PHYSICAL REVIEW B VOLUME 10, NUMBER 3 1 AUGUST 1974

Electron-paramagnetic-resonance investigation of the dynamic Jahn-Teller effect for Sc'+ in

BaF„SrF„andCaF,

J. R. Herrington~~

1nstitut fur Angewandte Festkorperphysik der Fraunhofer-Gesellschaft, 78 Freiburg im Breisgau, BundesrepublikDeutschland, West Germany

L. A. BoatnerAdvanced Technology Center, Incorporated, Dallas, Texas 75222

T. J. Aton*~ and T. L. Estle*Physics Department, Rice University, Houston, Texas 77001

{Received 25 February 1974)

The EPR spectrum of Sc'+ has been observed in single crystals of BaF„SrF„andCaF, atliquid-helium temperatures. At 1.2 K, the spectra were characterized by intense anisotropic hyperfine

patterns with partially resolved ligand hyperfine structure. The anisotropy, line shapes, and temperaturedependence of the anisotropic spectrum obtained for each host crystal were described within experimental

error by second-order solutions of an e6'ective Hamiltonian for an isolated vibronic 'F. state which is

split by large random internal strains. Coexisting with the anisotropic pattern was a weak nearly

isotropic hyperfine pattern with "conventional" line shapes. No ligand hyperfine structure was resolved

on this pattern. Intensity variations as a function of temperature imply that the nearly isotropicstructure results from averaging by rapid vibronic relaxation of a portion of the anisotropic pattern.This interpretation is further strengthened by the observation of a small predicted anisotropy. Observeddeviations of effective Hamiltonian parameters from the values predicted by crystal-field theory implythe existence of a weak-to-moderate vibronic interaction for these systems, i.e., a dynamic Jahn-Tellereffect. For d'-configuration ions in cubic symmetry, the effective Hamiltonian parameters aresummarized and discussed.

I. INTRODUCTION

The Jahn-Teller theorem' states that a symmet-ric nonlinear polyatomic complex in an orbitallydegenerate electronic state is unstable with re-spect to at least one asymmetric distortion whichremoves the orbital degeneracy. For ions in sol-ids, the fundamental mechanism for the Jahn- Tel-ler effect is the Coulomb interaction which couplesthe orbitally degenerate electronic states of thecomplex to 1.attice vibrations. '

The early lorn-temperature electron-paramag-netic-resonance (EPR) data for ions with orbital-doublet ground states were, in general, consistentwith the concept of a static Jahn-Teller effect. a'

Later, however, results were obtained at low tem-peratures for Cua' in MgO and CaO and forSc ' in CaF~ and SrF~ ' which mere not character-istic of the static Jahn-Teller effect. These datawere initially interpreted using a strong vibroniccoupling model mith tunneling through potentialbarriers separating the "static" Jahn-Teller con-figurations. Subsequently, Hams'3 proposed anem interpretation of the EPR spectra observedfor the CaF2: Sc ' and SrF2. Sc ' systems. He

demonstrated that these data did not necessarilyimply the large Jahn-Teller stabilization energiespreviously assumed, and that, in fact, the experi-mental parameters describing the Sc~' EPR dataimplied weak vibronic coupling.

Somewhat later Chase, ' in his study of theEPR spectrum observed from an excited orbitaldoublet of Eu ' in CaF2, found that when the vibron-ic coupling and random strain were sufficientlystrong, interaction with an excited vibronic singletcould selectively broaden certain features of theEPR spectrum. Chase suggested' that this selec-tive broadening might play a role in the CaF2: Sc 'spectrum.

In our EPR investigations ' of La ', Y ', andSc ' in SrClz, we found that the spectra observedfor these systems were adequately described byan isolated vibronic doublet formalism which im-plied weak-to-moderate vibronic coupling. ' Inview of the uncertainty in previous interpreta-tions ~ ~ ' ~' of the Sc + EPR data, the new theo-retical insights provided by Ham ' and Chase,and the experience gained from our earlier EPRexperiments, ' ' a decision was made to reinves-tigate Sc ' in CaFz and Sr' and, in addition, toinvestigate Sc ' in BaF&. These data permit an ex-tension of our systematic study ' of the dynamicJahn-Teller effect for divalent ions with d config-urations in eightfold coordination and extend theresults given in a preliminary report.

II. THEORY

The weak-to-moderate vibronic coupling modeland the associated effective Hamiltonian formalism

10 833

HE RRINGTON, BOAT NE R, ATON, AND EST LE 10

developed by Ham~'3 describe the line shapes,angular dependence, and temperature dependenceof the EPR spectrum for an isolated vibronic dou-blet. Ham ' has also given expressions for es-timating the Jahn-Teller stabilization energy andhence the strength of the vibronic interaction andhas demonstrated the importance of random inter-nal strains.

For completeness, the effective Hamiltonian foran isolated E state will be reviewed here, but thereader is referred to earlier papers3'3' 6 for amore extensive treatment.

