PHYSICAL REVIEW B 1 SEPTEMBER 1997-IVOLUME 56, NUMBER 9

Microscopic theory of magnetic-field-induced structural phase transitions

Michael D. KaplanPhysics Department, Boston University, 590 Commonwealth Avenue, Boston, Massachussetts 02215

and Chemistry Department, Simmons College, 300 The Fenway, Boston, Massachussetts 02115

George O. ZimmermanPhysics Department, Boston University, 590 Commonwealth Avenue, Boston, Massachussetts 02215

~Received 11 November 1996!

The microscopic theory of a magnetic-field-induced structural phase transition is developed. The mechanismof the phase transition is connected with the cooperative Jahn-Teller effect. For a set of microscopic parametersdetermined earlier the anomalies in the acoustic, magnetic, and magnetoelastic properties have been analyzedfor a wide range of temperature and external magnetic fields. Special attention is paid to the unusual giantdynamic magnetostriction.@S0163-1829~97!04633-X#

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INTRODUCTION

The understanding of the mechanism of the cooperaphenomena in condensed matter at a microscopic levemains of prime importance in many modern fields of scienincluding ferroelectricity, superconductivity, magnetismstructural transformations, polymerization, peptide foldinand others. In this list, the structural phase transitions arspecial and very visible significance for being tightly conected to the electronic structure and influencing the proties of the matter.

Nowadays the most developed microscopic theorystructural transformations is based on the analysis ofelectron-phonon interaction which in the case of the nKramers degeneracy of the electronic subsystem is ocalled the cooperative Jahn-Teller effect. The applicationthis theory to the modern problems mentioned above is unconsideration in numerous recently published articlesbooks.1–8 It is completely clear that if the Jahn-Teller effeis really responsible for the structural transition in the systunder consideration, the external magnetic field, while inencing directly only the electronic subsystem, can drasticeffect the geometry of the distribution of nuclei in space~thecrystal lattice in the case of solids!. Therefore the magneticfield influence on structural transitions can be consideredpositive test for the electron-phonon microscopic mecnism.

The changes in the properties of Jahn-Teller ferroelacrystals caused by the external magnetic fields were conered earlier in many papers~see Ref. 9, and referencetherein!. However in this paper we are analyzing a phenoenon of structural phase transition induced by the magnefield-stimulated cooperative Jahn-Teller effect.10,11 As con-trasted with the previously analyzed systems in the crysunder consideration no phase transitions take place withmagnetic field, so that there are no doubts as to the electphonon origin of the phase transitions. Recently such sttural phase transitions were observed in the experimentgiant magnetostriction12 and on Raman scattering.13 Below,the theory of the magnetic-field structural transitions is alyzed in detail by using the TmPO4 and TbxY12xVO4 crys-

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tals as examples. The results of numerous calculationdifferent crystal properties, already experimentally observor predicted, are presented.

THE HAMILTONIAN

The crystals under consideration have the tetragonalcon structuresD4h and the local symmetry of the rare-earJahn-Teller ions isD2d . The TmPO4 and TbxY12xVO4 crys-tals are the representatives of this class of materials.electronic structure of the Tm31 and Tb31 ions is describedby two singlets separated by a 2D gap, in the middle ofwhich there is a non-Kramers doublet. Other energy levare well separated from this ground group and are not csidered at the low temperatures of interest. This is dobecause at these temperatures only the induced strucphase transitions take place. In this case the higher enlevels are practically empty. At the same time, the mixingthese excited states with the ground quadruplet can benored because, at the magnetic field under considerationZeeman energy is much smaller than the energy gap betwthese two groups of levels.

The Hamiltonian of the crystal with this electronic bascan be written as8

H5Hel-ph1Hel-str1Hph1Hstr1Hcryst1HZeem, ~1!

where

Hel-ph5(mk

Vmkszm~bk1b2k

1 !,

Hel-str52g0AC0V

NUz(

msz

m ,

Hph5(k

hvkS bk1bk1

1

2D , Hstr51

2C0VUz

2, ~2!

5193 © 1997 The American Physical Society

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5194 56MICHAEL D. KAPLAN AND GEORGE O. ZIMMERMAN

Hcryst52D

2~11tz!sx ,

HZeem52gb(m

~HxSxm1HySy

m!.

