theseisB,whichisconnected (through anamplifier) totheY-plates of anoscilloscope; thisregisters avoltagebecauseoftheflowofsuch electrons throughtheresistance R.Thispulsedoesnotquitecoincide intimewiththepulseatA,butthetimedelayisthatrequired byan electromagnetic wavesetupbythedisturbance atAtotraveltoB, whichisoforder10-10secondsandquitenegligible. Theholesinjected atAaresweptbythefieldtowards B,andarriveatimet=Ljvlater, whereListhedistance between electrodes A,Bandvisthedriftvelocity inthesteadyfield.OnarrivalatBtheyappearasasecondvoltage pulseontheoscilloscope, andthetimeinterval tbetween thetwopulses canbedetermined fromtheoscilloscope tracebycalibrating thetime­ base.ThevoltageVbetweentheelectrodes A,Bduetothesteadyfield ismeasured independently. Sincethedriftvelocity v=uE=uVIL, thetimet=Llv=L2juVandthemobility isfoundfromtherelation u=L2jVt. (19.26) 556 SEMICONDUCTORS [19.5 Inatypicalexperiment thevalueoftisabout30JLsec.Thesecond pulseatBduetothearrivalofholesisbroaderandsmallerthanthe firstfortworeasons: (a)diffusion oftheholesinrandom directions; foraccurate resultstheholesmustbesweptfromAtoBinatimet shortenoughtomakediffusion effectssmall,andthevoltagepulse appliedatAmustbeshortcompared witht;(b)holesarelostby recombination withelectrons withinthespecimen. Tomeasure themobility ofelectrons, allthatwouldseemnecessary atfirstsightwouldbetousen-typematerial andapplyanegative pulse atA.Thiswouldinjectelectrons, creating alocalexcess,whichis, however, dissipated inanextremely shorttime(seeProblem 19.3), restoring theequilibrium concentration ofelectrons everywhere inthe semiconductor. Inthiscaseonlythefirstpulseisobserved atB,and nosecondpulse.Whenapositive pulseisappliedatA,withann-type semiconductor, holesarecreated,andalthough electrical neutrality is restored byaninflowofelectrons, wehavenowanon-equilibrium distribution withanexcessofholesandacorresponding excessof electrons. Equilibrium isrestored onlywhentheholesflowoutatB, orareannihilated withinthesemiconductor byrecombination with electrons. Thenetresultisthatwecanmeasure directlythemobility onlyof'minority carriers'; i.e.ofholesinann-typesemiconductor, or electrons inap-typesemiconductor. Byobserving thespreadinthe secondpulseatBthediffusion constant canbedetermined, andby observing itssizeasafunction ofLorofelectricfieldEtherateof recombination canbefound. Thequantity measured directly insuchexperiments iscalledthe 'driftmobility', butinmanysemiconductors itcannotbesodetermined becauseofrapiddiffusion ofthecarriers. Inthatcasethemobility, whenonlyonetypeofcarrierispresent, canbefoundfromtheconduc­ tivityandHalleffect,sincethenIRHI=BjnlelandIRHal=Bu.The quantity IRHulisoftenwrittenuHandcalledthe'Hallmobility' to distinguish itfromthedriftmobility. Variation ofmobilitywithtemperature Themobility ofelectrons orholesinsemiconductors islimitedby scattering processes whicharebasically similartothoseinmetals,but thetemperature dependence ofthemobility isverydifferent fortwo reasons. Inametalonlyelectrons attheFermisurfacecontribute to theconduction current,andtheirvelocity vissubstantially independent oftemperature; hencewedonotneedtotakeintoaccountanyvelocity 19.5] SEMICONDUCTORS 557 dependence inascattering cross-section. Similarly thefactthatthe scattering cross-section determines thefreepathl,whilethemobility depends ontherelaxation time 7=l/v,doesnotofitselfintroduce any temperature dependence. Inasemiconductor, however, theaverage kineticenergyofthechargecarriersis~kT,andthevelocityvariation asTisplaysanimportant role. Measurements ofmobility atvarious temperatures anddifferent impurity concentrations showthatitisafunction ofboth.Inapure material thechargecarriers arescattered bythelatticevibrations (phonons); atallbutthelowesttemperatures (whenimpurity scattering TABLE 19.2 Mobilities inelemental semiconductors Electrons H0168 Silicon Germanium(4'OX109)T-'"(2·5x108)T-2.• (3'5x101)T-H(9·1X108)T-2o ' Theunitsarecm2jvolt.sec, andthevaluesarequotedfromZiman(1960),Electrons andPhooons (O.U.P.). dominates inanycase)thescattering cross-section isproportional tothe meansquareamplitude ofthethermalfluctuations, andhenceispropor­ tionaltokT.Thisgivesameanfreepath1proportional toT-lwhichis thesameforallcarriervelocities, andtherelaxation time 7=l/vand hencealsothemobility shouldvaryasT-i.Table19.2showsthatthis lawisnotverywellobeyed, exceptforelectrons ingermanium. The discrepancies maybedueeithertoscattering byshortwavelength lattice vibrations whereadjacent atomsvibrateinanti-phase (theso-called 'opticalmodes'), ortothecomplicated bandstructure (seeFig.19.7), bothofwhichallowscattering processes withalargechangeinelectron wavevector(ordinary latticescattering bythelong-wavelength or 'acoustic' modesallowsonlysmallchanges inelectron momentum becausethephononmomentum issmall).Theobserved mobilities in siliconandgermanium atroomtemperature arefoundfromTable19.2 toliebetweenabout500and4000cm2fvolt-sec, andarethusconsiderably higherthanthoseinmetals(forcopperthevalueisabout40cm2fvolt­ sec).Thehighmobility ispartlyduetolowvaluesoftheeffective mass ofelectrons andholesinthesemiconductors. Thescattering cross-section duetoneutralimpurities isinversely proportional tocarriervelocity, givingarelaxation timeindependent ofvelocity andhencealsooftemperature. Charged (i.e.ionized) 558 SEMICONDUCTORS [19.5 impurities willscattercarriersbyaprocessanalogous toRutherford scattering ofalphaparticles; thecross-section isinversely proportional tothesquareofthecarrierenergyandhencevariesasT-2,sothatthe meanfreepathvariesasT2.Sincethemeanvelocity variesasTi,the relaxation timevariesasTi.Aspecialcaseofscattering bycharged particles isthemutualscattering ofelectrons andholes. TABLE 19.3 Summary ofdependence ofelectron(hole)scattering on velocityvandtemperature T CroBB-Bection FreepathRelaxation time Scattering mechanism a loca-I or=ltv Phonons (atordinary temperatures) P p-l p-t Neutral impurities v-I v constant Ionizedimpurities V-I v& v3==pi Thevelocityandtemperature dependence ofthesescattering mechan­ ismsaresummarized inTable19.3.Inafirstapproximation therates ofscattering bydifferent processes areadditive; thatis,wecanwrite !=2:.!-. (19.27) TiTi Atlowtemperatures thelatticescattering decreases asthelattice vibrations dieaway,andthemobility iseventually dominated bythe impurity scattering. Recombination anddiffusion Atanygiventemperature thereisanequilibrium concentration of electrons andholesinasemiconductor, thetwoconcentrations being equalinintrinsic material andgenerally unequalinextrinsic material. Intheexperiment ofShockley andHayneswehaveseenthatanexcess ofminority carrierscanbeinjectedatacontact,andtopreserve electrical neutrality therewillbeacorresponding injection ofmajority carriers, possiblyatanotherelectrode. Anyabnormal chargedistribution caused therebyvanishes inabout10-11sec(seeProblem 19.3),sothatwecan writeIJ.nh=IJ.ne,whereIJ.nh,IJ.nearethelocalexcesses inthenumber ofholesandelectrons respectively perunitvolume.If,vedidnothave thisequality, aspacechargep=e(lJ.nh-lJ.ne) wouldbesetup,which byPoisson's equation divE=pfEEO=(efEEo)(lJ.nh-lJ.ne) wouldgiverisetostrongelectricfields.Thesewouldcausecurrents to 19.5] SEMICONDUCTORS 559 flowwhichwouldneutralize thespacechargeinthetimegivenabove. Obviously thiscurrentflowconsists mainlyofthemorenumerous majority carriers, andthecontrolling factoristhedeparture fromthe equilibrium valueofthenumberofminority carriers. Thefactthat 1i.nh=1i.ne(inpracticetheequality isnotexact,butdepartures from itareverysmallandcanbeneglected forpresentpurposes) meansthat thefractional changeinthenumberofminority carriersmaybeappre­ ciablygreaterthanthefractional changeinmajority carriers. Anumber ofimportant devicesdescribed laterinthischapterdependonchanges intheminority carrierconcentration; suchchangesrepresent departures fromequilibrium, andthemechanisms bywhichtheydecayplayan important roleinthedesignofsuchdevices. Asmentioned in§19.1,thechargecarriersinasemiconductor have afinitelifetime, butthisvarieswidelywiththepurityofthecrystal. Simplerecombination ofanelectron andaholecanonlytakeplaceif certainrestrictions onmomentum andenergyaresatisfied, andmeasure­ mentsonverypuregermanium showthatthisprocesswouldgivea lifetime greaterthan10-2sec.Theobserved lifetimes aregenerally muchshorter, owingtothepresence ofchemical impurities which provide extralevels,knownas'traps'.Inann-typematerial, for example, electrons maydropfromtheconduction bandintosuchtraps; holesmaythencollidewiththeseelectrons togiverecombination. The energyandmomentum considerations involved inthis'indirect' process aremuchlessrestrictive thanforthedirectprocessofrecombination, andthelifetimeofthecarriersiscorrespondingly shorter. Thesepro­ cessestakeplaceinthebodyofthesemiconductor, butthediscrete levelsatthesurface(seep.564)mayalsoactastrapswhichpromote recombination through 'indirect' processes. Afurtherlossofminority carrierswillalsooccurattheelectrodes. Thechanceofaholeandanelectron recombining isproportional to theconcentration ofeachspecies; hencetherateofannihilation is -anhne,whereaisaconstant whichdependsonthemechanism involved. Inthermal equilibrium thelossbyrecombination isbalanced bythe creationofnewcarriersthroughthermalexcitation intotheconduction band;ifwedenotetherateofcreation byc,thenclearlyc=an~n~, where n~,n~aretheequilibrium concentrations. Whenadeparture fromequilibrium takesplacewehave dnhldt=dneldt=c-anhne=a(n~n~-nhne). Asmentioned above,thefractional changeinthemajority carrier 560 SEMICONDUCTORS [19.5 concentration ismuchsmallerthanthefractional changeintheminority carrierconcentration, andtoafirstapproximation theformermaybe neglected. Hence,takingforexample theholesinann-typematerial, wemaywrite dnh/dt:an~(n~-nh) =-J:i.nh/'Th' (19.28) showingthattherateofdecayoftheminority carrierconcentration is simplyproportional totheexcessofminority carriers. Thequantity 'Th isthe'recombination lifetime' oftheminority carriers. Recombination atthesurfaceorextraction atanelectrode causesa localdiminution intheexcessofminority carriers; thisiscounteracted bythemovement ofcarriersfromregionswheretheyaremorenumerous. Thisisaprocessofdiffusion, andforsmallfieldstrengths diffusion currents aremuchlargerthanconduction currents. Forsimplicity, we consider acasewheretheminority carrierdensitynh(takingagain holesinn-typematerial) variesonlyinonedimension. Thenthenumber crossingunitareapersecondis-Dh(onh/ox), andthenetrateofincrease inathicknessaxisd{-Dh(onh/ox)} =-Dh(02nh/ox2)dx,whereDhis thediffusion coefficient fortheminority carriers. Inthesteadystate thisequalstherateoflossbyrecombination, whichinthickness dxis {(n~-nh)/'Th}dx; henceweobtaintherelation o2nhnh-n~-2=--. (19.29)oXDh'Th Thishasasolution (writing J:i.nhfor(nh-nm J:i.nh=(J:i.nh)Oexp( -x/Lh), (19.30) where(J:i.nh)Oistheexcessconcentration atX=0,andLh=(Dh'Th)*is ameasure ofthemeandistance aminority carrierwillmoveunderthe actionofdiffusion beforeitislostbyrecombination. Itisknownas the'diffusion length'andisanimportant quantity intransistor design. Atypicalvalueforgermanium is0·1cmforbothholesinn-typematerial andelectrons inp-typematerial; typicalvaluesinsiliconaresmaller byafactorofaboutthree. 19.6.Metal-semiconductor junctions Toinvestigate theelectrical properties ofsemiconductors itisneces­ sarytomakeelectrical connexions tothem.Thebehaviour ofajunction between ametalelectrode andasemiconductor depends onthenature andgeometry oftheconnexion, aswellasontheproperties ofthemetal andsemiconductor, makingafulltreatment verycomplex. Themost important property ofsuchajunction isthatthecurrentflowforagiven 19.6] SEMICONDUCTORS 561 voltageisquitedifferent inopposite directions, sothatitactsasa rectifier. Whencontact ismadebetween ametalandasemiconductor, a potential difference issetupbetween thetwoinasimilarmannerto thatbetween twometals(thecontactpotential). Forann-typesemi­ conductor whoseFermilevelisabovethatofthemetal,electrons pass fromthesemiconductor tothemetaluntilthetwoFermilevelsareequal. Thisprocessisillustrated inFig.19.11.Theexcessnegative chargeon Conduction band IValence band. Semiconductor Metal-------------1---- - + I + Vl!' ++ Conductionband Semiconductor•Metal Beforecontact Mterequilibrium isestablished FIG.19.Il.EnergybandsandFermilevelsatametaltosemiconductor contact, before andafterequilibrium isestablished. themetalrepels electrons nearthesurfaceofthesemiconductor, creating alayerwhichisdepleted ofconduction electrons andsohasahigher resistance thanthebulkofthesemiconductor. Thislayerisknownas the'barrierlayer'andisaregionofpositive spacechargebecauseit contains theionizeddonorimpurities withoutthecompensating charge ofthenegative conduction electrons. ByPoisson's equation (equation (2.1»thepotential willvarythrough thespacechargelayer,sothat therewillbeapotential difference Yobetweenthepositionofthebottom oftheconduction bandatthesurfaceandinthebulksemiconductor; theenergybandsaretherefore distorted nearthesurface,asshownin Fig.19.11. Inequilibrium therewillbenonetcurrentflowacrossthebarrier, butthisisadynamic equilibrium between acurrent-Ill.ofelectrons flowingoutofthemetalintothesemiconductor, andanequalcurrent 851110 00 562 SEMICONDUCTORS [19.6 ofeleotrons (and,toamuohlesserextent,holes)leavingthesemi­ conduotor forthemetal.Thelattermaybewritten intheform loexp{-eVafkT), sinoethefractionoftheeleotrons whichhavesufficient energytosurmount thebarrier Vaisproportional toexp(-eVafkT). IfavoltageVisnowappliedwhichmakesthemetalpositive with respeottothesemiconductor, thisextravoltageappearsalmostentirely aorossthebarriersinoethishasamuohlargerresistance thaneither Conduotion band - -Fermilevel Valenoeband Valenceband I Valenceband Metal Semioonduotor Metal Semioonduotor Metal Semioonduotor Ca)Metalpositive (b)Noappliedvoltage (c)Metalnegative (forward direotion) (backward direotion) FIG.19.12.EffectofappliedvoltageVatmetal-semiconductor junction. In(a)afor­ wardvoltageVisapplied, reducing thebarrierheighttoVo-V,andgivingalargecurrent flowofelectrons fromthesemiconductor intothemetal;in(c)areversevoltage is applied, increasing thebarrierandreducing thecurrent flow.Notethechangewith voltageintheeffective thickness ofthebarrier(ll={2£EO~~:V)t),andthatthe Fermilevelinthemetalisdepressed whenthemetalismadepositive becauseofthe negative signoftheelectronic charge. themetalorthebulksemiconductor. Theourrentleavingthesemi­ conduotor thenbecomes loexp{-e(Va-V)fkT}, because thebarrier heightisreducedfromVatoVa-V;ontheotherhandthecurrentleaving themetalisstill-la,sincethebarrierwhichtheelectrons inthemetal havetosurmount isunaltered (seeFig.19.12).Since la=10exp(-eVafkT), thenetcurrentflowis 1=la{exp{eVjkT)-I}. (19.31) 19.6] SEMICONDu'CTORS 563 IfVispositiveandgreaterthan(kT/e),largeforward currents can flow,whileifVisnegative Iapproaches-Ia;thecurrent-voltage characteristic therefore hastheformshowninFig.14.15,andthe junction isanefficientrectifier. Thistreatment givesasatisfactory qualitative treatment ofthe rectifying properties, thoughagreement withexperiment isbynomeans exact.Asatisfactory featureisthatitgivestherightsignforthe forward direction; thatis,thatthedirection ofeasyflowofelectrons (forann-typesemiconductor) isfromsemiconductor tometal.Inan earlytheorytheflowofelectrons throughthebarrierwasascribed to thetunneleffect,sothattheeasyflowwasfrommetal(wherethe electron densityishigh)tosemiconductor. However, thetunneleffect isonlyappreciable whenthebarrierthickness iscomparable withthe electron wavelength inthemetal;i.e.lessthan10-7cm,whereas the theoryofabarrierduetoadepletion layergivesathickness oforder 10-6cm.Asimpleversionofatheoryforthebarrierthickness put forwardbySchottky isasfollows. Weassumethatallconduction electrons areremoved fromthebarrier layer,sothatthechargedensity p=eNa'whereNaisthenumberof donorsperunitvolume(forsimplicity wetakethesealltocarryunit charge,andalltobepositively ionized; i.e.weareintheexhaustion region). Letthesurfaceofthesemiconductor betheplanex=0, andthebarrierextendfromthesurfacetotheplanex=S,sothatthe spacechargebecomes zero forx~S.Thenthepotential isconstant forx~S,andtheelectricfieldvanishes atx=S.Inthebarrier, Poisson's equation reducesto V2V=d2V/dx2=-eNa/EEo. (19.32) Integration, usingtheboundary conditions dV/dx=0(noelectricfield) atx=S,andV=0atx=0,gives V=-(eNa/2EEo){(X-S)2_S2}. (19.33) Hencethedifference ofpotential atthesurfacex=0fromthatinthe interiorofthesemiconductor (x~S)is Yo=eNaS2/2EEo· (19.34) IfwetakeE~12(asinsilicon)andNa=1018cm-3(1024m-3),Sisfound tobeabout3X10-6cmifYoisabout1volt. AdifficultyintheSchottky theoryisthatYoshouldbeequaltothe difference intheworkfunctions ofmetalandsemiconductor, andhence la'whichisproportional toexp(-eYo/kT), shoulddependonthemetal used,whereas experimentally itdoesnot.Toovercome thisdifficulty 564 SEMICONDUCTORS [19.6 Bardeen putforwardtheideaof'surfacestates'.Atthesurfaceofthe semiconductor theatomicarrangement isdifferent because thereare noatomsontheonesidewithwhichtoformbonds.Thusthesurface atomshavedifferent energylevels,andtheselevelsarediscretebecause bandsareonlyformedfromthelevelsofatomswhichareidentical. Theremayalsobeimpurity atomsabsorbed onthesurface. Those surfacelevelswhichliebelowtheFermilevelofthesemiconductor will befilledbyelectrons whichdropintothemoutoftheconduction band, givinganegative chargeonthesurfacewhichrepelselectrons nearthe surface. Thisgivesadepletion layerjustinsidethesurface,whichacts asthebarrier.If~isthenumberofsurfacestatesperunitarea,and N(jthenumberofdonorsperunitvolume,thenthethickness 3ofthe barrierwillbedetermined bytherelation ~=N(j3ifweassumethat alltheconduction electrons foradistance 3aredrawnintothesurface states.Application ofPoisson's equation tothebarrierlayeryields equation (19.34)asbefore,butonwriting3=~/N(jwehave Yo=eN~/2€€oN(j' (19.35) Thisequation showsthatYodepends on~andNdforthesemiconductor andisindependent ofwhatmetalisusedtomakethecontact; infact thebarrierlayerexistsintheabsenceofanycontact. Thediscussion abovehasbeenonthebasisofann-typesemiconductor, butsimilararguments applytoap-typesemiconductor ifholesare substituted forelectrons (thereby reversing thedirection ofeasycurrent flow).Thusametal-semiconductor junctionwillactasarectifier. However, inordertouseittwojunctions mustbemadetocomplete a circuit,andsincetheforward direction willbeintheopposite senseat thetwojunctions, norectifying actionwillresultunlessthetwojunctions aredifferent innature.Atmicrowave frequencies onejunction mustbe verysmallincross-section, sinceotherwise thecapacitance betweenthe metalandbulksemiconductor acrossthebarrierlayeractsasashort circuit,thecurrentflowingas,displacement currentthroughthiscapaci­ tanceinsteadofrealcurrentthrough thebarrier. Thissmallcontact ismadebyathinmetalwhisker(usuallytungsten) pressedagainstthe semiconductor (seeFig.14.16);theothercontactissoldered andoflarge area,sothatitofferslittleresistance (andlargecapacitance) totheflow ofcurrentinwhatwouldotherwise beits'backward' direction. Suchfinecontacts havearelatively highresistance, sincethecurrent hastospreadoutfromafinepointthrough theinteriorofthesemi­ conductor, andtheycannotcarrymorethanafewmAofcurrent. At 19.6] SEMICONDUCTORS 565 powerfrequencies semiconductor-metal junctions oflargeareacanbe used;thesecondcontactissoldered, insuchawaythatthesemiconductor surfaceisdopedtomakealmostanohmiccontact. Inthepastboth Cu20andselenium havebeenusedforsuchpowerrectifiers, butsemi­ conductor-metal junctions forthispurpose havenowbeensuperseded byjunctions between twoparts,differently doped,ofasinglesemi­ conductor crystal. Conduction band Valenceband ./////////////////////////// FIG.19.13.Energy bandsandFermilevelsatap-njunction, beforeequilibrium is established. 19.7.Thep-njunction Singlecrystalsofasemiconductor (usually germanium) canbepre­ paredinwhichoneendisdopedtomakeitp-type,andtheotherend n-type,thechangefromp-typeton-typetakingplaceinaregionwhose thickness isoforder10-6em.Suchaunitiscalledap-njunction. The p-typematerial ismadebydopingwithaconcentration Naofacceptors, then-typebydopingwithaconcentration Ndofdonors;obviously there willbeanarrowregionatthejunction wherethedopingconcentration variesfromoneextreme totheother,butprovided thisregionissmall inwidthcompared withthethickness ofthebarrierregionestimated below,wecanregardthechangefromp-typeton-typeasdiscontinuous. Ingermanium dopedwithgroupIIIandgroupVimpurities theioniza­ tionpotential issosmallthatatroomtemperature wecanregardthe donorsandacceptors asfullyionized(i.e.weareintheexhaustion region). Theenergylevelsituation showninFig.19.13isnotstableandcould onlyexistifthen-typeandp-typematerial wereinseparate crystals. Thereisanexcessholeconcentration inthep-type,andexcesselectron concentration inthen-type,sothatwhentheyareinthesamecrystal holeswilldiffusetotherightandelectrons totheleft,eachgivinga 566 SEMICONDUCTORS [19.7 positive currenttotheright.Thisgivesapositive potential tothe n-typematerial, sothattheenergylevelsofitselectrons arelowered becauseoftheirnegative charge.Thisprocesscontinues untiltheFermi levelsofthetwohalvesareequalized, asinFig.19.14.Thedifference ofpotential between thetwohalvesmeansthatstrongelectricfields existnearthejunction, andthesesweepoutthemobilecarriersinthe p-type-------------1------ Vo Fermilevel---------------- ------ ------ n-type _____1~____ FIG.19.14.EnergybandsandtheFennilevelatap-njunction, afterequilibrium is established. Then.typebecomes positively charged, sothatitselectrons havelower potential energy. vicinityofthejunction, givingabarrierlayerofhighresistance. The wholeofthepotential dropoccursacrossthisbarrierlayer;removalof theholesinthebarrierlayeronthep-typesideleavesanegative space chargeof-eNaperunitvolume,andremovalofelectrons onthen-type sideleavesapositivespacechargeof+eNdperunitvolume,asshownin Fig.19.15.WemayapplyPoisson's equation tothebarrierlayer,mak­ ingthesamekindofsimplifying assumptions asinthetreatment ofthe metal-semiconductor junction. Letthechangefromp-ton-typeoccur discontinuously attheplanex=0,andletthebarrierthickness be8p and8noneachsiderespectively. Then,takingV=0,atx==Owehave, usingtheboundary conditions dVj&=°atx=-8pandatx=+8n: x<O x>O d2V=+eNa d2V=_~~L &2 ££0 &2 ££0 V=eNa{(x+8)2_82} V= _eNd{(x_On)2_8~} 2££0 p p 2££0 19.7] SEMICONDUCTORS 567 Thustheoverallpotential difference Yobetween then-typematerial andthep-typematerial isthedifference between Vn,thevalueofVat x=()n'and"fpatx=-()p'Thisis Yo=Vn-"fp=-2e {Nd()~+Na()~}' (19.36) €€o II:=-8, I I I+--­dVIij;=OI I III:=0 II:=8. r I ++++ ++++ P=e.N. ++++++++ p·typep=-elV;.1--. IdVIdx=0 on-type FIG.19.15.Spacechargedensityatap-njunction, onthesimplified modelusedin thetext. FromthefactthatdVjdxmustbecontinuous atx=0weobtainthe additional condition N()=N() (19.37) ap dn' Finally, comparison ofFigs.19.13and19.14showsthat e(~-"fp) =lJ'Fn-lJ'Fp ~lVy, (19.38) whereWFn,WFParetheFermilevelsinn-andp-typeasshownin Fig.19.13;intheexhaustion rangetheselieclosetothebottomofthe conduction bandandtopofthevalencebandrespectively, sothatthe energydifference isnearlyequaltotheenergygap.Ifwetake Na=Nd=1024m-3, thenforgermanium thewidthofthebarrier «)n+()p)isfoundtobe about5X10-6em. Thefactthatthebarrierlayerhasaverymuchhigherresistivity than thatofthebulkmaterial meansthatanyexternal voltageappliedappears almostwhollyacrossthebarrierlayer;thereislittlepotential variation inthebulkmaterial andcurrents nearthebarrierareduetodiffusion. 568 SEMICONDUCTORS [19.7 (19.39)Majority carrierswhichcrossthebarrierbecome, ofcourse,minority carriersonthefarside;forexample, holesleavingthep-typematerial ontheleftofFig.19.14becomeminority carriersonentering then-type material ontheright.Thiscreatesanexcessofminority carriersatthe barrieredge,givingrisetoadiffusion currentawayfromthebarrier whichisthecontrolling factorinthesteadystateforthenetcurrent flowacrossthebarrier. Fromequation (19.30),wehaveforholesinthe n-typeAnh=(Anh)OexP{-(x-on)jL h}, where(Anh)Oistheexcessconcentration atx=on'theright-hand edge ofthebarrier. Thediffusion holecurrentdensityis ih=-eDh(onh!ox) =e(AnhMDh!Lh)eXP{-(X-Dn)jLh} =e(AnhMD h!Lh) atthebarrieredge.Adetailed analysis showsthat,aswemightexpect, theholeconcentration atx=Onisproportional tothedensityofholes inthep-typematerial whichhavesufficient energytosurmount the barrier.Ifanexternal voltageVisappliedwhichmakesthen-type material lesspositive, thevoltageacrossthebarrierbecomes(Vo-V) andhence(nh)O=Aexp{-e(Vo-V)!kT}.Intheabsenceofanyexternal voltage (nh)Oisjustequaltotheequilibrium holedensity nhinthe n-typematerial, sothatnh=Aexp(-eVojkT) and (Anh)o=(nh)O-nh=nh{exp(eVjkT)-l}. Thisgives ih=e(nhDhjLh){exp(eVjkT)-l} andasimilarequation isobtained fortheflowofelectrons acrossthe barriertotheleft.Hencethetotalcurrentdensityacrossthejunction is . .+.{nhDhneDe}{eVI}J=JhJe=e--+-- exp-- ,LhLekT wherenhlDhlLhrefertoholesinthen-typematerial andne,De'Leto electrons inthep-type. Thisemphasizes theroleplayedbytheminority carriers. Thisequation, dueoriginally toShockley, issimilartothatobtained intheprevious sectionforametal-semiconductor junction, andap-n junction actsasarectifier.Itcanbecontrolled inproduction much betterthanametal-semiconductor junction becauseoftheabsenceof anexternal surface. Theforwardresistance islowerthaninathermionic vacuum tubediodebecauseofthehighdensityofmajority carriersin asolid,butthereisasmallback-current, andthejunction cannot withstand suchhighbackvoltages. Thisisbecauseathighfieldstrengths 19.7] SEMICONDUCTORS 569 thecarriersgainsufficient energyfromthefieldtoexciteelectrons from thevalencebandintotheconduction band,producing morecarriers. Thisisanavalanche process,andsimilartotheTownsend discharge in agas,sothatthebackcurrentincreases veryrapidlybeyondacertain voltage. Bothgermanium andsiliconareusedforp-njunctions. Germanium hasthesmallerenergygap,givingabiggercarrierdensity. Themobility isalsosomewhat higher,sothattheforward resistance islowerthan "'ndu........J""'"\I!I!I!I!II!I!1/I!I!I! • I!I!1//1/1/1/11!1/1//1. ---'-'-'-- ------- --'------ -----Fermi level .-• .~ /77777777//7//777777717777777777777777777///, Valenceband FIG.19.16.TheenergybandsandFermilevelinann-p-njunction withoutexternal bias. forsilicon,butthebackcurrentishigher(ingermanium thevalueoffa atroomtemperature isabout10-5A/cm2).Siliconcanbeoperated at highertemperatures beforethermal ionization acrosstheenergygap increases thebackcurrentanddropstherectification efficiency appre­ ciably.Theabsenceofaheatedcathoderequiring itsownpowersupply, betterreliability, andlongerlifegivesolid-state rectifiers agreatadvan­ tageovervacuumtubedevices. 19.8.Thejunction transistor Thejunction transistor isasinglecrystalofsemiconductor (usually germanium) inwhichdifferent regionsaredopedinsuchawaythata verythinlayerofp-typematerial isproduced between twon-type regions,orviceversa.Forconvenience weshallrestrictourdiscussion totheformer,butitappliesequallytothelatterifweinterchange the rolesofholesandelectrons. Atransistor mayberegarded astwop-n junctions backtoback,thep-typematerial inthethincentralsection (knownasthebase)beingcommontobothjunctions. Whennoexternal voltageisapplied,theelectron energydiagram isreadilyseenfromthe considerations ofthepreceding sectiontobeasillustrated inFig.19.16. Inoperation thedeviceisconnected tobatteries asshowninFig.19.17, 570 SEMICONDUCTORS [19.8 alargervoltagebeingappliedtothe'collector' side,sothattheelectron energydiagram nowbecomes asshowninFig.19.18.Thepotential acrosstheleft-hand junction isreduced, sothatalargecurrentof emitterjunction collector junction E..-----1~--- ..emittern-type 1..-------II-+------B~-----1III-+-· -_....I emitterbiasbattery collector biftsbattery FIG.19.17.Biasvoltages appliedtoann--p-njunction innormaloperation. Asshown, thecircuitappliestogrounded baseoperation, thesignalvoltagevebeingappliedatthe emitterelectrode togiveanamplified voltageacrosstheloadresistance Rinthecollector circuit.Thearrowsshowthedirection ofelectronflow. Fermilevelcollector n-type'Conduction hand emittern-type Valenceband, FIG.19.18.Electron energies inacorrectly biasedn-p-njunction.emitterbias.. ~~i;v'I-- collector biail ------''"-----------------+-- -_.- - - - - - - - electrons (themajority carriersinthen-typeregion)cancrossintothe centralp-typeregion.Theregiontotheleftisknownasthe'emitter', andthejunctiontotheleftasthe'emitterjunction'. Onarrivalinthe centralp-typeregiontheseelectrons become minority carriers, and diffusethrough theright-hand junction (the'collector junction') since 19.8] SEMICONDUCTORS 571 thepotential dropisintheirfavour,andarriveatthecollector electrode. Withsuitablegeometry inthebaseregion(e.g.thickness about10-3cm, cross-section afewmillimetres) practically alltheemittercurrentgoes ontothecollector, andverylittleflowstothebaseelectrode. Theemitterjunction isessentially ann-pjunction operated inthe forwarddirection. Thusitsinternalresistance islow,andasmallchange Veintheemittervoltageproduces anappreciable changeieintheemitter currentIe.Ifafraction 0:ofthiscurrentreachesthecollector, thenthe collector currentchangeisic=o:ie•Ifaresistance Risconnected in thecollector circuit,thevoltagechangeacrossitis va=Ric=o:Rie=o:R(dle/dYe)v e• Thusavoltageamplification isobtained provided thato:R(dle/dYe) is largecompared withunity.Since0:::::::0,98,thisrequiresthattheload resistance Rbelargecompared withtheinputresistance (dYe/dIe); it ispossible tofulfilthiscondition without materially reducing the voltageatthecollector provided thatRisnotasgreatasthecollector junction resistance, andsincethisisap-'T£junction working inthe backwards direction itsd.c.resistance ishigh.Typical valuesare dYe/dIe=40ohms,R=30000ohms,givinganamplification of about700. Incontrasttoavacuumtubetriode,theinputimpedance isquitelow. Ifweniaketheapproximation 0:=1,sothatnocurrentflowsoutat thebaseelectrode, thenthetransistor circuitwithgrounded basecon­ nexionisquitesimilartothatofFig.14.8,andthevoltageamplification isequaltotheratiooftheimpedances inthecollector andemitter circuits, sincethesamecurrentflowsthrough each.Inpractice the emitterimpedance isgenerally smallerthantheinternal impedance of thegenerator connected inserieswithit,sotheamplification ismore nearlyequaltotheratiooftheloadimpedance totheinputgenerator impedance. Thecommon baseconnexion circuitthatwehavedescribed hasacurrentgainic/ie=0:,whichis lessthanunity,butothercircuits arepossiblewitheithercommon emitterorcommon collector circuits. Thesehavecurrentgain,sincethebasecurrentisib=ie-ic=(l-o:)ie; thusthecurrentgaininthetwoalternative circuitsisic/ib=0:/(1-0:) orie/ib=1/(1-0:) respectively, beingintheregion50to100ineach case.Thecommon emittercircuit,inwhichthecollector circuitis connected backtothepointEinFig.19.17insteadoftothepointB, istheonegenerally usedinpractice, sinceitgivescurrent,voltage,and powergain. 572 SEMICONDUCTORS [19.8 Transistors canbeusedtoperform mostofthefunctions ofvacuum tubes,andhavemanyadvantages. Theyaresmaller(lessthan1cm3), morerugged,andhavelongerlife;theyrequirenofilament powersupply, andelectrode potentials ofafewvoltsratherthan102voltsareneeded, givingmuchlowerpowerdissipation; theuseofp-n-ptransistors aswell asn-p-nproduces greaterflexibility incircuitry. Theirdisadvantages are:smallerpowerhandling capacity, greatersusceptibility todamage through overloading, andgreatersensitivity toambient temperature (inparticular, boththereversecurrentflowacrossthejunctions dueto diffusion, andtheemitterresistance dYeldIeareconsiderably dependent ontemperature). Transittimeeffectsproduce appreciable phaseshifts intheregionof104cisratherthanat108cis,sothatthecircuitsused mustbedesigned tocopewithsucheffectsatmuchlowerfrequencies thanwithvacuumtubes.Evenataudio-frequencies transistor circuitry isconsiderably different fromvacuumtubecircuitry becauseofthelow inputimpedance ofthetransistor. GENERAL REFERENCES ROLLIN, B.V.,1964,AnIntroduction toElectroniCB (O.U.P.). SIMPSON, J.R.,andRICHARDS, R.S.,1962,Phy8ical Principles andApplications ojJunction Transistors (O.U.P.). PROBLEMS 19.1.Showthattheenergylevelsofan'atom'consisting ofapositive charge equaltothatoftheproton,withaneffective massmt,andanegative chargeequal tothatoftheelectron, withaneffective massmt,movinginamediumofdielectric constant £,arethosegivenbyequations (19.18)and(19.19). Showthatthebinding energyofadonorimpurity levelinsilicon(mt=00, mt/m=0,4,E=U'5)isabout0·041eV,andthattheBohrradiusisabout6·5 timestheinter·atomic distance (2,35A). 19.2.Inmanysemiconductors theenergygapvarieslinearlywithToverafair temperature range,thoughittendstoaconstant valueatlowtemperatures. ShowthatiflVg=WZ-aT, thenaplotofIn(RHTf)againstI/Tgivesastraight linewhoseslopegivesthevalueofWZ.Showalsothattheeffectofthetempera· turevariation oflVgonequation (19.22)istoreplace lVgbyWZandtoincrease theapparent valueoftheproduct (m:mt)byexp(2a/3k). 19.3.Bycombining equations (1.20),(3.3),and(3.5)showthatanyabnormal chargedistribution inaconductor ofconductivity 0',dielectric constant E,vanishes intimeasp=poexp(-t/7'), where 7'=££0/0'.Showthatforgermanium where £=16,0'= 10(ohm.metre)-l, thevalueof7'isabout1·4X10-11sec. Asampleofgermanium contains 1018holes/m3•Showthatthepresence ofa netspacechargeequivalent toIpercentoftheholeconcentration wouldgive risetoanelectricfieldgradient ofabout107V/m3• SEMICONDUCTORS 673 19.4.Whenanexternal biasvoltageVisappliedtoap--njunction withahigh resistance barrier,thevoltage acrossthebarrierinequation (19.33)becomes ~-v"=V+Vg,whereIeIVg=lVg.Showthatthetotalcharge±Qineachofthe spacechargeregionsofFig.19.15,whenNa=Nil=N[Jisgivenby Q=(EeoeN[)I(V+Vg)1 perunitarea. Thisisafunction oftheapplied voltage, andthebarrieractsasacapacitance fora.c.voltages ofmagnitude 0=(dQ/dV) =t(eeoeN[)I(V+Vg)-lperunitarea. Specialjunction diodesofthistypeareusedasvariable capacitors (since0is afunction ofV)intheparametric amplifier (§16.5). 19.5.Verifythatthecapacitance derivedintheprevious problem isthesameas thatofaparallel-plate capacitor withplateseparation equaltothebarrierthick­ nessandfilledwithdielectric ofrelativepermittivity e. 19.6.Inacrystalthevelocityofaparticleisgivenbytherelation 1/,v=gradkW, wheregradk=i.,(%k.,)+ill(%k,A+J.z(O/ok.), inwhichi."etc.,areunitvectors alongthex,y,zdirections. Showthatingeneralvisnotparalleltokunlessit isalongoneoftheaxes,foraparticlewhoseenergyWisgivenbyequation (18.14). Showthatwithrespecttoaxes(x',y',z')whicharederivedfromthe(x,y,z) axesbyarotation throughanangle0aboutthey-axis,equation (18.14)becomes W=lv.±l/i,2{k'2(COS20+sin28\+k'2/m*+k'2(sin28+cos28)+o.,m:mf} 1111•m:m: +2k~k~sin6cos6(~-~)}.m.,m. Henceshowthattheeffective massforaparticleforwhichkisalongthez'-axisis 1C2Wsin'8cos20 1/,2ok'2=m*+m*.• rJJ • (m*-1isatensorquantity, andcross-product termssuchask~k~,etc.,areabsent onlywhenasuitable choiceofaxes(usually dictated bythecrystalsymmetry) ismade.) 19.7.Forasemiconductor intheinfra-red, where W7"~1,theeffective conduc­ tivity(seeProblem 18.2)becomes u'=uo/(I+w2T2).Showthatanelectro­ magnetic wavewillfallinintensity insidethesemiconductor asW=lYoexp(-(XX). where ex== uo.neoc(l+w~2) provided thatk<{ninthecomplex refractive indexn-jk.Calculate thevalue ofexforasampleofgermanium inwhichn=4,uo=10(ohm-metre)-1, ata frequency where W7"=100. (Answer: a:=10-1permetre,approximately.) 20 THEATOMIC THEORY OFPARAMAGNETISM 20.1.Ageneralprecession theorem INChapter 8theoriginofparamagnetism wasdiscussed, anditwas showntoexistinsubstances containing permanent magnetic dipoles. Suchdipolemoments areassociated withmovingcharges, beingdue eithertothemotionofelectrons intheirorbitsabouttheatomicnucleus orthespinoftheelectron aboutitsownaxis.Fromobservations of hyperfine structure inatomicspectraitwasinferredthatthenuclei ofmanytypesofatomalsopossess'spin'duetorotation aboutan internalaxis,andthatamagnetic dipolemoment isassociated withthis spin.Inallthesecasesthemagnetic moment isassociated withsome unitsofangularmomentum, andthedirection ofthemagnetic moment misparalleltothatoftheangularmomentum vectorG,andpropor­ tionaltoit.Thuswemaywrite(cf.equation (8.2») m=yG, (20.1) whereyisaconstant whosereciprocal (1/y)isknownasthegyromagnetic ratio.Foranelectronofcharge-eandmassmomovinginanorbit, yisequaltotheclassical value-e/2mo;themagnetic moment asso­ ciatedwiththeintrinsic spinoftheelectron isanomalously large,the ratiobeinginthiscaseverynearlyequalto~e/mo'Theminussignin eachofthesecasesarisesfromthefactthatthechargecarriedbythe electron isnegative, andshowsthatthemagnetic moment isoppositely directedtotheangularmomentum vector.Inthecaseofthenucleus, theangularmomentum isofthesameorderasthatofanelectron, being eitherasmallhalf-integral orintegralmultiple ofn,butthemagnetic moment is~athousand timessmaller, corresponding tothegreater massoftheparticles (protonandneutron) inthenucleus. Thevalue ofyisthenUn(e/2M), whereMisthemassoftheproton,andUnisa number whichisoftheorderofunitybutisnotingeneralanexact integerorasimplefraction. Whenanatomornucleuswithapermanent magnetic dipolemoment misplacedinasteadymagnetic fieldB,acoupleisexertedonitwhich maybewritteninvectorformasm1\B.Theangularmomentum must therefore change(eitherinmagnitude ordirection) atarateequalto 20.1] THEATOMIC THEORY OFPARAMAGNETISM 575 thiscouple;thatis (20.3)Sincem isproportional toandparalleltoG,wehave G=yG/\B. (20.2) Thisisavectorequation whosesolutioniseal'lilyfoundbywritingdown itscomponents referredtoCartesian coordinates. Ifthemagnetic field isassumed toactalongthez-axisthecomponents are ~x=yBGy) Gy=-yBGx• G-0s- Integration ofthelastequation showsthatthecomponent Gsalongthe z-axisisaconstant.ItfollowsthattheangleexwhichGmakeswithB isconstant, andwemaywriteGs=Gcosex.Theequations forthex­ andy-components maybesolvedbydifferentiating oneofthemand eliminating eitherGxorGy•Onefinds Gx=yBGy=-(yB)2Gx withanidentical equation forGy•Thesolution isoftheform Gx=Acos(-yBt+E), andfromequation (20.3)wefind Gy=Asin(-yBt+E). Thusitwillbeseenthattheprojection ofGonthexy-plane isofconstant magnitude A=Gsinex,butrotateswiththeangularvelocity-yB, whichwemaywriteasWL'Thusoursolutionforthecomponents ofGis Gx=GSinexcOS(WLt+E») Gy=GsinexSin(WLt+E) . (20.4) Gs=Gcosex Themotionissuchthatthemagnitudes ofbothmandGremaincon­ stant,buttheirdirections 'precess' ataconstant angleaboutthe .direction ofthefieldBasinFig.20.1.Theangularvelocityofthepre­ cessiondepends onlyonthegyromagnetic ratioandthesizeofthe magnetic field.Thedirection ofprecession isthatofaright-handed screwprogressing alongBifyisnegative, andviceversa.Theangleex depends ontheinitialconditions prevailing whenthemagnetic field wasswitched on. Although wehavechosentosolveequation (20.2)bytheuseofa Cartesian coordinate system,wecouldhavederivedacertainamount 576 THEATOMIC THEORY OFPARAMAGNETISM [20.1 ofinformation aboutthemotionbyinspection ofthevectorequation (20.2)itself.Sincethevectorproductoftwovectorsisavectorperpen­ diculartoboth,itfollowsthatGisnormaltoGandtoB.Thus,in Fig.20.1,ifinstantaneously bothGandBareintheplaneofthepaper, themotionofGmustbenormaltothepaper.Thismeansthatifthe B GCOSI1.z )--_., FIG.20.1.Precession ofGaboutB.Thedirection ofprecession isthatforan electronic momentum (ynegative). momentum vectorGisdrawnfromafixedorigin,thenitstipmustmove outofthepaper,andsinceGalwaysremainsnormaltoG,thetipmust moveinacirclearoundB,i.e.theangularmomentum vectorprecesses aroundB. Thisprecessional motion, originally derivedinatheorem dueto Larmor,isaquitegeneralresult,depending onlyontheconnexion between angularmomentum andmagnetic dipolemoment. Inaquan­ tummechanical system,suchastheatom,theangularmomentum plays animportant role,asiswellknownfromatomictheory.Inthenext sectionweturntoconsideration ofthemagnetic moments ofsingle atoms,makinguseofourgeneralprecession theorem. Thechiefdiffer­ enceweshallfindfromtheclassical caseconsidered aboveisthatthe angleexisnowfixedbytherulesofquantization, onlyasmallnumber ofvaluesbeingpossible, insteadofanyvalue. 20.2.Thevectormodeloftheatom Anunderstanding oftheoriginofmagnetic moments inatomsispos­ sibleonlywhenonehasathorough knowledge ofthequantum theoryof thebehaviour oftheelectrons intheatom.Acomprehensive discussion ofthistheoryisfarbeyondthescopeofthisbook,anditistherefore necessary toassumethereaderisacquainted withatomicstructure to 20.2] THEATOMIC THEORY OFPARAMAGNETISM 577 theextentgiveninmostelementary textbooks ofatomicphysics. The outlinewhichfollowsisintended onlyasaresumeofthetheory,mainly intermsoforbitsratherthanwavefunctions. Theresultsquotedwill bethoseappropriate tothewave-mechanical theory,however, unless otherwise indicated. Thestateofanelectroninanatomisdefinedbyfourquantum numbers n,l,ml,and8,whosesignificance isasfollows. Theprincipal quantum numbernhasintegralvaluesfromunityupwards, andtheenergyofthe electron ismainlydetermined bythevalueofn.OntheoriginalBohr theorytheenergydepended onlyonthevalueofn,itsvaluebeing,for anatomwithonlyoneelectron, RZ2Jf.=--, (20.5) 11.n2 whereRisauniversal constant (Rydberg's number), andZethecharge onthenucleus. Thesameresultisobtained bywavemechanics fora one-electron atom,butthisresultdoesnotholdforatomswithseveral electrons owingtotheelectrostatic repulsion betweentheelectrons. For mostatomsitremains truethatelectrons withthelowervaluesofn havethelowerenergy,andthedifference ofenergyforsuccessive values ofndecreases asnbecomes larger(cf.Fig.20.2). Thequantum numberlisdefinedbythevalueoftheangularmomen ~ tumwhichtheelectronpossesses initsorbitaroundthenucleus,thisbeing equalto.J{l(l+l)}n, wheren=h/27T,andhisPlanck's constant. The allowedvaluesoflareintegral, from0upto(n-l)foranelectronwhose principal quantum numberhasthevaluen.Sincetheelectronischarged, itsmotioninanorbitisequivalent toacirculating current,andamag­ neticdipolemoment isassociated withtheorbitwhichhasthesame valueasthattobeexpected onclassical theory.Thatis,amoment m=(-e/2mo).J{l(l+l)}n =-(en/2m o).J{l(l+I)). Iftheangularmomentum isrepresented byavector1normaltothe planeoftheorbitoflength proportional to.J{l(l+l)}, thenthedipole momentm isparallelto1andproportional toit.Theminussignshows thattnand1haveopposite directions, owingtothenegative charge possessed bytheelectron. WeseealsothattheBohrtheorygivesus anaturalunitofatomicdipolemoment, equalto(en/2mo).Thisisknown astheBohrmagneton, anditsmagnitude is 0·9273X10-23ampere-metre2(0·9273X10-20e.m.u.); itwillbedenotedbythesymbolfl. 851110 Pp 578 THEATOMIC THEORY OFPARAMAGNETISM [20.2 Sinceeachorbithasamagnetic dipolemoment associated withit,the precession theorem of§20.1showsthatinthepresence ofamagnetic fieldthedipolemoment, andhence,also,theangularmomentum vector, willprecessaboutthedirection oftheappliedfield.Eachvectormakes aconstant anglewiththisdirection, andthecomponent oftheangular Sodium Hydrogen '811 df' spdj 0n;=:oo n=000 _-------------n=7n=7__~~==~~-------------n=611.=6---- ,.-------------n=5-,.~,. _____________ n=4 -1 11.=5·- / -10,000/ //-____________ n=3 / I n~4--/ I I -2;§I-20,000 I:0- ~I ~-'<> 'i' -3.wSI <> I.S .S Il>. -30,000 ~l>.I ~~.,I -4f"'I ~ f"'II I ~40,000I I -5 11.=3.-- FIG.20.2.Theenergylevelsofsodiumcompared withthoseofhydrogen. Forsodium thelevelsarethoseofthesingleelectron outsidetheclosedshellsls2,2s2,2p6.Forthe highervaluesofnthelevelsapproach closelythoseofthehydrogen atom.Thisis becauseatlargedistances fromthenucleustheelectricfieldisthatofthenuclearcharge +Zesurrounded by(Z-l)electrons, andhence(byGauss's theorern) isthatofunit positive charge. Orbitswithlowervaluesofnpenetrate theclosedelectron shelland sofeelagreaterpositive charge,givingalowerenergy. Thisismostmarked forthe 'penetrating' s-orbits. momentum inthisdirection istherefore constant. Onquantum theory themagnitude ofthiscomponent mustbeanintegral multiple ofn, anditiswrittenmin,wheremliscalledthemagnetic quantum number. Itmaytakeallintegralvalues(including 0)betwElen.+land-l,asin Fig.20.3.Sincethemagnetic moment isproportional totheangular momentum, itfollowsthatthemoment associated withtheorbithasa fixedcomponent paralleltothedirection ofthemagnetic fieldofmagni- 20.2] THEATOMIC THEORY OFPARAMAGNETISM 579 II: FIG.20.3.Qua.ntization oforbi­ talangular momentum. The figureisdrawnfortheclIBeof l=2.tude-m,f3,together withacomponent ofmagnitude {l(l+I)-mn 1f3 whichrotatesintheplanenormaltothefield.Theangularvelocity ofprecession w= -(ej2mo)B,whichisthesameastheclassical value givenbyLarmor's theorem. Notethatthe projection oftheangularmomentum onthe fieldhasthevaluemin,not.J{ml(m,+I)}n; itisageneralfeatureofwavemechanics .~ thattheabsolute magnitude oftheangularj2(A) momentum associated withanyquantum= numbersuchaslhasthevalue.J{l(l+I)}n, ~l(A) whilecomponents ofangularmomentum in~ agivendirection areoftheformmin,whereS m,istheassociated magnetic quantumaO(A)IE-------+I number. ~ Theelectron alsopossesses, inaddition§....toitsorbitalmotionaboutthenucleus, a.:;-l(A)= spinaboutitsownaxis,whoseangular§ momentum isequalto.J{s(s+I)}n, where S~-2(A) istheelectronic spinquantum numberand 0 isalwaysequaltot.Withtherotating chargeoftheelectron isassociated amag­ neticmoment -gs.J{s(s+I)} x(enj2mo) =-gsf3.J{s(s+I)}. Herethecoefficient gsisinserted becausetheratioofthemagnetic moment totheangular momentum differsfromtheclassical value (corresponding togs=I).Foralongtimeitwasthoughtthatthe valueofgsfortheelectron spinwasexactly 2,butithasnowbeen shownbothexperimentally andtheoretically thatthevalueis 2(1'001l60±0'000002): forourpurposeitissufficient toomitthecorrection andassumethat gBis2forelectron spin.Inanatomtherearerelativistic anddiamagnetic corrections toboththeorbitalandspinmagnetic moments (oforder10-6 to10-4),whichweshallneglect. Theminussignintheexpression for themagnetic moment showsthatitisoppositely directedtotheangular momentum vector,owingtothenegative chargeoftheelectron, asinthe orbitalcase.Inamagnetic fieldboththespinangularmomentum and itsmagnetic moment precessaboutthedirection ofthefield,asinFig. 580 THEATOMIC THEORY OFPARAMAGNETJt;M [20.2 20.4,thesteadycomponent oftheangularmomentum inthisdirection havingoneofthevalues±!