An effective Hamiltonian representing the ran-dom strain, Zeeman, hyperfine and quadrupole'interactions for a E state in cubic symmetry isgiven in Eq. (1):

H = q V {e )g + e,h, ) +gyp 8 ~ S+q +A. &I ~ SCq

+ 2qg21f. 2[{3—H, S, —H S)$2+ ~3{H+„-H„S„}$,]

+ 2qqA2[(SI+, - I ~ S)$2+ v 3 (I,S,- I„S~)Sq]

+ 2qQ[(SI, I, —I ~ I)g2+ v3 (I„I„-I,I„)gq]. (1)

Here, 8&, S~, and 8, are vibronic operators,

symmetrized strain components, H is the magneticfield, S is the electronic spin operator, and I isthe nuclear spin operator. The parameters, q V„g&p, ~, Az, 2qgap, z, ~@AD and ~qQ represent therelative strengths of the various terms, and thereduction factor ' q is included explicitly in theappropriate parameters.

When the strain interaction determines the com-position of the vibronic states, 3'3'7 ' the EPRtransitions occur within the two resultant Kramersdoublets. These EPR transition frequencies aregiven to second order in Eq. (2}:

}2

ttf f) ~ qtf -qf-)'t"')))I4(qQ)'A~

tA, q-,'qAQ, )' tqA )')t2)gslfeH gq )f s,H

(qA2)' f,M2, 8(qQ) (2f. f.)M3-where the two signs differentiate between thestrain-produced Kramers doublets. The functionsf» f2, and f2 are given in Eqs. (3)-(5):

f~= (3n —1)cosP+ M3(l —m )sintt),

f2= —,'(2- [(Sn2 —1)cosqf + v3 (l —m2) sintt)]

—[- (3n —1) cos2$ + a 3 {l —m )sin2$]),

f2=/ (4+ [(3n —1)cos)f)+ v 3 {l2—m2) sing]

+4[- (3n —1)cos2$ + )) 3 (12- m2) sin2$]] . (5)

Here, I, m, and n are the direction cosines of themagnetic field with respect to the (100) axes, and

tanqt) = e,/e2.When the composition of the vibronic states is

determined by the anisotropic parts of the Zeemanand hyperfine interactions and not the strain inter-action, then the frequencies for the two most-in-tense transitions are also given by Eq. (2), butwith the following definitions of f&, f„andf& whichapply only in (110}planes:

f, = 1 —3n (6)

f2= —', (1-n )(3n +1),f4= 2n'+42(I -n')'.

As before, n is the cosine of the angle between themagnetic field and the (100) axis in the particular(110]plane. The two signs in Eq. (2) now refer totransitions within the Kramers doublets producedby the anisotropic terms in the effective Hamilto-nian.

The inclusion of a weak isotropic ligand hyper-fine interaction in the effective Hamiltonian resultsin a splitting of each hyperfine transition into asymmetric series of transitions. This situation isillustrated in Fig. 1 for the MI = —-', hyperfine com-ponents. Assuming that all values of the strainangle ft) occur with equal probability, neglectingall second-order terms, and not including ligandhyperfine structure, the microwave absorptionpredicted by Eq. (2) is shown in Figs. 1(a) andl(b). The effect of a weak isotropic hyperf inc in-teraction with eight ligand nuclei of spin —,

' isshown schematically in Figs. 1(c) and 1(d). Eachof the hyperfine components such as those in Figs.l(a) and 1{b)is split into nine lines whose relativeintensities are given by the ratios1:8:28: 56: 70: 56:28:8:1. This results in nineoverlapping envelopes with the above relative in-tensities. A first derivative of the envelopes ofFigs. 1(c) and 1(d) will have partially resolved li-gand hyperfine structure near each of the two ex-tremes of the envelopes. However, it is evidentthat the ligand hyperfine structure patterns dependon the exact shape of the original hyperfine enve-lopes.

In second order, the line shape of the individualhyperfine components becomes asymmetric. "'For H II (100), this asymmetry is produced by thoseterms in Eq. (2) which are multiplied by f2 and bythe (qg2f, /g&) term in the expansion of I/(gzs-,'qg2f&), although these terms are small. Formagnetic field orientations away from H II (100),the (qA2) f2/Aq term will produce an effect larger

10 E PR INVESTIGATION OF THE DYNAMIC JAHN- TE LLER.

SECOIIN-OROER EFFECTS ON THE RESOLUTION OF LI SANO

HYPERFINE STRUCTURE

CaF: Sc22'v $ v

BaF2 Scv f v

(O) (b)

I I I I! I I I I

I I I I I I! 11! I I I I

I I I I I I I ~

te 4I' & ~ 65'

MAGNE T I C F IELO

FIG. 1. Shown in this schematic representation of themost extreme effects of second-order terms in Eq. (2)on the appearance of ligand hyperfine structure are: theline shapes predicted in first-order for the M&= —2 hy-perfine component of the EPB spectrum, Hll (110},for(a) CaF&. Sc ' and (b) BaF2. 8c '; the line shape predictedin first order including an isotropic ligand hyperfine in-teraction for the Mz=- 2 hyperfine component, H II (110},for (c) CaF2..Sc ' and {d) BaF2. Sc ', and the line shapepredicted in second-order for the M&= —

