zamar relaxation NMR paper

/3/1/3 /3/1/4 Brazilian Journal of Ph ysics/, v ol/. /2/8/, no/. /4/, Decem b er/, /1/9/9/8Molecular Motions in Thermotropic Liquid CrystalsStudied b y NMR Spin/-Lattice RelaxationR/.C/. Zamar/, C/.E/. Gonz/#13 alez and O/. MensioF acultad de Matem/#13 atic a/, Astr onom /#13 /#10a y F /#13 /#10sic a/,Universidad Nacional de C/#13 or dob aCiudad Universitaria /, /5/0/0/0 C/#13 or dob a/, A r gentinaReceiv ed /2/9 Octob er/, /1/9/9/8Nuclear magnetic resonance relaxation exp erimen ts with /#0Celd cycling tec hniques pro v ed tob eav aluable to ol for studying molecular motions in liquid crystals/, allo wing a v ery broadLarmor frequency v ariation/, su/#0Ecien t to separate the co op erativ e motions from the liquidlik emolecular di/#0Busion/. In new exp erimen ts com bining NMR /#0Celd cycling with the Jeener/-Bro ek aert order/-transfer pulse sequence/, it is p ossible to measure the dip olar order relaxationtime /#28 T/1 D /#29/, in addition to the con v en tional Zeeman relaxation time /#28 T/1 Z /#29 in a frequencyrange of sev eral decades/. When applying this tec hnique to nematic thermotropic liquidcrystals/, T/1 D sho w ed to dep end almost exclusiv ely on the order /#0Ductuation of the directormec hanism in the whole frequency range/. This unique c haracteristic of T/1 D mak es dip olarorder relaxation exp erimen ts sp ecially useful for studying the frequency and temp eraturedep endence of the sp ectral prop erties of the collectiv e motions/.I In tro ductionLiquid crystal molecules when ordered in a mesophaseundergo di/#0Beren t kinds of motions/. Some of themare individual molecular motions/, lik e anisotropic ro/-tational and translational di/#0Busion/. These are liquid/- lik e rapid random jumps whic h can b e c haracterizedwith a correlation time /#1Cc /, generally falling in the range/1/0 /, /8/, /1/0 /, /1/1s /. There are also slo w er reorien tationalmotions in v olving a large n um b er of molecules/, thatare c haracteristic of the liquid crystalline states/. Thesecollectiv e molecular /#0Ductuations/, called or der /#0Ductua/-tions of the dir e ctor /#28OFD/#29/, are describ ed through asup erp osition of a broad sp ectrum of individual purely dissipativ e mo des/, with relaxation times ranging from/1/0 /, /4to /1/0 /, /9s /. As a consequence of the div ersit yo fc haracteristic times of the /#0Ductuations in liquid crys/-tals/, the di/#0Beren t pro cesses are clearly visible in di/#0Ber/-en t time scales/. F or example/, in Zeeman order relax/-ation NMR exp erimen ts/, the lo cal motions dominate inhigh frequencies /#28/1/0 /6/, /1/0 /8Hz /#29 while the order /#0Ductua/-tions represen t the main mec hanism of spin relaxationin the lo w and medium ranges /#28/1/0 /3/, /1/0 /6Hz /#29/. The NMR relaxation metho d has b een used forman yy ears for studying the complex anisotropic re/-orien tation of molecules in the liquid/-crystalline state/.The measured quan tities are spin/-lattice relaxationrates whic h dep end on the in tra/- and in ter/-molecularin teractions of the n uclear spin magnetization with lo/-cal magnetic and electric /#0Celds/; these /#0Celds b eing mo d/- ulated b y the underlying molecular motions/./#5B/1 /#5DI t i sexp erimen tally observ ed that in con v en tional liquidsthe relaxation times are almost indep enden t of theLarmor frequency /. On the con trary /, in liquid crystalmesophases/, the presence of co op erativ e motions causesthe relaxation times to ha v e a mark ed frequency dep en/-dence/, sho wing usually a disp ersion of ab out t w o ordersof magnitude in the frequency range /1/0 /4to /1/0 /8Hz /.NMR studies of molecular motions in liquid crystalsare presen tly usually p erformed b y essen tially t w o dif/-feren t spin/-relaxation exp erimen ts whic h can pro vide asatisfactory n um b er of data to disen tangle the under/-lying sp ectral densities of the sup erimp osed molecular reorien tation/: On the one hand b y com bining high/-/#0Celd/, Zeeman relaxation /#28 T/1 Z /#29 with high/-/#0Celd dip olarand quadrup olar order relaxation /#28 T/1 D and T/1 Q resp ec/- R/.C/. Zamar et al/. /3/1/5tiv ely/#29 measuremen ts at constan t Larmor frequency /#17L /,in /1Ho r /2H as a function of temp erature/./#5B/2/#5D On theother hand b y frequency dep enden t T/1 Z studies o v er abroad Larmor frequency /, /#17/; range/, often applying fast/-/#0Celd cycling /#28F C/#29 tec hniques/./#5B/3/#5D No w/, with adequateinstrumen ts suc h frequency dep enden t measuremen tsare not necessarily restricted to T/1 Z /, but in principleare also applicable to T/1 D /; this ho w ev er has only b eentried rarely and not systematically /, mainly b ecause ofexp erimen tal limitations/.Due to the complexit y of the molecular motions/, andthe fact that the di/#0Busion motions mak e their maincon tribution in high frequencies/, it is di/#0Ecult to dis/-tinguish b et w een the co op erativ e and the lo cal mo/-tions from standard high /#0Celd NMR relaxation mea/- suremen ts/. Ho w ev er/, b y com bining /#0Celd cycling datasupplemen ted with con v en tional measuremen ts /#28for thehighest frequencies/#29/, it is p ossible to co v er a broad fre/-quency range /#28/1/0 /3/, /1/0 /8Hz /#29/, allo wing to distinguish themec hanisms unam biguously /.The basic theory for explaining relaxation exp er/-imen ts relates the relaxation times with the sp ectraldensities of the lattice/; that is/, with the nonspin v ari/-ables/. The second step demands a mo del for thesp ectral densities that relates them with the particu/- lar system /#28viscosit y /, elastic constan ts/, etc/#29 and withthe externally con trolled v ariables /#28temp erature/, fre/-quency /, orien tation/#29/. The theory of spin/-lattice re/-laxation for liquid crystals is based on the w eak/-orderassumption/,/#5B/1 /, /4/, /5/#5D that w as traditionally applied inliquids and solids/. The starting p oin t is the sto c has/-tic Liouville equation for the densit y op erator of thespin system/, whic hi s i n tegrated up to second orderin the p erturbation /#28the in teraction with the lattice/#29/.The spin system is considered as an ensem ble of iso/-lated spins/, the neigh b oring spins b eing considered asfar as they form part of the lattice/.In the calculation of the sp ectral densities of thelattice/, the molecular motions are generally mo deledb y sto c hastic Mark o vian pro cesses/. This is suitablefor liquid crystals/, b ecause of the lo w degree of p o/-sitional order of the molecular motions/. The mo delsfor rotational and traslational di/#0Busion are basically the same as those used in liquids/, with some re/#0Cne/- men ts to re/#0Dect the c haracteristic anisotrop y of the liq/-uid crystals/./#5B/6/, /7 /, /8/#5D The mo dels for the OFD are based on the h ydro dynamic theory /./#5B/9 /#5D It is assumed that themolecules follo w the director dynamics/; so doing the ef/-fects of the in termolecular short range correlation areneglected/./#5B/1/0/, /1/1/#5D This approac h yields sp ectral densi/-ties prop