/i,andthecorresponding steadycomponent ofthemagnetic moment havingthevalues ~tggfJ ~~:fJ.Notethat thesecomponents amounttooneBohrmagneton, thoughthespinishalf integral. Inanatomcontaining anumberofelectrons, thetotalangularmomen­ tumwillbethevectorsumoftheindividual momenta, bothorbitand z FIG.20.4.Quantization ofspinangularmomentum s ~. spin.Ingeneralthisvectorsumcanbeformedinanumberofways, withanumberofdifferent resultants. Toknowwhichoftheseiscorrect or,ifseveralareallowed, whichcorresponds tothestateoflowest energy (thegroundornormalstateoftheatom),weneedtoknowmoreabout themutualinteractions betweenthevariouselectrons. \Veshallseethat thesecanbeexpressed intheformofasetofrulesforcoupling together theangularmomenta informingthevectorresultant. Theserulesare subjecttooneoverriding condition, expressed inthewell-known Pauli principle: 'notwoelectrons inthesamesystemcanbeinstateswith identical setsofquantum numbers'. Whenappliedtoanatomthis meansthatnotwoelectrons canhaveidentical setsofvaluesforn,l,mi' andmg,wheremsisthemagnetic quantum numberassociated withthe electron spin.Sincemscanonlyhavethevalues±!,itmaybeomitted 20.2] THEATOMIC THEORY OFPARAMAGNETISM 581 ifwerestatetheruleas:notmorethantwoelectrons canhaveidentical setsofquantum numbers n,l,mi'Hereitmustbeunderstood thatany twosuchelectrons mustbeinthestatesms=+land-lrespectively. ThePauliprinciple showsthatthereisalimittothenumberofelectrons withanygivenquantum numberinagivenshell.Wehavealreadyseen thatonlytwoelectrons canhaveidentical valuesofn,l,mi'Sincem,can onlyhavethe(2l+1)valuesl,l-l,l-2,...,-(l-l),-l,only2(2l+1) electrons canbeinasubshellwithagivenvalueofl.Again,sincelcan onlyhavethevalues(n-1),(n-2),...,1,0,only 2{(2n-1)+(2n-3)+ ...+3+1}=2n2electrons canhaveagivenvalueofn.Whenever electrons occupyallthepossible statescorresponding toagivenn,wehavea'filledshell',andsimilarly whenallpossible statesforagivenn,lareoccupied, wehavea'filled subshell'. Theoccurrence ofthesefilledshellsgivesasimilarity between different elements, expressed inthe'periodic table'.Fromthepointof viewofmagnetism themostimportant property ofaclosedshellisthat itsresultant angularmomentum iszero,whichcanbeseenasfollows. Whenwehavetwoelectrons withms=+land-l,thetotalprojection oftheirspinmomentum onanyaxis(suchasthatsupplied byanexternal field)is+l-l=O.Theprecessing components alsovanish,aswe shouldexpectfromthefactthatthetotalangularmomentum should be.J{0(0+1)}/i =O.Similarly, foranygivenvalueofl,whenwehave electrons occupying allthestatesm,=l,(l-I),(l-2),...,-(l-l),-l, thetotalprojection onanyaxisaddstozero,andthetotalorbitalangular momentum isalsozero.Sinceinthecaseeitherofspinororbitthe associated magnetic moments areproportional totheangularmomenta, itfollowsthattheresultant magnetic moment iszerowhenwehavea closedsubshell(n,l).Thusthemagnetic momentofanatomisdueonly totheunfilled subshells. Ournextproblem isthatofhowtocoupletogether theangular momenta inapartly-filled subshell. Thisdepends onthemutualinter­ actionsbetween theelectrons, ofwhichthetwoprincipal typesareas follows: (a)Mutualrepulsion betweentheelectrons, duetotheirelectrical charge. Whenthisistreatedbywavemechanics anunexpected resultisfound. Theenergyofthesystemcontains twoterms,onecorresponding tothe classical coulomb interaction, theotherknownasan'exchange energy', becauseitappearstobeconnected withanexchange ofanypairof electrons between thestatesweassigned tothembeforeincluding the 582 THEATOMIC THEORY OFPARAMAGNETISM [20.2 effectoftheirmutualrepulsion. Theseexchange forceshavenoanalogue inclassicaltheory,butplayanimportant roleinatomictheory.Bymeans .ofthePauliexclusion principle, theireffectcanbeshown(see§21.9) tocorrespond toastrongcoupling between theelectron spins,this coupling beingsuchthatwithinanatomthestatewiththespinsparallel ismorestableandhasthelowerenergy. Theenergyofinteraction of thiscoupling maybewrittenintheformW=-2/..;Si'Sj'whereSi's; arethespinvectors, and/..jiscalledthe'exchange energy', being positive foranypairofelectrons withinagivenatombutvaryingin magnitude, depending ontheirorbitalquantum states.Thuswitha numberofelectrons, theprimary effectoftheexchange forcesisto coupletogether thevariousvectors Si's;toformaresultant S,which inthestateoflowestenergyhasthelargestpossible valuecon­ sistentwiththeexclusion principle. Theremaining orbitalmomenta arethencoupled together bytheelectrostatic forcestoforma resultant L,whichinthestateoflowestenergyagainhasthe largestpossible valueconsistent withtheexclusion principle. (These tworulesareknownasHund'srules.)LowervaluesofSandLarepos­ sible,butcorrespond tostatesofhigherenergy.Thismethodofcoupling theangularmomenta iskI;l.OwnasRussell-8aunders coupling. Onwave mechanics SandLarequantum numbers andtheabsolute magnitudes ofthetotalangularmomenta associated withthemare-y'{S(S+l)}/b and-y'{L(L+l)}/b; itiscommon practicetospeakjustofthevectorss,1, S,L,etc.(corresponding totheoldquantum theory),butitmustbe remembered thattheabsolute magnitudes associated withtheseare -y'{s(s+l)}, -y'{l(l+l)}, etc. (b)Magnetic coupling betweenthemagnetic moments oforbitandspin ('spin-orbit' interaction). Themotionofanelectron roundthecharged nucleusproduces afieldBwhichwecanestimate asfollows. Fromthe theoryofrelativity onefindsthatacharged particle movingwith velocityvthroughanelectricfieldEexperiences aforcewhichisequiva­ lenttoamagnetic fieldB=-(vAE)/c2,wherecisthevelocity of electromagnetic waves.InanatomwithnuclearchargeZeandasingle electronthefieldEatdistance rfromthenucleusisE=r(Ze/47rE or); inanatomwithmanyelectrons theelectricfieldisstillradialtoagood approximation (thisisthe'central-field' approximation usedinatomic theory),butthefieldisreducedbecauseofthescreening eflectofother electrons. Wecantherefore writeE=r(Z'e/41TE or),whereZ'eisthe nuclearchargewhichwouldgivethecorrectvalueofthefieldatdistance r.Thenthemagnetic fieldBexperienced bytheelectron throughits 20.2] THEATOMIC THEORY OFPARAMAGNETISM 583 (20.7)(20.6)motionthroughthefieldEis B=Z'e(r1\v)=/LoZ'eG=/LoZ'e_lil=/Lo2Z'fJI, 41T£Oc2r347Tmor341Tmor341Tr3 whereGistheorbitalangularmomentum whosequantized valueism. Itwas:firstfoundempirically andlatershowntheoretically thatthis formula shouldbemultiplied byafactort(thisisarelativistic effect associated withthemotionoftheelectroninacurvedpath).Wemust alsoaverageBoverthedistribution ofspinmoment, giving B=/Lo(Z../)fJl, 41Tr3 wherethebrackets<>meanthattheaveragevaluemustbetaken.The interaction withthespinmagnetic momentrnsistp.en 'ZI> -rns·B =YsfJ(s.B) =YS:;\r3fJ2(1.S)={(1.s). Theeffectofthisspin-orbit interaction istotendtocoupletogether thevectorssand1foreachelectrontoformaresultant j;thevarious valuesofjfortheindividual electrons wouldthenbecoupledtogether (vectorially) toformthetotalangularmomentum vectorJ.However, thespin-orbit interaction issmallerinmagnitude thantheexchange interactions betweeIithe spinsdiscussed in(a)above,exceptinthe heaviest elements. Weshalltherefore confineourselves toRussell­ Saunders coupling, wheretheindividual spinsarecoupledtoforma resultant S,andtheindividual orbitalmomenta toformaresultant L. Thespin-orbit interaction thencouplesSandLtogether withanenergy W=AL.S, (20.8) whichissimilarinformtoequation (20.7)(itcanbeshownthatthe relation between thetwoconstants isA=±'!2S,wheretllepositive signisrequired forashellthatislessthanhalf-filled, andthenegative signforonethatismorethanhalf-filled; thespin-orbit coupling para­ meterAvanishes forahalf-filled shell).Thiscoupling ofSandLgives aresultant vectorJ,ofangularmomentum {J(J+l)}lli. Thenumber ofpossible valuesofJiseither(28+1)or(2L+1), whichever isthe smaller. Lhasonlyintegralvalues,andthevaluesofJaretherefore integralorhalf-integral according towhetherthevalueofSisintegral orhalf-integral. Thelatterdependsonwhetherthenumberofelectrons involved isevenorodd. Thenomenclature usedtodescribe atomicenergystatesismainly deriyedfrompre-quantum attempts toanalyseatomicspectra, and 584 THEATOMIC THEORY OFPARAMAGNETISM [20.2 therefore doesnotpossessthesimplelogicalsequence whichquantum theorycouldgiveit.Singleelectronstatesforwhichtheorbitalquantum numberl hasthevalues0,1,2,3,4,5, ...arecalled8,p,d,f,(j,h,...states, andsimilarly thelevelsofamany-electron atomforwhichL=0,1,2, 3,4,5,...aredenotedbythesymbols S,P,D,F,G,H,....Thevalueof nforasingleelectronstateisgivenbythenumberpreceding thesymbol, i.e.18,28,2p,38,3p,3d,etc.Thenumberofelectrons withgivenvalues ofnisdenotedbyasuperfix; thus,3electrons withn=2,l=1appear as2p3.Thespectroscopic stateofthewholeatomisdefinedbythevalues ofS,L,andJ;thevalueofthespinmultiplicity 28+ 1isgivenbya superfix preceding thesymbolforL,andthevalueofJbyafollowing suffix.Thusthesymbol4FfmeansthatthestatehasS=I,L=3, J=I;theotherpossiblevaluesofJinthiscasearet,t,l,thusranging inallfromL-8toL+8. Thecoupling schemeforamany-electron atomorionmaybeillus­ tratedbyreference totheenergyleveldiagram fortheCr3+ion,shown inFig.20.5.Thetriplycharged chromium ionhastheconfiguration 182,282,2p6,382,3p6,3d3,withthreeelectrons inthepartlyfilled3dshell. ByHund'srulestheenergyislowestwhenallthreeelectrons have parallelspins,givingS=I.Theelectrons mustthenallhavedifferent valuesofm"bythePauliprinciple, butsincewehavefivepossible values(m,=2,1,0,-1,-2)thereare(5!/3!2!)=10possiblearrange­ ments.Thelargestpossible valueofML=!m,thatwecanhaveis 3=2+1+0, andthisbelongstoanL=3state.Thishas2L+l=7 valuesofML,whichtherefore takeupsevenofthepossiblearrangements ofelectrons inthem,states;theotherthreebelongtoastatewithL=1. ByHund'srule,theL=3stateswillhavelowerenergythantheL=1 states.BothareshowninFig.20.5,the4p(L=1)statesbeinghigher inenergythanthe4F(L=3)statesbyabout14000wavenumbers. Statesofstillhigherenergyareformedbyreversing onespin,giving S=!.Twoelectrons withopposite spincannowoccupythem,=2 state,sothatthegreatest possible valueofLz=MLis5=2+2+1, whichbelongstoa2Hstate.Altogether sixdoubletstates,2H,2G,2F, 2D(twice),2pareallowed; theyhaveenergies ranging fromabout 14000cm-Ito37000em-I.Allotherstatesaremuchhigherinenergy, the3d248configuration lyingabout100000cm-Iabovetheground state3d3,4F. Theseparation ofthevariousquartetanddoublettermsisdetermined byacombination oftheexchange interaction andcoulomb interaction arisingfromthemutualrepulsion oftheelectrons. Thespinorbit 20.2] THEATOMIC THEORY OFPARAMAGNETISM 585 coupling splitsthe4FtermintofourlevelswithvaluesofJranging fromIL-SItoIL+SI,andthe4ptermissimilarly split,theonly allowedvaluesofJinthiscasebeingt,t,andt.Fromtheenergylevels giveninFig.20.5itcanbeverified(seeProblem 20.7)thatthespin-orbit coupling constantAhasthevalue87cm-1forthegroundstatesofCr3+. Energy (em-I) 14481 14215 14072+ 956 561 244 oTermMultiplicity 'PSI2-:;:6t;;,X _ 'P'12--:,4",-X _ 'PlI2-1L 'F 1I2--:;8""X _ 4F SI2....;;6""X _Centreofgravityof'P 1 13,774cm-I(splitting duetoelectrostatic interaction) •Centreofgravityof'F FIG.20.5.Quartet energylevelsofthefreetriplycharged chromium ion,Cr'+,3d'. Theseareformedfromthethreeelectrons inthe3dshell:thesplitting between the4F and4ptermsisduetoelectrostatic repulsion between theelectrons; thesplittings between levelsofdifierent Jwithineachtermareduetospin-orbit coupling. Doublet termsformedfrom3d'lieintheenergyrange14000~37000em-I.Thenext lowestlevelsarethosebelonging totheconfiguration 3d"48,andlieabove100000em-I. Itcanbeseenthatthesplittings duetothe'magnetic' spin-orbit coupling areanorderofmagnitude smallerthanthosedueto'electro­ static'interactions. 20.3.Magnetic moments offreeatoms Whenweturntoconsider themagnetic properties ofatomswefind thattheproblem issimplified bythefactthatthesedependonlyonthe partlyfilledelectron shells,sincecompletely filledshells(andofcourse emptyshells)haveS,L,andJ=o.Themagnetic moment associated witheachelectron spincanbedescribed byavectorparalleltoand 586 THEATOMIC THEORY OFPARAMAGNETISM [20.3 proportional totheangularmomentum vectors..andonformingthe vectorsumSforanumberofelectrons themagnetic moments addin asimilarway,sothatthetotalmagnetic moment ofthespinisparallel toSandhasthesamefactorofproportionality toit.Thesameistrue ofthetotalorbitalmagnetic moment andthetotalorbitalangular momentum L.Whenwecometomakethevectoraddition ofSandL D A FIG.20.6.Vectorcoupling ofangular momentum vectorsL,S,J (represented byAB,BO,AO)andtheassociated magnetic moments mL'ms,m(represented byAB,BD=2BO,AD).AEisthepro- jectionmJofmonJ. theproblem isnotsosimplebecausethefactorofproportionality be­ tweenthemagnetic momentandtheangularmomentum isnotthesame forSandL.Thevectorrepresenting thetotalmagnetic moment will nottherefore beparalleltoJ;thisisillustrated bythevectordiagram inFig.20.6.Herethemagnetic moment vectorassociated withLis drawnofthesamelengthasL,butonthisscalethemagnetic moment vectorassociated withSmustbedrawntwiceaslongasS.Theresultant magnetic moment vectorIDistherefore atanangletoJ. Inconsidering thisquestion furtherwemustreturntothediscussion ofthespin-orbit coupling betweenLandS.Thisisprimarily magnetic inorigin,andarisesfromthemagnetic moments oftheorbitandspin. Eachoftheseproduces amagnetic fieldwhichinteracts withthedipole momentoftheother. Theinteraction energy,\I..Sisequivalent to-IDs.BLorto-IDL.Bs; i.e.toafieldBL=-ALjYsn actingonthespinmoment IDs=ysnS, 20.3] THEATOMIC THEORY OFPARAMAGNETISM 587 ortoafieldBs=->'Sj'YLli actingontheorbitalmomentmL='YLliL. Hencefromequation (20.2)theequations ofmotionwillbe L='YLLi\Bs ='YLL/\(->'Sj'YLli) =-(>.jli)(L/\S)}. .(20.9) S='YsS/\BL ='YsS/\(->'Lj'Ysli) =-(>'jli)(S/\L) Wenotethatthecouplesareequalandopposite, astheymustbesince noexternal coupleactsonthesystem. SinceL+S=J,and Li\L=S/\S=0, wehave ~=-(>'jli)(L/\S+L/\L) =-(Afli)(L/\J)}, (20.10) S=-(Afli)(S /\L+S/\S)=-(>'jli)(S /\J) showingthatLandSeachprecessaboutJwithangularvelocity Alli. Sincethemagnetic moments associated withL,Sareparallelto them,itfollowsthattheyandtheirresultant mmustprecessroundJ atthesamerate.Thusthetotalmagnetic moment oftheatomhasa :fixedcomponent, mJ,givenbytheprojection ofmonJ,andapre­ cessingcomponent. Ingeneralweshallbeinterested onlyinthe:fixed component, andthismaybecalculated bysimplealgebraifweremember thatthevaluesofthesquaresofangularmomenta associated withS, L,andJareS(S+I),L(L+l),J(J+1)(eachtimesli2).Theprojection ofmonJmaybefoundinthefollowing manner, usingthevector diagram ofFig.20.6.Themagnetic momentmLassociated withLis -ft{L(L+l)}l, anditscomponent onJismLcosBAG. Fromthe geometry ofthetriangle, _BAG_S(S+I)-L(L+l)-J(J+l) cos -2{L(L+l)J(J +1)}1 andtheprojection oftheorbitalmoment onJistherefore +fJ[S(S+I)-L(L+l)-J(J +1)] 2{J(J+l)}1 ' whereftistheBohrmagneton, asbefore.Similarly theprojection of thespinmoment onJhasthevalue mscosAGB =-2ft{S(S+I)}lcosAGB =+2ft[L(L+l)-S(S+I)-J(J +1)] 2{J(J+1)}1 ' wheretheextrafactor2appearstoallowfortheanomalous valueofthe moment associated withthespin.Thesumofthesetwocomponents is _QL(L+l)-S(S+I)-3J(J +1) mJ-fJ 2{J(J+1)}1 . 588 THEATOMIC THEORY OFPARAMAGNETISM [20.3 Byanalogywithourdefinitions ofthemoments associated withLand S,wedefinethemagnetic moment massociated withJas mJ=-gf3{J(J+l)}t, wheregistheLandefactor(namedafteritsoriginator) whosevalueis, fromcomparison ofthetwoequations formJ' _3+S(S+1)-L(L+1) (20.11) g-22J(J+1) . ItiseasytoseethatifSorLiszero,sothatJ=LorJ=S,then gis1or2respectively, corresponding tothecasesof'orbitonly'and 'spinonly'. Whenanatomsuchaswehavebeenconsidering isplacedinan external magnetic field,thebehaviour oftheangularmomentum vectors ingeneralwillberathercomplicated. Thereasonisthateachofthe magnetic moments associated withorbitandspinisactedonbythemag­ neticfieldduetothemagnetic moment oftheotheraswellasthe external magnetic field.Nosimpledescription ofthemotionispossible whenthesefieldsareofthesameorderofmagnitude, butwhenoneis muchlargerthantheotheranapproximate treatment ispossible. We shallconsider onlythecasewhentheexternal fieldisverysmallcompared withthefieldduetothespin-orbit coupling. ThevectorsL,Sthen precessroundJasinthecaseofzeroexternal field,butJisnolonger stationary inspace,itsmotionbeingaprecession roundtheexternal fieldB.Theprecession ofL,SaboutJisatamuchhigherfrequency thanthatofJaboutB,sincetheexternal fieldissmallcompared with thatduetothespin-orbit coupling, andwemaytherefore picturethe components ofthemagnetic moment precessing aroundJasaveraging tozero,leavingonlythesteadycomponent alongJ.Thisisactedon bytheexternal fieldtogivetheprecession ofJaboutB,atanangular velocity CAl=-g(-eJ2mo)B,wheregistheLandeg-factor. Thisis identical withthegeneralresultof§20.1,ifwetake y=-g(eJ2mo)=-gf3JI'i. Thequantization rulefortheprojection ofJonthefieldBissimilar totheprevious rulesforotherangularmomentum vectors. Theprojec­ tionhasthevalueMJI'i,whereMJtakesthevaluesJ,J-1,J-2,..., -(J-1),-J.Thecomponent ofthemagnetic dipolemoment ofthe atomparalleltothefieldthushasthevalue-MJg(3,andtheenergyis WMJ=-m.B=MJgf3B. (20.12) Thusthe2J+1levelswithdifferent valuesofMJaresplitinenergy 20.3] THEATOMIC THEORY OFPARAMAGNETISM 589 bytheapplication ofamagnetic field,buthavethesameenergywhen B=O.Inthelattercasetheyaresaidtobe'degenerate', andthe application ofafield'liftsthedegeneracy'. This'Zeeman splitting' is illustrated inFig.20.7forthe4FstatesoftheCr3+ioninafieldB=10 MJ_--+9/21-----+7/2 1----+5/21-----+3/2 --r-(~::JE=~+~I~/2~-'F'/I 6·7cm-1 -1/2 \-----3/2 \-----5/2 \-----7/2'-----9/2 395cm-1 26g=­21[+7/2 I+5/2 +3/2 +1/2 \1/2 ..c. ~-3/2 ~-5/2 -7/26·2cm-1 3g=­5 2g=­5317cm-1 ,..----+5/2 1-----+3/2 --t--t=J:==+~I/~2~-'F'/I 3·0cm-1 1/2 \-----3/2 '------5/2 244cm-1 ---Y=E+3/2 +1/2-IF --7--- 2'0cm-1 III _1/2 '------- 3/2 FIG.20.7.Zeeman splitting ofthe'PstatesoftheCr3+ion(see Fig.20.5)inamagnetic fieldB=10'gauss(lOweber/m2).Notethat theZeeman splittings areverymuchsmallerrelativetothespin-orbit splittings (separation ofstatesofdifferentJ)thanthefiguresuggests. weberjmetre2=105gauss.Thevalueofthespin-orbit coupling para­ meterAisabout87cm-1forthision(seeProblem 20.7),andthefrequency ofprecession ofL,SaboutJisAjh=2·6X106Mc/s.Incontrast the frequency ofprecession ofJaboutBintheJ=istateisonlyabout 6X104Mc/sinafieldofB=105gauss.Thusourassumption ofavery 590 THEATOMIC THEORY OFPARAMAGNETISM [20.3 fastprecession ofL,SaboutJ, andamuchslowerprecession ofJ aboutB,willbevalidforallfieldsofordinary magnitude. InFig.20.7 thiscorresponds tothefactthattheZeemansplittings betweenthestates ofdifferent MJ(butthesameJ)areverysmallcompared withthe separation between statesofdifferent J.Itisonlywhenthisinequality holdsthattheenergyofaZeeman sub-level islinearlyproportional to theappliedfield(equation (20.12)); whentheZeeman energy(.-tf3B) iscomparable withAthebehaviour oftheenergylevelsismorecompli­ cated.Intheopposite extreme whenf3B~A,thecoupling between L andSisbrokendownandeachtendstoprecessindependently about theexternal field;thisisknownasthePaschen-Back effect,andcan beobserved onlyinveryhighfieldsforlightatomswherethespin-orbit coupling issmall. 20.4.Themeasurement ofatomic ma~netic moments-the Stern-Gerlach experiment Thespatialquantization ofangularmomentum (thatisthefactthat MJcanhaveonlyadiscretenumberofvalues,andnotacontinuous range,asinclassical theory)wasdirectlydemonstrated inthecelebrated atomicbeamexperiment ofSternandGerlachin1922,whichalsomade possiblethedirectmeasurement ofthemagnetic moment ofanatom. Thoughthisexperiment hasbeensucceeded bymorerefinedandaccurate methods, itremainsanhistorical landmark, anddevelopments ofthis method, mainlyduetoRabiandhiscolleagues, havemadeexperiments withatomicandmolecular beamsthebasisoftheextremely accurate knowledge wenowpossessaboutatomicmagnetic moments, andtheir interaction withthemagnetic moment ofthenucleus. Amolecular oratomicbeamisabeamofmolecules oratomsmoving withthermalvelocities inagivendirection.Itisformedbyheatingthe substance inanovenuntilitsvapourpressure isabout10-2mmHg, theovenbeinginahigWyevacuated enclosure. Atomsormolecules effusethrough anarroworifice81(seeFig.20.8),andifthepressure is solowthatthemeanfreepathislargecompared withthedimensions of81nocollisions occurintheorificeandallmolecules willbemoving insubstantially thesamedirection. Theangulardiameter ofthebeam isthenfurtherlimitedbytheslits82,8a.Thetotalpathtraversed by thebeamintheapparatus maybeupto50cmandthepressure must besolow«10-6mm)thatveryfewcollisions occurtoscatterthe molecules outofthebeam. Inordertodetermine themagnetic moment ofanatom,Sternand 20.4] THEATOMIC THEORY OFPARAMAGNETISM 591 Dr I.Gerlachdeflected thebeaminaninhomogeneous magnetic field.This wasobtained fromamagnetwithonewedge-shaped polepiece,which givesafieldgradient 8B{0zinthez-direction, whichisperpendicular to thepathofthebeam.Thebeamtraversed theinhomogeneous fieldfor adistancelandthenstruckadetector, whichintheearlyexperiments wassimplyatargetcooledinliquidaironwhichthemolecules condensed. DetectorI 8. Magnet plateI IVI/II/III!!/JI:Itill//IIfIfI/1: I· IOven D~l FIG.20.8.Experimental arrangement fortheStem-Gerlach experiment. Mendviewofpolepiecesproducing inhomogeneous field. . Ddensityoftraceondetector plateforatomsindoubletgroundstate,e.g.Ag,'Sl' S18.S.collimating slits. Ifmzisthecomponentparallel to8B{0zofthemagnetic momentofthe atom,thentheforceexertedontheatominthez-direction ismz(8B{0z), andthedeflexion 8aftertraversing thefieldgradient foramolecule of massMandvelocity vis 8=1(l{v)2mz(8B{8z){M =l2mz(8B{0z){2Mv 2•(20.13) Toobtainappreciable deflexions (oftheorderofamillimetre) fieldsof about104gausswithgradients ofabout105gauss{cm arerequired. The deflexion isinversely proportional tothethermal energyiMv2ofthe molecules, andthetracesarethusspreadoutowingtothedistribution ofvelocities appropriate tothetemperature oftheoven.Thislimits theaccuracy ofthistypeofexperiment, butwithatomssuchassodiumor silver,whicharebothin281states,twodistincttraceswereobtained, with deflexions appropriate tovaluesofmzequalto±oneBohrmagneton. Thusboththeexistence ofspatialquantization corresponding toMJ=±1 andthemagnetic momentofoneBohrmagneton associated withelectron spinofinwereconfirmed. Latermodifications oftheseexperiments gavefairlyprecisevaluesofatomicmagnetic moments, butmuchhigher accuracy hasbeenobtained bythemagnetic resonance method,outlined inChapter23. 20.5.Curie'slawandtheapproach tosaturation Atheoretical derivation ofCurie'slawduetoLangevin wasgivenin Chapter8.Thiswasbasedonaclassicalapproach initsuseofBoltzmann 592 THEATOMIC THEORY OFPARAMAGNETISM [20.5 (20.14) (20.15)statistics, butusedtheideaoftheexistence ofpermanent magnetic moments offixedvalues.Thislatterassumption isnotinaccordance withclassical theory,forweshouldthenexpectacontinuous rangeof magnetic moments from-00to+00.Itwasshownindependently by BohrandbyMissvanLeeuwen thatifsuchacontinuous rangeisassumed, theparamagnetic anddiamagnetic contributions tothesusceptibility ofanysystemshouldbeexactlyequalandopposite, andthus,ina strictlyclassical calculation, thesusceptibility wouldbezero.The quantum mechanical approach outlined in§20.2showsthatfiniteper­ manentmagnetic dipolesdoexistinatoms,andwemustnowexamine howtheLangevin calculation mustbemodified totakeaccountofthe factthatonlyafinitenumberofprojections ofthemoment onan external fieldareallowed. Itwasshownin§20.3thatthepotential energyWofanatomina magnetic fieldisMJgf3B,whereMJisthemagnetic quantum number, andgtheLandefactorappropriate tothespectroscopic stateofthe atom.Asinclassical theory,theprobability ofanatombeinginastate withanenergyWisproportional toexp(-WfkT},andforagivenvalue ofMJthisistherefore proportional toexp(-MJgf3BfkT}. Thusthefrac­ tionofallatomsinthisstateisexp(-MJgf3BfkT}f"2, exp(-MJgf3BfkT}, wherethesummation isoverallvaluesofMJ•(Weassumethatallthe atomsareinthesamespectroscopic stateL,S,J,thisbeingtheground stateoftheatom.)Thecomponent oftheatomicmagnetic moment paralleltoBis-MJgf3,andthetotalmagnetic moment ofasystemof natomswilltherefore be -"2,(-MJgf3)exp(-MJgf3BfkT)nm=n ,"2,exp(-MJgf3BfkT) wherethesummation ineachcaseisoverallvaluesofMJfrom+Jto -J.Thisexpression isratherclumsytohandle,butitmaybeshown byanalgebraic reduction thatitreducestotheform(seeProblem 20.1) _ {2J+1 (2J+1) 1(y)} nm=ngJf3--coth--y--coth--,2J 2J 2J 2J wherey=Jgf3BfkT. Theexpression inbrackets inequation (20.15)is calledtheBrillouin function. WhenJbecomes verylargeitapproaches asalimittheLangevin function {cothy-(lfy)}, asweshouldexpect fromthefactthatasummation overalargenumberoftermscanbe replaced byanintegration, asusedinthederivation in§8.3. Atnormalfieldstrengths andordinary temperatures, thevalueofy isverysmall;atB=1weberfm2=104gaussandT=2900K,with 20.5] THEATOMIC THEORY OFPARAMAGNETISM 593 g=2andJ=l,yisabout0·002.Itisthenpossibletomakeaseries expansion ofeitherequation (20.14)or(20.15). Tothefirstorderthe formerequation becomes +J j+Jnm=-ng{3~MA1-M Jg{3BJkT) "!-.,(l-M Jg{3BJkT) ng2{32B ~ =(2J+l)kTLMj.-J Thesummation amounts to!J(J+1)(2J+1),andthesusceptibility is thus nmp-ong2{32J(J+1)0 X=H=3kT =T' (20.16) whichisthesameastheclassical expression (equation (8.13))ifwewrite m2=g2{32J(J+1). Thisisjustthevalueoftheatomicmagnetic moment whichweshould expectonthequantum mechanical theory,butthemagnetic moment ofthewholesystemisdifferent athigherfieldstrengths, corresponding tothedifference between theLangevin andBrillouin functions. Inpar­ ticular,thelimiting saturation moment reachedathighfieldstrengths andlowtemperatures (largevaluesofy)isng{3J,andnot nm=ng{3,J{J(J +1)}. Thisisbecausethegreatest component ofeachmoment paralleltoBis Jg{3,andtheactualmagnetic moment alwaysprecesses atafiniteangle tothefield.Thecorrectness oftheBrillouin function hasbeenverified inanumberofexperiments, representative resultsbeingthoseofHenry shownin"Fig.20.9.Notethatthecloseapproach tosaturation is obtained bythecombination ofhighfield(50kilogauss) andlowtem­ perature (4°Kandlower). 20.6.Susceptibility ofparamagnetic solids-the 4fgroup Thetheorygivenaboveappliesonlytoanassembly offreeatoms, andthesituation isratherdifferent whenoneconsiders matterinthe aggregated state,becauseofthelargeforcesexertedbytheatomson eachother.Thesearemainlyelectrical inorigin,andaregenerally far stronger thantheinteraction between themagnetic moment ofan atomandanexternal magnetic field.Wemusttherefore consider the effectoftheinter-atomic forcesfirst,andweshallfindthatwhereasmost freeatomshavepermanent magnetic dipolemoments, mostboundatoms donot.Thisisduetothefactthattheexchange forcesbetween electrons indifferent atomsarenearlyalwaysofopposite signtothosebetween 851110 Qq 594 THEATOMIC THEORY OFPARAMAGNETISM [20.6 Ielectrons inthesameatom,andtheytherefore tendtomaketheelectron spinslineupanti-parallel, givingnoresultant spinwhenever possible. Thusintheformation ofahomo-polar molecule suchasN2'thebinding electrons fromthenitrogen atomsaresharedbetweenthetwoatomswith 7-00,---,---::;;;;;~Jii""-'"T""~--" III6-001--- .........=t----I----I----I II 5·00r---¥--r--~::;;o.....c>_1jO--o-__t t4·001----I--J.'4-----l----j-----j oI'"3·00f--tJci>----j------:;;;;::!Ji;;ar-(>-t--<:......, ~2·001-RL--1Y-+-----l-~--j-----1 l·OOft'1J---t-----l----j-----1 -.Brillouin o1·300KA2·o0oK 0·00 e3'OOoKD4'21oKo 1 2 3 4 BjT;inunitsofweber.m- sdeg-I(lO'gauss deg-I) FIG.20.9.Plotofaveragemagnetio moment perionmagainstBIT for(I)potassium ohrOlnium alum(J=S=t),(II)ironammonium alum(J=S=f),and(III)gadolinium sulphate octahydrate (J=S=t). theirspinsanti-parallel. Theorbitsarealsoarranged sothattheelectrons havenoresultant orbitalangularmomentum; thetotalangularmomen­ tumistherefore zeroandthemolecule hasnopermanent magnetic moment, thoughtherewillalwaysbeaninducednegative moment when amagnetic fieldisapplied, givingrisetodiamagnetism. Inhetero­ polarbindingamolecule suchasNaOlisformedofthetwoionsNa+and 01-,bothofwhichhaveclosedelectron shells;thusagainthereisno resultant magnetic moment. Thoughthispictureofmolecule formation isoversimplified, andingeneralwehaveamixtureofhomo-polar and hetero-polar binding, thegeneralresultofnopermanent magnetic moment isstilltrue.Thustheonlycommon gaseswithpermanent moments areNO,whichhasanoddnumberofelectrons, sothataresul­ tantspinmustremain(thereisalsooneunitoforbitalangularmomen­ tum),and02'wheretwoelectron spinsareunpaired, givingoxygengas aparamagnetism appropriate toS=1,g=2. 20.6] THEATOMIC THEORY OFPARAMAGNETISM 595 Inthesolidstatemostsubstances consistofionswithclosedshellsof electrons andaretherefore diamagnetic. Themainexceptions tothis ruleariseincompounds oftheso-called 'transition elements', wherean electron shellisinprocessofbeingfilled.Suchelements aremarkedby theirpossession ofmorethanonechemical valency, andsome(ifnotall) oftheirionsofdifferent valency haveunclosed shells,andhencea permanent magnetic moment. Sinceitistheelectrons intheunclosed TABLE 20.1 Oomparison ojtheoretical andmeasured valuesojp2Jor trivalent rareearthions No.of Ground Average electrons spectro-Thooretical ValU68ea:perimental in4f scopic valueof shell Ion state SLJgpS=gSJ(J+l) pS 0La+++ ISo 000- 0 0 1Ce+++ sP.t3t.Q.6·43 6•2Pr+++ aHa 154t 12·8 12 3Nd+++ 'IIt6t-L 13·1 12 11 4Pm+++ 51, 264.l!. 7·2 - Ii 5Sm+++ 5Hi.!!.5.!!.; 0·71(2'5) 2·4••6Eu+++ 'Po33 0- 0(12) 12·6 7Gd+++ 8Si:z.0:z.2 63 63• •8Tb+++ 'P83 3 6A 94·5 92•9Dy+++ 8H'f .!!.5II A 113 110• •3 10 Ho+++ 518268.!!. 112 110&. 11 Er+++ 'It-A6II .Q.92 90• •Ii 12 Tm+++ 8H, 156• 57 52 8" 13 Yb+++ sPi-t3:z.-; 20·6 19 a 14 Lu+++ ISo 0 0 0 - 0 0 Thevaluesgiveninparentheses forSma+andEua+arethosecalculated byVanVleck allowing forpopulation ofexcitedstateswithhighervaluesofJ,atT=2930K. shellwhichdetermine themagnetic properties, weshouldexpectthe paramagnetism tobetypicaloftheion,nottheatom.Thusionswith thesameelectron configuration, evenifformedfromdifferent atoms, havesimilarmagnetic properties. Theseionsmaybelabelledbythe spectroscopic description oftheelectron shellwhichispartlyfull;these are,3d(irongroup),4d(palladium group), 4J(lanthanide group), 5d (platinum group),and5/(actinide group).Thetitlesinbrackets areoften usedasbeingmoredescriptive, thoughlessprecise. Weconsider firstthe4Jgroup,whoseparamagnetism inthesolidstate isclosesttothatofanassembly offreeions.Thespectroscopic statesof thefreeionsofthe4JshellareshowninTable20.1.Itwillbeseenthat theyconform toHund'srules,thevaluesoffirstSandthenLbeing thegreatest possibleconsistent withthePauliexclusion principle. The 596 THEATOMIC THEORY OFPARAMAGNETISM [20.6 groundstatehasthesmallest possible valueofJinthefirsthalf,and thelargestvalueinthesecondhalf,asthespin-orbit coupling interaction changes signwhentheshellismorethanhalffull.Theexperimental valuesofp2havethefollowing significance. Ifweassumethatthe susceptibility ofasubstance obeysCurie'slaw,wemaywrite X=p-onm2f3kT=P-onp2f32J3kT. (20.17) Herepiscalledtheeffective Bohrmagneton number, andbycomparison withequation (20.16)weseethatforanassembly offreeions p2=g2J(J+1). Itisconvenient togivetheexperimental resultsintermsofp2,sincethis facilitates comparison withthetheory,butitmustberemembered that thoughwecanalwayscalculate avalueofp2fromthesusceptibility at agiventemperature, ithaslittlesignificance ifthesusceptibility does notobeyCurie'slaw.Thelattercanbeestablished bymeasuring the susceptibility overarangeoftemperature. Inthisconnexion itmust beemphasized thatonlymeasurements on'magnetically dilute'salts aresignificant; bythisphraseismeantsaltswheretheparamagnetic ionsarefairlyfarapartsothatmutualinteraction between themmay beneglected (seeProblem 20.2and§21.1).Thiscondition isgenerally fulfilledforhydrated salts,andthevaluesofp2inTables20.1,20.2are forsaltswheretheeffectofmutualinteraction onthesusceptibility is appreciable onlyatverylowtemperatures. Thecalculated valuesofp2assumethatonlythegroundstateof angularmomentum Jisoccupied. Sincestatesofdifferent Jgenerally lieatseveralthousand oK,thisisagoodapproximation atroomtem­ perature, exceptfortheions4j5,4j8whereexcitedlevelswithhigher valuesofJareexceptionally low-lying, andwhosepresence cannotbe neglected. VanVleckhasshownthattheirinclusion givesmuchbetter agreement withexperiment, andhisvalues,calculated forT=2930K, areshowninparentheses. Theexperimental valuesofp2fortheotherionsareinfairagreement withthosecalculated foranassembly offreeionswithangularmomen­ tumJ,butinfactthesevalueshavemostlybeendeduced byfittingthe experimental measurements ofsusceptibility toaformulaofthetype P-onp2f32 X=3k(T+~)' (20.18) Thismodification ofequation (20.17)isknownastheCurie-Weiss law (see§21.1),butitisbettertoregarditasanexpression whichincludes atermin'1'-2andisthestartofaseriesexpansion ininversepowersof 20.6] THEATOMIC THEORY OFPARAMAGNETISM 597 T,ascanbeseenbywriting(20.18)intheform 11-0np2fJ2( ~)X=3kT1-fji+....(20.19) Theempirical valuesof~whichgivethebestfittothesusceptibility intheregionofroomtemperature areoforder10-20°K,butatlow temperatures thesusceptibility oftendepartsquitemarkedly fromany suchsimpleformula. Thereasonforthisisthatwecannotneglectthe influence ofthecharged ionswhichsurround eachparamagnetic ionin thesolidstate.Inamagnetically dilutesalttheseimmediate neighbours carrynopermanent magnetic moment (theyarediamagnetic ionssuch asF-,0=,etc.),buttheyareelectrically charged, andhaveanelectro­ staticinteraction withthe4felectrons whichareresponsible forthe paramagnetism. Toagoodapproximation the4felectrons canbe regarded asmovinginanelectricfieldsetupbytheneighbouring ions, knownasthe'crystalline electricfield'.Theenergyofinteraction with thisfieldissmaller,forionsofthe4fgroup, thanthecoulomb, exchange andspin-orbit interactions withintheparamagnetic ionitself,andgives risetoa'Stark'splitting ofthe2J+1levelsoftheion.Thisissimilar innaturetotheeffectofanexternal electricfieldonthespectrum ofan atom,firstinvestigated indetailbyStark,butisconsiderably more complex becausetheelectrostatic potential setupbytheneighbours variesinacomplicated wayoverthespaceoccupied bythe4felectrons. Theoverallsplittings ofthe2J+1levelsofa4fionaregenerally ofthe orderofafewhundred OK.AscanbeseeninFig.20.10,thesusceptibility isratherinsensitive tosuchsplittings, andapproaches thatofthefree ionattemperatures wheremostofthelevelsareappreciably populated. Atlowtemperatures whereonlytheverylowestlevelsarepopulated, thesusceptibility canbeverydifferent fromthatofthefreeion,andin asinglecrystalmaybehighlyanisotropic. Atfirstsightitmayappearsurprising thatthecrystalline electric fieldcanhavesuchamarked effectonthemagnetic properties. The basicreasonisthatthewavefunctions corresponding todifferent values oftheorbitalmagnetic quantum numberMLhavedifferent angular dependencies; i.e.foreachvalueofMLthedistribution ofelectronic chargehasadifferent shape,andhenceacquires adifferent electrostatic energyinthecrystalline electricfield.Thustheprimary interaction is associated withtheelectronic orbit,andtheinteraction iszero(except forasmallresidual effectduetoaslightdeparture frompureRussell­ Saunders coupling) foranionsuchasEu2+orGd3+withahalf-filled shell 598 THEATOMIC THEORY OFPARAMAGNETIRM [20.6 carrying noorbitalangularmomentum (L=0).Fortheotherionsthe coupling together ofLandSmeansthatstatesofdifferent MJhavea different chargedistribution, sothatthecrystalline electricfieldsplits the2J+1stateswhichotherwise havethesameenergyintheabsence ofamagnetic field.Animportant restriction onthissplitting occurs 100 Perpendicula.r 300 FIG.20.10.Thevaluesofp.(pa.rallel andperpendicular) forasinglecrystaloferbium ethylsulphate Er(CaHaSO,)a,9H.O. Thisformshexagonal crystals, andthesusceptibility issymmetrical aboutthehexagonal axis.ThegroundstateoftheEr3+ionis'bt,and thisissplitbythecrystalline electricfieldintoeightdoublets lyingat0,61,108,159, 249,301,375, and438OK. forionswithanoddnumberofelectrons, whichhavehalf-integral values ofSandhencealsoofJ;inthiscasethestatesoccuralwaysinpairs whichhavethesamechargedistribution anddifferonlyintheorienta­ tionofthemagnetic moment. Eachpairmustthusretainthesame energyinanelectricfield,thoughtheycanbesplitinamagnetic field. Thisresultwasprovedinatheorem ofKramers andthedouble degeneracy ofsuchstatesinanelectricfieldisknownas'Kramers' degeneracy' . 20.7.Susceptibility ofparama~netic solids-the 3dgroup Thespectroscopic groundstatesofthefreeionsofthe3dshellare showninTable20.2,whenceitcanbeverifiedthattheyfollowHund's 20.7] THEATOMIC THEORY OFPARAMAGNETISM 599 rules.Anassembly offreeionswouldtherefore giveasusceptibility corresponding top~=g2J(J+1),butacomparison ofthevaluesofthis quantity withthe'experimental valuesshowsastrikingdisagreement. Infacttheexperimental valuesliemuchclosertop2=48(8+1), the valuewhichwouldbeexpected iftherewerenoorbitalangularmomen­ tumandthemagnetism weredueentirelytotheelectron spin.Thisis TABLE 20.2 No.of electrons Ground pi in3d8hall Ion 8tate SLJftJ(J+l) (wper.) 48(S+I)---- 0K+,Cal-!-,ScH,ISO 000 0 0 0TiH,VO+ 1TiH,VH IDi2A2·4 2·9 3 3Ft I 2V8+I1 3 2 2·67 6·8 8 3VI-!-,Cr3+ 4F JI.3.lI.0·6 14·8 15 OD:3 3 4dW,MnH~2 0.0 (2~'3) 24 5 ~I,fo,Fea+ ··S."§.0§.35 34·0 35••6Fe* °D4<22 4 45 28·7 24 7Co* 4Ft.ll3J!.44 24·0 15••8Ni* 8F41 3 4 31·3 9·7 8 9CuS+ IDti2A12·6 3·35 3 I 10 Cu+,Znl+ ISO 000 0 0 0 Thevaluesofpi(at3000K)arefordoublesulphates ofthetypeM~(S04)I,6HIO or M"'M'(S04)I,12HIO (whereM#=divalent paramagnetic ion,MIlO=triva.lent paramagnetic ion,M'=monovalent diamagnetic ion).Inthesesaltsthedistance between nearest paramagnetic ionsisatleast6A,andinteraction between themisnegligible. Thevalue inparentheses isforCrS04,6HIO: nodoublesulphate ofera+hasbeenmeasured. clearlybroughtoutinFig.20.11,inwhichaverageexperimental values ofp2areplottedtogether withthequantities g2J(J+1)and48(8+1). Thisphenomenon, knownasthe'quenching' oftheorbitalmagnetism, isaresultofthecrystalline electricfield.Inthelanthanide group the4/electrons, whichareresponsible fortheparamagnetism, are fairlydeepseatedintheatom,butintheirongroupthe3delectrons areinanoutershellwhichhasaverymuchlargerinteraction withthe crystalline electricfiel4ofneighbouring charged ions.Ontheother handthespin-orbitcouplinginthe3dgroupisconsiderably smallerthan inthe4/group.Theresultisthatinteraction between theorbitand thecrystalline electricfieldisagooddealstronger thanthespin-orbit coupling for3dions,sothattheorbitalmomentum isprimarily coupled tothecrystalfield,anditisnolongercorrecttoregardLand8as coupledtoformaresultant J.Thequantitative expression ofthis situation isthatthe2L+lorbitalstatesaresplitinthecrystalfield, 600 THEATOMIC THEORY OFPARAMAGNETISM [20.7 andhaveenergies differing byabout10000em-I,whichismuchlarger thanthespin-orbit splittings (oforder100-1000 em-I)between the statesofdifferent Jinthefreeion.Thesimplest casetoconsider is thatwherethecrystalfieldsplitting oftheorbitallevelsgivesasinglet stateasthelowestlevel.Suchastatehasnomagnetic moment, andthe orbitalmomentiscompletely 'quenched'. Ontheotherhandtheelectron r-"", oAverageI\ I \40experimental valuesI \I \ \ \ \gIJ(J+l) 30\ \ \ pI\0\ \ 20 \ \ \ \ \ 10\ \ \ \ \ \ 0 0 4 6 10 FIG.20.11.Experimental (at3000K)andcalculated valuesofp'for the3dgroup. Inthesecondhalfofthegrouptheorbitalangular momentum is lesseffectively quenched thaninthefirsthalf,sothatthevaluesof p'lienoticeably abovethespinonlyvalues. spinhasnodirectinteraction withthecrystalline electricfield,and remains freetoorientitselfinamagnetic field.Thus,inthiscase,the susceptibility wouldcorrespond exactlytothe'spinonly'valueof p2=48(8+1) atalltemperatures suchthatthereisnoappreciable population ofanexcitedorbitalstate. ItcanbeseenfromFig.20.11thatthevaluesofp2donotfollow exactlythe'spinonly'values,particularly forionswithd6(Fe2+)and d7(002+)configurations. Thebasicreasonforthisisthatthecrystalline 20.7] THEATOMIC THEORY OFPARAMAGNETISM 601 electricfielddoesnotalwaysresultinasingletorbitalstateasthelowest state,butsometimes givesagroupoflow-lying orbitalstateswhichcan makesomecontribution tothemagnetic moment, thoughlessthanthe fullorbitalcontribution fromafreeion.Inprinciple wecouldcalculate thesplittingoftheorbitallevels,butinpracticethisisextremely difficult todo.However, thegeneralfeaturesofthemagnetic properties ofsalts ofthe3dgrouparewellunderstood, mainlythrough theworkofVan HsO~-----------¥ oHs FIG.20.12.Octahedron ofwatermolecules roundaparamagnetic 3dion. Vleck,andwewillnowattempttooutlinethemainresultsofthecrystal fieldapproach. Thesizeofanionofthe3dgroupissuchthatsixnegatively charged ionscanbepackedroundit;whentheseionsareidentical, theyare arranged intheformofanoctahedron whichisverynearlyregular.In hydrated saltsthesesixionsarecommonly theoxygens ofsixwater molecules, asshowninFig.20.12.Theseionsareknownasthe'ligand ions',andingeneralthereisasmallamountofhomopolar binding between themandthe3dion.Inthecrystalfieldtheorythisisignored, andthemagnetic 3delectrons areassumed tobelocalized onthe3dion, andtomoveintheelectrostatic potential ofthesurrounding charged ligandions.Ifthe3dionisassumed tobeatthepoint(0,0,0),theligand ionsmaybetakentolieatthepoints (±a,0,0),(0,±a,0),(0,0,±a) 602 THEATOMIC THEORY OFPARAMAGNETISM [20.7 (20.20)thusforming aregularoctahedron. Ifweassignacharge-2etoeach oxygenion,theelectrostatic potential nearthecentreoftheoctahedron (seeProblem 2.18)is V-12e 2e{35}(4+4+434) ------- --xyz-or, %£0a%£04a5 andthiswillchangetheenergyofanelectron onthecentralionbyan amountIif;*(-eV)if;d-r,whereif;istheelectronic wavefunction. This energychangeisaquantitative expression ofthefactthat,sincethe electrons onthemagnetic ionarenegatively charged, theywillhave alowerenergyinstateswheretheyavoidthenegatively chargedligand ionsasmuchaspossible, andahigherenergywhentheydonot,because oftheelectrostatic repulsion. Takinglinearcombinations togivereal wavefunctions, wecanwrited-orbitals asaradialfunction f(r)times thefollowing functions, whichexpresstheangulardependence inCarte­ siancoordinates insteadofthespherical harmonics of§2.2: r202,0=!(2z2-X2-y2)) 1 .J3 (dy) .J2r2(02,2+02,_2)=2(X2_y2) -!2r2(02,2-02,-2) =.J3xy !2r2(02,1+02,-1) =.J3yz (d£). -~2r2(02,1-02'-1) =.J3zx Thelastthree(knownasd£states)areeachzeroalongtwoofthe cubicaxes(seeFig.20.13),sothatthechargedensity(whichispropor­ tionaltothesquareofthewavefunction) isalsozeroalongtwoofthe axes.Thisgivesalowerenergyforthesethreestates(bysymmetry eachmusthavethesameenergyinacubicfield,sincex,y,zareall equivalent iftheoctahedron isregular)thanfortheothertwo(dy)states whichhaveafinitedensityalongallthreecubicaxes.Henceweget asplitting oftheD(d)stateasshowninFig.20.14ford1.Thesplitting issimilar,butinverted, ford9,whichisoneelectron shortofafilled d-shell.Afilledshellhasaspherical chargedistribution, andthecharge distribution ford9isequivalent toafilledshellplusa'positive hole',for whichtheelectrostatic energyinthecrystalfieldhastheopposite sign.A half-filled shellalsohasspherical symmetry, withL=0;forthisreason theligandfieldplaysvirtually noroleinaffecting theparamagnetism 11 -Ze -Ze-Ze (a)1p=(~)J(r) (b)1p=(X1;'1I)J(r) FIG.20.13.Angular variation ofdwavefunctions. (a)isadE-state (theothertwodE· statesaresimilarbutdifferently oriented); (b)and(e)aredy-states. dE'anddy-functions havedifferent symmetry properties: dy-functions donotchangesignonreflection inany oneofthecubicaxes(i.e.III-+-Ill,or11-+-y,orz-+-z),whilethedE·functions change signfortwosuchreflections butnotforthethird. Ix3x 3x ax 3X.IX3x (a) (b) (e) (d} FIG.20.14.Splittings ofDandFstatesinacrystalfieldofoctahedral symmetry. The overallsplittings liegenerally intherange10000-15000 em-I.Thestrongcoloursof manyparamagnetic compounds ofthe3dgroupareduetoabsorption bandsinornear thevisibleregionofthespectrum whicharisefromtransitions between thegroundstate andexcitedstatesshownabove,combined withvibrational effects. 604 THEATOMIC THEORY OFPARAMAGNETISM [20.7 ofd5ions,whosegroundstateis6S!.However ad6ion,withoneelec­ tronoutsidethehalf-filled shell,hasasimilarsplitting todt,whiled4 corresponds toapositive holeinahalf-filled shellandbehaves liked9 (seeFig.20.14).Thusdl,d4,d6,d9haveabasically similarsplitting pat­ ternbecauseeachisequivalent toasingleelectron orsingleholestate, asfarastheorbitisconcerned (theydonotallhavethesamespin). Similarly, theremaining ionsd2,d3,d7,dSareorbitally equivalent to two-electron ortwo-hole states(d7=half-filled shell+2 electrons; d3=half-filled shell+2 holes;dS_filledshell+2 holes).Theyareall inFstates,withL=3,whicharesplitbyacubiccrystalfieldinto asingletandtwotripletlevels,asshowninFig.20.14.Ind3anddS thelowestlevelisasinglet(itcorresponds toawavefunction xyz,which haszerodensityalongallthreecubicaxes),butind2andd7thesplitting patternisinverted. Usingthesecrystalfieldsplittings, wecandistinguish between two separate cases: (a)Whentheorbitalgroundstateisasinglet,ithasnocomponent ofangularmomentum alonganyaxis,sothatthemagnetism isdue primarily tothespin.Thesusceptibility followsCurie'slawveryexactly (forexample, thesusceptibility ofachromealumsuchasCrK(S04)' 12H20 doesnotdeviatefromCurie'slawbymorethan2percentbetween room temperature and2°K).Thereare,however, tworesidual effectsofthe spin-orbit coupling: (1)theeffective g-valuediffersfromthefreespin valuebyanamountoforderA/D.,whereAisthespin-orbit coupling and D.thesplitting between thegroundorbitallevelandtheexcitedorbital statesshowninFig.20.14.Thespin-orbit constant Aispositiveifthe d-shellislessthanhalf-filled, andnegative ifitismorethanhalf-filled. Itisalsolargerfortheionsattheendofthegroupbecauseoftheincreased nuclearcharge.Thustheeffective valueofgisabout1percentsmaller thanthefreespinvalueforCr3+,d3,but10percenthigherforNiH,dS• (2)Wherethespinis1ormore,the2S+1spinstatesmaybesplitby amounts oforder(A2/D.),whichisusuallyoforder0·1-10cm:'-l.This givesaspecificheatanomaly, ofwhichatypicalexample isshownin Fig.20.15. (b)Whenthegroundstateisnotasinglet,thereisafirst-order contribution fromtheorbittotheparamagnetism, butlessthanthat forthefreeion.Whenthegroundstateisatripletin:Fig.20.14,it behaves likeaP-statewithL=1,andaneffective gLwhichis-1 fordlandd6,and-iford2andd7•Thusitcaninteract withthe spinthroughthespin-orbit interaction, givingstateswithaneffective 20.7] THEATOMIC THEORY OFPARAMAGNETISM 605 Jof8-1,8,and8+1if8~I,or!and!if8=!.Thesplittings between theselevelsareoforder100cm-1(140°K),sothatCurie's lawisnotobeyedbecause excitedstatesbecomeoccupied asthetem­ perature israised.Thisisparticularly noticeable forcobalt(C02+,d7) salts,asshowninFig.20.16.Theionsd4,d9areexceptional becausethe _.... ;~at_....-­------ o0·4,1·2 0·8"1·6Cp .caldeg-1 mole-1 o 2 4 6 8 10 12 FIG.20.15.Magnetic specificheatanomaly ofaNiS04,6H.O(afterStoutandHadley, 1964).Theanomaly isassociated withthespintripletstates,whichlieat0,6,44,and 7·26°Krespectively. doubletorbitalstatesleftastheirgroundstatesbytheoctahedral field haveeffectively YL=o.Thustheybehaveratherlikecase(a).Figure 20.16showsatypicalcupricsalt,Cu2+,d9,whereCurie'slawisobeyed closely,buttheeffective y-valueismorethan10percenthigherthan thefreespinvalue,givingp2=3·76insteadofthevalue48(8+1) =3 wewouldexpectforasinglehole(8=i). Astrikingeffectinmanysinglecrystalsofparamagnetic substances ofthe4/groupisthehighanisotropy ofthesusceptibility; thisarises becausethesurroundings oftheparamagnetic ioninsuchcrystalshave onlyaxialsymmetry. Fortheregularoctahedron showninFig.20.12 therewouldbenoanisotropy, butinthe3dgroupthisoctahedron is normally somewhat distorted, andasaresulttheorbitalcontributions tothemagnetism dependonthedirection inwhichtheexternal fieldis appliedrelativetothecrystalaxes.Theanisotropy islargewhenthe 606 THEATOMIC THEORY OFPARAMAGNETISM [20.7 splitting ofthelowestorbitallevelsissmall;thatis,whentherearealso considerable departures fromCurie'slaw,asincobaltsalts. 