Volume55A, number6 PHYSICSLETTERS 12 january1976

RELAXATION OF INDIVIDUAL rn-STATES IN FERROMAGNETS: 153Eu IN EuO

D. FEKETE andN. KAPLAN

The RacahInstituteof Physics,TheHebrew University’ of Jerusalem,Israel

and

T.B. REEDLincoln Laboratory,M.I.T., Lexington.Mass.. USA

Received10 November1975

A modelcalculation of nuclearrelaxationin inhomogeneousferromagnetsis extendedto treat individualen m + I transitions.The relaxationof individual t3Eu transitionsin EuO is analisedin termsof themodel,yielding the separateZeemanand quadrupolecontributionsto theline width.

The presentstudy,in which transversenuclearrelaxationof individual m m + 1 transitionsin ferromagnetsis examined,wasmotivatedby the recentobservation[I] that 153Eu(I = 5/2) exhibits resolved5 line n.m.r.multiplet in magnetisedsingle crystalspheresof EuO.We shall extendexisting modelcalculations[2--SIso as totreatindividual transitionsin inhomogeneoussystemsandcomparethe model predictionswith ourexperimental

results.Unlessdefineddirectly,ournotationswill conform with refs. [3—-SIIn ahomogeneousnuclearsystem.the secondmomentcausedby theSuhi-Nakamura(SN) interaction in a

m ~ m + I transitionis [41

Mr(m, Horn.)= F(J, m) ~ B~

~ . (I)

En the presenceof microscopicinhomogeneityg(w), a straightforwardapplicationof ref. [31yields

M~]”~(m,Inhom.)= F(I, m)h~cg(w) ~ V~/~V~ (2)

wherec is theisotopic abundance.For a reasonablecharacterizationof the dynamicline-shape,at leastthe 4thmomentM

4 is also required.Noting that only thenucleartermsF(I, m) -- but not the geometrical~V~1term- changewhen the unsplit line is replacedby an individual transition,we can avoid theextremely tediousM4 cal-culationfor the latter. Instead,usingresultsfrom ref. [3] onecan approximatethe moment ratio for an individualtransition by *

~Mr(m, lnhom.)/( r(m, lnhorn.))

2 hg1(w)~3~ ~ --~~ ~hg~(w) ~ (3)

andusingref. [31 onceagain,we finally obtain

~ (~f3)3I2(~f5)~lI2 [F~I~m)]lI2 ~g(w)[A~]* (4)

* The expression18 — 31(21(1 + 1))l in ref. 131 is incorrect,andshould be replacedby 13 — 31(21(1+ 1))]. Our approximationisequivalentto neglectingthenuclearterm in the squarebrackets.

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Volume 55A, number6 PHYSICSLETTERS 12 January1976

In arealistic model,thecharacteristicwidth ~ of themicroscopicdistributiong(c~.,)is not an entirely free parametersinceboth Zeemanandquadrupolarenergyvariationscan contributeto g(w). In the simplestcaseof independentcontributionswe find

(5)

i.e.g(w) itself becomesa transition dependentfunctiong~(w)andmustbe insertedassuchin eq.(4).A similarsituationholds for the macroscopicbroadeningG(w), andsuchdependenceis indeedverified upon examinationof thestatic 5-lineprofile [1, fig. I]

Anotherpoint to considerin ourmodelis concernedwith the nucleardipolar interaction.As pointedout inref. [5] ,evenin the limit of extremeinhomogeneity,i.e. smallg(w)valueswherethe SN rate(eq.(4)) vanishes,theso called “unlike” If I)~termin the dipolar coupling remainsintact.The resulting“unlike” dipolar 2nd momentfor 153Euin EuO,which is transition independent,can be written as

M~~(Inhom.)= c153(~)2~I(I+ 1) ~ (d,~53)2+ C151(~)2~I(I+ 1) ~ (d/1

5~’53)2. (6)

Assumingtheexistenceof local spin packets[5] throughoutthe sample,with spin in eachpacketdistributedaboutw with a distributiong(w’ w), we note that for w’ = w the SN relaxationis dominant [5] ,while at thetail, where

theSN relaxation(eq.(4)) vanishesbecauseof g, themuchsmallerdipolar coupling (eq.(6)) takesover. Ignoringcrossterms [7] ,the line shapeduetoM~’~is gaussian[5] andthus the relaxationacrossa packetcenteredat

is w-dependent,proportionalto

t2M~(lnhom.) 1exp[_~—~ ~ — ~‘o) 2 S(w — w

0) (7)Am 2h

whereS(w — w0) variesbetween0 to I upon movingfrom thecenterto the tail endof the packet~‘. As afinalconclusion,ourmodel predictsthat theechoamplitudeobservedfor a m ‘~ m + 1 transition observedat a.frequency(0)0 andtime twill be given [5] by

+00 .—t t2M~(Inhom.)