& hyperfine com-ponent, Hll (110},for (e) CaF2. Sc ' and {f) BaF2

.. Sc '.The microwave absorption is plotted versus magneticfield with the individual transitions represented as &

functions. The solid diamonds represent the field posi-tions for Q = + $!!, + $w. The solid triangles representthe field positions for @=0, + 7I. Horizontal scales in(c) and {d) represent locations of the ligand hyperfinepattern at the two extremes of the absorption. The dis-placements of the diamond and triangle symbols in (e)and {f) indicated by horizontal lines show the differencebetween the prediction in first-order and those in second-order. The values of the strain angular parameter ft)

locating the low-field extremes of the absorption in (e)and {f) are shown. .

than the other second-order terms. For Sc ' inSr Cia, this (ques) fs/At term produced large effectson the line shapes.

To illustrate the most extreme influence of sec-ond-order effects on the ligand hyperfine structure„the line shapes (without ligand hyperfine structure)of the M&= —-', hyperfine components are shown inFigs. 1(e) and 1(f). ¹ne overlapping envelopesof the type shown in Figs. 1(e) and 1(f) would

greatly complicate the ligand hyperfine spectrumbecause of the three-peaked character of the ab-sorption. The solid triangles locate the positionsof the EPR transitions for (t) =0, +g, and the soliddiamonds locate the EPR transitions for (t) =

~2

The shifts produced by second-order termsfor P = 0, am and P = +-,'m are represented by thehorizontal displacements shown in Figs. 1(e) and

1(f}. The value of P for the low-field extreme isalso given. Second-order effects are much lessdrastic for other orientations and other hyperfinecomponents.

Rapid vibronic relaxation (i. e. , rhM s = 0) be-tween the two strain-produced Kramers doubletscan result in microwave absorption midway be-tween the positions of the two EPR transitionsfrom the two doublets. ' The intensities of these"average" transitions depend, in general, on theorientation of the applied magnetic field. ' Fora given magnetic field orientation, variations inintensities with Mz are also observed. ' In addi-tion, it has been shown that certain second-orderterms in Etl. (2) can produce small characteristicanisotropies for these transitions.

For ligand hyperfine structure to be resolvedon the averaged lines the following conditions musthold: (i) the linewidth of the individual EPR tran-sitions composing the envelopes must be smallerthan the spacing between adjacent ligand hyperfinelines {at high temperatures this linewidth may re-sult from spin-lattice relaxation}, and (ii) the vi-bronic relaxation rate must be much greater than2g4v where 4v is the maximum difference in fre-quency for the transitions resulting from the twostrain-produced doublets.

III. EXPERIMENTAL RESULTS AND DISCUSSION

Commercially available CaF&, SrF2, and BaF~single crystals were doped with Sc using the dif-fusion technique described by Hoehli. The diva-lent charge state of Sc was produced by either xor y irradiation at room temperature. EPR spec-tra were recorded using a back-reflection {Xband,9 GHz) spectrometer with backward-diode detec-tion. The microwave frequency was measuredelectronically, and magnetic field measurementswere made with an NMR gaussmeter. Precisionorientation of the magnetic field relative to thecrystallographic axes was achieved using indepen-dent orthogonal rotations of the sample and mag-net.

A. Low-temperature anisotropic spectra

The EPR spectra observed at 1.2 K from irra-diated CaFz'. Sc, Sr Fz.' Sc, and BaF3. Sc singlecrystals are shown in Figs. 2-4, respectively,for the applied magnetic field parallel to the threeprincipal crystallographic directions, i. e. , (100),(ill), and (110). For a genera. l direction of theapplied magnetic field, each EPR spectrum con-sisted of eight anisotropic hyperfine componentswhose line shapes resulted from large random in-ternal strains. The extremes of these line shapes

HERRINGTON, BOA TNE R, ATQN, AND 10

ANGULAR DEPENDENCE

l, 2 K

i = 8.9 GHz

CaF2. Sc2+

l

a l. Il,I ''I

I

I

I

t

l l

I

Il

Il

II

GAIN X 1(

I I'( II

l

I

Xl

s I .l I .( II" I " I ' I "I

MAGNETIC FIELD—

II

1

I

H ll &lao&

a.