20.8.Susceptibility ofparamagnetic solids-strongly bonded compounds Muchlessisknownindetailaboutthemagnetic properties ofsalts ofthe4dand5dgroups,butinmanycasesitappearsthatthebinding totheligandionsiscovalent ratherthanionicincharacter. Thisistrue 24 Coppersalt '-----:--_~-;;-------'--::-!:-:~-----__::_!,o n 100 200 300T(OK)-!..c= .S<:= CD S:='li! CD cil12....0 ~ ~,.Q 08<>... ~ ~ 4 0 0 FIG.20.16.Variation ofpIwithtemperature fortwoirongroupsalts. alsoforafewsaltsoftheirongroup,notablythecomplex cyanides such asKaFe(CN)6' Inthelatter,andinsaltssuchasK2IrCI6wherethe magnetic IrHionhastheconfiguration 5d5,theligandions(sixCN groupsintheformercase,sixCl-ionsinthelatter)areagainarranged intheformofaverynearlyregularoctahedron. Weshallconfinethe discussion tothistypeofcompound, asitaffordsaninteresting com­ parisonwiththepurecrystalfieldapproach. Inacovalent bondtheelectrons aresharedbetween thetwoions concerned, incontrast withapurelyioniccasewheretheelectrons are localized oneachion.Thelatterisanover-simplification, andinpractice thereisalwaysasmallamountofcovalent bonding, sothatwedistinguish onlybetween weakbonding, andstrongbonding. Inacomplex with octahedral symmetry, thedystateshavemaximum densityalongthe 20.8] THEATOMIC THEORY OFPARAMAGNETISM 607 cubicaxes(towards theligandions),andcanforma-bonds withthe ligandions,whilethedEstatescanonlyform1T-bonds. Intheformation ofabond,ad-stateonthemagnetic ioniscombined withtheappropriate bonding stateoftheligandion,andtheoverlapoftheelectronic wave functions givesasplitting ofthecombined levels,inthesamewayas pointedoutin§18.2.Theoverlapisgreaterforthedystates(forming a-bonds) thanforthedEstates(forming 1T-bonds), givingtheenergy leveldiagram showninFig.20.17.Thelowerbonding statesareall (dr.ligand)statesanti-bondingj availa.ble formagnetic electrons (ch,ligand)statesanti·bonding .states (ch,ligand)statesbonding (dr.ligand)statesbondingjfilIolwithbo_g~ FIG.20.17.Splitting ofthea-statesonthebonding model.Thelowest(bonding) states arefilledwithelectrons, andonlytheanti-bonding statesareavailable forthemagnetic electrons. Inthecrysta.lline electricfieldapproach thebonding statesplaynorole,and acubicfieldsplitting (seeFig.20.14)ofthea-statesisobtained similartothatforthe anti-bonding statesabove. filledwithelectrons, andbehaveasfilledsub-shells. Thusthestates available formagnetic electrons aretheanti-bonding states,whichare splitinthesamewayasbyanoctahedral crystalfield.Inweaklybonded compounds thissplitting isabout10000cm-I,asmentioned inthe previous section,butinthestrongly bondedcompounds itisverymuch larger,sothatthelatterbehaveasthoughsubjected toaverymuch stronger crystalline electricfield.However, theapproach fromthe bonding viewpoint ismorecorrect,sinceitallowsthemagnetic electron wavefunotions tospreadoutfromthecentralionontotheligandions, forwhichthereisdirectexperimental evidence frommeasurements of thehyperfine interaction betweenthemagnetic electrons andthenuclear moments oftheligandions. Inthismoregeneralapproach, allowing forbonding, thesplitting between thedEstatesandthedyanti-bonding statesisascribedtothe 'ligandfield',anditisinteresting tocontrastthetwocasesofsmalland largeligandfield.Herethecomparison iswiththeelectrostatic and 608 THEATOMIC THEORY OFPARAMAGNETISM [20.8 exchange energywhichisresponsible forRussell-Saunders coupling, andwhentheligandfieldsplitting islargecompared withthisexchange energywemustregardtheRussell-Saunders coupling asbroken. The wayinwhichthedEanddyanti-bonding statesareoccupied bythe magnetic electrons inthetwocasesisdetermined bythecompetition between theexchange energy(whichfavoursparallelorientation ofthe ,p(ionicorweak bonding)' (a) (b),p(strong"bonding) (0) FIG.20.18.Singleelectron modeloffillingofd·statessplitbyanoctahedral crystalfield. Theexchange energyformsparallel spinarrangements (subject totheexclusion prin­ ciple);thecrystalfieldsplitting favours electrons inthedEstates. electron spins)andtheligandfieldsplitting (whichfavours electrons goingintothedEstatesbecauseoftheirlowerenergy). Wemayrepresent eachorbitalstatebyapairofsquareboxes,asin Fig.20.18,intoeachofwhichwecanputoneelectron, withspinupor down.Withoneelectron, thestateoflowestenergywillobviously be whenthiselectron isinthedEstates.Whenfurtherelectrons areadded, theywillalsogointothedestates,butwithspinsparallelinorderto maketheexchange energyaminimum; uptothreeelectrons canbe accommodated inthisway.Thedeshellisthenhalffull,andbehaves likeastatewithzeroorbitalmomentum (corresponding tothesinglet orbitalgroundstateford3inFig.20.14)andS=!.Whenmoreelectrons 20.8] THEATOMIC THEORY OFPARAMAGNETISM 609 areadded,theycannotgointothed€shellwithparallelspin,because thiswouldviolatetheexclusion principle. Thereistherefore acom­ petition between theexchange energy,whichprefersparallelspin,and theligandfieldenergy,whichprefersthed€states.Inthehydrated salts,thelatterislessimportant, andthefourthandfifthelectrons go intothedystateswithparallelspin,makingdSastatewithL=0, S=i(ahalf-filled d-shell).Inthemorestrongly-bonded saltstheligand fieldsplitting issolargethatnoelectrons gointothedystates,since theyhavealowerenergybyoccupying thed€stateswithanti-parallel spin,asshowninFig.20.18(c).Thusastrongly bondeddSsaltbehaves asifithadoneholeinthed€shell;forexample, KaFe(CN)s, wherethe Fea+ionhasa dSconfiguration, hasmagnetic properties quitedifferent fromhydrated ferriccompounds, showing considerable departures from Curie'slaw,asusceptibility closetothatofasinglespin,andstrong anisotropy insinglecrystals. Withsixelectrons, dS,strongly bonded, thed€shelliscompleted, andtheionhasnopermanent magnetic dipole moment (KaCo(CN)s hasonlyasmalltemperature-independent para­ magnetism). Anyfurtherelectrons wouldhavetogointothedystates, butthesearesohighinenergythatsuchionsareusuallychemically unstable. However, whentheligandfieldislessstrong,thedystates areoccupied, asshowninFig.20.18(d)forahydrated Nj2+,dS,ion. Herethed€statesarecompletely filledandthedyhalf-filled, giving againagroundstatewithnoorbitalmomentum (cf.Fig.20.14(e)).On thissingleelectron picturewecanseethattheorbitalmomentum is effectively quenched whenever thetwosetsofd€anddystatesareeach eitherempty,half-filled, orcompletely filledwithelectrons; thereader canverifythatahalf-filled sub-shell, withallelectronspinsparallel, can beachieved byonlyonepossible arrangement oftheelectrons inthe variousboxes,andtherefore corresponds toasingletorbitalstate.On theotherhand,whenthed€statesareoccupied byoneortwoelectrons, orfourorfiveelectrons, therearethreeequivalent waysofarranging them,givingatriplydegenerate orbitalstate.Thiscorresponds tothe tripletstatewhichislowestfordI,d2,dS,d7inFig.20.14. 20.9.Electronic parama~netism-a summary Inconclusion wemaysummarize themagnetic properties asbeing theresultofcompetition betweentheelectrostatic (including exchange) interactions between electrons onthesameion,spin-orbit interaction between theseelectrons, andelectrostatic (crystal field)orcovalent bonding interaction withligandions.Inthe4/groupthelatteristhe 851110 Rr 610 THEATOMIC THEORY OFPARAMAGNETISM [20.9 weakestofthethree,givingsplittings oforder102cm-Iwhilethespin­ orbitcoupling is::::::103em-I.Inionic3dsalts,thespin-orbit coupling (::::::102em-I)istoosmalltocompete withthecrystalfield(104em-I), butthelatterissmallerthantheexchange interaction (e.g.themean separation between thequartet, S=t,statesandthedoublet, S=I, statesforCr3+isabout2X104em-I).Inthestrongly bondedsalts, interaction withtheligandsoutweighs theexchange andelectrostatic interactions between theelectrons withinthemagnetic ion,breaking downtheRussell-8aunders coupling. Inthe51,oractinidegroup,thebehaviour isgenerally similartothatof the41(lanthanide group),thoughonlythesaltsofthefirstmembers ofthegroup(U,Np,Pu)havebeeninvestigated inanydetailbecause ofthehighradioactivity oftheothermembers. Anexception isthe complex ionsU02,Np02'etc.,wherestrongcovalent binding exists between theactinide ionandthetwooxygenions. 20.10.Nuclear moments andhyperfine structure In§20.1itwasmentioned thatthenucleiofmanytypesofatoms possessangular momentum. Thisisassociated witha'spin'ofthe nucleusaboutaninternalaxis,andtheangularmomentum isquantized justasinthecaseoftheorbitalandspinangularmomenta oftheelectron. Thefundamental nuclearparticles ('nucleons') aretheprotonandthe neutron, eachofwhichpossesses aspinofIn,liketheelectron. All nucleiareregarded asassemblies ofprotonsandneutrons boundtogether; thenumberofprotonsisequaltoZ,theatomicnumber, sincethenuclear chargeisZe,andthenumberofneutrons N=(A-Z), whereAisthe atomicmassnumber. Thespinofanygivennucleus isdenoted by In,andIischaracteristic ofanygivenisotope. Thenumberofnucleons inanucleusisequaltoA,andthenuclearspinishalf-integral orintegral according towhether Aisoddoreven.Nosimplerulecanbegivenfor calculating thenuclearspinaprioriinaparticular case,thoughthe observed valuescanbefittedintoashellmodelnotgreatlydifferent fromthatusedinatomictheory. Themostimportant ruleisthatall nucleicontaining anevennumberofprotonsandanevennumberof neutrons haveI=0inthegroundstate.Thiscanbeunderstood in termsofa'pairingoff'ofthespinsofprotonsandneutrons similarto thatofapairofelectronsinans-state.Fornucleiwithanoddproton oroddneutronthevalueofIisattributed totheresultant oftheintrinsic spinofIfortheoddnucleonandan'orbital' momentum whosevalue isanintegralnumberofunitsofn,duetocirculation ofthisoddnucleon 20.10] THEATOMIC THEORY OFPARAMAGNETISM 611 withinthenucleus. Relatively fewstablenucleiexistwhichcontainodd numbers ofbothprotonsandneutrons (suchasiDandItN)butthese haveintegralvaluesofIotherthanzero,anexceptionally highvalue being1=7for176Lu.Thehighesthalf-integral valuesofarobserved ist.Thesevaluesareforthegroundstatesofnuclei.Investigation of nuclearstructure hasledtotheassignment ofspinvaluesformany excitednuclearstates,butthesehavenotbeenobserved directly, except inafewcaseswheretheexcitedstateshaveanabnormally longlife. Allnucleiwhichhaveanon-zero valueofthespinIpossessmagnetic moments, andthesearemeasured intermsofaunitcalledthe'nuclear magneton'. ThevalueofthisunitisPn=eli/2M,whichissimilarto thatfortheBohrmagneton exceptthatthemassinthedenominator is thatoftheprotoninsteadofthatoftheelectron.Iftheprotonobeyed asimilarwaveequation tothatfortheelectron, wewouldexpectitto possessamoment ofonenuclearmagneton associated withitsspin1, justastheelectron hasamoment ofoneBohrmagneton andspin1. Infactthemoment oftheprotonis+2·793nuclearmagnetons (n.m.), andtheneutron(which,beinguncharged, weshouldnothaveexpected topossessamagnetic moment) hasinfactamoment of-1'913n.m. Herethesignificance oftheplusandminussignsisthatthemagnetic moments arerespectively parallelandanti-parallel tothespin.Since themagnetic moment ofneitherneutron norprotonisanintegral numberofnuclearmagnetons weshouldnotexpectthemoments of morecomplicated nucleitobesimpleintegers. Theydo,however, followt;hetrendwhichthenuclearshellmodelwouldindicate (see Problem 20.5);ingeneralwewritethenuclearmagnetic moment as mn=UnPnI,whereUnisthenuclearmagnetogyric ratio. Interactions betweenanuclearmagnetanditssurroundings aresmall. Inamagnetic fieldeachofthe2mz+1statescorresponding todifferent orientations ofthenuclearmoment takesupadifferent energy (20.21) inafieldof1weber/metre2(104gauss)theseparation betweensuccessive levelscorresponds toafrequency oforder107cis.Inanatomorion whichhasapermanent electronic magnetic moment, thelattersetsup amagnetic fieldatthenucleuswhichmaybeasmuchas107gauss,being generally largerintheheavieratoms.Thismagnetic fieldispartlydue totheelectronic orbitandpartlytothespin,butformostpurposes we needconsider onlythesteadycomponent oftheelectronic fieldBe' whichisparalleltoandproportional totheresultant electronic angular 612 THEATOMIC THEORY OFPARAMAGNETISM [20.10 momentum vectorJ.Thuswehaveanadditional 'hyperfine' energy W=-mn.Be=AJ.I, (20.22) whereAisaconstant whoseorderofmagnitude canbeestimated as follows. Theelectronic fieldBeisoforderme<Re-3)=-yfiJ<R;3) , where<R;3)isthemeaninversecubeofthedistance oftheelectron fromthenucleus; sincemn=YnfinI,Aisoforderggnf3f3n<R;3). In frequency unitsA/hgenerally liesintherange108-1010cis,sothatthe hyperfine energymayapproach 1cm-1=1'43°K. Inaddition topossessing amagnetic moment, anucleusmayhave anon-spherical distribution ofelectriccharge.Itselectrostatic potential canthenbeexpanded asin§2.3,givinganenergyofinteraction with theelectrons oftheform(seeequations (2.30)-(2.34)) W=_1_IIPePnd'Ted'T n 47TEOIRe-rnI =-41{zeIPeRd'Te+~(-1)lmIA2,mB2,_m+etc.}. (20.23) 7TEO em=-2 Herethesubscripts e,nrefertotheelectrons andnucleirespectively; Ze=JPnd'Tnisthenuclearcharge,sothatthefirsttermisthecoulomb interaction duetoapointchargeatthenucleus, andthequantities in thesecondtermare A2,m=fPnr~C2,m(()n,epn) d'Tn, B2,-m=f(-1)lmlpeR;3C2,_m,(()e,epe) d'Te· Thistermrepresents theinteraction between theelectric quadrupole moments ofthenucleusandoftheelectrons, whosenatureisthatof atensor.Thechargedistribution isspherical fornucleiwithI=0ort, andforelectronic shellswithJ-'-0ort,sothatthequadrupole inter­ actionvanishes ineithercase.Sincethenuclearchargeissymmetric abouttheaxisofnuclearprecession, thenucleartermscanbeexpressed intermsofasinglequantity A-f.12(32())d-1Q{3mi-I(I-f-l)}(20.24)20-Pnyrncosn-1'Tn-yeI I.,, (2-1) where Q=~fPnr~(3cos2()n-1) d'Tn (20.25) iscalledthenuolearelectricquadrupole moment, andhasthedimensions ofanareaofthesameorderasthe(nuclear radius)2.Itisexpressed in termsofthe'barn',aunitequalto10-24cm2•ThesignofQispositive foraprolatespheroid, andnegative foranoblatespheroid, asillustrated 20.10] THEATOMIC THEORY OFPARAMAGNETISM 613 inFig.20.19.Theexpression inparentheses inequation (20.24)gives thevariation ofA2•0withthenuclearmagnetic quantum number m/> anditiseasilyverifiedthatinthestatesmr=±1,A2,o=leQ. Intheabsence ofanexternal magnetic field,theelectronic and nuclearangularmomentum vectorsJ,Iarecoupled together bythe magnetic hyperfine energy(equation (20.22»toformaresultant vector c:>••---- Axesofnuclearrotation----.~0 Positive quadrupole momentOblate spheroid Sphere--- J Negative quadrupole moment FIG.20.19.Representation ofnon-spherical chargedistribution innucleusas combination ofsphereandquadrupole. (20.26) WF=lA{F(F+I)-J(J +1)-1(1 +I)}F.Different valuesofFcorrespond todifferent energies, sincetheangle betweenJandIischanged, andfromthevectormodelitcanbeshown that sothattheenergiesofsuccessive statesformanarithmetical progression (cf.Problem 20.7,forthecorresponding caseofspin-orbit coupling). Thisrule,knownastheLandeinterval rule,nolongerholdswhenthe electricquadrupole interaction isincluded, butitcanbeshownthatthen wherew;=IAO+B 10(0+1)-1(1 +I)J(J+1) F Q21(21-I)J(2J-I) ,(20.27) 0=F(F+I)-I(1 +I)-J(J +1);BQ=2eQB2,o!41T€o=eQ(02V!OZ2), where82V!oZ2isthefieldgradient setupbytheelectrons atthenucleus. Theenergylevelsgivenbyequation (20.27)forthecaseofJ=I,I=f areshowninFig.20.20.IngeneralthesizeofBQiscomparable with thatofA,exceptinatomswhereJorIis0ort,andBQvanishes. 614 THEATOMIC THEORY OFPARAMAGNETISM [20.10 Nuclearspinscanbefoundfromobservations ofhyperfine structure in spectra,andvaluesoftheconstants AandBQareobtained fromthe separations ofthehyperfine levels.Inmagnetic resonance experiments (see§23.6)theprecision withwhichtheseconstants canbedetermined isveryhigh,andnuclear magnetic dipoleandelectric quadrupole WlI' r-r----::I~~L~:::::::=::==E~-=-==- !A+1Bo ---- ......-----.A.-B o ,..-----r------fAHB" Magnetic+quadrupole interaction FIG.20.20.Splitting ofgroundstateofanionwithJ=1,I=-!duetomagnetic dipole andelectricquadrupole interaction. Thefigureisdrawnforpositive valuesofbothA andB.Notethat(allowing forthemultiplicity 2F+1)thecentreofgravityofthelevels remains constant. moments canbeestimated withanaccuracy generally limitedbythe lackofexactelectronic wavefunctions fromwhichthequantities Bein equation (20.22)ando2VfoZ2mustbecalculated. Inthesolidstateanassembly ofnucleardipolesbehaves asasimple paramagnetic substance, contributing anamount(cf.equation (20.16)) I-'ong~~~1(1+1) (20.28) Xn= 3kT ' whichisonlyabout10-6ofthatofanyelectronic paramagnetic sub­ stance,sincethesusceptibility depends onthesquareofthemagnetic dipolemoment. Thenuclearcontribution hasbeendetected bystatic susceptibility measurements insolidhydrogen, wherethenuclear paramagnetism justoutweighs theelectronic diamagnetism atabout 10K(seeProblem 20.9). 20.10] THEATOMIC THEORY OFPARAMAGNETISM 615 Thenuclearsusceptibility followsCurie'slaw,equation (20.28),only attemperatures suchthatkTislargecompared withanysplittings of thenuclearlevels.Insubstances withoutpermanent electronic magnetio dipolesthismeanstemperatures downtoabout10-6OK,exceptwherethe nuclearlevelsaresplitthroughanelectricquadrupole interaction with theelectrostatic fieldgradient (the'crystalfield')setupbyneighbouring ions.Insuchcasesthegradient (o2V/oZ2) isfixed,unlikeinafreeatom whereitfollowstheprecessing electronic angularmomentum vectorfor theorbit,whichdetermines theorientation oftheelectronic charge cloud.Henceinasolidinwhichthelocalsurroundings ofanucleus havesymmetry aboutanaxis(whichwetaketobethez-axis)the nuclearlevelsmaybesplitaccording totheformula Jv.:=_1_A B =Q(02V/f)z2) 3m}-I(1 +1). (20.29) mJ417£0 2,02,0e 41(21-1) Thesplittings rangefromafewkc/suptoover2000Mcjsfor1271in12, solidiodine. Insubstances containing ionswithbothelectronic andnuclear magnetic dipolesthetwocontributions tothesusceptibility areadditive attemperatures suchthatkTislargecompared withanyhyperfine structure splittings (inpracticethisusuallymeansdowntoabout1°K). Suchsplittings arisefrombothnuclearmagnetic dipoleandelectric quadrupole interactions inthesamewayasforfreeatoms,buttheeffects aremorecomplicatedbecause ofthecomplex interaction oftheelectrons withthecrystalorligandfielddiscussed in§§20.6-20.9. Thehyperfine splittings usuallycorrespond totemperatures intherange10-3-10K, andaffectboththeelectronic andnuclearcontributions tothesuscepti­ bilityinthistemperature rangeandbelow. GENERAL REFERENCES COULSON, C.A.,1952,Valence (Clarendon Press). KOPFERMANN, H.,1958,NuclearMoments (Academic PressInc.,NewYork). KUHN,H.G.1964,AtomicSpectra(Longmans, Green&Co.Ltd,London). 616 THEATOMIC THEORY OFPARAMAGNETISM PROBLEMS wherey=JgflB/kT. Themagnetization Mforanassembly ofnsuchatomsisgivenbytheformula dM=nleTdB(logeZ). Usingthisformula, derivetheBrillouin function ofequation (20.15).20.1.Instatistical mechanics thepartition function Zisdefinedas Z=Iexp(-~/kT), i wherelfiistheenergyoftheithstate.Showthatforanassembly ofnon· interacting paramagnetic ions,eachofangularmomentum J,inafieldB Z_sinh{(2J+l)y/2J} -sinh{y/2J} , 20.2.Showthattheenergyofinteraction oftwomagnetic dipolesm adistance r apartisoftheorder/LomB/47T1"3. Inpotassium chromealum,eachchromium ioncarriesamagnetic moment of 3Bohrmagnetons, andthemeandistance apartisabout7·8X10-8em.Assuming thatseriousdepartures fromCurie'slawwilloccurwhentheinteraction energy between twoneighbouring dipolesis~leT,showthatthistemperature isapproxi­ mately0.010K.(Infactthelevelsofeachchromium ionaresplitbyabout0.20K through ahighordereffectofthecrystalline field,andthisismoreimportant than themagnetic dipoleinteraction between neighbouring ions;italsogivesaspecific anomaly atabout0.10KofthetypeshowninFig.20.15.) 20.3.ForaCu++ion,S=iandtheenergylevelsofthegroundstateinamagnetic fieldBareoftheform W=±igflB_!CXB2. Showthatinsmallfields (gflB/leT~ I)thepartition function Z=2+cxBB/kT+g2fl2B2/4(kT)2+ ..., andhencethat x.//Lo=n{g2f12/4kT+cx} =np2fl2/3kT. ThisshowsthatthetermB2inWgivesrisetoatemperature-independent contri. butiontothesusceptibility. Notethatp2isthenoftheformA+BT.(Bisvery smallforCu++,butCo++ionsobeythisrelation below1000K-seeFig.20.16.) 20.4.Showthatforasystemwherethemagnetic moment associated withorbital angularmomentum lliisgIlflandthatassociated withspinaliis(Issflandlands arecoupledtogether toformaresultantj, thegeneralized Lande'formula forthe g-factor is j(j+I)(gl+g.)+{l(l+ 1)-8(8+ I)}(gl-g.) g= 2j(j+I) . 20.5.Onthenuclear shellmodelthenuclearspinisduetotheoddneutron or protonwithspintmovinginanorbitwithinthenucleuswithanb'Ularmomentum Iii.Theobserved nuclear spinIiseitherl+torI-i.Applytheformula of thelastquestion tocalculate themagnetic moment, assuming thatforaproton gl=Iandg.=5,586,andforaneutron gl=0andg.= -3·826.Showthatthe magnetic moment m=gnfl..IofanucleusofspinIis (a)oddproton1=l+t, m=fl..(1+2·293), 1=l-t, m=fl..1(1-1'293)/(1+ I), THEATOMIC THEORY OFPARA:MAGNETISM 617 (b)oddneutron1=l+t, 1=Z-l,m=-1'913,81'1=ml'l' m=1.913,81'11/(1+1) =-mI'l1/(1+1). Theseformulae areknownastheSchmidt limits.Observed nuclearmoments followthetrendgivenbytheseformulae butgenerally liebetween theselimits. 20.6.Thearrangement ofparallel cylindrical conductors carrying equaland opposite currents ofProblem 5.3isusedtogivealargefieldgradient anddeflect atomsinanatomicbeam.Eachcylinder carriesacurrentof1000Aandthe axesofthecylinders are1emapart.Abeamofatomsinthe"S.statefromanoven at9000Ktravelsparalleltothecylinders atthepointwheretheinhomogeneous fieldisamaximum. Calculate theseparation between thetwocomponents ofthe beamaftertravelling adistance of20em. (Anewer: ~0·01em.) 20.7.Usethevectormodelasin§20.3toshowthat,asaresultofthespin-orbit coupling AL.S(==ALScosABOinFig.20.6),theenergyofastatewithtotalangular momentum Jis lV"=!A{J(J+I)-L(L+l)-S(S+I)}, sothatlV,,-lV,,-1 =,\J(thisisknownastheLandeinterval rule). Showfromthesplittings ofthe•Fmultiplet giveninFig.20.5fortheCr3+ion thatthevalueofAisabout87cm-1(slightly highervaluesareobtained fromthe •Pstates,buttheseareperturbed bydoubletstateswhicharenotfaraway). 20.8.Hydrogen molecules areoftwotypes:(a)ortho-hydrogen, wherethenuclear spinofinofeachprotonisparalleltotheotherandthenuclearspinforthe molecule is1=I;(b)para-hydrogen, wherethetwoprotonspinsareanti-parallel giving1=0forthemolecule. Showthatathightemperatures wheretheratio ofortho-topara-hydrogen molecules is3:1,thesusceptibility duetothenuclear paramagnetism isidentical withthatofthesametotalnumberofhydrogen atoms withindependent spin1=t.(Notethattheequilibrium ratioof3:Icorresponds tothefactthattherearethreequantum statesfor1=I,associated withthree possible orientations ofthespin,eachwiththesameaprioriprobability asthe singlestatefor1=0.) 20.9.Calculate theparamagnetic susceptibility ofagramme molecule ofhydrogen atJOKduetothenuclearmoments, assuming thattheortho-para ratioisstill3:1. Showthatitisofthesameorderasthediamagnetic susceptibility duetothe electrons, assuming thateachofthetwoelectrons isinanorbitforwhichthemean radiusistheBohrradius. (Anewers: 2·2X10-11and-2·0X10-11(m.k.s.).) 20.10.Thegroundstateofsodiumis"Sl>andtheyellowD-linesareduetotransi­ tionstothegroundstatefromthetwolowestexcitedstates"Piand"Pt.Show thattheLandeg-factors ofthesetwostatesareiand!respectively. HenceshowthatoneD-linewillbesplitinamagnetic fieldBintofourcom­ ponents, withfrequencies D1±iS,D1±!o;andtheotherD-lineintosixcompo­ nentswithfrequencies D2±!-S,D2±S,D,,±i-S, whereS=fJB/h,whenviewed normaltothefield.(OnlytheaM=±1components areseenwhenviewedparallel tothefield.) 21 FERROMAGNETISM 21.1. Exchan~e interaction between parama~netic ions INthediscussion in§§20.5-20.9 ofparamagnetism inthesolidstate, itwastacitlyassumedthatinteractions between different paramagnetic ionscouldbeneglected. Suchinteractions areoftwotypes:(a)magnetic dipole-dipole interaction, arisingfromthemagnetic fieldduetoone dipoleactingonanother; (b)exchange interactions betweentheelectrons indifferent paramagnetic ions,ofthesamenatureasthosebetween electrons withinthesameatom(givingrisetoRussell-Saunders coupling) orbetweentheelectrons ofdifferent atomsinchemical binding. Ofthese twotypesofinteraction, thelattergreatlyoutweighs theformerinordi­ narysubstances. Forexample, theCuriepointofnickelis6310K(see Table21.1).Thisisaroughindication ofthetemperature atwhichthe interaction between neighbouring nickelions(separation 2·5A)isof theorderkT,whereas (Problem 20.2)thepurelymagnetic interaction oftwoatomicdipolesatthisdistance wouldbeequivalent tokTwith Tlessthan10K.Exchange interaction decreases morerapidlythan magnetic dipoleinteraction astheatomicseparation isincreased, though nosimplelawcanbegivenforitsrateofdecrease. Asanexample, in paramagnetic saltsofthe3dgrouptheexchange interaction ismore important thanthemagnetic dipoleinteraction untiltheseparation betweentheparamagnetic ionsisgreaterthanabout6A,andthenboth aresosmallthattheyhaveanappreciable effectonthemagnetic properties onlywellbelow10K. Themechanism ofexchange interaction, asoriginally proposed by Heisenberg in1928,isoneinwhichtheforcesinvolved areelectrostatic inorigin,butwhich,becauseoftheconstraints imposed bythePauli exclusion principle, areformally equivalent toaverylargecoupling between theelectronspins,ofthetype W=-2/si.sj• (21.1) Thequantity /isknownastheexchange energy.Thoughseveraltypes ofindirectexchange interaction havesincebeensuggested (see§21.9), theyallleadtoabasiccoupling betweenthespinsofthisform,dependent onthecosineoftheanglebetweenthetwospinvectors. Fortwoseparate 21.1] FERROMAGNETISM 619 atomswithtotalspinvectors8i,8jwemayusethevectorsummations toshowthatthetotalinteraction energyis W=-2JIIsi,Sj =-2JIsi.ISj =-2f8 i.ISj=-2f8 i.Sjij i.i j (21.2) whichdepends onlyontherelativeorientation ofthetwototalspin vectors8i,8j•Animmediate resultofequation (21.2)isthatthe exchange interaction vanishes foranyclosedshellofelectrons, since then8=O.Thusweneedconsider onlythepartlyfilledshellswhich areresponsible forpermanent magnetic dipolemoments inatomsand ions. ForanioninwhichJisagoodquantum number(suchasionsofthe 4/group),wemustproject8ontoJ;thereasonforthisisthatJisa constant ofthemotion,andhencesoalsoistheprojection of8ontoJ. Thecomponents ofSnormaltoJareprecessing rapidly,sothattheir contribution tothescalarproduct 8i.8jiszeroonatimeaverage. Fromtheequivalences L+28=gJ(wheregistheLandefactor), L+8=J,wefindatoncethat8=(g-I)J; thisresultcanbederived inalengthier butmoresatisfying wayfromthevectormodel(see Problem 21.4).Thus,forapairofsuchions(assumed identical, with thesamevaluesofJandg)wehave W=-2f8i.8j=-2f(g-I)2Ji.Jj=-2f'Ji.Jj•(21.3) Thisgivesacoupling oftheangularmomentum vectorsofthesame formasequation (21.2),butwithamodified valueoftheapparent exchange energy. Inasolid,anygivenmagnetic ionissurrounded byothermagnetic ions,witheachofwhichitwillhaveanexchange interaction. Thetotal interaction foreachionwilltherefore beasumoftermssuchas(21.3) .takenoverallpairsofions;theenergyforatomiisthus Wi=-2Ji·I,fIiJ j•, Themagnetic dipolemomentofeachionisproportional totheangular momentum J,sincem=gfJJ,sothattheexchange energycanbe expressed intermsofthedipolemoments, giving assuming againthatallionshavethesameLandeg-factor. Inaferro­ magnetic substance, oraparamagnetic substance subjected toan external magnetic field,eachionwillhaveanaveragedipolemoment 620 FERROMAGNETISM [2I.l inthedirection ofmagnetization, together withfluctuating components inotherdirections whosetimeaverage iszero.Insumming overthe interaction withneighbouring ions,thatpartassociated withthe fluctuating components willtendtoaverageout,sinceatanyinstant thecontributions fromdifferent neighbours willbeasoftenpositive as negative. Toafairapproximation wecantherefore replacethevector sumovertheneighbouring dipolemoments byasumovertheaverage moment perneighbour mi,andifweassumefurtherthattheonly important interaction iswithzequidistant neighbours, eachhavingthe sameinteraction energyJ',wecanwrite w=-2(:;)'LJ'(~) =-2(:;)·t;;)J' (2ZJ')= -ng2f32m.M=-m.B int• (21.4) Herewehavedropped thesubscript i,sinceweassumeallionsare identical, andtheenergyisthesameforeach;andwehavereplaced themeanmomentperionbythemagnetization M=nm,wherenisthe numberofionsperunitvolume. Theresultisanequation formally identical withthepotential energyofadipoleminafield Bint=(2zJ'{ng2(32)M=AM; wemaytherefore represent theeffectoftheexchange forces,toagood approximation, byaneffective 'internal field'Bintwhichisproportional totheintensity ofmagnetization. Thisconceptwasfirstintroduced by Weisstoaccount fortheoccurrence ofspontaneously magnetized sub­ stances(ferromagnetics). Asapreliminary, weshalldiscusstheeffectofthisinternal fieldin aparamagnetic substance. Thetotalfieldactingonanionisthen Bo+Bint=Bo+AM, whereBoistheexternal field.Solongasthe magnetization issmallcompared withthesaturation valuewemay assumethatCurie'slawX=O{TstillholdsifwereplaceBinour earliertheorybyBo+AM. Thenwehave M=(O{T)B{JLo =O(Bo+AM){JLoT andhence X=JLoM{Bo=O{(T-AO{JLo) =O/(T-e). (21.5) ThisisknownastheCurie-Weiss law,andrepresents thebehaviour of paramagnetic substances attemperatures T>ewithfairaccuracy; 8=AO{JLoisoftencalledthe'Weiss'constant. Theformofequation (21.5)showsthatsomeradicalchangeinthe magnetic properties istobeexpected atthetemperature e,andwemay 21.1] FERROMAGNETISM 621 interpret theinfinitesusceptibility whichispredicted byequation (21.5) atthispointinthefollowing way.SinceX-1-'0M/Bo,andthemaximum valueofMisfinite,beinglimitedtothesaturation moment obtainable whenallthedipolesarealignedparalleltooneanother, wemustassume Bo=0;inotherwords,thesubstance ismagnetized evenintheabsence ofanexternal field.This'spontaneous magnetization', duetothe internalfield,isacharacteristic offerromagnetism, andthetemperature TABLE 21.1 Saturation momentandOuriepointofsome ferromagnetic materials Saturation momentat0°KCurie (a) (b) point Substance e.m.u.Jg BohrmagnetonsJatom (OK) Fe. 221·7 2·22 1043 Co(>6700K). 162·6 1·715 (>6700K).(167'3) (1'76) 1394 Ni. 57·6 0·605 631 MnBi 75 3·52 630 MilAs 146 3·40 318 FetOa 83·5 1·20(peratom 893 ofFe) Notes:Cobalthasaphasetransition atabout670°K,beinghexagonal instructure belowthattemperature, andface-centred cubicabove.Thevaluesinbrackets are .obtained byextrapolation. Inthem.k.s.system,thesaturation moment inampere·metret/kg isthesameasthe valuegivenincolumn(a);inanysystemthevaluesofM.,thesaturation moment per unitvolume, maybeobtained bymultiplying thevaluesperunitmassbythedensity. fJistheboundary between paramagnetic behaviour atT>fJandferro­ magnetic behaviour whenT<fJ.Thetemperature belowwhichspon­ taneous magnetization appears isknownastheCuriepoint,andthe experimental valuesforanumberofsubstances aregiveninTable21.1. The'ferromagnetic Curietemperature' isdefinedasthatbelowwhich spontaneous magnetization setsin,anditoftendiffersby10°or20° fromthevalueof()determined intheparamagnetic regionbyfitting theobserved susceptibility toequation (21.5).Thelattervalueissome­ timescalledthe'paramagnetic Curietemperature'. Onoursimpletheory thereisnodifference between thetwoCurietemperatures. SincetheCurieconstant 0=l-'ong2f32J(J +1)/3k,thevalueofthe Weissconstant inequation (21.5)is ()=>"0/1-'0=(2zf'/ng2f32) x{p,ong2f32J(J +1)/3k}71-'0 =2zf'J(J+1)/3k (21.6) 622 FERROMAGNETISM [21.1 andonsimpletheorythisisalsothevalueoftheCurietemperature To. Moresophisticated methods ofcalculation produceasomewhat different valueofthenumerical constant, andRushbrooke andWood(1958)show thattheresultscanbefittedremarkably wellbytheempirical formula To=:~(z-1){llJ(J +1)-1}. (21.7) Thispredicts somewhat lowervaluesfortheCuriepointthanequation (21.6),andconversely, giveshigherestimates oftheexchange inter­ action.Forexample, nickelhasitsCuriepointat6310K;itscrystal structure isface-centred cubic,forwhichthenumberofnearestneigh­ boursis12,whichwetaketobethevalueofz.Ifwemakethefurther assumption thatJ=S=t,thenwefindthatf'fkis1050Kfrom equation (21.6),and1500Kfrom(21.7).Thustheexchange energy (thereisnodifference betweenf'andfwhenwearedealingwith spin-only magnetism), isabout10-2electron volts.Themagnitude of thisinteraction canperhaps beappreciated bestbyexpressing itin \L,-termsoftheinternalfieldBintofequation (21.4),whichisfoundtobe ;/f\oforder107gauss(103weberfmetre2).Thisisover100timeslargerthan anyfieldwhichcaneasilybeproduced inthelaboratory, sothatexternal fieldswouldbeexpected tohavelittleeffectonthespontaneous mag­ netization belowtheCuriepoint. Equations (21.6)and(21.7)showthatthesignof()andToisthesame asthatoff'(andhencealsooff,solongaswearedealingwithidentical ions).Thusapositivevalueoftheexchange energyisrequired togive avanishing denominator intheCurie-Weiss law(equation (21.5»,and aco-operative stateinwhichtheelectronspinsareparalleltoeachother. Thisferromagnetic stateisadirectconsequence ofthefactthatthe exchange coupling (equation (21.1»givesalowerenergyforanypair ofelectrons whentheirspinsareparallel,provided theexchange energy fispositive.Ifitisnegative, thestateoflowerenergyisonewith anti-parallel spins;theWeissconstant isalsonegative, andthedenomi­ natoroftheCurie-Weiss lawdoesnotvanishatanyrealtemperature. Nevertheless aco-operative statedoesthenoccur,butoneinwhichthe basicarrangement isofanti-parallel spins.Thisphenomenon iscalled 'anti-ferromagnetism', andisdiscussed inChapter 22. 21.2.TheWeisstheoryofspontaneous ma~netization Sincetheinternalfieldinaferromagnetic substance issolarge,the magnetization willapproach thesaturation valueevenatordinary 21.2] FERROMAGNETISM 623 temperatures. Theassumption thatthemagnetization issmalland proportional totheeffective field,usedinderiving equation (21.5)for thesusceptibility abovetheCuriepoint,thuscannotbeusedbelowthe Curiepoint.Ifweretaintheconceptofaninternalfield,themagnetiza­ tionmaybecalculated usingtheBrillouin function (seeequation (20.15» whichmaybewrittenintheform M/Ms=ep(y). (21.8) Here~isthesaturation magnetization perunitvolume, andequals ngJf3,wherenisthenumberofatomicdipolesperunitvolume. The argument oftheBrillouin function maybewrittenas y=gJf3B/kT =~B/nkT andBmustbetakenasthesumoftheexternal fieldBoandtheinternal fieldAM.Hencewehave y=~(Bo+>'M)/nkT, whichmaybesolvedforM,giving M/Ms=y(nkT/AM:)-(Bo/~)'(21.9) (21.10) Thevalueofthemagnetization underanygivenconditions ofBoandT maybefoundbyeliminating theparameter ybetweenthetwoequations (21.8)and(21.10).Itisclearthatthiscannotbedoneanalytically, but thegeneralbehaviour ofthemagnetization canbefoundfromagraphical solution. Weshallbeginbyequating Botozero,andfindingthevalue ofthespontaneous magnetization Moinzerofield.Toobtainagraphical solution wethenplotthetwofunctions Mo/Ms=ep(y)(fromequation (21.8»andMo/Ms=y(nkT/>.M:) (fromequation (21.10»againsty,asin Fig.21.1.Thesecondfunction givesastraightlinewhichpassesthrough theoriginandintersects thecurveforep(y)atthispoint.Thusonepossible valueofthemagnetization isalwayszero.Ifthetemperature Tissuffi­ cientlyhigh,theslopeofthelineMo/~=y(nkT/>.M:) issogreatthat thisistheonlypointofintersection, andthesubstance musttherefore beunmagnetized. inzeroexternal field.Thiscorresponds tothepara­ magnetic behaviour abovetheCuriepoint,discussed inthelastsection. Asthetemperature Tfalls,theslopeofthelinegivenbyequation (21.10)decreases, untilatacertaintemperature Toitistangential to thecurve(a)attheorigin.Forsmallvaluesofy, ep(y)=M/~=y(J+I)/3J, andonequating thistothevalueofM/~givenbyequation (21.10)with 624 FERRO MAGNETISM [21.2 Bo=0,thevalueofToisfoundtobe T.=AM=(J+1)=>.ng2f12J(J+1)=0>./=e(21.6a)onlc3J 3lc /Lo, where()istheWeissconstant definedbyequation (21.6).Atstilllower temperatures, theslopeofthelineislessthantheinitialslopeofep(y), andtherewillbetwopointsofintersection, andtwopossible valuesof themagnetization, onezeroandtheotherfinite.Itiseasytoshowthat r(b) (e) FIG.21.1.Graphical solution oftheequations (21.8)and(21.10)for spontaneous magnetization. (a)istheBrillouin function ef>(y)(equation (21.8)); (b),(e),(d)arethestraight linesMIM. =y(nkT/~) fortempera­ turesT>To,T=To,andT<Torespectively, where'1'0isthe Curiepoint(allwithBo=0); (e)isthefunction inequation (21.10);anexternal fieldBoisapplied, withthetemperature thesamea,sfor(d). theformerisunstable andthelatterstable.For,ifweimaginethe magnetization atanyinstanttocorrespond tothepointQonep(y),then theinternalfieldproduced bythemagnetization corresponds tothepoint R,andthisfieldwillproduce thegreatermagnetization corresponding tothepointSonep(y).Thusthemagnetization willincrease untilthe pointPisreached wherethetwocurvesintersect. AboveP,thetwo curvescrossandanyfurtherincreaseinthemagnetization wouldproduce aninternal fieldinsufficient tosustaintheincreased magnetization. It thusappearsthatthestateofspontaneous magnetization corresponding tothepointPisstable,whiletheunmagnetized stateisunstable. Sincethevalueofthespontaneous magnetization isdetermined by theintersection withep(y)ofthelinecorresponding toequation (21.10) (withBo=0),andtheslopeofthislinedepends onthetemperature, it 21.2] FERROMAGNETISM 625 isobviousthatthewholeofthecurveef>(y)willbetracedoutaswelower thetemperature fromtheCuriepointtotheabsolute zero.From equation (21.6a),AM:fnk =30Jf(J+1),andhencewemayexpress equation (21.10)(withBo=0)intheform Mof~=y(~)(J3jl). FIG.21.2.Reduced equation ofstateforaferromagnetic substance. - -~-fromtheWeisstheory(equation (21.8))forJ=t.--experimental curvefornickel.-0-experimental curveforanickel--eopper alloy(76%-24%). (AfterOliverandSucklnnith, 1953.) Elimination ofybetween thisequation andequation (21.8)showsthat a"'reduced equation' maybefoundoftheform Mof~-f(TfO), (21.11) wherethefunctionf(TfO) isthesameforallsubstances withthesame valueofJ.Thisfunction isplottedinFig.21.2(brokenline)forthe specialcaseofJ=t;thecurvesforothervaluesofJlieslightlyinside thiscurveatintermediate valuesof(TIO):Theexperimental determina­ tionofMo/~andtheverification ofthis'LawofCorresponding States' willbediscussed in§21.6. Whenaconsiderable external magnetic fieldEoisappliedtheeffect onthemagnetization canbefoundbyagraphical solutionofequations 851110 Ss 626 FERROMAGNETISM: [21.2 (21.8)and(21.10),wheretheterminBoisretainedinthelatterequation. Thestraightlinecorresponding toaplotofMIMsagainst yisnowdis­ placedtotherightcompared tothatforBo=0atthesametemperature. Theintersection withMIMs=c/>(y)occursatthepointP'inFig.21.1, andthemagnetization isslightlyincreased overthatcorresponding toP, thevalueforzeroexternal field.Attemperatures wellbelowtheCurie pointMoisalreadyclosetoMsandc/>(y)increases onlyveryslowly,so thattheeffectofBoissmall.Attemperatures neartheCuriepointPis onthesteeperpartofthecurveforc/>(y)neartheoriginandtheincrease inMproduced byanexternal fieldismorenoticeable. Thetheoryoutlined aboveissimilartotheoriginaltheoryofWeiss exceptthattheBrillouin function hasbeensubstituted fortheLangevin function. Itsgreatsuccessliesintheexplanation ofthepresence of spontaneous magnetization inaferromagnetic substance, butthereare alsodifficulties. Thefactthattheunmagnetized stateisunstable appears tobecontrary toexperience, sinceitiswellknownthatapieceofiron canbedemagnetized bydropping it.Moreover, inasinglecrystalthe magnetization canberestoredbyapplying anexternal fieldoflessthan 1gauss,although theinternalfieldisabout107gausslWealsorequire someexplanation ofthehysteresis curve.Toovercome thesedifficulties Weissintroduced theconceptofdomains ofmagnetization withinthe specimen. Eachdomaincontains some1017_1021atoms,andapiece ofunmagnetized ironcontains manydomains allspontaneously mag­ netized,butthedirections ofmagnetization ofdifferent domains are orientedatrandom. Thetheoryofspontaneous magnetization appliesto asingledomain,butthemagnetization ofthewholespecimen depends onwhether thedomains themselves arealignedtowards thefieldor whethertheyarerandomly oriented. Thistheory,whichwasconceived beforethenatureoftheexchange interaction whichcausesthespon­ taneousmagnetization wasknown,isremarkably successful inexplaining themainfeaturesofferromagnetic substances. Theexistence ofdomains hasbeenconfirmed bytheexperiments ofBitter,brieflydescribed inthe nextsection,whereweshallfirstconsiderwhatfactorsdetermine thesize andshapeofthedomains. 21.3. Ferroma~netic domains Aconsiderable advance intheunderstanding offerromagnetism occurred whenitbecamepossibletoobtainsinglecrystalsofiron,cobalt, andnickelsufficiently largefortheirmagnetization curvestobemeasured. Ineachcaseitwasfoundthatthecrystalsareanisotropic; thatis,the 21.3] FERROMAGNETISM 627 magnetization dependsonthedirection thefieldmakeswiththecrystal axes.Fig.21.3showsthecurvesforiron,whichformsbody-centred cubic crystals. TheM-BocurveisfoundtorisemoresteeplywhenBoisparallel totheedgeoftheunitcube[100]thananyotherdirection, suchasa facediagonal [110]orabodydiagonal[Ill].Theenergyofmagnetization isfBodM,andisrepresented bytheareabetween themagnetization I M 400gauss 0·04weber/metret Bo~ FIG.21.3.Magnetization curvesforasinglecrystalofiron. Thedirections ofeasymagnetization arethecubeedges(e.g.[100]). Whenthefieldisnotalongacubeedge,theinitialprocessisofmag­ netization alongthecubeedgesindirections nearesttothatofthe field;hencethecurvefor[110]breaksoffroughlyatMo!"-'2,andthat for[111]atMo!"-'3,sincefurthermagnetization requires domainrota· tionagainsttheanisotropy energy. curveandtheM-axis(Bo=0).Thisenergyisleastwhenthesingle crystalofironismagnetized alongthe[100]direction (oritsequivalents, [010]and[001]),andtheseareknownasdirections ofeasymagnetization. Inthecaseofnickel,withaface-centred cubicstructure, thedirections ofeasymagnetization arethebodydiagonals, whileforcobalt,witha hexagonal structure atroomtemperature, thereisonlyonedirection of easymagnetization, thehexagonal crystalaxis. Theexcessenergyrequiredtomagnetize thesubstance inaharddirec­ tionisknownastheanisotropy energy.Itisclearthattheanisotropy energycannotarisefromtheexchange interaction, forthelatterdepends 628 FERROMAGNETISM [21.3 onlyonthemutualorientation ofthedipolesandnotontheanglewhich theymakewiththecrystalaxes.Itsoriginisthought tobesimilarto thatofparamagnetic anisotropy (seeendof§20.7),arisingfromthe combined effectofspin-orbit coupling andtheelectricfieldofthe neighbouring charged ions.Theanisotropy energyhasthesamesym­ metryproperties asthecrystal,andissmallest forcrystals ofhigh symmetry. Thusitislessforironornickel,whicharebothcubic,than r1N Ns r1r1r1 S (a)SN (b) (0)l/~~~~, (d). FIG.21.4.Possible domain structures inasinglecrystal, wherethedirections ofeasy magnetization arealongtheedgesofacube. (a)Singledomain; external linesoffieldrunfromnorthtosouthpoleandgivelarge external field. (b)Doubledomain, whereexternal linesoffieldrunmostlybetween adjacent northand southpoles,andtheenergystoredinexternal fieldismuchreduced. (e)Arrangement withnofreepolesandnoexternal field;thedomains withperpendicular magnetization attopandbottomarecalled'domains offluxclosure'. (d)Asin(e),butwithfurthersubdivision intosmallerdomains. forcobalt,whichhasonlyaxialsymmetry. Theanisotropy energyalso causesachangeoflengthonmagnetization (magneto-striction). Inzerofieldthespecimen, whetheritisasinglecrystaloranaggregate ofcrystals, willbeinequilibrium whenitspotential energyisaminimum. Inanunstrained crystaltheimportant contributions aretheexchange energy,theanisotropy energy,andthemagnetostaticenergy (theenergy storedinthemagnetic field).Ifthecrystalconsisted ofonesingledomain, asinFig.21.4(a),the'freepoles'attheendswouldgiverisetoalarge external magnetic fieldandtoalargemagnetostatic energy. Thisis reducedbyhavingtwodomains oppositely magnetized asinFig.21.4(b), whenthetwopolespartially canceloneanother.Iftherearenofreepoles onanysurfacethemagnetostatic energyisreducedstillfurther.Forthis tobethecase,thefieldBatthesurfaceofthecrystalmustalways 21.3] FERROMAGNETISM 629 beparalleltothesurface,andthenormalcomponent ofBmustbe continuous acrosstheboundary between twodomains.Ifthetwo domains aremagnetized inperpendicular directions thewallbetween themmustrunatanangleof45°toeachdirection ofmagnetization, andFig.21.4(c)showsapossiblearrangement. Thelittlesurfacedomains whichproduce aclosedcircuitofBarecalleddomainsofclosure,andare generally muchsmallerthantheinnerdomains. Thesizeofthedomains -Width ofwill~---- FIG.21.5.Variation ofspinorientation inaBlochwall. dependsverymuchonthesizeandshapeofthecrystal,andthisdepends ontheprevious historyofthesubstance. Thisfitsinwiththefactthat thehysteresis curveisverysensitive tothecomposition andstateofthe specimen, sincethedomainstructure mustdetermine theshapeofthis curve. Theconfiguration inFig.21.4(d)isanalternative tothatofFig.21.4(c) andonemightexpectthedomains alwaystobeverysmallinsizeand largeinnumber; butenergyisrequired toformtheboundary between twodomains, sincethemagnetization oneithersideisinopposite directions. Theboundary between twodomains isknownasa'Bloch wall'.Ithasafinitethickness, extending overanumberofatomswhose spinschangegradually indirection asweproceed through thewall (Fig.21.5). Fromequation (21.3)theexchange energybetween neighbouring identical spinsisapproximately ~lf,;=-2.1'J2costjJ,wheretjJisthe 630 FERROMAGNETISM [21.3 anglebetweenthedirections ofthespinmomentum vectors. Therefore, thetotalexchange energyingoingthroughthewallis JYe= -22/'J2cos,pii'i>i Ifthewallthickness extendsovermanyatoms,andtheanglebetween neighbouring spinsissmall,wemaywriteCOS,pij!:::! 