In Hamiltonian~1! the first two terms describe the interactioof the electrons with the phonons~the second quantizatiooperatorsbk ,bk

1! and the homogeneous strainUz , the nexttwo terms are connected with the free phonons and homneously strained crystal elastic energy, and, at last, thetwo terms represent the crystal-field splitting of the eletronic states~singlet-doublet-singlet structure! and the Zee-man interaction with thexia and yib(5a) components ofthe external magnetic field. It is important that the magnefield orientation at 45° to the crystala axis,

S Hx5Hy51

&H0D ,

will be under consideration in this paper. The magnetic fiHz perpendicular to theab plane of the crystal can onlysuppress the orthorhombic strain in theD4h-symmetry te-tragonal crystals.8 At Hi@110# or @010# the magnetic fieldforms the crystal strain ofB2g(D4h) symmetry due to theregular magnetostriction effect, according to symmegroup theory. That strain symmetry is the same as the sptaneous strain at the induced structural phase transitionother words, at this field orientation the strain exists at anonzero magnetic field, so that no phase transitions cancur. At Hi@110#, ~45° to the crystala-axis orientation!, theB2g(D4h) strain cannot be induced as regular magnetosttion, and its appearance is the result of the cooperative inactions. This is discussed below in greater detail in thecussion of magnetostriction.

The matrices of the electronic operators at a chosen bcan be represented as

sz5S 1000

0100

00

210

000

21D ,

1

2~11tz!sx5S 0

010

0000

1000

0000D , ~3!

Sx51

2~11sz!tx5S 0

100

1000

0000

0000D ,

Sy51

2~12sz!tx5S 0

000

0000

0001

0010D .

We note thats and t matrices are similar to the Dirac matrices for the electron, and obey the commutation rules14

e-st-

-

d

yn-Inyc-

-r-

s-

sis

s3s52is; t3t52i t; @s,t#50. ~4!

It is easy to see after the minimization of the crystal eneupon the spontaneousU strain that the last relation is proportional to the average of theSz electronic operator

Uz5g0A N

C0Vsz ~5!

~the spontaneous strainUz is the order parameter and has tB2g symmetry as the structural phase transition is fromtragonal symmetry to the orthorhombicD2h symmetry!.

For further analysis of the Hamiltonian~1! it is con-venient to subject it to a unitary shift transformatioH5exp(iR)H exp(2iR),15,16 where

R5(m

gmszm , gm5 i(

k

Vmk

hvksz

m~bk12bk!. ~6!

After the transformation the crystal Hamiltonian takes tform

H5Hel-str1Hstr1Hph1HZeem2 (mnk

Vmk* Vnk

hvksz

mszn

21

2D(

m~11tz

m!~sxm cos 2gm2sy

m sin 2gm!. ~7!

The fifth term of the transformed Hamiltonian~7! de-scribes the virtual phonon exchange of the Ising type respsible for the effective electron-electron interaction in tcrystal, the first four terms are unaffected by the transformtion as the corresponding operators commute with the soperator~the free-phonon operators describe the samequency phonons with the shifted equilibrium positions!. Thelast term of Eq.~7! has a complicated form~the price for theaccurate Hamiltonian transformation of the system unconsideration! and represents the dynamical coupling of telectron and phonon variables. Separating the electron vables from the phonon variables~the k-space area of strongdynamic interaction is much smaller than the size of the fiBrillouin zone! for the electronic part of the Hamiltonian onobtains

Hel52(mn

Amnszmsz

n21

2Dg~11tz

m!sxm

21

&gbH(

m~Sx

m1Sym!, ~8!

where

Amn5(k

Vmk* Vnk

hvksz

mszn1g0

2sz(m

szm ~9!

is the effective electron-electron interaction constant whthe contribution of the electron-strain interaction is takinto account. It is seen in Eq.~9! that the interelectronicinteraction in the Jahn-Teller crystals is the result of the vtual phonon exchange and the interaction of the electrwith the homogenous strains. In this case, the effective etron correlation radius depends on the wavelength of

O

m

dee

emin

byap-thec-

-

56 5195MICROSCOPIC THEORY OF MAGNETIC-FIELD- . . .

phonons which contribute to the interaction. In the TmP4and TbVO4 crystals, the greatest contribution toAmn comesfrom the long-wavelength acoustic phonons and the hogeneous strains~infinitely big wavelengths!. Therefore thecorrelation radius is big, and that is confirmeexperimentally.8 On the other hand, it is well known that thmolecular field approximation is valid if the crystal latticconstanta is smaller than the correlation radiusr . At a/r

an

ti

uchtha

ee

ed

o-

!1 the molecular field approximation describes the systquantitatively,~see Chap. 3 in Ref. 8, and references there!.

