schumacher NMR

Introduction to Modern Physics Monograph Series EDITOR: FELIX VILLARS, Massachusetts Institute of Te chnolog y ROBERT T. SCHUMACHER, C arne g ie- M e I lon Uniue rsit y Introduction to Magnetic Resonance: Pr inciples and Applicationsagnetic Resonance I' r i nciples and Applications ITOBERT T. SCHUMACHER ( drtu,g ie- MeIIon Unioersitv rt .l Benjamin, Inc.. New york. 1970 IN TROI)UCI ION TO MAGNETIC RESONANCEContents EDITOR'S FOREWORD P REFACE CHAPTER I Basic Principles l-1. Definitions l-2. Energy in an External Magnetic Field: Spatial l-3. Stern-Gerlach Experiment l-4. The Rabi Magnetic Resonance Experiment l-5. Applications and Literature Surveyvlt I I Quantization 3 4 7 l9 CHAPTER 2 Macroscopic Properties of Nuclear Magnetism 2-1. The Equilibrium Distribution 2-2. Energy, Magnetization, and Susceptibility 2-3. Response to an Alternating Field; Complex Susceptibilities 2-4. The Bloch Equations 2-5. Solutions of the Bloch Equations 2-6. Some Experimental Considerations 2-7. Conclusion and Literature Survey CHAPTER 3 Line Widths and Spin-Lattice Relaxation in the Presence of Motion of Spins 3-1. Introduction 3-2. Random Walk Calculation of Z 3-3. Very Short Correlation Times: äot" < |22 22 28 30 34 36 43 55 58 58 60 63 xii Contents ?-1 Random Frequency Modulation; Spectral Density3-5. Some Applicarions 3-6. Summary and Literature Survev CHAPTER 4 Nuclear Magnetic Resonance in Solids 4-1. Rigid-Lattice Hamilronian 4-2. The Method of Moments 4-3. Thermodynamics of Spin Systems 4-4. Nuclear Quadrupole Interaction 4-5. Spin-Lattice Relaxation in Soli<ls 4-6. Summary and Literature Survev CHAPTER 5 Magnetic Resonance in the Alkoli Metals 5-1. The Pauli Susceptibility of an Electron Gas 5-2. The Electron-Nuclear Interaction 5-3. Conduction Electron Spin Resonance (CESR) 5-4. Literature Survev CHAPTER 6 M isce llaneous Subjec ts 6-1. Cyclotron Resonance 6-2. Optical Pumping 6-3. Magnetic Resonance in Excited States 6-4. Literature Guide APPENDIX Some Quantum Mechanics of Spin I INDEX64 72 88 9l 92 97 100 ll0 122 t27 130 130 133 t42 150 r52 t52 t72 188 206 209 217CHAPTER I Basic Principles llistorically, experimental investigations into the quantum properties of angular momentum and magnetic moments followed the same course that now seems to be the most natural in introducing the subject conceptually. This first chapter is concerned with the concepts and the experiments on isolated atomic systems with angular momentum, which began with the molecular beanr experiments of Stern in thc 1920's and which lead naturally into lhe rnagnetic rcsonancc experiments of Rabi in the 1930's. The rnaterial is probably länriliar to all studcnts with thc background of an introductory course in modern physics. However, it is recommended that even students with confidence in their command of the subject study the chapter, if only to identify special terminology and points of view relied upon in later chapters. The student who finds the quantum mechanical references of Section l-4 somewhat obscure should repair to the brief Appendix for some help, at least in the mathematical manipulations of the quantum mechanics of the spin j system in magnetic fields. I-I. DEFINITTONS A system consisting of a mass undergoing circular motion about a fixed point in a plane has angular momentum. If the mass carries electrical charge, it has a nugnt,tit'tnoment that is proportional to the angular momentum. It is comforting to know that such simple statements are true in general for quantum mechanical systems, and that, for magnetic dipole moments, the proportionality factor is a scalar. The theorem stating this is an application of a powerful and ubiquitous statement known as the Wigner-Eckart theorem. We are concerned with angular momenta of various atomic, nuclear, and elementary particle systems. Table l-l shows the conventional symbols used for most of the systems in which we are interested. When the discussion is about an abstract angular momen- tum vector, we usually use the vector symbol J, which serves also as the total angular momentum of an atom. Introduction ttl Mlrgnctie llcsonltrtcc 'l able l-l ( ()n\cntr()nlrl Syrnbols for Angular Momcnta Systcnr Symbol srrrglc clcctrr)n spin clcctron orbit at()m n ucleus atom including nucleusS L J:(L+S) I F:I+J We also find it convenicnt to consider thc angular momentum vector symbols to be dimensionless, and to display thc units in which angular momentum is measured explicitly. fhc furrdamental unit is, of course, hl2n : fr, Planck's constant. Thus, the Wigner Eckart theorem states simply 11 : yhJ (l-l) where y, the gyromagnetic ratio (more rationally, the magnetogyric ratio) is the scalar promised by the theorem. Now, if one pursues the example of the opening paragraph, the factor 7 can be calculated immediately. The angular momentum is llrJl : lr x nlvl : ntr2e), where r is the orbit's radius, o the angular frequency, ä, the mass, and v the velocity. The magnetic moment, in Gaussian units, is 1r: iAlc, where I : rr2 is the orbit's area; the vector is perpendicular to the orbital plane, as in the case of the angular momentum. Thus, iA _inrz _q@r2 _Jhq cc2c2mc(r-2) If the particle is an electron with charge e: -4.g x l0-1o esu, and mass rn:9. I x l0-" g, the gyromagnetic ratio, y: ef2mc, is related to the Bohr magneton: P" : *: hY : -o'g27 x lo- 20 ergs/G For nuclei, it is convenient to define a nuclear Bohr'magneton I'":l#: 5'05 x lo-2a ergs/G where M is the proton mass.Basic Principles Electrons, protons, neutrons, and p mesons have intrinsic angular momentum. Atoms and nuclei of interest to us are compound systems, the total angular mornentum and magnetic moment of which are still proportional by the Wigner-Eckart theorem, but the proportionality factor of which depends on the details of the system. Those details are conventionally absorbed into a g factor, or spectroscopic splitting factor. This factor is gr, the Landö g factor for atoms, or g :2.000. . . , according to the Dirac equation, for electrons and p mesons. We define the nuclear g factor by analogy. For the most part, it remains an experimental parameter characterizing nuclear moments, since, in most cases, nuclear theory is not yet able to provide better than rough estimates of its magnitude. In general. then, the expression ( 1-3) lrt:{l rltol:y,hl gives the relation between p and J, or f, and it defines g. Equation (l-3) also defines the gyromagnetic ratio y, which is now g(el2mc) for a system with intrinsic angular momentum (spin). Tables, particularly the most commonly encountered tables of nuclear moments, publish a quantity called " the magnetic moment in units of the Nuclear Bohr magneton." The maximum projection of J along any axis occurs for the state Mr:J. The magnetic moment is p, : gBoJ, and the published number is g,/. I-2. ENERGY IN AN EXTERNAL MAGNETIC FIELD: SPATIAL QUANTIZATION The energy ä of a magnetic moment p in an external fieldr H is given by the familiar expression E: _F .H or, in terms of the angular momentum,(l-4)u,: n,(#lot : no"7 : v.hr ' E: _ gBoHml (l-5) . ' [n a vacuum it does not matter whether one uses H or B for the magnetic field ifthe quantities are expressed in Gaussian units. Strictly speaking, one siould use g, oul conventionally most of the literature uses f/, a practice that we follow. occasionallyr( ls lmportant to make the distinction in solid state physics applications, and then itts wefl established that the correctly calculated B is to be used. Ittltotltrrlt()n t() M:rgrrctie llcsonaltec JI - sßtt ltMr -J+l J+) J-l J liq. Ll [:nergy-lcvel diagranr for spin of angular monlentumfield H. where rlJ is the projection of J on H. Quantum mechanics restricts mrto the 2J + I integrar or halr'-integral varues. The energy-lever dragramcorresponding to Eq. (l-5) is shown in Fig. l_1. I-3. STERN-GERLACH EXPERIMENT The application of a magnetic fierd H removes the 2J + | degeneracy ofthe magnetic subrevels, as we have seen. Arthough these stites are nolonger degenerate, the energy differences between them are very small.In a field of l0a G, Eq. ( I -5) corresponds ro an encrgy separatron of, I cm _r or about l0-a ev for electron moments, l0-a cm in. rb , "v ro. nuclearm-oments' This energ_y difference must be pcrccivc<J against a backgroundof 200 cm-r or 0.025 ev of thermal "n.rgy at ro()l' tcrrpcrature andseveral erectron volts of energy for atomic transiti.ns. Until the early1920's, the consequences of spatial quantization had been manifestedprimarily through the Zeeman effect and the Faraday and other magnero-optic effects' The Zeeman effect was incompletery unaerstooJ frior to ttr"discovery ofelecrron spin, and the quantitative reiation ofspatiäl quantiru-tion to the Faraday effect was obscure. The reality of spatial quantization was demonstrated in a particurarly graphic fashion by the Stern-Gerlach experiment, successrulry performedin 1922. If a beam of neutral atoms pasies through u rron-'ogän'.ou, -ug-netic field, it is undeflected by that fierd, even though the l'agnetic de-generacy is lifted. But if the field is not spatiaily horiogene.rr,.,"th.." i, unet force on the moments in the beam thai is given by ,i" "^pr"rrronJ in magnetic F: (F' V)H: ,,U# * ,,# * u"# ( l-6)F:0 Fr: F": !" (l-8)Basic Principles There is a component of this force that is constant while the.rnoment is in the gradient, and it produces a deflection ofthe beam thatris propor- tional to ;r,. To see that, choose the following simplest pobsible field gradient. See Fig. l-2. The beam travels in the x direction with the r l il |JHtdz r'---lll | | I .'vcrrlli - 1 |l- l' I Inrgncr cull|lnJtlnglrt/llzltt\/ L-,tlx{ /DcJmt__J x section c - c' 1.,de tec tor (b) (a) Fig. l-2 Schenratic representation of Stern-Gerlach apparatus. (a) Arrange- ment of main components: oven, collimating slits, magnet, and detector. (b) Cross section c - c' of (a). (c) Enlarged view of beam and magnetic field in region of the beam. (d) Appearance of film deposited on substrate in original experiments of Stern. {iefd arranged so that H,:0. The field is principally in the z direction. All derivatives of // with rcspect to -r vanish, and, in the beam region, both V'H:0 and Vxtl:0 are satisfied. The components of Eq. (l-6) are F, : o F,: tt,* . L,,d# F,: tjy* . p,a# (1-7) Furthermore, let the beam lie in the symmetry plane y : 0, where Hr=0 (Fig. l-2c). Then AHrlöz:0, and, since V x H:0, )H,löy: itlrlöz:0. Since V.H:0, AH"l)y: -0H,102, Eq. (l-7) reduces to eH, 0zL- (d) (c) aH,- Fu -a-' .JZ Introduction to Magnetic Resonance In the next section, we emphasize in great detail that the magnitude of the field 11" produces a torque lr x H, causing a precession about H (which is virtually entirely H" at the beam coordinate) such that 1r, is constant and 1r" oscillates about an average value of zero. So the only component of the force that produces a net deflection is in the z direction, and it may be written in terms of the magnetic quantum number as { : mtgBo(öH,löz). The presence of m, means that the beam splits into 2J + I components. The first experiment was done on silver (partly because the deposit could be easily " developed "), and two components were seen, as illustrated in Fig. l-2d.2 We now know that 2J + | -- 2 requires J : ,, and that the ground state of the silvcr atttnr is an orbital S state with a single electron of'spin l. lhe lirst expcrlment was d<tne prior to the discovery of electron spin. but the result was not Inlerpreted as requiring half-integral spin since it was assumed silver had an orbital angular momentum L : I and the ntr:0 state was not allowed in old quantum theory. Even in its simplest form the experiment has several components, none trivial, so that molecular beam experiments have long been known as the most dlfficult in atomic physics. The experimental problems to be solved include a high enough vacuum so that a typical beam atom can traverse the apparatus without colliding with a rcsidual gas molecule in a meter or more of flight. The source, usually an oven with a small hole, must produce a well-collimated beam. The field gradient must be as large as possible, but the magnetic field itself cannot change too abruptly in time as sensed by the moving magnetic moment that passes from the fieldfree region to a region of maximum field gradient, and then out again to a fieldfree region before striking the detector. Finally, some device must detect, with considerable spatial resolution, the beam intensity. The modern solution of these problems is discussed in detail in the definitive monograph on molecular beams by Ramsey [2]. A somewhat briefer discussion appears in another standard reference in the field of magnetic resonance, Kopfermann's Nuclear Moments l3l. Among the refinements particularly useful when two Stern,Gerlach apparatuses are put in serres for the standard molecular beam resonance experiment (Section l-4) have been velocity selectors between the oven and the field region so all mole- cules in the beam receive the same deflection. Sophisticated universal detectors, which partially ionize the beam and send it through a simple mass spectrometer before it registers on the ultimate detcctor, have also 2 The student will find it an amusing exercise in the propagation of crrors to watch for illustrations such as Fig. l-2 in which the beam is traveling in the 7 dircction, rrans- verse to the long dimension of the apparatus. As far as I can determine, the first such incorrect illustration appeared in A. sommerfeld's Atombau und specrrallinien [l), in all editions subsequent to 1923. lt is reproduced in many texts oi rhat school, but it also still appears in texts published in the United States as recently as 1967.Basic Principles been developed. (See references listed at the end of the chapten for more discussion of experimental techniques.) The Stern-Gerlach technique, by itself, reached its pinnacle of usefulness under the direction of Stern, particularly with the aid of Otto Frisch and I. Estermann. Although the resonance method of Rabi did prove to offer unheard of precision compared to the nonresonant experiments, the basic technique did provide a few triumphs beyond the first demonstration of spatial quantization. one of these was the discovery of the anomalous g factor of the proton (i.e., that Qp:5.59... rather than g:2.000..., as expected from the Dirac theory of a spin ] particle). The initial report of this work appearedin Nature in 1933 [4], and it is a model of elegant brevity. The student who understands it, sentence by sentence, has a good working grasp of many of the necessary fundamentals of modern physics. It should be emphasized that the Stern-Gerlach apparatus is a very useful practical example of a quantum mechanical state selector, or beam polarizer. The separated beams of moments of that energy from the field gradient region, each characterized by its own Drr, r,rc polarized. The apparalus may be reversed in function and a partially or l'ully polarized beam sent in. Its trajectory in the apparatus is determined by the state function (i.e., the ntrlevel) of the constituents of the beam, so the appararus now functions as an analyzer. These functions are important to undcr- stand and distinguish in following the magnetic resonance experimcnt of Rabi. A comprehensive discussion is given in volume 3 of the f'c'1,nmun Lectures on Physics l5l. I-4. THE RABI MAGNETIC RESONANCE EXPERIMENT If a Stern-Gerlach apparatus can be a state selector, it can also be an analyzer. What could be more natural than to put two of them in series, with some experiment in between? In the 1930's, Rabi, who had done postdoctoral work under Stern at Hamburg, performed the first magne(ic resonance experiment and made the first precision nuclear magnetic moment measurement, in a homogeneous magnetic field between a polarizer and analyzer. Figure l-j shows two inhomogeneous fields, produced by the conventionally designated ,{ and .B malnets, with the homogeneous C magnet between them. In the C region, the magnetic resonance experiment causes transitions between magnetic quantum levels. Consider a J: I system. Figure l-3 shows the polarizer and analyzer with field gradients in the same direction. Also, care is taken that the direction of the field ä itself always points in the same direction. At the end of the polarizer, one of the two separated beams may be deflected or stopped by a baffie, leaving a beam of pure mt : I particles, for instancc, Introduction to Magnetic Resonance H ,1 dlt a ldzttIlttItn Idll s ldz I I rnagnet( rrtagncl B nragnct Fig. 1-3 Fields and beam trajectories in Rabi resonance experiment. The solid curved line is a greatly exaggerated trajectory for a spin + system that undergoes artransition from mt : L to mr: - | in the resonance region R in the homogeneous C magnet. The dotted line in the B magnet represents the path of a molecule fhat does not undergo a transition and is prevented by the baffie from reaching the detector. to enter the C magnet. lf nothing is done to them there, they enter the sdcond Stern-Gerlach apparatus where they are further deflcctcd. lf the detector is placed to detect a beam that comes through undeviated, it will detect no signal. If the beam in the C region is manipulated so that some or all of the moments are put into the mt: -I state, then in the B field their deflection will be down, and, if the I and .B magnets are identical, then the B magnet will reverse the deflection produced by the A magnet, and the beam will hit the detector. Some of the preceding details are arbitrary and of no particular im- portance. The experiment as described is known in the molecular beam trade as a" flop-in " experiment: the change