1-IM/2, andthe totalincrease inexchange energybecausethespinsarenotexactly parallelis JYe~/'J22,p~i' Forawallwhichformstheboundary between twodomains wherethe spinsareanti-parallel, thetotalchangeinangleingoingthroughthe wallis2,pij=1T.Ifthereisalineofn'atomsinthethickness ofthewall, and,pijisthesameforalladjacent pairsofatoms,n',pii=7Tand JYe~n'/'J2{1TJn')2 =7T2/'J2Jn'. Thisequation showsthattheexchange energyisreducedbymaking n'large,anditwouldseemthatthewallshouldbeinfinitely thick. Thiswouldincreasetheanisotropy energylfa,however, sinceanumber ofspinsinthewallarepointingatanangletothedirection ofeasy magnetization, andthisnumberincreases withthewallthickness. Thus lfaisproportional ton',andthetotalenergyperunitareaofwallinthe substance is JYe+lfa=7T2/'J2Jn'a 2+Kn'a, (21.12) whereKisaconstant roughlyequaltotheanisotropy energyperunit volume. aisthelatticeconstant ofthesubstance, sothat,forasimple cubiccrystal,thereareIJa2atomsperunitareaofwall,andn'aisthe thickness ofthewall. Theformofequation (21.12)showsthat:therewillbeaminimum value ofthetotalenergyforsomevalueofn',whichbydifferentiation isfound toben'={7T2/'J2JKa3)1.Fornickel,J=t,/isabout10-14ergs, Kisabout105ergsJcm3,anda3isabout10-23cm3•Hencen'isofthe orderof100atoms,andthethickness ofthewallisafewhundred Ang­ strOmunits.Substitution oftheoptimum valueofn'inequation (21.9) givestheexpression 27T{/'KJ2Ja)1 forthewallenergyperunitarea, whoseorderofmagnitude isfoundtobeaboutanergJcm2• Asthedomainwidthdecreases inthefluxclosurearrangement shown inFig.21.4(d),thenumberofwallsperunitareaofthecrystalsurface increases, withacorresponding increase intheenergy. Theenergy required toformawalltherefore tendstokeepthedomains smallin number, andlargeinsize.WhenKislarge,particles ofabout10-4em diameter arefoundtoconsistofasingledomain, becausetheenergy 21.3] FERROMAGNETISM 631 required toformawallismorethanthereduction inthemagnetostatic energywhichwouldresultfromthesubdivision intodomains. Inlarge crystalsanotherfactorwhichentersintothedetermination ofdomain sizeisthatthedomains ofclosureinFig.21.4(d)mayrequiretobe magnetized inaharddirection, therebyincreasing theanisotropy energy. Thevolumeoccupied bythedomains ofclosuredecreases asthewidth ofthedomains decreases, andtheanisotropy energytherefore tendsto reducethedomainsize,whilethewallenergytendstoincreaseit.The optimum domainsizeisdetermined byacompromise between these twoeffects. Themoststrikingevidence fortheexistence ofdomains isprovided bytheBitterpatterns whichareobtained whenfinelypowdered ironor cobalt,orcolloidal magnetite, isspreadonthesurfaceofthecrystal. Thesurfacemustbeverycarefully prepared andelectrolytically polished toremoveirregularities. Theparticles depositthemselves alongthe domainboundaries sinceheretherearestronglocalinhomogeneous magnetic fieldswhichattracttheparticles. AtypicalBitterpatternis showninFig.21.6;the'fir-tree' effectisobtained whenthesurface makesasmallangleof2or3degreeswiththetrue(100)crystalplane. Thebranches ofthetreearethedomains ofclosurewhichclosetheflux circuitovertheprimary domains below.Onlookingthrough amicro­ scopethepatterns canbeseentochangeasamagnetic fieldisapplied. Thedirection ofmagnetization inadomainisfoundbymakingatiny scratchonthesurfacewithafineglassfibre.Ifthescratchisparallel tothemagnetization thepatternisunchanged, butifitisnormalto itthepatternisdistorted. Thisisbecauseascratchparalleltothefield behavesasalongnarrowcavity,withnofreepolesattheends;ascratch perpendicular tothefieldwillhaveinducedpolesonitssides,andthere willbeastrongfieldinthecavitysothatthepatternisdistorted. Experiments ofthistype,andothers,inwhichthescattering ofbeams ofelectrons orpolarized neutrons havebeenusedtoinvestigate domain structure, showthatthetheoryoutlined aboveiscorrectinitsmain features. Thechangesinthedomainstructure whichoccurwhenamagnetic fieldisapplied,andthecorrespondence between thesechangesandthe variouspartsofthemagnetization curve,havealreadybeenoutlined in§8.4.Theinitialportions ofthemagnetization curveareassociated withmovements oftheBlochwalls,whicharereversible insmallfields butirreversible afterlargerfieldshavebeenapplied. Wherethereare strainsorinclusions ofimpurities theenergydependsonthepositionof .-- b b.. I1 Ib 1 b +- 0·01em. FIG.21.6.Domain patterns onademagnetized singlecrystalofsilicon-iron (thesurface isverynearlya(100)crystalplane). Themagnetization isnormaltothefinescratches visibleonthesurface, andisdirected asshowninthekeydiagram above.Domain wallslabelled aformtheboundary between domains magnetized indirections differing by90°,andthoselabelled bareboundaries between domains differing by180°.The'fir-tree' closuredomains arisebecause the surfaceisnotexactly acrystalplane.Twodifferent typesofclosuredomain (labelled 1and2)canbeseenonthe90°wall. (Photograph byL.F.BatesandA.Hart.) 21.3] FERROMAGNETISM 633 thewall,ascanbeseenfromconsidering theeffectofasmallparticle embedded inthematerial. Suchaparticlewillbeasmalldomainmag­ netizedinoneofitsowneasydirections ofmagnetization, whichdonot ingeneralcoincide withthoseofthesurrounding material, oritmaybe aparticleofanon-ferromagnetic substance. Inthelattercasethere sS S S NNNN (a) Domain boundary (b) FIG.21.7.Effectofanon-magnetic inclusion. (a)Inthemiddleofadomain. (b)Whenintersected byadomainboundary. willbefreepolesonitssurface,asinFig.21.7(a),andthefieldofthese polesgivesextramagnetostatic energy.IfaBlochwallintersects the particle, asinFig.21.7(b),thisenergywillbereduced, justasinthe caseoffreepolesonthesurfaceofaferromagnetic substance inFig. 21.4(a,b).Thisgivesaminimum ofenergywhenawallintersects asmanyinclusions aspossible. Inasmallexternal fieldthewallis displaced slightlyawayfromtheminimum energy,butreturnswhenthe fieldisremoved; thisgivesareversible wallmovement. Inlargerfields thewallmaybeshiftedtoamoredistantposition wheretheenergy curvehaspassedthroughamaximum andthendiminished; onremoving thefieldthewallcannotcrosstheenergymaximum andsoisunableto returntoitsinitialposition. Thedisplacement isthenirreversible. The morefreethematerial isfromstrainsandinclusions, thegreaterthesize ofreversible wallmovements, andthelowerthefieldrequired toproduce amovement, thusgivingalargeinitialpermeability, anda'soft'mag­ neticmaterial. Withlargestrainsandmanyinclusions thesmalleris thepossibility ofboundary movement, andthehigherthecoercive force. 634 FERROMAGNETISM [21.4 21.4.The~yroma~netic effect Itwaspointedoutin§20.1thatthemagnetic moment ofanatomis proportional tothetotalelectronic angularmomentum oftheatom.For amacroscopic system,thetotalmagnetic moment Mandthetotal electronic angularmomentum Geareformedbysimilarvectoraddition oftheindividual components, andtheyshouldtherefore berelatedin thesameway.Thuswehave MjGe=Y=-g'(ej2m o), whereg'isaneffective Landefactor.Itfollowsfromthisthatifwe couldmeasure insomewaythechangeinelectronic angularmomentum associated withthechangeinmagnetization ofaspecimen, thevalue ofg'wouldbedetermined. Sinceg'differsbyafactorof2according towhether themagnetic moments areassociated withorbitalorspin angularmomentum, thisaffordsamethodofverifying ourassumption thatferromagnetism inthe3dgroupisassociated withtheelectronic spins. Sincenoexternal coupleisexertedonaspecimen bytheactofchanging itsmagnetization, thetotalangular momentum ofthesystemmust remainunaltered. ThechangeflGeintheelectronic angularmomentum musttherefore beaccompanied byanequalandopposite change flGlattice= -flGe intheangularmomentum ofthe'lattice', definedastherestofthe specimen, apartfromtheelectrons responsible forthemagnetization. Itisthislatterchangeinangularmomentum whichisobserved, butit isverysmall.Inacubiccentimetre ofnickeltherearesome1023electrons whoseindividual momenta canbechangedby1i:::::::10-34newton-metre (10-27dyne-cm) byreversalofthespin.Thetotalangularmomentum thusimparted tothelatticeisonlyabout10-11newton-metre (10-4 dyne-em). Avarietyofexperimental methods havebeenusedtodetermine y, butonlyashortaccountwillbegivenhere(moredetailsaregivenby theauthorstowhomreferences aremadeinthissection). Themethods fallintotwoclasses.Inone,anunmagnetized specimen issetinto rotation andtheresultant magnetization ismeasured. Thisisthe Barnett effect,andtypicalexperiments arethoseofBarnett (1944). Evenwithalargespecimen, themagnetic moment induced isvery smallowingtothelimitedvelocities ofrotation whichcanbeem­ ployed.Inthesecondclass,themagnetization ischanged byaknown amountandthechangeinangularmomentum isdetermined; thisis 21.4,] FERROMAGNETISM 635 knownastheEinstein-de Haaseffect,thoughfirstsuggested byRichard­ son.Thismethod hastheadvantage thatresonance canbeusedto enhance theeffect.Aferromagnetic rodissuspended insidealong solenoid supplied withalternating currentwhoseperiodisequaltothe torsional oscillation periodofthesuspended rod.If~isthemoment ofinertiaoftherod,bthedamping constant, cthetorsionconstant of thesuspension, andMsinwtthemagnetic moment ofthespecimen at anyinstant,theequation ofmotionis d2()d() dO1 ~T2+bT+c() =-:1=-wMcoswt.wtwt wt'Y Atresonance, theamplitude oftheangleofrotation is(1jy)(MIb);bis foundfromthelogarithmic decrement, andMmustbemeasured independently. Thismethodofmeasuring ywasemployed byScott(1951)usinga modification oftheapparatus builtforthedetermination ofelmofthe carriersofelectriccurrent(see§3.1).Thespecimen, intheformofa rod,issuspended asatorsional pendulum. Acoiliswoundontherod, andbyreversing acurrentinthiscoilthemagnetization oftherodcan bereversed. Thechangeinmagnetization ismeasured byanullmagneto­ meterplacedhalf-way between therodandastandard coilcarrying a steadycurrentwhichissimultaneously reversed withthatinthespeci­ men.Thissteadycurrentisadjusted untilabalanceisobtained. The magnetometer isfittedwithamirror,andthelightreflected fromitfalls onatwinphotocell feedinganamplifier, adevicesimilartothatusedto amplifygalvanometer deflexions. Bythismeansanullmagnetometer ofgreatsensitivity isproduced. Acorrection wasmadeforthenon­ uniformity ofmagnetization oftherod,andtheearth'sfieldwasneutral­ izedbyasystemofHelmholtz coils.Theperiodofoscillation oftherod was26sec,anditsrotation wasobserved byreflections ofabeamof lightfromamirrormounted immediately abovethespecimen. The procedure usedwastoreversethemagnetizing currentatamoment whenthespecimen passedthroughthecentreofitsswing.Thedirection ofreversal waschosensothatfor60currentreversals theamplitude wasincreased, andthenfor60reversals itwasdecreased. Withsmall damping, theprogressive changeinamplitude wasverynearlylinear, andtheamplitude changeforonereversal wasobtained fromthetwo slopesoftheplotfor120reversals. Anumberofexperiments ofhighprecision havebeencarriedoutusing boththeEinstein-de HaasandBarnetteffects,andameanoftheresults 636 FERROMAGNETISM [21.4 obtained between 1944and1960isgiveninasurveybyMeyerandAsch (1961),whoshowalsothatthereisgoodagreement withresultsofferro­ magnetic resonance experiments usingmicrowave radiation (see§23.7). TheresultsareshowninTable21.2,andareexpressed intermsof twoquantities gandg',obtained fromferromagnetic resonance andgyro­ magnetic experiments respectively. Whenwehaveamixtureoforbit andspin,themagnetic momentandangularmomentum maybewritten asM=ML+Ms=(ef2mo){GL+2Gs},G=GL+G S'andtheratiois y.__M~e{GL+2Gs}~g'(e) (21.13) ~G~2moGL+Gs~2mo' The'spectroscopic splitting factor'gmeasured inaferromagnetic resonance experiment hasbeenshownbyKittelandVanVlecktobe definedby M~~e~{GL+2Gs}=g(~), (21.14) Gs2moGs 2mo fromwhichitfollowsthat 1 1g+?=1. (21.15) TheresultsgiveninTable21.2showthatthisrelationisfulfilledwithin theexperimental errorforironandnickel,andMeyerandAschshow thatthisistruealsoforawiderangeofalloysofthe3dgroup.The factthatg,g'aresocloseto2showsthatthemagnetism oftheferro­ magnetic metalsofthisgroupisalmostentirelyduetospin.Thisresult TABLE 21.2 Somevaluesofthequantities g'andg Thequantity g'isderivedfromgyromagnetic (magneto-mechanical) ex­ periments, thequantity gfromferromagnetic resonance experiments; the valuesquotedarethemeansofanumberofexperimental results,givenby MeyerandAsch(1961). 1I Substance g' g g+g; Iron 1·928±O·OO4 2·094±O·OO3 O·996±O·OO4 Cobalt 1·854±O·OO4~ ~ Nickel 1·840±O·OO8 2·185±O·OlO 1·OOI±O·OO9 issimilartothatfoundfortheparamagnetism ofsaltsof3dgroupions (Chapter 20),andthereislittledoubtthatitisdueessentially tothe samecause,'quenching' oftheorbitalmoment byelectrostatic inter­ actionwiththeneighbouring (ligand)ions. 21.5J FERROMAGNETISM 637 21.5.Thermal effectsinferromagnetism Whenasubstance ismagnetized, withalltheelectron spinspointing inonedirection, itisinastateofgreaterorderthanwhenitisunmag­ netized,withthespinspointing inrandomdirections. Themagnetized stateistherefore oneoflowerentropythantheunmagnetized state,and inpassingfromtheformertothelattertherewillbeanincreaseinthe entropyofthespinsystem.Ifthetransition isaccomplished byheating aferromagnetic substance throughitsCuriepoint,theentropy change appearsasananomaly inthespecificheat.Ifitisaccomplished bythe sudden(adiabatic) removal ofamagnetic field,theentropy change appears asafallinthetemperature ofthesubstance; thisisknownas themagneto-caloric effect.Boththiseffectandthespecificheatanomaly havebeenusedtoobtaininformation abouttheferromagnetic state. Thespecificheatofasubstance is0=T(dSjdT), wheretheentropy changedSisgivenbytherelation TdS=dU-!BdM; (21.16) P dUisthechangeintheinternal energy,and-BdM istheincrease inthemagnetic potential energywhenthemagnetization isincreased bydMatconstant fieldB.Thedensity pappears becauseMisthe magnetization perunitvolume,whilethespecificheat(andotherthermal quantities) areperunitmass.Inaferromagnet, B=Bo+AM, andthe specificheatisthus 0=TdS=dU_(Bo+~ (dM)dT dT P-JdT =OM_(Bo~AM) (:~. (21.17) BelowtheCuriepointanyexternal fieldBoisverysmallincomparison withtheinternal field>tM,sothatwecanwrite o=0_!(~)d(M2). (21.18) M2PdT HereOMisthespecificheatofthesubstance atconstant magnetization, whilethesecondtermarisesfromthechangeinmagnetization with temperature. SinceMfallswithincreasing T,itgivesapositive con­ tribution tothespecificheat(asthetemperature rises,thedegreeof orderinthemagnetic systemdecreases, andtheentropyassociated with themagnetization increases). Reference toFig.21.2showsthatthe rateofchangeofMwithtemperature isgreatest justbelowthe Curiepoint,andtheanomalous specificheatarisingfromthemagnetic 638 FERROMAGNETISM [21.5 properties shouldbegreatestatthispoint,followed byasharpdrop abov~theCuriepointwhereMbecomes zero. Anexperimental curveshowing thevariation of0withTfornickel isgiveninFig.21.8.Theanomalous specificheatisappreciable only neartheCuriepoint,butthedropabovetheCuriepointspreadsovera rangeoftemperature, insteadofappearing asasharpdiscontinuity. In 9 3 -200 o 200 400T{OC.) FIG.21.8.Themolarheatofnickel,fromthemeasurements ofGrew,1934. ordertoobtainavaluefor'\fromthespecificheatanomaly, OMmustbe estimated andsubtracted fromthemeasured specificheat,sothatonly themagnetic contribution remains. Measurements aremadeatconstant pressure, sothatwecanwriteOM=0v+(Op-Ov)+Oe' 0visobtained byextrapolation, usingtheDebyeformula, frommeasurements atlow temperatures; (Op-Ov)maybefoundfromtheexpansion coefficient and compressibility usingastandard thermodynamical formula; Oeisthe electronic specificheat.Thisisabnormally largeinaferromagnetic metalanddifficulttoestimate sinceitisassociated withahighelectron densityinthe3dband(see§18.4).dM2JdTmustbefoundby:plotting M2asafunction oftemperature, andthen,\isobtained. Thisisnota veryaccurate methodoffinding,\andthevaluedoesnotagreetoowell withthevalueobtained fromthemagnetization curve,probably because oferrorsinOe.However, thegeneralformofthespecificheatcurveis notincompatible withtheoryandthisalsoappliestoironandcobalt, although themeasurements onthesemetalsarelesscertain(forexperi­ mentaldetails,seeGrew,1934).Thegeneralformofthespecificheat 21.5] FERROMAGNETISM 639 (21.19)anomaly ('lambda type')istypicalofa'co-operative' transition from anorderedtoadisordered state. Ifafieldisappliedtoamagnetic substance; thereisingeneralan increaseinmagnetization, andthisresultsinastateofgreaterorder thaninzerofield.Inotherwordstheentropy ofthesystemhas decreased, andthelossof(magnetic) potential energyofthedipolesin turningtowardsthefieldappearsasheatofmagnetization. Ifthefield isswitched offisothermally, heatisabsorbed.Ifthefieldisswitched offadiabatically theentropyofthesystemmustremainconstant; the increase inentropy duetoincreased disorder ofthedipolesisthen compensated byadecrease intheentropy associated withthermal agitation, andthereistherefore afallintemperature. Thisisthebasis ofthe'magnetic cooling'methodforobtaining temperatures below10K usingparamagnetic substances. This'magneto-caloric' effectalsohas applications toferromagneties. SincedB=0inareversible adiabatic process, wehavefromequation (21.17) dT={Bo+AM}dM. PCM AbovetheCuriepointsaturation effectsarenegligible andM/Boisa constantatagiventemperature, sothatinafinitechangeofthemag­ netization wehave (21.20) (21.21) !:i.T=2Ac!:i.(M2).PMBelowtheCuriepointwecanneglectBoincomparison withAM,and weobtain Iftheexternal fieldisinitially zero,sothatthemagnetization ofeach domainhasthespontaneous value.J4,thetemperature riseonapplying afieldis (21.22) If!:i.TisplottedasafunctionofM2,acurveoftheformshowninFig.21.9 isobtained.Itbecomes astraight lineintheregionwheretheexternal fieldislargeenoughtochangethemagnetization ofthedomains, with acurvedtailatlowerfieldswherethemagnetization ofthesubstance is mainlyduetowallmovements ortherotationofdomains. Extrapolation ofthestraightportiontotheaxis!:i.T=0givesM~fromtheintercept. Themagneto-caloric effectmaybeusedforanumberofpurposes, such asinvestigation ofthehysteresis curve,oneofthemostimportant being 640 FERROMAGNETISM [21.5 thedetermination ofthespontaneous magnetization MoneartheCurie point.Agooddescription ofexperimental technique isgivenbyOliver andSucksmith (1953)inworkona copper-nickel alloy(24%eu;76%Ni). ------.~M2 FIG.21.9.Curveshowing thevariation oft!TwithM2inthe magneto· caloriceffect. t!T=(A/2pGM)(M2_M~). 21.6.Measurement ofthespontaneous magnetization Moasa function oftemperature In§21.2itwasshownthatthespontaneous magnetization ofasingle domainshouldobeyanequation ofstatewhichdepends onlyslightlyon J(seeFig.21.2).Inordertotestthisrelationitisnecessary todetermine thevalueofMoforasingledomainatzerofieldoverawiderangeof temperature. Sinceinpractice anyspecimen consistsofanum.berof domains randomly oriented, sothat(apartfromremanence) thenet magnetization willbezero,itfollowsthatthespontaneous magnetiza­ tionofasingledomaincannotbedirectly measured. Ifweapplya sufficiently strongfield,however, thevariousdomains willrotateuntil theypointinthedirection oftheexternal field,andtheresultant magnetic moment willbeclosetothespontaneousmagnetization ofthe individual domains. Itwillslightlyexceedit,sincethemagnetization undertheseconditions corresponds, nottothepointPinFig.21.1,but tothepointP',thestablestateinthepresence ofamagnetic field.In ordertofindthevaluecorresponding toP,wemustmakemeasurements ofMforarangeofvaluesoftheexternal field,andthenextrapolate back tozerofield.Sincethefieldswhichareappliedaresmallcompared with 21.6] FERROMAGNETISM 641 theinternalfield,thepointP'isneverfarfromP,andtheextrapolation required isnotverygreatattemperatures wellbelowtheCuriepoint. NeartheCuriepointthemagneto-caloric effectisusedasdescribed in theprevious section. Inanumberofmagnetic materials thenucleusofthemagnetic ion possesses anuclearmagnetic dipolemoment mn,whichinteracts with themagnetic fieldBeoftheelectrons (see§20.10).(ThefieldBeisthe actualmagnetic fieldatthenucleusgenerated bythemagnetic electrons, andisnothingtodowiththeeffective molecular fieldBintintroduced byWeisstoexplainferromagnetism.) Theinteraction energy W=-mn.Be givesahyperfine splitting ofthenuclearlevels,fromobservation of whichBecanbefoundifthenuclearmoment isknown.Inaferro­ magnetic substance Beisparalleltothemagnetization, anditstime averagevalueisproportional totheaveragemagnetic moment oneach ion(apartfromsomesmallcorrections). ThusBeisproportional tothe magnetization, andobservation ofthehyperfine structure separation as afunction oftemperature givesaconvenient andaccurate methodof determining thesaturation magnetization curve.Thiscanbedonein zeroexternal field,sinceitisnotnecessary tolineupthedomains, and manyofthedifficulties ofdirectmeasurement ofthebulkmagnetization areavoided. Themagnitude oftheelectronic fieldBeliesgenerally between 105 and107gauss.ThenuclearlevelshaveenergyWmI=-YnfJ",mIBe, wheremIisthenuclearmagnetic quantum number, andareequally separated byanamountcorresponding toafrequency of10L1010cIs. Twomethods areavailable formeasuring thisseparation overarange oftemperature. OneoftheseistheM6ssbauer effect,inwhichalow­ energyy-rayisemittedfromanucleusinanexcitedstateandabsorbed byanucleusinthegroundstate.Ay-rayphotonofenergyhvcarries momentum hvlc,sothattheemitting orabsorbing nucleusisgivena recoilmomentum, andhencetakesupenergywhichreducesthephoton energy.Ifthenucleiareinasolidtherecoilmomentum isgenerally takenupbythesolidasawhole,andtheenergytakenfromthey-ray isnegligible. Thusinasolid,unlikeagas(seeProblem 21.3),thereis nospreadinenergyofthephotonduetothevaryingamounts ofenergy takenupbytherecoil. Onlythosey-rayswhichareextremely narrowareofuse,sincethe widthofthey-raymustbesmallerthanthehyperfine splitting. For 861110 Tt 642 FERROMAGNETISM [21.6 magnetic purposes, the14·4keYtransition between theexcitedstate (1=-!)andthegroundstate(1=!)oftheisotope57Fehasbeen especially useful.Thisgivesalinewidthofabout3Mc/s,andthe hyperfine levelsandstructure oftheM5ssbauer gamma-ray areshown inFig.21.10.Thestructure isexactlyanalogous totheZeeman effect 1=! 1=1L IHkeV' ,+1 -1 -l -1 +1~J Unsplitline Linesplitinmagnetic fieldB~. FIG.21.10.Hyperfine splitting ofthenuclearstatesof67Feinamagnetic field.The groundstate1=1hasUn=+0,18,andtheexcitedstate1=thasUno~-0·010; the allowedtransitions arethoseforwhichti.mI=0,±1.Notethegrossdisparity inscale; thehyperfine splittings areabout10-11to10-12ofthe,,-rayfrequency. Insomesub· stancesthereisalsoanelectricquadrupole interaction intheI=!state. inanatomictransition. Thesplittings areaverysmallfractionofthe y-rayfrequency, andtheiranalysis ismadebymeansoftheDoppler effectproduced byarelativemotionofthesourceandabsorber.Itis convenient touseasourcewithnohyperfine structure, suchasli7Fe (derived fromthenucleardecayof5700)instainless steel,which is non-magnetic, sincethisgivesasingleemission line,asshownonthe leftofFig.21.10.Forthistobeabsorbed byali7Fenucleusinamagnetic substance, wheresixtransitions areallowed withslightly different frequencies asontherightofFig.21.10,aDoppler shiftisneededof thecorrectvelocitytobringoneofthetransitions tothesamefrequency asthesinglelineontheleft.Thustheentirehyperfine patterncanbe scannedbysystematically changing therelativevelocity ofsourceand absorber (thevelocity required isoforderafewmm/sec). Ofcourse sourceandabsorber canbeinterchanged, andthechoiceisdetermined byexperimental convenience. Thehyperfine fieldinmetallicironhasbeendetermined asafunction 21.6] FERROMAGNETISM 643 oftemperature bymeansoftheMossbauer effect;theresults(see Fig.21.11)ofNagle,Frauenfelder, Taylor,Cochran, andMatthias (1960) showcloseagreement withthesaturation curvedetermined bycon­ ventional means.Atsufficiently lowtemperatures forthemagnetization toreachthesaturation value,thehyperfine fieldis330kilogauss. On applying anexternal fieldparalleltothemagnetization Hanna,Heberle, ARun1 eRun2 0·21·01::-------- o 0.2 0·4 0·6 0·8 1'0 TIT. FIG.21.11.Thehyperfine magnetic fieldata67Fenucleusinmetallic iron,relativeto thatatroomtemperature, plottedagainstthereduced temperature TITc'Theexperi­ mentalpointsaremeasured bytheMossbauer effect,thesolidlineindicates therelative saturation magnetization asdetermined byabulkmeasurement (seeNagleetal.,1960). Perlow,Preston, andVincent(1960)foundthatthenetfieldatthe5?Fe nucleuswasreduced, showingthatthehyperfine fieldwasintheopposite sensetotheexternal field,andhencealsotothemagnetization. Thehyperfine splitting ofthegroundnuclearlevelsinaferromagnetic substance hasalsobeenmeasured bythemethodofnuclearmagnetic resonance (see§23.5).For5?Fethisgivesadirectobservation oftransi­ tionsbetweenthestatesmI=+1and-1ofthegroundstateI=t, atafrequency suchthat hv=W-i-W H=gnfJnBe, (21.23) wheregnisthevalueforthegroundstate1=t.Thisgivesamore precisemeasurement ofBethantheMossbauer method(inwhichthe 644 FERROMAGNETISM [21.6 linewidthcannotbelessthanthatdetermined bythelifetimeofthe excitedstate),andtheresonance frequency, about46Mc/sinmetallic ironnear00K,canbefoundwithinafewkc/s.Benedek andArmstrong (1961)havemadeacarefulstudyofthepressure andtemperature dependence oftheresonance frequency iniron,andhaveshownthat theresonance frequency isnotquitelinearlyproportional tothemag­ netization athighertemperatures, butthedeparture islessthan1per centat3000c. 21.7.Foundations ofthetheoryofferromagnetism Thebriefdescription ofthechiefproperties offerromagnetic sub­ stancesgivenaboveshowsthatwepossessafairlygoodqualitative understanding ofthebasicphenomena. Thereisnodoubtthatferro­ magnetism isduetoexchange forces,butthequantitative theoryof ferromagnetism contains manydifficulties andcanbetreatedonlyby approximate methods. Wemaydistinguish between twoseparate problems: (a)thenatureofthemechanism givingrisetoexchange forces;(b)development ofmethods oftreating theproblem ofan assembly ofmagnetic particles subjecttoexchange interaction. We shalloutlinetheprincipal approaches to(b)first,andpostpone con­ sideration of(a)to§21.9. IntheoriginalHeisenberg model,themagnetic electrons areregarded aslocalized oneachatom.Thisisclearlyagoodapproximation inan ionicsolid,suchastheparamagnetic substances discussed inChapter20. Inthe3dgroup,withwhichweareprincipally concerned, thecrystal fieldinteraction effectively 'quenches' theorbitalmagnetism, leaving onlythatduetotheelectron spin.Thespinsonadjacent atomsthen interact through theexchange interaction. Onthisbasisweshould expectthesaturation moment ofaferromagnetic tocorrespond toan integralnumberofspinsperatom,andsinceagvalueof2isassociated withthespin,weshouldexpectanintegralnumberofBohrmagnetons peratom.Reference toTable21.1showsthatthisisbynomeansthe case.Nickelhasasaturation moment corresponding to0·6magnetons, iron2,22,andcobalt1·72magnetons. Thesesubstances are,ofcourse, metals,wherethesuccessofthebandmodelfortheconduction electrons suggeststhatitshouldbeusedasthebasisofatheoryofferromagnetism. The'collective electron' modelhasbeeninvestigated principally by Bloch,Slater,Stoner,andWohlfarth. Asinthetheoryofmetallic conduction, theelectrons obeytheFermi-Dirac statisties, andthe allowedenergiesfallintobands.Theexchange interaction isintroduced 21.7] FERROMAGNETISM: 645 asaninternalfieldAM,proportional tothemagnetization, asintheWeiss treatment. Thisgivesadifference inenergybetween spindipolespointing parallelandanti-parallel totheinternal field,whichwemayrepresent bydividing theenergybandintotwohalvesasinFig.18.13,butwith theimportant difference thattheeffective fieldisnowtheinternalfield Bintandnottheexternal field.Hencetheenergyseparation ofthetwo halvesofthebandis2fJBint=2fJ(AM)=2fJ(>"2xofJ)=4xo>"fJ2,where Xo isthenumberofelectrons transferred fromonehalf-band totheother, givinganexcessof2xointhe'parallel' orientation andanetmagnetiza­ tionof2xofJ.Thustheenergyseparation isitselfproportional tothe numberofelectrons transferred. Reference to§18.7showsthatthe extrakineticenergyrequiredbythexthelectrontotransferittoavacant levelisapproximately 2xw=4X{g(W)F}-1, where{g(W)F}isthenumber oflevelsperunitoftranslational energyatthetopoftheFermidistribu­ tion.Hencethetotalkineticenergyrequired totransfer Xoelectrons is z.I4X{g(W)F}-1 dx=2x~{g(W)F}-1. Thechangeinmagnetic energyis o -lMBint=-lAM2=-1>"(2xofJ)2=-2xpfJ2. Hencethenetchange inenergyofthesystemis 2X2{_>.p+_l_}, o g(W)F andthiswillbenegative provided that >"fJ2g(W)F> 1. Ifthechangeinenergyisnegative, itfollowsthatthemagnetized state isoneoflowerenergyandistherefore thestablestate;iftheenergy changeispositivetheunmagnetized statewillbestableandtherewill benospontaneous ferromagnetism. Itturnsoutthatthevaluesof>.. aresuchthatferromagnetism ispossible forbandswhichhaveasmall energywidth,andhencealargevalueofg(W)F' Inasimplecase,suchassodium, wemayusetherelation g(W)F=3n/2Jlj;. givenbyequation (4.13).Thenwehave 3>"fJ2n>1 (21.24) 2WF asthecondition forferromagnetism. Forsodium,thisrequires avalue ofAaboutahundred timeslargerthanthatobserved iniron.Inthe 3dgrouptheoverlapping 3dand48bandsproduce amuchhighervalue ofg(W)Fthanthatforsodium, andtheeffectoftheoverlapping is 646 FERROMAGNETISM (21.7 enhanced bythefactthatn,thenumberoffreeelectrons perunit volume, isalsolarger.Itturnsoutthatg(W)Fisgreatest whenthe3d bandisalmostfilled,asituation reachedbyiron,cobalt,andnickel,so thatthecondition forferromagnetism issatisfied. ThenthevalueofXo willriseuntilanyfurtherincrease wouldraisethetotalenergyinstead (a)iEnergy + FIG.21.12.Schematic diagram ofenergybandswithexchange interaction. Inboth(a)and(b)equalnumbers ofelectrons havebeentransferred fromtheanti· parallel(-)orientation totheparallel(+)orientation. Asaresultthe(+)bandis loweredinenergyrelativetothe(- )bandineachcasebythesameamount, determined bythesizeoftheinternal fieldsetup.Incase(a),withawideenergyband,thetop oftheFermidistribution inthe(+)bandcomesabovethatinthe(-)band,showing thatthisdisplacement hasahigherenergythanifthenumbers inthetwobandswere equal.Incase(b),withanarrowenergyband,thetopofthedistribution ishigherin the(-)band,andmoreelectrons willtransfer tothe(+)band,incroasing thenet magnetization stillfurther. Hence(b)givesspontaneous magnetization while(a)doesnot. oflowering it.Suchanequilibrium stateispossiblebecausetheeffective valueofg(W)Fchangeswhenwetransferanappreciable fractionofthe totalnumberofelectrons fromoneorientation totheother,asonehalf­ bandisbeingemptied andtheotherfilled(compare Fig.21.12).Itis obviousthatthereisnoreasontoexpectthatthesaturation magnetic moment, determined bytheposition ofthisequilibrium, shouldcorre­ spondtoanintegralnumberofelectron spinsperatom. Ferromagnetic substances haveabnormally largeelectronic specific heats,aswouldbeexpected ontheenergybandpicturefromthelarge 21.7] FERROMAGNETISM 647 valuesofg(W)Frequired forferromagnetism (see§18.4).The'col­ lectiveelectron model',asitiscalled,alsoaccounts forthedifference between theferromagnetic andparamagnetic Curiepoints,andforthe considerable difference intheeffective Bohrmagneton numberderived fromtheslopeoftheCurie-Weiss lawintheparamagnetic regionfrom thevaluegivenbythesaturation magnetic momentintheferromagnetic region. The'collective electron' modelofferromagnetism hasbeencriticized byVanVleck(1953)onthegroundsthatitneglects theelectrostatic repulsion oftheelectrons, whichformspartofthe'correlation energy' mentioned attheendof§18.2.Inasimplemetaltheenergybandsare ratherwide,andthecorrelation energyhaslittleeffectontheordinary conduction properties. However, ferromagnetism canoccuronlyifthe bandsarerathernarrow; thisrequiresd-(orj-)electrons, whichare moretightlyboundandforwhichtheenergyneededtochangethestate ofionization ishigher.Thusstatesofexcessive ionization, whichare allowedundueweightonthefreeelectron theory,areveryimprobable, andfluctuations inthechargedensityoneachionarerelatively small. Inthisrespectalocalized electron modelmayformabetterstarting point,andVanVleckhasputforward ageneralization ofthismodelin whichthereisonaverageanon-integral numberofspinsperatom,the spinsbeingcontinually redistributed between different sites.For example, thelowmomentofnickelcanbeunderstood ifweassumethat 40percentofthenickelionsareinthenon-magnetic 3dlOstate,with closedshells,whiletheremaining 60percentarein3d9states,each contributing oneBohrmagneton tothetotalmoment. Sinceallnickel atomsareidentical thereisnothingtodetermine whichatomsshould beinthe3dlOstateandwhichin3d9;anygivenatomfluctuates rapidly fromonetotheother,sothatonaverageitisina3dlOstatefor40per centofthetime,and3d9for60percent.Infactneutron diffraction measurements yieldascattering patterncorresponding toeachatom carrying anidentical moment, sothatfluctuations between statesof different ionization mustaverageoutinaveryshorttime.VanVleck hasrefinedthismodeltoincludestatesofhigherionization, typical valuesbeing53%dlO,35%d9,10%dS,11%d7,0·1%d6,givingan averagevalueof3d9•4•Thisgivesa'minimum polarity' fortheions,in contrast withtheexcessive polarity allowedinthefreeelectron model. Theproblem oftheco-operative magnetic stateinaconducting solid withitinerant electrons isextremely difficulttohandlemathematically, andexisting methods startfromover-simplified modelswhichare 648 FERROMAGNETISM [21.7 progressively refined. Ideallyanytheoryshouldincludenotonlythe exchange interaction, butalsotheligandfieldinteraction (§§20.6-9), which is primarily responsible for'spinonly'magnetism intheiron group,andthespin-orbit interaction, whichmakestheg,g'values discussed in§21.4different fromthefreespinvalueof2.Considerably moreprogress hasbeenmadeonthemagnetic properties ofelectrically insulating materials, wherethetheoretical problems aresomewhat easier,andmagnetic resonance experiments providemuchmoredetailed information; suchcompounds aremostlyanti-ferromagnetic orferri­ magnetic, whoseproperties arediscussed inChapter 22.Thereis, however, onetheoretical technique, originally developed forlocalized electrons, butwhichhassincebeenshowntobevalidforitinerant electrons; initthecollective excitations oftheassembly ofmagnetic carriers, knownas'spinwaves',arehandledbymeansofexpansions validatlowtemperatures intheco-operative state.Abriefoutlineof thismethod, firstformulated byBloch(1932),isgiveninthefollowing section. 21.8.Spinwaves At0°K,wherethemagnetization hasthesaturation value,allthe spinsarerigorously parallel,butthisisobviously notsoatanon-zero temperature wherethemagnetization issmaller. However, itwould beincorrect toregardthereduction inmagnetization asduetothe reversalofanygivenindividual spin,becauseanysuchdeviation would bepassedontoneighbouring spinsthrough theexchange interaction inatimeoforder{/'(Ii),sothatitwouldnotremainlocalized onany givenatom.Infacttheaverage deviation ofeachspinfromexact parallelism issmall,andcanbeanalysed intermsofsinusoidal spatial variations throughout thecrystal,knownas'spinwaves'.Inaspin waveofwavevectorks'theanglebetween adjacent spinsadistance ro apartisks.r0'andsincetheexchange energyvarieswiththecosineof thisangle,theextraenergyrequired toexciteawaveisproportional to/'(l-cos ks.ro)=IJ'{ks.ro)2=If'k;r~ COS2()k,ro'where ()k,roisthe anglebetween ksandro.Thismustbesummed overallneighbours, giving ~=liwk=Dks2• (21.25) Thisisthedispersion relationconnecting thefrequency andwavelength ofaspinwave.Disproportional totheexchange energyf',andin acubiclatticewithzequidistant neighbours withangularmomentum J. D=t;"zJr~. (21.26) 21.8] FERROMAGNETISM 649 Theenergyrequired toexciteaspinwaveisproportional tok2,and atlowtemperatures onlylongwaveswillbeexcited; inthelimitat 0°Ktheonlywavepresentwillbek=0,whichcorresponds toallthe dipolesbeingparallel. Asthetemperature risesmorespinwavesof shorterwavelengths willbeexcited, andtheenergyrequired forthis showsupasanadditional terminthespecificheat.Thisisquitedistinct fromtheabnormal electronic specificheatdiscussed above,andwould bepresentinaferromagnetic insulator.Itispurelymagnetic inorigin, andconstitutes thelow-temperature tailofthemagnetic specificheat anomaly discussed in§21.5.VanKranendonk andVanVleck(1958) haveshownthataspinwavebehavesformally likeaharmonic oscillator, itsmeanenergybeing - liwk _ JJie= =nliwk' (21.27)exp(liwk/kT)-1 whereii={exp(liwk/kT)-I}-l isknownasthe'occupation number'. Thenumberofspinwavesofwavevectorks(wehaveusedkstoavoid confusion withBoltzmann's constant kwhichoccursinequation (21.27) isg(ks)dks=(V/271"2)k:dks'ascanbesoonfromequation (4.8)bysubsti­ tutingks=(271"/>").Hencetheinternal energyattemperature Tis U=fii(liwk)g(k s)dks co VJ(Dk~)k~dks =271"2 exp(Dk~/kT)-1 oco VD(k~~Jx4dx =271"2D)exp(x2)-I' o wherewehavemadethesubstitution x2=(Dk~/kT). Theintegralcan betakentoinfinityatlowtemperatures, andistherefore justanumerical constant. Differentiation withrespecttotemperature gives Ov=dU/dT=c(kT/D)!, (21.28) wherecisanumerical constant whosevaluedepends somewhat onthe crystalstructure; forasimplecubiclattice(z=6)wehave(permole) Ov=0·1l3kV(kT/D)! =0·1l3k(V/r3HkT/2/'J)! =0·1l3R(kT/2/'J)!. (21.29) Fortheordinary ferromagnetic metalsthismagnetic specificheatis smallanddifficulttomeasure inthepresence oftheabnormally large electronic specificheat,andthelatticespecificheat.Ithasbeendetected insomenon-conducting ferrimagnetic compounds (soo§22.3). 650 FERROMAGNETISM [21.8 Thedeparture ofthemagnetization fromtheabsolute saturation value~canbecomputed similarly, usingthefactthateachspinwave reducesthemagnetic moment byanamount gf3ii.Thus ~-Mo =gf32ii=(gf3Vf27T2 )JexP(~i~~ZT)-1 '" =(gf3Vf27T2)(kTfD)iJX(2~~1.expx- Onsubstituting forDandusingthefactthatgf3(VfrX) =Ms'inasimple cubiclattice,oneobtainstherelation Mof~=1-a(kTf2f'J)I, (21.30) (21.31)wherea=0·059fJinasimplecubiclattice.Thisresultisthefirstterm ofapowerserieswherethenexttermsareinT!,Tf,andT4.Theterms inT!andTihavebeenverifiedinaspecialcasewheretheyareunusually large(seeGossard, .Taccarino, andRemeika, 1961),usingnuclearmag­ neticresonance ofthe53CrnucleusinCrBr3'wherethenuclearresonance frequency isaccurately proportional tothemagnetization. Ingadoli­ niummetal,wherethespontaneous magnetization hasbeenmeasured byElliott,Legvold, andSpedding (1953)bythebulkmagnetization method, theT!lawholdscloselyalmostuptotheCuriepoint,aresult thathasbeenexplained asduetoanearcancellation ofthehigherterms (Goodings, 1962).Theseexperimental resultsforthevariation ofthe magnetization withtemperature confirmthevalidityofthespinwave method; incontrast, thecollective electron modelpredicts avariation oftheform whilethemolecular fieldmodelgivesanexponential term(seeProblem 21.1).Directexperimental confirmation oftheexistence ofspinwaves isobtained frommagnetic resonance experiments inthinferromagnetic films(see§23.7). 21.9.Mechanisms ofexchan~e interaction In§21.1itwasstatedthatferromagnetism isduetoexchange inter· action;certainly weknowofnootherinteraction ofthecorrectform whichislargeenoughinsizetoproduce ferromagnetism. Thoughitis generally agreedthatferromagnetism isduetoexchange interaction, attempts tomakenumerical calculations ofitssizehaveprovedvery difficult; morethanonemechanism ofexchange interaction hasbeen proposed, eachofwhichnodoubtplaysarole,buttheabsenceofreliable 21.9] FERROMAGNETISM 651 quantitative information makesanassessment oftheirrelativeimpor­ tancestillratherspeculative. Theoriginaltreatment ofHeisenberg andDiracdealswiththeinter­ actionbetween twoelectrons onthesameatom.Ifthetwoelectrons didnotinfluence oneanotherthesolution ofthewaveequation would beasimpleproductofthetwosolutions forasingleelectron, oftheform .1._..1.(1)..1.(2)'t'1-'t'k't'm' Thephysical interpretation ofthisisthatelectron (1)isinorbitalk, andelectron (2)isinorbitalm.Sincethetwoelectrons areequivalent, theenergyisunchanged ifthetwoelectrons areinterchanged, giving anothersolution .1._..I.(2)..1.(1)'t'll-'t'k't'm' Ingeneral,anylinearcombination ofthesetwosolutions isallowed,the correctcombination beingdetermined whenweincludetheelectrostatic energye2jr12(wherer12isthedistance between thetwoelectrons) of repulsion between thetwonegatively charged electrons. Thecorrect solutions arethenthesymmetric andanti-symmetric combinations !/JSym=(2)-1(!/Jr+!/Jll)' !/Jant=(2)-1(!/Jr-!/Jn)' Thesetwosolutions nolongerhavethesameenergy,becausethesym­ metrical solution allowsthewavefunction tohavealargeamplitude ifthetwoelectrons areatthesamepoint,whiletheanti-symmetric wavefunction thenvanishes because !/Jr=!/In.Thustheelectrostatic repulsive energybetween thetwoelectrons islargerinthefirstcase thaninthesecond. Sofarithasnotbeennecessary toincludetheelectron spin.With twoelectrons, thespinstates(twoforeachelectron, hencefourinall) aredividedbetween thetripletstates(8=1,Ms=1,0,-1)andthe singletstate8=0,according towhether thetwoindividual spinsare paralleloranti-parallel. Thetripletstatesaresymmetrical withrespect tointerchange ofthetwoelectrons, thesingletstateisanti-symmetrical. Sinceonlystateswhoseoverallsymmetry isanti-symmetrical are allowedinnature,thespintripletstatescanonlybecombined withthe orbital!/Jant'andthesingletspinstatewiththeorbital !/JSym.Thusthe difference inelectrostatic energybetween thesymmetrical andanti­ symmetrical orbital st~tescarrieswithitacorresponding energydiffer­ encebetweenthespinsingletandtriplet.Thisisformally similartothe introduction ofacoupling energybetweentheelectron spinsofthetype assumed in§21.1. 652 FERROMAGNETISM [21.9 Fortwoelectrons withinthesameatom,theexchange energyis alwayspositive, sothatthestateoflowerenergyiswiththespins parallel. Thiscoupling is'ferromagnetic' innature,andisthejustifica­ tionforHund'srule(§20.2)whichmakesthegroundstateoftheatom theonewithmaximum multiplicity inthespin.Inthephenomenon offerromagnetism proper,however, weareconcerned withexchange interaction between electrons ondifferent atoms,andtheelectrostatic energyinvolved contains termsarisingbothfromtherepulsive forces between thetwonucleiandbetween thetwoelectrons andfromthe attractive forcesbetween anelectron ononeatomandthenucleusof theotheratom.(Themuchlargerenergyofattraction between an electronandthenucleusofitsownatomhasalreadybeenincluded in thewaveequation foreachelectron.) Insimplemolecules likeH2the overallexchange termiseasilyshowntobenegative, inagreement with thegroundstateofthemolecule beingasinglet,butwithmorecomplex ionssuchasthe3dgroup,opinions havedifferedwhether theneteffect (whichobviously varieswithinteratomic distance) wouldbepositiveor negative attheionicseparations typicalof3dgroupmetals. Since improved wavefunctions havebecomeavailable fromelectronic com­ puters,attempts havebeenmadetocarryoutcalculations whichmight bereasonably realistic. StuartandMarshall (1960)obtained apositive energy,thoughtwoordersofmagnitude toosmall,butFreeman, Nesbet, andWatson(1962)findanegative energy.Thus'directexchange', due todirectoverlapoftheelectronic wavefunctions, appearsincapable of accounting forferromagnetism. Anegative exchange energywouldgive riseto'anti-ferromagnetism' (seeChapter 22),whereneighbouring spins arearranged anti-parallel ratherthanparallel. Inconducting solidsanalternative mechanism hasbeenproposed, whichinvolves exchange interaction between theferromagnetic 3d electrons (largelylocalized) andtheitinerant conduction electrons. A conduction electron isspin-polarized byexchange interaction withone ion,thenmovesawaytointeractwithanotherion,carrying thememory ofitspolarization withit.Thisgivesrisetoanindirect exchange interaction between thetwoions,involving theexchange interaction fttmbetween theconduction electrons (or's'electrons) andtheferro­ magnetic ('m')electrons, andtheenergyJJi,.attheFermi ~mrface. The resultant interaction between theionsisproportional to(fsm)2/~, sincetheexchange interactionf ttmisinvolved twice,andthepolarization oftheconduction electrons isinversely proportional totheFermienergy (see§18.7).Sincethewavefunction ofaconduction electron spreads 21.9] FERROMAGNETISM 653 throughout themetal,atfirstsightthisexchange interaction appears tobeindependent ofdistance, andnecessarily ferromagnetic since "t:mIW Fmustbepositive. However, apropercalculation showsthat itcanbeofeithersignanddoesfalloffwithdistance (thoughmuchless rapidlythandirectexchange), beinganoscillatory function ofx=2kFR, wherekFisthewavevectorofaconduction electron attheFermi surfaceandRtheinter-ionic distance. Inthismechanism theconduction electrons serveasamedium through whichaninteraction istransmitted between spinswhichare localized. Infactitwasfirstproposed fornuclearspins,thehyperfine energyofinteraction betweenthenuclearspinandaconduction electron appearing insteadoftheexchange interaction"t 8mabove;theoscillatory natureoftheinteraction wasdeduced byRudermann andKittel(1954). Theexchange mechanism forelectrons wasproposed byYosida(1957) andKasuya (1956);itisgenerally thoughttoprovidethemechanism forferromagnetism inthelanthanide metals(see§22.4),wherethe magnetic 4/electrons arequitelocalized. Howlargearoleitplaysin the3dmetalsisuncertain. Exchange interaction ininsulators Inelectrical insulators noconduction electrons existtoprovidean exchange mechanism ofthetypejustconsidered, butexchange inter­ actionsofconsiderable magnitude arefoundincompounds wherethe interionic distance issolargethatdirectexchange between electrons localized onthemagnetic ionsmustbenegligible. Asanexample we consider asimplecompound suchasanoxideMO,whereMisadipositive ionofthe3dgroup(e.g.Mn2+,Fe2+,C02+,NiH).Thesehaveaface­ centredcubicstructure similartothatofNaCl,atypicalplaneofatoms beingshowninFig.21.13.Herethecirclesaredrawninproportion to theionicradii,anditcanbeseenthatthemuchlargeranionsseparate almostcompletely eventhenearest neighbour cations. However, neutron diffraction resultssuggestthatthestronger exchange inter­ actionsarebetween ionsonnextnearestneighbour sitessuchasA,B ratherthanbetween nearestneighbour sitessuchasA,C.Direct overlapbetweenthewavefunctions ofmagnetic electrons onthecation sitesisverysmall,butapurelyionicmodelwherethemagnetic electrons arelocalized onthecationsisanover-simplification. Somedegreeof covalent bindingisalwayspresent, whereby thewavefunctions ofthe magnetic electrons spreadoverontotheadjacent anions.Directoverlap ofthemagnetic electrons fromneighbouring cationscantherefore take 654 FERROMAGNETISM [21.9 placeontheintervening anions.Ofcoursethedegreeofoverlapdepends ontheamountofcovalent bonding, andsoalsodoesthesizeofthis 'indirect' exchange interaction; itisverymuchsmallerforionsofthe 4/group,whichtakealmostnopartincovalent bonding, thanforions ofthe3dgroup(forexample, thetransition temperature ofthelanthanide oxidesisbelow100K,whilethoseofthe3dgroupoxidesareover1000K). FIG.21.13.AplaneofatomsinanoxideMO,whereM++isa dipositive ionofthe3dgroup.Theionsaredrawnapproxi­ matelytosize,showing howthesmallcations(M++)arcwell separated bythelargeanio(0-). Although thissolvestheproblem ofinteraction between ionsat relatively largedistances, therearetwodifficulties. Thefirstisthatthe potential energyduetotheelectrostatic repulsion betweenthemagnetic electrons intheoverlapregionleadstoaferromagnetic interaction (as forelectrons withinthesameatom,whereitresultsinHund'srules-see §20.2),whereasthevastmajority ofinsulating magnetic compounds have ananti-ferromagnetic interaction. Second,theinteracting magnetic electrons formapartly-filled band,whichaccording totheclassification setoutin§18.3shouldmakethesubstance anelectrical conductor. However, theenergybandisnarrow,andthefallinkineticenergyof anelectron (cf.Fig.18.2)ontransferring fromalocalized state(which corresponds inenergytothecentreoftheband)toaconduction state atthebottomofthebandisonly::::::leV.Ontheotherhand,because oftheelectrostatic repulsion between theelectrons, theirpotential energyisleastwhentheyareuniformly distributed, givingeachion thesamenumberofelectrons. Thispotential energyispartofthe 'correlation energy'mentioned attheendof§18.2.Fors-electrons, and 21.9] FERROMAGNETISM 655 toasmallerextentforp-electrons, thecorrelation energyissmall because theirwavefunctions areextended andthechargedensity withintheatomissmall;however, theextended wavefunctions give alargeroverlapandgreaterbandwidth inthesolid(cf.Fig.18.7).Thus suchelectrons havealowerenergyoverallwhentheyarenon-localized, andbecome conduction electrons. Ford-electrons thebandwidth is smallerandthecorrelation energygreater, makingthelattercorre­ spondingly moreimportant; the'minimum polarity' modelofVanVleck discussed in§21.7fornickelisanattempttoallowforthis.Incom­ poundsthedisparity isevengreater;thed-electron bandwidthisnot morethanaboutIeV,whileabout10eVisrequired totransferan electronfromoneiontoanother(i.e.tocreateapairofionsindn-1,dn+1 statesfromapairbothindnstates).Thusitisenergetically favourable fortheelectrons toremainlocalized andthesubstance isanelectric insulator. Asaresultoftheseconflicting energyconsiderations, thelocalization isnot,however, absolutely complete.Ifbisthereduction inkinetic energywhichwouldresultfrommovingfromsitetosite,whileUisthe potential energyrequired toovercome theelectrostatic repulsion, the equilibrium stateisonewherethechanceofsuchamovement isoforder btU,andthenetreduction inkineticenergyisoforderb2/U.Through theexclusion principle thispossibility ofmovement toadjacent sitesis restricted nearlyalwaystoelectrons withanti-parallel spin,whichcan therefore acquirealowerenergythanthosewithparallelspin.Thisis equivalent toananti-ferromagnetic exchange energyoforderb2/U, whichis oforder10-2to10-1eV(afewhundred OK). Thefirstexplanation ofhowexchange interaction couldarisebetween ionsattheratherlargeinter-ionic distances foundincompounds was putforwardbyKramers (1934),andanumberofsubsequent attempts weremadetoarriveatmoreexplicitinterpretations. Thetheoryout­ linedaboveisduetoAnderson (1963),andthoughdifficulttoexplain insimpleterms,appearstobethemostsatisfactory initsgeneral approach. REFERENCES ANDERSON, P.W.,1963,Advanc. SolidStatePhys.14,99. BARNETT, S.J.,1944,Phys.Rev.66,224. BENEDEK, G.B.,andARMSTRONG, J.,1961,J.Appl.Phys.32,106s. BLOCH, F.,1932,Z.Phys.74,295. ELLIOTT, J.F.,LEGVOLD, S.,andSPEDDING, F.R.,1953,Phys.Rev.91,28. FREEMAN, A.J.,NESBET, R.K.,andWATSON, R.E.,1962,ibid.125,1978. GOODINGS, D.A.,1962,ibid.127,1532. 