ortional to /#17 /, /1 /= /2and /#17 /, /1for nematics /#5B/1/2 /,/1 /3 /#5Dand smectics /#5B/1/4 /,/1 /5 /#5D resp ectiv ely /. In the case of nemat/-ics/, this la w has b een con/#0Crmed b y /#0Celd cycling exp er/-imen ts of Zeeman relaxation of protons and deuteronsin n umerous nematogens/./#5B/3 /,/1 /6 /#5DI tw as observ ed thatthe OFD are dominan t in the frequency range /1/0 /3/-/5 /#02 /1/0 /5Hz/. There are also clear NMR F C examplesof b oth smectic thermotropic phases and smectic/-t yp ely otropics displa ying the men tioned linear frequencydep endence/./#5B/1/8 /, /1/9/, /2/0/, /2/1/#5DOne of the main h yp otheses of the traditional the/-ory of relaxation concerning the correlation time of the/#0Ductuations is that of the fast motion/: /#17SI /#1Cc /#3C /1 where/#17SI is the c haracteristic frequency of the spin in terac/-tion /#28dip olar or quadrup olar/#29 and /#1CC is the correlationtime of the /#0Ductuation/. A consequence of this assump/-tion is that the spin in teractions can b e neglected fromthe spin dynamics in the microscopic time scale/. /#5B/5 /#5DThough the former condition is not ful/#0Clled b y the slo w/-est comp onen ts of the OFD/, this fact do es not consti/-tute a di/#0Ecult y in high /#0Celd Zeeman order relaxation/.In these cases the picture of a single spin in teractingwith a random bath is adequate/, hence the details ofthe spin in teractions during the lifetime of the /#0Ductu/-ations can b e neglected/. The analysis of exp erimen/-tal T/1 Z within this theoretical framew ork sho w ed to b econsisten t/. /#5B/2 /,/3 /, /2 /2 /#5DHo w ev er/, this w ould not b e the case of dip olar or/-der relaxation/, where the relaxing magnitude is thedip ole/-dip ole energy /. In fact/, recen t results of T/1 Din thermotropic nematic liquid crystals/, as a functionof the Larmor frequency in /#0Celd cycling exp erimen tsevidenced clear discrepancies with the standard w eak/-order theory /#5B/2/8/#5D/. The OFD are the leading relaxationmec hanisms ev en at high frequencies/, ha ving a largerw eigh t than the predicted b y the standard approac h/.This fact indicates the presence of additional relaxationmec hanisms/, driv en b y the OFD/, whic h are not tak enin to accoun tb y the traditional approac h/. Due to itshigher sensitivit y to the co op erativ e motions/, T/1 D isap o w erful to ol for in v estigating sp ectral prop erties of /3/1/6 Brazilian Journal of Ph ysics/, v ol/. /2/8/, no/. /4/, Decem b er/, /1/9/9/8h ydro dynamic /#0Ductuations in mesophases/.The p ossibilit y of measuring sev eral indep enden t re/-laxation times/, re/#0Decting di/#0Beren t features of the molec/-ular motions/, together with the existence of p o w erfulexp erimen tal tec hniques/, mak e NMR relaxation a v eryimp ortan t to ol for c haracterizing molecular motions inmesophases/.In Section /2 w e presen t a short review of the the/-oretical bac kground necessary to in terpret the exp eri/-men tal results/. Some examples illustrating the p ossibil/-ities of /#0Celd cycling NMR exp erimen ts/, in protons anddeuterons/, are sho wn in Section /3/. This selection do esnot attempt to b e a review of this v ast /#0Celd/. Finallyin Section /4 some new/, con tro v ersial results on dip olarorder relaxation are presen ted/, and discussed in termsof the existen t theory /, together with an analysis of ap ossible w a y to understand the disagreemen t/.II Theoretical bac kgroundNMR relaxation exp erimen ts deal with the n uclear spindegrees of freedom and pro vide information on the hostlattice through the time ev olution of the spin system/.It is usual to think on the spin system as a thermo dy/- namic system in con tact with a reserv oir /#28the lattice/#29ha ving a h uge n um b er of degrees of freedom/. In a t yp/-ical exp erimen t/, energy in the form of radio frequencypulses/, is deliv ered to the spins in order to prepare it /#28in the quan tum mec hanical sense/#29/. The system is thenallo w ed to dev elop in con tact with the lattice duringthe ev olution p erio d/. The last step of the exp erimen ti sjust to pic k up the result of the ev olution/. The reco v/-ery of the equilibrium state o ccurs with a c haracteristictime called spin/-relaxation time/, conditioned b y the ef/-/#0Cciency of the spin/-lattice relaxation mec hanisms/.The Hamiltonia n describing the spin system and thelattice isH /= HS /+ HL /+ HSL /; /#28/1/#29HS and HL are the spin and the lattice Hamiltoni/-ans resp ectiv ely /, and HSL represen ts the spin/-latticein teraction/./#5B/1 /#5D The spin Hamiltonian con tains the en/-ergy of in teraction with the external magnetic /#0Celd/#28Zeeman energy/#29 and the a v erage energy of the spinin teractions/. F or the usually a v ailable magnetic /#0Celds/,the principal spin in teractions are the dip ole/-dip ole andquadrup olar energies/. F or example/, in the case of thedip olar in teraction b et w een lik e spins the spin Hamil/-tonian is/:HS /= HZ /+HSI withHZ /= /, /#0D /~ H/0 Pj Ijz /; andHSI /= /1/2 Plk Pq A qlk F qlk /; /#28/2/#29/#0D is the gyromagnetic ratio and the upp er bar meansa v erage o v er the lattice ensem ble/. A qlk and F qlk are spinand lattice op erators asso ciated to the dip olar spin in/-teraction/:cA /0lk /= /, /#0D /~ f/, I zl I zk /+ /1/4 /#28 I /+l I /,k /+ I /+k I /,l /#29 g F /0lk /= /1r /3lk /#28/1 /, /3 cos /2/#12lk /#29A /#06 /1lk /= /, /3/2 /#0D /~ /#28 I zl I /#06k /+ I /#06l I zk /#29 F /#06 /1lk /= /1r /3lk sin /#12lk cos /#12lk e /#06 i/#1ElkA /#06 /2lk /= /, /3/4 /#0D /~ I /#06l I /#06k /; F /#06 /2lk /= /1r /3lk sin /2/#12lk e /#06 /2 i/#1Elk/: /#28/3/#29d/#12 and /#1E are the spherical angular co ordinates of the in/-tern uclear v ector rlk resp ect to a frame whose z /-axis isparallel to the magnetic /#0Celd H/0 /. In the quadrup olarcase the in teraction is that of the n uclear quadrup olarmomen t with the electric /#0Celd gradien t of the electronicc harge distribution/. The angular dep endence refers tothe orien tation of the principal axis system of the elec/-tric /#0Celd gradien t tensor with resp ect to the lab oratoryframe/./#5B/2 /#5D The spin/-lattice in teraction Hamiltonian re/#0Dects the/#0Ductuations of the spin in teractions caused b y the lat/-tice motions/. F or example/, for dip olar in teraction/:HSL /=/#01 HD /= /1/2 Plk Pq /=/0 /#06 /1 /#06 /2 A qlk /#01 F qlk /;/#28/4/#29where/: /#01 F qlk /#28 t /#29 /#11 F qlk /#28 t /#29 /, F qlk /:The densit y op erator of the whole system /#28spins and R/.C/. Zamar et al/. /3/1/7lattice/#29 satis/#0Ces the Liouville equationd/#1Adt /= /, i/~ /#5B H /;/#1A /#5D /: /#28/5/#29In NMR exp erimen ts only the spin system is detected/,therefore/, the relev an t statistical op erator for calcu/-lating the time ev olution of the observ ables asso ci/-ated to the spin system is the r e duc e d densit y op erator/#1B /#11 TrL f /#1A g /, where TrL stands for the trace o v er thelattice v ariables/. The time ev olution of this op erator isdescrib ed b y /#5B/2/3 /#5Dd/#1Bdt /= /, i/~ TrL /#5B H /;/#1A /#5D /: /#28/6/#29A t this p oin tt w o basic assumptions are made/: i /#29The coupling b et w een the spin system and the lattice is w eak enough for allo wing to factorize the total den/-sit y op erator in the second mem b er of Eq/./#28/7/#29 as a /#0Crstappro ximation/.ii /#29 The lattice is considered as a quan tum dissipa/-tiv e system ha ving so man y degrees of freedom thatdissipates the energy transferred b y the spins in a shorttime of the order of the lattice correlation time/, whic his m uc h shorter than the in terv al where the observ edmagnitude v aries appreciably /. This means that the lat/-tice can b e considered alw a ys at the equilibrium state/.Under these conditions/, the master equation in the in/-teraction picture adopts the t ypical form of Mark o vianpro cesses/: d/#1B /#03/#28 t /#29dt dep ends only on /#1B /#03/#28 t /#29/, with time in/-dep enden t co e/#0Ecien ts/:/#5B/1/, /2/3 /#5Dcd/#1B /#03/#28 t /#29dt /= /, /~ /, /2 Z/1/0 dt /0TrL /#5B H /#03SL /#28 t /#29 /; /#5B H /#03SL /#28 t /0/#29 /;/#1B /#03/#28 t /#29 /#1AL /#28/0/#29/#5D/#5D /; /#28/7/#29where /#1AL /#28/0/#29 is the equilibrium canonical densit y op erator of the lattice and /`/*/' indicates that op erators are expressedin the in teraction picture/.Eq/./#28/7/#29 can b e rewritten in a more con v enien t form for studying longitudinal relaxation/. F ollo wing Abragam /#5B/1 /#5Dd/#1B /#03/#28 t /#29dt /= /, /~ /, /2/2 TrL nR/1/,/1 dt /0/#5B H /#03SL /#28 t /#29 /; /#5B H /#03SL /#28 t /0/#29 /;/#1B /#03/#28 t /#29/#5D/#5D /#1AL /#28/0/#29 /,/, R/1/,/1 dt /0/#5B H /#03SL /#28 t /0/#29 /;/#1AL /#28/0/#29/#5D /#1B /#03/#28 t /#29 H /#03SL /#28 t /#29/+/+ H /#03SL /#28 t /#29 /#1B /#03/#28 t /#29 R/1/,/1 dt /0/#5B H /#03SL /#28 t /0/#29 /;/#1AL /#28/0/#29/#5D o/: /#28/8/#29dThe /#0Crst term has the structure of an ensem ble a v er/-age o v er the the lattice v ariables and coincides formallywith the semiclassical master equation as deduced fromthe sto c hastic Liouville equation/. The traditional w eakorder master equation is obtained from Eq/./#28/8/#29 b y as/-suming high lattice temp erature /#28 h/#17 /=KB T /#1C /1/#29/, andthat the spin system is nev er v ery remote from a state with equal p opulations of all spin energy lev els/#28 /#1B /#03/' I /= A /, I b eing the iden tit y op erator in the Hilb ertspace of the spin system and A b eing the n um be r o fdegrees of freedom of the spin system/#29/./#5B/1 /,/2 /4 /#5D The re/-sulting equation coincides essen tially with the /#0Crst termof Eq/./#28/8/#29 but including the correct stationary state/:cd/#1B /#03/#28 t /#29dt /= /, /~ /, /2/2 Z/1/,/1 dt /0/#5B H /#03SL /#28 t /#29 /; /#5B H /#03SL /#28 t /0/#29 /;/#1B /#03/#28 t /#29 /, /#1B/0 /#5D /: /#28/9/#29d /3/1/8 Brazilian Journal of Ph ysics/, v ol/. /2/8/, no/. /4/, Decem b er/, /1/9/9/8This k ey equation allo ws the calculation of thetime dep endence of an y observ able of the spin systemd hO /#03idt /= Trs /#5B d/#1B /#03/#28 t /#29dt O /#03/#28 t /#29/#5D/. It describ es the irrev ersibleev olution of a macroscopic quan tit y and allo ws the cal/-culation of the transp ort co e/#0Ecien ts/. In NMR relax/-ation suc h co e/#0Ecien ts are the relaxation times/; theycan b e related to the sp ectral densities/, that is theF ourier transforms of the time correlation functions ofthe /#0Ductuating lattice op erators/, ev aluated at the Lar/-mor frequency /, /#17 /.F or example/, the time ev olution ofthe z /-comp onen t of the magnetization /#28parallel to theexternal /#0Celd/#29/, due to dip olar coupling in a system of t w o lik e spins isddt h Iz /+ I /0z i /= /1T/1 Z fh Iz /+ I /0z i/, h Iz /+ I /0z ieq uil g /; whereT /, /1/1 Z /= /9/8 /#0D /4/~ /2f J /1/#28 /! /#29/+ J /2/#28/2 /! /#29 g /#28/1/0/#29is the Zeeman order relaxation time and /! /=/2 /#19/#17 /. Thesp ectral densities J q/#28 /! /#29 are de/#0Cned asJ q/#28 /! /#29/= Z/1/,/1 F qF /, q/#28 t /#29 exp i q/!tdt /:In T able /1 w e summarize the standard theoreticalexpressions for the most commonly measured relaxationtimes/. There w e sho w the observ able corresp onding toeac h case/, the kind of /#0Ductuations resp onsible for dis/-sipating the initial condition/, and the relaxation timesform ulae in terms of the sp ectral densities/. These ex/-pressions are the outcome of the traditional w eak ordertheory /#5BEq/./#28/9/#29/#5D/.cT able I/: Relaxation rates within the w eak order theory of spin relaxation/. In the three /#0Crst cases the observ able isthe energy of in teraction of isolated spins/. F or T /, /1/1 D the observ able is the secular dip ole/-dip ole energy /, and a mo delof isolated spin pairs is assumed/. Here CD /= /9/8 /#0D /4/~ /2/, CQ /= /3/2 /#10e /2qQ/~ /#11/2/, and Q is the n uclear quadrup ole momen t/.Observ able Relaxation time SL in teraction Refs /:Ph Izj i proton Zeeman order T /, /1/1 Z /= CD Plk J /#28/1/#29lk /#28 /! /#29/+ J /#28/2/#29lk /#28/2 /! /#29 dip olar /#5B/1/, /4/#5Dh Iz i deut/. Zeeman order T /, /1/1 Z /= CQ /#02J /#28/1/#29/#28 /! /#29/+ /4 J /#28/2/#29/#28/2 /! /#29 /#03quadrup olar /#5B/2 /#5Dh H /0Q i quadrup olar order T /, /1/1 Q /=/3 CQ J /#28/1/#29/#28 /! /#29 quadrup olar /#5B/2/5 /, /2/6/#5Dh H /0D i dip olar order T /, /1/1 D /=/3 CD J /#28/1/#29/#28 /! /#29 dip olar /#5B/3/3 /, /5/0/#5DdThe form ula for proton Zeeman relaxation is a di/-rect generalization of Eq/./#28/1/0/#29 for the case of man y non/-in teracting spins /#5B/1 /#5D/. The deuteron Zeeman relaxationtime describ es the relaxation of the magnetization of aquadrup olar n ucleus due to /#0Ductuations of the electric/#0Celd gradien t of the c hemical b ond b earing the reso/-nan tn ucleus/. This is a single/-spin description/, and isadequate when the dip olar in teraction with neigh b or/-ing spins can b e neglected/./#5B/2 /#5D T/1 Q and T/1 D refer to therelaxation of quadrup olar or dip olar order created b yapulse sequence /#28Jeener/-Bro ek aer t exp erimen t /#5B/2/7 /#5D/#29 thattransfers the magnetic Zeeman order to the quadrup o/-lar or dip olar reserv oirs/. The three /#0Crst lines of T able /1represen t situations whic h can b e adequately describ edb y a picture of isolated spins/, acted on b y a random p er/-turbation coming from the neigh b oring spins/. On the con trary /, the case of dip olar order relaxation is basi/-cally di/#0Beren t b ecause in this case the observ able hH /0D iin v olv es the in teractions among man y spins/. Here w eincluded the t w o/-spin form ula for T/1 D rep orted in theliterature/. Nev ertheless/, a recen t /#0Celd cycling studysho w ed that this form ula is incomplete for describingthe exp erimen tal data in nematics /#5B/2/8 /, /2/9/, /3/0 /#5D /#28see Sec/-tion IV/#29/.III ApplicationsA/. The Field/-Cycling T ec hniqueMeasuremen ts of the relaxation times /#28 T/1 Z /, T/1 D orT/1 Q /#29 pro vide/, in principle/, a means of determining thesp ectral densities of the molecular motions/. Accord/-ing to T able I/, b y com bining data of di/#0Beren t relax/- R/.C/. Zamar et al/. /3/1/9ation exp erimen