of state in the C region causes the beam to " flop-in " to the detector. A different position of the detector, or reversal of the gradient in the .B region, could result in a " flop-out " experiment. What is not unimportant is the maintenance of a magnetic field oriented in the same direction through the apparatus, or, if the field does reoilent, it must do so slowly, as seen by the moment as it moves through the various regions. The first restriction is required so that the beam remains in the same quantum mechanical state unless a transition to another state is deliberately produced in the C region. The second restriction is required so that no " accidental " transitions occur between magnets.Basic Principles 9 We turn now to the theory of the magnetic resonance experiment. The quantum mechanical and classical equations of motion of a spin in a magnetic field are identical: the latter equations are for classical angular momenta and magnetic moments; the former are for expectation values of angular momentum operators. It is often sulncient to consider only the classical case. Figure l-4 shows an angular momentum J, moment p, I J,F Fig. 1{ Magnetic moment p: yhJ precessing in a constant ficld llu. Ihc figure is drawn for positive y. inclined at an arbitrary angle with respect to the z axis, thc liclcl dircction. The classical equation of motion ist ( l-')) The change in J during dt, dJ : yfiJ x Ho dr, is perpendicular to thc planc defined by the vectors J and H6. The motion is a precession, with J defining a cone with the axis Ho. The angle between J and H,, rcnlatns constant. The precession frequency,@o:7f1o, is known as the Larnror frequency. In the molecular beam resonance method, äo is a spatially homoge ncous field produced by the C magnet. In addition, a transverse alternltrrlg field f1,(t) is applied at frequency ro in the C magnet region. Hr(l) : 2lH, cos at ( l- lO) In the presence of the alternating field, the equation of motion is,#:FxHo:yfrJxHe # : ,t x (käo + lzH tcos cof) (r-il) l0 Introduction to Magnetic Resonance The problem is to find J(l). Approximate solution to (l-ll) can be obtained easily by transforming to an appropriate rotating coordinate system. There is a theorem, proven in all classical mechanics courses, to the effect that the time derivative of J as viewed from a rotating coordinate system, öJl0t,is related Io dJldt, the time derivative as viewed from a stationary coordinate system, by the expression dJ AJ --:-+c)XJdt dt AJ ;+orxJ-7JxHool H.rr: Ho 19(l-12) where o is a vector whose magnitude gives the angular frequency of rotation of the rotating system and whose direction is the axis about which the system rotates. Some necessary insight is obtained by sub- stituting (l-12) into (l-9): (l-l 3) That is, as far as the rate of change of J is concerned, transforming to a rotating reference system at o is the same as adding an effective field <rr/7, and considering the motion in the effective field ( 1-14) It follows immediately that J is time independent in a rotating coordinate system such that H.r:O, or ot: -?Ho. The result, and the trans- formation, are intuitive for this case. The sense of rotation of J, as seen in the laboratory (i.e., stationary) frame in Fig. l-4, is just the sense of rotation of the rotating coordinate system necessary to ,. stop " the motion. The solution of the problem with alternating (or, conventionally, radio frequency or rf) field, Eq. (l-ll), can be obtained from the rotating co- ordinate transformation after observing the following: decompose I1r(r) into two circularly polarized components of equal amplitude rotating in opposite directions in the xy plane.ui: rt ' (". * ?) H'(t): H" + Ho (r-1s) (1-18)Basic Principles I I where H. : l/1(l cos @t + t sin ctrt) and Ho : llr(l cos cot - J sin cr.rt) (l-16) Simple inspection shows that Ho is rotating in the same sense as the moment in Fig. l-4, and Ho in the opposite sense. A rotating coordinate transformation to a system defined by or : f<o leaves the field fl, stationary in the transformed system, but the component rotating in the opposite sense rotates at 2a in the rotating coordinate systern To see that we may neglect Ho under most circumstances, examine the effect of Ho alone. Figure l-5 shows the fields Ho and H, as they appear in the rotating system Flg. 1'5 Fields and angles in the coordinate system rotating at r^r: -ft<,r. in which Ho appears stationary, that is, the rotating system defined by the transformation or : -ficr.l. The effective field in the z direction is k1.ffo - coli from Eq. (l-14). The total effective field is given by Hcrr: f.(*rr - 9) + rH, \ 7/ The angle 0 is defined by tan 0: Hl Ho - aly(1-17) 12 Introduction to Magnetic Resonance The motion of a magnetic moment initially along the z direction consists, in the rotating system, of precession about H.o with an angular frequency @"n: lH,n with the angle 0 constant. That is, the vector lr precesses on the surface of a cone having angle 6 and axis H.6. A brief digression from the main line of the development is necessary to dispose of the counterrotating component IIa. The fields we have considered produce their effect by virtue of being static in the rorating frame. H" rotates at 2o in that frame, . rhich means that whereas it acts to produce the same general effect as H, when it is approximately parallel to it, it produces the opposite effect at a time nl2at later, when it is directed opposite to Hr. On the average, we expect its effect to be zero. This result has several consequences. First, it is only necessary to apply a linearly polarized rf field in the C region, lbr we shall be certain, regardless of the algebraic sign of 1,, that one of the rotating colnponents will produce some precession of the moment away from the z direction. We arbitrarily chose the magnitude of the linearly polarized field to be 2Hr, so that the rotating component's amplitude would be the conventional il,. We could have applied a circularly polarized rf field, and that procedure is the one commonly used to determine the sign of 7. The arguments we have used to dispose of the counterrotating com- ponent were qualitative ones. Quantitatively, the neglect of the counter- rotating component is valid only if (HtlHol < l. [--xperiments have been done in fields for which this inequality is not well obeyed. The result is a shift in the radio frequency that produccs the rnaximunr elfect in tipping the moment to the -z direction. lt goes by the name "Bloch-Siegert shift." For future work, it is important to notice here the obvious fact that as the moment begins to precess around fl.tr, it acquires a component in the xy plane. In the laboratory fiame, that component is rotating at the angular frequency ro. To make the connection with the magnetic res- onance experiment done in the C-field region, between two Stern-Gerlach apparatuses, we must discuss transitions between ,r?r states, something the preceding discussion of a classical moment seemingly contains no hint of. Let us specialize the discussion to the case of a spin + system. The orienta- tion of the classical moment along the z direction is related to the prob- ability that the spin is in the + ] or - ! mr state. If the spin wave function is lX): alt> + bl-t>, then P(]), the probability the spin is in the l|) state, is l(1lr)l': lal2 and P(-+): l(*]lr)1, :lbl2. We consider the function. cosa:lol'-ltlt 0<a<z (l-le) with the subsidiary normalizing condition that lal2 + lblt : I (there is certainty of finding the spin in one or the other of the states). The functionBasic Principles 13 cos a has, in fact, the properties of the projection of the spin on the z axis, and it is through cos d that we make the connection to the classical cal- culation. Again, defining cosd: tt.Hollpll}Iol, we find, after some three-dimensional geometry, where we assumed the boundary condition a : 0 at / :0. Using the normalizing condition lal2 + lblz: l, we can also write (l-19) as cosd:1-2lbl Comparison of (l-20) and (l-21) yields immediately(l-21)cos d : I - 2 sin2 g rinz^lHttt2 P(-i) : lblt : sin2 0 sin2 Ü:! 2(l-20) (r-22) for the probability that, at time r, the spin is in the state nrr : - *, having started in state mt : L at t : 0. Equation (l-22) checks the result obtained by inspection of Fig. l-4; namely, that for 0:9O", the spin precesses to the -z direction in a time t : nf a"p : nlyH.r. Of course, 0 : 9O" corresponds to the condition of exact resonance. a : u)o: lHo (l-23) It is clear from (l-21) and (l-22) that the probability of a spin being in either of the rn, : * l states varies periodically, with a full cycle com- pleted with angular frequcncy yll.o . What is less clear, since we have not done a full quantum mechanical treatment but have only made a plausible connection to quantum mechanics, is that the amplitudes c(l) and ä(r) of the m1 : ** and Dt1 : - | states, respectively, vary in time in a coherent fashion. The significance of that statement is that the transverse com- ponents of angular momentum operators, ,/, and ,I' have nonzero but time dependent values. For our future discussion of transient methods in magnetic resonance, as well as the discussion of the Ramsey modification of the molecular beam resonance method, it is most interesting to imagine doing the experiment at exact resonance (sin 0 : l), and turning off H, when cr.r.6: : nl2. Then P(]): P(-+) : +. In rhe rotating frame, the amplitudes a and b are constant and equal in magnitude (but, since they 14 Introduction to Magnetic Resonance are complex, not n€cessarily equal in phase). We can write the spin wavefunction l1) tobe l1) : (2)-trz tll) + e*'o l-])1. Theexpectation value of .1, is, if we suppress the time dependence from the Larmor pre- cession. (/,) : *t(*l + e-tö <-ylJ,|+) + e'0l-t>) : 1t(1U,1 -I)e'$ + (-llJ,ll) e-tof (t-24) where @ is the phase difference between a and b. It would, in fact, be zero for the example we have used. The important point is that, just as in the classical case, the angular momentum has been turned over toward the xy plane. Viewed in the laboratory frame, it is precessing at crro , and will continue to do so indefinitely with the sante phase as long as nothing perturbs it. The point to emphasize is that the transverse components of J, J, and J, arise from coherent linear superpositions of state functions of the various mr states. Later, when we deal with ensembles of spins, the question of the relative coherence of the superposition from one spin to the next will be crucial. The Rabi rr.clecular beam magnetic resonance experiment is done by applying a transverse rf magnetic field in the C magnet. The original, and still common, method of applying the rf field is by means of the "hairpin," shown in Fig. l-6. The intensity of the beam at the detector is monitored as a function of the radio frequency <r-r. A typical intensity 1(<o) versus frequency curve has the bell-shaped appearance of Fig. l-7, ,-/"" beam Flg. 1{ The " hairpin," a method of applying the rf field in the C magnet of a resonance exDeriment._cosd 2/(c,r )Basic Principles l5 Fig. 1-7 Beam intensity at the detector as a function of the frequency of the rf field in the C magnet. which has been drawn for a hypothetical " flop-out " experiment. The exact shape of the curve is of no particular interest to us, but there is something to be learned by examining the origins of its width, Äro. Since the first object of a beam resonance experiment is to find a.ro , it is important to make Acrr as small as possible. Experimentalists sometimes use the rule of thumb that the frequency @o can be located to within Äco divided by the signal-to-noise ratio. (Of course, we showed no noise in Fig. l-7, but it is inevitably there in experimental data.) In the natural effort to make the signal as large as possible, to obtain the optimum " flop-out,,' the experimenter adjusts the magnitude of H, so that, in language appropriate to aJ : I system, the opposite spin state has the largest possible amplitude at ro : @o. From Eq. (l-22), that occurs when 7-F1rr : z. The time I here is the time of flight of the moment through the hairpin. Approxi- mately half the amplitude of the signal occurs when ar is such that p(]) : P(-t) : lt. We assume t and H, are fixed by the criterion Hrt : nfy. Then the half-maximum intensity occurs for cos a :0 in Eqs. (l-20) or (l-21), or P(-]) : lbl' : I in (l-22). We leave it as an exercise for the reader-an exercise involving primarily the solution of a transcendental equation-to show that P(+):P(-+):I when H.n:1.275Hb or 0:51.5o, and (f/e -ra.l:D:0.8I1r. It follows that the full width at half-maximum intensity, the Äro of Fig. l-7, is Äar : l.6yHt. When it is coupled with the other requirement, which we have expressed as yHrt : n, we obtain Lrlot : l.6n 0-25) 16 Introduction to Magnetic Resonance Equation (l-25) looks very much like the uncertainty principle, which relates the precision with which the energy may be established to the time over which the measurement is made. That is exactly what Eq. (l_25) is, for t is the time during which the system is in the probing fierd H,. Correct use of the uncertainty principle argument would have obviated the some- what tedious calculation of the line width, but the effort was worthwhile since it was instructive. To achieve greater precision, the experimenter must increase r, the time in the apparatus. That goal can be accomplished by selecting the slowest molecules coming from the oven-but with the certain result of a loss rn intensity-and also by increasing the length in the beam direction of the c-field magnet and the hairpin. The limitation rn atr.r, soon reached, is not from the preceding considerations, but rather fr.rn the inhomogeneity of the c field. The larger the magnet the harder it is t. pr.clucc o.e with a homogeneity of field that is less than the natural lir.irarions .l'L,q. ( l-25). To overcome this limitation, Ramsey devisetr a very beautiful technique, which we examine briefly for its own sake and for what it can contrrbute to the understanding of later magnetic resonance experiments. Consider a monochromatic beam (i.e., constant velocity u) so that each molecule in the beam spends the same time in the hairpin. Ramsey split the hairpin into two parts, both driven by an rf oscillator such that thephase of the rf field is the same in each. They are both in the c maenet. but situated at opposite exrremiries of rhe homogencous field. rigurä t-s l*- , -*l C magnel Fig. 1-8 Ramsey split rf field experiment.r hairpirt /tt lvlttlll tt--^ ll -Pg_____J/ l-- / ---J rlHoBasic Principles 17 shows the arrangement and defines some of the quantities we need. The length L is much greater than /. The latter is adjusted so that y H rt I : n12, where t,-- lf u. If the beam enters the hairpin in the nrr: l statc, it leaves in an equal admixture of nt, : ] and - 1 states, with phasc co- herence in the admixture. That is, the angular momentum is in the transverse plane and precesses freely about Ho at a:.lHo. In the time t": [,f1t it takes for the beam to reach the other hairpin, it precesses an angle <D in the ,r.r'planc. We dcline the average field H-o and thc rverlgc frequency o-o by the relatton rD: öo tL: yHotL ( I -ltr) The average -Eo is a spatial average of the fields that the beanr secs. (Wc have assumed for simplicity that the beam has zero transverse ditnctlstotts. so that each moment in the beam samples the samc l/,,.) Adlust thc frequency of the oscillator driving the hairpins to be cractlv rrt,,. .l hcrr the rf phasc ot'll, at thc scconcl htrirpirt is thc satttc lts thc phrtsc ol- tltc transverse c()lnp()llcnt ol'.1 . tlrlrt ts. thcy lltlc thc sittnc sprttiltl ()nctttlttt()ll. so that in the l'rlttilc r()tatltlg lrl 0r,, llrr nl()r)r('rrt L()nlilluc\ tllc pr('r'c\\l()ll from thc +z to thc : dircetiott tltirl tt \llrrtc(l trr llrt ltt:l lt:ttrpttl Figure l-9 illustrates thc prcccssiorts scltcrtt:rlrrrtllv ttt lltc tol;ttttrg tclct- ence system. The extent of thc irnprr),vcnrcnl ol tlrrs tccltnrrlrtc orcr tltc single rf field region method can be cstttttrttcd by thc ill)pr()l)rr.ltc trrr certainty principle argument. The line width Aor*,,,,,", wottltl llil\c t() ()[)cv Ä,rr"o-r." tt=2n (l-17) by analogy with ( l-25). Hence. Au)*un,,."/Aoconvenr(,..,r t,'t r. l'l . I o see that t, is indeed thc appropriate "measurement tinlc" t() irl\clt rr'l rrtl uncertainty principlc arguntcnt. examine what happcrts wltctt lltc tltrlt,r frequency is not qtritc r,r,,, but is rrr,, * Atr-rp. We now clclittc Alr* ptc- cisely by noting. as suggcstecl in Fig. l-9b, that if. during tltc tttttc /,.lltc rf oscillator accuntulatcs phase O * n, then the H, seen by thc s1'ritrs ttl lltc second hairpin is in thc l direction in the frame rotatltlg rll rr),r I hc precession of thc spins about that field undoes the work of thc lirst ltrttrprtt. so the spins precess up to the +z direction again. lf precession to tlrc -z direction gives a nraxinrum at the detector, then this sccotttl .rst' corresponds to an absolute nrinimum in the signal. The rl' osctllrttot frequency corresponding to this minimum signal is just (rr-ro + Ärr-l^)lr_ : rD * n Lttt^ : + '1"(l-ls) l8lntroduction to Magnetic Resonance (e, (t ) Fig. l-9 Rotating frame precession of the magnetic moment in the Ramsey split rf field experiment. (a) Precession in the first hairpin. (b) Average orientation of J between hairpins. (c) Motion in second hairpin. (d), (e), and (f) Same as (a), (b), and (c) except for reversal of rf phase at second hairpin. See text.