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01 Nov 1994

Nature of the Ferroelectric Phase Transition in PbTiO₃ Nature of the Ferroelectric Phase Transition in PbTiO

Noam Sicron

Bruce D. Ravel

Yitzhak Yacobi

Edward A. Stern

et. al. For a complete list of authors, see https://scholarsmine.mst.edu/matsci_eng_facwork/1752

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Recommended Citation Recommended Citation N. Sicron et al., "Nature of the Ferroelectric Phase Transition in PbTiO₃," Physical Review B (Condensed Matter), vol. 50, no. 18, pp. 13168-13180, American Physical Society (APS), Nov 1994. The definitive version is available at https://doi.org/10.1103/PhysRevB.50.13168

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PHYSICAL REVIEW B VOLUME 50, NUMBER 18 1 NOVEMBER 1994-II

Nature of the ferroelectric phase transition in PbTiOs

N. SicronRacah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, Israel 9190$

B. RavelPhysics Department FM 15, -University of Washington, Seattle, Washington 98195

Y. YacobyRacah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, Israel 9190$

E.A. SternPhysics Department FM 15, U-niversity of Washington, Seattle, Washington 98195

F. DoganDepartment of Materials Science and Engineering FB 10, Uni-versity of Washington, Seattle, Washington 98195

3.J. RehrPhysics Department FM 15, Univ-ersity of Washington, Seattle, Washington 98195

(Received 1 September 1993; revised manuscript received 28 March 1994)

We have measured quantitatively the temperature dependence of the local distortions of PbTi03crystals below and above the structural ferroelectric phase transition, using x-ray absorption fine-

structure (XAFS) measurements. Two probe atoms, Pb and Ti, were used. The results were ana-

lyzed by fitting parametrized theoretical XAFS spectra to the experimental results. These measure-

ments provide quantitative "distortion parameters" defined as the difFerence between the distanceof the probe to its nearest neighbor measured in the actual structure and that in a centrosymmetricstructure with the same unit cell dimensions. At low temperatures the Pb-edge spectra were fit using

four shells and included single, double, and triple scattering configurations. The high-temperaturefits included only two shells. The Ti-edge spectra were fitted with the first shell only. The Pband Ti distortions vary relatively little with temperature below the transition and decrease fasternear the transition temperature. Above the transition temperature, the Pb and Ti distortions atthe peaks of their distortion distribution functions (DDF s) are both more than 70'%%uo of the corre-

sponding low-temperature values. These results show that an essential element of order-disorder is

present even in this nominally pure ferroelectric crystal which displays a soft mode and a dielectricconstant typical of displacive-type ferroelectrics. The presence of the local distortions suggests thatthe displacements should have at least two correlation length scales, one associated with the thelocal distortions, the other with the order parameter.

I. INTRODUCTION

In materials undergoing structural phase transitions,the average distortion from the high-symmetry structureis represented by the order parameter. However, the ac-tual local distortion can be quite different. One can con-sider two extreme cases. If the local distortions and theorder parameter have siTni&ar temperature dependenciesin a range including the transition temperature the tran-sition is considered to be "displacive. " On the other hand,if the size of the local distortions does not change withtemperature across the transition, the transition is of the"order-disorder" type. In this case only the orientationsof the local distortions change with temperature. Belowthe transition there is a preferred orientation. Above thetransition the orientations are disordered. The distinc-tion between the two is not absolute. Local distortions in

a displacive crystal undergoing a second-order transitionwill not become zero just above the phase transition be-cause of critical fluctuations. Similarly, the magnitude ofthe local distortions in an order-disorder crystal is neverprecisely constant because of elastic interactions. How-

ever, these two types of transitions involve different basicmechanisms. In the displacive case, an instability result-ing from long range cooperative interactions produces thelocal distortions. In the order-disorder case the local dis-tortions are due to local instabilities. Thus, it is of funda-mental interest to our understanding of phase transitionsto determine which mechanism dominates.