ts/, it is p ossible to determine J /1/#28 /#17 /#29and J /2/#28 /#17 /#29 separately /./#5B/2/8 /#5D Then/, the comparison of theirdep endence with the magnetic /#0Celd and temp eraturewith the theoretically predicted dep endence for the in/- v olv ed molecular /#0Ductuations/, allo ws to obtain informa/-tion ab out the di/#0Beren t motional pro cesses in v olv ed/.F or example/, con v en tional NMR temp erature dep en/-den t exp erimen ts of Zeeman and quadrup olar align/-men t relaxation in selectiv ely deuterated liquid crystals/,allo wt o c haracterize in tramolecular motions and alsogiv e some consisten t information ab out the order /#0Duc/-tuations/. /#5B/3/1 /,/3 /2 /#5D The dynamics of the aliphatic c hainsin nematics w as studied in detail b y using this metho dand comparing with theoretical mo del for molecularp oten tials/./#5B/2 /, /3/3/#5DHo w ev er/, since the con v en tional NMR studies pro/-vide the T/1 Z pro/#0Cle only in a rather narro w sp ectro/-scopic windo w/, the /#0Ctting of data with form ula fromT able I and Eq/./#28/1/1/#29 b ecomes am biguous and it is dif/-/#0Ccult to separate the con tribution from the di/#0Beren trelaxation mec hanisms/. Due to this w eakness/, someearly high frequency measuremen ts /#5B/3/4 /, /3/5/, /3/6/#5D b eliev edto ha v e found the theoretically predicted square ro otfrequency dep endence c haracteristic of the OFD in ne/-matics /#5Bsee Eq/. /#28/1/2/#5D/.This apparen t result is incorrect b ecause the re/-p orted /#0Cts in the MHz range do not coincide with sub/-sequen tl o w frequency mesuremen ts when extrap olated/.The mistak em a y b e originated in the fact that theother relaxation mec hanisms p ossess a narro w rangewhere the corresp onding sp ectral densities can b e ap/-pro ximated b y J /#28 /#17 /#29/= a/#17 /, /1 /= /2/+ b /.A more comprehensiv e relaxometric study can b ep erformed b y extending the measuremen ts to the kHzdomain/. Ho w ev er/, standard NMR sp ectrometers can/-not p erform lo w frequency measuremen ts/, b ecause thesignal amplitude strongly decreases at lo w er externalmagnetic /#0Celds/. The adequate exp erimen tal metho d tosolv e this di/#0Ecult y is the /#0Celd/-cycling tec hnique/, sinceit allo ws to enlarge the range of the motional sp ectrumscanned b y longitudinal spin/-lattice relaxation exp eri/-men ts/, b y man y orders of magnitude/. Details of the ba/-sic principles of the tec hnique can b e found in Reference/#5B/3/#5D and /#5B/3/7/#5D In few w ords/, the metho d is based on theconcept of adiabatic demagnetization/. The spin systemis p olarized at high /#0Celds /#28 frequency in the MHz re/- gion/#29/. Then/, the external /#0Celd is adiabatically switc hedto a selectable lo w er lev el/. The spin system ev olv es /#28re/-laxes/#29 under these conditions/, and after a giv en in terv althe /#0Celd is again adiabatically switc hed to its originalv alue/, where the magnetization is detected b y applyinga radiofrequency pulse /#28see Fig/. /1a/#29/. So doing allo wsto deal with a MHz qualit y signal but no wk eeping theinformation of the spin/-lattice relaxation at lo w /#0Celd/. Figure /1/. a/#29 T ypical mo dulation of the Zeeman /#0Celd in a/#0Celd cycling NMR exp erimen t/. b/#29 Sc heme of the Jeener/-Bro ek aert pulse sequence com bined with a /#0Celd cycling/.During the p olarization p erio d t w o phase shifted pulses areapplied to create dip olar order/. Subsequen tly the spin sys/-tem ev olv es in a di/#0Beren t magnetic /#0Celd during the relax/-ation p erio d/. Finally /, after switc hing bac k the Zeeman /#0Celdto the detection v alue HD /, a read pulse is applied and thedip olar signal is acquired/.B/. Selected Exp erimen tsA systematic study of the Larmor frequency dep en/-dence of T/1 Z in thermotropic liquid crystals w as car/-ried out mainly b y the group of F/. Noac k /#28Univ ersitatStuttgart/#29/. P articularly /, the nematic phase has b eenextensiv ely studied/. /#5B/3 /, /3/8/, /3/9/, /4/0/#5DAn illustrativ e example is the textb o ok comp oundPA A /#28/4/,/4/' /- dimeth ylo xy azo xyb enzene/#29/. Fig/. /2 sho wsthe proton relaxation disp ersion T/1 Z /#28 /#17 /#29 for nematicand isotropic phases/. The isotropic phase sho ws thec haracteristic b eha vior of common liquids/. On thecon trary /, relaxation in the nematic phase exhibits aw ell dev elop ed /#17 /1 /= /2dep endence in the frequency range /3/2/0 Brazilian Journal of Ph ysics/, v ol/. /2/8/, no/. /4/, Decem b er/, /1/9/9/8/1/0 /3/, /1/0 /6Hz /, rev ealing the o ccurrence of order /#0Ductua/-tions of the director/. Figure /2/. Proton relaxation disp ersion T/1 Z /#28 /#17 /#29 for nematicand isotropic P AA/. The nematic phase sho ws a square ro otfrequency dep endence/, that is absen t in the isotropic phase/.The solid lines are mo del /#0Cts includin g OFD/, self/-di/#0Busion/,molecular rotation and a lo w frequency cut/-o/#0B/. Data fromRef/./3/9/.T ypically /, in thermotropic liquid crystals the sev/-eral mec hanisms con tributing to spin relaxation can b econsidered as statistically indep enden t pro cesses/. Un/-der this condition/, the sp ectral densities can b e cal/- culated as a sum of separate con tributions from themolecular rotational tum bling /#28R OT/#29/, traslational self/-di/#0Busion /#28SD/#29 /#28lo cal motions/#29 and the OFD/:J q/#28 /#17 /#29/= J qOF D /#28 /#17 /#29/+ J qSD /#28 /#17 /#29/+ J qRO T /#28 /#17 /#29 /: /#28/1/1/#29The three sp ectral densities are/:/1/. Or der Fluctuations of the Dir e ctor/1a/. Nematics /#5B/1/2 /,/1 /3 /#5DJ /1OF D /#28 /#17 /#29/= A/#17 /, /1 /= /2/; /#28/1/2/#29withA /= kB TS /2/2 /#19K r /6 /#12K/#11 /+ D /#13/, /1 /= /2/#28/3 cos /2/#0B /, /1/#29 /2/4 /; /#28/1/3/#29where kB is the Boltzmann constan t/, K is the a v er/-age F rank constan t/, /#11 is the viscosit y /, r the in terprotondistance and /#0B stands for the angle b et w een the in ter/-n uclear v ector and the molecular axis/./1b/. Sme ctics /#5B/1/4 /#5DJ /1OF D /#28 /#17 /#29/= B/#17 /, /1/; /#28/1/4/#29 withB /= kB TS /2/#19K/1/1 /#18r /6 /#28/3 cos /2/#0B /, /1/#29 /2/4 /;where K/1/1 is the spla y elastic constan t /#28assumed to b eequal to the t wist constan t K/2/2 /#29 and /#18 is the coher/-ence length in the direction p erp endicular to the smec/- tic la y ers/. The sp ectral densit y J /2OF D is assumed to b enegligibly in the small amplitude appro ximation/. /#5B/1/0 /#5D/2/. R otation /#5B/2/#5DJ qRO T /#28 /#17 /#29/= /3/2 BR /#1CR q /2/1/+ /#28 q/#19/#17 /#1C/? /#29 /2where BR is a relaxation amplitude factor dep endingon b oth bulk and molecular prop erties and /#1CR is thecorrelation time of the reorien tation ab out the shortmolecular axes/./3/. Self/-Di/#0Busion /#5B/6 /#5DJ qSD /#28 /#17 /#29/= C/#1CD q /2x /4 /#5B u /+/#28 u sin x /+ v cos x /#29 e /, x/#5Dwhere u /= /,x/2 /, /1x /#01/;v /= /,x/2 /+ /1x /+/2 /#01/; and x /#11pq/#19/#17 /#1CD /, C is the di/#0Busion amplitude factor/, and /#1CDis the correlation time of the translational molecularjump/.The mo dels describ ed ab o v e allo w a quan