Volume60A, number3 PHYSICSLETTERS 21 February1977

NUCLEAR ACOUSTIC RESONANCE ABSORPTION AND DISPERSION

V. MULLERInstitut fürAtom-undFestkOrperphysik(B), Freie UniversitätBerlin, 1000Berlin 33, Germany

Received30 November1976

A generalizedform of the “Kubo susceptibility” is presentedcorrelatingNAR absorptionandNAR dispersionwith the soundinducedperturbationHamiltonianh (t). To illustrate the advantageof NAR susceptibility,wegive thefirst quantumtheoreticaltreatmentof dipole NAR in metalswhoseresultsarein agreementwith experiment.

To date,Kubo’s methodof “generalizedsuscepti- wherethe “memory operator”W(r) is independentofbility” [ii is oneof the mostpowerful toolscorrelating f(r) and W(r) = 0 for r < 0. By aid of theconvolutionan externalperturbationHamiltonianh(t) with relevant theoremequation(3) canbebroughtin a morecon-experimentalquantities.Thismethod,however,success- venientform yieldingfullyusedaswellinNMR[2j asinNAR [3j failsif h ~ = - 4theperturbationHamiltonianh(t) is notof the formh’t~= Wf’t~ where W(~)is equivalentto the Laplacetransformof

‘- ‘ / ‘ ‘ ‘ W(t) (sinceW(t) = 0 if t <0). Startingfrom the masterwheref(t) is an “externalforce” and W is an operator equationandkeepingonly first order termsin h(t),which is assumedto be independentof f(t) and con- the Fourier transformof p(t) isstantwith respectto the timevariable t. In nuclear . —

(5)acousticresonanceh (t) generallyis not of the form(1) and therefore,withouta generalization,Kubo’s wheretheorycannotbe appliedto NAR.

Providedthat theperturbationis weakand taking f dt e_~te_Hoth(cz)e~~~’ot, (6)mto accountonly termslinear in f(t) the Kubo sus- 0

ceptibility” is defmedby —Tr {p(~)W} = x(~2)f(&T~)or accordingto (1) andp0 is thestatisticaloperatorat t = —00 whichis

Tr ~ ‘&2~h ~ diagonalin the energybasis.Sinceh (&2) is linearly re-= — ~ f(~-~\ , (2) latedto f(~),accordingto eqs.(4) to (6) the Fourier

~ ‘- -“ ‘ ‘ transformp(~)of the densityoperatoralsomustbewheretheargument&2 standsfor the Fouriertransform a linearfunction of f(~).with respectto thetime variablet andp(t) is the den. Extendingthe definition(2) to the linearresponsesity operator.AnalyzingKubo’s theoryonefinds that Hamiltonian(3), asa consequenceof eqs.(4) and (5)themain premiseis thevalidity of linearresponseof the it is evidentthat the resulting “generalizedsuscepti-densityoperatorp(t) to the “externalforce”f(r). In bility” — just asthe “Kubo susceptibility”— is inde-the frameworkof linearresponsetheory,however, pendentoff(&2). Since in NAR theexternal“force” istherestrictionto a perturbationHamiltonianof the the nonzerocomponente(r, r) of the acousticstrainform (1) is notnecessarysincethemostgeneralpertur- tensorandh (t) is thesoun4inducedperturbationbation Hamiltonianwhich is compatiblewith theas- Hamiltonianof the nuclearspinsystem,accordingtosumptionof linearresponseis of the form definition(2) we term

1 — Tr{p(~Z)h(cZ)} 7h(t) ~- f dr W(r)f(t — r), (3) XNAR( ) - — p C

2f(V)d3P[e(r, ~)]2’ ~

— the generalizedNAR susceptibility.In this formulaCa

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Volume60A, number3 PHYSICSLETTERS 21 February1977

is the phasevelocity of sound,p~themassdensityand where~a= Ea(Vs)/Ea(V) is the acousticfilling factorV~the volumeof thesamplecontainingthe nuclei, andEa(Vs)is that partof the totalacousticenergy

In NAR thedevice undertestis thesamplewith Ea(V)whichis storedin the specimencontainingthetransducerforming a compositeresonatorandthenon- nuclei.Theacousticfilling factorcanbe assumedto bezerocomponente(r, t) of theacousticstraintensor of theorder 1 if themassof the transducerplusbondis of theform is smallas comparedto themassof thespecimen.