(b)Basic Principles 19 The full width between the minima is Äcop: X2nltr. Although aco^ calculated this way is almost the same as the " blind " extension of (l-25), not too much should be made of the fact since different quantities are being calculated, which depend in detail on the shape of the tlsonarr"e line. The shape of a Ramsey curve for a " flopin " experiment is shown in Fig. l-10. The width ar.rr is roughly the width obtained if the split rf fields are put together. Fig. 1'10 Detector intensity as a function of frequency for the Ramseyexperiment. I-5. APPLTCATIONS AND LITERATURE SURVEY For precision measurements of magnetic moments, the simple stern- Gerlach apparatus was almost totally eclipsed by resonance methods. The molecular beam techniques by themselves have been used. however. in some important investigations. one of the most obvious is the in- vestigation of the velocity distribution of the beam emitted from the hole of an oven at temperature ?". The last publication by Stern before his retirement was an investigation of this subject, which, incidentally, was the topic that prompted his interest in molecular beams in the first place. In the paper by Estermann et ar. [6], the analysis of the velocities of the molecules is done by measuring their fail, or downward deflection, in the earth's grauitational field ! one rarely encounters any practical conse-quence of an atomic particle's grauitational mass in laboratorv atomicphysics. The combined Stern-Gerlach and magnetic resonance experiments of Rabi gave the first precise measurements of nuclea, -o-"ntr, and they are still used for this purpose, particularly, in recent years, to measureI(t:) 20 Introduction to Magnetic Resonance nuclear moments of radioactive nuclei. The number of applications is so large as to defy even reasonable enumeration. The student would be advised to read the elementary review articles of Frisch [7] and of Kusch [8], as well as to look into some of the comprehensive tomes, such as Ramsey []. Reading the original literature in this field provides a palatable introduction to scientific literature in journal form. Much of it appeared in the 1930's, when brevity to the point of total obscurity was not yet the hallmark of most papers in contemporary journals. The first comprehensive discussion by Rabi et al. l9l of the molecular beam, magnetic resonance method, and the first measurement of the anomalous moment of the electron il01 both are relatively readable by upper-division students with vector-model type command of atomic physics. Some further applications of particular importance will be discussed in Chapter 6. I cannot resist concluding this chapter by remarking on the central position occupied by the Stern-Gerlach experiment in modern quantum physics. The reason seems not to be any particular uniqueness or pro- fundity of the technique, but rather the simplicity of the technique as an example of the problem of state preparation in quantum mechanics. It serves not only as a favorite pedagogical vehicle (see Feynman, vol. 3 [5])but also as a source of " gedanken experiments " for weighty discussion of such vexing questions as the problem of measurement in quantum mech- anics. For an example of the latter, see wigner, symnretries antl Re- fections ill], particularly p. 160, where the student should expericnce a shock of recognition if he has read footnote 2. There is no doubr that the Stern-Gerlach experiment, and the Rabi resonance experiment, represent the most elegant, simple, and yet most profound physics experiments of our century. Problems 1-1. (a) Calculate the vacuum required (in mm Hg) for an atomic beam expcri- ment if the distance from oven to detector is I m. Express the result in terms of the cross section o for collision between a beam at.nr and amolecule of residual gas. Assume that o: l0-r5 cm2 to 6btain aquantitative answer. (b) The atoms in the beam emitted from the oven do not havc the samevelocity distribution as the atoms in the oven. Sh.w that the beamatoms have a verocity distribution proportionar to r.,r exp[ 3nu2lzkr),where u is the velocity, rn is the atomic mass, /< -. l.3g .r lO-,u ergs.iis Boltzmann's constant, and r is the absolute temperature. Find themost probable velocity and the velocities u, and u, such that half theatoms in the beam have verocities between rrr ä'd üu . Let T - 500,K.l-2. I -3. t-4.Basic Principles 2l (c) Assume the field gradient in the magnet is r0r G/cm, and that themagnet is 40 cm long. Assume further that the detector is 50 cmbeyond the end of the fietd gradient region. Find the separation ofthe two beams of silver atoms for the atoms with the most probable velocity in the beam. Assume r: 500"K, and that the width of eachbeam is as determined in part (b). Let the hairpin of Fig. r-6 be lo cm long, the separation between thewires 3 mm, and the wires I mm in diameter. Find the rf current in thewires necessary to produce a transition from the m, - ü to mr: _t state of the ground state of silver for a beam atom of velocity l0r tm/sec.what is the width of a resonance line in an apparatus operated under theseconditions ? How far below line of sight does a cesium atom of most probabre velocityfall in a 2-m horizontal atomic beam apparatus if the oven temperature is100'c? In our discussion of the Ramsey split field modification of the molecurar beam resonance experiment, we assumed the beam molecules to have asingle velocity. f)iscuss quaritatively the c.nsequences to the line shape(Fig. l-10) if thc bcan'r is not monochronratic. Do not forget there are twotransit times that may have somewhat diffcrent consequences: the timcspent in the constant field region between the split rf netas, and the timespent in the rf field region. References A. Sommerfeld' Atombau und spectrailinien,4th ed., fr. vieweg und Sohne, _B-raunschweig, Germany (1924), p. 145, or 8th ed. (1960), uollt, p. t:S.N. F. Ramsey, Molt,cular Beams, Oxford University press, Lonion (1955).H. Kopfermann, Nuclear Montents, Academic press Inc., New york ileSSl.I. Estermann, O. R. Frisch, and O. Stern, Nclr/re 132, 169 (1933). R. P. Feynman, R. B. Leight.n, and M. Sands, Ile Feynman L"rtrr", unPhysics,Addison-wesrev, pubrishing co., Reading, Massachusetts irs6sl,vols. l. 2. 3. I. Estermann, O. E. Sinrpson, and O. Stern, phys. Reu. 71,23g (1947). O. R. Frisch, Contentp. phvs. 1,3 (1959). P. Kusch, Ph1,s. Todal'19, No. 2, 19 fi96il. L I. Rabi, S. Millrr-an, p. Kuseh, and J. R. Zacharias, phys. Reu. SS,526( r e39). P. Kusch and l{. M. Foley, phys. Ret,.74,2SO (194g). E. P. wigner, si.t'rttntetrie-s and Reflections, rndia.na university press, Bloomington (1967). Aduances in Atomic ancl Molecular physics, D. R. Bates and I. Estermann, Eds., Academic Press lnc., New york, vol. I (1965);vol.2 (1966); vol- 3(1967); and vol. 4 (l968).l. 2. 3. 4. 5. 6. 7. 8. 9. 10. ll. 12. CHAPTER 2 Macroscopic Properties of Nuclear Magnetism The enormous expansion of the basic ideas of the magnetic resonancetechnique beyond the molecular beam experiments occurred when it waslearned how to do experiments on macioscopic quantities of magneticmoments as found in solids and liquids. There is a considerable differencebetween flipping an isorated spin from up to down in a molecular beamexperiment and doing,the analogous thing all at once on 1022 spins inpunderable matter. we introduce in this "hupt". the statistical mechanicalsteps necessary to describe macroscopic magnetization and its interactionwith external electromagnetic fields ano wiitr the materiar in which it isimbedd.ed. The important new idea will be the concept of complexsusceptibility, and the practical achievement will be the Bloch .quutionsfor the behavior of nuclear magnetism in liquids and some discussion ofthe apparatus of nuclear magnetic resonance. 2-1. THE EQUILIBRIUM DISTRIBUTION chapter I has described the basic technique for producing transitionsbetween mr states of isorated spins. The apprications Jf magneticresonance in chemistry and solid state physics are based on those methods,but they emproy different concepts to produce polarization and to detectthe resonance. In this section we sha[ set forth the basic considerationsthat govern the relative popurations of the different m, levels when alarge number of identical magnetic moments interact with a heat reservoir(usually called a " lattice," even when the moments are in a riquid or a gas)' we shall be able to make some very general statements about theway thermal equilibrium is attained, ano we shall also derive a few of themacroscopic magnetic properties of the sample, such u, it, .ug*tizationand magnetic, or Zeeman, energy. For convenience, we.discuss nuclear magnetic moments, which can be nuclei of atoms in a solid, a liquid, or u gurl Much of the discussion willMacroscopic Properties of Nuclear Magnetism 23 apply also to electron paramagnets, but they have some properties thatpresent complications requiring rather more specialized äiscussion, forwhich we refer the reader to pake [l]. In the beginning, we further idealize the system of interest by assuming we can neglect the interaction of the moments with each other. we also make a more significant approximation that renders irrelevant the " statistics " of the paiticles- whether Bose-Einstein or Fermi-Dirac-by requiring the deniity of the system to be low enough to allow the use of Maxwell-Bortzmann siatistics.As a result, we specifically exclude from consideration in this chapterconduction electrons in metals or in liquid .He at low temperatures. Bothsystems require the Fermi-Dirac distribution function at low temperatures.otherwise, for nuclei, the density of ordinary solid matter is easily smallenough to allow the Boltzmann distribution to be valid. _ We start as simply as possible. Consider ,A/ spin ] nuclei iir a magneticfleld r1o, applied in the traditional z direction. The population of themr: XI states are N* and N_, with N* + N_ : N, Now the nuclei,whether in solid, liquid, or gas, have translational degrees of freedom_kinetic and potential energy. calr these degrees of freedom the rattice.Let the lattice be homogeneous; characterize it by a single constanttemperature r' our only other assumption about the rattiie is that itsheat capacity is large compared with the magnctic energy of interactionof the nuclear moments with the magnetic field. ror täw temperaturesand high fields' this assumption is, in practice, rather restrictive, and thetheory developed here must be redone to treat that case. Just how low atemperature and how high a fietd will be the subject of a problem. Figure 2-l shows the energy-level diagram and herps define relevantquantities. The Zeeman energy of each spin is If y > 0, the nt, - -+ state is higher in energy. To say anything more,we must assume that the spin system (i.e., the totality of .|y' spins, each mt=-) mr=+ä Fig. 2-l Energy levels of are ordered for y > 0.p: -yhHoml(2-t) "rhHo a spin I in a magnetic field I1o. Energy levels 24 Introduclion to Magnetic Resonance interacting with the field F/o) can exchange energy with the Iatticc. we need not specily the mechanism to make general statements about some of its properties. The lattice must be regarded as a quantum rnechanical system with states we label by Greek letters e, ß, .. . The states are to be thought of as simple harmonic oscillator levels for atoms bound rn a solid. The relative occupation of two levels a and B of energy E,and E, is proportional to exp[(E, - E,)lkTl. lt the lattice energy consists of the kinetic energy of translation in a gas, for example, then the energres Eo and Eo refer to the kinetic energy, and the preceding expression is Boltzmann's generalization of thc Maxwcll vclocity <iistribution. The combined system can bc labeled in tcrnrs ol poprrlutions of'thc various states of the subsystenrs. 'l'hc populati.'s arc spcciliecl by (,,V*, N ;N", Ne, ) Since the combined systcnts. Zcenran plus lattice, are assumed isolated fronr the rest ol'thc universe. an increase in Zeeman energy must be accompanied by an equal decrease in Iattice energy. Figure 2-2 shows two lattice state populations that differ by the Zeeman N -No NT- 1y, _1 _ f{o+l - '\'.+l Nd I state ( 2) (2-2)-r TltHsI state ( I ) Fig. 2-2 Energy levels for system of spin I and a pair of lattice levels withsame energy difference. States ( I ) and (2) of the combined system have the sameenergy. energy hyHo, and it indicates schematically two populations of the com- bined system that have the same energy. part (l) has the higher zeeman energy of the two; (2) has the higher lattice energy. we postulate that there is a rate process, determined by a transition rate ll, which connects pairs of states the Zeeman populations of which differ by one spin being turned over. we write the following equations for the time rate of .hunn" of the spin populations: +:-N+w(+--)+ N-W(--a1 ,lN-:_dN* dt dtand (2-3)N*o -(8, - E_) *yhHn N:b: exp kT : exP kt---:(2-e)Macroscopic Properties of Nuclear Magnetism 25 since N* * N- : N. Remember that the transition rate W(* - -), for example, involves implicitly a transition from (2) to (l) of Fig.2-2, including the redistribution of lattice-level populations. The thermal equilibrium condition is dN*ldt:0, from which we conclude that where the superscripts indicate thermal equilibrium. On the other hand, the total number of transitions per second of the entire system from (l) to (2) is given by trans/sec from (l) to (2) : N - Now (2-5) where n' rs a quantum rnechanical transition probability involving only the squares ol' rnatrix elcments. clcnsitics r>f states, and constants. The imp'.,rt of thc consitution of- '' is that it is nricroscopically reversible; w appears in the equivalent expression for the total numbcr of transitions per second from (2) to ( | ): trans/sec from (2) to (l): N*Nnw (2-6) In thermal equilibrium, the quantities calculated in (2-5) and (2-6) are equal, from which we concludeNro W(- - +) Nl: t(+ *) N*o Nf N,"- : N;(2-4) (2-7) (2-8)That is, the Zeentan state population ratio is the same as the population ratio of any pair of lattice states separated by the Zeeman energy. This latter ratio is, except lbr the case of very low temperatures alluded to earlier. N, _ exp(- EßlkT) , ^-^ -(Ep - E") Nn : exP --17- From Eqs. (2-l), (2-7), and (2-8), and from Fie. Q-2), we have 26 Introduction to Magnetic Resonance That is, the lower energy level of the spin system, mr : *1, is more highly occupied in thermal equilibrium. Moreover, from Eq. (2-3) we get W(- - +) thHo--::-: e.\D - lt(+ + -) kT(2- r 0) (2-n) (2-t2a) (2-r2b) (2-14) (2- r 5) (2-16)Downward transitions are more probable; indeed, you can rook atEq. (2-10) as providing the mechanism whereby the equilibrium population ratio (2-9) is produced and maintained. we may now return to (2-3) and tark about the approach to equilibrium from a nonequilibrium initial condition. Define the population äiff...n.. n: N* - N_ and rewrite N* and N_ in terms of n and N: r*: j(N+n) N_: j(N_n) In terms of y'/ and n, Eq. (2-3) becomes dn at : rtlW(- - +)- W(+ - -)l _ nfw(+- _) + W(_ - +)) (2-l 3) or, by factoring out flV(+ -- -) + W(_ - +)f, dn dtno-tl Tl where ^,w(--+)-w(+--)xo:/Y' : ' W(+ - -) + llrl- -, a; and I 7:w(+*-)+w(--+)I where no is the equilibrium population difference, as substitution of (2-10)and (2-l l) into (2-15) will verifv:l\1.r, r , "., .,pic PropertieS Of Nuclclr Nt,rltrct rrlr .' : ..W(+-' )[crP1 ,'/1 ;1.,4/)-ll .y1fill,,i,?o:N:-- ,i:Ntanhlfrf t:-tlt" W(f +-)[cxp(,'/r//,,4/tr rr \L^t / The time ?"r, defined by (2-16), rs thc s1'rrrr lirtlrt'c rcl:rxation time;it is the time constant of the approach of thc sprn \v\tcnr to tlrcrnurl equilibriunr with the lattice. lf (2-14) is solved with n(o) ();rs thc rrrrti:rl condition, as would be the case if the field were switchctl on sutlrlcrrlv ut / 0 al'tcr having always been zero before, we find that n : nofl - exp(- tfr)l lt is appropriate at this point to put in perspective what wc havc beerr doing, and perhaps even allow a glimpse of a skeleton in a closct. Equations (2-3), (2-5), and (2-6) are examples of the princlple o./' detailed balance, which was first used by Einstein in his l9l6 rederivation of'the Planck radiation formula. Given t--q. (2-2). which is also known:rs u master equotion, the irrcversibility of'thc approach to ccquilrhrrunr is already determined, even though (2-2) nrukcs cxplrtrt urc ol thc nrrr'ro- scopically reversible quantum mechanical trarrsrtiorr proh;rhrlrty rr', introduced in Eq. (2-5). The justilication, or, it'yotr ursh. tlcrrr'irtrorr ol the master equation is the central problem of nonccprlibrrrrnr stltrslrtirl mechanics. Magnetic resonance experiments on nuclcar sprrr svstcrrrr rrr solids and liquids have in recent years provided intercsting antl trlr.ruhlc model systems for which specific derivations of equations such us I,q t2-.1; could be tested, and the limitations understood. Although it is rather far afield from our main purpose, sonrcthirrg rrrorc than the preceding mysterious remarks can be made with rcgarrl to tlrc origin of the irreversibility. In terms of the coefficients a and ä introtirrccrl in Chapter I for the spin wave functions lf): al]) + äl - l), ir is clcar that(2-2) is an equation in lal2 and lö12, since the probability of occupuriorr of a given stage for a single spin is essentially (l/N) times the occup3rioll of that state in the ensemble of N identical spins. The informarion rhul is missing is the relative phase of the l!) and l-]) states, which, it'wc remember Chapter I , is related to the transverse component ol t lrc magnetic moment. In equilibrium there is no transverse macr()sc()prc magnetic moment, not even a coherent alternating one. From thut, rvc reason backward to the conclusion that the relative phase of the rr, r i states from spin to spin must be a random quantity, so that thc totrrl transverse moment vanishes. The reasoning is purposely circular, btrt rt points to the crux of the problem and reintroduces the idea that a nlrcro- scopic transverse magnetization, such as produced in a magnetic rcs()n.lncc experiment, has to do with a coherent admixture, from spin to spin, ol'rhc magnetic states m/ . 