As the systems under consideration are characterizedthe long-range radius of correlations the molecular fieldproximation is good enough for a quantitative analysis ofproperties of the crystals. In this approximation the eletronic order parameters is described by the following equation:

sz5Asz /Aa1Ab~11h0

2/Ab!sinh~Aa1Ab/kT!1As/Aa2Ab~12h02/Ab!sinh~Aa2Ab/kT!

cosh~Aa1Ab/kT!1cosh~Aa2Ab/kT!, ~10!

where

a5A2sz21D2g21

1

2h0

2; b5D4g411

2h0

2~D2g214A2sz2!, h05gbH. ~11!

Taking into account that 2(a1Ab) is the gap between the ground and the excited singlets and 2(a2Ab) is the splitting of thedoubly degenerate~at H50! excited state, Eq.~10! can be written as

sz5P1Asz~11h0

2!tanh~Aa1Ab/kT!

Aa1Ab1P2

Asz~12h02!tanh~Aa2Ab/kT!

Aa2Ab, ~12!

where

P15cosh~Aa1Ab/kT!

cosh~Aa1Ab/kT!1cosh~Aa2Ab/kT!~13!

is the total population of the singlet states andP2512P1 is the total population of the doublet components.If the electronic order parameters from Eq. ~10! or ~12! is not zero the spontaneous strainUz according to Eq.~5! is also

nonzero and the structural phase transition takes place. It is easy to see that in the case ofHx5Hy50, Eq. ~10! and thepopulationP1 are given by expressions discussed earlier17

sz5Asz /~AA2sz

21D2g2!sinh~AA2sz21D2g2/kT!1sinh~Asz /kT!

cosh~AA2sz21D2g2/kT!1cosh~Asz /kT!

, ~14!

for-

an-me-lleran-of

era-als,etronthen,gye

thebe

een

P15cosh~AA2sz

21D2g2/kT!

cosh~AA2sz21D2g2/kT!1cosh~Asz /kT!

. ~15!

The transcendental equation~14! at H50 was analyzed inRef. 14, where it was shown that forD/A.1.20 only onesolution exists ats̄z50. This is the case of the TmPO4 crys-tal whereD/A51.33 and no phase transitions take placeall temperatures. This also holds for different compositioof sufficiently diluted TbxY12xVO4 crystals. However, wehave found that in the presence of sufficiently big magnefields oriented at 45° to the crystala axis, sÞ0 solutionsexist. Moreover, at some values of magnetic fields two strtural phase transitions take place. The second reentrant ptransition corresponds to the return of the crystal fromorthorhombic phase to the higher-symmetry tetragonal phwith the lowering temperature.

The calculations of the crystal phase diagrams depdence upon the external magnetic fields were done in R18. Here we will focus mostly on the physics of the induc

ts

c

-aseese

n-f.

structural transformations and accompanying these transmations, anomalies of different crystal properties.

The microscopic mechanism of the structural phase trsitions under consideration can be understood in the frawork of the general theory of the cooperative Jahn-Teeffect. According to that approach the structural phase trsition is the result of the electron-phonon interaction andthe corresponding virtual phonon exchange in the cooptive systems. However, in the pseudo-Jahn-Teller crystwhere the vibrations mix initially split energy levels, thphase transition takes place only when the electron-eleccorrelations are bigger than the initial energy gap. InTmPO4 crystal the virtual phonon exchange interactiowhile not zero, is not sufficiently great to overlap the energap ~A522.5 cm21, D530 cm21! and as a result there arno phase transitions. But if the external magnetic fieldHx

5Hy is not zero, the phase transitions can be induced iffield value is greater than some critical value. This caninterpreted in the following way. The externalH field affectsthe system in two ways: it changes the energy gaps betw

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5196 56MICHAEL D. KAPLAN AND GEORGE O. ZIMMERMAN

the electronic states and it changes the electronic wave ftions mixing each of the singlet components with one ofdoublet components. This can be easily seen from the stture of the electronic operators of the magnetic momeMx5gbSx andM y5gbSy . Calculations show that the mixing of the wave functions is very important for the effectivness of the electron-electron interaction.

It is worthy of note that such magnetic-field influencesystems with structural phase transitions is quite unusuaall previously analyzed systems under the external magnfield, the field could only reduce the phase transitionssmear out them. Which effect actually took place depenupon the symmetry properties of the solids, on the magnfields, and upon the commutation rules of the operatorsthe electron-phonon and Zeeman interactions. The magnfield-induced structural transitions are connected with a nsituation that will be discussed in the section dealing withmagnetoacoustic properties.