Perovskite crystals with an AB03-type molecularstructure have long been considered as displacive typeferroelectrics. The main evidence for this type of be-havior has been the existence of a Brillouin zone cen-ter soft mode both below and above the transitiontemperature and the large size of the dielectric con-

0163-1829/94/50(18)/13168(13)/$06. 00 50 13 168 Q~1994 The American Physical Society

50 NATURE OF THE FERROELECTRIC PHASE TRANSITION IN PbTi03 13 169

stant. Soft modes have been observed in many per-ovskite ferroelectrics. s In some the soft mode is un-

derdamped even up to a few degrees from the transi-tion temperature. s Excluding critical phenomena, thedielectric constant of crystals undergoing a second-orderor weak first-order transition follows a Curie-Weiss law

oc~& &

~

where T is the actual or extrapolatedC

second-order phase transition temperature. It turnsout that the proportionality constant C is much largerin displacive than in order-disorder-type ferroelectrics.The coefficient C of perovskite crystals has values typ-ical of displacive-type ferroelectrics. The existenceof soft modes and the size of the dielectric constantled theoreticians to describe perovskite ferroelectrics asdisplacive. s s s According to these models, the atoms vi-brate about center of inversion positions in the cubicphase. Below the transition temperature the structuredistorts and the distortion increases with decreasing tem-perature.

During the last two decades much evidence ac-cumulated indicating that local distortions in per-ovskite crystals also exist above the phase transitiontemperature. io ii The first to notice this were Comeset al. i2 They found difFuse x-ray lines above the tran-sition temperature to the cubic phase indicating the ex-istence of some form of structural disorder. Many typesof optical experiments followed. These include Ramanscattering, s M hyper-Raman, ir infrared refiectivity, isand optical refractive index. is These experiments showthat local distortions are possibly present at tempera-tures far above the phase transition temperature. How-ever, none of these experiments provide direct quantita-tive information on the magnitude and temperature de-pendence of these distortions. The first experiments toprovide direct quantitative information on the tempera-ture dependence of the distortions below and above thetransition were x-ray Absorption Fine-Structure (XAFS)measurements20 on KTaOs. Nb crystals. zi s XAFS ex-periments showing disorder above the transition in nnxednonperovskite crystals were also reported. 242s XAFSmeasurements on KNbOs also indicate distortions aboveT, but the results in this case were not analyzed indetail. 2s In the KTaOs. Nb experiments, the positionsof the Nb atoms were measured relative to the oxygenoctahedra, to the K atoms at the corners of the cubeand to the first Ta neighbors. The results show thatthe Nb atoms are displaced by about 0.145 A. in the[111]direction relative to the surrounding atoms. Fur-thermore, the magnitude of the displacements changesgradually by only 12%%uo &om about 16 K below the tran-sition (T,=85.6 K) to 215 K above it. These resultsshow conclusively that the transition has at least par-tially an order-disorder character. Obviously, a pureorder-disorder-type transition involving the Nb ions alonewiH not account for some of the experimentally observedfeatures in this material such as the soft phonon2~ anda large inverse temperature coeKcient of the dielectricconstant. It seems that, in this material, the hostKTaOs provides a displacive component which, combinedwith the Nb ions, may account for the properties of thissystem. The question is what happens in a pure per-

ovskite which has a very well-behaved soft mode and alarge dielectric temperature coefficient characteristic ofan ideal displacive ferroelectric. To answer this questionwe investigated PbTiOs which has been described as aclassic case of a displacive ferroelectric.

Above 763 K PbTiOs is cubic. 29 At this temperature itgoes through a weak first-order transition to a tetragonalphase with a c/a ratio of 1.075 at 100 K.M In the tetrag-onal phase the crystal is ferroelectric. Both Pb and Tiatoms are displaced from their respective oxygen planesby 0.49 and 0.32 4, respectively. The displacements arein the [001] direction. At 183 K the crystal undergoesa transition to an orthorhombic phase with a very smallchange in the lattice parameters, si but remains ferroelec-tric. In addition an anomaly in the temperature depen-dence of the Pb displacement was observed at ~400 K.This anomaly is not accompanied by a visible change inthe lattice parameters. s2

A schematic representation of the low-temperaturephase with the Pb and Ti atoms at the center is shown inFig. 1. In the following we shall use the crystallographicnotation and refer to the oxygens in the plane above andbelow the Pb as 02 and 02+, respectively, and the oxy-gens which are approximately in the Pb plane as Oi.