titativ edescription of relaxation disp ersion/. As an example/,Fig/. /3 sho ws the T/1 Z /#28 /#17 /#29 pro/#0Cle for the nematic HpAB/#28/4/,/4/' /-bis/-hept ylo xy azo xy/-b enzene/#29 with the con tribu/-tions from the di/#0Beren t relaxation mec hanisms/./#5B/2/8 /#5D Thisis a c haracteristic pro/#0Cle of thermotropic nematics/. Itis clear again that the OFD dominate the relaxation in the KHz regime/. F or the /#0Cttings it w as also consideredal o w frequency cut/-o/#0B /#17c as predicted b y R/. Blinc/./#5B/4/1 /#5D R/.C/. Zamar et al/. /3/2/1 Figure /3/. Proton relaxation of the Zeeman and the dip o/-lar order in nematic HpAB sho wing the three individualcon tributions for T/1 Z /#28 /#17 /#29 together with the rotation termfor T/1 D /#28 /#17 /#29 /#28self/-di/#0Busion do es not con tribute to dip olar re/-laxation /-see text/#29/. Rotations con tributes appreciably toZeeman order relaxation in the MH z regime/, but are neg/-ligible for dip olar order relaxation in all measured range/.Data from Ref/./2/8/.The F C measuremen ts allo w ed to o v ercome someam biguities in the ev aluation of the parameters in v olv edin the /#0Cttings/, namely A /, B /, BR /, C /, /#17c /, /#1CR /, /#1CD /.P artic/-ularly /, the exp erimen tal amplitude factor A can b e ob/-tained with go o d precision/. The consistency of the /#0Ct/- tings is supp orted b y reasonable magnitudes of the pa/-rameters and systematic parallels for all the measuredcomp ounds/. F or instance/, the /#0Ctted amplitude factorsA ha v e a go o d agreemen t with the theoretical predic/-tion from Eq/./#28/1/3/#29/, for man y studied comp ounds/./#5B/3 /, /4/2/#5DMoreo v er/, the obtained v alues for the self/-di/#0Busioncorrelation time /#1CD can b e compared with di/#0Busionconstan t measuremen ts b y means of a sp ecial com bi/-nation of NMR Field/-Cycling with pulsed /#0Celd gradi/-en ts tec hniques/./#5B/4/3/#5D Assuming the relation /#5B/6 /#5D /#1CD /= b /2/6 D/#28with the distance of closest proton approac h for di/#0Bu/-sion and D /= /1/3 /#28 Dk /+/2 D/? /#29 an e/#0Bectiv e di/#0Busion con/-stan t/#29/, the calculated correlation times coincide in orderof magnitudes with the /#0Ctting parameters/. /#5B/4/3 /#5DAnother in teresting feature arising from F C mea/-suremen ts is the o ccurrence of the lo w frequency cut/-o/#0B t ypically ab out few KHz /#28 Fig/. /3/#29/. A t the presen t/,t w o p ossible explanations for this b eha vior ha v e b eensuggested/: the /#0Crst one states that/, since the minim um/#0Celd whic h could giv e rise to Zeeman order cannot b eless than the lo cal /#0Celds pro duced b y the neigh b oringspins/, it w ould b e exp ectable that for frequencies of the order of few KHz /, T/1 Z b ecame frequency indep enden t/;the second one refers to the /#0Cniteness of the nematicorder correlation length/, /#5B/4/1 /#5D /#18 /, due to the presence ofdisclinations and other defects/. In suc h a case only or/-der /#0Ductuations with a w a v elength smaller than /#18 cantak e place/, giving rise to a minim um cut/-o/#0B frequency /,through the /`disp ersion relation/'/#17c /= /2 /#19K/#11/#18 /2 /; /#28/1/5/#29where K is the e/#0Bectiv e elastic constan t and /#11 is thee/#0Bectiv e viscosit y /. Although most w ork ers fa v or thesecond h yp othesis/, a /#0Cnal exp erimen tal con/#0Crmation isstill lac king and the sub ject is no w ada ys b eing in v esti/-gated theoretically and exp erimen tally /./#5B/4/4 /#5DThe thermotropic smectic mesophase has not b eenstudied so extensiv ely as the nematic one/. Nev ertheless/,the a v ailable exp erimen tal results sho w clearly the pres/-ence of collectiv e motions/. /#5B/1/9 /,/4 /5 /#5D Fig/. /4 sho ws the T/1 Zpro/#0Cle for TBBA in the smectic A phase/./#5B/1/9/#5D It can b eappreciated the predicted /#17 /1/-la w for order /#0Ductuationswithin a smectic la y er /#5BEq/./#28/1/4/#29/#5D/. This linear dep endenceis eviden t in the frequency range /1/0 /3/, /1/0 /5Hz /, sho wingagain that the OFD are the dominan t relaxation mec h/-anism for the Zeeman order only at lo w Larmor fre/-quencies/. Tw o additional relaxation mec hanisms /#28self/-di/#0Busion and anisotropic rotations/#29 are necessary to de/- scrib e the exp erimen tal results/.Recen tly /, other w ork ers /#5B/4/6 /#5Dh a v e measured a tran/-sition from /#17 /1to /#17 /1 /= /2in the T/1 Z frequency dep endenceof smectic /8CB /#28/4/- cy ano/-/4/' /-/8/-alkylbiphen yl/#29 at /2/3 oK/,i/:e/: close to the smectic A/-nematic transition/. A similarresult w as observ ed in the smectic HpAB at /8/2 oK/, butno w far from the transition/. /#5B/4/7 /#5D Conceiv ably /, this fea/-ture w as not detected in earlier measuremen ts due tothe lo w densit y of exp erimen tal p oin ts/. A t the presen t/,it is not clear if this is a pretransitional e/#0Bect or if it re/- /#0Dects an in trinsic c haracteristic of the collectiv e /#0Ductu/-ations in the smectic phase/. Eviden tly /, more systematicmeasuremen ts are needed/. /3/2/2 Brazilian Journal of Ph ysics/, v ol/. /2/8/, no/. /4/, Decem b er/, /1/9/9/8 Figure /4/. Proton spin relaxation disp ersion/, T/1 Z /#28 /#17 /#29/, forTBBA in smectic A and smectic C phases/. The solid lineis the /#0Ct according to T able I and Eq/. /#28/1/1/#29 includin g againOFD/, molecular rotations and self/-di/#0Busion/. Data from Ref/./1/9/. Figure /5/. Proton relaxation disp ersion T/1 Z /#28 /#17 /#29 for the micel/-lar/, hexagonal/, cubic and lamellar phase of a ly otropic p otas/-sium laurate/-D/2 O mixtures/. The micellar and the hexago/-nal system sho w the /#17 /1dep endence t ypical of smectics/. Thelinear pro/#0Cle is absen t in the isotropic phases/. Data fromRefs/. /3/,/1/8/.The linear frequency dep endence is b etter dev elop edin ly otropic liquid crystals/. As p oin ted b y Kuhner etal /, /#5B/4/8 /#5D this feature is exp ectable b ecause the smec/-tic t yp e la y ers are almost decoupled b y the w ater in/-terface/. In this w a y /, eac h one constitutes basically at w o/-dimensional system/. An example are lamellar orhexagonal p otassium laurate/-w ater mixtures /#28see Fig/. /5/#29/, where the /#17 /1/-la w is observ ed appro ximately in therange /1/0 /3/, /1/0 /5Hz /.An indep enden t exp erimen t pro viding additionalexp erimen tal evidence of the presence of collectiv e /#0Duc/-tuations in nematics are NMR /#0Celd/-cycling studies of deuteron spin/-lattice relaxation/. Con trarily to protonNMR sp ectra/, the deuteron sp ectra sho w without an yspin decoupling tec hnique man yw ell/-resolv ed line dou/-blets/, whic h can b e assigned to non/-equiv alen t siteson the molecule/. /#5B/1/7 /#5DA t presen ti t w as not y et at/-tained enough signal qualit y and resolution to separatethe corresp onding relaxation times for all the observ eddoublets/, due to the unfa v orably smaller gyromagneticratio /#28a higher detection /#0Celd is needed/#29/. Nev erthe/-less/, it is p ossible to measure the spin lattice relaxation rates for some a v erage spin p ositions/, namely for thering windo w and the c