ORDER PARAMETER

The order parameters is described by Eqs.~12! and~14!which are valid at all temperatures and all values of the mnetic fields. These equations are transcendental equa

FIG. 1. The order parameter, dynamic magnetostriction, magtization, and magnetic susceptibility as a function of temperatureparametersD530, A522.5, andh517.7. All units are in cm21.The parametersD andA correspond to TmPO4.

FIG. 2. The order parameter, dynamic magnetostriction, magtization, and magnetic susceptibility as a function of the magnfield h, for parametersD530,A522.5, andT58.7. All units are incm21. The parametersD andA correspond to TmPO4.

c-ec-ts

Inticrdicofic-we

-ns

and their numerical solutions are represented in Figs. 1–can be seen that depending upon theD/A ratio, temperature,and magnetic field, one, two, or no structural phase trations are found. In some cases thes̄zÞ0 solution for sometemperatures appears only atH0Þ0. This corresponds tofield-induced transitions. The two transitions correspondthe reentrance of the crystal from a low-symmetry to a higsymmetry phase as the temperature of the system is loweSuch a possibility for diluted TbxY12xVO4 crystals was dis-cussed earlier.14 We show here that the reentrant phase trsitions can also be induced by the external magnetic fielFor analysis of the magnetic-field influence on the orderrameter let us consider the order parameter equationT50. As it follows from Eqs.~13! and~14!, at zero tempera-ture P151, P250 and

s̄z5As̄z

Aa1AbS 11

h02

AbD . ~16!

From this equation it is easy to find the critical magnetfield value at which the structural phase transition takes plat T50. For h0 crit we find the equation

e-r

e-ic

FIG. 3. The order parameter and the modulus of elasticity afunction of temperature for parametersD530, A522.5, andh517.7. All units are in cm21. The parametersD andA correspondto TmPO4.

FIG. 4. The order parameter and the modulus of elasticity afunction of the magnetic field for parametersD530, A522.5, andT58.7. All units are in cm21. The parametersD andA correspondto TmPO4.

q.

g-vbeise

ono

rueto

alth

eldnta-lityon-theonetic

-–8

ninge

ld isonof aeenced

tionare, 2,asesin

nfo

net

s a

s a

56 5197MICROSCOPIC THEORY OF MAGNETIC-FIELD- . . .

1

A5

112h0 crit2 /D2g2A112h0 crit

2 /D2g2

Ah0 crit2 /21~D2g2/2!~11A112h0crit

2 /D2g2!. ~17!

From Eq.~17! it is seen that ath0 crit50 the structural phasetransition is possible only ifA5Dg. However if the mag-netic field is present the solutions of Eq.~17! exist even forDg.A as is shown in Figs. 1–8. In part, it follows from E~17! that at h0→` the solution of Eq.~17! exists if A51/2Dg. The meaning of this result is that while the manetic field makes possible the structural phase transition eat Dg.A, nevertheless the crystal field splitting cannotbigger than double the molecular Jahn-Teller field. If itthe phase transition does not take place at any magnetic fi

MAGNETOSTRICTION

It is well known that magnetostriction is the phenomenof inducing or changing the strain of a sample by meansan external magnetic field. In that sense the induced sttural phase transition is a magnetostriction effect. Howevit is a very unusual magnetostriction. As a rule the magnestriction is quadratic in magnetic fields when they are smThis is a direct result of the symmetry of the strain and of

FIG. 5. The order parameter, dynamic magnetostriction, magtization, and magnetic susceptibility as a function of temperatureparametersD59, A58.65, andh51.44. All units are in cm21. TheparametersD andA correspond to TbxY12xVO4.

FIG. 6. The order parameter, dynamic magnetostriction, magtization, and magnetic susceptibility as a function of the magnfield h, for parametersD59, A58.65, andT51.53. All units are incm21. The parametersD andA correspond to TbxY12xVO4.

en

,ld.

fc-r,-

l.e

magnetic-field orientation. The square of the magnetic-fisymmetry representation contains the symmetry represetion of the strain that causes the coefficient of proportionabetween the strain and the magnetic field square to be nzero. However this is not the case in the phenomenon ofinduced structural phase transitions. For crystals with zircstructure the square of the representation of the magnfield oriented at 45° to the crystala axis ~A1g1B1g repre-sentations! does not contain theB2g-symmetry representation of the spontaneous strain. As it is seen from Figs. 1the strain does not depend upon the magnetic field remaizero if H0,Hcrit , and even at the bigger fields, when thstrain is induced, the dependence upon the magnetic fienot quadratic. Moreover, the regular magnetostriction upthe field dependence is characterized by the presenceknee before the striction saturation starts. As it can be sfrom our calculations no knees are present at the industructural phase transition phenomenon.