28 Introduction to Magnetic Resonance 2.2. ENERGY, MAGNETIZATION, AND SUSCEPTIBILITY The magnetic energy, or Z.eeman energy, of the spin system is given for a general spin 1 by I E: I E(m)N(m) üt = - I(2-18) It is convenient to define the zero of energy E(mr) for each spin to be at mr : O for 1even, and midway between the m,: * j energy levels for 1 an odd half-integer. Then Eq. (2-l) is the appropriate expression for E(m,) in Eq. (2-18). For N(m,), we shall simplif y matters a little by using an expression that will be valid in the high temperature limit only: yhHo 4 kT. Är,,...\ N'*!!:l't''-LII:llI) -. N -hvm,llu,'\,n,) =ffi)= 11 aexp - fri-" (2- le) The first equation in expression (2-19) is, of course, exact. The de- nominator is that fundamental expression of statistical mechanics, the " sum over states," or partition function. The replacement ol the denominator by 2l + I in the second equarion in (2-19). although retaining the full exponential expression in the numerator. is eorrvcntional but inconsistent. lf yhHom,lkT < I, the exponential in the denominator may be expanded: i .*o =#t : er + D -y!:;2 i^,.:(#)'t^, * The second term in the sum is zero since I mr : 0, but the next term is not. Thus, the denominator's lowest term in the expansion parameter is quadratic, but the numerator's lowest term is linear. To be consistent, we must take two terms of the numerator and one of the denominator. The exponent in the numerator has been retained at this stage because one so often is concerned with population ratios, in which case it is somehow easier always to write exp(E"i k I) (E" - F.r) "*pte utn: exP k7 -- = I -FE,_ EP KT , E"- Ep -kTthan itffi=('*#)('-#) ='Macroscopic Propertics ttl Ntt, lt.tt l'\l'rfilr('ll\r)r 29 Equation(2-19) satisfies Eq. (2-9), agrees with(2-17)rrr tlrc lttglt tcttrPt't:tltttc timit, ana satisfies conservation of spins, lr-, N(nrr) N. trt tltrtt lrrttrt a|so.Expansionof(2-18)with(2-19)inthehightempcraturclrtttttytcIt|s _#^,i) : - -r {11\' f ^,, 2l+l\kTl -t It is easily verified that lr-, m,2 :SI1l + l)(2|+ l)l/3' so thatE = #i*u"^,(t f,-Ny2h2t1t * l)Hu2(2-20) 3kr As an aside, it is interestrng to c()nlpilre tlrc lirrrrrLrl:r r2-20) with thc classical formula in thc sanre linlit: L -. (p ' Il,,)N ,1tl/,, N(tos ll; (2-21) where ( ) indicates the thernral llvcrallc ot'lltc qrtitttttly irrsrtlc il' crrlttrlittctl according to Boltzmann statistics. 'I'o find (cos (,1), ()nc rlltrsl wctSht cos 0 byihe probability that the montent 7r is.ricntc6 rrt rtrrglc () t"t ll', f^ I cos 0 exp(gHo cos 0/AT) rio J61 wheretheintegrationisoverthesolidanglethectlmplcterangetll.whlclr is the denominator 4n. Expand the exponential. The lirst term vanishcs and the second Yields: IrHo Ll'',/g.o.' o sin o ::* kT 2Ju"" 3 kT Substituting back into (2-21), we obtain(cos 0) : E:- N p'Ho' 3kr The quantum mechanical equivalent of p2 is y'h2t1t + l)" and the llrctor 'Occasionally one sees the quantity lg2I(l t l)l'i'] defined 1o q". tl". rrrill{Ircrr( nloment, rather than 91. iftt iottnt' is moit frequently found in the oltler lrtcraltlrt {rrr magnetlsm. 30 Introduction to Magnetic Resonance of 3 in the denominator of (2-20) is the quantum average of m,2,just as it is the average of cos2 0 in the classical calculation. (This equivalence was known as the "principle of spectroscopic stability" in the early days of quantum mechanics.) The magnetic moment per unit volume may be defined by the expression El V : - Mo . Ho . From (2-20), we see that '.:#,!iF}r" (2-22) The same expression is obtained from Mr: no!,1V, where no is given by the high temperature expansion oi (2-17). The static magnetic susceptibility 1 is defined by Mo:.XoHo, so we obtain the well-known formula (.x)-"* - (x)"r p (x)-or", : (t),"r u where p in (2-24a) is the density, and u in (2-24b) is the molar volume, u: NtMlp, where N,, is Avogadro's number, and M is the atomic mass. 2-3. RESPONSE TO AN ALTERNATING FIELD:COMPLEX SUSCEPTIBILITI ES In the Rabi magnetic resonance experiment, isolated spins interacted only with the static and alternating fields. Earlier in this chapter we introduced the concept of spin-lattice interaction. we now need to(2-23) A word about units is in order. In the Gaussian system, which is still largely used in research physics even though it is no longer the system of choice in elementary courses, the dimensions of M, B, and ll are the same. Hence, 1o, from the defining expression, is dimensionless. As defined here, it is often referred to as the uolume susceptibilit.t', however, to distinguish it from mass or molar susceptibilities. These quantities arise l'rom a definition of Mo in which the number of moments contributing is not the number in cubic centimeters, asin(2-22), but the number in a gram or the number in a mole. with such a definition, since Mo and .Flo must still have the same dimensions, the quantity xo is not necessarily dimensionless. The mass and molar susceptibilities are related to the volume (i.e.. dimen- sionless) susceptibility by (2-24a) (2-24b)Macroscopic Properties of Nuclear Magnetism 3l combine them all and discuss, as we did before, the components of the magnetic moment in the presence of these interactions. Now, however, the magnetic moment to consider is the macroscopic magnetic moment of the entire sample. We shall begin by discussing the z component, followed by the transverse components M, and M,. There is a subtle difference between the effect of the rf field on the isolated atom in Eq. (l-22) and its effect in the presence of spin-lattice interaction. The origin of the distinction lies in the fact that with spin-lattice inter- action the states mr ^re not quite eigenstates of the total Hamiltonian. When one speaks of the states in the approximate language of m,, one must then absorb the approximate nature of the description into a lack of sharpness of the energy level, and one speaks of the level as having a breadth given, in fact, by the uncertainty principle argument LE > hlTt. In discussing M in the presence of the rf field, we are particularly interested in small deviations of M from Mo . The rf field is regarded as a perturba- tion that, in the language of Chapter l, is to produce only a small amplitude ä in a state which initially had lal' : l. Thus, in Eq. (l-22), P(r) ( I in the perturbation thcory limit, if lolt -- I is to continue to be true. P(-]) is small for small times / alter the perturbation is turned on. and one sees immediately that P(-11= 12 for intervals of I such that P(-l) < t. Fortunately for the preservation of a linear theory, the quadratic de- perrdence on / is not correct for the problem with which we are now concerned, and the correct time'dependence for the probability of the mr: -I state being occupied is a linear rather than a quadratic one. The apparent paradox is resolved by noting that P(-]) was computed for exact eigenstates nr, : + ], whereas these are not exact eigenstates in the presence of spin-lattice relaxation. The final result must involve an In- tegration over all the states making up the " level " of nonzero width. The details are carried out in Abragam's treatisc Nut'lear Magnetism l2l (see Chapter 2). The probability of a transition from the state rrl, to the state m, - | t's proportional to time. Unlike the case of spin-lattice relaxation, however, the transition rate for a spin in state mt - | to rrl is the same as the transition rate for a spin in state lrr lo mr - l. The processes are to be contrasted: the spin-lattice relaxation process drives the population differ- ence n toward the thermal equilibrium value n6;the rf field drives n toward zero. The latter statement is easily verified: ldN- r l-l : -N, W,t * N -W,r: -nW,r \ ttt /,t and ldN -\ /dN. \l-', - I : - l-' ,:- I\ dt tt \ dt /,r 32 Introduction to Magnetic Resonance Hence, (2-2s) Equilibrium occurs when n : 0, Q.E.D. The transition probability per unit time W,, is properly computed by the standard time dependent perturbation formula (the so-called " golden rule ") of quantum mechanics. The perturbation is the interaction of the magnetic moment and the rf field, -p. H,;we need remark here only that Vl/,, is proportionalto H12. The total rate of change of n is the sum of (2-14) and (2-25)(#) ,,: -2w,,n 4!: -2w.,n+!:-u lt"Tl In the steady state dnfdt: 0, and (2-26) shows that(2-26) (2-27) (2-28) (2-29a) (2-2eb)Equations (2-26) and (2-27) may be rewritten in terms of M,: nyhl2 simply by substitutinE M, for r and Mo for no. Equation (2-27) shows that the rf field does not appreciably disturb M, from Mo as long as zw,rTr <1. When this inequality is not satisficd, n < n., and the resonance is said to be saturated. Additionally, the rate of energy absorption from the rf field may be calculated liom (2-26),| + zW-,Tl M,:2x.'Hr Mr:2x"Ht{: lt.nw-, - nohow'r dt " 1+2WnTl Note that dEldt becomes independent of W,, as W6 exceeds j?"r.2 The discussion of the transverse components of M in the steady state situation must be phrased in rather general terms. It is again convenient to work in the frame rotating with the rf field, angular frequency ro. Let H, be along the x axis in the rotating frame. Define susceptibilities 1' and 1" by the relations Equations (2-29) are, first of all, linear. That is, the transverse magnetiza- tion is proportional to the first power of the perturbing rf field Hr. Note that x' has been defined to be the proportionality factor between H, and 2 The failur"e of the first attempts to observe nuclear magnetic resonance in solids,by c. J. corter in the late 1930's, can be traced to the use öf a sampte with a r, tnatp'obably was several hours, so that 2w,rTt was undoubtedly very laige and the powerabsorbed from the rf field by the sample was very small.Macroscopic Properties of Nuclear Magnetism 33 the component of M parallel to, or in phase with, 11, in the rotating frame. The out-of-phase or orthogonal component is determined by X'. Necessary additional insight is obtained by transforming bdck to the laboratory frame, X, Y, Z: z, where we assume that the rotating.Ef, is produced by a linearly polarized rf field in the X direction: H,(t) : 2H, cos at In the laboratory frame, then, the X component of the magnetization is, from Eq. (2-29), Mr(t) : (X' cos @t + X' sin cot)2H, (2-30) Equation (2-30) may be expressed more compactly, and more convention- ally, as the real part of a complex quantity. We define the complex quantities ffx:2Hfi't and ilx:zXHGt-t. Their real parts are the physical field and magnetization, respectively. Comparison with Eq. (2-30) shows that M, : Re "// x only if 1 is the complex quantity x: x' - ix' (2-3r) The linear relation between the complex driving term -/f xe) and the com- plex response function .// r(t) has many familiar parallels in physics. Probably the first one encountered by most students is the generalized Ohm's law from ac circuit theory, {:-f t, where lt isthe complex impedance. The same care must be used in calculating quantities with complex '// and t(, as with complex current, impedance, and voltage. Thus the instantaneous power absorbed from a generator is (R e ,//) (Re /f), not Re(J/lf ). The average power absorbed by a unit volume of material is ,:+!,*"* ""(#)0,(2-32) where r :2nla is an rf period. The only term in the scalar product in the integrand is the X component, so P : + f"rrn rr' cos or(-rox' sin cr.rr * ax" cos .llt) dt :4Ht '-x"(2-33)3'-r"(+J'.or' at at) :2H r 3comparison,with Eq. (2-28) would allow immediate determination of y, if r/,1were known. we shall not pursue that course, since we shall eventually get ntir,for theparticular situation (2-28) represents (lifetime broadened levels) withoui ising luantummechanical perturbation theory. see the discussion at the end of section 2-5.' 34 Introduction to Magnetic Resonance The analogy between the impedance and r cannot be made blindlv. The real part of I determines the loss, and the imaginary part determines the nature of the periodic but lossless exchange of energy between the circuit and the generator. The terms "real" and "imaginary" must be inter- changed in the preceding discussion because the voltages that interact with currents in the magnetic system are induced; they are determined by d'/lldt: ia'//. The phenomenon of induction accounts for the factor of ar in (2-33) and the appearance of x" instead of x' in the expression for the absorbed power. without further assumptions about the details of the system, except for the all-important one that the " cause must precede the effect," one can establish that y' and 7' are not independent of each other but are related by integral relations known as the Kramers-Kronig relations, which were independently derived with reference to optical absorption by Kramers and Kronig in 1926. A thorough discussion and derivation of these relations may be found in the text by Slichter [3], in which there is a precise mathe- matical formulation of the somewhat enigmatic statement made previously about cause and effect. Relations such as those of Kramers and Kronig exist between the real and imaginary parts of the complex linear response functions of physical systems, whether they be magnetic or electric suscep- tibilities, impedances of passive electrical circuits, or reactions in elemen- tary particle physics. 2-4. THE BLOCH EQUATTONS Nuclear magnetic resonance in condensed material (namely, hydro- genous materials such as wat€r or paraffin wax) was first observed in 1946 independently by Professor Felix Bloch and coworkers at Stanford. and Professor E. M. Purcell and coworkers at Harvard.a In conjunction with the experiments at Stanford, Bloch proposed phenomenological equations of motion for the macroscopic magnetization vector M, which ,"ir" u".y well to describe magnetic resonance experiments in liquids, gases, or " liquidlike " solids. It will be one of the tasks of this boot to enable the student to comprehend in physical terms the limitations of the Bloch equations, but first we must set them down and explore their solutions. The object is to express the interaction of the magnetization 'ith the external fields (static and alternating), with the lattice, and to write a term that expresses the interaction of the magnetic moments with each other and with other internal magnetic fields in the sample. we have already accomplished the first two tasks, and we have a major portion of the Bloch 'Bloch and Purcell shared the 1952 Nobel prize for their work, the third Nobelprize awarded for work described in this book. Stern won the prize for his molecularbeam work in 1943, and Rabi for the magnetic resonance method in l9zr4.Macroscopic Properties of Nuclear Magnetism 35 equations if we assemble Eqs. (l-l l) and (2-26) in slightly altered and compatible form. The effect of the interaction of the moments under- going resonance with each other and other magnetic momedts in the iample, via the dipole-dipole interaction or through more esoteric quantum mechanical effects called exchange interactions, is contained within the Bloch equations by a single parameter that affects only the transverse magnetization, the components of which are M, and Mr' (We shall use lowircase subscripts for the laboratory coordinate system in this section, and identify equations written in the rotating system explicitly') Bloch assumed that the internal interactions of spins with each other could be expressed by the equation 1Mx,y _ _M',, 0t T2 (a) (b) Fig. 2.3 Rotating frame view of decay of M, Partial decay (t - T). (c) Total decay (l ) G).