hain windo w as sho wn in Fig/./6/. It is in teresting to note that/, while the disp ersionpro/#0Cle of the c hain deuterons rev eal an appro ximatesquare/-ro ot la w in the same frequency range than theproton data/, the rings deuterons exhibit a w eak fre/-quency dep endence/. This fact can b e qualitativ ely un/-dersto o d b y realizing that/, since the relev an t orien ta/-tion for quadrup olar coupling is the angle b et w een thecarb on/-h ydrogen b ond and the static magnetic /#0Celd/,the con tribution from the molecular rotations alongthe long axis of the molecule is signi/#0Ccan tly greater fordeuteron than for proton spin relaxation/./#5B/1/7 /#5DFinally /, in Fig/. /6 w e also sho w a comparison b e/-t w een proton T/1 Z of the nematic /5CB /#28/4/- cy ano/-/4/' /-/5/-alkylbiphen yl/#29 and the alkyl c hain deuteron T/1 Z for thep er/-deuterated nematic /5 CB /, d/1/9 /. As can b een clearlyseen the collectiv e motions dominate in coinciden t fre/-quency ranges/.IV Dip olar order relaxationField Cycling exp erimen ts are able to rev eal the fre/-quency range where co op erativ e motions dominate re/-laxation as w ell as giving an appro ximate v alue for therelativ ew eigh t of the di/#0Beren t relaxation mec hanismsin a n um b er of nematics and smectics/.The NMR proton sp ectra of liquid crystals generallypresen t unresolv ed broad lines/, due to the strong dip o/-lar in teractions/. The measured relaxation times repre/-sen t all spins in the molecule/, and the con tributions tothe sp ectral densities from di/#0Beren t kinds of sites aresup erimp osed/. When the molecular motions are com/-plex/, T/1 Z /#28 /#17 /#29 alone is insu/#0Ecien t for disen tangling the R/.C/. Zamar et al/. /3/2/3di/#0Beren t con tributions to the sp ectral densities/. Thenit w ould b e adequate to complemen t the data of Zee/-man relaxation with another exp erimen t/. Under theassumption that the /#28w eak order/#29 form ulae from T a/-ble I hold in the whole frequency range/, measuring thefrequency dep endence of di/#0Beren t relaxation times/, lik eT/1 D /#28or T/1 Q /#29 together with T/1 Z comes out to b e a moredescriptiv e exp erimen t/. The sp ectral densit y J /1/#28 /#17 /#29 canb e directly determined through the /#0Crst one and then/, b y using it in the form ula for the latter/, J /2/#28/2 /#17 /#29 can b ecalculated/. Figure /6/. Comparison of the frequency dep endence of T/1 Zin protons of /5CB /#28triangles/#29/, c hain deuterons of /5CB/- d/1/9/#28op en squares/#29 and ring deuterons only /#28circles/#29/. Deuteronmeasuremen ts allo w to select a separate frequency windo wfor di/#0Beren t sites in the molecule/. Data from Ref/. /1/7/.The /#0Crst step in this direction w as the /#0Celd cyclingmeasuremen t of b oth proton T/1 Z /#28 /#17 /#29 and T/1 D /#28 /#17 /#29i n afamily of thermotropic nematic liquid crystals/./#5B/2/8 /, /4/0/#5DThis metho d for measuring T/1 D com bines the /#0Celdcycling tec hnique with the Jeener/-Bro ek aert pulse se/-quence for creating dip olar order/. Figs/. /#28/1b/#29 and /#28/1c/#29 sho w a diagram of the tec hnique/. Fig/. /#28/3/#29 sho wsthe disp ersion of T/1 D and T/1 Z in the nematic phaseof HpAB/. The /#0Crst outstanding feature is that T/1 D /#28 /#17 /#29follo ws the t ypical trend of the OFD in the whole fre/-quency range /#28ev en for /#17 /#1D /1/0 MHz/!/#29/. This b eha vioris v ery di/#0Beren t from that of T/1 Z where/, as a rule/, theOFD dominate/, at most up to h undreds of kilohertz/.Another in teresting feature is the o ccurrence of a cut/-o/#0B frequency ab out /5 /#02 /1/0 /4Hz/, whic h is an order ofmagnitude higher than the one of Zeeman relaxation time/.The t w o/-spin mo del for dip olar relaxation /#28see T a/-ble /1/#29 predicts that T/1 Z /=T/1 D /#14 /3o v er all the frequency range/, since the sp ectral densities are p ositiv e quan ti/-ties/. Con v ersely /, the exp erimen ts sho w that this ratiois greater than three in b oth HpAB and /8CB/, as dis/-pla y ed in Fig/. /7/. Also/, as can b e seen in Fig/. /8/, there isa noticeable gap b et w een the T/1 D calculated with theisolated phen yl spin pairs mo del and the exp erimen taldata/; the di/#0Berence is frequency dep enden t and pro/-p ortional to /#17 /, /1 /= /2/,e v en at high frequencies/. Similarc haracteristics w ere found in other sev eral comp oundsin the nematic phase/, /#5B/2/9/,/4/0/#5D where it w as realized thatthe gap increases with the size of the molecule/. Figure /7/. The quotien t T/1 Z /=T/1 D as a function of the Lar/-mor frequency in the nematic phase of HpAB and /8CB/.This ratio exceeds the v alue of three in discrepancy withthe semiclassical t w o/-spins mo del/. Data from Ref/./2/8/.As can b e seen in Fig/./#28/3/#29/, the con tribution fromthe OFD to T/1 Z deca ys rapidly for increasing external/#0Celd/, and it generally b ecomes comparable to the sp ec/- tral densities of rotational and translational di/#0Busion for frequencies higher than /5 /#02 /1/0 /5Hz/. The situation israther di/#0Beren t for dip olar relaxation/, since it is mainlydriv en b y the OFD mec hanism even at high /#0Celds/. Itcan b e clearly appreciated in Fig/./#28/3/#29 the larger in/#0Duence of the OFD as compared with the molecular rotationaldi/#0Busion/. It can b e a/#0Ermed that the rotations are prac/-tically negligible in T/1 D /.These exp erimen tal /#0Cndings are c hallenging sincethey clearly p oin t out some de/#0Cciencies of the existen ttheoretical approac h/, but/, also rev eal T/1 D /#28 /#17 /#29 as a mag/-nitude that re/#0Dects the OFD almost exclusiv ely in thea v ailable frequency range/.The fact that T/1 Z /=T/1 D /#15 /3 indicates that the ob/-serv ed dip olar order relaxation is faster than the onepredicted b y the traditional mo del/. On the con trary /3/2/4 Brazilian Journal of Ph ysics/, v ol/. /2/8/, no/. /4/, Decem b er/, /1/9/9/8T/1 Z /#28 /#17 /#29i s w ell describ ed b y the form ula from T ableIi n n umerous liquid crystals in the high /#0Celd limit/#28 H/0 /#3E/#3E Hlocal /#29/./#5B/2 /,/3 /,/3 /1 /,/3 /2 /,/4 /9 /#5D Therefore/, the fail/-ure w as assigned /#5B/2/8 /#5D to the standard t w o/-spin /#28phen yl/#29approac h for dip olar order spin/-lattice relaxation rate/. Figure /8/. F requency dep endence of the dip olar order re/-laxation in the nematic phase of HpAB and /8CB/. Thefull line corresp onds to the prediction from the t w o/-spinmo del and the dashed line is the b est /#0Ct with the functionT /, /1/1 D /=/3 CD J /1/#28 /#17 /#29intr a /+ a/#17 /1 /= /2/+ b /. Data from Ref/./2/8/.It should b e k ept in mind that the usual mo del fordip olar relaxation in thermotropic liquid crystals/#5B/1/4/, /5/0/#5D starts from the w eak order master equation /#5BEq/./#28/9/#29/#5Dand considers the spin system as an ensem ble of isolatedspin pairs/. Only in tramolecular con tributions are k ept/,arguing that the rapid di/#0Busiv e molecular motion a v/-erages out the in termolecular con tribution/. The c hainprotons are not considered either/, due to their high mo/-bilit