FIG. 2. EPR spectrumobserved (v = 8. 9 GHz) at1.2 K for a reduced CaF& ..Sc2' sample. The first-derivative presentation ofabsorption is shown versusmagnetic field for: (a)Hll (100), (b) Hll (111), and

(c) H ll (110}. The extremesof the anisotropic hyperfinecomponents are designatedby broad vertical bars.The dashed vertical lineslocate the positions of theeight hyperfine lines ob-served for H ll (111). Thepositions of the eight fea-tures produced by averag-ing by rapid vibronic re-laxation are indicated in4,) and (c) by small verti-cal lines

were described, in first order, by the cubic anisot-ropy function [1 —3(l m +m n +n l )]2 2 2 2 22]1/2 The importance of second-order terms is indica e yt db the

II ~111observation that the line positions for H II ~

{indicated by dashed vertical lines in Figs. 2-4)differ from the average of the two extremes for thecorresponding hyperfine components for Hli (100)or H II (110). In Fig. 2, the average field positionsof the extremes of the eight hyperfine componentsfor H II (100) and H II (110) are indicated by narrowvertical bars on the horizontal scales below thetraces. The extremes of the components in Figs.2-4 are indicated by broad vertical bars on thehorizontal scales below the traces.

Ligand hyperfine structure was partially re-solved near the extremes of each hyperfine com-ponent although it is better resolved on one of theextremes. For HII (ill) [see Figs. 2(b), 3(b, and4(b)] as many as nine separate ligand hyperfinelines were resolved with relative intensities whichare given approximately by the1:8:28: 56: VO: 56 28:8:1 ratios expected for aligand hyperfine interaction with eight equivalentnuclei of spin ~. In the fluorite structure, tligands for a substitutional ion are not all magnet-ically equivalent for H II (111), but the data, suggestthat the anisotropy in the ligand hyperfine inter-action may be neglected for the purposes of this

work. As the direction of the magnetic field wasvaried, the ligand hyperfine splittings were ob-served to be approximately constant, but the rela-tive intensities of the ligand hyperfine lines did

ary The observed ligand hyperf ine splittingsle I.~ for the three systems are listed in TableLH FS

aF:Sc 'The decrease in AL»s in the order a 2. cr 2.' c, anS F 'S ' and BaF:Sc ' is expected since the2'

host lattice constant (and hence the nearest-neigh-bor distance for these hosts) increases in thatol der.

The angular dependence of the central line ofthe ligand hyperfine structure at each extreme (oran estimate of its position when ligand hyperfinestructure was not resolved) is shown in Figs. 5-7for CaF2:Sc ' SrF Sc2' and BaF2. Sc ', respec-tively. The measured field positions are repre-sented by open circles. The solid curves in eachf re computed using the second-order solu-igure ation of the effective Hamiltonian for an isola eE state in the large random strain approximation,i. e. , Eq. (2), and the appropriate parameters list-ed in Table I. The error limits listed in Tablewere chosen to reflect the precision with whichthe extremes having no resolved ligand hyperfinestructure could be measured.

From Figs. 2{b), 3(b), and 4(b), it is apparentthat the linewidths of different hyperfine lines for

10 E PR INVE STIGATION OF THE DYNAMIC JAHN- TE LLER ~ . .

aweeiae eeaieawca1.2 N

p .e.ls 811'

r- 8—r- -~r

~ I I II I l

Maeef. TIC flfle

FIG. 3. EPR spectrum observed (~=8.8 GHx) at1.2 K for a. reduced SrF2 .. Sc2' sample. The first-deriva-tive presentation. of absorption is shown versus magneticfield for: {a) HlI (100), (b) Hll (111), and (c}HII (110).The vertical bars locate the extremes of the anisotropichyperfine components. The dashed vertical lines locatethe eight hyperfine lines observed for H ll (111).

the same system are unequal for H I1 (111). Al-though this could result from mosaic structure, 'it may be the result of a variation in the magnitudeof the ratio, q V, (ez+e, ) /(qg~sH+qkgfz). Ifall complexes do not satisfy the large randomstrain condition, then second order terms will re-

suit in a distribution of line positions. This inten-sity variation has been observed in SrCl3

..La,SrCl~'. Y, SrC1~. 8c ', CaF2. 8c ', SrF2.'Sc ', and

BaF~:Sc '.For H ti (111), seven additional weak features

were observed at 1.2 K. These additional linesoccurred approximately halfway between adjacenthyperfine lines. Their positions and intensitiesare characteristic of bM& = +1, AMi= +1 transi-tions and, as previously reported, ' imply the ex-istence of a weak nuclear electric quadrupole in-teraction for the isolated E state in cubic symme-try.

B. Temperature dependence between 1.2 and 4.2 K

The temperature dependences of the EPR spec-tra observed for CaF&. Sc, Sr'. Sc ', and

BaF&.Sc ' are shown in Figs. 8-10, respectively.As seen in these figures, a weak eight-line patternwas observed to coexist with the anisotropic pat-tern discussed above. These lines are located inFigs. 8-10 by dashed vertical lines, and they wereobserved to increase in apparent intensity (thepeak-to-peak amplitude in a first-derivative pre-sentation) as the applied magnetic field was rotatedin a (110jplane toward a (111)axis. The intensityof each individual line depended on its relative po-sition in the eight-line pattern. As the tempera-ture was increasedfrom l. 2to4. 2 K, the intensitiesof each of these lines increased approximately lin-early with temperature, and this increase was ac-companied by the gradual broadening of the ligand

ANGULAR OEPENDENCE

1.2 K

i =8.9 GHz

I

I I

I I

lI

IGAtN X 2

I ~ i I + I . 1 I1 II I I I I

II II II

I

I

BaF:Sc

Hll &Ioo&

a.