Im(t, w0)ct f Gm(w— w0)g~(w— w0) exP~~~m(w— w0) 2h

2 S(w — w0)]dw

[ t2M~’P(Inhom.) 1

= exp[_ 2h~~ s’J f G~(W w0)g~(w w0) exp[_ -~ g~(w— wo)]dw (8)

whereS’ is somenumberbetween0 and1.The experimentalset.upis similar to that in ref. [1] . A non-conductingcrystalof EuO,labeled“type B” in

ref. [9] , wasexamined.The echodecayat “-134.5 MHz and4.2 K wasrecordedfor H0 valuescorrespondingtoeachmaximumof the5-line multiplet [I, fig. I] . H0 wasappliedalong 1111] to ensuremaximumline separation[1] .The resultsfor 3 transitionsareshownaspoints in fig. 1. Eq.(8)wasthen fitted (e.g.thesolid curvesin fig. 1)to theexperimentalpoints by adjustinggandS. A “best-fit” wasobtainedforg~(~— w0) = exp[—(w w0)

2/2~],

with ~m (eq.(5))givenby 5mag= 17.2 gauss,5q = 10.5 gaussandfor S’ 0.5. Thechoice ofg~provedextremelysensitivein fitting the initial andmiddle partsof thedecaycurves,whereastheparticularchoiceof S’ wasimportantonly in orderto fit the tail-endof thecurves.Very recently,the tail-end of a decaycurvefor 153Euin EuO wastreated,in a morequalitativemanner,by Hohnet al. [10] with similar conclusions.

* Thesignificanceof S is partly treatedby Walstedt[8]. A detaileddiscussionis outside thescopeof thepresentletter.

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Volume 55A, number6 PHYSICSLETTERS 12 January1976

sot ~ RELAXAT 105 0F INDIV DUAL M 55

THEORY

200 - • EXPERIMENTS /2-5/2

jioo — —

1/2.- 3/250 -

00

I

20 1/2 ~- 1/2

~ 10 -

I

5

1 I I _______ I0 200 400 600 800 1000 1200

2t (psec)

Fig. 1. Experimentaland theoretical‘53Eu spin echodecaycurvesfor individual m ~ m+ 1 transitions.

The agreementbetweenourmodel predictionsandtheobservation,whichwasobtainedfor all 5 transitions,forvariousH

0 valuesand for all t valuesby adjustingessentiallyonly two parameters,6magand~q, seemsto demonstrate

theusefullnessof the techniquein studyinglocal environmentswithin ferromagnets.In particular,we may concludethat thepresentrelatively puresampleexhibits adistinctly different typeof local inhomogeneitycomparedwiththepowdersampleof a presumablymetal.richEuOin whicha lorentiang(w) with ~ = 6 gausswasreported[51.Sinceno separationbetween and waspossiblein thelatter, quantitativecomparisonsbetweenthe twosampleswerenot feasible.However,from thepresentstudy it appearsthat onemay gain valuableinformation bystudyingthe relaxationsin varioustypesof single crystalsamples,i.e. insulating,stoichiometricandmetallic,ofEuO,

We wish to thankF. Waldnerfor avery usefulcommentconcerningthepresentstudy.

References

Ill D. Fekete,N. KaplanandT.B. Reed,So!. State.Comm. 15 (1974) 1827.[2] H. Suhi,J. Phys.Rad.20(1959)333;

T. Nakamura,Prog.Theor. Phys.20 (1958) 547.[31 D. Hone,V..Jaccarino,T. Ngwe andP. Pincus,Phys.Rev. 186 (1969)291.[41 H. Yasuoka,T. Ngwe andV. Jaccarino,Phys.Rev. 177 (1969)677.15] J. Barak,I. Siegelstein,A. GabaiandN. Kaplan, Phys.Rev.B8 (1973) 5282.[61 J. Barak,unpublishedPh. D. thesis.[7] N. Kaplan,P. PincusandV. Jaccarino,J. Applied Phys.37 (1966) 1239.[81 R.E. Walstedt,Phys.Rev. B5 (1972)41.[9] M.R. Oliver, J.O.Dimmock, A.L. McWhorterandTB. Reed,Phys.Rev. B5 (1972) 1078.

[101 H.G. Bohn,R.R. Arons, H. LUtgemeierandK.J.Fischer(preprint).

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