(2-34) Equation (2-34) defines the parameter Tr, known variously as the trans- u"rr" o, spin-spin relaxation time. (The partial derivative has been written only to call particular attention to the existence of the other terms that catse M,,, to change.) Equation (2-34) is the equation for a magnetiza- tion that ä".uyr exponentially to zero. Suppose that in the rotating frame at aro : 7i1o the transverse component M, is created at t : 0' In the context of Chapter I it persisted indefinitely. What can cause it to decay ? One obvious cause would be an inhomogeneous magnetic field across the sample so distributed that the Larmor frequencies of the spins in the various parts of the sample differ sufficiently so that in time 12 they would get out äf phu." with each other enough to diminish the initial M,to l/e of its value. Although it would take special field inhomogeneity to make the magnetization decay exactly exponentially, the point is worth illustrating with a figure. Figure 2-3 shows M,(O), in the rotating frame' Imagine M,(0) (c) (a) Initial condition.Mr(Tz) (b) 36 Introduction to Magnetic Resonance M * made up of small magnetization vectors that precess at a variety of frequencies differing by small amounts either way from the average frequency <rro . Then in a frame rotating at 0)6, they precess one way or the other until by t : Tz, their vector sum is still in the x direction but has diminished to M,(Olle. One internal physical process that diminishes the transverse magnetiza- tion is the spin-lattice relaxation time Z1 . Any other process, such as field inhomogeneity, only adds to the rate at which M'., diminishes, so that T, < Tr. The most interesting Ir process is the interaction of each spin with internal fields. By analogy with the external field inhomogeneity mechanism, we should guess that the internal helds that act to dephase M,,rare those parts of the total internal fields which are in the z direction and which are static and quasistatic. That the internal fields should manifest themselves just as single parameler T, in an equation of the form of Eq. (2-34) is, in fact, a result of rather special circumstances which we shall explore later. Combining Eq. (24q with the torque and relaxation equations, we get the Bloch equations: where rt:kF10+lHr(r) 2-5. SOLUTIOI.JS OF THE BLOCH EQUATIONS Solutions of Eqs. (2-35) are not difficult to obtain for a few special experimental conditions that are also of particular interest in practice. We have actually already discussed, in two separate parts, one particularly lnteresting solution that we can do without mathematics; thus we begin with that example. Free induction decay In making the plausibility argument for the form of the Trterm, we began arbitrarily with the magnetization in the x direction in the rotating frame. The subsequent exponential decay with time constant ?", , determined by Eq. (24\, appears in the laboratory frame asdM" Mo-M" , ^ ":T +7(MxH)' {. : _++ }(M x H),,, tlt T2 M-(t):M,ocosa)orc -r'xp T,(2-35a) (2-3sb) (2-36)Macroscopic Properties of Nuclear Magnetism 37 We have two problems: how to produce M"(0) and how to detect M,(t). Producing M,(O) may be accomplished by a "90" pulse," a transverse rf pulse of magnitude 11, in the rotating frame at @o : lHo, which acts for a time r such that yHrr : nl2. From Chapter I we see that if 11, is station- ary in the y direction in the rotating frame, M, : M o precesse s about ,F1, atyHt: arr until, at t: nl2yHr, it is pointing in the -x direction. The experimental arrangement to do this is shown in Fig. 2-4. We have neglected one thing of great importance. The pulsed rf field that pro- duces the 90' rotation of M o in the rotating frame must act in the presence of Zt and I, processes, rather than in their absence, as in Chapter l. Consequently, the 90" nutation of Mo is only an approximate description of what happens, and we must find how good an approximation it is. If the major torque on the magnetization is to be from I/, during 0 < / < r, then the relaxation toward äo (in time ?"r) and the dephasing of the transverse magnetization (in time Zr) must not be important compared to the precession about 11, during r: r ( Tr<Tr. The requirement on 11, is thus (nl2yHr) ( ?"2 , or fr lHr) nLr2(2-37) It is the same result we get if we assume, plausibly, that I/, must be much larger than the internal fields or external field inhomogeneities described by Tz. A word is in order about the magnitudes involved for a nuclear resonance experiment. The largest internal fields in ordinary substances (think of NaCl, for example) are caused by the nuclear magnetic dipole-dipole interaction. The magnetic field produced by one dipole a distance r from another is on the order of p/r3. Typically, since p - yh: l0-23 ergs/G, and r-2x l0-8 cm, AFl:plri =19-zr73x l0-2a.:l G. Although the Bloch equations are not, in fact, generally valid for solids, they do provide a framework for rapid estimates of upper or lower limits. To satisfyourinequalities, 11, Z l0Gisrequired,and r < nl(2 x l0a x l0) r: l0 psec. The Z, for this case is about Z, - lly L,H: 100 lsec. These calculations provide upper limits on fields and lower limits on times for this particular substance, because, as we shall see in the next chapter, the effect of the nuclear motion that occurs in a liquid or gas is to decrease the effective dipole-dipole interaction for ?", processes, often by several orders of magnitude. How large is the induced signal ? The precessing magnetization in the xy plane in the laboratory frame is of initial magnitude Mo: ysHs, where 1o is the static susceptibility, Eq. (2-23). Referring to Fig. 2-4,let the coil that produced the 90'pulse of11, serve also to pick up the induced I H"l /W\ -H, t' \ (a) field geometry (c) Flg. 24 Schematized apparatus for observing nuclear free induction decay. (a) Field geometry. (b) Electronics. (c) Rotating frame field and magnetizationA\_1/ nput (b) electronics Mo(t = 0)Macroscopic Properties of Nuclear Magnetism 39 signal caused by the rotating magnetization immediately after the 90" pulse. If the coil is of unit volume, cross-section area A, and has n turns, then the induced voltage will be Yo: -ndÖ c'dt-! utrn# : -! qxnn,qa*to (2-38) where { is the flux linking the coil: ö : Bl, and I :4nM, and 4 is a factor between zero and one, the " filling factor," which takes into account incomplete flux linkage between sample and coil. A problem at the end of the chapter will show that the magnitude of V can be as large as several millivolts for nuclear systems if the coil is part of a resonant circuit of reasonable Q.5 Several millivolts is, of course, a very easily detectable signal at radio frequencies. Equation (2-38) gives the signal Zo immediately after the 90' pulse, where the full equilibrium magnetization Mo is turned over into the transverse plane. The transverse magnetization, as we have seen, decays exponentially with a time constant Ir. Since 7", \ 2nlao, the signal is contained within a slowly varying envelope. The envelope decays according to the expression Zo exp(-l/?"r), which is known as the "free induction decay." Steady stote solution The next type of solution of the Bloch equations to investigate is the steady state solution in the presence of continuous l/,. We shall express the solutions of (2-35) in terms of the susceptibilities 1'(o-l) and 1"(ro). To begin, we rewrite Eqs. (2-35) in component fornr in the rotating frame of the rf field, which, in the laboratory frame, is H,(t) - l2II, c.ts <,tt The transformation is delincd by aer = rotating component, as in ('haptcr l.,,rL. Wc ignore the counter- dM, dtdl'rt-. i: -YM,II, ,Mr-M,-r, -') - Y,;(2-39a) (2-3eb) - M' (2-39c1 T2_ y,v,(rr, +: -rl*,u, - u.(tr, ;)] 5 The results of Eq. (2-38) will be in thc (iaussr;rn units, where I/ is expressed in statvolts. The result must be multiplicd by Jü) to cxprcss the results in volts. 40 Introduction to Magnetic Resonance In the steady state, the left-hand sides of Eqs. (2-39) vanish. From Eq. (2-39a) we see that (M o - M ") is proportional to M " H t. We expect, from the definitions of the susceptibilities X' and y", that Mrwill be pro- portional to Hr, so (Mo * M":) is of the order Hr2. As long as we restrict ourselves to terms linear in Hr,we can replace M,by Mo in Eqs. (2-39b) and (2-39c). The algebra is simplified by introducing .,//*: M,+ iM,. Add (2-3gb) to r/ -l : i times (2-39c): diln ,lt : -'// - The steady state solution tofll-+lT.,r("" - ?)l* iyMoH, (2-40) Eq. (2-40) is uy'y'* -iyMo H , tlTz + iy(Ho - uly) We define @o:lHo, substitute Mo:Xo-F10, and then real and imaginary parts to get(2-41) separate rnto (2-42a) M,: xoul,r, fi]_ffi- tr, M, : XoaoTz j* (ar - oo)T2H | \2-42b) From the definitions of X' and X", Eq.(2-30), we identify the components of the complex susceptibility: (2-43a) (2-43b)7,:xo@;rz i##ü ,,, _Xo@_oTz =___l__2 | *(ar_ ao)2T22 In Fig. 2-5, X' and y" are plotted versus (roo - a)Tr. - The components of the complex susceptibilities have some interesting features. The full width at half-maximum of the absorption x,, is 21'2, and the peaks of the dispersion X, are at o: an t llTr. Equations(2'43) are called Lorentz curves, after the rine shape ofthe opticat ausorp-tion and emission predicted by Lorentz using a damped simpte harmonicMacroscopic Properties of Nuclear Magnetism 4r Fig. 2-5 The rf susceptibilities X' and y" as a function of radio frequency c,-'. The vertical scale is in units of 1o c,ro Tzl2; the horizontal axis is in units of l/Tz . oscillator model of the atom. The power of the resonance technique is illustrated by the observation that the maximum value of 7", XoaoTrf2, may be written X,,HnlLH, where Aff :2lyTz is the full width of X" in field units. In liquids, ?", can be on the order of seconds, and hence the factor oro Trl2 or ttolLH can be as high as l0E. Thus, although 1o might easily be l0-rr, and hence the static magnetization XoHo: l0-? G, the dynamic susceptibility at resonance may be l0 a. Of course, the probing field Il, is necessarily small (to be discussed), but the magnetization is at roo rather than zero frequency, so the relatively simple techniques of rf signal amplification and detection are used. Had we not neglected the differcnce between Ms and M" in F'q. (2-39c), the algebra leading to Eq. (2-43) would have been more complicated, but far from intractable. The result would have been to add the term yt Hr'TrT, to the denominator of Eqs. (2-43a) and (2-43b), so that (2-43b), for example, would read Xo@oTzx: z l+(@-.,ro)(2-44) TtT,Hr T2+y The term 5: y2HrzT,?", is called the saturation factor. Regarded as a function of S, the absorbed power, 2Hr22q'as, is a maximum when S: l. For S) l, X" tends to zero; the resonance is said to be saturated. The solution for M" including saturation is also interesting: M":XoHol+Tr2(cr.o- co)2(2-4s) | + Trz(ao- (D)'+y'Hr'TrT, It is worth displaying (and worth the student's time obtaining), because it can be used with previous general remarks to calculate l/,1 . 42 Introduction to Magnetic Resonance Recall F,q. (2-27) for the excess population n: n: fro | + 2W,rTl Using no yh : 27o Hs, and M, : lnyh, we equate obtain(2-27) (2-27) and (2-45) to **::G##?s (2-46) Note the rapid diminution of the transition probability as the rf frequency or differs from a;o. Of more interest is W* at @:@o. To generalize as much as possible, we must consider the line shape for an absorption experiment. It is, of course, X", but we wish to express that shape as a function of v : al2n, g(v), which is normalized to unity: f*so)dv:r The correct 9(v) for " ror"o,, ttin. l, 2T, sQ): Value of 9(v) is 2712 at y : yo. Returning to Eq. (2-46), we then can substitute for I, the quantity Ig(vJ, and we get, for v: vo, ll'u: lyzHrzg(vo) (2-47) This formula is, in fact, the quanturn mechanical result (the .. golden rule ") for the transition probability of a spin I system, where the quantity g(vo) is the " density of final states," which explicitly expresses the width of the level via the shape and normalization. The quantity y2Hr2 14 is obtained from the square of the matrix element of the perturbation - p, Hr between the initial and final states: lGlyHJ,l - +>l' Finally, to complete our earlier discussions, we can equate (2-33), P :2Ht2X"a, and (2-28), P : floh<olft6lll + 2W,tZ,l, and solve for 1", which turns out to be (2-M). we conclude that the Bloch equations are consistent with our general discussions about energy absorption based on detailed balancing.Macroscopic Properties of Nuclear Magnetism 43 2-6. SOME EXPERIMENTAL CONSIDERATIONS A great variety of experimental arrangements to observe nuclear mag- netic resonance have been successfully used. Descriptions of some of the earlier ones are in the book by Andrew [4]. Although Andrew's book is old, most of the basic techniques still in use are described there. We want to analyze the experimental problem in a general way to establish the understanding with which the student can investigate on his own any particular circuit he may wish. Although our discussion will be couched in the language of radio frequencies---coils and capacitors are prominent- rather than microwave frequencies where resonant cavities are usually used, the analysis is really sufficiently general to apply to the microwave case also. Q-meter detection Consider a coil the inductance of which in the absence of a sample is l,o . lf a sample of permeability 4 occupies all space threaded by the magnetic lines of force generated by a current through the coil, then the inductance of the coil becomes L: FLo: (l + 4trX)L1) (2-48) If the sample does not fill all space, the susceptibility X must be multiplied by the filling factor 4, as in Eq. (2-38). The quantity 1 in Eq. (2-48) is the complex susceptibility defined in (2-31). Since no coil exists without resistance, unless it is made of superconducting wire, we must really always specify the coil's resistance Äo as part of the coil. Our problem is to calculate the coil impedance t t: Ro + i@9. Since l/ is complex, icr.rJf has 4 real part (assume 4: l): t,. -- Ro + ico[l + 4n(7' - iX"))Lo Ro + 4naLoX" + iU * 4nX'laL6 To design an experimental method of detecting 7' and X", we must know their size relative to other terms in Eq. (2-a9). Let us use, for numerical purposes, a fairly typical sample having a resonant frequency of l0 MHz in lOa G, and with a static susceptibility, )6 of l0-tr; Tr: Tz: l0-r sec. Then 421 has a maximum value of l0-7. A coil of wire that is useful at l0 MHz has an inductance of, say, a nricrohenry, so otLo=60 O, and a reasonable Ro would be I O. Therefore, the reactive (imaginary) part of t t changes by one part in 107 as we go through resonance, and thc resistive part changes by the fractional amount 4taL6X" lRo : l0- r. (lt Introduction to Magnetic Resonance is convenient to identify the ratio aLolRo: Q as the "quality factor." See a text on ac circuits if you are unfamiliar with the definitions and uses of this parameter in resonant circuits.) We conclude that a necessary feature of any circuit design is that it be particularly sensitive to small changes in the real or imaginary part of 9'r- With few exceptions, the coil L is used in a resonant circuit by placing a capacity C6 in parallel with L such that the Larmor frequency, 7H6, is the same as the resonant frequency ,,oo: ll(LoCo)tt'' The simplest con- ceivable circuit for observing a nuclear resonance is the so-called " Q- meter" circuit, shown in Fig.2-6. The oscillator and large resistor Ä Fig.2-6 Block diagram of Q-nleter nuclear nlagnetic resonance detector. form a constant current generator of current /,,. The parallel resonant circuit, tuned to cDo, presents an impedance that, at resonance, is real: Zo: Qa6Ls. The voltage across tt, Vo: IoZo, is amplified, and its magnitude is detected by the peak-reading voltmeter. From Eq' (2-49), we see that as we go through resonance by changing äe, for example, the real part of I changes fractionally by the amount 4nX" Q, so that the voltage at the amplifier input changes by A,.1. : Io4n/'Q2aLo or by the fractional amount y : I o 4!x_Q2_aLo : 4nx,, eVo IyQaLo(2-4e) We shall now give a word of explanation as to why we were able to ignore the change in reactance of the resonant circuit. For future uses it is best explained with a diagram, rather than analytically. Figure 2-7 is a complex plane diagram, sometimes called a "phasor" diagram, in which the magnitude and phase of the off resonance voltage across the sample isrf - level me terMacroscopic Properties of Nuclear Magnetism l<o - ool= llTz Fig. 2-7 Magnitude and relative phase diagram for voltages in Q-meter detection. given by Vo,the "signal " voltage by L7', and the total voltage by Vs * L'l', the magnitude of which is detected by the peak-reading voltmeter. The complex signal Ay'' is given by L{ : -Vs(4n7' * i{ny')Q (2-50) Reference to Eq. (2-50) and Fig. 2-7 shows that only the component of L{ in phase with Zo is effective in determining the length of Vo + Llr. Since Zo is real, because Zo : Qoolo is real, the Q-meter circuit detects only y" to first order. The imaginary component, - iVt) Q4nX', is said to be in quadrature and affects the length only to second order in X'. The analysis of the "series-parallel " resonant circuit of Fig. 2-8a is (a) (b) Fig. 2-E (a) Series'parallel tank circuit. (b) Equivalent parallel circuit. algebraically clumsy. Equation (2-50) is an approximation not only to first order in 1, but also to the extent that <r.rs : ll(LoCo)t/' is the resonant frequency only in the approximation Q ) l. It is more convenient to analyze ttre admittance a./ of the equivalent parallel circuit, Fig. 2-8b. The fictitious resistance R, is related to R", in the high p approximation, by Q: aLl R": RploL. Then it is easy to show that 0!: l/Rp(l + 4nxQ), and, by virtue of the smallness of 1, g : llg:.,Rp(l - 4"xQ), when <o6 : ll(LoCo)'/t-hen.e Eq. (2-50). The discussion of the " O-meter " detector. so named because the 46 Introduction to Magnetic Resonance detected voltage is proportional to the change in Q ofthe circuit, hence to X", was phrased to make it clear that the working of the circuit depends on the selection of the component of the signal in phase with a much larger voltage, in this case Yo. A clear grasp of this concept is necessary to appreciate the operation of bridges in magnetic resonance detection. Figure 2-7 and the subsequent discussion make the point that the Q-meter circuit picks out X" because Vo and nQX'Vo: Re(Ä/') are in phase. A large number of bridge circuits have been devised to accomplish this purpose, both in the microwave and rf regions. The archetype of the rf phase reference or coherent detection technique can be summarized in the block diagram of Fig. 2-9. In the discussion, it is understood that in Flg. 2-9 Schematic archetypical bridge circuit. the output of the oscillator Vo et'', the ei't is suppressed. We assume Zo is real for simplicity. The "bridge" is a device which, when balanced, has zero output, and which is unbalanced by the change in sample circuit impedance on resonance. The only output is the signal voltage L,7'. The phase shifter in Fig. 2-9 shifts the phase of the oscillator voltage by @ and supplies a reference signal to the adder, whose output is Voeio + L7/. The detector is just the "peak-reading voltmeter" as before, and the amplitude lVoe'Ö + L{l is, to first order in A"y'', just Zo plus the component of L{ in phase with the reference signal. One may write down the result more easily by pretending we shifted the phase of Ll. by (-@) instead of the phase of Vo by ( + d) and by computing the real part of A,{ e- io : R:e(L{e-'01 : 4rVo QX" cos $ - 4nVo Qx' sin $(2-5 l ) Choosing d : 0 with the phase shifter gives X", ö :90'chooses X', and, as advertised, any admixture may be chosen as well./oeio + AV lVoeio + AYI Vol phase I Voei,Macroscopic Properties of Nuclear Magnetism 47 In real experimental arrangements, the functions of two or more of the separate components of Fig. 2-9 are usually combined into one device. If a real rf bridge is used in place of the box marked " bridge " in Fig. 2-9, it may be balanced by dividing Voand sending half of it through an arm, after which it recombines with the half that went through the signal arm with, of course, equal amplitude and opposite phase. If the bridge is operated in this fashion, completely balanced (off resonance, anyway), then the reference part of Fig. 2-9 is needed. Often, the bridge is unbalanced either in amplitude or phase (or a combination. of the two), so that the steady unbalance signal is much larger than the signal. Then the external reference channel in Fig. 2-9 is unnecessary, and the unbalanced bridge has also performed the function of the adder. On the other hand, if the bridge is balanced, the "adder" and detector may be combined, as is often done in modern instrumentation, by using a device called a " mixer," which takes two signals applied to two inputs and puts out of the third port the sum and difference frequencies. ln our case, the difference frequency is zero, and its magnitude is given by Eq.