Other much more drastic changes of the magnetostricunder consideration in comparison with the regular oneconnected with the dynamic magnetostriction. Figures 15, and 6 show a strong maximum near the structural phtransition. From the first point of view this result is not asurprising as earlier, the giant dynamic magnetostriction

e-r

e-ic

FIG. 7. The order parameter and the modulus of elasticity afunction of temperature for parametersD59, A58.65, and h51.44. All units are in cm21. The parametersD andA correspondto TbxY12xVO4.

FIG. 8. The order parameter and the modulus of elasticity afunction of the magnetic fieldh, for parametersD59, A58.65, andT51.53. All units are in cm21. The parametersD andA correspondto TbxY12xVO4.

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the

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etic

5198 56MICHAEL D. KAPLAN AND GEORGE O. ZIMMERMAN

the paramagnetic ferroelastics was predicted theoreticand observed experimentally. However, in comparison ewith the giant dynamic magnetostriction of the cooperatJahn-Teller crystals, the magnetostriction under considation is qualitatively different. The dependence of the widof the peak of the dynamic striction upon the field depedence is very narrow, much narrower than in the earlier dcussed situations. For example, in the same TmPO4 andTbxY12xVO4 crystals under external magnetic fieldHia, thediscussed peak is broadened as a result of the smearinof the structural phase transitions when they exist withoutfield. But at the 45° orientation of the field no smearing o

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of the transition takes place and that is why the peak ofdynamic magnetostriction is narrower.

MAGNETIC PROPERTIES

Magnetic properties of the Jahn-Teller crystals with strutural phase transitions are very specific. It is reasonableexpect for these properties some specific anomalies cnected with the induced phase transitions.

In the framework of the developed approach the magnmoment M parallel to theH0 field is proportional to theaverage of the electronic operatorS0 which is described bythe formula

S̄051

2h0

@11~4A2s̄z21D2g2!/2Ab#/Aa1Ab sinh~Aa1Ab/kT!1~124A2s̄z

2/2Ab!/Aa2Ab sinh~Aa2Ab/kT!

cosh~Aa1Ab/kT!1cosh~Aa2Ab/kT!,

~18!

althe

ongon-turethena-

isu-

the

il-

is

with

ed

-

wherea and b were given in Eq.~11!. Before we start todiscuss the results of numerical calculations at different teperatures and magnetic fields let us consider the less comcated case ofT50. The simplified formula forS̄0 takes theform

S̄05h0

2Aa1AbS 11

4A2s̄z21D2g2

2AbD . ~19!

From this formula it can be seen that the presence ofelectron-electron correlations (AÞ0) caused by virtual phonon exchange leads to an additional nonlinear dependenthe magnetic moment upon the field becauses̄z is a compli-cated function uponH. As a consequence of that, an anomlous ‘‘knee’’ appears in theM (H) dependence. It is easy tsee that ifA50 the regular saturation of a magnetic momeof a paramagnetic material under high fields takes placethis situation of no structural phase transitions, the magnmoment is described by

S̄05h0~11Dg/AD2g212h0

2!

2Ah02/21~D2g2/2!~11A112h0

2/D2g2!, ~20!

from which we obtain the result that at very big magnefields (h0@D)S̄0→&/2. As is seen from Figs. 1, 2, 5, andthe anomaly connected with the structural phase transitioquite small and could be more easily observed at the msurements of the magnetic susceptibility that was calculafrom Eq. ~18!. As it is shown in Figs. 1, 2, 5, and 6, a smabut sharp jump atT5Tc andH05Hcrit appears.

ELASTIC PROPERTIES

The elastic susceptibility is the reaction function conected with the order parameter at the structural phasesitions. Since the field-induced structural transitions unconsideration are of ferroelastic type the elastic susceptibis expected to diverge atT5Tc and H5Hcrit . The corre-sponding modulus of elasticityC is supposed to go to zer

-li-

e

of

-

tInic

isa-d

-n-r

ty

@C(T5Tc)50#. At the structural transitions from tetragonto orthorhombic phase, induced by the magnetic field,soft modulus of elasticity isC5C66. This modulus deter-mines the velocity of the transverse mode propagating althe @100# direction with the polarization vector parallel t@010# direction. As it is known from the cooperative JahTeller effect theory, the measurements of the the temperaand field dependences provide a lot of information aboutsystem under investigation and in part allow the determition of the electron-strain interaction constant. The lastespecially important as this interaction is the main contribtion to the electron-electron interaction responsible forphase transition in the TmPO4 and TbxY12zVO4 crystals.