Hll &IIo&

FIG. 4. EPR spectrumobserved {v=8.9 GHz) at1.2 K for a reduced BaF2.Sc ' sample. The first-derivative presentation ofabsorption is shown versusmagnetic field for: (a)Hll (100), (b) Hll (L11}, and

(c) Hl! $10). The verticalbars locate the extremesof the anisotropic hyper-fine components. Thedashed vertical lines locatethe eight hyperfine linesobserved for Hll (111).

I

I I eI

I I

I

iGAlN X 2I,' I,' Ie

' I ' l I

MAGNETIC. F IE LD—'* 506

838 ~RRyN GTON BOAT NE R

mj] toni

ATPN

oman parameters i

AND E ST LE

qQ(10+ cm ')

o 12{8)b,~—O. 12(5)b'—O. 12(5)b"—O. 2O(5)"

A)(1O-' cm-')

—es. 3(8)~—ev. 5(5)'—e5. s(5)'—e2. v(s)'+24. 3{8)'—11O(2)b'—119.5(S)~

qA&

{1O"cm-')

—24. 5(S)'—24. 3(5}'—24. 1(5)'—31.5(5)'

+ 5. o(s)'

LHPS

(G)Vg2

—o. 03o9(s)'—0. 0240(5)—0. 0211(5)—0. 0442 (8)—o. o4vs(s)'

~ ~ ~

—o. oesv(s)'

3.8(5)'4. v(3)'5. 0(S)'

BaF "Sc"SrF, :"Sc"CaF, :"Sc'+SrCl, :"Sc"SrC12. Y 'SrCl . La'SrC12. ~39La '

1.9555(8)1.9e4o {5)'1, 9719{5)1.9530.)'1.929(1)'

~ ~ ~

1.881{1) +0. 15(5) '—18.8{8)

TABLE I. Effectiect in alkaline-ea

1 ltl

the last di

1' it

Host: Ion

parameter.e in parentheses

10

This paper. bReference 18. cRReference 19 Reference 1V

hyperfine structupic pattern. At ttemperature

em domines near '77

o the aniso-

anisotro iinates the EPR

opic pattern haspectrum

as broadened barac ter istic

ne beyond dthe ori in

tyon 18 hat the positiostent

positions of these

lines foror H II (100) diffH II (ill). Th ese deviations

yperf' e spl'tt ngs, are ll

the weakSc ' h th poe corn

zeal

1 s or H II (111). Th'f/

ANGULAR DEPENOIENCE: 2: c:CaF: Sc2+

3600 3500

3500 3400Il

e 3400—

IdC

4.

3300—

F

X 3200—

3300—LU

O 32004JKC9C

3100—

C——

3(00— 3000—

3000—I

Oa

[Ioo]

I I I i I

75~ 904[o ii]

2900—

I

Ll ooj

I I I I I I

60'[iii] [0 I i]

FIG. 5. A ngular de ett obmsotropic

pendence of thdf C

re experimenr aF2 Sc2» opr and the solid

Sc ' and E

bility.e /we re assumed

v ues oof the strain.sn angu-equal proba-

FIG. 6. Angular de eru

* att obru sotropicpendence of th

re experimenr rF2. Sc2'.

ental measup g

nd the solid

q. (2), All 1p rs in Table I

ability.wj. equal prob-

10 F THF DPN AMIC JAHN- TE LLER ~ATION 0 839

5700

3600

ANGULAR OEPENOENCE: 8|IF2:Sc + is relatively large for the S y

NS OF THE WEAK-TO-MODERATEIV. IMPLICATIONS OFVIBRONIC COUPLING

5500

~ 5400—LLI

U.

I-~~ ~SOO-C5

x

3200 —c—

5I 00 —c

5000—

I I

[I I I] LO I IjI lf001

endence of the extremes for the" .h"-rn observed for BaF2. Sc . eanisotropic pattern o sents and the solidex erimental measuremen. s

bluted using the param(2) All 1

ves were computed g h am

to o th 1 ob-lar parame ert P were assumed to occur wability.

s ectra reported here are character-n isolated F. state in cu ic

db'"-".-.i." -y.—state is predicte y cod

th o d bo'dination. Ham has s own

r classification of e grthe symmet ytronic E state wi

'th weak-to-

u lin is alsovidence for the existence o a

od atevb o o p' g'

th b tibronic interaction is e oto-moderate vieters are reducedn ex erimental parame erstha cert' p

s redicted in e a. , by conven zonabronic coupling, x. e. ,

f this section toeor . It is the purpose o ifield theory.compare ethe observed reductions inters mea sured for Sc ' in a 3, z,

e im lied strengths of the vi-and to and&cate that the impheou li are p

s resented ' this section areu h and are presented mereextremely rough an

imental data andconsistency between the experimen a ab '

coupling mod-'s weak-to-moderate v~brom. c coHaml re orted ' for1 ' The data previous y r pel.