( \ — ~( \ ~ Combiningeqs.(12)and(9), and using thedefinitione~r,t, — ~ , ~ — - ,, 1

XNAR — XNAR — ‘XNAR uie 5lLiI L Or uie piionon i055wheref6(t) is a dimensionlessharmonicfunctionof ratehr (NAR absorption)andthe acousticphasetime. If the electricvoltageat the transduceris suffi- velocity Ca (NAR dispersion)becomes~N 1/r

ciently small, linearresponsemaybe assumedandthe- ~NhIT-w XNAR, ANC —(c /

2)XNAR, (13)compositeresonatorcanbe describedbyan electric fl a aimpedanceZ which changesdueto nuclearspintran- thusrelatingmicroscopicto experimentalquantitiessitions. In the vicinity of an acousticstandingwave which are relevantto NAR.resonancefrequencywn/211the relativeimpedance To illustrate the advantageof generalizedNAR sus-changeis [4] ceptibilitywe give a quantumtheoreticaltreatment

~~NZ Q(fl) of NAR if the spin-phononinteractionis due to the= fla~ {1~N(hIT)— 2i~Nwfl}, (9) couplingbetweenthemagneticdipole momentsof the

0 nucleiand theultrasonicallyinducedmagneticfield

wheretherealquantityZ0 is thespinindependentpart which is considerablelargein metals.Hence,theper-

of the electric impedanceat w = w~,Q~h1~is the acoustic turbationHamiltonianbecomesquality factor, 1 /r the phononlossrate and ~a theacousticfilling factor. h(t) = hD(r) = — f d

3rm(r)b(r, 1) , (14)If P(t) is the electricpowerfed intothe transducer (V)

and1(t) theelectriccurrentwhich is assumedto be a S

harmonicfunction of time, that is J(~)= 1(d){6(fZ—w) wherem(r) is the operatorof thenuclearspinmagneti-+ &(d + w)} with I(—dZ) = i*(d), by aid of the convo- zationandb(r,t) thesoundinducedrf magneticfield.lutiontheoremandU(d) = Z(d)I(dz) it canbe shown Inserting(14) into (7) andusingfor b(r, r) theexpres-that theelectricimpedanceZ is relatedto theelectric siongiven in ref. [5j, the NAR dipole susceptibilitypowerby becomes

Z(w) = 2 .P(2w)/(f(w)I(w)), (10) (J~) = B~cos2Osin24

wherethe argument2w standsfor the Fouriertrans- NAR 120PsCa2 \ 1+ j32 /form at d = 2w.Hence,the contributioni~NZ(w)of x{(1 — 132)+2i13}x(d), (15)thenucleito theelectricimpedancebecomes

where0 is theanglebetweenthe acousticwave vector= ~ (11) and thedc magneticfield B

0, ~ the anglebetweenthewhereP0(r) is theelectricpowerwithoutspin transi- polarizationvectorandB0,120 thevacuumpermeabi-tions,PN(t)= Tr~ (t) ah/at} is thepowerabsorbedby lity, 3 d/(j.z0ac~),athe dc conductivity,thenucleiandPN(2w) P(2w) —P0(2w).It is worth Tr{p(d)hD(d)}mentioningthat thelatterequationonly combines x(d)= — 3 2 (16)reversiblepowerterms.Hence,this equationgenerally f(V~ ?‘ [b1(T,d)] /120doesnotholdin thetime space.Restrictingourconsi- is the conventionalnuclearmagneticspin susceptibilityderationsto piezoelectrictransducerswherethe electric familar in NMR andb1(r, fZ) is the Fouriertransformcurrentis found tobe proportionaltodf~/dt, by aid of thecomponentof the rf magneticfield (perpendicularof eq.(4) to (8) and(11)we obtain toB0) whichis relevantto nuclearmagneticdipole

transitions.Combiningequations(15)and(13)we find~NZ(w)/Zo = iflaQ~~)XNAR(w), (12) thesamemathematicalexpressionsfor theshift of the

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Volume60A,number3 PHYSICSLETTERS 21 February1977

phononloss rateandacousticphasevelocity asobtained [2) R. KuboandK. Tomita,J.Phys.Soc.Jap.9 (1954) 888.by aid of a classicaltreatment[5—7]. [3] A.R. Kessel,thesis,University of Kasan(1962);A.R. Kessel,AkustischeLermrespmamz(AkademieVerlag

This work was supportedby theDeutscheForschungs- [4] V.M(iller,J.Phys.E:Sci.Instr. 8 (1975)127. In formulagemeinschaft(SFB 16h).The authorwould like to (2) thetermi(~w~—~c~) mustbe replacedby i

2(~~n

thankProf. Dr. S. Wilking for helpful discussions. &~)andin all relatedformulas, containing&.., ora factor2 mustbe inserted.

15] V. MUller, G. Schanz,E. Fischer,andE.J.Unterhorst,Phys.Stat.Sot. (b), submitted.

References [6] i. Buttet,Solid StateCommun.9 (1971)1129.[71 P.A. Fedders,Phys.Rev. 87 (1973)1739.

[1) R. Kubo, I. Phys.Soc.Jap.12 (1957)570.

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