(2-51). Every statement previc.rusly made for rf techniques (frequencies up to, say, 100 MHz) has its equivalent in microwave techniques. The tuned LRC circuit is replaced by a resonant cavity, which has a " Q" and to which the language of impedances, voltages, and currents may also be applied. Mic;owave bridges are in many ways easier to understand than the older rf ciruuits, and in recent years there has become available an rf version of the time-honored component of the microwave bridge, the hybrid junction or " magic tee." It is now possible to make a nuclear resonance spectro- meter that is an exact copy of the microwave equivalent, even to the extent of the language used to describe it. All the original nuclear resonance by the Harvard and Stanford groups used bridge techniques. The Stanford group lead by Bloch used a device, the crossed coil spectronreter, which is perhaps the instrument of choice if space around the sample permits. A sketch of the essentials of the arrange- ment is shown in Fig. 2-10. Unlike other methods the bridge balance is achieved by geometry. The field 11, is applied in the y direction in the laboratory by the split coil called the transmitter coil, and the signal is induced in the orthogonal coil, the receiver coil, in the x direction of the laboratory frame. Remember that the induced magnetization rotates in the x-y plane; thus either coil sees a magnetization varying at frequency <r.r. The bridge is balanced because, in principle, the lines of flux from the transmitter do not link any receiver coil turns. The unbalance is achieved by the natural lack of complete orthogonality, and control of the unbalance can also be achieved by mechanical means with " paddles," devices that " steer" the flux by mcans of currents induced in them by the field. Since the balance is achieved mechanically, it is somewhat broad band, and the 48Introduction to Magnetic Resonance Fig. 2-10 Crossed coil geonretry: 7' is the split tra.snrittcr coil and ,R the receiver coil containing the san-rplei Hu is applied pcrpenclicular to the page. Not shown are flux steering arrangenrents (paddles). resonant frequencies of the transmitter and receiver can be swept syn- chronously if necessary. one can also balance as well as possible mech- anically and achieve the rest of the balance and the reference through an electronic phase shifter, as in Fig. 2-9. There has even been a microwave version of the crossed coil spectrometer, with two resonant cavities coupled by the sample. one somewhat academic differcnce exist.s betw,cc' what is measured by the crcssed coil apparatus and the others we havc discussetl. Strictly speaking, the complex susceptibility 1is a tensor rather than a scalar quantity, and the single coil measures the response M,to a field applied in the x direction: we should write M, : x,, Hr,. The ,:rossed coil apparatus measures x,r, srnce the receiver coil is in the x direction and the transmitter coil is in the y direction M,: x,yHty. There is no case in pure nuclear magnetic resonance in which the distinction is important, since the pre- cessing magnetization is circularly polarized. In the case of pure qua- drupole resonance (see Chapter 4), the magnetization is actualiy linearly polarized, and X,r:0;the crossed coil technique does not work! Marginal oscillators Bridge and Q-meter circuits separate the function of oscill.tor and receiver. The separation of these functions has the advantage that the oscillator contribution to the noise of the device can be made negligible, so that the entire noise generation is in the receiver ancl sample circuits. The separation of these functions has the disadvantage that it is awkward to sweep the frequency rather than the external field in displaying the reso-nance or in searching for it. That objection is less true of the crossedcoil circuit, previously mentioncd, bctirtrrc llrr lrtt,lle I'llnn, r ,lc1x'rr,l. ,'rr geometry rather than on a nuntbcr ol lrctlrrcrrt ! rf rrrrlr\r ( il{ url rlrtrlcill'. Still, it is quite awkward to sweep ()vcr l lutlor ul llrlre rtr lrcrlrrcrrr \ unless the generator and receiver l'unctiorrs urc torrrhtlrerl I lrnt olrlctt is achieved if the sample tank circuit is nradc l pnrl of lhc ortrllrrlor l,rnl The device is then operated so that the voltagc lcvel o[ orltllntor tlcpcrrrlr critically on the tank circuit Q. Although thc tcrrrr 4at'Q rlro ullctts thc frequency of oscillation, normally only the levcl o[ oflcfllton n nr(,nrt()rc(|, so the circuit detects 1". The circuits thus use<.1 arc al lcrrl 6 lrrr'lor rrl lwo less sensitive than bridge circuits (because of the conlrrbulrrtrr ol oscrllutor noise), and they become very poor at low rf levels, wherc thc n{r\c pr()pcr- ties of the combined oscillator-receiver system bcconr Jxror llut thc convenience of being able to sweep frequency by adjustin3 olrly tlrc cnprcl tor in the tank circuit has made these circuits very populnr Robrnrorr has described a clever combination of Q-meter and murgrnrl orullulor circuits that seems to have all the advantages of Q-metcr nnd rrrurgrrrnl oscillator circuits, plus the ability to operate with very low rf lcvclr (to 100 pV) on the sample coil (Robinson [5]). Detection techniques We have progressed gently in this chapter from elementary statisticll mechanics increasingly toward experimental considerations. This is as it should be, since the numerical magnitudes of magnetization, the dy- namical behavior of that magnetization as it interacts with the externally applied fields with the Iattice, and with itself (7"r), all force onto the experimentalist the techniqtres he uses. To conclude this chapter, wc examine some of the methods commonly used-and used not only in magnetic resonance to display with greater clarity, with as much freedortr from noise interference as possible, the magnetic resonance signal. A few words about noise in general will have to suffice to establish the motiva- tion for the techniques used;lbr even a modest step beyond mere intro- duction to the theory and practice ofnoise on electronic signals, the studcnt will have to consult a text on the subject (particularly recommended is thc book by Robinson [5]). In discussing the various nuclear resonance techniques, we have slorrghctl over a number of crucial points in the interest of moving the narrltivc along. Let us remedy the fault by a rhetorical question: What arc lhc major sources of noise in typical magnetic resonance experintents'l 'l hc answer is oscillator noise, source noise, receiver noise, detector noisc, atttl microphonics. The marginal oscillator circuits are particularly srrs ceptible to oscillator noise; the bridge circuits and Q-metercircuits trtrty bc operated so that oscillator noise is not a factor. Receiver noisc is llrc Introduction to Magnetic Resonance best kind to haüe limiting your experiment because it can be combatted by application of design skill and/or money, and it will always be a limita- tion until it is smaller than the inherent noise of the source. The detector, the device that converts the radiofrequency into direct current, has one important property. It is usually a diode, so it converts to direct current efficiently only when the radiofrequency applied to it is, say, greater than + V. That is another rather good reason for applying a sizable reference radiofrequency in addition to the large one required by our analysis of phase detection. Now, if the device labeled " peak-reading voltmeter " in Fig. 2-9 is actually a dc instrument, one is at the mercy of the instabilities of dc amplifiers, since very frequently rf amplification preceding detection is insufficient to provide a signal of convenient size. The problem is over- come by modulating the signal at an audiofrequency la, so that the detected signal is at frequency y- , not zero, and may be amplified further. lf the signal is somehow modulated, the detected signal is at v. and noise near vd is of importance. Diodes contribute noise with a l/v frequency spectrum down to quite low frequencies. This restriction is particularly important for microwave diode detectors. Microphonic noise may, in principle, be overcome by good experimental design, but the ioeal is often hard to achieve in practice. Those with long experience with microphonics seem to agree that a noise spectrum of liv nray not bc bad approximation in practice. The source noise remains. A parallel tuned circuit on resonance looks like a resistance Ä,: QaoLo. The unavoidable noise presented to the input terminals of the receiver is called Johnson noise, and is given by the Nyquist formula VN2 : 4R"k sT L,v (2-s2) where the noise resistance Vn is in volts, R, is in ohms, I is in degrees Kelvin, ku, Boltzmann's constant, is 1.38 x l0-2r J/K, and Av, the bandwidth, is in sec-t. Note that Eq. (2-52) depends on Av, but is independent of v, the center frequency. Put together all these factors and one concludes that the game to play is to modulate the signal at as high a frequency as possible and to use as narrow a bandwidth as possible, dictated by Eq. (Z-52). Figure 2-ll shows schematically the most common method for impressing a modulation on a signal. The ordinate is 1" and the abscissa is äo , which is varied around the mean value by an amount H^, at frequency r,_. Thus, H(r) : Ho + H^ cos 2/ryn t, say. If 11. is smaller than I lyTr, the rf signal at the detector will be V(t):(Vo * Z-cosa.r-l)cos(Dot, where Z_ is propor- tional to dX"ldHo. The last clause follows from Fig.2-ll, and would beMacroscopic Properties of Nuclear Magnetism 5l Fig.2-ll Graph to shou that lock-in dctccts dy"illl. exactif y" were exactly straight between Hu - H^ and Ho * H^: the output signal V(t) is amplitude modulated with a modulation depth proportional to the derivative of X", at least to first order in an expansion parameter lTz H^. The frequency spectrunr of the amplitude modulated rf signal has a main peak at r,,, and sidebands at lu * r,.. The signal pou,er i:, in the sidebands. After detection (now we had better say after the first detector), the signal power is at a frequency ln. r,,n is usually a low audio frequency for nuclear resonance, but may be as high as 100 kHz in some commerical electron rpin rcsonance spectrometers. 1,,, must satisfy the requirements v^4liTr. so that the magnetization can follow the field and always be in the steady state corresponding to our solutions of the BIoch equations. The ultimate signal we wish to record is to be at direct current, that is, at co : 0. Clearly the job can be accomplished by the same method by which the rf signal was rectified, and, for much the same reasons, a phase- sensitive detector is employed. Since the phase of the audio signal carrying the signal power is known, phase-sensitive detection at the audio frequency may be used to discriminate against noise at v,, not in phase with the signal at r'-. The device that accomplishes this task, the second detector, is often called a lock-in. The terminology originated in radar work during World War ll. Finqlly, the bandwidth Är'is normally limited not by the bandwidth of an rf amplifier at vo , or the audio amplifier at r'., but by an RC network arranged as a simple " integrator" at the output of the lock-in detector. 52 Introduction to Magnetic Resonance Figure 2-12 shows dx"ldHo of the 27Al resonance in aluminum metal, taken with the bridge, rf amplifier, and modulation lock-in technique described in this chapter. There are many circumstances in which the signal is strong enough that signal modulation is neither desirable nor even possible. The only thing to be done is to sweep through the resonance with the external field fairly rapidly, making sure the subsequent amplifiers and detectors have enough bandwidth to reproduce the signal faithfully. During this rapid sweep, it may easily turn out that dHoldt may be too large to be ignored in the Bloch equations. Their solution then becomes much more difficult than our steady state solution; some of the details were worked out in tir- first in- vestigation by Bloch et al., since their experiments were done that way. Figure 2-13 shows a proton resonance in water. A problem at chapter's end deals with some of the aspects of that signal. 68 l0 Fig;,2-12 Nuclear magnetic resonance signal, rty', ldH, of ,'Al in Al metal.Macroscopic Properties of Nuclear Magnetism 53 II Fig. 2-13 Proton resonance in water, showing the phenomenon of " wiggles " (after Abragam [2], Figure III, 15). Spin echoes We conclude this chapter on some of the macroscopic aspects of magnetic resonance, and on the Bloch equations, by describing a pheno- menon that appears, at lirst glance, to be too special, too clever a trick, to merit description in a text with an aim as general as ours. But the phenomenon of the spin echo, of a physical system emitting spontaneous iignals if suitably prepared, has been exhibited in a sufficient variety of physical systems to make clear that it is a general property of systems with the sort of nonlinearity exhibited by the Bloch equations. The physical systems in which the spin echo has been seen, beyond the original nuclear magnetization system, include electron spin systems, atomic systems with induced electric dipole moments (at optical frequencies-the " photon echo "), and plasmas. The spin echo is most easily seen in the following sort of system' Let Trand?", be rather long, but let ?"2 be the parameter characterizing internal spin-spin interactions. Place ti; sample in a somewhat inhomogeneous external field H,, + H(r), so the variation of the external field over the sample, describecl by H(r), can be described by a T{ 47.2' The co- ordinate r ranges over the sample. The following sequence of rf pulses is applied to the sample: (a) at t:0,a 90'pulse; (b) at l: x,& 180'pulse, *here Z, > t > Tl . The result is a spontaneous signal of magnitude Voexp(-2rll2) appears at t:2t. Examine the process with the aid of Fig.2-14. The 90' pulse tips Mo into the x direction of the coordinate system rotating 4t ron: yllo (see Fig. 2-l4a). The " isochromats" precess in both directions relative to the x axis until they have fanned out completely, in t>Ti = l/yI(r), where a(r; is some average deviation of the field from l/e (2-l3b). Between lhct Introduction to Magnetic Resonance "-+YMacroscopic Properties of Nuclear Magnetism 55 end ofthe 90'pulse and the beginning ofthe 180" pulse at t: r, a repre- sentative magnetization vector will precess relative to the x axis of the rotating frame by the angle $t: yH(r,)t. The index i labels a particular small region of the sample. lt is convenient to label the precession angle relative to the y axis: @, : (n 2) + {1, where t'j is illustrated in Fig. 2'14b, for time t just before the 180" pulse. The 180' pulse flips the entire "pancake" about the 1'axis. The position of the magnetization that had precessed t', is shown in Fig. (2-l4c). lt is now at Öi:@ 2) - Öi. It continues to precess in the same sense during the subsequent time, so that, at 2t, it has accumulated another Ö, : ("12\ + 0', . The total phase accumulated at 2r, including the 180' pulse, is thus ("12) - $', + (rl2) + öi : n. The accumulated angle is independent of the labeling index i; therefore, all regions of the sample contribute to a signal at t :2t, and all magnetization vectors add to form a macroscopic vector in the x directron in the rotating frame. The shape of the echo may be described as two free induction decays back-to-back, as shown in Fig. 2-14e. The signal is attenuated from the initial magnitucle of the free induction decay, Vo, by real T, processcs. which result in ttnrc'coverablc loss of phase coherence, as distinct f'ronr losses caused by static magnetic lield inhomo- geneities. lf the echo height is nreasured as a fut.tctionof r,the 90 to t80' pulse separation, the echo height will follow the exponential exp( -2tiTr). The preceding example of a spin echo is the barest introduction. Many variations in pulse length, sequence, and number are possible. The tech- nique is frequently used in the study of liquids and solids, where vartous contributions to the line widttrs can often be unraveled. There are many modern applications of the basic idea in seemingly remote fields, such as nonlinear optics and plasma diagnostics. 