The soft modes of this material had been investigatedby a number of groups as a function of temperatures ss

and pressure. r The soft mode frequency decreases asthe transition temperature is approached. It is under-damped at all temperatures and can be followed up to thefirst-order transition temperatures where its frequency isabout 55 cm i. Near the transition temperature a cen-tral peak is observed. s4 The area of this peak seems togrow as the transition temperature is approached. Thisresult is interpreted as a manifestation of some disorderpresent in the system close to the transition tempera-ture on both sides. Furthermore, the dielectric constantin this material is found to be even larger than that ex-pected from the Lydane-Sachs-Teller relation, suggest-ing that even the ordinary displacive-type model wouldnot be sufhcient to account for the dielectric constant.Fontana et al. s4 suggest that there is a crossover fromdisplacive to order-disorder behavior near the phase tran-sition temperature. Neutron dHFraction experiments re-ported recentlyM'I also suggest that PbTiOs has a cer-tain degree of disordered distortions above the phasetransition. These results will be discussed in more de-tail in Sec. IV.

In this paper we report the temperature dependenceof local distortions in PbTiOs obtained &om (XAFS)measurements. Previous XAFS measurements onPbTiOs have been reported but have not been analyzedin any detail. ss XAFS is particularly suitable to pro-vide quantitative local structural information for severalreasons. It can provide information on the structuresurrounding difFerent types of atoms in the systein ex-tending out to fourth-nearest neighbors providing thepair correlation function with very high resolution. Indisordered systems it is better than any other presentlyavailable tec&nique. It is not affected by long range orien-tational disorder and its efFective measuring time is veryshort (on the order of 10 is sec).

13 170 SICRON, RAVEL, YACOBY, STERN, DOGAN, AND REHR

(a)0

O~Ti-

(b)

~ Pb

FIG. 1. Schematic structure of PbTi03around each one of the probe atoms. (a) Ti,(b) Pb.

Oo„0&+

iles

In this study, we used both Pb and Ti as probe atoms.The Pb XAFS provides us with the displacements of thePb atoms relative to the oxygen octahedra and the Tiatoms. Analyzing the first shell of the Ti XAFS pro-vides direct information on the Ti displacement relativeto the oxygen octahedra. In addition, it provides directinformation on the distortion of the oxygen octahedrathemselves. Thus, the information obtained from the twoprobes is complementary.

Our results show that large local distortions exist even190 K above T, . Furthermore, the individual oxygen oc-tahedra are tetragonally distorted also above T, . But, incontrast to the KTaOs. Nb case, the size of the distortiondoes decrease close to the phase transition.

In Sec. II we discuss the sample preparation and theXAFS experiments. In Sec. III we present the experi-mental results. The analysis of the results is describedin Sec. IV and the physical meaning of the results is dis-cussed in Sec. V. Section VI summarizes and concludesthe paper.

II. SAMPLE PREPARATION ANDEXPERIMENTAL CONSIDERATIONS

Pure lead titanate was prepared from an aqueous so-lution of nitrates of lead and titanium. The solution wascoprecipitated by altering its pH with ammonia. Theprecipitate prepared in this fashion was a molecular mix-ture of lead and titanium. It was freeze dried then heattreated in air at 750'C for 6 h. The resulting powderwas checked by x-ray diff'raction at room temperature toassure its crystal structure. It was found in the desiredtetragonal phase. Scanning electron microscopy foundthat the typical dimension of the crystallites was of theorder of a micron.

For the measurements on the lead I gyes edge, the pow-der was inixed with graphite in proportions that as-sured an edge step of nearly imity. 2o Since the absorptionlength of the material is 11 pm at this edge and the mate-rial was well dispersed in the graphite, we were confidentthat thickness effects would. not distort our data. 4o Thegraphite and PbTiOs ~ixture was dry pressed, producinga pellet that could be easily handled and placed withinour experimental apparatus. The data for the lead edgeXAFS were obtained in transmission at bea~hne X11-A

at the National Synchrotron Light Source.Transmission XAFS was not an option for the tita-

nium edge measurements. The absorption length of thematerial at the Ti X edge is 2 ym. Since the crystalliteswere not small compared to this length, thickness effectswould be introduced. 4o We therefore chose to prepare asample for fiuorescence. This was done by dry pressingthe pure sample and sintering it at 1000 ' C for 30 min inambient air. The final product was a yellow disk 15 mmin diameter and 1 min thick. The fiuorescence data were

also collected at Xll-A.In both experiments a Si(111) double crystal

monochromator in the nondispersive mode was used.The monochromator was detuned to 75% of the maxi-mum intensity for harmonic rejection. This was partic-ularly important for the titaniu~ edge experiment be-cause the rejected third harmonic could excite the leadL gyes edge.