656 FERROMAGNETISM GOSSARD, A.C.,JACCARINO, V.,andREMEIKA, J.P.,1961,Phys.Rev.Letters, 7,122. GREW,K.E.,1934,Proc.Roy.Soc.A,145,509. lIANNA, S.S.,HEBERLE, J.,PERLOW, G.J.,PRESTON, R.S.,andVI~CENT, D.H., 1960,Phys.Rev.Letters,4,513. KASUYA, T.,1956,Prog.Theoret. Phys.16,45. KRAMERS, H.A.,1934,Physica, 1,182. MEYER, A.J.P.,andASCH,G.,1961,J.Appl.Phys.32,330. NAGLE,D.E.,FRAUENFELDER, R.D.,TAYLOR, R.D.,COCHRA~, D.R.F.,and MATTHIAS, B.T.,1960,Phys.Rev.Letters,5,364. OLIVER, D.J.,andSUCKSMITH, W.,1953,Proc.Roy.Soc.A,219,1. RUDERMANN, M.A.,andKITTEL, C.,1954,Phys.Rev.96,99. RUSHBROOKE, G.S.,andWOOD,P.J.,1958,Molec.Phys.1,257. SCOTT,G.G.,1951,Phys.Rev.82,542. STUART, R.,andMARSHALL, W.,1960,ibid.120,353. VANKRANENDONK, J.,andVANVLECK,J.H.,1958,Rev.Mod.Phys.30,1. VANVLECK,J.H.,1953,ibid.25,220. YOSIDA, K.,1957,Phys.Rev.106,893. PROBLEMS 21.1.Showthatforasubstance consisting ofatomsorionsinthe2S!state,the Brillouin function becomes M/Ms=tanhy, wherey=(JB/kT. Showthatforsuchaferromagnetic substance atlowtemperatures, whereyis large,theWeissinternal fieldtreatment of§21.2leadstotheformula Mo/Ms=1-2exp(-2AM oMs/nkT) forthespontaneous magnetization Moinzerofield.Notethatthisdoesnotlead toasimplepowerlawsuchasinequations (21.30)or(21.31). 21.2.UsingtheresultofProblem 6.11,showthatthemagnetostatic energyof asmallspherical particle ofnickel,ofradiusbandmagnetized tosaturation (Ms=5'1105ampere/metre =510e.m.u./cm3),isapproximately 2X105b3joules (binmetres). Fromtheresultsof§21.3,theenergyrequired toformaBlochwallincreases withb2(fornickelthewallenergyisaboutanerg/cm2).Henceshowthatfor particles whoseradiusislessthanabout10-8metres,thereduction inmagneto­ staticenergyobtained bydivision intotwodomains islessthantheenergyrequired toformawall. 21.3.ThenucleusofanatomofmassMmovingwithvelocity vemitsay-rayof energyhvintheforward direction. Showbyconsidering thechangeinmomentum andenergyoftheatomthatthey-rayenergyisincreased byafraction (v/c), provided thatthey-rayenergyissmallcompared withtherestmassoftheatom [hv/Mc2~1].Notethatthisisthesameastheclassical Doppler shift. 21.4.Show,fromthevectormodeldiagram ofFig.20.6,thattheratiotoJofthe projection ofSonJis {S(S+l)}!cosAOB/{J(J +I)}!=g-1. 22 ANTI-FERROMAGNETISM AND FERRIMAGNETISM 22.1. Anti~ferroma~netism INaparamagnetic substance thedipolesarefreetoorientthemselves atrandom, andthereisacorrespondingly highentropy; ifthereare 2J+1levelshavingthesameenergyinthegroundstate,theentropy isRln(2J+1).Thesubstance wouldobeyCurie'slawdownto0°K ifthegroundstatecontains twoormorelevelswiththesameenergy intheabsenceofanexternal field,butthiswouldbeaviolation ofthe thirdlawofthermodynamics, bywhichtheentropyinasubstance in thermodynamic equilibrium mustbezeroat0°K.Inpracticethereis alwayssomemutualinteraction between thedipoles(eitherthrough exchange ormagnetic dipolarinteraction), suchthattheinternalenergy Uofthesystemislowerwhenthedipolesareoriented inanorderly arraythanwhentheyarerandomly oriented. Thus,attheabsolute zero,wherethefreeenergyF=U-TSisequaltoU,theequilibrium stateoflowestfreeenergywillbetheorderedstatewiththelowest internalenergy.Atasufficiently hightemperature, ontheotherhand, theparamagnetic state,withitshigherentropy corresponding tothe random orientation ofthedipoles,willhavethelowerfreeenergy becauseofthesecondterminF=U-TS,andwillthusbetheequili­ briumstate.Asthetemperature falls,anysubstance wherethedipoles stillhavesomefreedomoforientation (thisexcludes thoseparamagnetic substances whichhaveasingletgroundstateandatemperature­ independent susceptibility) willmakeatransition fromadisordered phaseintoanorderedphase.Theferromagnetic statediscussed inthe lastchapter,inwhichallthedipolesareoriented paralleltooneanother at0°K,isthestateoflowestenergywhentheexchange energyJ'has apositive sign.However, ferromagnetism isexhibited byrelatively fewsubstances, thoughtherearemanycontaining transition groupions. Itwassuggested byNeel(1936)thatinmanysubstances theexchange interaction islargebutnegative, resulting inanorderedstatewhere neighbouring dipolesarealignedinananti-parallel arrangement. Such anarrangement forasimplecubiclatticeisshowninFig.22.1;the dipolesatadjacent cornersofeachcubiccellpointinopposite directions. 851110 uu 658 ANTI-FERROMAGNETISM ANDFERRIMAGNETISM [22.1 Another simplecaseisthebody-centred cubiclattice,withanionat thecentreofeachcubeaswellasatthecorners; herealltheionsatthe cornershavetheirdipolesparalleltoeachother,butanti-parallel tothe ionsatthecentres. Ineachcaseagivendipoleissurrounded bya numberofequidistant dipolesallpointing intheopposite direction, whilethenextnearestneighbours pointinthesamedirection again. R~---....fI"~---+-""'" Q~----\-a::-----I-Y / / / /P/ FIG.22.1.Anti-ferromagnetic arrangement ofdipolesina .simplecubiclattice. Thesystemmaybethoughtofasconsisting oftwointerlocking sub­ lattices, oneofwhich is spontaneously magnetized inonedirection, whiletheotherisspontaneously magnetized intheopposite direction. Asinferromagnetism, thisspontanoous magnetization ofthesub-lattices setsinonlybelowacertaintransition temperature, generally knownas the'Neeltemperature'. Abovethistemperature thedipolesareran­ domlyoriented, andthesubstance isparamagnetic, obeying aCurie­ WeisslawwiththeWeissconstant ofopposite signtothatfoundin ferromagnetism, asweshouldexpectfromthereversed signofthe exchange energy.Theonsetofspontaneous magnetization inthesub­ latticesasthesubstance iscooledthrough theNeeltemperature is accompanied byaspecificheatanomaly oftheco-operative type,as illustrated inFig.22.2.Thesubstance asawholeexhibits nospon­ taneousmagnetization inzerofield,sincethetwosub-lattices areequally andoppositely polarized. Whenanexternal fieldisapplied, asmall magnetization occursgivingapositive susceptibility; thegeneral behaviour ofthesusceptibility canbeexplained quitewellonamolecular fieldmodel,asoutlined below. 22.2] ANTI-FERROMAGNETISM ANDFERRIMAGNETISM 659 22.2.Themolecular field-two sub-lattice model Letthetwosub-lattices bedenotedbyAandB.Thenadipolein latticeAissubjecttoanexternal fieldBoandaninternalfieldpropor­ tionaltothemagnetization ofsub-lattice Bwhichwemaywriteas 5 FIG.22.2.SpecificheatofNiCla,6HaOatlowtemperatures, showing the.\-typeanomaly attheNeeltemperature, 5·34°K(afterRobinson andFriedberg, 1960).Theentropy intheanomaly isRIn3,corresponding tothethreefold degeneracy oftheS=Iground stateoftheNi++ion.Theriseathightemperatures isduetothelatticespecificheat. Notethedifferent shapeoftheco-operative anomaly fromthatduetoasimplelevel splitting inanother nickelsalt(Fig.20.15). -AMB,wheretheminussignappears becauseofthereversed signof theexchange integral. Theeffective fieldactingonadipoleinAis therefore B-B_'11..- ~- 0IUY.LB}(22.1)Similarly BB=Bo-A.1lfA• Athightemperatures wherethedipolesarerandomly oriented the magnetization ofeachsub-lattice shouldobeyCurie'slawifwetake theeffective fieldinsteadoftheexternal field.Thuswehave A{".=iOBA!l-'o T, (22.2) where°istheCurieconstant perunitvolume,andthefactoriappears because onlyhalfofthedipolesareinagivensub-lattice. Thetotal 660 ANTI·FERROMAGNETISM ANDFERRIMAGNETISM [22.2 (22.3) Hencemagnetization istheno 0M=1l44+MB=---p{2Bo-,\(1l44+MB)} =--p(Bo--t'\M). 2~o ~o ~OM 0 0 X=Bo=T+('\Of2~O) =T+O' Thisequation forthesusceptibility abovetheNeelpointissimilarto thatfoundinferromagnetism exceptforthereversed signoftheWeiss constant O. D FIG.22.3.Graphical solution ofequations forspontaneous magneti. zationofananti·ferromagnetic substance. CurveOBOAD istheBrillouin function q,(y),BOAisthestraight line relation between Mandywhentheexternal fieldiszero,andB'OA' isasimilarlineforthecasewhenanexternal fieldisappliedparallel tothedirection ofthespontaneous magnetization. Inordertoinvestigate thebehaviour atlowtemperatures wecannot assumeCurie'slawtohold,sincealargespontaneous magnetization willbesetupineachsub-lattice bytheinternal field.Wemustuse, insteadofequation (22.2),modified formsofequation (21.8): 1144 lngf3J~(YA)},(22.4) MB=lngf3J~(YB) wherey=(gf3JfkT) X(effective fieldonsub-lattice), and ~(y)isthe Brillouin function givenbyequation (20.15).Thefactor1appearsagain becauseonlyhalfofthedipolesareineitherlattice;wehavealsoassumed thatthemagnetism isduetodipoleswithangularmomentum J. Intheabsenceofanexternalfieldthemagnetization ofeachsub-lattice isgivenbythesolutionofequation (22.4)with BA=-AMB=-'\(-Mo)=+AMo, BB=-~=-,\(+Mo)=-AMo• (gfJJ\Mo=!ngfJJef>kT>.Mol"22.2J ANTI.FERROMAGNETISM ANDFERRIMAGNETISM 661 sincethetwosub-lattices AandBwillhaveequalandopposite magneti­ zation,+Moand -Morespectively. Theequation maybesolvedgraphi­ cally,asinferromagnetism. Thespontaneous magnetization isgivenby thepointsAandBinFig.22.3,whichcorrespond tothestablecondition, whiletheotherpossiblesolution MA=MB=0isunstable. Thevalue ofMoatanytemperature isgivenbytherootofthetranscendental equation Asthetemperature risesthelineBOAbecomes steeper,andthepoints A,Bmovebacktowards theorigin;thespontaneous magnetization TABLE 22.1 Neel temperature Substance TN(OK) 8/TNXO/XTN Cr 311 - - alpha-Mn 100 -- MnF. 67 1·2 0·76 FeF. 78 1-5 0·72 CoF. 50 - - NiF. 73 -- MnO 116 4to50·69 FeO 198 3 0·75 CuCl.,2H.O 4·3-- NiCl.,6H.O 5·3-- disappears attheNeeltemperature TNwherethelineABistangential totheBrillouin function attheorigin.Sinceef>(y) =y(J+1)/3Jfor smallvaluesofy,wehave Mo=IngfJJ[(~j(J3~1)>.MoJ. or TN=l>..ng2fJ2J(J +1)/3k=>..O/2p,o=8. Thusonthissimpletheory,duetoVanVleck(1941),theNeelpointTN shouldhavethesamevalueastheWeissconstant 8.Thevaluesofboth 8andTNaregiveninTable22.1foranumberofsubstances nowestab­ lishedasbeinganti-ferromagnetic. Itwillbeseenthatingeneral8and TNaredifferent, andthiscanbeaccounted forbyanextension ofthe theorygivenabovewhereinteractions withnextnearestneighbours belonging tothesamesub-lattice areincluded (VanVleck,1951;see Problem 22.1).Inaddition othertypesofarrangement ofthedipoles, wherenotallthenearestneighbours areanti-parallel, arepossible. Whenanexternal fieldisappliedatatemperature belowtheNeel 662 ANTI-FERROMAGNETISM ANDFERRIM GNETISM [22.2 (22.5)MA=Mo+oM }, MB=-Mo+oMpoint,apositivemagnetization resultswhosemagnitdecanbeestimated fromthetheorygivenabove.Ingeneral,theeffectofapplying afield istochangethemagnetization ofeachsub-lattice lightly,sothatwe maywrite wherethesemustbetakenasvectorequations ifthexternal magnetic fieldisappliedatanarbitrary angletothedirection ofthespontaneous magnetization Mo.IfBoisparalleltoMo,soalsowibe8M,and,ifwe x~Of2TN,l---~--'"""'-- X i T.., FIG.22.4.Variation ofXIIandX..Lonthesimpletheryofanti­ ferromagnetism. returntothegraphical solutionofourtranscendent Iequation, wesee thatthemagnetizations ofthesub-lattices willbeivenbytheinter­ sectionofthedottedlineB'A'inFig.22.3witht eBrillouin curve. Fortheeffective fieldsbecome, usingequation (22.1), BA=Bo-;\.(-Mo+oM} =AMo+(B o-oM), BB=Bo-;\'(+Mo+8M)=-AMo+(B o;\'8M). Theresulting netmagnetization 20Mwilldepend theslopeofthe Brillouin function (forsmallfields)atthepointMo.AsMoincreases, thisslopedecreases, reaching zeroatsaturation. Ifollowsthatthe susceptibility XII(inthedirection paralleltoMo)dcreasestozeroas thetemperature fallstozero,asillustrated inFig.22..Theexactshape ofthecurvedepends onlyslightlyonthevalueofJ,inthecaseofthe saturation curveforaferromagnetic substance. Iftheexternal fieldisappliedperpendicularly to0'wecanevaluate thesusceptibility moreeasily.Inthiscasethemagntizations MA,MB ofeachofthesub-lattices turnthrough asmallangl extowards Bo,as showninFig.22.5.Theeffective fieldonadipolenowhasthetwo 22.2] ANTI·FERROMAGNETISM ANDFERRIMAGNETISM 663 components Bz=±AMo(theuppersignforlatticeA),and Bx=Bo-)'3M. Forsmallexternal fieldstheanglea:issmall,andtheratioofthemagneti­ zationcomponents inthetwodirections willbe 3MBxBo-)'3M Mo=Bz=AMo "':1:I I MBM..{ t,5M at at r • -Mo +Me Z FIG.22.5.Effectofapplying afieldBoperpendicular tothespontaneous magnetiza. tionMeinananti-ferromagnetic substance. (22.7) andweshouldexpectfromwhich23M=BolA,andthesusceptibility willbe Xl.=2fL03MIBo=fLo/).=GI2TN• (22.6) ThusXl.shouldbeconstant belowtheNeelpointandequaltothevalue attheNeelpoint,asshowninFig.22.4.Forapowdered specimen con­ sistingofmicro-crystals withrandomorientation, wehave X=!(XII+2Xl.)' X(T=O) _2. X(T=T) 3 Thevaluesofthisratioforanumberofpowdered anti-ferromagnetics arealsogiveninTable22.1.Amoredirectcheckofthetheoryisobtained frommeasurements ofthesusceptibility ofasinglecrystal;Fig.22.6 showstheexperimental pointsofStoutandTrapp(1963)forMnF2• HeretheMn++ionisinaoS!state,andthesusceptibility showsnegligible anisotropy abovetheNeelpoint,aswouldbeexpected. BelowtheNeel pointXIIfallsrapidlyandapproaches zero,whileXl.remains almost constant, aspredicted bythetheory. 664 ANTI-FERROMAGNETISM ANDFERRIM GNETISM [22.2 Thetwosub-lattice modelisvalidformanyanti-erromagnetic sub­ stances,butinsomecasestherearemore(inaface-cntredcubiclattice therearegenerally four).Asinferromagnetism, t eexchange inter­ actionitselfgivesnopreference toanyparticular orientation ofthe spinsrelativetothecrystalaxes;thisarisesfromthenisotropyenergy. Inasimpletetragonal crystalsuchasMnF2'thespinarealignedalong 28 •24• • ~-;•••'i"""..•20 Susceptibility 16 perg/mole(inunits of.l0-lle.m.u.) 12 8••.•.••.•• •Xu•••••••••• •...o'--'.'-.--'---L,..----''---:-':---=--'-o--c-!:-:--+-~-_..,_'_-''7:'_::___::_::_:!o FIG.22.6.Principal susceptibilities ofasinglecrystalofMnF(afterC.Trappand J.W.Stout,1963). thetetragonal axisinasimpletwosub-lattice anti-pa llelarrangement, butmuchmorecomplicated: arrangements areposibleinwhichthe vectorsumofthedipolemoments iszero-the distintivefeatureofan anti-ferromagnetic. 22.3.Ferrimagnetism Thetechnical importance ofmagnetic materials inlectricalindustry hasincreased continuously, theidealsubstance beinonewithalarge magnetic moment atroomtemperature, whichisalsoanelectrical insulator. Ferromagnetic metalsandalloyshavebeewidelyexploited, buttheirhighelectrical conductivity isaserious andicap inradio­ frequency applications becauseoftheeddycurrenlosses.Forthis reasonanumberofmagnetic oxides('ferrites', ofhichmagnetite, Fes04,isthemostfamousastheoriginal 'lodestone') becameofgreat technical interest becauseoftheirlowelectrical coductivity. They showspontaneous magnetization, remanence,and otherproperties 22.3] ANTI-FERROMAGNETISM ANDFERRIMAGNETISM 665 similartoordinary ferromagnetic materials, butthespontaneous moment doesnotcorrespond tothevalueexpected forfullparallel alignment ofthedipoles. In1948Neelputforward atheoryforsuchmaterials; hesuggested thattheycontaintwosub-lattices inwhichthemagnetizations are oppositely directed, butwhichgiveanetmoment because thetwo sub-lattice moments areunequal. Forthisphenomenon hecoinedthe word'ferrimagnetism'. Itcanarisefromanumberofarrangements, ofwhichthesimplest areillustrated inFig.22.7.In(a)allthedipoles areequalinmagnitude, buttherearemoreononesub-lattice thanon 11 FIG.22.7.Threepossiblearrangements ofthedipolemoments inaferrimagnetic material. (a)Unequal numbers ofidentical moments onthetwosub-lattices. (b)Unequal moments onthetwosub-lattices. (0)Twoequalmoments andoneunequal. theother;themostnotableexample isyttriumirongarnet(YIG).The simplearrangement (b)withionsofunequal moments occursrather rarely.Thearrangement (c),withtwoequalandopposite moments, and athirdmoment ononesub-lattice istypicalofferrites suchasMnFe204• Ferrites Thesehavethetypical formula M++Fet++Oi- (equivalent to MO,Fe203),whereM++isadipositive ion,commonly Mn++,Fe++, Co++,Ni++,Cu++,Zn++,orMg++;othertripositive ionssuchasCr+++ canreplaceFe+++.Thecrystallographic structure iscubicandsimilar tothemineralspinel(MgAl204),andtheunitcell,witheightformula units,equivalent toMsFe16032' contains eightcationsiteswithtetra­ hedralcoordination (tofouroxygenions)andsixteencationsiteswith octahedral coordination (tosixoxygenions),knownastheAandB sitesrespectively. Thedivision ofcationsbetween thesesitesisnot unique,thelimiting casesbeing: 'normal' structure ,inverse' structureA8it68(8) 8M++ 8Fe+++BBit68(16) 16Fe+++ 8M+++8Fe+++ 666 ANTI-FERROMAGNETISM ANDFERRIM GNETISM [22.3 Intermediate arrangements arealsofound,andwewillconsider only theinversestructure. EachFe+++ionisina6St,stat ~withamomentof 5Bohrmagnetons; however, themoments ontheAandBsitesare anti-parallel. IfmMisthemoment oftheM++ion,thenetsaturation momentat0°KfortheunitMFe204willbe m=mM+(mFe)B-(mFeLl =mM+5f3- 5t=ml\I' (22.8) Themoments calculated thus(assuming thattheMi~nhasa'spinonly' moment) arecompared withtheobserved moments nTable22.2.The agreement issatisfactory; someorbitalmoment woW-dbeexpected in theFe++,00++,Ni++,Ou++ions,andinmagnesium f~rritethestructure isnotcompletely inverse. TABLE 22.2 MmM(spinonly)Observed moment TCK) Mn 5 4·4-5·0 573 Fe 4 4·0-4·2 858 00 3 3,3-3,9 793 Ni 2 2·2-2·4 858 Ou 1 1,3-1,4 728 Mg 0 0,9-1,1 - Themagnetic moments areinBohrmagnetons perunitMF82°., Neelsuggested thatalltheinteractions intheferritesareanti­ ferromagnetic insign,butthattheA-Binteraction isconsiderably stronger thantheA-AorB-Binteractions. Thusittheinversestruc­ turethedominating A-Binteraction makesthespinlwithineachgroup parallel, despitetheirmutualanti-ferromagnetic irteraction. Thisis supported bythefactthatZnFe204,whichhasthenormalstructure, hasnonetmoment. HeretheAsitesareentirelyoc(upiedbyzincions, withnomoment, sotheA-Binteractions arezero.Theferricionson theBsitesarethenalignedanti-parallel throughtheanti-ferromagnetic B-Binteraction, inequalnumbers, sothattheompound isanti­ ferromagnetic. ItsNeeltemperature (9°K)isquitlow,aswouldbe expected iftheB-Binteractions areweak. Garnets Thesehavethetypicalformula MaFe5012(ofwIUchtwounitsare equivalent to5Fe20a,3M20a),whereboththeMcatonandtheFeare tripositive ions;theM+++ioniscommonly yttrium ~ramemberofthe 4/transition group.Thecrystallographic structure i~cubicandsimilar tothemineralgarnet,thoughthishascationsofot~ervalencies. The 22.3] ANTI·FERROMAGNETISM ANDFERRIMAGNETISM 667 unitcelliscomplex, containing eightunitsofMaFes012;forsimplicity weshalldiscussmainlyyttrium irongarnet,wheretheY+++ionhas aclosedshellandcarriesnomagnetic moment. Theferricionsoccupy twotypesofsite;ineachunitYaFeS012twoFe+++ionsoccupy'a'sites, coordinated tosixoxygenions,andthreeFe+++ionsoccupy'd'sites, coordinated tofouroxygenions.Themagnetic moments ofthetwo 'a'ionsareantiparallel tothoseofthethree'd'ions,givingthearrange­ mentshowninFig.22.7(a);thenetmoment perunitYaFeS012isthus thatofoneFe+++ion,or5Bohrmagnetons (thebestexperimental value is4'96,8).TheNeeltemperature is5450K. Amongst otherferrimagnetic materials wemention onlyBaFe12019 (equivalent toBaO,6Fe20a).Thishasahexagonal structure, witha numberofinequivalent sitesfortheferricions.Ofthetwelveferricions performulaunit,themoments ofeightareanti-parallel totheremaining four,givinganetmoment of4X5=20Bohrmagnetons; theNeel temperature isabout8200K.Bariumferrite,asitisfrequently called, hasahighvalueof(BH)max andisusedasapermanent magnetmaterial (cf.Chapter 8).Beinghexagonal, ithasahighanisotropy energy;itis usedintheformofpressedoriented fineparticles. Discussion Neel'stheoryofferrimagnetism hadconsiderable successinexplaining theanomalous behaviour ofthesusceptibility abovetheNeelpoint. Usingamolecular fieldapproximation withthreeconstants representing theA-B,A-A, andB-Binteractions hededuced therelation 1T1a-=-+---- (22.9)X0XoT-f) forasubstance whereallthemagnetic ionshavethesamemoment, such asYIG,orMFe204whenMcarriesnomoment. Here0istheusual Curieconstant, buttheotherparameters arefunctions ofthemolecular fieldconstants andthenumbers ofionsineach ~ub-Iattice. Thegeneral behaviour oftheinversesusceptibili'(jy givenbyNeel'srelationasfitted toexperiments onYIGisshowninFig.22.8.Thetheoryalsoexplains qualitatively thecomplex behaviour ofthespontaneous magnetization curvebelowtheNeeltemperature. Themagnetization doesnotalways increase monotonically asthetemperature falls,andinferrimagnetic compounds containing morethanonetypeofmagnetic ionwhose spontaneous magnetization variesindifferent wayswithtemperature a'compensation point'maybeobserved, wherethemagnetization of thetwosub-lattices isequalandopposite. 668ANTI-FERROMAGNETISM ANDFERRIMAGNETISM [22.3 Themagnetization curveofgadolinium irongaret,Gd3Fel;012'is showninFig.22.9,together withthatofY3FeIi012' Thelatterisnot unlikethatofaferromagnetic, butatlowtemperat estheformerhas amuchhighermagnetization, fallingtozeroatthecmpensation point atabout295°K.At0°KwewouldexpecteachG+++iontohavea momentof7Bohrmagnetons; ifthesearemutually paaIlel,butopposed 60 1T-=-+30·5 _-X50__-.--­..-~. .."",..... . oExptlpointsforYIG__Fittetheoretical ~urve__Highemperature asymptote; 1500 FIG.22.8.Inversemagnetic susceptibility oftheferrimagnetic sbstanceyttrium iron garnet,whichhasthreeandtwoFe+++ionsonthetwosub-Iatices(arrangement (a) inFig.22.7). tothenetferricmoment, wewouldexpectanoveramoment forthe unitGd3FeIi012of(3x7-{3X5-2x5}),8 =(21-5),8 =16,8;thisis closetotheobserved moment. Asthetemperature riesthemagnetiza­ tionofthegadolinium ions,whicharesubjected t acomparatively weakinteraction withtheferricions,fallsmuchmorerapidlythanthat oftheironlatticewithitsstrongmutualinteractions etweentheferric ions.Infactthebehaviour oftheGdionsisnotfarromthatofpara­ magnetic ionswithS=i,subjected toaninternal eldgenerated by theironlattice. TheNeeltemperature ofGd3FeliO2(564°K)isnot appreciably different fromthatofY3FeIi012(545°K),aswouldbe expected onthisbasis. Apartfromtheirtechnical importance, ferrimagne icmaterials have playedamajorroleinadvancing ourunderstanding fmagnetic prob­ lems;forthispurposethegarnetsaremorefavoured thantheferrites, 22.3J ANTI-FERROMAGNETISM ANDFERRIMAGNETISM 669 sincethestructure isuniqueandtherearenouncertainties concerning thesitesoccupied bythemagnetic ions.Theabsence ofconduction electrons isagreatasset,notonlytechnically butalsoscientifically. Ontheonehandwearedealingwithlocalized magnetic moments, so thetheoryrestsonamuchfirmerfoundation; ontheothermany important experiments canbecarriedouttocheckthetheorywhich '"'+15=0..,,"=~=... .J:l+~O0 J:l=l .S..,="=0=+5 t:l.i=~ ~ 0600 _5b-------- FIG.22.9.Variation ofthespontaneous moment withtemperature forGd.Fe60n (GdIG) andY.Fe60n (YIG)inBohrmagnetons performula unit. wouldotherwise beimpossible. Anobvious example ismagnetic resonance experiments (cf.Chapter 23)inthefrequency range101°_ lOlacis,whichhavebeenaveryfruitfulfieldbothforferrimagnetics andforanti-ferromagnetics. Theabsenceofconduction electrons plays alessdirectbutnolessimportant roleinmeasurements ofthemagnetic specificheatcontribution predicted byspinwavetheory.Atermpro­ portional toTiwasfirstconfirmed byKouvel(1956)usingFea04(ithltS alsobeenmeasured incompounds suchasYIG),whereasintheordinary ferromagnetic metalsitisobscured bytheelectronic specificheat. Another experimental achievement istheopticaldemonstration ofthe presence ofdomains, usingtherotation oftheplaneofpolarized light; propagated paralleltothedirection ofmagnetization (theFaraday effect);whenathincrystalofYIGisplacedunderapolarizing 670 ANTI-FERROMAGNETISM ANDFERRIM GNETISM [22.3 microscope thedomains arevisibleaslightanddrkregionswhose motioncanbeobserved undertheactionofanappledfield. 22.4.Thelanthanide ('rareearth')metals Measurements ofthesusceptibilities oftheIanthaidemetalsathigh temperatures givegenerally aCurie-Weiss lawwhrethesizeofthe Curieconstant agreeswellwiththatexpected fort etripositive ions. Therearetwonotableexceptions tothis:europium mtalandytterbium metal,whicharecubicinstructure withanionicizeindicating the presence ofdipositive ions.Inaddition, ceriumtenstoshowaphase transition atlowtemperatures toacubicstructu withCe4+ions, dependent onthethermalhistoryofthespecimen. heCe4+andYb++ ionshaveclosedshellsandnomagnetic moment, 0theyarenotof interesthere;theEu2+ionhasahalf-filled shell,grodstateS=t,but themagnetic behaviour ofthemetalshowsunexpec edcomplications. Weshalltherefore restrict ourselves tothemetscontaining tri­ positive ions,dataforwhicharegiveninTable22.3 Nometalshowsaco-operative stateaboverootemperature, so thatexchange interactions aresmallcompared wththespin-orbit coupling. Wemaytherefore regardthespinand0bitascoupledto givearesultant angular momentum J,asinthearamagnetic salts (cf.§20.6).Onthisbasisthesaturation moment peronat0°Kshould bl3g.JBohrmagnetons, wheregistheLandefactorppropriate tothe groundstateJofthefreetripositive ion.Values fgJaregivenin column 2ofTable22.3,andaregenerally substantiat dbythemagnetic evidence forgadolinium andtheheaviermetals. Inconsidering theexchange interaction, wehave 0projectthespin vectorSontothetotalangularmomentum vectorJaspointedoutin §21.1.Theexchange interaction -2,1Si,Sj betwenthespinsthus becomes equivalent toacoupling -2,1'Ji'Jjbetwee thetotalangular momenta, withJf' =(g-1)2Jf, asgivenbyequation (21.3).IfJfwere thesameforallthelanthanon metals,weshouldexpettheCuriepoints tovaryas(g-1)2J(J +1),fromequation (21.6).Thisuantityislargest forGd+++,withahalf-filled shell,andthismetalsowsco-operative effectsatahighertemperature (290°K)thananyotherlanthanide metal.Reference toTable22.3showsthatthetemeratureatwhich co-operative effectsappearvariesqualitatively inacordancewiththis relationinthesecondhalfofthegroup.However, hesemetalsshow morethanoneorderedphase,beinganti-ferromagne icathighertem­ peratures andferromagnetic atlowertemperatures. hiseffect,which 22.4] ANTI-FERROMAGNETISM ANDFERRIMAGNETISM 671 appearstorequireareversal in-signoftheexchange interaction asthe temperature falls,wasforalongtimeverypuzzling. The4felectrons inthelanthanons belongtoaninnershell,andtheir wavefunctions aremuchlessextended thanthoseofd-electrons. For TABLE 22.3 Magnetic dataforthelanthanon metals,assuming Ln+++ions.Forthe valuesofg,JseeTable20.1.ThevalueofgJgivesthemomentperionat 0°Kassuming theionsarenotsubjecttoanycrystalfieldeffects.Pmhas beenomittedforlackofdata(ithasnoradioactively stableisotopes); europium metalbecomesanti-ferromagnetic below87°K(andpossibly ferromagnetic atalowertemperature), andappearstocontainEu++ions, withahalf-filled shellandS=i;ytterbium metalcontainsYb++ions withafilledshellandnomagnetic moment. gJ(g-I)2J(J+I) TN(OK)Tc(OK)--La. 0 0 - - Ce 2·14 0·18 12·5 - Pr 3·2 0·8 25- Nd 3·17 1·84 7,18- Sm 0·71 4·5 14 - Gd 7 12·25 - 290 Tb 9 10·5 228 220 Dy 10 7·1 179 85 Ho 10 4·5 125 40 Er 9 2·55 80 20 Tm 7 1·17 50 20 Lu 0 0 - - thisreasondirectexchange, involving overlapoff-electron wavefunc­ tionsonadjacent ions,isunlikelytobeimportant, andtheoriginof exchange interaction observed inthelanthanon metalsisascribed to thesecondmechanism discussed in§21.9,theconduction electrons beingpolarized byexchange interaction withthe4fshells,andserving asamedium whereby theorientation ofthemoment ononeioncan influence thatonneighbouring ions.Wecanthusregardthemetalsas consisting ofionswithwell-localized moments duetotheir4fshellsin aseaofconduction electrons formedfromthevalenceelectrons, which contribute littletothemagnetic properties directly butprovidethe medium forexchange interaction. Electrostatic interaction between the4felectrons onagivenionand thechargeontheadjacent ionsprovides a'crystalfield'interaction in themetalswhichwouldbeexpected tobeofthesameorderasthatin 672 ANTI·FERROMAGNETISM ANDFERRIM GNETISM [22.4 saltsofthelanthanide group.Directevidence fothiscomesfrom Schottky-type anomalies inthespecificheatsofthfirstmembers of thegroup;theexcess'magnetic' specificheatduetcrystalfieldsplit­ tingsoftheJ=4stateofPr+++inpraseodymium metalisshownin Fig.22.10.Theoverallsplitting produced bythecstalfieldisinthe regionofafewhundred oKforCe,PI',Nd,andS ;sincethevalues 1·2/1\ 1·0I\0·8\CjR ~\ 0·6 0'.J I 0·2//' 2 I) 200 FIG.22.10.Themagnetic specificheatpermoleofpraseod iurnmetal,dueto crystalfieldsplittings ofthe8H,groundstateofthePr8+io(Bleaney, 1963). of(g-1)2J(J +1)areratherlow,theexchange iteraction issmall compared withthecrystalfield.Aco-operative phseisfoundonlyat temperatures below25°K,andinpraseodymium, wherethecrystal fieldsplitting leavesasingletasthegroundstate,thco-operative state hasonlyasmallmoment. Thelanthanon metals(apartfromthosewithC+,Eu++,orYb++ ions)allformhexagonal crystals, butthestructure hangesslightlyat gadolinium. Inthisandtheheaviermetalsthecrytalfieldissmaller thaninthefirstmetalsofthegroup,whilethevaluof(g-1)2J(J +1) tendstobelarger;thusexchange interactions prepoderateovercrystal fieldsplittings, andtheco-operative phasesetsinattmperatures where crystalfieldeffectsarerelatively lessimportant. S heffectsdo,how­ ever,playamajorpartindetermining themagne'cstructure inthe co-operative phase.Theyproducean'anisotropy eergy'whichvaries withpowersofthemagnetization uptothesixtdegree,andisa complex function oforientation ofthemagnetic moent,reflecting the hexagonal symmetry ofthelattice.Thisanisotro yenergyfavours 22.4]ANTI·FERROMAGNETISM ANDFERRIMAGNETISM 673 orientation ofthemomentsincertaincrystallographic directions, while theexchange interaction favoursasimpleparallelorientation. Atthe lowesttemperatures gadolinium andtheheaviermetalshaveaferro­ magnetic phaseinwhichthedirection ofmagnetization isdetermined bytheanisotropy energy. Thelatterislarge,andthemetalsaremag­ netically hard,exceptinthecaseofgadolinium. TheGd3+ion,witha half-filled 4jshellisinanSStstatewithnoorbitalmoment; ittherefore hasnofirstorderinteraction withthecrystalfield,andtheanisotropy energyisrelatively small. Athighertemperatures theco-operative phaseofterbium andthe following metalschangestoanantiferromagnetic state,withnoresultant magnetization. Insomecasesthemoments lieinaspiralarrangement wheretheanglebetween successive layersisafunction ofthetempera­ ture,andinothersthemoment liesalongthehexagonal axisbutshows aspatialoscillatory variation inmagnitude. Theequilibrium stateis thatwiththelowestfreeenergyF=U-TS, andthesecomplex arrangements havealowerfreeenergyathighertemperatures because oftheirhigherentropy. Atstillhighertemperatures theparamagnetic phasebecomes theequilibrium phase.Table22.3givesboththeNeel temperature TNatwhichtheanti-ferromagnetic phasesetsin,andthe Curietemperature To. 22.5.Neutron diffraction Aproperdescription ofthetheoryandpracticeofneutrondiffraction isoutsidethescopeofthisbook,andonlyabriefoutlinecanbegiven ofthemajorroleithasplayedinestablishing thestructure oftheordered stateofamagnetic compound. Associated withaparticlewhosemomen­ tumispisawavelength A=hlp,wherehisPlanck's constant; for neutrons ofthermal energies, thiswavelength isoftheorderofan Angstrom unit(forneutrons inthermalequilibrium withatemperature of0°C,thewavelength is1·55A).Thenucleioftheatomsinacrystal latticescatterneutrons, anddiffraction patterns areformedinasimilar waytothoseforX-rays. Neutrons canthusbeusedforthedetermina­ tionofcrystalstructures inmuchthesamewayasX-rays;inparticular thepositions ofhydrogen ions(which,beingjustprotonswithnoelec­ trons,havevirtually zeroscattering powerforX-rays)incrystallattices canbedetermined accurately. Inmagnetic materials, thepermanent electronic magnetic moments give anadditional scattering mechanism forneutrons whichoftenoutweighs thenuclearscattering, through theinteraction between theelectronic ~lla xx 674 ANTI-FERROMAGNETISM ANDFERRIMA NETISM [22.5 magnetic moment andthenuclearmagnetic moment ftheneutron.If theelectronic moments arerandomly oriented, asiaparamagnetic substance, thescattered neutrons areincoherent inpaseandtheresult isanaddition tothegeneralbackground ordiffusescatering.Although somemagnetic information canbeobtained fromcarfulmeasurements 800K 293°KI (311)nucl 20°(311)magn (333)mag ,j. (511) ~(311)magnt(ll1)magn .j. 5°Residual short-range magnetic ~rdering100 80 60 ......40., ~ .S20S ~ Po '"=e~gs t'ioo '01=., d80.... 60 40 20 10° 15° :Braggangle8 FIG.22.11.Theneutron diffraction patterns ofMnOat80°K,290K(bolowandabove theCurietemperature of1160Krespectively). Thelow-temp raturepattern shows extraantiferromagnetic reflections whichcanbeindexed inteofamagnetic unit withdimensions twicethoseofthechemical unitcell.(ShandSmart,1949.) oftheadditional diffusescattering, neutron diffractonprovides much morestriking andusefulinformation whenthemgneticdipolesare oriented inanorderedstructure, asinmostspontan ouslymagnetized substances. Anorderedarrayofelectronic dipolesgvesrisetodiffrac­ tionpeaksinspecificdirections determined bythethree-dimensional structure ofthearray.Ifthemagnetic unitcellhast esamedimensions asthechemical unitcell,thecoherent magnetic diffrtionpeaksappear 22.5] ANTI·FERROMAGNETISM ANDFERRIMAGNETISM 675 atthesameangularpositions asthepeaksduetothenuclearscattering. Inananti-ferromagnetic thedimensions ofthemagnetic unitcellmay differfromthoseofthechemical unitcell;forinstance, inFig.22.1,the magnetic patternonlyrepeatsinthedistancePR,whereasthechemical patternrepeatsatthedistancePQ.Extrapeakstherefore occurinthe diffraction patternoftheorderedarray(seeFig.22.11),whichareabsent inthedisordered array(theparamagnetic state).AswithX-rays, neutron diffraction canbeobserved usingpowdered orpolycrystalline substances, butfullermagnetic information isobtained withsingle crystals. Suchinformation includesthesymmetry ofthemagnetic array (Le.thesizeofthemagnetic unitcellandtherelative orientation of themoments withinit),theactualorientation ofthemoments relative tothecrystalaxes,andthesizeoftheindividual moments. Though detailsofthemagnetic structure canoftenbeinferred fromother magnetic measurements, onlyneutron diffraction givesadirectdeter­ mination. Themorecomplicated themagnetic structure, thelesslikely itcanbededuced indirectly; anobviousexample isthehelicalmoment structure ofsomelanthanon metalsandothersubstances (infactMnAu2 wasthefirstsuchstructure discovered byneutron diffraction, in1959). Manystructural determinations ofasimplernaturehavebeencarried out,ofwhichonlytwomaybementioned briefly. (a)Although some suggestions ofaferrimagnetic structure havebeenputforward for iron,neutron diffraction showsthateveryironatomappears identical andcarriesthesamemoment, atanyrateonatimeaverage; (b)the seriesofcompounds MnF2,FeF2,CoF2,whicharetetragonal, have beenshowntohaveasimpleanti-parallel arrangement ofspinsoriented alongthetetragonal axis,butinNiF2thespinsarecantedawayfrom thisaxisbyabout10°,givingaweakferromagnetic moment (Le.the spinsareoriented asinFig.22.5,butthrough theanisotropy energy, notbyamagnetic field). REFERENCES BLEANEY, B.,1963,Proc.Roy.Soc.A,276,39. KOUVEL, J.S.,1956,Phys.Rev.102,1489. N:EEL,L.,1936,Ann.Phys.Paris,5,256. ROBINSON, W.K.,andFRIEDBERG, S.A.,1960,Phys.Rev.117,402. SHULL,C.G.,andSMART,J.S.,1949,ibid.76,1256. TRAPP,C.,andSTOUT,J.W.,1963.Phys.Rev.Letters,10,157. VANVLECK,J.H.,1941,J.Ohem.Phys.9,85.--1951,J.Phys.Radium, 12,262. 676 ANTI-FERROMAGNETISM ANDFERRIMA NETISM PROBLEMS 22.1.Thetheoryofanti-ferromagnetism canbeextended bassuming thatthe molecular fieldactingoneachsub-lattice contains atermd etoexchange inter­ actionwithionsonthesamesub-lattice aswellasatermdutoionsontheother sub-lattice. Showthatintheparamagnetic statetheequatons MA=(0/2/-,0T)(Bo-AMB-).'M A), MB=(0/2/-,0T)(Bo-AMA-).'M B) leadtoaCurie-Weiss lawforthesusceptibility (equation ( 8=(0/2/-,0)().+).'). 22.2.TheNeeltemperature TNcanbefoundbyputtingBo()inthepreceding question, andfindingthecondition thatthepairofequatinsforMA,MBstill haveasolution. (TNisthetemperature atwhichavanishin lysmallmagnetiza­ tioncanexistwhenBo=0,andtheBrillouin function sapproximated by Curie'slaw.)ShowthatthisgivesTN=(0/2/-,0)().-).'). (23.2)23 MAGNETIC RESONANCE 23.1.Themagnetic resonance phenomenon ITwasshownin§20.1thatwhenanatomornucleuswitharesultant angularmomentum Gandmagnetic momentmisplacedinasteady magnetic fieldBotheequation ofmotion(obtained fromequation· (20.2)bymultiplying by1')is dm/dt=ym/\BOl (23.1) whereI'=miGisthemagnetogyric ratio.Themotionrepresented by thisequation consistsofaprecession oftheangularmomentum vectorG andhencealsoofmaboutthedirection ofBowithauniform angular velocity-yBo,whichweshalldenotebyWL'Ifthesystemisun­ disturbed itwillcontinue indefinitely inthisstateofuniformprecession withmatafixedangletoBo,anditisconvenient tomakeuseof rotating coordinate systemsinconsidering thismotion.Itisshownin §A.lOthattherateofchange(dm/dt)ofanyvectorquantity suchas minthelaboratory coordinate systemisrelatedtotherateofchange (Dm/Dt) inasystemrotating withangularvelocity 00relativetothe laboratory system,bytheequation dm/at=Dm/Dt+w/\m. Substitution ofthisinequation (23.1)gives Dm/Dt =I'm/\Bo-w/\m =ym/\Bo+m/\c..> =ym/\(Bo+~). Thisresultshowsthatintherotating coordinate systemtheapparent magnetic fieldis(Bo+w/y), andtheapparent precession velocity is -y(Bo+w/y) =WL-W. Thustheapparent angular velocity isde­ creasedby00,aswouldbeexpected fromsimpleconsiderations of relativeangularvelocity.IfwewriteB'=-00/1',theapparent field intherotating systemis(Bo-B'), anditisreducedifB'ispositive (Le.00hasthesamesignaswL)asshowninFig.23.1.Clearly,if B'=Bo,theapparent field(Bo-B') andprecession velocity-y(Bo-B') 678 MAGNETIC RESONANCE [23.1 B'=-ro11arebothzero,andthevectormisstationary inthertatingcoordinate system. Weshallnowconsider theeffectofapplying asm11oscillating mag­ neticfieldB1coswtitheplanenormal tothedirection oft esteadyfieldBo• Thisoscillating fielmaybeplane polarized orcircular! polarized; inthe lattercaseB1issimpy avector,normal toBo,whichisconsantinlengthbut whichrotatesabouBowithangular velocity oo.Ifthescillating fieldis planepolarized, witacomponent say inthex-direction (tkingBoalongthe z-axis),itmaybedeomposed intotwo vectorsrotatingin0positesenses;thus thereisnolossofgenralityinconsider­ ingonlythecirculalypolarized case. Ifwenowtransfe fromthelabora­ torysystemtoasytemrotating with theangularvelocity 00,thenthevector B1isstationary intissystem,andcan FIG.23.1.Eftectivefields inarotating berepresented byaonstantvectorB1 coordinate system.asshowninFig.231.Inthissystem, theatomornucleusfeelsanapparent magnetic fiel(Bo-B') parallel tothez-axis,together withthefieldB1normaltohisaxis;thusthe resultant fieldinthisrotating systemisthevectorsumofthesetwo fields,whichisdenoted byBerrinFig.23.1.Toaobserver inthis system,thedipolemomentmwillappeartoprecssaboutBeffwith angular velocity-yBerr,anditsprojection onBo'llchangeastime goeson.Ifmwereinitially paralleltoBo(aswehouJdexpectina macroscopic system), itwouldprecessaboutBeffndatsomesub­ sequent timewouldreachamaximum angle2()wih Bo,where tan()=B1/(Bo-B'). IfBo-B' =0,Berr=B1and()=17T,sothatmillturnrightover toreachaposition anti-parallel toBobeforecomencingtoreturn. Thisoccursonlywhen 00=-yB'=-yBo=oov sothatthefrequency oftheappliedfieldisthentheameasthatofthe Larmorprecession. Thustheprecession aboutBeffisaforcedresonance 23.1] MAGNETIC RESONANCE 679 phenomenon, whoseamplitude isgreatest whentheappliedfrequency wisequaltothenaturalfrequency WL' Inthecaseofanatomicornuclearsystem,theangularmomentum isquantized, andsoareitsprojections onBo,sothattheenergy W= -m .Boisalsoquantized. Weshallconsider firstthenuclear case,assuming anucleusofspinangularmomentum Hi,andmagnetic momentm,wherethemagnetogyric ratioy=Yn(eI2M). Thenthe potential energyinastatewhosemagnetic quantum numberismis Jv,n=-m.Bo=-y1iI.B o=-y1imBo' (23.3) Undertheinfluence ofanoscillating magnetic fieldpolarized intheplane normaltoBo,transitions between stateswithdifferent valuesofmmay takeplaceaccording totheselection rule11m=±1.Thequantum of energyrequired is (23.4) (23.5)Thisisthesameforalltransitions, asshowninFig.23.2.Fromthisit followsthatthefrequency oftheradiation mustbe v=_.2.Bo=+wLI27T.27T Thisistheresonance condition, whichisthesameasthatgivenbythe classicaltreatment above.Theminussignissignificant onlyifcircularly polarized radiation isused.Ifasystemofnucleiwithapositive value ofyistoabsorbenergyfromanappliedoscillatory field,theselection ruleforabsorption is11m=-1,andthevectorB1mustrotateinthe left-hand senseaboutBo,whileifyisnegative, thereverseholds.This givesamethodofdetermining thesignofy,butformanypurposes this isimmaterial andlinearly polarized radiation maybeused.Since thiscanberegarded ascomposed oftwocircularly polarized components rotatinginopposite senses,transitions canbeinducedwhatever thesign ofy.Intheusualspectroscopic terminology theseare'magnetic dipole' transitions, corresponding tothefactthattheyarecausedbytheinter­ actionofanoscillatory magnetic fieldwiththemagnetic dipolemoments ofthesystem. Thisphenomenon isgenerally knownas'magnetic resonance', anditoffersamethodofdetermining ydirectly froma measurement ofafrequency andamagnetic field.Theorderofmagni­ tudeofthefrequencies required maybefoundfromthespecificcharge (elM)oftheproton,ifweassumethatYnisaroundunity.Thevalue ofelMfortheprotonisnearlyequaltotheFaraday, i.e.itisabout 680 MAGNETIC RESONANCE [23.1 108coulombs/kg (104e.m.u./g). Hencethefrequenc v=-gn(~)Bo 27T2M is~107cisinafieldof1weber/m2(10kilogauss inthecaseofa nuclear magnetic moment. Electronic magnetic omentsaresome 2000timeslarger,owingtothesmallermassofthelectron,whilethe B +2--r-~~- ±(m-2) +(m-l)+3.....,....L..r------------+1.~~~------------ +m+4_..L.....l.. ---------m -1-Z..,...-J"-r---- ---------3~..,-L...r- =(m~) -2--r-~~-- =(m~)m=-4--r--r--- - - - - - - - -~-- Energylevels 0--r~r"""- ~m=±l FIG.23.2.Digram showing thenineenergylevelsandtheallowdtransitions between themforanuclearspinI=4inafieldBo•Transitions canbeinucedbyanoscillating fieldoffrequency vifv=-(y/21T)B o• associated angularmomentum isofthesameorderasnthenuclearcase, sothatthevalueofyandofthefrequency required a ehigherinpropor­ tion.Foranatomwithamagnetic moment dueontoelectron spin, thewavelength oftheradiation required forresoanceinafieldof 10700gaussis1cm(afrequency of3X1010cis). Themagnetic resonance phenomenon hasbeensedtoinvestigate systemsofbothatomicandnuclearmagnetic moents,andweshall discussfirstthelatter.Itmakespossibleadirectdtermination ofthe valueofy,andhence,foranucleuswhosespinIisknwn,ofthenuclear magnetic moment. Thechiefexperimental difficulty iesinthesmallness oftheeffect,andweshalldescribe firstaningenious ethodduetoRabi, wherethephenomenon isdetected byitseffectonthpathofamolecule inamolecular beam. 23.2] MAGNETIC RESONANCE 681 23.2.Molecular beamsandnuclearmagnetic resonance Theuseofatomicbeamsforthemeasurement ofatomicmagnetic moments hasbeenmentioned in§20.4.Themethoddepends ondeflect­ ingtheatomsbypassingthemthrough aninhomogeneous magnetic field;thedeflexion isproportional totheprojection ofthemagnetic moment onthedirection ofthefieldgradient, andtheinitialbeamis splitinto(2J+1)beamsifthetotalelectronic angularmomentum has quantum numberJ.Themagnetic moment canbecomputed fromthe FIG.23.3.Rabi'smolecular beamapparatus. ThebrokencurvesintheBmagnetshow thepathsofmolecules whichhaveundergone atransition inthefieldBooftheGmagnet duetother.f.fieldappliedatFperpendicular toBo• sizeofthedeflexions ifthemagnitude ofthefieldgradient isknown. Themaindifficulty inachieving highprecision isthespreadinvelocity oftheatomsinthebeam,sincethedeflexion isinversely proportional tothesquareofthisvelocity. Application ofthismethodtothedetermination ofnuclearmagnetic moments demands greatrefinements, sincethesizeofthemoment is some2000timessmallerthanthatofanelectron, andthedeflexion is correspondingly smaller.Foradirectmeasurement molecules suchasH2 orNaCI,withnoelectronic magnetic moment, mustbeused.Owingto thegreatdifficulty ofworking withpurelynuclearmoments, methods weredevisedusingatomswithahyperfine structure duetomagnetic interaction between theelectronic moment andthenuclearmagnetic moment. Thesearerathercomplicated, andsuchdeflexion methods havenowbeensuperseded byothersmakinguseofmagnetic resonance. ThefirstofthesewascarriedoutbyRabiandhiscolleagues (1939). Aschematic diagramoftheapparatus isshowninFig.23.3.Molecules fromanoven0emergethrough anarrowslit,movingatsmallangles withtheaxisoftheapparatus. Theyenteraregionofinhomogeneous 682 MAGNETIC RESONANCE [23.2 magnetic fieldintheA-magnet, andaredeflected byanamountpropor­ tionaltotheprojection oftheirmagnetic moments onthedirection of thefield;thussomeofthemwillpassthrough thecollimating slitS. Neglecting forthemoment theO-magnet, wefolowthemolecules throughtheB-magnet, whichproduces aninhomog neousfieldexactly likethatoftheA-magnet exceptthatthegradient iseversed. Theforce onthenuclearmagnets inamolecule istherefore alsoreversed (provided thattheorientation ofthesemagnets isthesameasiwaswhenpassing through theA-magnet), andthemolecules arehereforedeflected 100 ~ 'iiJ=.,.., .S90 S.e., P=l 80 _.