2.1. CONCLUSION AND LITERATURE SURVEY Most of the matcrial in this chapter is covered in every text and revlew article that discusses magnetic resonance. The material requiring statisti- cal mechanics may be lbund in Feynman [6] and Reif [7], and, of course, all more advanced treatmcnts of statistical mechanics. Treatments of the Bloch equations more or less on a somewhat more advanced level than found here are in the article by Pake [8], the books by Andrew [4], Kop- fermann [9], Slichter [3], and Abragam [2]. Discussions of experimental Fig. 2-f4 (a) Effect of the 90' pulse on the magnetization vector Mo. (b) "Pancake" formed in xy plane of rotating frame after r > Tl. (c) Effect of the 180' pulse on the ith magnetization vector. (d) Precessing magnetization vectors as echo is forming, corresponding to t'of (e). (e) Pulse sequence and signals seen at various times. r'corresponds to vector diagram of (d). Dotted line traces echo envelope as r is varied. 56 Introduction to Magnetic Resonance techniques may be found in Andrew and Kopfermann. A totally ex- haustive compendium of everything done until 1966 in the area of micro- wave instrumentation is in the book by Poole [10]. The preViously mentioned monograph by Robinson [5] on noise in rf circuits gives an ex- cellent and clear discussion of elementary principles, with applications in the last chapter to magnetic resonance instrumentation. We should perhaps conclude by emphasizing that much of our discussion in this chapter is applicable beyond nuclear magnetic resonance, even though the language of NMR was used for convenience. The Bloch equations are not valid generally for nuclear or electron spins in solids (except conduction electrons in metals), but they are correct for liquids, and do serve to provide an introduction to the phenomena involved. It is also useful pedagogically to have them available fbr calculating X' and x", in order to provide concrete examples of these important but slightly abstract functions. We remind the student that the chapter has been macroscopic-we moved as quickly as possible to macroscopic magnetizations and response functions for macroscopic samples. The " model theory " we used, the Bloch equations, was entirely phenomeno- logical, and also dealt only with macroscopic magnetizations and fields. The observed signals are also macroscopic voltages, and the experimental problems of measuring them formed an important part of the chapter. Problems 2-1. Find the correct approximation to Eq. (2-17) in the limit yhHsl2kT4l. What temperature T must be reached for protons (y:2.6 x lOn G-r sec-') in a field of lOa G before the high temperature approximation is wrong by l0f ? Find the same quantity if the magnetic moment is that of the electron (y" : |.74 x lO1 G-' sec- t). 2-2. Obtain Eq. (2-23) directly from Eq. (2-17) in the high temperature limit of problem 2-1. Find Xo for protons in water at room temperature. 2-3. From the Bloch equations and Eq. (2-33), obtain the maximum power absorbed per unit volume from the protons in water at 60 MHz. Use an Hr that makes the saturation parameter S : y'Ht'TrT2: I in Eq. (2-44). Assume T, : Tt: 3 sec. 2-4. Find the approximate maximum voltage at the input of an rf receiver produced by the free induction decay after a 90'pulse applied to Na23Cl. Assume a receiver coil of 5 turns. cross section I cm2. and a resonant frequency of l0 MHz. 2-5. Find the signal at the receiver input from a spin echo in water under the following experimental conditions: coil, l0 turns, area I cm2; pulses, 90 to 180" sequence, r - 2 sec, |, :3 sec.Macroscopic Propcrtics of Nttclcltr Mitgttctr"rrt 2-6. Invent other pulse sequences involvin-g more than two pulses that gtvc rtrc to other echoes. (There are almost limitless possibilities') 2-7. Figure.2-13 shows the proton resonance in HtO in a relativcly hotrt<t- g"i""ui n"ro. The beating phenomenon' known as " wiSgles"' artscs because the exteriäl rnugt"iiJ field is changing rapidlv. enough that thc field is off resonance before the transversJ mägnetization has decaycd' Since the -ugn",,täiion pt*tt"t at a frequency proportiona.l to thc licltl' the signal beats w;ih the constant oscillator frequency' and.the.c()nstantly changingdifferencefrequencyappearsinthedetectedsignalas..wrgg|cs''' The sweep in Fig. 2-13 is tineär at 0'l sec and 5 mG per division' frstirnatc it t orn ih" figu.. and the field inhomogeneity at the sample' 2-8.Thecw(continuouswave)orsteadystateresonancesigna|inapartictrlnr tiquiO ,u*pte "onrirt, of two nearby Iines of equal intensity^ and lransvcrsc relaxation time f, ' Calculate the fre'e induction decay following a (X) pulse if a transient L-p"iit*tt is performed' If the lines arc scparatcd bv angular frequency Ao in the cw experiment' show that an approxirnatcly 90'pulse can Ue applied to both with a singlc prrlsc applrc<l at.a l-rcqucncy midway between-ihe lines if the go pulsc contlitron rs satrslicd, and lhc pulse length satrsfies tfie inequatity 7 '1 15ro) ' (Satisf'yinB thc q) pulsc condition simultanc<.rusly meäns ll, >. \atly, wlrcrc '\r,r/7 rs lhc.scpllroll()lt of the lines in lield units' Such an //' is sai<t to bc sullicrcnt t() "c()vcr" the lines. ) 2-g. Discuss the shape of the echo formed after two pulses of problenr 2-8' References l. G. Pake, Paramagnetic Resonance'W' A' Benjamin' I-nc'' New York (1962) 2. A. Abragam, r,ii,ipiu.t of Nuctiar Magnetism' Oxford Universitv l)rcss London (1961). 3. C. P. Slichter , Principles of Magnetic Resonance' Harper & Row' Ncw Yor I (1e63). 4. E. R. Andrew, Nuclear Magnetic Resonance' Cambridge Universitv Pres' Cambridge, England (1955)' 5. F. N. H. noui.,sänl ;;;it;';' Electrical Circuits' oxford Universitv l'rcr' London (1962). 6. R. P. Feynman, R. B. Leighton, and Matthew Sands' Tfte Feynmun I'ccturt on Phvsics, noOitä"-Witf-"v'iuUfitti"g Co'' Reading' Massachttsctlr vols. 1,2, 3 (1965). 7. F. Reif, statiriclat Phvsics, McGraw-Hill Book Companv' Ncw Yot (1967). 8. G. Pake, "Nuclear Magnetic Resonance"' Solid Srate Physics' l; licrt and D. ru.tunii, Lis',lc'cao"tic Press inc'' New York' v.l' 2 (les(r pp. l-91' 9. H. Kopferma nn, Nuclear Moments'Academic Press Inc'' Ncw York ( | 91ti 10. C. Poole, nrcriÄ--iii" Ä-"onon"':..A Comprehensiue Trt'atist' on I t'1tt't mental fecnniquJs, iJt"tt"i"n"" Publishers' New York (1967)' Line vVidths and Spin-Lattice Relaxation 59 To complete and sharpen the paradox, we write down the magnetic field produced at a distance r by a point magnetic dipole. CHAPTER 3 Line Widths and Spin-Lattice Relaxation in the Presence of Motion of Spins This chapter will be concerned v"11t. providing a microscopic model with which to estimate the parameters of the Bloch equations Q and Tr. To do so, we shall introduce the complementary concepts of random fre- quency modulation and random walk. The main use of these concepts will be to achieve a clear understanding of the motional narrowing of nuclear magnetic resonance lines, but the ideas are also important in the understanding of other experiments in modern physics, which we shall discuss in the latter part of the chapter. 3-I. INTRODUCTION From a strictly economic point of view, magnetic resonanc€ owes a great deal to its usefulness in chemistry. It is useful in chemistry because the nuclear magnetic resonance line widths in liquids are so narrow that resonant frequencies differing by as little as a part in 108 may often be resolved. It has been found that the frequency of nuclei in different chemical surroundings depends on the details of the surroundings. For example, the resonance frequencies of protons in ethyl alcohol, CH3CH2OH, are divided into three groups, corresponding to the protons in CH., in CH2, and in OH. Tl:re chemical shifts caused by the different chemical environments are small, however, compared with the magnetic dipole-dipole interaction between the protons in the molecule, which corresponds to a magnetic field of l0 G produced on one proton by another an angstrom away. Our first guess ought to be that the resonance lines would be about l0 G wide, precluding a resolution of better than one part in l0a. We shall be concerned with the solution to this apparent paradox in this chapter.(3-l) The field Ha has the {ämiliar dipole shape; the interaction energy of one dipole at the origin with another dipole at r has a rather complicated angular dependence, a llrr radial dependence, and it depends as wetl on the relative orientation of the dipoles. Thus the dipolar field varies from site to site, and cannot be exactly the same for each nucleus. The simple estimate of lHTl - l0 G between protons, for example, is calculated by using p/r3, where r is the nearest neighbor distance, and is to be taken as an estimate of the rough magnitude of local fields in hydrogenous solids. Thus we present the paradox as follows. The widths of nuclear reso nance lines in a liquid can be a fraction of a cycle per sccond in the presence o[ local dipolar interactions as Iargc as 50 kHz. It is our purpose in this chaptcr to resolve thc paradox both quantitative- ly and qualitatively, to discuss the Bloch equation relaxation times T, and T, as a function of the resonance field or frequency, and to broaden the discussion to include the phenomena of "exchange narrowing" and "exchange broadening" in nuclear and electron paramagnetic resonance, the Mössbauer effect, and the intensity of Bragg reflections in X-ray crystallography. We also hope to make clear the distinction between "motional narrowing" (a name for the effect we want to discuss) and the "pressure broadening" of Iines in atomic spectra. It must be admitted at the outset that the way of understanding the phenomena that we shall develop is a natural one for magnetic resonance but not so natural lor some of the other phenomena. Nevertheless, it is important to grasp pheno- mena from as many viewpoints as possible, so it is worth the strain placed on our method to do so. The resolution of the paradox involves the recognition that in a liquid, a gas, and even in solids under some circumstances, the resonant spins can move substantial distances relative to their average spacing in T, , and even in a Larmor period in some circumstances. Thus, the local fields with which we are dealing are not static but rather time dependent, most often in a random way, and we must develop ways to understand how the time dependence affects the resonance experiment. To begin with, we shall treat the problem temporally; that is, we shall examine the precession of a typical spin as a [unction of time. Although we shall thus provide ourselves with a useful formula with which we can estimate T, in a widc variety of cases, we shall not have grasped the significance of thc tcrrrtHa:-{+:Prt.r- r' 60 Introduction to Magnetic Resonance " motional narrowing " until we have reexamined the problem in fre- quency space. To do that, we use some of the concepts and language of frequency modulation of a classical oscillator. The language will be classical throughout, and we shall deal in quali- tative estimates. Such an approach does the subject something of an injustice, since the phenomena are susceptible to quite precise and elegant formulation via the density matrix of quantum statistical mechanics. we henceforth banish from these pages any serious reference to the density matrix, and guide more ambitious and sophisticated readers to the text by Slichter []. 3-2. RANDOM WALK CALCULATION OF T2 we imagine ourselves sitting on a proton in water, for example, respond- ing to the various magnetic fields in the sample. The strongest of these is the external field Ho, and we can dispose of it by looking at the world from a reference frame rotating at lHs : ar. The internal fields caused by the magnetic dipoles of other protons are random in orientation and time dependent. The z components of the local dipole fields add to or subtract from l/o and cause a more rapid or less rapid precession than cr;o about the z axis. In the rotating frame, these z components are responsi- ble for the only existing precession about the z axis. Looking back at the discussion surrounding the Bloch equations in Chapter 2, the student should be able to recognize that these fields contribute to a ! process. we shall defer until later the discussion of T, processes, but it is appro-priate here to identify their source. precession of the spin away from the z axis is caused by transverse fields that are static in the rotating frame. Thus we expect local fields having components transverse to the z direction and frequency components at the Larmor frequency to contribute to ?,processes. There is more to it than that, and we shall return to the Tlproblem later. Let us construct a model of the longitudinal (z direction) local fields, and compute T2. Let us presume that each spin in the sample sees a constant local field hr4 Ho in the z direction for a time t", after which it may or may not, with equal probability, reverse itself. In these terms. theproblem can be phrased in terms of the famous " random walk " problem.r In that problem, the mean square distance traveled in the x direction, (x2), after n steps, each of length Ä in either the + -r or -.r direction, is (x2) : nL2 (3-2) I For a derivation using arithmetical induction, see Feynman [2], vol. l, p. G_5.Line Widths and Spin-Lattice Relaxation 6l In our problem, the unit of length is phase of precession about the z axrs in the rotating frame, and the step length is yhrr". After time r, the number of steps n rs tf r"; therefore, from Eq. (3-2), the mean square phase accumulated is (O(r)') : ! gt 6,)2 : y2hr2r"t 11--l) ac It is perfectly within the spirit of the Bloch equations to identily 'Il as the time such that (Ot) : I (rad)2.2 This criterion yields ( l-4) where örr.r : lh t. The particular problem we have "solvetl " seems lr vcry nrtificiul nrrxlcl for the behavior of internal liclds in a liquirl. 'l lrc tletinrtronr ol rlor lrrtl t.can be sharpencd a grcat dcal but at thc cxpcnsc ot lontc rrtllherrrnlrtul complexity. As it stands, Hq. (3-4) providcs un cxtrcmely urcful erlrnrltc of 7, in a liquid. Il'you wish to think ol'motions as bcrn3 morc "flurtl." lcss jerky than the model suggests, then rc may bc rcaurtled rt lhc ltnrc Curing which the local field changes by an amount compllrtrh lo rtr magnitude-a rather vague concept, to be sure, but onc which nrrllrr satisfy one's feeling that molecules are in continuous motittn, Aclurllr', the 'Jump " model has been shown by NMR techniqucs or wcll rr hr neutron diffraction to be a fair description of a liquid. rn whrch I ltvcrl environment around a given molecule pcrsists for r",lirllowed hy r chrnlr to another conliguration, wrth thc tlurltion of the changrnj ltmo hrrrrp much less than r... ll'onc rcgirrtls thc constant t" to be an nvctt;l, tnrt tlrc single local field to hc an avcragc ()vcr local fields causcd by rll Jxrmrl'lc local arrangcnrcnls ol'ncurly nr:rgncll( ll)()ments, then the prcturc mry n,r look so unrealistic. Antl rt rhoultl krok quite good lirr dcx.rlbln; rhc eflect on the nuclcar rcs()nrn(c lurc wrrtth of diffusion in rolrlr Equation (l-4) rs u rclsorrublc cstnnate of T, as long lr /kor, . I ll t. is long, thc stcp length ,lr,r rs rtscll, when divided by y, the wldth of rlrc resonance linc. So rrr thc nrc\cn(.c ol rr changing local envrtonmanl, tlrc Iine width is narrowcr thnn tlrc rtirtrt linc width öat, as lonl rr lhe hx.rl environmenl changcs' nrprrlly ( ()rnl):lr(.(l with l/ö@. Equation (3-4) rs uscf'ul rrr u r+,rtlcr vuriety of circumstnncct lhan tlr. student can prcscnlly Inlnßurc Wc tligress for a paragrlph flonr tlrr 2 The usc of tllc n(!trtron I j rnrtcrrl of /, u,ill be explained later, whrn 11 frtrrlrxrare discusscd. I lrc rtrrtrrrr tt,ut lrt$.rn t , ,rnJr I ] is made herc to indrerta lhlt ra nrr not havc inclutlctl lll p,rrlhlr r,,urr r1 ,,1 /, rrr tlris discussion.fr: {u.)'"" 62 Introduction to Magnetic Resonance main line of thought to show the application of measurements to the measurement of the diffusion constant. Let t" be the mean time between jumps or changes in the local environment. Imagine that a spin jumps spatially an interatomic distance 40 each time. It proceeds, then, by random walk (in three dimensions) and travels in time t a mean square distance The process described is called diffusion. lt is described in continuum theory by a differential equation in the concentration c of the diffusing constituent:.