The modulus of elasticity can be calculated if in Hamtonian~2! the applied uniaxial stressP of B2g(D4h) symme-try is included. The corresponding term of the Hamiltonian

HP52g0

AC0VNP(

msz

m . ~21!

The equation for the order parameter has the same form,the only difference being that allAs̄z are replaced with

As̄z1g0P

AC0VN.

Taking into account that the modulus of elasticity is definas

C215S ]U

]P DP50

~22!

and

U5g0A N

C0Vs̄z , ~23!

the formula forC5C(H,T) can be determined. For simplicity the formula forT50 only will be shown

thaipthon

re-

heralac-

eri-ged

aliessults

56 5199MICROSCOPIC THEORY OF MAGNETIC-FIELD- . . .

C

C05

~12A/Aa1Ab!F0

@12~A2g02!/Aa1Ab#F0

, ~24!

where

F0511h0

2

Ab2s̄z

2F122a1Ab

Ab3

h04

~11h02/Ab!2G . ~25!

The results of the calculations of the temperature andmagnetic-field dependences for the modulus of elasticityshown in Figs. 3, 4, 7, and 8. The modulus of elasticity dto zero at the critical temperatures and critical values ofexternal magnetic field. If the reentrant phase transiti

n

ereses

take place the modulus of elasticity becomes zero, twicemaining small enough but not zero atTc1,T,Tc2 .

CONCLUSIONS

In the framework of the cooperative Jahn-Teller effect tmicroscopic theory of the magnetic-field-induced structuphase transitions is developed. The coupling between mroscopic and microscopic parameters is found. The expments on giant magnetostriction12 and on Raman scatterinof light confirm the theoretical prediction of thphenomenon.13 Different magnetic, magnetoacoustic, anelastic properties are investigated and a series of anomconnected to the phase transition are predicted. These reare waiting for further experimental verification.

v,

n,

R.

n

tt,

1T. G. Bednorz and K. A. Muller, Z. Phys. B64, 436 ~1986!.2K. H. Johnsonet al., in Novel Superconductivity, edited by S. A.

Wolf and V. Z. Kresin ~Plenum, New York, 1987!, p. 563;Physica C162-164, 1475~1989!.

3T. Riceet al., Phys. Rev. Lett.67, 3452~1991!.4M. Schluteret al., Phys. Rev. Lett.68, 526 ~1992!.5I. B. Bersuker and V. Z. Polinger,Vibronic Interactions in Mol-

ecules and Crystals~Springer-Verlag, Berlin, 1989!.6I. Miloshevichet al., Phys. Rev. B47, 7805~1993!.7A. J. Millis, Phys. Rev. B53, 8434~1996!.8M. D. Kaplan and B. G. Vekhter,Cooperative Phenomena i

Jahn-Teller Crystals~Plenum, New York, 1995!.9G. A. Gehring and K. A. Gehring, Rep. Prog. Phys.38, 1 ~1976!.

10B. G. Vekhter, V. N. Golubev, and M. D. Kaplan, JETP Lett.45,168 ~1987!.

11M. D. Kaplan and A. V. Vasil’ev, Physica B179, 65 ~1992!.12B. G. Vekhter, Z. A. Kazey, M. D. Kaplan, and Yu. F. Popo

JETP Lett.54, 578 ~1992!.13M. D. Kaplan, C. H. Perry, D. Cardarelly, and G. O. Zimmerma

Physica B194-196, 453 ~1994!.14R. J. Elliott, R. T. Harley, W. Hayes, and S. R. P. Smith, Proc.

Soc. London, Ser. A328, 217 ~1972!.15R. J. Elliott, in Proceedings of the International Conference o

Light Scattering in Solids, Paris, 1971, edited by M. Balkanski~Flammarion, Paris, 1971!, p. 354.

16B. G. Vekhter and M. D. Kaplan, Phys. Lett.43A, 398 ~1973!.17R. T. Harley, W. Hayes, A. M. Perry, S. R. P. Smith, R. J. Ellio

and I. D. Saville, J. Phys. C7, 3145~1974!.18M. D. Kaplan and G. O. Zimmerman, Phys. Rev. B52, 1 ~1995!.