__L 1800 1900 2000 Magnetic fieldingauss' FIG.23.4.Resonance curvefortheF'·nucleusinN Fobtained byRabiwiththeapparatus inFig.23.3. upwards andreachadetector. Theprovisoabouttheorientation is important, becauseifitchanges betweenAandB,thenthedeflexion produced byBisdifferent fromthatinA,andthemolecules willnot reachthedetector. Thisgivesameansofdetecting aagneticresonance phenomenon, sinceitcanbeusedtocauseachangeintheorientation afterleavingAandbeforeenteringB.TheO-magnet roducesauniform fieldBo,andbetween itspolefacesisaconducting Iopcarrying anr.f. currentwhichproduces anoscillating fieldB1coswtinadirection per­ pendicular toBo•Whentheresonance condition isflfilled,transitions areinducedwithintheZeemanlevelsofthenuclear omentinthefield Bo,sothattheorientation ofthenuclearmoment iscanged.Thebeam currentatthedetector thenfalls,atypicalexampebeingshownin Fig.23.4.Heretheradiofrequency iskeptconstant andthemagnetic fieldBoisvariedthroughtheresonance. Thevaluefyisfoundfrom theobserved valuesofBo(atthecentreofthersonance) andthe frequency. Fromthewidthoftheresonance curvenFig.23.4itcan 23.2] MAGNETIC RESONANCE 683 FIG.23.5.Cross·section oftheAand Bmagnets usedinRabi'sapparatus ofFig.23.3,normaltothepathofthe beam(markedB inthefigure).The curvedpartsofthepolepiecesare cylindrical, buttheradiusofcurva· turefortheupperpoleislargerthan forthelowerone.z 1 Av=±y(Bo-B')/21T=±1/2t, (23.6) showing thatthelinewidthissimply relatedtothetimetforwhichthedipole moment issubjected totheoscillatory field. Anexperiment ofthiskindisbyno meanseasy,ascanbeseenfromthefact thatthedeflexion ofamolecule isonlyabout0·05mminamagnet withafieldgradient oftheorderof105gauss/cm. Thisisusinga magnet50cmlongwithpolefacesoftheshapeshowninFig.23.5;the curvature isadjusted togiveauniform valueofdB/dzoverthewidth ofthebeam.Becauseofthesmalldeflexions thedefining slitsatSand thedetector mustbeverynarrow (~0·01mm)andthebeamintensity atthedetector isverysmall,beingdetermined bythesolidanglewhich thedetector slitmakeswiththeoven,someIimetresaway!Originally thedifficulty ofdetecting abeamofuncharged molecules limitedthe methodtohydrogen, deuterium, andthealkalimetals,whosenuclear moments weremeasured withaprecision ofafewpartsperthousand (seeTable23.1).ThespectraofH2andD2aremore complicated than thoseofheaviermolecules, forarotating H2molecule hasarotational magnetic moment ofthesameorderasthenuclearmoment; inaddition thereisanother interaction intheD2molecule, between theelectric quadrupole moment ofthedeuterium nucleusandtheelectricfield gradient oftheelectrons.sothatbeseenthattheaccuracy isverymuchgreaterthancouldbeobtained byasimpledeflexion method. Anestimate ofthewidthcanbeobtained asfollows.Ifthenuclear moment isinitially paralleltoBo,thenatresonance itprecesses about B1intherotating coordinate framewithangularvelocityyBllandin atimetitwillexactlyreverseitsorientation provided thatyBlt=TT. Thiscorresponds tothemaximum intheresonance curve.Whenwe areoffresonance, themoment precesses aboutBeu,andifBeuisatan angle (J=45°toBothemomentwillonlyreachamaximum angleof 2(J=iTTtoBo•Ifwetakethistodefine thehalfintensity pointsonthereson­ ancecurve,theycorrespond to Bo-B'=±B1, 684 MAGNETIC RESONANCE [23.2 Present methods ofdetection dependonionizatipn ofthemolecule byelectron bombardment; theresultant ionistheIpassedthrough a massspectrometer toseparate itfromthebackgoundioncurrent. Finallyitisaccelerated ontothefirstplateofandectron multiplier, inwhichthesecondary electrons ejectedbyitsimpactareamplified as inaphoto-multiplier tube(§4.4). TABLE 23.1 Nuclearspinsandmagnetic moments ofsomeco~monisotopes Magnetic moment (nuclear nagnetons) Molecular Nclear Nucleu8 Spin beamvalue rMonaneevalue neutron t-1,913 - 1Rt +2·789 +·7927 oR 1 +0·856 +p·8574 'R i- +~'9788 Alkalimetal8: 6Li 1 +0·821 +~'8220 7Li ! +3·253 +~'2563 o'Na ! +2·215 +·2175 ,oK! +0·391 +·3915 uK i +0·215 +·2154 s5Rb ! +10340 +·3527 s7Rb ! +2·733 +·7505 13"OS t +2·558 +·5789 Halogens: 1°F i +2·62 +·6285 35CI! +0·819 +·8218 37Cl i +0·681 +·6841 7°Br ! +2·UO +·1056 81Br i +2·271 +·2696 1071 ! - +·8090 Anexperiment ofthiskindwascarriedoutbyJlvarezandBloch (1940)todetermine themagnitude ofthemagneticdi!olemomentofthe neutron. Inthiscaseitisnotpossibletousethemetodofdeflexion in aninhomogeneous fieldbecausethenumberofneutrons reaching the detector, withthenarrowslitsrequired, istooSlIallfordetection. Instead, usewasmadeofthefactthattheabsorptiop. ofneutrons ina ferromagnetic material, suchasironmagnetized tosa~uration, isdiffer­ entforneutrons whosespinisparalleltotheelectron S1insinthematerial fromthatforneutrons withanti-parallel spins.Tw(]suchmagnetized blocksaretherefore usedas'polarizer' and'analysr'insteadofthe inhomogeneous fieldmagnetsAandB.Fromthefirstftheseapartially 23.2] MAGNETIC RESONANCE 685 polarized beamofneutrons entersahomogeneous fieldC,wheretransi­ tionsareinducedbyther.f.magnetic fieldatresonance. Whenresonance isachieved, itisdetected byadropintheneutron countofthebeam emerging fromthesecondblock,sincesomeoftheneutrons havemade transitions totheorientation havinggreaterabsorption intheiron.In laterworktheprecision hasbeenimproved bymeasuring thefieldofthe C-magnet byprotonresonance (see§23.3),sothatbymeasuring the radiofrequencies required formagnetic resonance oftheprotonand neutroninthesamefield,theirrelativemagnetic moments areimmedi­ atelydetermined (bothneutronandprotonhaveI=I).Highprecision canonlybeobtained withnarrowresonance curves,andfromequation (23.6)thisrequires alargevalueoft;i.e.alongpaththroughtheC-field. Thecorresponding requirement ofhighuniformity intheC-fieldismade lessrigorousbytheuseoftwoseparate oscillatory fields,oneateachend oftheC-field.Withthisarrangement, duetoRamsey (1949),onlythe averagevalueoftheC-fieldoverthewholepathisrequired tobethe sameasthatattheposition oftheoscillatory fields.Bymeansofthis andotherspecialtechniques Cohen,Corngold, andRamsey (1956) obtained theresult(p.p.m.=partspermillion) magnetic momentofneutron/magnetic moment ofproton =0·685039 (±25p.p.m.). Byapplying anelectricfieldofabout2X105V/cmparalleltoandin thesameregionastheC-field,assuggested byPurcellandRamsey (1950),ithasbeenshownfromtheabsenceofanyeffectduetoprecession intheelectricfieldthattheupperlimitofanyelectricdipolemoment ontheneutron islessthanthechargeontheelectron multiplied bya length5X10-20em(Smith,Purcell,andRamsey, 1957). 23.3.Nuclear magnetic resonance inbulkmaterial Themolecular beammethodofdetecting nuclearmagnetic resonance isexperimentally verydifficult,butitwasusedbecauseatthetimethere seemednoprospect ofdetecting theresonance phenomenon directly;that is,byobservation oftheeffectofemission orabsorption ofquantaonthe oscillatory field.Atradiofrequencies therateofspontaneous emission ofquantaisnegligibly small,andspectroscopic linescanbeobserved onlyinabsorption. Themagnitude oftheabsorption innuclearmag­ neticresonance isverysmall,ascanbeseenfromthefollowing estimate. Fromthetheoryofanomalous dispersion (§17.4)theimaginary partof thesusceptibility X"atthecentreofanarrowlineisrelatedtothestatic 686 MAGNETIC RESONANCE [23.3 susceptibility Xobytheformula (seeProblem 23.1) x"w v-=-=-, (23.7) Xo2dw2dv wheredvisthedistance fromthecentreofthelinet apointwherethe intensity hasfallentohalfitsmaximum value.No /-tong~fJ;1(1+1) Xo= 3kT ' (23.8) d(l/Q)=X"/{l+X') ~10-5•andinafavourable case,suchastheprotonsinwaer,thevalueofXo atroomtemperature isapproximately 10-8m.ks./metre a(~10-9 e.m.u./cm3).Toestimate X"weneedanapproximat valueofdv.The maincauseoflinebroadening inourcaseistherandmmagnetic fields ofneighbouring nuclearmagnetic moments. Theeectoftheseisto changetheactualfieldatagivennucleusbyanamuntdepending on theorientation andnumber ofneighbouring maeticdipoles: this causesaspreadinthemagnetic fieldactingondierentnuclei,and sogivesafinitelinewidth.ThespreadinfieldBisoftheorder of/-tom/41Ti},a, wheremisthedipolemoment ofaneighbour, andi}, itsdistance. Themeanvalueofi},aisjusthalft eaverage volume occupied bythetwoprotonsinawatermolecule, whcedB~1gauss. Nowv/dv=Bo/(dB), sothatatafieldof2000gauss,X"wouldbeabout 10-8(2000/2) ~10-5m.k.s./metre3•Forprotons, heresonance fre­ quencyatthisfieldwouldbe~8·5Mc/s,andthiwouldcausethe powertransmitted through 10kmofthesubstance toallbyonly1·8per cent(seeProblem 23.3),andsoitisquiteoutofthquestion tousea transmission method. Insteadthesubstance isisertedinther.f. magnetic fieldgenerated inthecoilofatunedcircuifor8·5Mc/s.The magnetic resonance phenomenon willthencause changein(l/Q), whereQisthequalityfactorofthecircuit,ofmanitude(seeProb­ lem23.2) Thesmallest valueof(l/Qo),thereciprocal oftheqalityfactorofthe circuitintheabsenceofresonance, thatwecanexpecisabout5X10-a, sothattodetecttheresonance requires ameasurem ntofachangein Qoflessthan1percent. ThecircuitusedforthispurposebyBloembergen, urcell,andPound (1948)isshowninFig.23.6.Powerfromasignalgeeratorisfedtoa lowresistance R,andthenthrough asmallcapacitnee04toatuned circuitwithaQoofabout150.Thissmallresistancand capacitance ensurethatasmallconstant currentisfedtothetundcircuit,andthe 23.3] MAGNETIC RESONANCE 687 voltageacrossthetunedcircuittherefore variesdirectlywiththeQ. Thisvoltageisnotmeasured directly, however, butisbalanced against thevoltageacrossanexactlysimilarcircuit,shownenclosed inbroken lines.Thesampleisplacedinthecoilofonetunedcircuit,andthiscoil isplacedbetweenthepolesofanelectromagnet, theaxisofthecoil(and hencethedirection oftheoscillating fieldB1)beingperpendicular tothe oo I L I I I IL . ~r---------------l [ I I I Signal generator FIG.23.6.CircuitofBloembergen, Purcell, andPound(1948)forthedetection of nuclearmagnetic resonance. R,Rlowresistances (50ohm). 0.,0,capacitors coupling powertothetunedcircuitsL,O. 0a,Oacapacitors coupling powertother.f.amplifier whichdetectsthedifference in thevoltages acrossthetwotunedcircuits. steadyfieldBo•Adjustment ofoneorbothofthecapacitors 04now bringsthevoltages acrossthetwotunedcircuitsnearlytoequality, so thatthedifference voltage, appliedtotheinputofanr.f.amplifier, is smallenoughnottooverload it.ThesteadyfieldBoisthenvaried,and atresonance theQofthecircuitcontaining thesampleisslightly diminished, sothatthevoltageinputtother.f.amplifier alters.This changecouldbeobserved onavacuumtubevoltmeter afterdetection, butforittobeappreciable thebalancebetweenthetwocircuitswould havetobesteadyenoughtoreducethedifference voltagefedtothe receivertolessthanone-thousandth ofthatacrosseithercircuitover thetimeoccupied invaryingBotofindtheresonance. Toovercome thisdrawback, asmalllowfrequency modulation ofafewgaussis 688 MAGNETIC RESONANCE [23.3 superimposed onBo•IfBoisadjusted sothatthisodulation sweeps overaresonance line,thenthechangeinQandhneeinthevoltage inputtothereceiverrecurseachtimethelineistraersedandsogives alowfrequency modulation onthisinput.Afteraplification thiscan bedetected, andapplied (ifnecessary withfurterlowfrequency amplification) totheY-plates ofanoscillograph TheX-sweep is derivedfromthelowfrequency modulation, sothatgivenX-deflexion FIG.23.7.Nuclear magnetic resonance signalfromprotonsinliq'dwator,displayed on oscilloscope. Thevertical deflexion isproportional tothestrengt oftheabsorption and thehorizontal defiexion tothevariation intheapplied agneticfield. (Photograph byR.A.Kamper.) corresponds toagivenchangeinBocausedbythemoulation. Atypical resonance curveobtained inthiswaywithliquidwaerasthesampleis showninFig.23.7.AsthemeanvalueofBo(i.e.theeldcorresponding tothemid-point ofthemodulation) isslowlyvaried,theresonance line appearsatoneendofthetrace,movesacross,anddisappears atthe otherend. Thefirstexperiment ofthiskindwascarriedoutyPurcell,Torrey, andPound(1946).Simultaneously, thenuclearresoancewasobserved independently byBloch,Hansen, andPackard (196)usingaslightly 23.3] MAGNETIC RESONANCE 689 different principle, knownasnuclearinduction. Tounderstand thiswe shallreturntotherotating coordinate systemusedinFig.23.1.Ifthere issomedamping mechanism bywhichenergycanbetransferred from thesystemofnuclearspinstotheoutsideworld,theequilibrium state willbeonewherethenetmagnetization Mwillbeparalleltothesteady fieldBo.Whenarotating fieldB1isapplied,themagnetization vector precesses atanangle8aboutBowiththeangularvelocity wofthe appliedrotating field.Thusitisaconstant vectorintherotating coordinate system,thoughnotcoplanar withBoandB1(thisisthe steadystatesolution oftheforcedprecession ofthedamped system). Theoryshowsthat,ifMisthelinewidth,8isgivenby t8 B1 an={(Bo-B')2+LlB2}i Hence8isgreatestatresonance (Bo=B'),andtherotating com­ ponentMsin8isthenalsoamaximum, leadingtherotating fieldB1 byanangle!1T.Thiscomponent willinduceavoltageinacoilplacedat rightanglestothemaincoilproducing thedrivingfieldB1coswt.The induced voltageisanoscillatory onewiththesamefrequency, andthe detector coilmusttherefore becarefully oriented toreduceasfaras possibleanydirectpick-upfromthedrivingcoil.Hereagainthesteady fieldBois'wobbled' atanaudiofrequency, sincethisdifferentiates between suchstraypick-up, whichwillnotbemodulated atthewobble frequency, andtheresonance effect,whichis. 23.4.Relaxation effectsinnuclearmagnetic resonance ThePurcellmethodisgenerally knownas'nuclearresonance', andthe Blochmethodas'nuclear induction'; theyarealternative methods of detecting thesamephenomenon, andhavethesameultimate sensitivity. Bothhavebeenpushedtothelimitsofsensitivity inapplications such asmeasurement ofthevalueofyforrareisotopes. Inthisconnexion linewidthisofgreatimportance, sincetheintensity atthecentreofan absorption linevariesinversely asthelinewidth(seeequation (23.7», andtheaccuracy withwhichthepositionofthecentreofthelinecan bedetermined isalsohigherforanarrower line.Itturnsoutthatthe linewidthvariesveryconsiderably withthenatureofthesample,and weshalldiscussthisbrieflyfirst. Intheestimate ofthelinewidthforH20madein§23.3thefield duetooneneighbouring protonwasfoundtobeaboutonegauss.This wasanunderestimate ofthewidthtobeexpected, sincethereismorethan oneneighbour, andwewouldexpectthefullwidth(2LlB)tobeabout 851110 YY 690 MAGNETIC RESONANCE [23.4 10gauss.Infactitisfoundtobeabout16gaussinceatlowtempera­ tures.Inliquidwater,ontheotherhand,theresoanceisextremely narrow; sonarrow,thatitsactualwidthisverydiculttodetermine, asvariation ofthefieldproduced bytheexterna magnet overthe volumeofthesampleisusuallythelimiting factorndetermining the breadth. Byworkingatlowfieldstrength, however, rownandPurcell (1949)wereabletoshowthattheoverallwidthwasIesthan0·007gauss. Theexplanation forthisstrikingdifference fromthwidthiniceisas follows.Inwaterthemolecules arenotstationary, utontheaverage changetheirpositions onceevery10-11secorso,thifigurebeinggiven bytherelaxation timeoftheDebyeabsorption discusedin§17.7.This meansthatatagivennucleusthefieldofaneighburingnucleuswill notbeconstant, butwillchangeitsvalueevery10-11sec.Thisisavery muchshortertimethanthatoftheprecession piodinafieldof, say,2kilogauss, whichis~10-7sec.Atfirstsightthrapidfluctuations oftherandomfieldsoftheneighbours mightbeexpetedtobroadenthe line,butinfactthenucleuscannotrespondtochagingfieldswhose duration islessthanT2=(~w)-l=(y~B)-:t, theinerselinewidth:in thewordsofPurcell(1948),'thenucleusridesoutthstormlikeawell­ balanced gyroscope onperfectgymbals'. Itturnsutthatthemore rapidthefluctuations themorecloselydoestheireffctaveragetozero. Iftherateoffluctuation islower,ontheotherhandasitisinaliquid ofhighviscosity suchasglycerine, thelinewidthinotsoeffectively reduced, andincreases rapidlyiftheviscosity isinceasedbylowering thetemperature. WhentheDebyerelaxation tiebecomes much longerthanthecharacteristic timeT2,thefulllinewithisattained. In manysubstances thelinewidthdoesnothavetheepectedwidthdue totherandommagnetic fieldsofothernucleardipolmoments evenin thesolidstate.Thisisattributed tointernal motiwithinthesolid lattice,which'averages out'thefieldsoftheneighours,andnuclear resonance hasbeenusedtoinvestigate suchinterna motions inmany cases. Asecondquestion ofconsiderable importance istherateatwhich energyistransferred fromthenuclearspinsystem 0thelatticecon­ tainingthenuclei.Intheabsenceofanexternal rnaeticfieldBo,the nuclearspinswillpointinrandomdirections, andXow·llbezero.Ifafield Boisnowapplied,theenergylevelscorresponding tdifferent nuclear spinororientations willbesplit,asshowninFig.238.Forsimplicity, aspinI-1isassumed, givingjusttwolevels.Orignally,thepopula­ tionsofthesetwolevelswereequal,andafterBoisswtchedontheywill 23.4] MAGNETIC RESONANCE 691 remainsountilanumberaretransferred fromtheuppertothelower statetogivetheequilibrium Boltzmann distribution inwhich n2=n1exp(-WfkT). Thisinvolves atransferofenergyfromthesystemofspinstothelattice, andthemagnetization approaches itsequilibrium valueMoaccording to B Energylevels AW FieldzeroI A tn{l+exp( -W/2kT)} Population ofeachlevel I= ~--------"'l ::e:::==~;::::- --- --;- --=;::{l-exp( _W/2kT)} I ~Tl--.1 I I IFieldon __ --'-I ----.~Time FIG.23.8.Theenergylevelsandtherelative populations ofanucleusofspinI=t beforeandafteramagnetic fieldisswitched on.Thepopulations approach thenew equilibrium valuesexponentially, withtimeconstant TI•the'relaxation time'. theexponential law(compare thecorresponding equation forelectric polarization in§17.7) dMfdt=(Mo-M)fT1, whichgives Mo-M =Moexp(-tfT1) (23.9) ifM=0att=O.Theparameter T1isknownasthespin-lattice relaxa­ tiontime,sinceitdetermines therateatwhichenergyistransferred from themagnetic dipolesofthesystemofnuclearspinstothecrystallattice inwhichtheyareembedded. Suchatransfer requiresthattransitions beinducedbetweenthevariousnuclearlevelscorresponding todifferent orientations, andthesecanonlybecausedbythepresence ofanoscillating magnetic fieldwhosefrequency satisfiesthecondition forresonance. Furthermore, thisoscillating fieldmustoriginateinthethermalmotion ofthesurroundings ofthenucleus. InaliquidtheBrownian motionof neighbouring molecules causesthelocalmagnetic fieldsoftheirnuclei tofluctuate rapidly,andthisrandomfluctuation contains components oftherightfrequency tocausetransitions. Fordistilledwaterat20°C, 692 MAGNETIC RESONANCE [23.4. Bloembergen, Purcell,andPound(1948)foundthvalueofPltobe about2sec.Thiscanbeshortened bydissolving aparamagnetic salt inthewater,sothattheprotonsinteractwiththemuhbiggerelectronic magnetic moments. Another featureisthatonewoulexpectthedesired component oftherandomly fluctuating magnetic fiIdstobegreatest (andhencePltobeshortest) whentheDebyerelaxati ntimeoftheliquid Tisoftheorderl/wo,where Woistheangularfrequecyofthemagnetic resonance. ThiswasverifiedbyBloembergen, Purell,andPoundby measuring Plinglycerine overarangeoftemperat res,whichgivesa widerangeofviscosity andhenceoftheDebyereIxationtimeT.In solidsatlowtemperatures, wherethereispractically 0thermalmotion, weshouldexpectPltobeverylong,butinpracticitturnsouttobe nothinglikeaslongasthevaluespredicted bytheor.Thisisascribed tothepresence ofparamagnetic impurities, wheretheelectron spins turnoverandsogiveafluctuating field.Owingtotirlargermagnetic moments, veryfewsuchimpurity ionsarerequire,andthethermal contactbetweentheelectronspin(ororbit)andtheItticeismuchmore intimate thanthatbetween anuclearspinandtheattice(see§23.7), sothatatransfer ofenergytotheelectron spinsromthenucleiis effectively atransfertothelatticeitself. Thevariation ofPlwiththeDebyerelaxation tieTforthreesub­ stancesisshowninFig.23.9.Inethylalcohol, T~(110)andPldecreases asTincreases. Inglycerin Plpassesthrough aminiumandincreases again,whileinice,where T~(1/wo),Plisrisingwithincreasing T.This variation ofTl(withacorresponding increaseinthe. ewidthtowards thevaluecalculated foranassembly ofstaticdiplesasTincreases) confirms theinternal molecular motioniniceindictedbytheDebye relaxation. Internal motions existinmanyothersolds,andhavebeen investigated bymeasurement oftherelaxation time 1aswellastheline widthinnuclearmagnetic resonance. Suchexperientshaveadded considerably toourknowledge ofthesolidstate. 23.5.Applications ofnuclearresonance Themostobviousapplication ofnuclearresonanc istothemeasure­ mentofyforallpossibleisotopes. Adirectmeasure entofyinvolves aprecisemeasurement ofthefrequency, whichiscmparativelyeasy, andofthemagnetic fieldBo,whichisdifficult. Fort .sreasonitisusual tomeasure insteadtheratioofyfortheunknown istopetothatfora standard isotopesuchaslH,theproton. Thismabeaccomplished bymeasuring thetwofrequencies ofmagnetic resoanceinthesame 23.5] MAGNETIC RESONANCE 693 magnetic field.Toachievehighaccuracy, asinglesampleisused,suchasa solutionofasubstance containing theunknown isotopeinwater.From thediscussion oflinewidthin§23.4itwillbeappreciated thatasolution givesthenarrowest lines,andhencethegreatest accuracy aswellasthe greatest intensity atthecentreoftheline.Thesampleissurrounded bytwocoils,oneforeachofthetworesonance frequencies, whichare PI=Nuclearrelaxation time inseconds 10-4L.....-,-,----__-L-=-- -'-:- ---'--::-__---,-__=' 10-11 10-9 10-7 10--6 10-3 T=Debyetimeinseconds FIG.23.9.Thethermal relaxation timeTIforprotonsinethylalcohol, glycerin, andice, measured at29Mc/s,plottedagainsttheDebyerelaxation timeT,obtained fromdielectric dispersion data.Notethelogarithmic scales.Theslopeofthesolidlinesforalcoholand iceandtheshapeofthesolidcurveforglycerin havebeendrawninaccordance with theory. usuallyarranged tobemutually perpendicular soastominimize the mutualinductance between them.Theaccuracy whichcanbeachieved inthiswayisillustrated bytheratioofthenuclearmoments ofthe deuteron andtheproton,foundbyWimett (1953)tobe mn!mH=O·307012192±O·000000015. Themeasurement wasmadewithcompressed HDgasinordertoavoid difficulties with'diamagnetic shielding' (seebelow),andcertainother smallcorrections. Atableofnuclearmoments forhydrogen, thealkalimetals,andthe halogens isgiveninTable23.1;acomplete listwouldoccupymanypages. Theearliervaluesobtained bytheuseofmolecular beamsareshown forcomparison; inthecaseoftheneutronthevalueisthatobtained by specialbeammethods outlined in§23.2.Theprotonvalueisobtained fromanabsolute measurement ofy,described below,andothermag­ neticmoments fromthemeasured ratiosofytothatfortheproton. 694 MAGNETIC RESONANCE [23.5 Theabsolute measurement ofyforonenucleusiobviously ofcon­ siderable importance, theobvious choicebeingtheroton(whichhas ahighmagnetic moment andgivesalargesignal)inliquidwater.The principal difficulty isthatofmeasuring themagnetic eldwithsufficient accuracy, whichcanmostconveniently becarriedutinastandards laboratory. Thefirstsuchmeasurement wasthat0Thomas, Driscoll, andHipple(1950)attheUnitedStatesBureauofStndards,whoused amagnetic fieldof4700gauss,determined bymensoftheCotton [B'o •••••••••• water••••••••••~ lOOV \,...-....,.111111L-.---...I. .coilforpolarization B anddetectionLT'/...=':..~!'l'......",t+ 0 .....-. FIG.23.10.Planoftheapparatus usedbyVigoureux (1962)foreasuring thefrequency ofnuclear precession oftheprotoninaweakmagnetic field.T efieldBoisprovided bycoilsofknowndimensions carrying acurrent Imeasured bthovoltage Vacross astandard resistorR.Thepulseofthepolarizing fieldB~isprvidedbyasubsidiary coil,inwhichthesubsequent freeprecession induces avoltage hichisamplified and whosefrequency isdetermined bytheperiodcotel'. balance (see§8.6).Latermethods haveusedfieldsoforder10gauss produced byastandard solenoid whosedimensi nsareaccurately known,sothatthefieldcanbecalculated fromthsedimensions and thecurrent. Toobtainsufficient signalatsuch10frequencies, the methodof'freeprecession' isused.Bymeansofsubsidiary coil,a fieldBoof100-1000 gaussissetupattheprotonsapIeinadirection normaltothestandard fieldBo•Thesubsidiary fielB~ismaintained forashorttime(>T1),longenoughforthesampletacquireanuclear magnetization paralleltotheresultant field(Bo+Bo)ndproportional to it.Onremoving Bowehavetherefore acomparative ylargemagnetiza­ tionM=(xlfLo)(Bo+ Bo)whichisalmostnormaltothstandard fieldBo, andtherefore precesses aboutBowithangularfrequncywp=-YPBo• 23.5] MAGNETIC RESONANCE 695 Duringthisprocessthemagnitude ofMdecaystowardsthevalueMo appropriate tothefieldBo(inthermal equilibrium Mowillalsobe paralleltoBo).Theprecessing magnetization inducesanalternating voltageinasuitably oriented coil,whosefrequency (Wpj27T)canbe determined withanaccuracy limitedbythenumberofcycleswhich elapsebeforethesignalbecomes toosmalltobeobservable. TABLE23.2 ValuesofYP=-wpjBoasdetermined atsomeStandards laboratories. Thelastresultisreferredtotheunitofelectriccurrentmaintained atthe N.P.L., ratherthantheabsoluteampere,andoncorrection agreeswiththe measurement ofDriscollandBenderwithin1·1p.p.m.Thevaluesquoted donotincludethediamagnetic correction Method N.M.R. (Thomas. Driscoll, andHipple, 1950) Freeprecession (Driscoll andBender, 1958) Freeprecession (Vigoureux, 1962)'l'P(inunitsof(gaU88-8ec)-1) 26752·3 (±22p.p.m.) 26751·3(±7p.p.m.) 26751·71 (±2p.p.m.) AplanviewofthecircuitusedbyVigoureux (1962)oftheUnited Kingdom National Physical Laboratory isshowninFig.23.10.The currentIthroughthefieldcoils(solenoid) isequaltoVjR,whereVis astandard cellandRastandard resistor; toobtainIintermsofthe absolute amperetheratioVjRmustbedetermined bymeansofa currentbalance. Theresultsofthreeindependent determinations ofYP (inliquidwater)aregiveninTable23.2,together withtheestimated errorinpartspermillion(p.p.m.). Thetwomorerecentresultsquoted agreewithinaboutIp.p.m. Theabsolute determination ofYPisimportant notonlybecauseof itsuseindetermining nuclear moments, butalsobecause nuclear resonance offersasimpleandaccurate laboratory methodofmeasuring amagnetic field.Onlyafrequency measurement isrequired, whichcan readilyachieveanaccuracy ofIpartin104ifasimplefrequency meter withaquartzcrystaloscillator checkisused,orhigherifafrequency standard isavailable. Themainrequirement isthatthemagnetic field mustbeuniform overtheprotonsampleinordertoavoidbroadening theresonance, butthisisnotoftenaseriouslimitation. Thefreepre­ cessionmethodcanbeusedforverysmallfields,suchastheearth'sfield, whichcanbemeasured toabout1partin105•Thisaccuracy iscom­ parablewithothermethods suchastheearthinductor whichhavethe disadvantage ofmeasuring onlyacomponent oftheearth'sfield,80 696 MAGNETIC RESONANCE [23.5 thattheorientation ofthesensingelementmustbeknownaccurately (theadvantage inthisrespectofa'protonmagnetom ter'infindingthe earth'sfieldatthebottomof,say,theAtlantic Ocancanbereadily appreciated). Incomputing anuclearmagnetic moment from'tsprecession fre­ quencyinanexternal fieldBoacorrection mustbappliedfor'dia­ magnetic shielding'. Thiseffectiscloselyalliedto'amagnetism, and arisesfromtheprecession oftheclosedshellsofelctronsaboutBo, whichsetsupasmallfieldatthenucleuswiththeopositesensetoBo, thusmakingtheactualfieldactingatthenucleusslihtlysmallerthan theexternal field.Theapparent valueofy,ifnocoectionismade,is therefore lessthanthetruevaluebyafractional amountofabout 2·8X10-5forhydrogen, risingto10-2fortheheavistelements. This correction hasbeencomputed, withaprobable errrrisingtoabout 5percent(whichisgreaterthantheexperimental or)intheheavy elements. Thisshielding effect,whichmakesthefielintheinteriorof anatomdifferent fromthatoutside, mustbedistiguished fromthe 'demagnetizing field'(§5.4)andthe'localfield',whiccanheevaluated sufficiently accurately forthispurposebythemethdduetoLorentz (cf.§17.2).Eachofthesefieldsisproportional totheulkdiamagnetism ofthesample,andinaspherical samplethedemagne izingfieldandthe Lorentz fieldjustcancel,sothattheaveragelocaleldisthesameas theexternal field,apartfromthediamagnetic corretion. Theshiftduetodiamagnetic shielding depends 0thelocaldensity ofelectrons, andmaythusvaryfromcompound tocompound; in addition theremaybeshiftsduetotheinducedparmagnetic moment incompounds whichhavetemperature-independe tparamagnetism suchasKaCo(CN)6; theseeffectsareknownas'cheicalshifts'.The diamagnetic shielding mayalsovarybetween differntnuclearsitesin thesamecompound; forexample, inCH3CH20Ht eresonances from protonsintheCH3,CH2,andOHgroupsaresepaatedinfrequency byabout1partin106.Inveryhighresolution n..r.(1partin108 maybeobtained byspecialmethods) furthersplitingsduetointer­ actionsbetween neighbouring protonscanalsobersolvedinliquids, withimportant chemical applications. Suchpossibilities arisebecause thefrequency 0nuclearmagnetic resonance isameasureofthemagnetic fieldatthepoitinthecompound occupied bythenucleuswhoseresonance isbeingobsrved.Largeshifts areobserved instrongly magnetic solids,making nuclear magnetic resonance animportant toolintheinvestigation ofmneticcompounds. 23.5] MAGNETIC RESONANCE 697 Ingeneraltheelectronic magnetic dipoleschangetheirorientation very rapidly,eitherthroughrelaxation effectsduetothethermalfluctuations ofthelatticeorthroughinteraction withneighbouring spins,andsolong asthisreorientation occursmanytimesinatimeT2=(YnLlB)-lcharac­ teristicofthenuclearresonance linewidthLlB,relatively narrown.m.r. linesareobtained, shiftedinfrequency byalocalfieldwhichispro­ portional tothetimeaverageoftheelectronic magnetic moment. Thus theshiftistemperature dependent andproportional totheaverage electronic magnetization; inaparamagnetic substance wecanwritefor thenuclearprecession frequency (a)=-Yn(Bo+aM) =-YnBo(1+aX/JLo), showingthatthefractional changeinfrequency isproportional tothe susceptibility. Inanorderedmagnetic substance, wherethemagnetiza­ tionisfiniteintheabsenceofanexternal field,nuclearmagnetic reso­ nancecanbeobserved atafrequency whichisverynearlyproportional tothemagnetization inaferromagnetic substance, ortothesub-lattice magnetization inananti-ferromagnetic orferrimagnetic compound. Thenuclearmagnetic moment of19Fisquitelarge,andsignalsof highintensity havebeenobserved fromnuclearmagnetic resonance of thisnucleusinanumberofmagnetic compounds. Itwasfoundby Shulman andJaccarino (1956)thatinMnF2intheparamagnetic state, the19Fresonance wasshiftedbyanamountproportional totheelectronic paramagnetic susceptibility. However, themagnitude oftheshiftwas greaterthanwouldbeexpected fromthedipolarmagnetic fieldatthe fluorinesiteduetotheelectronic magnetic moments, assuming themto belocalized ontheMn2+ions;thelargerfieldisconsistent withaspread ofthewavefunctions ofthemagnetic electrons ontotheF-ions,due toasmallamountofcovalent bonding. Intheanti-ferromagnetic state theelectronic moments arefixedinorientation, andthefieldwhich theyproduceattheF-nucleusisquitehighevenwhenBo=O.At 0°KinMnF2theprecession frequency of19Finthiselectronic fieldis 159·99Mc/sfalling,asthetemperature rises,tozeroattheNeelpoint, 67·3°K.Acomparison ofthenuclear resonance frequency, which shouldbeproportional tothesub-lattice magnetization, withthe magnetization calculated fromthemolecular fieldtheory,isshownin Fig.23.11. 23.6.Electron magnetic resonance inatomicbeams Magnetic resonance experiments involving electronic magnetic moments canbecarriedoutinanalogous waystoexperiments with 698 MAGNETIC RESONANCE [23.6 nuclearmagnetic moments. Ingeneral, experime tswithelectronic moments areeasier,becausethemoments aresomuhlargerandmag­ neticresonance iscorrespondingly easiertodetect. ntheanalogue of Rabi'sexperiments (§23.2),beamsofatomswitheectronic magnetic dipolemoments areused,forwhichappreciable deflexions inthe inhomogeneous magnetic fieldsA,B(seeFig.23.3canbeachieved usingrelatively shortmagnets. Themaininterest 0suchexperiments 120' 6667 68 52 57585960616263 Temperature indegreesKelvin FIG.23.11.Temperature dependence ofthe19Fnuclearmagneti resonance frequency inMnFsbetween 520KandtheNeelpoint,67.30K.Thelowerc eshowsthevariation expected ifthesub-lattice magnetization, relativetothatat00 ,followed thecurve computed frommolecular fieldtheoryforS=t.(HellerandBnedel<,1962.)Inthis andothersubstances theresonance frequency variesnearNas(TN-T)t.103080 2090 ~ ::a70 .$ ~60 ~ t1'50£ 40 arisesfromthehighprecision whichcanbeobtained byincreasing the lengthofthea-field,sincefromequation (23.6)thewidhoftheresonance curvedepends onthetimetduringwhichtheatomisubjected tothe radio-frequency field.Ifthevelocityofatomsintheeamis105em/sec, andthelengthofthea-fieldis3em,thevalueofti 3X10-5sec,and thecorresponding linewidth 2~visabout30kc/s. 0achievesucha narrowlinethea-fieldwouldhavetobeveryhomgeneous, sincea 23.6] MAGNETIC RESONANCE 699 variation ofthefieldbyaslittleas10-2gausswouldchangetheelectronic magnetic resonance frequency ofanatomwithg=2by30kc/s.The requirement ofsuchhighhomogeneity isavoidedbyuseofthetwosep­ aratedoscillatory fieldsmethodofRamsey (1949),asmentioned in§23.2. Foranatomwithanelectronic magnetic moment butnonuclear moment themagnetic resonance transitions occuratafrequency (cf.equation (23.5» (23.10) anddetermination ofvinaknownfieldBogivesaprecisemeasurement ofgJ.Asmentioned in§20.2,therearecorrections toequation (20.11)for gJduetodiamagnetic shielding (cf.§23.5)byotherelectrons andthe relativistic increaseinthemassoftheelectron, whichamounttobetween 10and100p.p.m.Amorefundamental correction isduetotheintrinsic magnetic moment oftheelectron spinbeingslightlygreaterthanOIie Bohrmagneton. Firstindications ofthiswereobtained fromatomic beammeasurements ofthehyperfine structure ofhydrogen (seebelow), andfrommeasurements oftheratioofthevaluesofgjfortwostatesof thesameatomwiththesame8,lbutdifferentj (seeProblem 23.6).The accepted valueofUsisinagreement withthatcalculated usingquantum electrodynamics bySommerfield (1957): gs=2(I+o:/27T-0·328o:2/7T2+...)=2{1'001I596), (23.1I) where 0:isthefinestructure constant. Animportant application ofatomicbeammagnetic resonance isthe precisemeasurement ofhyperfine structure inatoms.Aspointedout in§20.10,inanatomwithbothelectronic andnuclearmoments the nuclearmagnetic moment precesses intheelectronic magnetic fieldBe, andtheelectronic momentinthenuclearmagnetic fieldBn,theresult beingaprecession ofeachmoment abouttheresultant angularmomen­ tumvectorF.Thisgivesrisetoasetofhyperfine energylevels,as illustrated inFig.20.20,wheretheeffectofanuclearelectricquadrupole interaction isalsoshown.Inmagnetic resonance thetransitions for whicht1F=±1haveanintensity associated withanelectronic mag­ neticdipolemoment, andcanreadilybeobserved; frommeasurements ofthefrequencies oftwosuchtransitions thevaluesofthehyperfine constants AandBQcanbedetermined. Theserequirenomagrtetic fieldintheO-magnet, butinpractice asmallfieldBoisgenerally used, whoseeffect(seeFig.23.12)istospliteachsetofstateswithagiven valueofFinto2F+1levels,provided thattheZeemanenergy(lJf3{J .Bo) issmallcompared withthehyperfine energies. Thiscorresponds toa 700 MAGNETIC RESONANCE [23.6 precession ofthevectorFaboutBoatarateslowcmparedwiththe precession ofJ,IaboutF.Theallowedtransitions rethenthosefor whichI1F=0,±1;I1mF=0,±1,butthe11m=transitions occur onlyiftheoscillatory fieldhasacomponent paralleltoBo(the11m=±1 transitions requireacomponent Blperpendicular to0'asinFig.23.1). 1350kc/s per~&uss ]mp 43------ ----­2----- ----- 1---- ----- o ---- -1 -2- =:1 -3 -2- -1 o 1 2 3F=3 atransitions 7Ttransition l!J'=±l, l!J'=±l, Amp=0 Amp=±1 FIG.23.12.Thehyperfine structure of133CS,S=t,I=t,shoingtheZeeman effect inasmallfield(notethedifference inscalebetween theZeeman slittingsandtheoverall splitting) andtheallowedtransitions (afterEssenandarry,1957).magneticfield-----.1'.~"92""I- energy,W Theatomicbeammagnetic resonance techniqu hasbeenwidely appliedtomeasure theconstants A,BQ,andina£atomsafurther veryweakinteraction between anuclearmagnetic octupole moment andthesecondderivative oftheelectronic fieldBeasbeendetected. Itcanalsobeusedwithradioactive isotopes, whichc nbemeasured by meansoftheirradioactive emission afterbeingcollecedatthedetector. Onlytwoexamples willbediscussed here;thehyprfinestructure of caesium, whichgivesanatomicstandard offrequenc ,andthehyperfine structure ofhydrogen, becauseofitsfundamental iportance. Thegroundstateofthecaesium atomis28t,a dtheonlystable isotope l33eshasnuclearspinI=t.Sincetheelectonicgroundstate is8=I,thereisnoelectricquadrupole interaction, a dinzeromagnetic fieldtherearetwosetsofhyperfine levelscorrespon 'ngtoF=3and 23.6] MAGNETIC RESONANCE 701 F=4,asinFig.23.12.Caesium isfairlyvolatile, sothatanatomic beamofsufficient intensity canbeemittedfromanovenat2000C,and thiscombined withtheratherhighatomicmassgivesalowthermal velocityandincreases thetimerequired totraverse theC-field,where adistance ofabout50cmisusedbetween theseparated oscillatory fields(seeEssenandParry(1957». Asmallfieldof0·05gauss(this requires cancellation oftheearth'sfield,whichisabout0·5gauss)is maintained astheC-field,andtheresonance observed isthetransition (F,mF)=(4,0)~(3,0),whichhasonlyasecond-order Zeeman effect v=vo+426Bgcis(Boingauss). Following collaboration betweentheStandards laboratories oftheU.K. andtheU.S.,thevalueofVoisfoundtobe(seeMarkowitz, Hall,Essen, andBarry(1958» Vo=9192631 770±20cis(Ephemeris time)at1957·0. Theatomicfrequency standards inthetwocountries havebeenfound toagreewithin1partin1010incomparisons madeoverseveralyears, andthisaccuracy (Icis)ishigherthanthatwhichcanbeobtained in determining themeanrateofrotationoftheearth.Thevariation inthe lengthoftheday,asmeasured attheNational Physical Laboratory over aperiodofsevenyearsagainstacaesium 'atomicclock',isshownin Fig.23.13;theannualvariation isabout1millisecond perday(about 1partin108)andthereisasuggestion ofalongertermchangealso. Aninternational recommendation in1964makesthecaesium 'clock'the newstandard offrequency, thesecondbeingdefinedasthetimeinter­ valcontaining exactly9192631770 cyclesofthecaesiumhyperfine fre­ quencyinzeromagnetic field. Thegroundstateofahydrogen oralkalimetalatomis28:.,sothatthe electronic fieldatthenucleusisdueonlytotheelectron spinofthe oddelectron, whichisinans-state.Theelectron densityinsuchastate isspherically symmetric, withamaximum atthenucleusandfalling exponentially withdistance. Themagnetic moment duetotheelectron spinissimilarly distributed, andwecanregardtheatomaspossessing amagnetization which(because ofthenegative electronic charge)is anti-parallel tothespin,anddistributed inaspherically symmetric fashion. Thisisequivalent toaferromagnetic spherewhichisevery­ wheremagnetized inthesamedirection butwithavaryingintensity M=-gs,BSltflI2, wheretflisthewavefunction forthes-state. Thenuclearmagnetic 702 MAGNETIC RESONANCE [23.6 moment isthenduetoasmallcurrentloopimmesedinthisferro­ magnetic mediumatthecentre.Theinteraction beteenthetwogives anenergyW=-mn.Be.Incalculating Beathecentreofthe spherical distribution wenotethatthefieldatt ecentredueto auniformly magnetized spherical shelliszero(seeProblem 5.9),so thatthewholecontribution toBecomesfromtheagnetization Mo 2·0 ~~ ~ ~ ~ ~ FIG.23.13.Variation inthelengthoftheday,asmeasured b clock(courtesy oftheDirector, National Physical LI, ,b""1>..... ""~..,"0> 0>>-,.... .... ~ ~te>-, atthecentre.Fromequation (5.34),intheabsence 0anexternal field, wehaveH=-lMo,sothatBe=JLo(H+M o)iJLoMo.Hence W=-~.Be =-(YnPnI).(-iJLoYsPSIif1( )12) =iJLoYnYsPPnltft(O)12S.I AS.I. (23.12) Foratomichydrogen intheIs-state, 1if1(O)12=I!'TTa8whereao=Bohr radius,beingthefractionoftheelectrontobefounderunitvolumeat thenucleus. Afterinserting smallrelativistic andreucedmasscorrec­ tions,thebestvalueofAJhobtained fromtheOohe,Dumond, et.al. (1955)valuesoftheatomicconstants is(assuming g=2exactly) AJh=I4I8·90±2 p.p.m.McJs andsinceS=f'I=!thisshouldbeequaltothfrequency ofthe singletransition between thetwostatesF=0 dF=Iofthe 23.6] MAGNETIC RESONANCE 703 hyperfine structure. Thebestexperimental valuesare 1420·40573±0·035 p.p.m.Mc/s(Kusch, 1955), 1420'40580±0'04 p.p.m.Mc/s(WittkeandDicke,1956), wherethefirstresultwasobtained byanatomicbeammethodandthe secondbyanothermethod. Thediscrepancy withthetheoretical value iswelloutsidetheexperimental error,anditwasthefirstdiscovery of thisdiscrepancy in1947-8thatledtothesuggestion thatgsisgreater than2. Adetailed theoretical treatment showsthatsomeothersmallcorrec­ tionsbesidesthatforgsarerequired intheformula forA,andthe discrepancy withexperiment hasbeenremoved. Atalmostthesame timeasthisdiscrepancy wasdiscovered ananomaly intheseparation oftheelectronic 28and2pstatesofatomichydrogen wasestablished bytheexperiment ofLambandRetherford (1947).Thishasalsobeen explained byquantum electrodynamics. Experiments ofthekindmen­ tionedaboveandothersarediscussed inTheSpectrum ofAtomic Hydrogen, byG.W.Series. 23.7.Electron magnetic resonance insolids Magnetic resonance experiments onsubstances containing permanent magnetic dipolemoments duetoelectrons canbecarriedoutina manneranalogous tothoseonnucleardipoles,butthereareanumber ofsignificant differences. Iffieldsofafewkilogauss areused,the resonance frequency forelectrons isinthevicinityof1010cIs,corre­ sponding towavelengths ofafewcentimetres. Frequencies ofthisorder andhigherareinfactusedforanumberofreasons: (a)thesensitivity ishigh;thisispartlythrough equation (23.7),and partlybecauseabetterfillingfactorcanbeobtained fromasmall samplebyusingitinatunedcircuit(acavityresonator) which hassimilardimensions; (b)theelectronic levelsmayhavesplittings oforder0·1cm-1ormore duetocrystalfieldeffects(see§20.7); (c)linewidthsinthesolidduetomagnetic fieldsofneighbouring ions areoforder102-103gauss,andtoachievereasonable accuracy in determining thecentreofalinemeasurements mustbemade usingexternal fieldsaslargeaspossible. Linewidthandshape arealsoaffectedbyexchange interaction between theions;this canbeavoidedbymakingmeasurements on'diluted'crystals­ crystals inwhichmost·oftheparamagnetic ionshavebeen 704 MAGNETIC RESONANCE [23.7 (23.13)replaced bydiamagnetic ions. Ie,acrystalof K2Zn(S04}2,6H20 containing afewtenthsofpercentofCu++ (3d9)ionsreplacing Zn++(3dIO)ionsgivesaliewidthofabout 10gauss;thisresidualwidthisduemainlytothnuclearmagnetic moments oftheprotonsinthewaterofcrystllization, andcan befurtherreducedbygrowing crystalswith20insteadofH20 becauseofthesmallernuclearmoment ofthedeuteron. Anotherimportant featureofelectronspinresonan einparamagnetic substances isthatspin-lattice relaxation timesmayeextremely short. Obviously theelectron, withitslargermagnetic oment,willbein moreintimate contactwiththelatticevibrations thn anucleardipole, butamuchmoreimportant effectisthatthelatticvibrations distort thelocalsurroundings ofaparamagnetic ion,andsoprduceafluctuating modulation ofthecrystalelectricfieldortheliganfield.Theextent towhichthisaffectsthemagnetic dipoledepend onthedegreeof 'quenching' oftheorbitalmoment: foranioninS-state, suchas Mn++(3d5,6S.)orGd3+(41',SSt),thereisnoorbialmoment except through smalldepartures fromRussell-Saunders oupling, andthe spin-lattice relaxation timeTIvariesfrom~10-6seatroomtempera­ tureto10-3secatliquidheliumtemperatures. orotherionsthe valuesofTIareverymuchsmaller,andinmanyinsofthe4/group TIissoshortandthelinessobroad (~w=Til)thatagneticresonance isunobservable, exceptatliquidheliumorliquidydrogen tempera­ tures.ThevalueofTIalwaysincreases asthetempeaturefallsbecause thelatticevibrations dieout,thevariation beingineneraloftheform 1T.=aT+bTn+cexp(nw/kT}. 1 ThefirsttermaTisdueto'directtransitions' inwhihmagnetic quanta areexchanged withlatticevibrations ofthesamfrequency asthe magnetic resonance frequency; thesecondtermbit(wheren=5,7, or9according tothetypeofmagnetic ioninvolved) isdueto'indirect' or'Raman' processes inwhichanytwolatticevibrtionsareinvolved whosefrequency difference isequaltothemagnetic rsonancefrequency (wlattice=Wlattice±wresonance); theexponential terisduetolattice vibrations whosequantanwcoincide withthefference inenergy between thegroundstateandanexcitedstateofthemagnetic ion.The secondtwoprocesses areweakerthanthefirst,a datliquidhelium temperatures thefirsttermalmostalwayspredo.ates. Aspin-lattice relaxation timeT1willproduce alinewidthof2~v 23.7] MAGNETIC RESONANCE 705 between thehalf-intensity pointsofaline,where2miv=Aw=TIl, andmeasurements oflinewidthcanbeusedtofindTlwhenthisisthe dominant effectinthelinewidth.Forotherpurposes theneedforhigh resolution inordertoobtainaccurate measurements makesitdesirable, however, toworkattemperatures wherebroadening duetospinlattice relaxation isnegligible, andatypicalapparatus forlowtemperature workisoutlined inFig.23.14.Powerfromamicrowave oscillator pen recorderPhase sensitive deteotorAmplifier Magnet pole. pieoeMonitor &nd frequency meter Ma.gnet pole. pieoeDewarvessel;Klystron osoilla.tor Singlecrysta.l_pIe FIG.23.14.Outlinediagram ofanelectron spinresonance apparatus. Powerfroma klystron oscillator isfedthrough awaveguide toalooselycoupled resonant cavity containing theparamagnetic sampleandimmersed inarefrigerant between thepoles ofanelectromagnet. Asmallfraction ofthecavitysignalisfedthrough asecond waveguide toasilicondiodedetector. Themodulation duetotheabsorption isamplified anddisplayed onanoscilloscope orfedthrough aphase-sensitive detector toapen recorder. (usually aklystron) iscarriedbyawaveguide orco-axial cabletoa cavityresonator contained inadewarvesselandplacedbetween the polesofanelectromagnet. Whenthelatterisadjusted toresonance, powerisabsorbed intheparamagnetic sample,whichisplacedinside thecavityinapositionofmaximum oscillatory magnetic field.This additional power~lossinthecavityproduces achangeinthesignal reflected fromthecavity,orinthesignaltransmitted through the cavitytoanotherwaveguide orco-axialline,whichisdetected bya siliconcrystalrectifier. Normally thefieldoftheelectromagnet is 861110 zz 706 MAGNETIC RESONANCE [23.7 Inodulated atanaudiofrequency, givingacorrespo dingmodulation ofthesignalwhenthefieldformagnetic resonance itraversed; after deteotion, thismodulation isamplified anddisplayed nanoscilloscope orarecorder. Thesensitivity achieved isquitehigh,andsignalsfrom asfewas1013electronic dipolescanbeseenifthelinearenarrow. +1 -I+1 ~t~<;CC±l- ....t. FIG.23.15.Energylevelsandabsorption curveatconstant freuencyforanionwith S=!andacrystaJfieldsplitting: W=g,8SzBz+D{~-lS(S+I)}; whenthemagnetic fieldisparaJ1eltotheprincipal axis(z.axis) 0thesplitting term(for otherdirections thelevelsdonotdiverge linearly withfield).Theintensity ofthe Sz-(Sz-l)transition isproportional to{S(S+I}-Sz(Sz-I}}, 'vingthe3:4:3inten· sityratioshowninthefigure. Theresultsofelectron magnetic resonance in havegreatlyadvanced thedetailed understanding theproperties of paramagnetic ionssubjected toligandfieldinteracionsinsolids.In generaltheresonance spectrum isveryanisotropic, epending strongly ontheanglebetween theexternal fieldandthecstalaxes;forthis reasonsinglecrystalsmustbeused.Indilutesaltst esplittings ofthe levelsduetotheexternal fieldasafunction ofanIecanbestudied, together withanycrystalfieldsplittings ofthesamerderasthemicro­ wavefrequency (seeFig.23.15).Whenthenucleus 0theparamagnetic ionorofaligandionhasanuclearmoment, ayperfine structure 23.7] MAGNETIC RESONANCE 707 Btl=Bo-JLoDz~' (23.15)maybeobserved, asinFig.23.16.Anumberofnuclearspinsand moments havebeendetermined fromhyperfine structure inelectron magnetic resonance, andthedegreetowhichthewavefunctions ofthe magnetic electrons overlapontotheligandionsbecauseofcovalent bonding effects(see§20.8)canbeestimated fromthehyperfine struc­ tureduetointeraction withthedipolemoment oftheligandnucleus. Inmoreconcentrated saltstheeffectofmagnetic dipoleandexchange interaction between neighbouring dipolescanbestudied,givingoneof thefewdirectmeasurements ofexchange interaction. FIG.23.16.Hyperfine structure oftheliMn(1=!)nucleusintheelectronspinresonance spectrum ofaMn++ion(8=t,transition 8.