=t lt=-ao DY2c -a;:, ,=n(3-s) (3-6) (3-7 )The constant D is the diffusion coefficient. If Eq. (3-6) is solved for simple initial conditions and one-dimensional geometries, the root mean square distance the concentration spreads in time / from an initial con- centration is approximately (2Dt)| t2. Thus in Eq. (3-5) we identify Since ao is related to the density, we have in Eq. (3-7) an expression for t" in terms of macroscopic quantities that can be determined by quite different, nonresonance experiments. Alternatively, we see from Eq. (3-4) that T, measurements can provide a value for the diffusion constant D, a number frequently hard to get if one is concerned with the self-diffusion c,f a molecule surrounded by identical molecules-H2O in HrO, for example. To summari ze Eq. (3-4) and some of the subsequent discussion, we plot logT, versus t. in Fig.3-1. The abscissa might easily be llD or 4lT, where 4 is the viscosity and Tthe absolute temperature. For justification of the last clause, see the reprint of Bloembergen's thesis [3], the original work in the field. Note in Fig. 3-l the leveling off of 7r to öco-r at a value of t" on the order of öco-r, something we have justified only by a plausibility argument so far.Line Widths and Spin-Lattice Relaxation 63 I l6co I /c.:o I /6c,: Fig. 3.'1 LogT, versus logz.. Plor of Lq. (l-4) in rcgion of its validity, öcoz. ( l 3-3. VERY SHORT CORRELATION TIMES: oo?" ( I Suppose aor"4l, and suppose the medium is spatially isotropic in the sense that the fluctuating intepal fields point in no preferred direction. Then the magnitudes of the fields parallel to and transverse to l/o are the same, and their frequency properties are also the same. As a result, the random walk in the transverse plane in the rotating coordinate system, produced as described in the last section, also occurs av'ayfrom the z axis. This longitudinal relaxation, a T, process, is produced by transverse local fields s/alion ary in the rotating frame, that is, at coo . The two independent and orthogonal random walks occur at exoctly the same rate if the magni- tude ofthe fluctuating fields at or near zero frequency is the same as at the Larmor frequency, rou. The simplest and most plausible description of the random internal field has that property if aror. ( L Since there are two orthogonal transverse components of the local field in the rotating frame, and since they act independently, the mean square angular migration of a spin av'ty from the z direction is (dr, (r)') :2y'h12 t,t lust twice as great as in Eq. (3-3). Again settinE ör, (Tr)2 :1,(3-8) we obtarn (t-9)l,: zrda"": + 64 Introduction to Magnetic Resonance At this point, we see the reason for the introduction of the notation T) before Eq. (3-a). If we wish to calculate the parameter Tr , which characterizes the line width of a Lorentz line, we must include not only the inhomogeneity of the quasistatic local fields (so-called secular broaden- ing), but also lifetime, or nonsecular broadening. Reexamination of the Bloch equation for M, and a little thought produces the following con- clusion. A I process is one that causes precession of M, away from the x axis of the rotating frame. Quasistatic fields in the z direction do this, as do fields in the y direction at roo. Fields in the x direction produce no effect because they exert no torque on Mr. Hence, we should have (3- l 0) where the first term is from Eq. (3-a) and the second is half of Eq. (3-9). Comparing Eqs. (3-4), (3-9), and (3-10), we see that Tr: Tz (3-l l) The equality holds in the limit aor"4l and depends, we reiterate, on spatial isotropy of the local fields. We have also filled in more of Fig.3-l if we regard it as a plot of Tj'or T, t versus r., rather than (T.j)-r versus r"; namely, the 7, and T, curves for r. ( l/c.ro coincide. To fill in the 7, curve for longer r" we require more general analytical tools, the develop- ment of which we shall indicate in the next section. 3-4. RANDOM FREQUENCY MODULATTON; SPECTRAL DENSITY In this section we shall make heavy use of the analogy between a magnetic moment precessing at the frequenc! v: (yl2n)lHo + äL(r)l and a classical, frequency modulated oscillator transmitting, for example, classical music at a center frequency of 97.8 MHz. We shall begin by considering a proper mathematical description of frequency modulation. The simplest situation to treat is r- sinusoidal frequency deviation; that is, the angular frequency of the t.cillator is given by @(t) : @o + L@ cos qt (3-12) where Ac-r is the frequency deviation and q is the frequency of modulation. We now need an expression for A(t), the amplitude of the fm oscillator as a function of time. One approach might be to writelll _-_!_ T2 Ti ' 2Tl A(t) : ,4ofcos rr-r(t)l]Line Widths and Spin-Lattice Relaxation 65 We shall not number that equation, since it is wrong, just as it is wrong to write that the distance a car travels in t to be d : n if t' is not a constirnt. The argument of a trigonometric function is a phuse, and the phasc accumulated between /':0 and I with frequency given by Lq (1-12) is et L.ot Ölrl: I r'r(t') tlt' : utol 1 - sin r;f Jo .l Now we can safcly writc, complete with equation nu'trbcr, f A,ol IA(t): .4o cos loo I + - sin Al I(r-|r1 (3-r7)(:1-14) The ratio Lolqis an extremely important pilralnctcr lirr lirturc tltscttsstotts t\t,t( l_ t5) q where nr is clllctl lltc nnxlulutiorr indr.t. lls strc rclultvc l(t ttrttly wrll bc crucial in many applit ttltolts. We wish to know thc f'rcquency spcctrtllrl ol l'{, (l-14) l trrt, rt is more convcnicnt to rcwritc I:q (,1-14) usrlrg the lrlSonotttctttt ttlcttttty cos(a * ä) : cos a cos ä - sin (J sin /r: A(t) :,4,,[cos (r)r) t c()\(rtt \ttl {t} \ltl .rrr, rllt(rt rrrr r7l}l (]-16) From immecliatc inspcttrr)n wc (itrl rcc lltnl lltc ntnnr f f ctltrctttl corttponents are at ots, antl lhc olltcr sJrtltrtl tottlcltl tt hlltrcrl lry the terms cos(z sin q/) antl sirt(nl srrr rlt). I hcy ntc ;xttrxltt t.rrtlr l)cr rctl 2nfq; that is, the argumcnt ol tllcsc lcrttts fcll1.rllr rtrlf wrllr tlt'rt Pcttotl' That fact makes them prirrrc tittttlttlittcr lot ct1rlltrloll ltl | (rrll rct scrics' Consider the cosine lunctiort We can use only ctlsittc lcrltlr ttt llrc ctplnll'rl \ltr(c the cosine is an even function and the cocltictctttr rtl tlttc lettttr tl'ttl,l rrt'.cssarily vanish' Thc coefficients a^(m) irrc lirtrrrrl by ntullrplyrnl lr,,tlt sttlcs of Eq. (3-17) by cosn'qt and averaging ()ver n Srrt,xl I ll t,rtt do so, you find thc following integral exprcs\r()n :cos(rn rttt ,rrl - L rt.(ttt)tor trrlt '_ll a^.(trr\ l .1,,.,t. n .r, (,r"( rrr rrrt ql) Introduction to Magnetic Resonance This expression may be developed in a power series in rn sin qt, which becomes, upon integration, a power series in ,|1. Fortunately, the labor has already been done; consultation of a complete set of mathematical tables [4] under "Bessel Functions" shows the coemcients of Eq. (3-17) to be Bessel functions of integral order: cos(m sin qt): Jo(m)+ 2 i Jrr(m)cos2kqt (3-l8a) and sin(nr sin qt) : 2i t ro*,(m) sin[(2k + l)qt) (3-l8b) The functio ns J^(m)"* ,"rr., *nctions of integral ortler o[ the first kind. Although they are surely less familiar than trigonometric functions, the student should not be put off by them. A graph of the first three is shown in Fig. 3-2' Note "Io(0): l, and /'(0):0, n:0' Also note that ,/o(n) deviates as ra2 from its value at m : 0 for small m, and the others begin from zero as mn. A qualitative look at Fig. 3-2 and these remarks are all we require o[ the Bessel functions. The expressions (3-18) must be put back into (3-16): A(t) : ./6(trt) cos u)o t t"/1n1 cos(rr;s + nq)t(sgn n) (3-19) \- L Fig. 3-2 Bessel functions of the first kind, "/,(rn), versus m, for n :O, l, 2.of the powerLine Widths and Spin-Lattice Relaxation 6J The notation (sgn n) means to multiply by (+ l) when r > 0 and by (- l) when n < 0. Again, Eq. (3-l9) is most valuable to use in graphical form, as in Fig. 3-3, which gives the frequency spectrum of the power, pro- portional to A(t)2. The graph has been presented with A(I) : constant, t..uu.. in our applications, the local field, which causes Ao, is a fixed characteristic of thc substance, whereas q m^y often be changed within a given sample, by changing the temperature, for example' (We shall pt.r..tt one situation, howcver, in which it is 4 that is fixed by the nature of the substance ancl A(, that is changed by the experimenter.) The thing to notice about Fig. 3-3 is that the power spectrum is mostly con- tained within Arrr of @6 , and when l?? < l, which means q > L@, the power is mainly in the center frequency <.cro. The sidebands are still spaced by the modulation frequency 4 but are small in amplitude. If you try to change the frequency back and forth too rapidly, the major effect is not to change it at all. Although thcrc is only a qualitative rcscmblancc bctwecn the problem of sinusoidal f'rcqucncy nloclullttion llnd ottr problcrn of nrotional nilrrow- ing, the results ol thc prcceding analysis arc itlrc:ttly srrggcstivc. If we make an arbitrlrry cotrnccti()n bctwcctr thc spcctrttttr ol'I ig. 3-3 lrnd the line width in a nuclear resonance, we can sere the origins ot thc brolrtlening that takes place as r" increases and becomes on the order tlf l/öto in Fig. 3-1. For all values of r" such that äorr" < l, the " modulation index" is less than I ; that is, öc,.rt" is roughly the same as the modulation index. There is, of course, a vast difference between the case of sinusoidal modulation and the random fluctuations of the local field in a liquid. To q q m=5.0ll rr i ,l-i,11m=3.O t- Fig, 3.3 Fourier components ,n 0.5. 1.0. 3.0. 5.0.Acl z 6(,) spectrum of Eq. (3-19) for 68 lntroduction to Masnetic Resonance sharpen the distinction, we make the following analysis. Let hr(t) be the local field seen by a nucleus at some arbitrary time f. Form the product hLQ)hLQ * z) and average over all time for a particular nucleus. That process defines /(r): -f(rl:<hLQ)hLU+r\> (3-20) where the brackets ( ) indicate average over time t and f(r) is the auto- correlation function of hr. It is independent of t since the sample is assumed to be homogeneous and in thermal equilibrium; there is nothing special about any particular time. Consider what we expect of /(r) as r becomes large. If the sample is large, if the local field is produced by many nuclei undergoing random motions. then we ought reasonably to expect no rclation between lrr(t) and h,,(t + r1. Since /rr. can be positive or negative, and since, in fact, the temporal randomness of ft. means <hL(tl> : 0. we expect lim/(r) :0 (3-21 ) to be reasonable behavior for f(r) at large t. Contrast that behavior with the autocorrelation lunction of Eq. (3-12): (co(t)@(t + r)): ((<,.r0 * Acocos qt)latu * Aocos q(l + t)l) cos 4r: (t)u' * L@z --------:- L (3-22) The leading term, c,.re2, is as expectetl, but the second term certainly does not vanish. Therefore, it is clear that a simple modulation such as Eq' (3- l2) fails to satisfy one's intuitive requirements for random modulation. The simplest form of f(r) that satisfies all the requirements is an exponential: "f G) : (h,(t)') exp - lrl/r. (3-23) Equation (3-23) reintroduces the correlation time r.. The average over time of the square of the local fielcl, (i.(l)2), is the same as the average of h,,2 over all the nuclei in the sampre by a fundamental hypothesis of: roo2 * o-'(*J'"'0"o, qt cos q(t+ tl ar)Line Widths and Spin-Lattice Relaxation 69 statistical mechanics [51. We see that we are able to specify the local field, and hence the ranclom ticquency modulation ol' each " nuclear oscillator" by that local field, less precisely than we did in the simple case of sinusoidal modulation, but we do preserve some points of similarity' These include the correspondcnce between (yhr\2 and Ato2, and the simi- larity between the paramcter 4 of Eq. (3-22) and l/t. of Eq. (3-23). At this point, our resort to plausibility arguments must come to an end, because the subscquent developnrcnt relies on theory that uses the density matrix of quanturn statistical nrechanics and time dependent perturbation theory. Thc l-ull theory is nothing less than a derivation of the Bloch equations from the basic principles of quantum statistical mechanics. Our previous paragraphs have attempted to establish a climate of acceptance in the student's mind for the results. The central formula of the theory is the Fourier transform of the correlation function.fG), which isj(a.r)' the spectral density function : Equation (3-24) is one of a pair of integrals known as the Wiener- Khintchine relations (see reference [5], p. 586). In the case of the par- ticular form of /(t) given by Eq. (3-24),j(r,r) : l[- ,<t r1rltrL(t + 'c)>e-i-' dr i,pktl: I l* *t',"{tlnr(t + t)e-i-' dt(3-24) (3-2s) (3-26)The spectral density function determines the effectiveness of the local field in producing transitions between quantum mechanical spin states. The relaxation rates l /7, and I lT, are determined byi(ro)' To write down the final results, we must specify 7(a;) more completely' Since the local field has the usual -r, -r', and z components, a more general j(o) may be written where a, ß : x, y, t. If the local field is truly random, then (ft.(l)är(t + t)) :0, for af B, and the only components of inu are i,,(a), rr(<o), and j,,(a). In terms of the spectral density function, the complete equation for T, I that replaces Eqs. (3-4) and (3-10) is Ti, : y2u,,(0) + jrr(as\l (3-27) 70 Introduction to Magnetic Resonance The first term is the same as Eq. (3-4), which we obtained by the random walk argument, and the second is the " lifetime broadening " effect we discusseä prior to Eq. (3-10). ln terms of /r and t" , from Eq. (3-25) we get T r' : f(rn: Sr" + (h,2) T .*V*) The assumption of spatial isotropy, made to obtain Eq' (3-10)' is that (h,'): (hrt): (h"2) =h"2. The result for T, r is Tr' :72[j,,(@o) + jrr(rrs)] :p^r, WecannowcompleteFig.3-lbyinc|udingtheportionoft|reT'curve for r,> llla,r. The full curve is shown in Fig' 3-4' At r.: llao' T1 goes ihrough a minimum, then increases, whereas 7i continues to decrease. Ätthough these features are an obvious analytical consequence of log I1 ,log 12(3-28) (3-2e) I l6<,: I I <^:o I l6u Fig.3-4 Log f', log T. versus logz. for Eqs. (3-28)' (3-29)Line Widths and Spin-Lattice Relaxation 7l Eqs. (3-28) and (3-29), it is instrr.rctivr- to sce how they can be remembered easily by examining one further property of j(rr-r), Eq. (3-25): I ,'' ' Ttlt't ,'4 dx ft ,rf i(ut\dttt:l .--, ,:l :- ,:. (3-30) hL':o .'u I f u)'t.' Ju I + xt 2 The area under.T(r,-,) \'L'rsu\ (, is a crrnstant, independcnt of r.. Several j(or) curves with this propcrty arc plottcd rn Fig.3-5. When r. is short [curve (l) oi Fig. 3-5], y(t,.ru) :/(0), and fr : I:. 7(too) is largest for curve (2)-hencc thc maximum in the relaxation rate there, or the ?l minimum of Fig. 3-4. For long r,, j(log\ decreases again as the Larmor frequency falls far out in the tail of j(o), accounting for the rise in [ for longer r.. Nothing in Eq. (3-28) explains the constancy ol T, when r" is longer than llyhr. At this point, the theory breaks down, but the reason for it and the rcsult can bc sccn in frig. 3-3. Thc spcctrurn firr rrr > I covers only the range A(rr ils thc ntoclulation inrlcx incrcuses. Thc local field is essentially static. IJntlcr lhcsc conrlitions, thc 1]loeh cqultiorrs are not valid, the line shapc ol'thc rcsornncc, 7"(r,r). is rrot l.ore rrtziun. Wc shall leave this case to thc ncxt chaptcr. log l(c,;) Fig. 3-5 (l) f(-) versus o, z. lcss tltatt <,,,, ( J ) ', t't1tt.tl l" "' greater lhan oq . 72 Introduction to Magnetic Resonance 3-5. SOME APPLICATIONS eualitatively, the effect of rapid nuclear motion is to average out internal fielis, or so the previous parts of this chapter would lead us to believe. In fact. we must be very careful to make a distinction between internal fields that can be averaged out and those that cannot. Equation (3-l) gives the expression for the field at a vector distance r from a point dipole Ha: -5* rä' (3-1) consider, for example, the z component of the field produced at the center of a unit sphere by a dipole on the surface. If the dipole points in the z direction, then H": -p(l -3cos20)r-r where g is the polar angle in the conventional spherical coordinate labeling. If the dipole is allowed to roam over the surface of the sphere, the average field at the center is since (cos2 0> : + when averaged over the solid angle. The demonstra- tion of the equivalent result for other components of Ho and arbitrary orientation of p is tedious, and, in fact, unnecessary, but the result still holds. Another type of local field that can be reduced by rapid motion is the inhomogeneity of the magnet. In this case, the inhomogeniety of 11, is all that counts. Suppose, as is likely to be the case in the typical electro- magnet, the inhomogeneity has cylindrical symmetry, and suppose the maximum deviation from the average i1o is AI1. If a given nucleus can be forced to sample the range of fields over the sample volume rapidly enough, the total effect of the local field will be reduced according to Eq. (3-a). The nuclei may be caused to sample the magnet's range of fields by spinning the sample about an axis perpendicular to the field direction. A typical geometry is shown in Fig. 3-6. To achiev. narrow- ing, cer must be so large that the modulation index, m : (y AI{a-r), is less than one. If each spin samples essentially the same magnetic fields during each revolution, the frequency modulation produced by the spin- ning is periodic, although probably not sinusoidal. Our original analysis of frequency modulation is useful, though, and we see that the criterion nr < I is sufficient to produce spinning sidebands which are farther away! l'" oo f" (r - 3 cos2 e) sin o do: o +74 J0=O JA=OFig. !6 Spherical sample in an inhomogeneous field. sample is spun atangular frequency co about an axis perpendicular to the page. than y aH. For a typical field inhomogeneity of l0-a G, such as isfound in electromagnets made for chemistry applications, a spinning frequency (.,l2n) of a few cycles per second is sufficient. If turtulence within the sample causes a given spin to sample the available fields in amore random fashion, we require the width produced by the field inhomo-geneity after narrowing to be comparable to, or less than, the naturalwidth. The requirement from Eq. (3-9) may be expressed | _r (yLH)' -1- T2 fi o) For typical hydrogenous liquids, where ?i -5 sec, and with a magnctinhomogeneity of lO-a G, ro again must be a few cycles per second. Thatspinning rate is easily achieved, and commercial apparatus is r.utrncrysupplied with sample spinning attachments.Line Widths and Spin-Lattice Relaxation IJ (3-3 I )