and SrC13. Sc are in-SrClp,'La ', SrC13. , afor comparison purposes.

r stal-field theory predicts theConventional crystal-fxeo ow' - x ressions for efollowing first-order exp

I II tI I

TEMPERATURE DEPENDENCE

H is IQ' from &)00& m I I l 0 l

90H

I

IQAIN X 2. 5

GAIN X 2.5

Pl—--'-'+- ~-

MAGNETlC FlEl 0—

2+CaF2, Sc

4.2 K

I.6 K

C.

l. 50G

FIG. 8. Temperaturedependence of the EPRspectrum observed from areduced CaF2 -. Sc~' samplefor (a) T = 4. 2 K, (b) T= 2. 7 K, and (c) T = 1.6 K.The dashed vertical lineslocate the eight lines dueto the averaging of a por-tion of the anisotropic pat-tern produced by rapid vi-bronic relaxation. The ex-tremes of the componentsof the anisotropic patternare designated by verticalbars. eTh first derivativeof absorption is shownversus magnetic field forthe magnetic field oriented10' from a (100) axis zn

a {110jplane.

840 HE RRINGTCN, BOATNER, ATON, AND EST I.E

TEMPERATURE DEPENDENCE

HIl 400& s F,:s '

(c)

tI~ f 5

MAGNETlC FlELD

l. 2 K

I~ 5 ~

:=406

ters, g„gz,A&, A3 and Q for a d'-configurationion in eightfold cubic coordination:

g~ = 2. 0023 —4X/6,

g2= —4X/6 =gi —2. 0023,

A, = (- 2pep (r ')/I)(K+ 4X/n),

A, = (- 2p.,g (r-')/I) ($+~7~/a),

2e Qg 3VI(2I —1)

(9)

(10)

(11)

(12)

(13)

The parameters X, b, , pe, p, I, (r ~), K, e, and

Q„arerespectively the spin-orbit coupling con-stant, the cubic-crystal-field splitting 10Dq, theBohr magneton, the nuclear magnetic dipole mo-ment, the nuclear spin, the one-electron expecta-

FIG. 9. Temperature dependence of a portion of theEPH spectrum observed from a reduced SrF&. Sc ' sam-ple for: (a) 4. 2 K, (b) 1.6 K, and (c) 1.2 K. The firstderivative of absorption is shown versus magnetic fieldfor the magnetic field aligned parallel to a (100) axis.The dashed vertical lines locate the components of the"averaged" pattern while the solid vertical lines denotethe positions of the hyperfine lines observed for H tI (111).The extremes of the components of the anisotropic pat-tern are designated by vertical bars.

tion value of r, the Fermi contact parameter, theelectron charge, and the nuclear-electric quadru-pole moment.

The reduction factor q, included explicitly in theparameters, qg» qA» and qQ in Table I, may beestimated from the experimental parameters, g,and qg2. The reduction factors for the varioussystems are listed in Table Q. They are relatedto the strength of the vibronic interaction andhence the Jahn- Teller stabilization energy F».In particular a decrease in q from unity toward —,

'corresponds to an increase in the ratio E»/hd.Estimates of the ratio, Ezr/If+, were made andare listed in Table G. Using the value 300 cm 'for Id (the approximate average optical-phononenergy for the alkaline-earth halides), the esti-mates of the vibronic coupling strength parameterV~ listed in Table D were obtained.

The relatively small variations in q (and hencein Ezr and Vzr) implied by the E PH data for theSc ' ions in the fluoride hosts and the lack of adefinite correlation between these reduction fac-tors and the fluoride lattice constants, indicatethat the vibronic interaction itself is essentiallyindependent of the nearest-neighbor spacing forthese systems. However, the appreciable differ-ence between q for Sc ' in SrClz and the q's forSc in the fluoride hosts implies that though thevibronic interaction does not depend strongly onthelattice constant of the host, it does depend on theelectron affinity of the anions.

Dividing the parameters qgz, qA~, and qQ listedin Table I by the appropriate reduction factor q,one obtains estimates for the parameters g3, A»and Q. These estimates, together with the pa-rameters gj and A, given in Table I, can be usedin Eqs. (9)-(13) to calculate the values for z,(r ), p, and Q„given in Table Il. The system-atic variations in these parameters are physicallyreasonable, and the predicted values for the nu-clear moments agree remarkably well with cur-rently accepted values which are shown in paren-theses in Table II.

The measured EPR parameters reported hereand elsewhere' ' for SrClz ..Se ', CaF2. ScSrF2:Sc, BaFz.'Sc ', SrC13.Y ', and SrCl~: Ia 'are all consistent with the following model, whichis illustrated schematically in Fig. 11: (1) TheD free-ion term of the nd configuration in eight-

fold coordination is split by the eubie-crystal-fieldinteraction into a ground F. electronic state and aTz excited electronic state separated by energies

on the order of 10 cm . (2) The ground vibronicstate resulting from a weak-to-moderate linearvibronic interaction (Err/II&o =0.05 to 1.0) is a Estate with the first excited state on the order of10 cm above the ground state. (3) Owing tolarge random internal strains (on the order of

10 EPR INVESTIGATION QF THE DYNAMIC JAHN- TE LLER. . . 841

TEMPERATURE DEPFNDENCE

sI

MAGNETIC FIELD—

l ' l~—l =50G

3.0 K

b.