=t_-t).Theresonance condition is hv=gfJ(Bo+B n),whereBnisthemagnetic fieldduetothenucleus.Infirstapproxima­ tionBnisproportional tothenuclearmagnetic quantum numberI.;thisgivesapattern of21+1=6lines,equallyspacedandofequalintensity, sinceallnuclearorientations areequallyprobable atthetemperature oftheobservation. Thelineshapeisthe derivative oftheabsorption curve;itisobtained byasinusoidal modulation ofBowith amplitude smallcompared withthelinewidth.Thisgivesacorresponding modulation ofthesignal(measured byaphasesensitive detector) whoseamplitude isproportional totheslopeoftheabsorption curve. Ferromagnetic reaonanoe Insubstances wheretheexchange forcesarestrongmagnetic resonance maybeobserved intheco-operative statebelowthetransition tempera­ ture.Sinceallthedipolesarecoupledtogetherbytheexchange forces itisconvenient toworkintermsofthemagnetization M,whichisthe vectorsumoftheindividual dipolemoments m.Byperforming this vectorsumoverbothsidesofequation (23.1)weobtaintheequation ofmotionforthemagnetization dM/dt=yMAB, (23.14) wherewehaveassumedthatalldipoleshavethesamevalueofy.Here wehavewritten,notBotheexternal field,butBthefieldwithinthe sample,sinceinaferromagnetic substance demagnetizing fieldsmay bequiteimportant. WeshallassumethatBoisalongthez-axis,and confineourselves tocertainsampleshapessuchthatwecanwritefor thecomponents ofB: Bz=-l-'oDzM z; 708 MAGNETIC RESONANCE [23.7 (23.16)Onsubstituting intoequation (23.14)weobtain dMxldt=y.My{Bo+1Lo~(DII-Dz)} dMlIldt=-yMx{Bo+1LoMz(Dx-Dz)} , dMzldt=rlJLoMx.My(Dx-D II)} whicharenolongerlinearinM.However, theequatonscanbesolved inthelimitofsmallamplitudes ofprecession, whent eproductMxMII becomes vanishingly smallandcanbeneglected. hend~/dt=0, andMzisconstant, itsvaluebeingequaltothestat0magnetization; thisislargeinaferromagnetic substanoe, andthecoectionstoBoare important indetermining theresonance frequency. Bysolvingthe equations forMz'.Myitiseasilyshownthattheprecesionfrequency is Therearethreesimpleoasesofinterest: (a)asphere,forwhichDz=DII=Dz;theprecssionvelooity is WL=-yBo,thesameasiftherewerenodemgnetizing fields; (b)athinplanefilmnormaltoBo,forwhichDz=II=-1,Dz=0, giving WL=-y{Bo-1Lo~}; (c)athinplanefilmparalleltoBo,forwhichDz= (assuming thefilmtobenormaltothey-axis), WL=-y{Bo(Bo+1Lo~)}l. Thistreatment assumesthatthemagnetization (ieludingthepre­ cessingcomponents) isuniform throughout thesapIe;thisrequires thatthedimensions besmallcompared withtheavelength inthe sample,andinaconducting samplethismeans.small ompared withthe skin-depth. Hencespherical samplesofmetalmusteverysmall,and colloidal samples (wheretheparticles areassume tobespherical because ofsurfacetensioneffectsinformation) avebeenused. Mostworkhasbeendoneonthinplanesamples, whihareattached to (butinsulated from)onewallofthemicrowave cavi.Theequations showthatthemagnetization mustbeknowninordrtodetermine y; forsimplicity itisusualtoworkatsuchhighfieldsthathemagnetization isequaltothesaturation value.Thephenomenon offerromagnetic resonance wasdiscovered experimentally byGriffiths 1946);thetheory givenaboveisduetoKittel(1948).Somevaluesfgmeasured by ferromagnetic resonance aregiveninTable21.2. 23.7] MAGNETIC RESONANCE 709 Spinwavere80nance inferromagnetic films Theuniform precession modeassumed above corresp~mds toaspin wavewithks=O.Itispossibletoexcitespinwavesforwhichks=1=0; sinceeachspinwavecorresponds toaunitchange1iinangularmomen­ tum,andhencetoachangeofgf3inmagnetization, theenergyrequired toexciteaspinwaveinafieldB(inthesample)is 1iw=gf3B+Dk:, (23.18) wheretheconstantDisthesameasthatinequation (21.25).Inathin filmofthickness l,theboundary conditions (assumed identical atthe twofacesofthefilm)limittheallowedvaluesofkstothoseforwhich thefilmthickness isanintegralnumberofhalf-wavelengths; thatis, ks=p1Tll,wherepisaninteger.Ifmagnetic resonance isobserved at constant frequency, thevalueoftheresonance fieldBisfoundfrom equation (23.18)abovetobe B=Bk'=O-(~;:)p2, (23.19) sothataseriesofresonances corresponding todifferent valuesofp shouldbeobserved onthelowfieldsideoftheordinary ferromagnetic resonance fieldBk.=o'Aspinwaveresonance curveisshowninFig.23.17 forathinfilmofcobaltmetal,ofthickness approximately equalto 6000A.Thisissmallcompared withtheskindepth,sothattheoscilla­ toryfieldisuniform withinthesample,andtheresonance intensity dependsonthenetmagnetic momentinthedirection oftheoscillatory field.Thisisproportional to z zfsinhdx=fsin(P1Txll) dx=(llp1T)(I-coSp1T). o 0 Thisvanishesfor evenvaluesofp,anddecreases asIIpforoddvalues, givingtheintensity changeshowninFig.23.17.Theresonance field decreases accurately asp2(seeFig.23.18),andthevalueofDcanbe foundifthethickness 1isknown. Femmagnetic andanti1erromagnetic resonance Thepresence oftwosub-lattices inthesesubstances makesthe magnetic resonance phenomena ingeneralmuchmorecomplicated. Onesimplecaseoccursinferrimagnetic substances withstrongly coupledsub-lattices; thetwosub-lattices canthenprecesstogether in suchawaythattherelativeorientation oftheirtwomagnetic moments remains unaltered. This occursatanangularvelocity w=-YelfB; hereBisthefieldinthesubstance andYe1fisanaveragevalueobtained 710 MAGNETIC RESONANCE [23.7 fromtherelation M=IMi=I'YiGi='YeffIGi='YfiG, i wherethesummation isoveralltheindividual ions.nbothferri-and anti-ferromagnetics morecomplicated modesofprcessionoccurin 17 18 19 Magnetic field(kilogauss) FIG.23.17.Ferromagnetic spinwaveresonance inathinfilm6000Athickness) of cobaltmetalatroomtemperature anda.frequency of9370Mc/s(3·2emwavelength). Thelineshapeisthatcorresponding tothedifferential ofthesorption curve.The intensity ofresonance decreases towards lowerfieldstrength (highervaluesofp) irregularly becauseoflackofuniformity inthefilmthickness (P.ipsandRosenberg, 1964).Boisnormaltothesurfaceofthefilm,sothatratherhighaluesofBoareneeded tosatisfytheresonance condition wL=-y{Bo-p.o M.}. whichtherelativeorientation ofthesub-lattice maeticmoments is notpreserved; thefrequency ofprecession thendepndsonanumber ofparameters, including theexchange andanisotrop energies. 23.8.Cyclotron resonance withfreecharged paticles Whenacharged particleofmassMandchargeqismovingina uniform magnetic fieldB,itsequation ofmotionis F=qv/\B. 23.8] MAGNETIC RESONANCE 711 Sincethisforceisalwaysnormaltoitsinstantaneous velocityv,the particlewillmoveinacircleofradiusrintheplanenormaltoBwith angularvelocity givenbytheequation Mw~r=qwerB, i.e. We=(q/M)B. Thusifitispossibletodetermine theangularvelocity Winaknown fieldB,theratioofchargetomassoftheparticlemaybefound.The 21 o 100 200 300 400 p. FIG.23.18.Plotshowing thelinearrelation between magnetic fieldandp.forferro­ magnetic spinwaveresonance inathinfilmofcobalt(afterPhillipsandRosenberg, 1964). frequency Wc/27Tisoftencalledthe'cyclotron frequency' sinceitisthe frequency ofther.f.electricfieldrequiredtoaccelerate chargedparticles inthecyclotron. Thesuccessofthisdevice,whichdependsonresonance between thefrequency oftheoscillating electricfieldandthefrequency ofrotationoftheparticles inthefieldB,suggeststhatasimilarprinciple maybeusedtodetermine theratioofqtoM. Webeginbyinvestigating themotionofachargedparticlestarting fromrestundertheactionofauniform induction B(whosedirection wetaketobethez-axisofasystemofcartesian coordinates) andan oscillating electricfieldoffrequency W/27Tpolarized sothatthelinesof electricfieldareparalleltothex-axis.Thentheequations ofmotionare Mi=qECoswt+qYB} My=-qxB. (23.20) Mz=O 712 MAGNETIC RESONANCE [23.8 Thelastoftheseequations showsthatthez-compon ntofthemotion willbeindependent ofEandB,anddoesnotapearintheother equations. Thesecondequation canbeintegrated 0cegiving My=-qxB, wheretheconstant ofintegration hasbeenequatedtozro,corresponding totheassumption thattheparticlestartsatrestfromtheorigin.ymay nowbeeliminated fromthefirstequation giving x+w~x =(qjM)Ecoswt, where We=(qjM)B. Thegeneralsolutionofthiseqationis (qEjM)coswt+ 0t+D.x= 2 2 cosWeSInct.We-W Iftheinitialconditions arex=0,x=Oatt=0,theunownconstants aredetermined andwehave (qEjM)(coswt -coswet)x=..:..=...---'-----'-':..-..,.---;;---~ w~-w2 _2(qEjM)sin!(w e+w)tsin!(w e-wt - (we+w)(we-w) qE. 't(sin!~wt) (23.21)=Mw,slnw ~w' wherew'=!(we+w), ~w=we-w.If~w~We't efactor sin!~wt variesveryslowlywithrespecttotimecompared witsinw't,andw'is veryclosetoWe'sothatusingtherelationy=-Wewefindapproxi. mately qE, (sin!~wt)y=Mw'cOSWt~w. (23.22) Examination oftheequations forxandyshowsthtthepathofthe particleisaspiralwithangularvelocity w'andradis (~~,)(sinl~wt). If~w=°(Le.W=we)thenthevalueofthefactor(int~wt)j~w isIt, showingthattheradiusincreases linearlywitht.0theotherhand, if~w=1=0,theradiushasamaximum valuero(whethesineisunity) equaltoqEjMw'l~wl, whichisverynearlyequatoqEjMwel~wl when ~wissmall.Hence,ifacollector isplacedatdistance Rofrom theorigin,onlythoseionswillreachitforwhichroRo,or l~wl~qEjMweR o=EjBRo• Thisisameasure oftheprecision withwhich We'a dhenceqjM,can 23.8J MAGNETIC RESONANCE 713 bedetermined. The'resolving power'willbe wc/ldwl=(qBIM)/(E/BR o)=qB2Ro/ME. (23.23) Hence,foragivenionandagivenfieldB,theprecision isincreased by usingasmallamplitude ofoscillating electricfieldEandalargevalue ofRo•Itcanbeshownthatourexpression fortheresolving poweris equaltoL/2Ro(seeProblem 23.4),whereListhetotalpathtraversed P- -- p,..--......./", ..../. "I/-- "­ II".","'\ \ \+"t..:;IJ"\~// 1 .......//--- / / 1.....~/ vf"\.,r.f.voltage FIG.23.19.Apparatus formeasuring thecyclotron resonance frequency oftheproton. Bisnormaltotheplaneofthepaper. Iioncollector. Visasteadyvoltageofabout0·1voltforfocusing theionbeam. Rpotentiometer systemforguardrings. Gguardrings. P,Pplates. Eoscillatory electricfield. bytheioninitsspiraljourneyfromtheorigintothecollector. Thus theresolving powerprimarily depends onthenumber ofrevolutions whichtheionsmakeontheirjourneytothecollector. Theapparatus usedbySommer, Thomas, andHipple(1951)isshown inFig.23.19.Anoscillatory voltageisappliedbetween twoparallel platesP,Pofsize3emX5em,andseparation 2em,withanumberof parallelguardrings.Theseringsareequallyspaced,andbymeansofa potentiometer systemRafractionofthevoltageproportional tothe distance fromoneendplateisappliedtothemsothatauniform r.f. fieldisobtained. Asmallsteadypositive voltageofabout0·1Vis 714 MAGNETIC RESONANCE [23.8 appliedtotheguardringsrelativetotheendplates 0astoretardthe driftofpositiveionsinthedirection paralleltothefildB.Themagni­ tudeofBisdetermined byanuclearmagnetic resnanceexperiment, usinganr.f.coilcontaining asampleofoil.Ionsareroducedalongthe axisoftheapparatus byfiringinanarrowbeam0electrons ofabout 70Venergy,whichcauseionization bycollision wihtheresidualgas. Thepressure mustbekeptlow(::::::10-6mmHg)nordertoprevent scattering oftheionsbycollision. Thewholeasseblyisenclosedina glasstubeof4·7emdiameter, whichfitsbetweenthpolesofanelectro­ magnet. Inatypicalexperiment B=4700gauss,andt eoscillatory fieldE isabout0·1VIcmatafrequency ofabout7Mcls£,rtheH+ion.With Eo=1em,theionsmakeabout7000revolutions ndattainanenergy ofabout1000eVbeforereaching theioncollecto whichisconnected toanelectrometer. Theioncurrentatthepeakofresonance isabout 3X10-14Awhilethebackground fluctuations a eabout4X10-16A. Owingtothesmallpositive voltageontheguarrings,andtospace charge,asmallradialelectricfieldexistswhichsplacestheresonant frequency slightly. InaradialfieldE'theequat'nofmotionis Mw2r+qE'=qwrB, whenceapproximately w=w[1-E'M]. (23.24)crqB2 InpracticeitturnsoutthatE'increases linearlywithr,andhencethe shiftintheresonance isindependent ofr,butprortionaltoM.Thus bymakingmeasurements bothwithH+andHtions(H+andDtions werealsocompared) thesizeoftheshiftcanbeetermined. Similarexperiments havebeencarriedoutusigan'inverted' cyclo­ tron,inwhichuseismadeoftheionswhichareretrdedbytheoscillatory fieldacrossthe'dees',ratherthanthosewhicha eaccelerated, inorder toobtainlongerpathlengthsandhigherresoltion.Theseionslose energyandspiralinwardsuntiltheyreachadetetor.Thismethod,first usedbyJeffries(1951),hasbeenimproved bytheuseofamodified systemofdecelerating electrodes inwhichtheioapproach anasymp­ toticorbitinwhichtheenergylossbecomes zer(Sanders andTurber­ field,1963).Thehighresolving powerthusobtaiedisfurtherimproved byusingoscillatory fieldsattheeighthorsieenthharmonic ofthe cyclotron frequency. Analternative approach, usedbyBoynendFranken (1961),is todetectthepowerabsorbed bytheionsfrmtheoscillatory field, 23.81 MAGNETIC RESONANCE 715 asinanuclearresonance experiment. Thishastheadvantage thatlow valuesoftheoscillatory fieldcanbeused,sothattheioncloudisnot appreciably disturbed bythepowerabsorption. BoyneandFranken mademeasurements onHiionsatfieldsbetween 8and12·5kilogauss, andcorrected forelectrostatic fieldsbyusingequation (23.24)and plotting enbacktoIjB2=O. Oyclotron re80nance forfreeelectron8 Analogous experiments canbecarriedoutwithelectrons, themain difference intechnique beingduetothefactthatinafieldofafew kilogauss theresonance frequency isnowatabout1010cjsinsteadof about107cjs.Thefirstpreciseexperiment (Gardner, 1951)wasbasedon thefactthatelectrons movinginanarrowbeamparalleltothemagnetic fieldmaygainenergyfromthecyclotron resonance effectandspiral outwards sothattheyfailtopassthrough anarrowslitguarding the collector. Thusthecollector currentshouldfallatresonance, butit wasfoundthatsuperimposed onthisdipincurrentwasamuchsharper maximum, associated withspacechargeeffects.Inlaterexperiments cyclotron resonance hasbeendetected throughtheabsorption ofenergy byfreeelectrons fromtheoscillatory magnetic field,inacavityresonator. Sanders, Tittel,andWard(1963)usedacurrentofabout1p,Aaccelerated through about1 Vfromatungsten filamentatoneendofthecavity. Frequency shiftsduetoradialelectricfieldsarisingfromspacecharge wereeliminated byextrapolating theresonance frequency tozero current. LiebesandFranken (1959)carriedoutasimilarexperiment usingsome10"-105freeelectrons ofabout1eVenergy,produced by photo-emission fromathinlayerofpotassium; theyworkedatfield strengths between 750and1700gauss,andextrapolated theresonance frequency toinfinitefield,asinthecorresponding protonexperiment. Re8ults Inallsuchexperiments theratiooftwofrequencies inthesamemag­ neticfieldisdetermined-the cyclotron resonance frequency ofthe electron (ve)orproton(vo)'andthenuclearmagnetic resonance frequency (vp)ofprotonsinwateroramineraloil.Themainresultsaresummarized inTable23.3,whichgivestheratiosmeasured forprotonsandelectrons, together withthequantity (vejvo)obtained bycombining apairofthese ratios,whichshouldbeequaltoMjm,theratioofmassesoftheproton andelectron.Ifacorrection of28p.p.m.isappliedfordiamagnetic shielding oftheprotonsinthewaterormineraloilsample,themeasure­ mentsgivethevalueofthenuclearmagnetic moment oftheprotonin 716 MAGNETIC RESONANCE [23.8 nuclearmagnetons, since wp=21TVp=gn(eI2M)B =!gn We21TVe(eIM)B andthenuclearspinoftheprotonis!.Ifthecycotronresonance of theelectron isusedinsteadofthatoftheproton,tIenuclearmoment oftheprotonisfoundintermsoftheBohrmagnete n.Apartfromthe TABLE 23.3 Measurements of:column1,ratioofprotonmagnetic 'resonance frequency Vp(inH20)toprotoncyclotron resonance frequency v("column2,ratioof electroncyclotron frequency Vetovp"column3,ratioifvetoVo=ratioof massesofprotonandelectron, obtained fromprecedin'rJ ratiosonthesame line. Vp/V. Ve/Vp vefve=,M/m 2'79265(10) J1951 2'79268(6) STH1951 657'475(8) G1951 1836'12(5) 2'79283(6) BF1961 657-462(3) LF1959 1836'22(4) 2'79268(5) ST1963 657-462(2) STW19631836'08(3) Thenumberinparentheses givestheprobable erroriJthelastdigit; e.g.1836'12(5) =1836·12±O·05 (theaccepted value's1836'12(2». ReferenceB : J1951Jeffries, 1951. STH1951Sommer, Thomas, andHipple(1951). G1951Gardner (1951). BF1961BoyneandFranken (1961). LF1959LiebesandFranken (1959). ST1963Sanders andTurberfield (1963). STW1963Sanders, Tittel,andWard(1963). cyclotron resonance experiment ofBoyneandFranken, whichgivesa ratherhighvalue,theresultsagreeclosely,themEanvaluebeing magnetic moment ofproton=2'79276(7),8n =1521043(6) X10-3,8. Theratiosdetermined abovemaybewrittenas WeelM Weelm-=--, -=._, wpYP wpYP showingthatbyusingtheabsolute valueofYPmeasllred attheStandards laboratories (see§23.5)theygivethespecific charg~oftheproton(elM) andelectron (elm)respectively. Multiplication o~theformerbythe isotopicmass(1'00728) oftheprotonalsogivesthealueoftheFaraday, thechargerequired toliberateunitmassofanion-.;hoseisotopicweight isunity.Theresultsareallingoodagreement withtheaccepted values. 23.9] MAGNETIC RESONANCE 717 23.9.Cyclotron resonance ofchar~ecarriers insemiconductors ItwaspointedoutinOhapters 18and19thattheequations ofmotion ofelectrons (andholes)intheperiodicpotential ofacrystallatticeare similartothoseofafreeparticle, provided thataneffective massm* isusedinsteadofthetruemass.Thisholdsalsoformotioninamagnetic field,andthecyclotron resonance frequency therefore becomes We=(q/m*)B, iftheeffective massisisotropic. Determination ofthisfrequency is thusofgreatimportance sinceitgivesadirectmeasurement ofm*. Inprinciple, theexperiment issimilartothosedescribed intheprevious section:anoscillatory electricfieldisappliednormaltothesteady magnetic field,andeitheritsfrequency orthestrengthofthemagnetic fieldisvariedwhilethepowerabsorbed ismeasured. However, the chargecarriersinasolidmakecollisions ataratewhichisusually comparable with(andoftenmuchhigherthan)thecyclotron resonance frequency; thisgivesaveryimportant damping term,andtheequation ofmotionmaybewrittenas(cf.Problem 3.9) m*{~;+~v}=q{E+v/\B}, (23.25) whereq=-8forelectrons and+eforholes. Tosolvethisequation weassumethatBisalongthez-axisofa Oartesian coordinate system,andEisanoscillatory fieldalongthe x-axis.Wetherefore writeEx=Eoexp(jwt), andlookforthesteady statesolutioncorresponding tothedrivenmotionatangularfrequency w;wecanthenreplaced/dtbyjw,andtheequations become ~w+~)Vx=:!*(Ex+vyB) (jw+~)Vy =-;"*vxB (23.26) (jw+~)Vz =0 Thelastequation showsthatanymomentary currentinthez-direction diesawayexponentially through collisions, andwemayeliminate vy between thefirsttwoequations inordertofindtheoscillatory velocity Vxinthedirection oftheappliedelectricfield.Thisgives {2 2 12jw}qE(1.)VxWc-W+:;:2+--;;:- =m*x:;:+Jw, 718 MAGNETIC RESONANCE [23.9 (23.27)wherewehavewritten Wefor(q/m*)B,thecyclotron resonance frequency. Theconductivity ofthesolidatangularfrequency Winthex-direction isax=nqvx/Ex'wherenisthenumberofchargecarriersperunitvolume ofmassm*,andisgivenbytherelation nq2{jw+l/T } AX=m*(w~-w2)+I/T2+2jw/T {l+jwT }=0'0 ,I+2}WT+T2(W~-W2) 1·0 0·75 CUT=5 (23.28)O!.-----:;;-L;;---------;-L;c------;;-'-;;-----2~-m,rro FIG.23.20.Plotoftheratioofther.f.conductivity tothed.c.con­ ductivity against (wclw).Cyclotron resonance measurements are usuallymadeatconstant wandvariable field;since Wcisproportional toB,thecurvesshowtheconductivity againstB (onareduced scale). Wellresolved resonance curvesareobtained when WTisrathorgreater thanWlity. where 0'0=n(q2/m*)Tistheordinary conductivity ofthesubstance at zerofrequency intheabsenceofamagnetic field.Thisequation shows thatthehigh-frequency conductivity iscomplex; onsolvingforthereal part o'~oftheconductivity, wefind u~ I+T2(w~+W2) 0'0={1+T2(w~-w2)}2+4w2T2· Thepowerabsorption perunitvolumeofthesampleis!a~E~;sinceit isusualtoworkatfixedfrequency wandmeasurethepowerabsorption asB(i.e.wc)isvaried,itisusefultoplotthequantity (a~!O'o)asafunction 23.9] MAGNETIC RESONANCE 719 of(wc{w)forvariousvaluesoftheparameter WT.Thisisshownin Fig.23.20.When WTisappreciably lessthanunity,themeantime between collisions isasmallfractionofanr.f.period,andlittlechange occursuntilWTapproaches unity.However, when WTisrathergreater thanunity,adistinctresonance effectisobserved, withmaximum power absorption atapointclosetothecyclotron resonance frequency. Inasemiconductor ormetalatroomtemperature thevalueofTis about10-12to10-14seconds, sothatevenatawavelength of1em, samplemounted oninsulating supportatoentreofcavity.rectangular wave-guideModulated lightItc:>... """t-:I~-4--coupling iris t"T onehalf wavelength1 FIG.23.21.Waveguide cavityresonator usedincyclotron resonance experi­ ments,showing thesamplemounted atthecentreofthecavitywherethe oscillatory electricfieldisamaximum. Carriers canbeexcitedinthesample bylightpasseddownthewaveguide andthroughthecoupling irislinkingthe cavitytotheguide. whereamagnetic fieldatresonance ofabout104gausswouldbeneeded ifm*=m,thevalueofWTisabout2X10-1to2X10-3•However, the electron scattering ismainlyduetophonons, andisreducedatlow temperatures. Inasemiconductor (see§19.5) TshouldvaryasT-i,and afactorof103isgainedingoingfrom3000to30K,making WT"""2to 200.AscanbeseenfromFig.23.20,thisissufficient forafairlyaccurate determination oftheresonance frequency. However, thenumberof chargecarriersn,which(see§19.5)variesasTiexp(-tVg{2kT) fora puresemiconductor, becomes vanishingly smallatheliumtempera­ tures.Dexter, Zeiger,andLax(1956)overcame thisdifficulty by irradiating thesamplewithlightofsufficiently shortwavelength tolift electrons acrosstheenergygapfromthevalencetotheconduction band, thuscreating bothholesandconduction electrons. Themainfeatures oftheirapparatus areshowninFig.23.21.Thesample,intheformof athindisksome3mmindiameter and0·5mmthick,ismounted at apointinawaveguide cavitywheretheoscillatory electricfieldisas 720 MAGNETIC RESONANCE [23.9 largeaspossible without producing seriouscarrierheating effects through acceleration ofthecarriers. Thiscavityterminates awave~ guide,andthechangeinthesignalreflected bythecavityisameasure oftheincreased powerabsorption inthesample. Thecavityisimmersed inliquidheliuminadewarvesselplacedbetween thepolesofan electromagnet. Themostsatisfactory methodofdetection istomodulate thelight beambypassingitthrough arotating diskpiercedbyalargenumber ofholes.Thelifetimeofthecarriersisshortandtheyarepresentonly fortheduration ofalightpulse;thereflected microwave signalisthere~ foremodulated atthesamefrequency (usually 100to1000cis).Instead ofusingirradiation bylight,carrierscanalsobecreatedthroughioniza­ tionofimpurity levelsbyapplication ofanelectricfieldacrossthe sample,orbytheoscillatory microwave electricfield.Thelattermethod givesdistorted line-shapes, however, sincethenumber ofsecondary carrierscreateddepends onthecarrierenergyandthisisamaximum atresonance. Ithastheadvantage thatonlyelectrons arecreatedin n-typematerial, andholesinp-type, sincethemicrowave energyisonly smncient tocauseionization acrossthesmallgap(,,-,0-01eVinGe) ofimpurity levels,andnotacrossthemaingap~. Inmanysubstances theeffective massisanisotropic (see§18.2),and theratioofcyclotron resonance frequency tomagnetic fieldisafunction oftheorientation ofthefieldrelativetothecrystalaxes.Forthisreason asinglecrystalmustbeused,witheitheraspecialdeviceforrotating itinthecavity,orforrotating theexternal magnetic field,sothata wholeplaneofdirections relativetotheexternal magnetic fieldcanbe explored. Anabsorption curveforagivenorientation ofgermanium is giveninFig.23.22;itisduetoDresselhaus, Kip,andKittel(1955),who madethefirstobservations ofcyclotron resonance insemiconductors in 1953.Whenanisotropy ispresent,theequations ofmotionaremodified andmustbesolvedtofindtherelationbetween thecyclotron resonance frequency andtheeffective massparameters; thefollowing methodfor thisisduetoShockley (1953). Whentheenergysurfaces arenotspherical ink-space,theycanbe approximated nearthebandedges(see§18.2)bytherelation W=In2{k~+k~+k~}=~{p~+p~+pl}, mi£mymz2mi£myrnz provided thatthedirections oftheX-,y-,z-axesarechosencorrectly. Alongtheseaxesthecomponents oftheequation ofmotionhavetheir 23.9] MAGNETIC RESONANCE 721 usualform,sothatinamagnetic fieldwithcomponents Bx'By,Bzwe have m",(dv",/dt) =q(vyBz-vzBy),etc. Tofindthecyclotron resonance frequency weassumethatthemotion issinusoidal withangular frequency WC'Wecanthenreplacethe 4000 o 1000 2000 3000 Magnetic fieldingauss FIG.23.22.Absorption curveforcyclotron resonance inasingle crystalofgermanium, at24000Mctsand40K.Thestaticfieldisin a(llO)planeat600froma(l00]axis(afterDresseIhaus, Kip,and Kittel,1955). differential operator d/dtbyjwc'givingthesetoflinearequations jwcmxv:c-qv yBz+qvzE,g=0, jWemyvy-qvzB:c+qva;Bz =0, jwcmzvz-qv:cB1I+qv1lBa; =0, whichhaveanallowedsolution onlyifthedeterminant jwcma;-qBzqB1I qBzjwCmy-qB", =0. -qBgqBa;jwcmz Thiscondition giveseitherWe=0,or 2 w~=q(m:c.m+myB~+mzB:). (23.29)m:cmgmz Thisequation showsthatthecyclotron resonance frequency dependson 861110 3A 722 MAGNETIC RESONANCE [23.9 theorientation ofthemagnetic fieldwithrespecttothecrystalaxes; inanygivenplaneaplotofw~againstanglegivesa(cosine)2 variation betweenthemaximum andminimum values.WhenBisdirectedalong oneoftheprincipal axes,suchasthez-axis,theresonance frequency is simply(wc)z=qBj(mxm y)!;thusbymeasurement alongeachaxisin turn,theprincipal valuesmx'my,mzoftheeffective masscanbedeter­ mined.Theresultsfortheelemental semiconductors Si,Ge,together withthoseforindium antimonide areshowninTable23.4.Forsilicon TABLE 23.4 Effective massesinsomesemiconductors determined bycyclotron resonance, relativetothefreeelectronmass Electrona Holes Sub8tanoo m1 m~ 'light' 'keavy' Si 0·98 0·19 0·16 0·5 Ge 1-64 0·082 0·044 0·3 InSb 0·014(isotropic) 0·02 0·4 References : Si,GeR.N.Dexter,H.J.Zeiger,andB.Lax,1956,PhY8.Rev.104,637. InSbElectrons-various authors. Holes-D. M.S.Bagguley. M.L.A.Robinson, andR.A.Stradling. 1963. PhY8.Letter86,143. andgermanium twooftheprincipal valuesoftheeffective massatthe bottomoftheconduction bandareequal;thisisknownasthe'trans­ versemass', m~,whilethethird(unequal) massiscalledthelongitudinal mass,ml.InInSbtheminimum oftheconduction bandoccursatk=0 (see§19.4),andtheeffective massisisotropic. Thepositionatthetopofthevalencebandismorecomplicated. Two energysurfaces coincideatk=0,andatpointsnear-byink-spacethe energysurfacesforSi,Gearegivenbytherelation W=Ak2±{B2k4+02(k~k~+k~k~+k~k~)}i, (23.30) whichisalsoapproximately correctforIII-Vsemiconductors. If0is smallthetwosurfacesarenearlyspherical, butwithdifferent curvature, corresponding totwodifferent effective massesknownasthe'light'and 'heavy'holesrespectively. ThesemassesareshownalsoinTable23.4. 23.10.Azbel-Kaner resonance inmetals Whenaspectralline duetomovingparticles isobserved, itisbroadened throughtheDoppler effect,byanamountwhichisproportional tothe randomparticlevelocity. Inasemiconductor atlowtemperatures, the 23.10] MAGNETIC RESONANCE 723 electrons orholeshaveordinary thermal velocities corresponding to energies oforderkT,andbroadening bytheDoppler effectisnot important. Inametal,ontheotherhand,theelectron velocityisthat attheFermisurface;incopper,assuming m*/m=1·5and~=4·7eV, thisvelocity isabout106m/sec,whilethephasevelocityinthemetal ofanelectromagnetic wavewithafree-space wavelength of1cmis onlyabout4X104m/sec(fromequation (10.30)itisequaltow~,where ~ istheskindepth).Broadening throughtheDopplereffectthusmakesit impossible toobservecyclotron resonance inmetalsbymethods similar tothoseusedforsemiconductors. Itcan,however, bedetected bya different method, originally duetoAzbelandKaner(1957,1958). Asbefore,areasonable degreeofresolution isobtained onlyifWT>1. Thismakesitessential toworkatliquidheliumtemperatures, using verypuresamples inwhichtheresidual resistivity duetoelectron scattering byimpurities andimperfections isassmallaspossible (10-3 to10-5oftheroomtemperature resistivity). Incoppertheradiusof theelectron orbitinthemagnetic fieldrequired tomakethecyclotron resonance frequency equalto3X1010c/sisabout5X10-6metres,and themeanpathlengthoftheelectrons mustbeofthisorderinorderto makeWT>1.Thisrequires aconductivity oforder4X1010(ohm­ metre)-l, andthe'classical' skindepthgivenbyequation (10.31)is about5X10-9metres,whichissmallcompared withthemeanpath length.Thisistheregionofthe'anomalous skineffect',wherethe conductivity iseffectively reducedbecauseonlythoseelectrons moving atasmallangletothesurfacesuchthattheirfreepathsliewholly withintheskindepthcontribute fullytotheoscillatory current. How­ ever,evenallowing forthis,the'anomalous' skindepth(seeProblem 18.5)isabout2X10-7metre,whichisstillsmallcompared withthe radiusofthecyclotron orbit.Ifthenamagnetic fieldBisapplied paralleltothesurfaceofthemetal,acertainnumberofelectrons moving inhelicalorbitsaboutBwillentertheskin.depth regiononcepercycle, andwhileinthisregiontheycanbeaccelerated bytheoscillatory electric fieldcomponent normalorparalleltoB.Thelattergeometry is illustrated inFig.23.23;animportant difference fromtheconventional cyclotron isthatacceleration occursonlyonceperrevolution instead oftwice.Electrons willgainenergysteadilyifthefrequency ofthe electromagnetic waveincidentonthesurfaceofthemetalissynchronous withthecyclotron resonance frequency, orisanintegralmultiple ofit. Hencetheresonance condition is W=pw()=p(q/m*)B, (23.31) '124 MAGNETIC RESONANCE [23.10 wherep=1,2,3,etc.Itisusuallyconvenient toworkatafixed frequency, makingthemetalsampleoneendofacavityresonator as inferromagnetic resonance (butwithBnormalorparalleltotheoscilla­ toryelectricfieldinsteadofnormaltotheoscillatory magnetic field), FIG.23.23.Geometry ofthesteadymagnetic fieldB,theoscillatory electricfieldand thecyclotron orbitsinametalforAzbel-Kaner resonance. Theelectrons areaccelerated bytheelectricfieldonlywhentheirorbitstakethemintotheskindepth. ,/, ",, '.Cu[1001m*=1·38m B(kG) FIG.23.24.Azbel-Kaner resonance at4°Kinasinglecrystalofcopper(afterKoch, Stradling, andKip,1964).Themagnetic fieldisparallel tothesurfaceandalonga [100]direction; thefrequency is67kMc/s(4'5mmwavelength). (23.32)andmaxima intheabsorption ofenergythenoccuratvaluesofB givenbytherelation B=.!.(m*w). Pp q Theabsorption islargestforp=1,anddecreases aspincreases, since theelectrons areonlyaccelerated everypthcycle,givingacurveofthe formshowninFig.23.24. 23.10J MAGNETIC RESONANCE 725 Samples ofhighpurityareneededtogivegoodresolution; ideally theymustbesoflatthatsurfaceirregularities aresmallcompared with theanomalous skindepth.Iftheeffective massisanisotropic, single crystals mustbeused,cutinspecialorientations sothatcyclotron resonance canbeobserved inalltheprincipal directions. Thesteady magnetic fieldBmustbeaccurately paralleltothesurface,orelectrons willmoveawayfromthesurfacebecauseoftheirvelocity components paralleltoB.GrimesandKip(1963)havefoundthattheeffective mass isisotropic insodiumandpotassium, withvaluesofm*1mequalto 1·24±0·02 and1·21±0'02respectively. Incopper(Koch,Stradling, and Kip,1964)thepredominant absorption isduetoelectrons withm*lm aboutequalto1,4,withonlyslightanisotropy, butotherveryanisotropic valuesrangingfrom0·4to6arealsoobserved, showingthattheFermi surfaceisrathercomplicated. REFERENCES ALVAREZ, L.W.,andBLOCH,F.,1940,PhY8.Rev.57,Ill. AZBEL,M.YA.,andKANER,E.A.,1957,SovietPhY8.J.E.T.P. 5,730.----1958,J.PhY8.Ohem.Salida,6,U3. BLOCH,F.,HANSEN, W.W.,andPACKARD, M.,1946,PhY8.Rev.69,127. BLOEMBERGEN, N.,PURCELL, E.M.,andPOUND,R.V.,1948,ibid.73,679. BOYNE,H.S.,andFRANKEN, P.A.,1961,ibid.123,242. BROWN, R.M.,andPURCELL, E.M.,1949,ibid.75,1262. COHEN,V.W.,CORNGOLD, N.R.,andRAMSEY, N.F.,1956,ibid.104,283. COHEN,E.R.,DUMOND, J.W.M.,LAYTON, T.W.,andROLLETT, R.S.,1955, Rev.Mod.PhY8.27,363. DEXTER, R.N.,ZEIGER,H.J.,andLAX,B.,1956,PhY8.Rev.104,637. DRESSELHAUS, G.,KIP,A.F.,andKITTEL, C.,1955,ibid.98,368. DRISCOLL, R.L.,andBENDER, P.L.,1958,PhY8.Rev.Letter81,413. ESSEN,L.,andPARRY,J.V.L.,1957,Phil.Tram.A,250,45. GARDNER, J.H.,1951,PhY8.Rev.83,996. GRIFFITHS, J.H.E.,1946,Nature,Lond.158,670. GRIMES, C.C.,andKIF,A.F.,1963,Phys.Rev.132,1991. HELLER, P.,andBENEDEK, G.B.,1962,Phys.Rev.Letters8,428. JEFFRIES, C.D.,1951,Phys.Rev.81,1040. KITTEL, C.,1948,ibid.73,155. KOCH,J.F.,STRADLING, R.A.,andKIP,A.F.,1964,ibid.133,A240. KUSCH,P.,1955,ibid.100,U88.--andFOLEY,H.M.,1948,ibid.72,1256;74,250. LAMB,W.E.,andRETHERFORD. R.C.•1947,ibid.72,241. LIEBES, S.,andFRANKEN, P.A.,1959,ibid.116,633. MARKOWITZ, W.,HALL,R.G.,ESSEN,L.,andPARRY,J.W.L.,1958,Phys.Rev. Letters1,105. PHILLIPS, T.G.,andROSENBERG, H.M.,1964,Phys.Letters8,298. PURCELL, E.M.,1948,Science, 107,433.--andRAMSEY, N.F.,1950,PhY8.Rev.78,699.--TORREY, H.C.,andPOUND,R.V.,1946,ibid.69,37. 726 MAGNETIC RESONANCE RABI,I.I.,MILLMAN, S.,KUSCH,P.,andZACRABIAS, J.R.,1939,ibid.55,526. RAMSEY, N.F.,1949,ibid.76,996. SANDERS, J.H.,TITTEL,K.F.,andWADD,J.F.,1963,Proc.Roy.Soc.A,272,103. SANDERS, J.H.,andTURBERFIELD, K.C.,1963,ibid.79. SHOCKLEY, W.,1953,Phys.Rev.90,491. SHULMAN, R.G.,andJACCARINO, V.,1956,ibid.103,1126. SMITH,J.H.,PURCELL, E.M.,andRAMSEY, N.F.,1957,ibid.108,120. SOMMER, H.,THOMAS, H.A.,andHIl'FLE,J.A.,1951,ibid.82,697. SOMMERFIELD, C.M.,1957,ibid.107,328. THOMAS, H.A.,DRISCOLL, R.L.,andHIPPLE,J.A.,1950,ibid.78,787. VIGOUREUX, P.,1962,Proc.Roy.Soc.A,270,72. WIMETT, T.F.,1953,Phys.Rev.91,499. WITTKE, J.P.,andDICKE,R.H.,1956,ibid.103,620. GENERAL REFERENCES ANDREW, E.R.,1955,NuclearMagnetic Resonance (C.U.P.). INGRAM, D.J.E.,1955,Spectroscopy atRadioandMicrowave Frequencies (Butter­ worth). SANDERS, J.H.,1961,TheFundamental AtomicOonstants (O.U.P.). SERIES, G.W.,1957,TheSpectrum ojAtomicHydrogen (O.U.P.). PROBLEMS 23.1.Inasubstance wherethesusceptibility issmallandtheLorentz internal fieldcanbeneglected, showthatequation (17.12)canbewrittenintheform ,•IInoel 1 X-JX='!nEo(w~-wl)+2jw Aw' IfXoisthestaticsusceptibility, andxPistheimaginary partofthesusceptibility whenw=wp,provethat X~/Xo=wp/(2Aw)=vp/(2Av), wherevpisthefrequency atthecentreoftheabsorption line,andAv=Aw/27r. Although thisformula wasderivedforelectricsusceptibility itisequallyvalid forthemagnetic case. 23.2.Theworkdoneperunitvolumetoincreasethemagnetization ofasubstance bydMinafieldBisdW=BdM.IfBisanalternating field B1coswt=Bl{B1exp(jwt)}, themagnetization maybewrittenas M=Bl{(X'-jX")(B1/JLo)exp(jwt)}, where(X'-jX")isthecomplex susceptibility. Showthattherateofdoingwork perunitvolumeis Bd:=dW/dt=-wx'(Bi/JLo)coswtsinwt+wx"(Bi/fLo)cos2wt andthemeanpowerdissipated perunitvolumeislwxIlBi/JLo' Usethedefinition (f)forQgivenin§9.3toshowthatl/Q=X"/(l+X') for acoilcontaining amagnetic substance inatunedcircuitwithnootherlosses. MAGNETIC RESONANCE 727 23.3.AdapttheresultsofProblem 10.6tothecaseofthemagnetic substance of thelastproblem (notethatx"/(l+x') isequivalent toE"/E'),andshowthatthe powerinanelectromagnetic wavepassingthrough suchamedium wouldfall according tothelaw WIJVo=exp(-2mx"xl>") ifX'~I. Verifythefiguresgivenin§23.3,thatfor>..=35metresandX"=10-5,the powerwillfallbyabout1·8percentinadistance of10kIn. 23.4.Show,fromequations (23.21)and(23.22)thattheinstantaneous velocityof thechargedparticleinitsspiralorbitis(qEIMAw)sin(tAwt). Henceshowthatthe totallengthofpathtraversed bytheparticleinreaching itsmaximum radius RowhenAw=1=0isL=2qEIM(Aw)2, andverifythatLI2Roisequaltothe resolving powerWeiAw. 23.5.AdapttheformulaofProblem 20.4tofindthevalueofgpwhenJand1 arecoupledtoformaresultant F,assuming thatthenuclearmagnetic moment canbeneglected. ShowthatintheZeeman splitting ofFig.23.12, gF~'=-gF~8=!gJ=t· 23.6.KuschandFoley(1948),usingtheatomicbeammethod, determined the ratioofthevalueofgJinthe2Pfand2Ptstatesofthegalliumatom,andfound ittobe2(1·00172±O·00006). Show,bywriting gz=1+8zandg8=2(1+8.), thattheratioisequalto2{1+!(88-8z)},andhencethattheirresultagreeswithin theexperimental errorwiththeaccepted value88=0·001160if8zisassumed to bezero. 23.7.Athinspherical shellofradiusr,thickness drofelectricchargedensityp rotateswithangularvelocity (J)aboutadiameter. Showthatthemagnetic field dBatthecentreis-i!Lopwrdr. Usethisresulttoshowthatthecorrection atthenucleusofahydrogen atom inamagnetic fieldBduetodiamagnetic shielding (see§23.5)is 8B !Loe2 B=-127l'mao' giventhatthechargedensityatdistance ris p=(-e/1TaZ)exp( -2rjao). 23.8.Showfromequation (23.28)thattheconductivity atzerofrequency inthe direction normaltoamagnetic fieldBvariesas q~jqo=Ij(l+aB2), wherea=(e7jm·)2. Thisisthemagneto-resistance effect,whichbecomes appre­ ciableonlyatlowtemperatures where 'Tincreases. (Whenonlyonetypeofcarrier ispresent, theeffectvanishes becausethesideways forceduetothemagnetic fieldisexactlynullifiedbythatduetotheHallvoltage; whenmorethanonetype ofchargecarrierispresentthiscancellation doesnotoccur.) 728 MAGNETIC RESONANCE 23.9.Showthatinacyclotron resonance experiment where WT;»1,thevalue ofO'~atresonance (welw=1)approaches !aD,andthatthelosstangentofthe specimen isthen(writing liweforT-1) neS tanSe=2*limW€€o We UsetheresultofProblem 23.1toshowthatinanelectron spinresonance experi­ mentthemagnetic losstangentatresonance forasystemofnelectrons with S=t,g=2is H Qa'"XpfLon,..WtanOm=I+Xo~2kTliwm andhenceprovethat(takingm*=m,liwe=liwm) tanSe4mc2kT tanSm=€(fiw)2• Henceverifythattheinherent sensitivity ofacyclotron resonance experiment is verymuchhigherthanthatofaspinresonance experiment, sothatfewerelectrons areneeded. Discusswhether theimaginary partoftheconductivity (seeequation (23.27» canjustifiably beneglected intheformula fortanSe. 24 UNITS 24.1.Unrationalized c.g.s.systems INmechanics threequantities arerequired todefineasystemofunits: standards oflength, mass,andtime.Inthec.g.s.systemthesestandards arethecentimetre, thegramme, andthesecondrespectively. Many alternative non-metric systemsareineveryday use,butnotinscientific use;itisobviously possible, however, tousedifferent metricunitsasthe standards, andthem.k.s.systemisbasedonthemetre,thekilogramme, andthesecond.Formechanical purposes eithersystemwilldo,andunits inonesystemarereadilyconverted intothoseoftheothersystem(they differonlybypowersof10).Inelectricity, afourthstandard quantity mustbedefined,andthemultiplicity ofsystemsofunitsisduetothe varying choicesofthisstandard whichareingeneraluse.Twoalter­ nativesystems havesurvived, onebasedontheJawofforcebetween electriccharges,andtheotheronthecorresponding lawbetweenmagnetic poles.Bothofthesesystemsarec.g.s.systems, sincetheirunitsoflength, mass,andtimearethesame,centimetre, gramme, andsecondrespec­ tively.Allmechanical quantities, suchasforceorwork,havethesame unitsineithersystem,buttheelectrical unitsarequitedifferent. Many oftheunitsareofunsuitable sizeforordinary work,andsoanothersetof units,thepractical system,hasalsocomeintocommon use.Thisisnot ac.g.s.system,sincemechanical quantities suchaspower,obtained from theproductofcurrentandvoltagemeasured inpractical units,arenot inc.g.s.units.Thebasisofthesethreesystems ofunitsisoutlined below. Unrationalized electrostatic units(e.s.u.) Inthissystemthec.g.s.unitsoflength,mass,andtimeareused,and afourthunit,thatofelectrical charge,isdefinedbymeansofCoulomb's law(invacuo) F=qlq2/r2, (24.1) wheretheunknown constant 0whichappearsinequation (1.1)hasbeen setequaltounity.Fromequation (24.1)theelectrostatic unitofcharge isdefinedasthatchargewhich,placedadistanceofonecentimetre away invacuofromanexactlyequalcharge,repelsitwithaforceofonedyne. 730 UNITS [24.1 Theunitofelectricfieldisthendefinedbytheequation F=qE (24.2) asthatfieldwhichexertsaforceofonedyneononeunitofcharge. Again,thepotential atapointBisoneelectrostatic unithigherthan thatatapointAifoneergofworkmustbedonetomoveunitcharge fromAtoB.Sincethecapacitance ofacapacitor istheratioofthe chargeonittothepotential difference between theplates,itfollows thatacapacitor hasunitcapacitance if,whenunitchargeisplacedonit, unitpotential difference issetupbetween theplates.Sincepotential hasthedimensions of(work/charge), capacitance hasthedimensions of (charge)2/ work,andfromequation (24.1)thisreducessimplytoalength. Hencethee.s.u.ofcapacitance isthecentimetre. Thee.s.u.ofelectricdipolemoment isdefinedasunitchargetimes unitdistance (centimetre), andthepotential whichadipolepproduces atadistancerisV=pcos8/r2•Polarization Pisthedipolemoment perunitvolume,andhencehasthedimensions (charge)/(length)2, which arethesameasthoseofelectricfield.Intheabsenceofanypolarizable medium, Gauss'stheorem (equation (1.7b»becomes ine.s.U. IE.dS=41T!q, (24.3) orindifferential form divE=41Tp, (24.4) where!qisthetotalchargeinthevolumeoverwhichtheintegral istaken,andpisthechargedensity. Whenapolarizable medium is present,thevolumechargedensitybecomes, onincluding thepolariza­ tioncharge, p~divP,andhenceGauss'stheorem takestheform IE.dS=41T(p-div P), orI(E+41TP). dS=ID.dS=47Tp, (24.5) sothattheelectricdisplacement Disdefinedas (24.6) Weseethatthefactor(41T),whichdoesnotappearinCoulomb's law, nowappearsinGauss'stheorem, andintherelation between D,E, andP.Also,theunitsofD,E,andPallappeartobethesame;this istrueonlyine.s.u.andisnottrue,forexample, ine.m.u.Theelectric susceptibility Xeanddielectric constant €aredefinedbytherelations P=XeE, (24.7) D=EE, (24.8) 24.1] UNITS 731 sothattherelationbetween themis E=1+47rXe. (24.9) Thedielectric constant isthesameasinthem.k.s.system(itissimply theratioofthecapacitance ofacapacitor filledwiththedielectric tothat ofthesamecapacitor invacuo),butthesusceptibility differsbythe factor 411".Inthem.k.s.systemthesusceptibility ofasubstance (per unitvolume=permetre3)isanumberafactor (411")largerthanthe corresponding numberine.s.u.(perunitvolume=percm3). Sinceelectriccurrentistherateatwhichchargeflowspastagiven point,thee.s.u.ofcurrentisequl11toaflowofonee.s.u.ofchargeper second.Incurrentelectricity andmagnetism itiscustomary towork ine.m.u.insteadofe.s.u.,andweshallnowdiscussthissecondc.g.s. system. Unrationalized electromagnetic units(e.m.u.) Originally theelectromagnetic systemofunitswasbasedonCoulomb's lawfortheforcebetween twomagnetic charges(ormagnetic poles)and theunitofpolestrength wasdefinedbysettingtheconstant inthe equation equaltounity,sothat(invacuo) F=m1m2/r2• (24.10) Thusunitmagnetic poleisthatwhichexertsaforceof1dyneonasimilar poleadistanceofonecentimetre awayinvacuo.Thelawsofmagneto­ staticsarethendeveloped formally inthesamewayasthoseofelectro­ statics,theqUl1ntities B,H,M,m,Xm'andf.Lplayingsimilarrolesto thoseofD,E,P,P,Xe'andE.Forexample B=H+47rM (24.11) and f.L=1+47rXm. (24.12) Hencethemagnetic volumesusceptibility ofasubstance inthem.k.s. systemisanumber largerbyafactor(47r)thanthecorresponding numberforthevolumesusceptibility inthee.m.u.system. Theconnexion withelectriccurrentismadeeitherbymeansofthe magnetic fieldproduced bythecurrent,orbydefiningtheequivalent magnetic dipolemomentofasmallcoilofareadScarrying acurrentIas m=IdS. (24.13) SincetheunitsofmanddSarealreadydefined,thisfixestheunitof current. Alternatively, theelectromagnetic systemofunitscouldhe developed fromthesamestarting-point asusedinChapter5,theexperi­ mentsofAmpere. Thentheequivalent ofequation (5.2)fortheforce 732 UNITS [24.1 between twocurrentelements wouldbe dF1=ItIz{dS1/\(dsz/\r)}/r3 (24.14) andtheunitofcurrentcouldbedefinedbymeansoftheforcebetween twoequalcurrents inparallelconductors, asin§5.1. Wehavenowtwoalternative unitsofcurrent,thee.s.u.andthee.m.u., whichwehavenoreasontosuppose bearanysimplerelationtoone another.Ifweassumethattherearenodimensional constants in equations (24.1)or(24.10),wecanworkoutthedimensions ofelectrical quantities intermsoflength,mass,andtime(theyappearratherqueer, involving (mass)!forexample), andthedimensions ofcurrentinthetwo systems willalsobedifferent. Theratioofcurrentine.s.u.tocurrentin e.m.u.hasthedimensions ofavelocity, anditturnsoutthattheratioof thequantities inthetwosystems isjustthevelocity ofelectromagnetic wavesinvacuo,c(inc.g.s.units).Hence number specifying currentine.s.u. _ _31010( ) b'f. . - c-Xapprox..numerspeClymgsamecurrentme.m.u. Inbothe.s.u.ande.m.u.theproductofcurrentandpotential ispower inerg/second, andhence number specifying potential ine.s.u. _11 number specifying samepotential ine.m.u.-c. Hencethederivedunits,resistance, inductance, and(capacitance)-t, all ofwhichhavetheratioofpotential tocurrent(apartfromadimension oftime)allchangeinthesameway;thatis,as number specifying resistance ine.s.u. _11z numberspecifying sameresistance ine.m.u.~c. (24.15)divB=0 leDcurlH=41TJ+-­cat =~(4?TaE+~~)c atUnrationalized mixedorGaussian units Inelectromagnetic theoryMaxwell's equations involvebothelectrical andmagnetic units,andinthec.g.s.systemtheyaregenerally written inmixedorGaussian units.Electrical quantities E,D,p,andconduc­ tivityaareine.s.u.,whilemagnetic quantities H,Bandcurrentdensity Jareine.m.u.ThuswemustwriteJ=(aE)lc,andthefundamental equations are divD=47rp, 1eBcurlE=----,cet 24.1] UNITS 733 (24.16)Elimination ofeithertheelectricormagnetic fieldleadstoawave equation V2(EH)=Elk~(EH), e 2Bt2' , showingthatthevelocityofelectromagnetic wavesinvacuoise. 24.2.Practical units Theelectrostatic unitsofchargeandcurrentandtheelectromagnetic unitofpotential areinconveniently smallforpractical use,andthe coulomb, ampere, andvoltareusedinstead. Originally thesewere definedinanarbitrary mannerlikethemetreandthekilogramme (the coulomb wasdefinedintermsofthemassdeposited inelectrolysis of TABLE24.1 Toconvert aquantity inpracticallUlits toaquantity ine.s.u.(ore.m.u.)multiply bythecorresponding factorgivenincolumnI(orII) Practical (I) (II) Quantity unit 6.8.U. 6.m.u. Charge coulomb 3xl0· 10-1 Current ampere 3x10'10-1 Potential volt 1/300 108 Power watt 101 101 Resistance ohm 1/(9xIOU)10' Inductance henry 1/(9X1011)10' Capacitance farad 9x101110-' Inthistabletheratiosareexactwheretheyaresimplepowersof10,butelsewhere thefactorchasbeentakenas3 X1010;moreaccurate valuesareobtained bytakingc asthevelocity oflight(inc.g.s.units)giveninAppendix C. acertainsolution), buttheseold'international units'havenowbeen replaced by'absolute. units',relatedbypowersof10totheelectro­ magnetic units,whichdifferfromtheinternational unitsbyamounts insignificant exceptinveryaccurate work.Thefactorsrequired to convertaquantity giveninpractical unitstotheequivalent quantity in e.s.u.ore.m.u.arelistedinTable24.1.Thesecanallbederivedfrom thefundamental relations onecoulomb ofcharge=10-1e.m.u.ofcharge, onevoltofpotential =108e.m.u.ofpotential, together withthefactthattheratiosofthesequantities ine.s.u.to e.m.u.areeandlierespectively. 