y /. Another imp ortan th yp othesis is the one kno wnas /`fast motion/'/, that is /#17D /#1Cc /#1C /1/#28 /#17D at ypical dip o/-lar frequency and /#1Cc a lattice correlation time/#29/. Thiscondition allo ws to neglect the spin/-spin in teractions incalculating the time ev olution of the spin op erators inthe in teraction picture/. /#5B/5/#5DThe observ ed frequency dep enden t gap b et w een the/-ory and exp erimen tal data indicates that the mo delfor T/1 D in liquid crystals is o v ersimpli/#0Ced and has ne/-glected imp ortan t mec hanisms of relaxation/, driv en b ythe OFD/. F or instance/, k eeping only t w o spins a v oids the presence of sp ectral densities ev aluated at zero fre/-quency or frequencies of the order of magnitude of thedip olar coupling/, J /0/#28 /#17D /#29/, b ecause the co e/#0Ecien ts asso/-ciated to these terms in v olv e spin traces whic h are iden/-tically zero for t w o spins/. The ph ysical reason of thisis that these co e/#0Ecien ts represen t /`/#0Dip/-/#0Dop/' transitions/,that do not in v olv ec hanges in the dip olar energy of onlyt w o spins/. Also sp ectral densities with q /=/2 /, J /2/#28/2 /#17 /#29/,do not app ear due to similar reasons/. According tothe proton sp ectra in nematics/, there is an appreciabledip olar con tact among the core and c hain protons/, giv/-ing the p ossibilit y of some e/#0Bects asso ciated to in terpairin teractions/. This imp oses a revision of the v alidit yo fthe h yp othesis in v olv ed in the mo del for dip olar relax/-ation/.A partial answ er came after a recen t exp erimen/-tal and theoretical in v estigation of T/1 Z /#28 /#17 /#29 and T/1 D /#28 /#17 /#29in P AA and in the meth yl/-deuterated P AAd /6 /./#5B/2/9 /,/5 /1 /#5DDeuterating the meth yl c hain yields a proton systemre/#0Decting only the dynamics of the c or e /. The frequencydep endence of the relaxation times in b oth comp ounds sho w ed discrepancies with the t w o/-spin theory /, similarto those previously observ ed in HpAB and /8CB/. Fig/./#28/9/#29 sho ws the obtained results for the frequency dep en/-dence of T/1 Z and T/1 D at /1/3/0 /0C/. The /#0Cgures also includea plot of the underestimated dip olar relaxation rate as giv en b yT able /1/. The exp erimen tal data of the dip olarrelaxation time can b e w ell represen ted b y the functionT /, /1/1 D /=/3 CD J /1/#28 /#17 /#29intr a /+ a/#17 /, /1 /= /2/+ b /,d o wn to /1/0 /5Hz/.The /#0Crst correction term has the w ell kno wn frequencydep endence of the OFD/, /#5B/1/2 /#5D and the parameter a is/5 /: /3 /#02 /1/0 /3s /, /3 /= /2and /4 /#02 /1/0 /3s /, /3 /= /2for PA A and PA Ad /6 /,resp ectiv ely /. The frequency indep enden t term is relatedto the sp ectral densities with q /= /0 whic h app ear if themo del of isolated spin pairs is abandoned/. In this case/,the constan t are mainly asso ciated to the dip olar in ter/-action b et w een the c hain and the core/. This term is notdominan t/#28 b /=/0 /: /7 /#02 s /, /1for PA A and b /=/0 /: /2 s /, /1forPA Ad /6 /#29/, and con tribute appreciably only in the MH zrange/. F or instance/, at /1/0 MH z /, the constan t term rep/-resen ts the /7/#25 of the total relaxation rate for PA Ad /6 /.In relation with the theory /, follo wing the lines ofthe semiclassical formalism of spin/-lattice relaxation/#5B/5 /#5Da general high /#0Celd expression for T /, /1/1 D in nematic liq/-uid crystals v alid for an arbitrary n um b er of spins w as R/.C/. Zamar et al/. /3/2/5deduced giving the result/: /#5B/2/9 /#5DcT /, /1/1 D /= T /, /1/1 D cor r /+ /2/7 /#0D /4/~ /2/8 N Plk J /1lk /#28 /#17 /#29 F /0lk /2Pj F /0lj /2 /+ /4 /#0D /4/~ /2/3 N Pj F /0lj /2 Plk Pq J qlk /#28 /#17q /#29 /#02na q Pm /6/= k /#10F /0lm /2/+ F /0km /2 /#11/+ b q Pm /6/= k F /0lm F /0km o/; /#28/1/6/#29d Figure /9/: Zeeman and dip olar relaxation rates/, measuredwith the /#0Celd cycling tec hnique in nematic PA A and ne/-matic P AAd /6 at the same temp erature/. The circles andsquares are exp erimen tal T/1 and T/1 D resp ectiv ely /. The up/-p er full lines are the /#0Ctting to T/1 Z with T able I and Eq/./#28/1/1/#29/.The dashed lines are the semiclassical t w o/-spin mo del forT/1 D from T able I/. The lo w er full lines are the b est /#0Cts withthe function T /, /1/1 D /=/3 CD J /1/#28 /#17 /#29intr a /+ a/#17 /1 /= /2/+ b /#28see the text/#29/.Data from Ref/./2/9/.The /#0Crst term corresp onds to in terpair cross corre/-lations/, while the others in v olv e uncorrelated in terac/-tions/. All terms ha v e similar structure/: they are pro/-p ortional to angular a v erages of the t yp e F /0lj /2/. Thein termolecular con tributions cancel due to the rapid rel/-ativ e motion of the molecule/. Accordingly /, the transla/-tional self/-di/#0Busion do es not con tribute appreciably todip olar order relaxation in nematics/.This general result allo ws to in v estigate the roleof in terpair correlation of protons of the core and thec hains/. When particularized to P AAd /6 for calculatingthe con tribution to T/1 D b y including second neigh bo r protons/, the calculation yielded a negligible correctionto the usual t w o/-spin prediction/. This fact/, whic hi sconsequence of the r /, /3dep endence of the dip olar en/-ergy in teraction/, is consisten t with NMR line shap esecond/-momen t studies/#5B/5/2/#5D that c haracterize P AAd /6 asat w o/-spin system/.According to the former results/, the dip olar orderrelaxation of P AAd /6 /, although seeming to b e a t w o/-spinproblem/, cannot b e describ ed b y the usual form ula forisolated spin pairs/!/. This means that the observ ed gaphas to b e asso ciated to some other t yp e of pro cesseswhic h are not considered b y this approac h/.A p oin t still requiring discussion is the fast motionh yp othesis/, con tained in the basic w eak/-order theory /.T ec hnically /, this amoun ts to calculating the ev olutionof the spin op erators in the in teraction picture with theop erator exp it/= /~ HZ/#28a simple rotation around the direc/-tion of the magnetic /#0Celd/#29/. F or the slo w est comp onen tsof the OFD the condition /#17D /#1Cc /#1C /1 is not ful/#0Clled/; thenneglecting the spin in teractions during the lifetime ofthe /#0Ductuations ma y b e inadequate/. In suc h cases/, thetime ev olution has to b e calculated using the completeop erator e it/= /~ /#28 HZ /+ HD /#29in order to the in tro duce the in/-teractions/. Ho w ev er/, this correction to the standard ap/-proac h did not in tro duce signi/#0Ccan tc hanges either/./#5B/3/0 /#5DThat is/, this pro cedure do es not lead to a frequency de/-p enden t correction as needed for explaining the T/1 D /#28 /#17 /#29disp ersion/.The insensibilit y sho wn b y the w eak/-order result tothe inclusion of the spin in teractions in the microscopictime scale/, can b e in terpreted as follo ws/: As sho wn inSection I I/, the w eak order master equation is obtainedb y neglecting terms of quan tum mec hanical c haracterin Eq/./#28/8/#29/. The relaxation rates calculated in this limitare formally equiv alen t to those calculated directly