C.

FIG. 10. Temperaturedependence of the EPRspectrum observed from areduced BaF&. Sc ' samplefor: (a) T=4. 2 K, (b) T=3.0 K, and (c) T=1.2 K.The first derivative ofabsorption is shown versusmagnetic field for the mag-netic field aligned parallelto a (100) axis. Thedashed vertical lines lo-cate the components of the"averaged" pattern whilethe extremes of the compo-nents of the anisotropicpattern are designated byvertical bars. Note thedifference in. gain used fortrace (c).

10 to 10 ), the E vibronic state is split into twoKramers doublets which are separated by about1 cm . This strain splitting varies from ion siteto ion site because of a distribution in strain mag-nitude. (4) The Kramers degeneracy is removedby the applied magnetic field, and the magnitudeof the Zeeman splitting depends on the ratio of thestrain components, ez and e, and on the orientationof the magnetic field. (6) The microwave absorp-tion consists of complex envelopes of transitions(indicated in Fig. 11 by solid arrows). The shapesof these envelopes depend on the distribution ofthe strain angular parameter, $, and all values of

this quantity occur with equal probability (6) . Ow-ing to rapid vibronic relaxation (indicated in Fig.11 by wavy arrows), a portion of these complexenvelopes is averaged.

V. SUMMARY AND CONCLUSIONS

EPR spectra have been observed for BaF3.ScSr F3.Sc, and CaF2. Sc2'. The spectra observedfrom CaFz.'Sc and Sr'.'Sc are similar to thosepreviously reported by Hochli and Estle, ' andHochli. Each EPR spectrum consisted of threedifferent patterns: (a) an intense eight-componentanisotropic pattern, (b) a weak eight-line "nea, rly"

TABLE II. Summary of parameters calculated from the experimental effective Hamiltonian parameters in Table Iusing the weak-to-moderate vibronic coupling model. Numbers shown in parentheses are the currently accepted valueswhich are to be compared with the calculated values.

Host . Defect

CaF, :Sc'+

SrF2. Sc 'BaF, :Sc'SrC12. Sc2'SrC12. Y 'SrCl&. La '

(cm-')

79797979

300640

gb(cm-'} q'

10 000 0.698 000 0. 637 000 0. 666 000 0. 86

15 000 0. 6520 000 0. 57

z»/a '0.40. 50. 50.10. 51.0

&»'(cm-')

4000 1.15000 1.15000 1.12000 1.15000 2. 17000 2. 4

(&-3 )(10 G /erg)

0. 50. 50. 50. 50. 60. 7

p (nuclearmag netons)

+ 5(+4.75)++5(+4.75)+5(+4.75)+5(+4.75)

—0. 2 (- 0. 14)+2. 5(+2.76)

@N

(b)

—0.30(- 0.22)—0.30(- 0 ~ 22)—O. 30(- 0.22)—O. 30(- 0.22)

+ 0. 30 (+ 0.20)

Computed from the &5g~- D3y2 free-ion. spin-orbit splitting listed in. NBS Circ. No. 467.Computed from g& in Table I and the free-ion & using Eq. (9).

'Computed from g, and g2 in Table I using Eqs. (9) and (10).Extrapolated from Fig. 5 of Ref. 3.Computed from E~/%co assuming Kcu =300 cm ~ using Fq. 2. 3.3 of Ref, 3.

HERRING TON, BOATNER, ATON, AND EST LE 10

Free- ionGround Term

Crystol fieldSplitting

2Tp

Vibronic

Coupling

2E

Strgln

SplittingZeemon

Split ting

FIG. 11. Schematic energy-level diagram implied by the in-terpretation of the EPR param-eters. The D free-ion term issplit by the cubic crystal field intoE and T2 states with the E state

lowest. A weak-to-moderatevibronic interaction results in aseries of vibronic states and theground vibronic state is also a Estate. The first excited vibronicstate is sufficiently far from theground state to enable one to treatthe ground state as an isolated Estate. This state is split by ran-dom internal strains into twoKramers doublets. The Kramersdegeneracy is removed by an ap-plied magnetic field and EPHtransitions (represented bystraight arrows) are induced bythe microwave magnetic field.The wavy arrows represent vi"bronic relaxation processes whichcan produce an averaging of a por-tion of the anisotropic EPH, pattern.This type of relaxation should notbe confused with spin-lattice re-laxation.

isotropic pattern, and (c) a very weak seven-com-ponent pattern observed only for H!t (111).