24.3.Therationalized m.k.s.system Sincethepractical unitsarethoseineveryday use,itisconvenient tomakethemthebasisofasingleconsistent system. Thisisachieved 734 UNITS [24.3 (24.17) (24.18)inthem.k.s.systembyadopting themetre,kilogramme, andsecondas thefundamental mechanical units,together withafourthunittodefine theelectrical quantities. .Thedefinition ofunitsofothermechanical quantities followstheusualrules.Theunitsofvelocityandacceleration arethemetre/second andmetre/second2respectively; unitforceisthat forcewhichgivesunitmass(1kg)unitacceleration (1m/sec2).Itis calledthenewton,andinmagnitude isequalto(103X102)=105dynes. Unitpowerisdeveloped byunitforcemovingitspointofapplication withunitvelocity; hencetheunitisnewton-metre/second withmagni­ tude(105X102)=107erg/sec. Henceitisidentical withthewatt,and theunitofworkisthewatt-second orjoule. Asthetheoryofelectricity andmagnetism hasbeendeveloped inthis bookinrationalized m.k.s.units,itisunnecessary todomorethanpoint outthedifference between thissystemandtheunrationalized m.k.s. system. Theequations expressing Coulomb's lawinelectricity and magnetism, inourunits,are F=q1q2/(bEor2), F=l1-om1m2/(417r2), whileAmpere's lawofforcebetween twocurrentelements is dF1=11-01112{ds11\(ds2I\r)}j(47Tr3). (24.19) Theseequations differfromthecorresponding equations inunrational­ izedc.g.s.units(equations (24.1),(24.10),and(24.14»)notonlyinthe introduction ofunknown constants EO'11-0butalsointhepresence ofthe factors 417.Theintroduction ofthisfactorintheseequations, which makesitdisappear fromotherequations suchastheequivalents of equations (24.6),(24.9),(24.11),and(24.12),constitutes theprocessof 'rationalization'. (Itcanbeappliedalsotoc.g.s.units,butasrational­ izedc.g.s.unitsarenotincommon useweshallnotdiscussthem.)We cannotjustlumpthefactor417intotheconstants EO'11-0sincethe constants EOand11-0appearwithoutanyfactor417accompanying them inequations suchas(invacuo) D=EOE, (24.20) B=l1-oH. (24.21) Rationalization givesagreatersimplicity tovariousequations, notably thoseinelectromagnetic theory,butithasthedrawback ofmakingthe definingequations different fromthoseinanunrationalized system,and hencethefactorsrequiredtoconvertaquantityintherationalized m.k.s. systemtotheequivalent quantityinanunrationalized c.g.s.systemare 24.3] UNITS 735 notjustsimplepowersof10.Theconversion factorsrequired, together withtwoillustrations, aregiveninTable24.2and§24.4. Asecondimportant difference between them.k.s.systemandthe olderc.g.s.systems isthattheconstants £0'f'oareallowedtohave dimensions. Thereasonforthisisthatafourthunit(thecoulomb (or ampere}) isintroduced, whichretainsthedimension ofcharge(orcur­ rent),whereastheabsenceofanydimensional constant inequations (24.1)and(24.10)madeitpossibletoderiveapparent dimensions for anyelectrical quantity intermsofmass,length,andtime.Suchderiva­ tionsarenotveryilluminating, sincetheyinvolvehalfintegralpowers, andthedimensions ofanygivenelectrical quantity aredifferent inthe twoc.g.s.systems. Dimensions areveryusefulinchecking anyphysical formula, andinelectricity itissimplertoemployasystemoffour dimensions, suchasthemetre,kilogramme, second,andcoulomb, than athree-dimensional system. Thusifwewishtochecktheequation U=!D.E byverifying thattheproduct(DE)hasthedimensions ofenergy/volume, weproceedasfollows:fromequation (1.19)(Gauss's theor~m) Dhasthe dimensions ofcharge/area, whilefromtheforceequation (1.3)Ehas thedimensions force/charge. Hence(DE)hasthedimensions force/area =energy/volume anditsunitsarejoule/metre3• Thedimensions ofthequantity £0arereadilyfoundfromthefact thatitisequaltotheratio(D/E).Usingthealternative dimensions of(potentialflength) forE,wehave (DIE)=(charge/areaH-(potential/length) =(charge/potential)flength =capacitance/length. Hence £0ismeasured inunitsoffarad/metre. Ifwetreatedthecoulomb asastandard ofchargearbitrarily defined likethemetre,kilogramme, andsecond,thenbotht"oandfl-owouldbe constants tobedetermined byexperiment, thoughtheywouldstill belinked(fromelectromagnetic theory)bytherelation £ofl-o=1/c2 sothatonceoneismeasured thevelocityofelectromagnetic wavescan beusedtodeducetheother.Inpracticewewishourunitstobesimply relatedtotheolderunits,andwetherefore take f'o=4'IT10-7henry/metre (exactly). 736 UNITS [24.3 Thesizeofthecoulomb (orampere) isthenfoundbymeansofexperi­ ment.Infactallthequantities ineveryday usearethenmeasured in thepractical unitslistedinTable24.1. Thefactthattheunitofenergyinthem.k.s.systemisthejouledoes notmeanthatanenergyshouldneverbequotedinergs.Similarly there isnoreasonwhyamagnetic fieldshouldnotbequotedingauss, insteadofweber/metre2,inabookinm.k.s.units.Thisisonlypractical wherethequantities inthetwounitsbearasimpleratiotooneanother, andwehaveavoideddoingthiswheretheconversioninvolves afactor (4?r)aswellasapowerof10.Asfaraspossiblewehaveendeavoured to makethetextsimpletofollowforapersonpreviously conversant only withthec.g.s.systems, andinthefollowing sectionsadditional tables aregivenforassistance. 24.4.Conversion factorsfromrationalized m.k.s.system Sincequantities suchassusceptibility aregenerally givenintables intermsoftheunrationalized c.g.s.systems, alistofconversion factors isgiveninTable24.2bymeansofwhichthevalueofaquantitygivell intherationalized m.k.s.systemcanbemultiplied tofindtheequivalent quantityinanunrationalized c.g.s.system,andviceversa.Becauseof thechangeinthedefining equations consequent uponrationalization wecannotsimplyemploya'ratiooftheunits',andwegivetwosimple examples showinghowtheconversion betweenquantities inthedifferent systemscanbeaccomplished. (a)Intherationalized m.k.s.systemtheformula forthemagnetic fieldHatthecentreofacircularcoilofradiusawithoneturnis H=I/2a. Henceafieldof1A/metreisproduced byacurrentof1Aflowinginsuch acoilofradiusimetre. Intheunrationalized e.m.u.system,thecorresponding formulais H=27TI/a. Withthesamecurrentandradiusasbefore,wehaveI=10-1e.m.u., a=50em,andhencethesamefieldine.m.u.hasthevalue (27T10-1/50)=47T10-3e.m.u. Thustheunitofcurrenthasincreased byafactor10,andtheunitof lengtl).by10-2;butowingtothedifference inthedefiningequation the factorbywhichwemustconvertthequantity isgivenbytheequivalence afieldof1A/metre=afieldof47T10-3e.m.u. 24.4] UNITS 737 (b)Anexperiment isperformed withGouy'sapparatus (§8.7)to measurethedifference inthevolumesusceptibility ofaluminium andair. Inthee.m.u.system,theforceFonarodofcross-section Awithone endinafieldHandtheotherendinzerofieldis F=!(XI-X2)AH2. InafieldH=4000oersted,aforceFof4·92dyneismeasured ona rodforwhichA=1cm2•Hence XI-X2=0·615x10-6e.m.u. Inourrationalized m.k.s.system,theexpression fortheforceis F=l/-'o(XI-X2)AH2. Intheexperiment (usingtheconversion factorsofTable24.2),theforce F=4·92X10-5newton,A=10-4metre2,H=4000/(4rr 10-3)=106/1T A/metre. Hencethedifference ofsusceptibility is _ _ 2(4'92X10-5)_. -6 XlX2-(41T10-7)(1O-4)(106/1T)2-41T(O615X10)m.k.s. Thusweobtainaquantity inthem.k.s.systemwhichisgreaterbya factor(41T)thanthecorresponding quantity inthee.m.u.system.These arevolumesusceptibilities, andtheunitofvolumeisthemetre3inthe m.k.s.systemandthecentimetre3inthec.g.s.system. Theconversion factorformasssusceptibility isnotjust(41T),becausetheconversion factorfordensityinthetwosystemsisinvolved. Since Xmll8S= Xvolume!density, theconversion factorforXmassgoingfrome.m.u.tom.k.s.is (41T)/(103)=41T10-3, or103/41Tgoingfromm.k.s.toe.m.u. 24.5.Equivalent equations inunrationaIized c.~.s.systems Rationalization isthemaindifficulty whichprevents simplerules beinggivenforobtaining theequivalent equations inc.g.s.unitsto replacethoseinthetext.Inthefollowing tablesmethods foreffecting thetransition aregivenforeachchapter. Theseapplyonlytothe numbered equations, butotherexpressions inthetextmayreadily bemodified bytheirhelp. 861110 3:B 738 UNITS TA.BLE 24.2[24.5 ]i'actorbywhich quantityinm.k.s. systemmUBtbe multiplied to Unitinm.k.s. Unitinc.g.s. converttoc.g.s. Quantity system system sYstem Length Metre Centimetre lOs Mass. Kilogramroe Gramme 103 Time. Second Second 1 Density kg/metre 3 g/cm3 10-3 Force Newton orkg-Dyne 106 metre-sec-l Couple Newton-metre Dyne-em 10- Work Jouleornewton- Erg 10- 'metre Power Watt,joule-sec-lErg-second-l 10- orvolt-ampere Chargeq Coulomb e.m.u. lo Currenti Ampere e.m.u.J..-10 Potential V Volt e.m.u. 10' Electric displacement D Coulomb-metre- se.s.u. 12'lTX106 Electric intensity E Volt-metre-lore.s.u. iXlQ-4 newton·CQulomb-1 Electric polarization P . Coulomb·metre-Se.s.u. 3x106 Inductance L Henry e.m.u. 10' Resistance R Ohm e.m.u. 10' Capacitance 0 Farad e.s.U. 9X1011 Magnetic fieldB . Weber·metre-Se.m.u. 1Qfo Magnetic fieldH . Ampere·metre-le.m.u. 47Tx1(r8 Magnetomotive force Ampere e.m.u. 47T/10 Magnetic flux:N . Weber e.m.u. 10· Intensity ofmagnetization MAmpere-metre-le.m.u. 1(r3 Magnetic moment m Ampere·metre" e.m.u. 103 Volume susceptibility X m.k.s./metre" e.m.u./cm31/4'lT Masssusceptibility m.k.s./kg e.m.u./g lOS/4'lT Gramme·molar susceptibility m.k.s./g-mole e.m.u./g·mole 108/47T Whereafactor3appearsinthetable,itinvolvestheapproximation c=3X1010CDl/sec. Tothesameapproximation, sincebydefinition /La=4'lT10--henrymetre-I, 1wehave £0=36rr10-'faradmetre-I, Zo(freespace)=12O?rohm. Ohapter 1 Toobtaintheequivalent equations ine.s.u.replace EOby1/411'in: 2,~4,5,7a,7b,7c,8,9,10a,10b,11a,11b,12,17,24,25,29,30, 31,37. Thefollowing equations areunchanged ine.s.u.: 3,6,13,14, 15,26, 27,32, 33,34,35. UNITS 739 (1.16) (1.18,1.22) (1.19) (1.20) (1.21) (1.23)(1.28)Theotherequations, expressed ine.s.u.,become P=XE D=E+41TP =eE JD.dS=JdivDdT=4'lTJPdT divD=4'lTp E=1+4'lTX EE=D=(q!r3)r D=EE=41TCT andtheright-hand sidesofequations 36,38, 39,and40mustbe multiplied by(4'lT)-1. Ohapter 2 Toobtaintheequivalent equations ine.s.u.replace EOby1/4'lTin: 1,3,4,5,6,29,30,31,47,48,49,50,51, 52, 53, 54, 55. Allothernumbered equations areunchanged. Ohapter 3 Allnumbered equations areunchanged except D=eE,J=CTE ) Ediv(grad V)=0,.CTdiv(grad V)=0JD.dS=41TQ,JJ.dS=I and R=e/41TCTO.(3.10) (3.11) Ohapter4 Allnumbered equations unchanged except:replace EOby1!41Tin 22,43,45, 46,47,48. Chapter 5 Theequations ine.m.u.dependontheway.thetheoryisdeveloped. Usingaparalleltreatment tothatinChapter 5,toobtaintheequiva­ lentequations ine.m.u.replace fLoby4'lTinthefollowing equations: 2,3,4,5,16,17,19, 23,39,46,4~4~ 50,53,54,58; replace fLoby1in: 10,II,15; replace 41Tby1in: II,12,13, 14,35, 36,37, 38,48,59; multiply by4'lTontheright-hand sidein: 21, 22,26,33,55, 56,57. 740 UNITS [24.5 (5.20) (5.30) (5.31) (5.34)Thefollowing equations areunchanged ine.m.u.: 1,6,7,8,9, 18,24,25,27, 28, 29, 32,40,41,42,43, 44,45,51,52, 60,61, 62, 63. Theremaining equations become: B=H+47rM B=pH J.L=I+41rX HI=Ho-47rMJ3 01uJ,pter6 Thenumbered equations areunchanged ine.m.u.except: 11,12, 15, 16, 17, 18(replace /-toby417). Also U=;17f(H.B)d-r (6.44) 3U=fJ.3Ad-r =;17fH.3Bd7' (6.45) Ohapter 7 Allnumbered equations unchanged ine.m.u.except: 13(replace /-toby1). Ohapter 8 Toobtaintheequivalent equations ine.m.u.: replace /Loby1in: 7,13,24,25; replace47rby1in: 21,22. Thefollowing equations areunchanged ine.m.u.: 1,2,3,4,5,6,9,10,11,12,14,15, 16, 17. Theremaining numbered equations ine.m.u.become: X=-2,83X1010L<r2> (8.8) 47TnI=JH.ds=Hda+Hmd m (8.18) 47TnI=BAa[da+dm] (8.19) Aa/LAm B=H=417Mssin2epcoseploge(bJa) (8.20) W=~fHdB (8.26) 24.5] UNITS '41 Ohapter10 Thetransition toc.g.s.unitsinthischapterisverycomplex because oftheuseofmixedunits.Thequantities D,E,p(chargedensityor resistivity) andu(conductivity) arethenine.s.u.,whileB,H,andJare ine.m.u.Wewilldealwiththevarioussectionsseparately. §§10.1to10.5inclusive. Wheredifferent fromthetext,thefundamental equations become divD=4rrp (10.1) 1curlE=--(oB/ot) (10.3)c curlH=41rJ' divJ=_!(op/Ot)c J'=J+_1(aD/at)4rrc 1curlH=-(41ruE+oD/ot)c curlE=-!(oBjot) =-(!:(oH/ot)c c 1 EcurlH=-(oD/ot) =-(8E/ot)c c cN=-(E/\H)4rr(10.4) (10.5) (10.6) (10.7) (10.10) (10.11) (10.23) w=lc2J~8p=c2H~p132rr28 (10.34a) Oftheotherequations whicharedifferent, theequivalent equations canbefoundasfollows: replace EO,floby1in: 8,9,19; replace (EO'flo)byl/c2in: 12,13, 14,15,17; replace EOby1/41r,floby41r/c2in: 24,26,28, 29,31, 33. Thefollowing equations remainunaltered: 2,16,18,22,27,30,34. Themodifications required inequations (10.25),(10.25a) arereadily foundfromequations (10.3),(10.7)above. 742 UNITS [24.5 Thesituation asregardsZoisrathercomplex, sincethisquantity isnotusuallydefinedinthec.g.s.systems. Inviewoftheequations V=JE.ds,1=(41T)-1JH.ds,itwouldbenaturaltodefineZoas Zo=4?T(E1I/~) (allquantities ine.s.u.oralline.m.u.). InmixedunitsZo(likep,0')shouldbeine.s.u.,andthen Zo=(41T/C)(E1I/HII)(Zo,E1Iine.s.u.;HIIine.m.u.). Foraplanewaveinanon-conducting medium, Ey=(fLMs~in mixedunits,andhence Zo=(4?T/c)(fLMi (Zoine.s.u.). (10.20) Theninequation (10.21)Zomustbereplacedby(fLMsinmixedunits, butequation (10.32)givesZoine.s.u.ifp,0'areine.s.u. §10.6.Allequations canbeusedinmixedunitsbyreplacing Zby (fL/E)i. §10.7.Iftheconductivity andresistivity aregivenine.s.u.,itis simplesttoworkine.s.u.,whenonlythemodifications Zl=41T/C, t=4?T0'3!c arerequired. Thusequation (10.56)becomes IAA'12-_l_2(fL!)t v(0'ine.s.u.). §10.8.Unaltered. §10.9.Thesignificant alterations are loAE=-cat-grad V(10.61) divA=_EfLoV (10.62) CBt H=(81o!r2)sin8cosw(t-r!c)-(21T810/r~)sin(Jsinw(t--rlc). (10.68) Replace EOby1/4?T,fLoby4?Tin: 63, 64,65,66, 67,73,74. Replace Zoby41TCtoobtainequation inc.g.s.u.(10ine.m.u.)in: 69,70. Replace Zoby4?T!c, EOby1!41T,fLoby41T/C2toobtainequation in e.s.u.(Rr,Poine.s.u.)in: 71,72. 24.5] UNITS 743 Chapters 9,13,14, 15,16,18 Thenumbered equations arevalidine.m.u.orpractical units,except: validonlyine.m.u.(mechanical quantities inc.g.s.u.): Chapter 14 9,10; Chapter 1610,11,12,13, 14,15; Chapter 18 21,25; validine.s.u.(writing Zo=4rr/e): Chapter 16 19,20; replace JLoby1toobtainequations ine.m.u.in: Chapter18 27,28, 29,30. Chapter11 Inequations 29,30,31replace EOandJLobylIetoconverttomixed units. Inequation 40replaceZlby(fL/E)ltoconverttomixedunits.Other numbered equations areunchanged. Chapter12 Allnumbered equations unchanged, exceptthatEOshouldbereplaced byIf4rrin3and7. Chapter17 Allnumbered equations areunchanged ine.s.u.exceptthatwhereit occurs EOshouldbereplacedbyIf4rr,andthefollowing equation becomes: D=Eo+4rrP=EEo(17.2) andinequations 33and34theequivalent expressions ine.s.u.arefound byreplacing Zoby(4rr/e). Chapter19 Replace EObyIf4rrtoobtainequations ine.s.u.in: 32, 33,34,35, 36. Chapters 20,21, 22,23 Replace fLoby1toobtainequations ine.m.u.(exceptin20.6:replace fLoby4rr). Replace EObyIf4rrtoobtainequations ine.s.u. APPENDIX A VECTORS A.1.Definition ofscalarandvectorquantities MANYphysical quantities arecompletely definedbymagnitude alone.Examples aretemperature, time,orlength. Thesearecalledscalarquantities. Theyobey theordinary lawsofalgebra,andarerepresented inthetextbyasymbolprinted initalictype. Otherphysical quantities, suchasvelocity, force,oracceleration arenotcom­ pletelydefined unlessthedirection aswellasthemagnitude isgiven.Such quantities arecalledvectors. Vectorquantities areprintedinbold-face typein thetext;abriefsummary followsofthevectorproperties whicharenecessary fortheunderstanding ofthetext. / - 0 (a) ~ (b)/---------/' / / / ~/-0 (c) aCd?:i /1 I QFIG.A.I A.2.Vectoraddition andsubtraction Avectormayberepresented graphically byanarrowpointing inthedirection of thevectorandoflength equaltoitsmagnitude. InFig.A.I(a),PandQaretwo vectors. Theaddition ofPandQiseffectedby drawing themasinFig.A.I(b),inwhichthe vectorsformtwosidesofaparallelogram. The vectorRdefinedbytheequation P+Q=Ris thediagonal ofthisparallelogram, anditsmagni­ tudeanddirection canbefoundbytrigonometry ifPandQareknown. Similarly, thevector FIG.A.2.HQ,RareatrightanglesD=P-Qisobtained fromFig.A.I(c).Inthe thenQ=Pcos9,R=Psin9.specialcaseonly,thatthevectorsP,Qare parallel, thenRisequaltothescalarsumofP andQ,andDisequaltothescalardifference, andRandDareparalleltoPandQ. Theconverse processisoftenuseful.Thatis,avectorP(Fig.A.2)canbe resolved intotwovectorsQandRsuchthatPisthediagonal ofaparallelogram, A.2] APPENDIX A 745 andQandRaretwoadjacent sides.Generally, QandRarechosentobeat rightangles,sothattheparallelogram isthenarectangle. QandRarecalled thecomponents ofP.Pmayberesolved intothreecomponents paralleltothe axesofCartesian coordinates ::c,y,andz. I I,,0. r O.I-------t-7 II,/0;'itI I Ip. I p. I --------------T---------+~1J~~_ I ~~ II ~....... II (JP.t.jp.............. liP. If)-_ I1-_I-..vA.3.Multiplication ofvectors (a)Multiplication ofavectorPbyascalarquantity mchangesthemagnitude ofthevectorbythefactorm, butthedirection isunaltered. Multiplication by-m givesavectorofmagnitude mPintheopposite direction, thatisthevector -mP.Ifi,j,andkarevectorsofunitlengthparalleltotheaxes::c,y,andz,we canwrite P=iP.,+jPII+kPz, z FIG.A.3 whereP."PII,and.Pzarescalarquantities givingthemagnitude ofthecomponents ofPparalleltothethreeaxes(seeFig.A.3).SinceQ=iQ.,+jQII+kQ. itfollows thatP+Q=i(P.,+Q.,)+j(PII+QII)+k(P'+Q.). (b)The8calarproduct. ThescalarproductoftwovectorsPandQiswritten P.Qandisascalarquantity numerically equaltothemagnitude ofonevector multiplied bythecomponent oftheotherparalleltothedirection ofthefirstone. Iftheanglebetween PandQis8 P.Q=PQcos8=Q.P and P.(Q+R+S+ ...)=P.Q+P.R+P.S+ .... Thescalarproductoftwoperpendicular vectorsiszero.Therefore, fortheunit vectorsi,j,andk,wehave I.j=j.k=k.i=0 and i.i=j.j=k.k=1. Anexample ofascalarproduct istheworkdWdoneonachargeqinmoving a.distance dsinaregionwheretheelectricfieldisE,whichis dW=-qE.ds. (A.i) 746 APPENDIX A [A.S Also, PA(Q+R+S+ ...) =(PAQ)+(PI\R)+(l>AS)+ .... Theformula foravectorproduct intermsofthe vectorcomponents maybeconveniently expressed as adeterminent. Fortheunitvectorsalongasetof right.handedCartesian coordinates, wehave iAi=jAj=kAk=O, iAj=k=-jAi,etc. PAQ=(iP",+jPlI+kP,)A(iQ",+jQI/+kQ~) =i(P1/Q.-Pz;Q~)+j(Pz Q",-P.,Q.)+k{P",QI/-P vQ,,), whichcanbewrittenas i jk Pi\Q= P",PI/Pz. Q",QI/Q• .Anexampleoftheuseofavectorproductistheequation fortheforcedFonan elementdsofawirecarrying acurrentIinamagnetic fieldB.Theforceisnormal todsandtoB,andofmagnitude IdsBsin8.Itisspecified bothinmagnitude anddirection bythevectorequation dF=I(dsI\B).Hence"(eJ'ThevectorprodUct. ThevectorproductoftwovectorsPandQisdl'lfineq as'a.veotorperpendicular tobothPandQofma.gnitude PQsinO.where8isthe anglebetween PandQ.IfPisperpendicular toQ,the vectorproductisPQbutifPandQareparallelthe vectorproductiszero. Thedirection ofthevectorproduct (PAQ)istha,t P1\Q inwhicharight-handed screwwouldmoveifturned fromthefirstvectorPtowardsthesecondvectorQ., asshowninFig.A.4.Hencewehave (PAQ)=-(QAP). FIG.A.4.ThevectorPAQ isnormaltotheplanecon­ tainingPlindQ.' Pl'oduots ofthreevectorsareoccasionally metwith,andcanbeevaluatedfroIri theforegoing rules.Thescalartripleproduct P .(QAR)=scalarproductofPand(QAR) isascalarquantity equalinmagnitude tothevolumeoftheparallelepiped whose sidesareconstructed fromthethreeveCtorsP,Q,R(seeFig.A.5).Clearly, P.(QAR) =(PAQ).R andthisisoftenwrittensimplyasPQR.Wehave PQR=QRP,=RPQ= -PRQ= -QPR= -RQP. Thechangeofsignoninverting theorderofanytwoofthevectorsfollowsalso fromthedeterminantal form Pll:PlIp,. PQR=Q",QIIQ. Rll:RlIR. Theformula forthevectortripleproductmaybeexpressed intheform PI\(QAR) =Q(P.R)-R(P.Q). A.3] APPENDIX A Thismaybeverifiedbyexpressing thevectorsintermsoftheircomponents aJong threeCartesian axes. A.4.Differentiation andintegration ofvectors Vectorquantities areoftenexpressed asfunctions ofscalarvariables. For example, theelectricfieldEcanbeexpressed asafunction oftheposition coordi­ natesx,y,andz.Thevectormaybedifferentiated andintegrated withrespect FIG.A.5.ThescalartripleproductP.(QI\R)isnumerically equaltothevolumeof theparallelepiped whosesidesarethevectorsP,Q,andR. B FIG.A.6tothesevariables. Thedifferential ofPwithrespecttoascalarvariable "is definedas q;p=limP(u+A.u)-P(u). du.Au-+O A.u WhenaforceFactsforasmalldistanceds,theworkdoneisdW=F.dsand ifthetotalworkdoneovera.finitedistance isrequired, wecanwrite W=IF.dB=IFcos9dB, wheredBisthecomponent ofdsparalleltoFatanypoint. Thisintegral occursfrequently andiscalledthelineintegralofFalongthe curve.ThelineintegralalongthecurveAB isillustrated inFig.A.6.Iftheintegration iscarriedoutroundaclosedpath,returning totheoriginalpointA,itiswrittenfF .ds. ThesurfaceintegralJF.dSisalsoim­ portant. F.dSisthefluxthroughtheele­ mentofarea.dSduetothefieldF,andthe..4. integral overasurfacegivesthetotalflux throughthatsurface.IfthevectorF=v represents thevelocity offlowofafluid,Jv.dSgivesthetotalvolumeoffluid passingthroughthearea.Sinunittime.IfFistheelectricdisplacement D,the integraJ givesthenumberoflinesofdisplacement crossingthesurfaceS. '148 APPENDIX A [A.4 Inmanyproblems inphysicsascalarquantity isusedwhichisasingle.valued function oftheposition coordinates ofthesystem.Forexample, inelectrostatics theelectricpotential Visafunction ofx,y,andzinaCartesian coordinate system. Thechangeinpotential corresponding toaninfinitesimal displacement dsisgiven byTaylor's theorem, thatis elV=(aVjox)dx+(oVjoy)ely+(8Vjoz)elz and ds=idx+jely+kelz. TherateofchangeofVwiththedisplacement sisexpressed intermsofanew quantity gradVwhichisdefinedbytheequation elV=(gradV).ds, (A.2) where gradV=i(8Vjox)+j(oVjOy)+k(oVjoz). gradVisavectorquantity andisanabbreviation for'thegradient ofV'.When gradVisparalleltods,dVisamaximum, sothatgradVisinthedirection of thegreatestrateofchangeofVwithrespecttothecoordinates, andisnormaltoan equipotential surface.Fromequation (A.l)theworkdoneonunitchargeinmoving adistance dsinafieldEis-E.dsandthisisequalto-dV.Therefore wehave E= -gradVandtheelectricfieldisequaltothegradient ofthepotential at anypoint,andisinthedirection ofthemaximum rateofchangeofpotential withrespecttothespacecoordinates. Theoperator i(ojox)+J(ojoy)+k(ojoz) isoftendenotedbythesymbolV(pro­ nounced 'del'),sothat gradV"=VV. (A.3) Theoperator Vcanberegarded asavectoroperator, whiohoperates onbothscalar andveotorquantities, andformsscalarandveotorproducts. Thusequation (A.2)oanbewritten elV=(VV).ds. (A.4) Ingeneral,anyscalarpotential funotion tf>,whichisfinite,single-valued, and freefromdisoontinuities (theseoonditions mustapplyalsotothefirstandsecond derivatives oftf>w.r.t.thespaoecoordinates), canberelatedtoafieldofforceF, where F=-gradtf>, 80thatoncetf>iseverywhere determined, Fisknownatallpoints.Also,the lineintegralofFbetween anytwopointsAandBisindependent ofthepath takenbetween thosepointssinoe B B B IF.ds= -I(gradtf».ds = -fdt/>=tf>.A-CPB ~ A A byanalogy withequation (A.2).Similarly, thelineintegralroundaclosedpath iszero. A.S.ThedJvergence ofavector Thedivergence ofavectorPiswrittendivP.Itisanoperator usedtodescribe theexcessfluxleavinganelementofvolumeinspace.Thefluxmaybeflowof liquidinhydrodynamics, heatinathermalfieldofvaryingtemperature, orelectric flux.Inthelattercase,theexcessfluxleavingthevolumeelement isrelatedto the.totalchargeenclosed byGauss's theorem. InFig.A.7thereisavarying electricchargedensity pthroughout space.Gauss's theorem isappliedtoa volume element:dxelydzatthepoint(x,y,z)inaCartesian coordinate system. A.5] APPENDIX A 749 Thetotalchargeenclosed ispdxdydz. Thetotalfluxthrough thefacesnormal tothex-axisis [{Dz+:X(Dz)dx}-Dz]dydZ =o~Zdxdydz. whereDzisthecomponent oftheelectricdisplacement paralleltothex-axisat thepoint(x,y.z).Writing similarexpressions forthefluxthroughtheothertwo I;."z)__ :'». / }----------------------., FIG.A.7 pairsoffaces,Gauss'stheorem becomes aD",aDI/aD.a;+ay-+a; =p, whereDz'DI/'andD.arethecomponents oftheelectricdisplacement alongthe threeaxesat(x.y,z). Theexpression ontheleft-hand sideofthisequation iswrittendivD,andis thedivergence ofthevectorDatthispoint. Nowusingtheoperator V,wehave V.D=(i:x+j~+k:z) .(iDz+jDI/+kD.) _aD",aD"+aD.-ox+fJyoz' Therefore V.D==divD. (A.S) Thedivergence ofavectorisascalarquantity, sinceitrepresents thenetamount offlux,orthenumberoflinesofinduction, comingoutofavolumeelement. If divD=0,thetotalfluxentering theelementdxdydzisbalanced bythatleaving it.Avectorsatisfying thiscondition issaidtobe8olenoidal. 760 APPENDIX. A [A.6 A.6.•Thecurlofavector Thecurl(orrotation) ofavectorPiswrittencurlP(orrotP).Itarisesiri. problems wherealineinNalofavectorroundaclosedpathisrelatedtothe fluxthrough thesurfaceenclosed bythepathofthelineintegral. Forexample. Ampere's lawforthemagnetio fieldduetoacurrentis fH.ds=fJ.dS. Letusapplythisequation toanelementdydz atthepoint(x,y,z)inaCartesian coordinate system(Fig.A.s).For"thex-component ofthecurrent, J""theline z w...'"/dY Jz }-------_._---,--------. y X FIG.A.S.Application ofAmpere's lawinCartesian coordinates. integralofHinthey,zplaneispositive inananti-clockwise direction. andwe have ,lzdydz=[HII-O:II~]dy+JR;+o::.d:]dz-[HII+O:II~]dy-[H.- ~~ldZ =(0:::-0:11)dydz. whereHIIandR;arethecomponents ofHparalleltothey-andz-axesrespectively. Therefore oR;oHII.,lz=----.oyOZ andsimilarly J.=oH",_oH.andJ.=oHII_oH",. 11ozox•oxoy Theseequations arewritten J",=curl",H, JII=curlllH. J.=curl.H orsimply curlH=J. wherecurlHisavectorquantity whosecomponents areexpressed bymeans·of thedeterminant :~~o curlH= - - - (A.6)axOyOz' H",By A.61, i.e. AlsoAPPENDIX A curlH=i(():;-()~lI)+ie~,"_I1:;)+ke:;-()~'")· (()118)VI\H=ioX+iay+koZl\(iH..+iHy+kH,,) =i(8H._()HlI)+i(8H.._8H.)+k(8H lI_8H,")ay8z 8z8xOxay =curlH.751 (A.7) A.7.Laplace's operator Another operator whichoccursinLaplace's andPoisson's equations inelectro· staticsistheoperator divgrad. IfVisascalarfunction, divgradV::;:V.{VV),and fromequations (A.3)and (A.5) (8·80)(8VOV8V\8sV81VI1IVV.{VV) =TOx+jay+kaz· ia;+iay+ka;} =iJxS+oyl+ezS' ButV.(VV)=V.V{V)=VIV,treatingVasaveCtor.Theoperator divgrad is therefore equivalent to 01OS()I VI=8xZ+ays+OZI' whichiscalledLaplace's operator (pronounced 'delsquared'). Byexpressing theoperators div,grad,andcurlintermsoftheoperator V, anumberofusefulrelations canbeestablished. Thereadershouid verify·for himSelfthoselistedbelow(remember thattheorderofanoperator anditsoperand mUstnotbealtered). ;curlgradV=V1\(VV)=O. graddivP ==V{V.P). divcurlP ==V.(VI\P) =0(cf.thesealartripleproduct iszeroiftwoofthe .vectorsareidentical). curlcurlP ==VI\(VI\P)=graddivP-VIP. divmP=mdivP+P .gradm wheremisascalar. curlmP=mcurlP-PI\gradm. div{PI\Q)==V.(PI\Q)=Q.curIP-P .curIQ. A.S.Stokes's theorem InFig.A.9thelineintegral ofthevectorHistakenroundaclosedpath bounding anunclosed surfaceS.Thisintegral isfH.ds.Ifthesurface is dividedupintosmal1elementsofares.dS;thenfrom§A.6 fH.d!=eurIH.dS. wherefH.dlisthelineintegralofHroundonesmal1elementofareadS.If this.equation isnowsummed overalltheelementary areas,alltheboundaries withinthesurfacewillcanceloutontheleft-hand side,andtheresultistheline integralroundthecircuitbounding thesurface. Therefore fH•ds=feurIH.dS. (A.S) ThisisStokes's theorem. Itisnecessary forHanditsderivatives tobewell- 752APPENDIX A [A.S behaved continuous functions, butinthecasesnonnally arisinginelectro­ magnetism, theseconditions aresatisfied. Conversely, ifthelineintegralofHroundaclosedcurveisequaltothesurface integralofPoverasurfacebounded bythecurve,irrespective ofwhatcurveor surfaceareused,thenP=curlH. FIG.A.9.Illustrating Stokes's theorem. A.9.Thedivergence theorem SinFig.A.lOisaclosedsurfaceinaregionwherethereexistsavectorfieldF. ThefluxthroughanelementofareadSisF.dS,andthetotalfluxthroughthe surfaceisSF.dS.Thetotalfluxdiverging fromanelementofvolume dTinside Sis,from§A.5above,divFdT, whereFisthevalueoftheforcefieldatthis FIG.A.IO.lliustrating thedivergence theorem. (A.9) fdivFdT=fF.dS,point.TheintegralSdivFdTthroughout thewholevolumeenclosed bySmust givethetotalfluxthroughthesurface,sinceforanytwoadjacent volumeelements thefluxthrough acommon facegivesequalpositiveandnegative contributions. Hence whichisthetheorem ofdivergence. Again,thevectorfieldFmustbeawell­ behaved function. Conversely ifthesurfaceintegralofavectorFisequaltothe A.9] APPENDIX A 753 volumeintegralofascalarfunction Poverthevolume enclosed bythesurface, whatever thesurface,thenwemayconclude that P=divF. FIG.A.H.Change SAin timeIltofavectorArota­ tingwithvelocityw.SA______ ~!S:A / / / / / /A.IO.Transformation fromarotating coordinate system Whendealingwiththeeffectofanappliedmagnetic fieldonanatomicsystem itisoftenconvenient totransform toarotating coordinate system. Vector methods makethistransformation simple,ascanbeseenfromthefollowing treatment. Suppose weareconcerned withsomevectorquantity A,whichtostartwith wewillsuppose tobefixedintherotating coordinate system(alineonaspinning topisanexample, butwedonothavetorestrictAto besimplyaradiusvector). Theangularmotionofthe coordinate systemisrepresented byavectorw,whose magnitude isequaltotheangular velocity andwhose direction isparalleltotheaxisofrotation. Itssense isthatinwhicharight-handed screwwouldadvance ifrotatedinthesamesenseastheangularmotion.If Aisfixedintherotating system,theninatimeStthe endpointofthevectorisdisplaced byanamount SA relativetoafixedcoordinate system,asshowninFig. A.H.Itisclearthatthemotionoftheendpointisa simplerotation abouttheaxisdefinedbyw.Hence SA=(wSt)AsinB=(WI\A)St. Hencethevelocity ofArelativetothefixedsystemis lim(SAjSt) =(dAjdt) =(wI\A). (A.IO) Ilt~ IfwenowsupposethatAisnotfixedintherotating system,buthasavelocity (DAjDt) relativetothatsystem,thenwehave,fromthevectoraddition ofthe twovelocities dAjdt=(DA/Dt)+(wI\A). (A.H) ThisrelationmaybeappliedtoanyvectorA,andhenceitmaybeappliedto thevector(dAjdt)tofindtheseconddifferential ofA.Retaining thenotation that(djdt)referstorateofchangeinthefixedcoordinate system, and(DjDt) torateofchangerelativetotherotating system, wehave(sincewisaconstant) dlAd(dA)(D )(dA)(D )(DA )dtl=dedi=Dt+wl\ dt=Dt+WI\ Dt+wl\A DIA(DA)=Dtl+2wl\Dt+wl\(wI\A). (A.12) A.It.Larmor's theorem Suppose thatachargeqismovinginafieldofforce(suchastheattraction of apositively-charged nucleus) whosevalueatanymoment isdescribed bythe vectorF.Whenamagnetic fieldisapplied, theequation ofmotionis dlrmdtl=F+qvI\B, (A.13) wheremisthemassassociated withthecharge,andvistheinstantaneous velocity (=drjdt).Thisistheequation ofmotioninvectorforminasetofaxesatrest 861110 3C 764 APPENDIX A [A.ll (A.I4)withrespecttotheobserver. Letusnowchangetoasetofaxesrotating with angularvelocity waboutthedirection ofB.Intransforming torotating axes(see §A.IO)wehavetherelation d2r D2r[Dr]mdt2=mDt2+2mw/\Dt+m[w/\(w/\r)], whereD2r/Dt2,Dr/Dtaretheacceleration andvelocity intherotating coordinate frame,andwistheangularvelocity e:lijPressed asavectorparalleltotheaxisof rotation, thedirection ofB.Thesecondtermontheright-hand sideisthe'Coriolis force'whichappearsiftheparticleismovingintherotating system,andthelast termisthecentrifugal forcenormaltotheaxisofrotation.If(w;\r)issmall compared withDr/Dt(asweshallshowbelowtobethecase),thecentrifugal force willbesmallcompared withtheCoriolis force,andinthefirstapproximation equations (A.13)and(A.I4)give D2r DrmDt2=F+qv/\B-2mw/\ Dt=F+qv/\B+2mv/\w, (A.15) wherewehaveneglected thesmalldifference between vand(DrjDt) (thevelocity intherotating frame)sincetheydifferonlybythequantity (w/\r)whichwehave alreadyassumed tobesmallincomparison. Itisapparent thatifwechoosethe rateofrotation oftheaxessuchthat w=-(q/2m)B (A.16) thelasttwotermsin(A.I5)willvanishandtheequation ofmotionisthesameas ifthemagnetic fieldwereabsent.Thustoanobserver rotating withtheangular velocity givenby(A.16)themotionofthechargeappearstobethesameasit wouldtoastationary observer intheabsenceofamagnetic field.Hencewemay regardthemotionofanelectronofcharge- einthefieldBasunchanged except foraprecession withangularvelocity w=+(e/2m)B abouttheaxisofB.This iscommonly knownasthe'Larmor precession'. Thefactthatitisjustifiable toneglectthelastterminequation (A.14)canbe seenasfollows. Whentheelectron isboundintheatom,itexecutes aperiodic motioninitsorbitwhosefrequency isofthesameorderasthatofvisiblelight. Thiscorresponds toanangularfrequency Wooftheorderof1015radians/sec. The termsD2r/Dt2andDr/Dtarethenoforderofmagnitude w~rand Worrespectively, sothatsuccessive termsinequation (A.14)decrease inmagnitude bytheratio (w/wo).Sincewisonlyabout1011radians/sec ~veninafieldof1weber/metre2 (10000gauss),thecentrifugal forcetermisanorderofmagnitude smallerthan theCoriolisforce.Inotherwords,theforceontheelectron duetothefieldBis smallcompared withtheforceexertedbythepositively-charged nucleus; ifit werenot,itwouldteartheatomapart. Thecentralforceassumed aboveisthatresponsible fortheorbitalmotionof anelectron inanatom,andtheangular velocity givenbyequation (A.16)is identical withtheangularvelocityofprecession (see§20.1)ofanelectronic orbital magnetic momentinamagnetic fieldB. • APPENDIX B THEUNIQUENESS THEOREM THEuniqueness theorem statesthatifapotential function VIisasolution of Laplace's equation whichsatisfiestheboundary conditions, thenitistheonly solution. Ifthiswerenotso,thenthereexistsanother solution 11;andwewrite V=17;.-11;,where,since17;.and11;areeachasolution, Valsosatisfies Laplace's equation.IfVcanbeshowntobezero,thenVI=11;andisauniquesolution. Consider therelation (cf.§A.7) -div(VE) =V.(VVV)=(VV)I+VV2V intheform f(€€ogradV.gradV)ch=fdiv(V€€ogradV)ch-fVdiv(€€ogradV)ch, wheretheintegral istakenoverallspaceoutsidetheconductors wherethereare nofreecharges. Thenthesecondintegral ontheright-hand sideiszeroby equation (2.1).Thefirstintegral isequaltoS(V€€ograd V).dStakenoverthe surfacesoftheconductors andthelimiting sphereatinfinity. Ontheconductors eitherV=0orS€€o(oVjor).dS =0,sinceeitherthepotential orthechargeon eachconductor mustbefixed.Forasetoffiniteconductors, V-+0asr-+00; thusVmustvaryasrftwheren;;;..1,andgradVasr-1,sothatSVgradV.dS variesasr2ft+!andvanishes atr=00. ThisshowsthatS(€€ogradV.gradV)ch=0,sothatgradV=0sincethe integrand isalwayspositive, HenceV=constant=17;.-J';.But17;.=VI=0 atr=00;whenceV=0andVI=11;everywhere, showing that17;.isaunique solutionofLaplace's equation. { APPENDIX C NUMERICAL VALUES OFTHEFUNDAMENTAL CONSTANTS (TOFOURSIGNIFICANT FIGURES) Asrecommended bytheCommittee onFundamental Constants oftheNational Academy ofSciences-National Research Council, U.S.A.(1964) ovelocity oflightinvacuo NAvogadro's number kBoltzmann's constant f3Bohrmagneton nuclearmagneton finestructure constant permittivity offreespace=(P-o(2)-1eelectronic charge melectron restmass Mprotonrestmass M/mratioofprotontoelectron mass hPlanck's constant nPlanck's constant/2'lT FFaraday's constant (Ne) elmcharge/mass forelectron e2/m aoBohrradius RoRydberg constant X0 RRydberg constant EO 47rEo P-opermeability offreespace(bydefinition) Zointrinsic impedance offreespace eVelectron volt kTenergyforT=290°K 1electron voltisequivalent to: wavelength A=1·240x10-6m frequency v=2·4161014sec-1 wavenumberv=8·066103cm-1 temperature T=1·161X104oK energyW=1·60210-19joule 1cm-1isequivalent to: wavelength ,\=1em temperature T=1·439°K2·998X108m/sec 6·0231026(kgmole)-1 =6·0231023(gmole)-1 1·60210-19coulomb 9·10910-31kg 1·67310-27kg 1·836103 6·62610-34joulesec 1·05510-34joulesec 9·649107coulomb/kg 1·7591011coulomb/kg 2·81910-8coulomb2/kg 5·29210-11m 3·2901015sec-1 1·097107m-1 =1·097105cm-1 1·38010-23joule/deg 9'27310-24A m2 =9·27310-21e.m.u. 5·05110-27A m2 (137'0)-1 8·85410-12farad/m 107/c2=10-9/9approximately 47rX10-7henry/mexactly 3·767102ohm 1·60210-19joule 4·00310-21joule APPENDIX D SOMEATOMIC FORMULAE INM.K.S.UNITS Rydberg's constant Xc Bohrradius Finestructure constant Bohrmagneton Nuclear magnetonme4me4 Rc--------=--o~-8€~h3-641T3€~nl 47r€on2 ao=me2 e2 ex=41T€OnC R_en t'-2m en fin=2M INDEX A,magnetio veotorpotential, 143,161,281. Absorption, non-resonant, 497. -resonant, 486. Aooeptor level,537. Admittanoe, 234. Ammeter, 181. Ampere, 130. Ampere's law,135, 137. -theoryofmagnetism, 195. Amplifioation factorofvacuum tube,340, 342. Amplifier, audio-frequenoy, 351. -effioienoy of,357. -power,355. -push-pull, 356. -radio-frequenoy, 359. Amplitude modulation, 375. Anderson bridge,428. Anisotropy energy, 628. Anoderesistanoe ofvaouum tube,342. Anti-ferromagnetism, 657. Atomiobeam,681. -clook,701. Attenuation onfilter,294. -ontransmission line,312. -inwaveguide, 318. Azbel-Kaner resonanoe, 722. B,magnetio field,126-30. Bandtheory,506-10. Barn,43. Barnett effeot,634. Baseeleotrode, 571. BiotandSavart's law,142. Bittermagnet, 213. -patterns, 631. Blookwall,628. Bohrmagneton, 577. Bolometer, 420. Boundary oonditions forDandE,20. -forBandH,138. Brewster's angle,273. Bridge,alternating ourrent, 424. -Anderson, 428. -Hartshorn mutualinduotanoe, 429. -Sohering, 426. -Wien,435. Brillouin funotion, 593,623. -zone,513. Brownian motion, 452. Capacitanoe, 22. -ofsphere,23.Capacitan(le oftwoinfiniteoylinders, 54. Capacitor, 22. Cathode-follower, 388. Cathode, oxide-ooated, 330. Cathode rayosoillograph, 415. Cavityresonator, 325. Characteristio ofvacuum tube,340, 348, 349. Charge, eleotrio, 4. Child'slaw,333. Clausius-Mossotti formula, 479. Coeffioient ofooupling k,163,246. Coeroive foroe,205. Colleotive electron modelinferromag- netism, 644. Colleotor junotion. 570. Conductanoe, 234. -input,fortube,390. Conduotion band,537. Conduotion current, 257. Conduotivity, electrioal, 521-8. -extrinsio, 538. -intrinsio, 536. -speoifio, 64. -thermal, 521-8. Contaot potential, 95. Continuity, equation of,63,257. Coriolis foroe,754. Correlation energy, 514,647. Corresponding states,lawof,625. Coulomb, unitofoharge,4. Coulomb's lawofinversesquares, 3,19. - -experimental proofof,10. Coupled oircuits, 243. Coupling ooeffioient k,163,246. -Russell-8aunders, 582. Crystaldiode,411. Curie,methodofmeasuring Xm'217. Curie'slaw,201,593. Curie-Weiss law,203,620. Curlofaveotor,750. Current balance, 192. -generator oircuit,354. Cyolotron resonanoe, 710. - -forelectrons, 715. - -forprotons, 713. - -insemi-oonduotors, 717. CylindrioaJ harmonio funotions, 47. Damping, ofgalvanometer, 184. Danielloell,113. deBroglierelation, 89,505. Debyeabsorption, 497. 760 INDEX Debyeunit,476. deHaas-van Alpheneffect,532. Demagnetizing factor,141. -field,209. Detection, 376. Detector, crystaldiode,411. -diode,375. -standing-wave, 432. Diamagnetism, 195,198-201. -ofconduction electrons, 529. Dielectric constant £,19. --measurement of,442. --theoryof,16,475. --variation withfrequency, 483. ---temperature, 480. Diffusion length,insemi-conductor, 560. Diode,thermionic, 331, -transistor, 565-9. Dip,angleof,223. Dipole,electronic, 13,39. -magnetic, 196. -radiation, 281. Discriminator, 385. Dispersion, 483-8. Displacement current, 257. -electricD,18,19. Divergence ofavector,748. Domain, ferromagnetic, 206,626. Donorlevel,537. Drude's theory, 85. Dynamometer, 183,419. elmmeasurement forcurrent carriers, 62. -forelectrons,~· elMmeasurement forproton, 716. Earnshaw's theorem, 31. Effective mass,510. --measurement of,717, 722. Einstein-de Haaseffect,635. Electrical conductivity, 521-8. Electrochemical equivalent, 110. Electrolyte, 110. Electromagnet, 210. Electromagnetic balance, 215. -units,731. -waves,256-87. - -impedance of,262. - -propagation of,inconductors, 265. ---indielectrics, 260. --reflection andrefraction of,269. --velocity of,259,445. Electrometer, 32. Electromotive force,66. Electron, 1. -elm,62. -inmetals,classical theory, 85. - --quantum theory, 88. -magnetic resonance, 698,703.Electron optics,75-81. -volt,89. Electrostatic units,3,72!l. Emitter junction, 570. Energybands,506-10, 516.-ofcurrentcircuit,143,144,172. -ofelectromagnetic wave,263. -ofelectrostatic field,26. ---systemofcharges, 25,42. --magnetic dipole,143. ---field, 175. Equipartition ofenergy, 452. Equivalent circuit,343. Exchange interaction, 582,609, 618, 630, 650. Exciton, 550. Exhaustion range,538. Farad,23. Faraday constant, Ill. -lawsofelectrolysis, 110. - -ofelectromagnetic induction, 158. Feedback, negative, 354. -positive, 364. Fermienergy, 91, 94. -surface, 513. -level,ofmetal,94. --ofsemiconductor, 545. Ferrimagnetic resonance, 709. Ferrimagnetism,664. Ferrites, 665. Ferromagnetic resonance, 707. Ferromagnetism, 195,618-55. -classical theory,204-7. Field,electric, 4,19. -emission, 99. -magnetic, 137. --measurement of,214,695. - -production of,207./ Filters,289-301. -band-pass, 299. -high-pass, 298. -low-pass, 297. -m-derived, 301. Flip-flop circuit,389. Flux,electric, 19. -magnetic, 143. Fluxmeter, 186. Foner'smagnetometer, 219. Force,onmoving charge, 152. -between currentcircuits, 174. Frequency, changing, 380. -measurement, 441. -modulation, 383. -resonant, 229, 236, 240. -standard, 701. Fresnel's formulae, 275. INDEX 761 Galvanometer, 179. -ballistic, 186. -damping, 184,454-6. Garnets, 666. Gauss'theorem, infreespace,8. --indielectrics, 18.--in e.s.u.,730. g,Landefactor,588. g.forelectron, 579,699. gnfornucleus, 611. Gouy,methodofmeasuring Xm'218. Gradient ofavector,748. Grid,339, 347. Gruneisen's formula, 523. Guidedwaves,315-26. Gyromagnetic, ratio,575. -effect,634. H,magnetic field,136. Half-power points,237,495. Halleffect,528,552. Harmonic generator, 375. Hartley oscillator, 369. Heisenberg modelinferromagnetism, 618, 644. Helmholtz coils,156. Henry,162. Hole,positive, 512, 529, 536. Hund'srules,582,652. Hyperfine structure, 612. Hysteresis, 205,221-3. Images, electrical, 48. Impedance, 229. -characteristic, 297,304. -offreespace,262. -ofametal,267. -input,fortransmission line,309. --fortriode,344. Impurity level,537. Inductance, mutual, 161,429. -self,161,428. Intensity ofmagnetization, M,135. Ionization potential, 121. Isotope, 1. Iterative impedance, 296. Johnson noise,454. Junction transistor, 569. k,space,513. -wavevector,89. Kelvin's bridge,83. Kipprelay,370. Kirchhoff's laws,68. Klystron oscillator, 400. -reflex,405. Kramers' theorem, 598.Landeg-factor, 588. Langevin, theoryofparamagnetism, 201. Lanthanide metals, magnetic properties of,670. Laplace's equation, 33,66,140. Larmor's theorem, 199,576,753. Lecherwireoscillator, 396. Legendre, equation of,35. -associated functions, 36. Lenz'slaw,158. Limiter, 385. Linecharges, 52. -offorce,7. Logarithmic decrement, 171, 187. Lorentz, theoryoflocalfield,478. Lorenzforce,152,529. -number, 524. Losstangent, tanIl,236. Magnetic field,duetocurrentcircuits, 148. --measurement of,214-16, 695. --production of,207. -focusing ofions,153. -induction, 126. -moment, 131,195,574. --offreeatoms,585,593. --nuclear, 610. -permeability, 139. -resonance, 677. -shell,130. -susceptibility, 139. -vectorpotential, 143. Magnetism, terrestrial, 223. Magnetization, M,139. Magneto-caloric effect,639. Magnetogyric ratio,196. Magnetometer, Foner's, 219. Magnetomotive force,134. Magneton, Bohr,577. -nuclear, 611. Magnetron, 405. Massspectrometer, 154. Maximum powertheorem, 66,356. Maxwell stresstensor,28. Maxwell's equations, 256. Metal-semiconductor junctions, 560. Mho,235. m.k.s.units,4. Mobility, 65. -ofelectrons inmetals, 86. --insemiconductors, 553. -ofgaseous ion,lI8. -measurement of,555. -variation withtemperature, 556. Modulation, amplitude, 375. -frequency, 383. -index,384. Molecular beam,681. 762 INDEX Momentum space,90. Mossbauer effect,641. Multipole expansions, 39. Multivibrator, 372. Mutualconductance ofvacuum tube,342. Mutualinductance, 162. - -bridge,429. n-typesemiconductor, 537. Neeltemperature, 658. Negative feed-back, 354. -resistance, 366. Neumann's formula, 162. Neutron, 1. -diffraction, 673. -magnetic moment, 611,684. Noisefigure,462. -Johnson, 454. -measurement of,470. -shot, 462. Nuclear induction, 689. -magnetic moments, 610,616. -resonance, 685. Ohm'slaw,64. Onsager, localfieldtheory, 491. Orbitalquantum number, 577. Oscillator, Hartley, 369. -power,368. -quartzcrystal, 437. -strength, 486. -tuned-anode, 364. -tuned-grid, 367. p-njunction, 565. p-typesemiconductor, 537. Paramagnetism, 195. -classical theory,201-4. -ofconduction electrons, 529. -Pauli,532. /' Paramagnetic resonance, 705. Parity,40. Partition function, 615. Paschen's law,122. Paschen-Back effect,590. Pauliexclusion principle, 88,90,580. -paramagnetism, 532. Peltiereffect,103. Pentode, 349. Permeability, offreespace, fIoo,129. -magnetic, p.139. Permittivity offreespaceeo,4. Phase-shifter, 254. Phonon, 522. Photo-conductivity, 551. Photoelectric emission, 97, 99. Piezo-electric effect,437. Planok's oonstant, h,89.Planewaveinconductors, 265. - -indielectrics, 259. - -reflection andrefraetion of,269-278. Plasmaosoillations, 123. -frequency, 124. Poisson's equation, 33. Polargases,480. -liquids, 491. - -radio-frequency dispersion in,493. -molecule, 109,480. Polarizability, 17,478. Polarization, electric, 17,478. -magnetic, 135. Positive hole,512. Potential, eleotric, 4-6. -magnetic vector,143-9,161,175,281. -magnetostatio, 134. Potentiometer, 73. Powerfactor,230. -amplifier, 355. -oscillator, 368. Poynting vector,.263. Practical units,733. Precession, 199,574,677, 708. Pressure ofelectromagnetic radiation, 278. Proton, 1. -eIM,716. -magnetic moment, 611. Quadrupole, electric, 14,30,43. -nuclear, 612. QualityfactorQ,171,237,242. - -oftransmission lino,315. - -ofwaveguide, 325. Quantum number, 577. Quarter-wave line,310. Quartzorystaloscillator, 437. Quenching oforbitalmomentum, 600, 609. Quincke, methodofmeasuring Xm'219. Rabi,moleoular beamapparatus, 681. Radiation resistance, 285. Radioreceiver, 386. Raman effect,502. Rareearthmetals,magnetic properties of, 670. Rationalized units,733. Rayleigh scattering, 500. Reactance, 232. Reciprocity theorem, 70. Rectifier, diode,336, 338. -p-njunction, 568. -semiconductor-metal, 564. Reflecting film,311,327. -power, 275, 277. Reflection coefficient fortransmission line, 307. - -ofplanewave,269-78. INDEX 763 Reflection coefficient, totalinternal, 276. Refraction atdielectric boundary, 21. -ofplanewave,269-78. Refractive index,260. --variation withfrequency, 487. Relaxation timeofelectrons inmetals, 84. ---inparamagnetic solids,703. --ofnuclei,690. Reluctance, magnetic, 210. Remanence, 205. Resistance, absolute measurement of,190. -highfrequency, ofwire,269. -negative, 366. -ofradiating dipole,285. -residual, 523. -specific, 64. -temperature coefficient of,64. Resonant frequency, 229,236,240. Resonance potential, 119. Ripplevoltage, 337. Rotating cordinate system, 753. Russell-Saunders coupling, 582. Scalarproduct, ·745. Scattering ofelectromagnetic waves, 49s-:.501. -ofelectrons inmetals, 522. Schering bridge,426. Screengrid,347. Secondary emission, 101. Seebeck effect,103. Selectivity Q,171. -oftransmission line,315. -ofcavityresonator, 325. Semiconductor, 65,536. -absorption edge,548. -degenerate, 547. -Fermilevel,545. -n-type, 537. -non-degenerate, 547. -p-type, 537. Shotnoise,462. Sidebands, 384. Skindepth8,267. - -anomalous, 528, 534. Snell'slaw,270. Spacecharge, 332. - -smoothing factor,464. Specificheatofconduction electrons, 517. - -ofaferromagnet, 637. Spherical harmonic functions, 36, 38. - - - expansion of,40. Spin-orbit interaction, 583. Spinquantum number, electronic, 579. ---nuclear, 610. Spin.waves, 648,709. Stern-Gerlach experiment, 590. Stokes's theorem, 751.Stress,atsurfaceofqielectric, 26. -tensor,28. Superconducting magnet, 214. Superconductivity, 527. Susceptance, 234. Susceptibility, electric, X.'17,19. -magnetic, Xm'139,200. --ofconduction electrons, 529. -measurement ofXm'216-21. ----~ Terrestrial magnetism, 223. Tetrode, 347. Thermal conductivity, 521-8. Thermionic emission, 97. Thermocouple, 107. Thermoelectricity, 103-9. Thevenin's theorem, 83. Thomson effect,103. - -measurement of,108. Three-halves powerlaw,332. Time-base, 417. Townsend discharge, 121. Transformer, 163. -highfrequency, 243. -lowfrequency, 247. -quarter-wave, 311. -transmission line,310. Transients, 165-72. Transistor, 569-72. Transittime,393. Transmission line,302-15. Travelling wavetubes,412. Triode, 339. Tunedcircuits, 236-42. Uniqueness theorem, 34,755. Units,729-43. Vacuum-tube voltmeter, 336,422. Valence band,537. VanLeeuwen's theorem, 592. Vectormodelofatom,576, 586. -product, 746. Velocity ofelectromagnetic waves,259. -ofwaveontransmission line,304,312. -measurement of,445. -ofwaveinwaveguide, 318. Volt,7. Voltage amplification, 343. Voltage standing waveratio,308,431. Voltmeter, 181. -vacuum.tube, 336,422. Watt,67. Wattmeter, 183,420. Waveequation, 259. -vector,k,89. -velocity, 259. 764 Wave-guides, 321-6. Wave-meter, 437. Weber, 143. Weissconstant, 203,620, 661. Wheatstone's bridge,71.INDEX Wiedemann's law,116,ZOO,524. Wien'sbridge,435. Workfunction, 92, 95. Zeeman effect,588. OXFORD BOOKS IONIZED GASES ByA.VONENGEL Secondedition1964 OISEINELECTRICAL CIRCUITS ByF.N.n.ROBINSON (OxfordLibraryofthePhysical Sciences) ANINTRODUCTION TOELECTRONICS ByB.V.ROLLIN THEFUNDAMENTAL ATOMIC CONSTANTS ByJ.H.SANDERS (OxftJrJIJibfflrynflhePhysical Sciellcu) THETHIRD LAWOFTHERMODYNAMICS ByJ.WILKS (OxJvrd1..ibmryvI/hePhysicol StieflcPs) OXFORD UNIVERSITY PRESS \851110/5/651ELECTRICITY AND MAGNETISM BLEAl\"EY AND BLEANE\ SECOND EDITION tilOOM1'1:"'~1j;;oj,NVSTlOILLY..Ell OXFORD