from /3/2/6 Brazilian Journal of Ph ysics/, v ol/. /2/8/, no/. /4/, Decem b er/, /1/9/9/8a sto c hastic Liouville equation/, where the lattice v ari/-ables of the spin/-lattice Hamiltonian are giv en randomtime functions/. By represen ting the lattice v ariableswith arbitrary random functions of time/, the bac k/-reaction of the lattice on the spin system during the lifetime of the /#0Ductuations cannot b e retained/.In summary /, the former example allo ws to safelyconclude that the w eak/-order theory do es not con tainall the ingredien ts necessary to explain the dip olar or/-der relaxation in systems ha ving slo w /#0Ductuations/, lik enematic thermotropics/. According to this outstanding conclusion/, the failure m ust b e sough t in the v ery basicassumptions of the w eak/-order relaxation theory /.In order to include the correlation b et w een a manyb o dy system and the lattic e in the micr osc opic timesc ale /, it is necessary to k eep the quan tum terms ofEq/./#28/8/#29/. Namely /, when dealing with the relaxation ofthe dip olar order/, the spin system should b e treated as an op en quan tum system in thermal con tact with aquan tum mec hanical lattice/. This can b e done b y elim/-inating the w eak/-order assumption/./#5B/3/0 /, /5/3/#5D Under thisgeneral condition/, the master equation in the form of Eq/./#28/8/#29 is not easily applicable to the calculation of thetime dep endence of spin observ ables/. A more tractableequation follo ws b y p erforming a series expansion of themaster equation in op erator form/./#5B/5/4 /#5D The lo w est orderterm of the expansion coincides with the /#0Crst term ofEq/./#28/8/#29/, while the higher order ones represen t the quan/-tum mec hanical part/. In fact/, the new con tributionscancel if the lattice is represen ted b y a sto c hastic pro/-cess/. The formalism is/, at this p oin t/, adequate for in/-tro ducing a /`trial/' solution/. The trial densit y op eratorcommonly used in NMR is the one corresp onding to the spin temp erature assumption/. Ph ysically /, it consists inassuming that the relaxation can b e describ ed as a suc/-cession of semi/-equilibrium states/./#5B/5/5/, /5/6/#5D Doing this/,it is straigh tforw ard to calculate a relaxation time interms of the sp ectral densities/./#5B/3/0/#5DUsing the Pincus/-Blinc mo del for the nematic h y/-dro dynamic motions/, leads to a correction term to the usual expression of the dip olar order relaxation rate/,ha ving the form a/#17 /, /1 /= /2/. This result agrees with thephenomenological expression used in /#0Ctting P AAd /6 andP AA data/, and other thermotropic nematic liquid crys/-tals/. Accordingly /, the discrepancy b et w een exp erimen tsand the w eak order theory can b e explained in terms of m ultispin pro cesses ha ving a quan tum mec hanicalc haracter/.The existence of the anomalous lo w/-frequency cut/-o/#0B can also b e discussed in terms of the quan tum me/-c hanical approac h/. When the external magnetic /#0Celdis lo w ered so that the Larmor frequency is comparablewith the linewidth /#28/1/0 /5Hz/#29/, the description is made interms of the dip olar and not the Zeeman energy lev els/.Under this condition the Larmor frequency ceases to b ea collectiv e parameter of the spin system/.V Final comm e n tsF requency dep enden t T/1 Z exp erimen ts using the /#0Celdcycling tec hnique ha v e sho wn useful in c haracterizingthe OFD/. This is sp ecially clear in nematics/, where thet ypical frequency b eha vior dominates in a broad fre/-quency range and less eviden t in smectics/. The factthat smectics ha v e slo w er lo cal molecular motions thannematics has a double e/#0Bect in reducing the e/#0Bectiv e/-ness of T/1 Z /#28 /#17 /#29 for displa ying unam biguously the con/-tribution of the collectiv e motions/. On the one handindividual molecular motions /#28rotation and di/#0Busion/#29ha v e a greater relativ ew eigh t in smectics/. On the other/,the frequency dep endence of the OFD is steep er/. This amoun ts in a narro wing of the frequency range wherethe OFD dominate relaxation /#5Bsee Fig/./#28/4/#29/#5D/.Due to the strong spin/-spin in teraction/, the protonZeeman relaxation time is an a v erage parameter repre/-sen ting all the protons placed in di/#0Beren t kind of sites/#28for example in the c hain or the core/#29/. In moleculesha ving man y protons this pro duces a certain degree ofam biguit y in the ph ysical parameters obtained from the/#0Cttings/, due to the large n um b er of parameters thatha v e to b e determined/. More informativ en uclei w ouldb e deuterons /#28D/#29 or carb ons /#28 /1/3C/#29 b ecause they sepa/-rate individual atomic sites on the molecules b y a gen/-erally w ell/-resolv ed sp ectrum/. Ho w ev er/, suc h exp eri/-men ts imply imp ortan t tec hnological di/#0Eculties/. Thislimitati on can b e partially eliminated/, in a more ac/-cessible w a yb y com bining exp erimen ts of relaxation inprotons of partially deuterated molecules/; so doing the di/#0Beren t relaxation mec hanisms can b e separated b y al/-ternativ ely deuterating the c hains or the cores/. Also/,these exp erimen ts can b e complemen ted with angular R/.C/. Zamar et al/. /3/2/7dep enden t measuremen ts/, whic h pro vide additional ex/-p erimen tal information /#5B/1/4 /, /2/2/, /5/7 /#5DIn Section /#28IV/#29 w e sho w ed some examples wheredip olar order relaxation re/#0Dects the OFD almost exclu/- siv ely /. This outstanding feature mak es T/1 D /#28 /#17 /#29 an opti/-m um parameter for studying slo w collectiv e molecularmotion in mesophases/. Accordingly /, it could b e help/-ful for testing theoretical mo dels for the h ydro dynamic/#0Ductuations in mesophases ha ving di/#0Beren t dimension/-alit y /.Due to the di/#0Beren t frequency resp onse of b oth re/-laxation times/, exp erimen ts of T/1 D /#28 /#17 /#29 in conjunctionwith T/1 Z /#28 /#17 /#29m a y pro vide v aluable exp erimen tal to olsfor the study of molecular motions in mesophases/. P ar/-ticularly /,i t w ould allo w to clearly discern the frequencydep endence of the collectiv e motions in smectics/. Ho w/-ev er/, for these studies to b e fruitful/, a thorough the/-oretical revision of the e/#0Bect of the slo w /#0Ductuationsof the director on the dip olar relaxation rate b ecomes necessary /, since a comprehensiv e expression of T /, /1/1 D interms of sp ectral densities is still lac king/.P arallel to the /#0Celd of the applications/, from a basicp ersp ectiv e/, dip olar order relaxation also pro vides anin teresting example of irrev ersible pro cesses where thespin/-spin in teractions o ccurring during the microscopictime in terv als seem to pro duce observ able e/#0Bects in thetransp ort parameters/.VI Ac kno wledgm en tW e are grateful to Dr/. D/. J/. Pusiol for helpful discus/-sions/. This w ork w as partially supp orted b y CONICORand SECYT from C/#13 ordoba/, CONICET and F undaci/#13 onAn torc has from Argen tina/. The authors dedicate thepresen tw ork to the memory of Prof/. F/. 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