The line shapes, anisotropy, and temperaturedependence of the strong eight-component aniso-tropic pattern mere described within experimentalerror by second-order solutions of an effectiveHamiltonian for an isolated E state. Random in-ternal strains mere dominant in determining thecomposition of the vibronic states. I igand hyper-fine structure was observed with eight F nucleiof spin —,'. Distortions of the line shapes of thehyperfine components produced by second-orderterms in the solution of the effective Hamiltonian

may influence the resolution of the ligand hyper-fine structure. The analysis and the experimentalparameters describing the spectra were found tobe consistent with the suggestion by Ham ' thatthe vibronic coupling was not sufficiently strong tonecessitate the use of the tunneling model.

Weak, nearly isotropic hyperfine lines were ob-served to coexist with the intense anisotropic pat-tern at low temperatures. Variations in the inten-sities of these weak lines as a function of tempera-ture, the observation of a small predicted anisot-ropy in their field positions, and the total lack ofresolved ligand hyperfine structure, imply thatthey result from averaging of a portion of the an-isotropic pattern by vibronic relaxation.

The seven very weak lines observed for 8!!(111)have been discussed previously and were shownto be ~~ =+ 1, ddMI = + 1 transitions. Variationsin the positions and intensities of these lines im-ply the existence of a nuclear electric quadrupoleinteraction for cubic-symmetry defects exhibitinga dynamic Jahn- Teller effect.

In conclusion; the principal features of the EPRspectrum due to Sc ' in CaF3, SrFz, and BaF3 maybe described within experimental error by an iso-lated vibronic doublet. Observed reductions ofcertain parameters from the values predicted byconventional crystal-field theory imply the exis-tence of a weak-to-moderate vibronic interaction,i. e. , a dynamic Jahn-Teller effect.

ACKNOWLEDGMENTS

We gratefully acknowledge the valuable technicalassistance of E. G. Clardy and the useful com-ments of Dr. R. W. Reynolds, Dr. B. Dischler,G. G. Setser, and R. Micr-Maza on the manu-script. In addition, we thank the office of NavalResearch for the contribution of helium used inthe initial stages of this work. Also, we wouldlike to thank Dr. R. J. Shalek and the M. D. An-derson Hospital of the University of Texas for per-mitting our use of their ' Cs source for sampleir rad iation.

EPR INVESTIGATION OF THE DYNAMIC JAHN- TE LLER. . . 843

Work supported in part by the National Science Founda-tion and the National Aeronautics and Space Administra-tion.

~A portion of the work was performed while at Rice Uni-versity, Houston, Texas 77001.

IPresent address: Physics Dept. , University of Illinois,Urbana, Ill. 61801.

H. A. Jahn and E. Teller, Proc. R. Soc. Lond. A 161,220 {1937).

A review of both the experimental results and theoryrelated to the Jahn-Teller effect is contained in thearticle by M. D. Sturge, in Solid State Physics, editedby F. Seitz, D. T'urnbull, and H. Ehrenreich {Academic,New York, 1967), Vol. 20.

3The theory of the Jahn-Teller effect and the experimen-tal results obtained by electron spin resonance havebeen. reviewed by F. S. Ham, in E/ectron ParamagneticResonance, edited by S. Geschwind {Plenum, New York,1972).

4R. E. Coffman, Phys. Lett. 19, 475 (1965).R. E. Coffman, Phys. Lett. 21, 381 (1966).R. E. Coffman, J. Chem. Phys. 48, 609 (1968).

'R. E. Coffman, D. L. Lyle, and D. R. Mattison, J.

Chem. Phys. 72, 1392 (1968).U. T. Hochli and T. L. Estle, Phys. Rev. Lett. 18, 128(1967).

~U. T. Hochli, Phys. Rev. 162, 262 (1967).I. B. Bersuker, Zh. Eksp. Teor. Fiz. 43, 1315 (1962)[Sov. Phys. -JETP 16, 933 (1963)].I. B. Bersuker, Zh. Eksp. Teor. Fiz. 44, 1239 (1963)[Sov. Phys. -JETP 17, 836 (1963)].M. C. M. O' Brien, Proc. R. Soc. Lond. A 281, 323(1964).

'BF. S. Ham, Phys. Rev. 166, 307 (1968).L. L. Chase, Phys. Rev. Lett. 23, 275 {1969).

' L. L. Chase, Phys. Rev. B 2, 2308 (1970).J. R. Herrington, T. L. Estle, L. A. Boatner, and

B. Dischler, Phys. Rev. Lett. 24, 984 {1970).YJ. R. Herrington, T. L. Estle, and L. A. Boatner,Phys. Rev. B 3, 2933 (1971).

~~J. R. Herrington, T. L. Estle, and L. A. Boatner,Phys. Rev. B 5, 2500 (1972).J. R. Herrington, T. L. Estle, and L. A. Boatner, Phys.Rev. B 7, 3003 (1973).

20T. J. Aton, J. R. Herrington, T. L. Estle, and L. A.Boatner, Bull. Am. Phys. Soc. 16, 386 (1971}.