Care was taken at every stage of the experiment toavoid systematic distortions to the data. The sample was

pure and homogeneous. A homogeneous spot in the beamwas chosen and the width of the slits defining the beamwas optimized for resolution. Gases were properly cho-sen for linear response in both detection chambers. Thesample was repositioned in the beam frequently duringthe course of the experiment. This frequent realignmentcompensated for any changing conditions in the experi-ment or in the beam.

III. EXPERIMENTAL RESULTS

The Ti and Pb XAFS spectra were measured at 13and 10 different temperatures, respectively, spanning thephase transition temperature. A large density of TiXAFS measurements were done near the transition tem-perature. Examples of the extended XAFS spectra atseveral temperatures are shown in Figs. 2 and 3 for thePb and Ti edges, respectively. For the sake of clarity,the spectra are presented after background removal, mul-

tiplied by the photoelectron wave number A:, and as afunction of k. The background removal and the deter-mination of the threshold energies, Eo, are discussed inthe next section. The important points to notice here

are that the Pb spectra at low temperatures have a very

good signal-to-noise ratio whereas the Ti data are noisy

50 NATURE OF THE FERROELECTRIC PHASE TRANSITION IN PbTi03 13 171

0

8 10

k (A'}l2 l4 l6

analysis is based on fitting the Fourier transforms of the-oretical calculations to those of experimental spectra, ina specified range in r space, where r represents the dis-tance to the probe. As a starting point for the theoreticalcalculation of the XAFS spectrum we used the room tem-perature structure of PbTiOs. Seen from the Pb probe(see Fig. 1), the first oxygen shell splits into three sub-shells with four oxygens each. The second, Ti, shell splitsin two with four atoms each. The Pb shell splits into twosubshells, one with two the other with four atoms. Fi-nally, the fourth (oxygen) shell splits into four subshells.The basic model extends up to 5.38 A. from the probeand includes 51 atoms with ll single, 34 double, and 14triple different scattering paths. The theoretical XAFSspectrum was expressed in the following form:4~

FIG. 2. The normalized Pb-edge XAFS multiplied by thephotoelectron vrave number I}: as a function of k for severalrepresentative temperatures. The vertical marks indicate thelimits of useful XAFS data (see text).

even at room temperature. As the temperature increasesthe intensities of the Pb spectra decrease remarkably andthe signal-to-noise ratio at large k decreases to a pointwhere the data are no longer useful. The limits of theuseful k range are indicated in Fig. 2. Except for changesin intensity the data do not display visible qualitativedifferences when going from temperatures far below totemperatures significantly above the phase transition.

IV. DATA ANALYSIS AND THE RESULTINGSTRUCTURAL PARAMETERS

A. Pb edge

The method of data analysis has been described byMustre et al.4~ 22 and by Hanske-Petitpierre et al.2s The

I I I I

II I I I

I

I I I I

I

I I I I

I

I I I I

I

I I

I I I I I I I I I I I I I

2 4 8 IO

FIG. 3. The normalized Ti-edge XAFS absorption multi-plied by the photoelectron wave number k as a function of kfor several representative temperatures.

y(k) = Im) n„SoE„(k)

x exp{i[kL„+8„(k) —s4k Cs„j}x exp( —2kios).

Here the summation index g goes over all differentscattering configurations; n is the number of equivalentpaths, L is the path length, S~~ is an overall amplitudefactor, o2 are the mean squared relative displacements,and Cs is the third cumulant of the probability distribu-tion function, 2o and is equal to the mean cubic relativedisplacement.

The scattering amplitude function E(k) and phaseshift 8(k) were calculated by the ab initio FEFF5 codeof Rehr et al.44 These functions depend on the specificgeometry of each configuration and on the participat-ing atoms. In order to calculate the XAFS spectra forstructures which differ slightly from each other we alsocalculated the first derivatives of the functions F(k) and8(k) with respect to each one of the geometrical param-eters that define the configuration. Using these functionsour programs fit the XAFS spectra for any structure thatdoes not deviate too strongly from the initial structure. 4~

The distortions from the room temperature structure in-volved in the present problem are small enough for thispurpose.

The model included four groups of parameters.(a) Structural parameters. These include two tetrago-

nal lattice parameters a and c and two distortion param-eters drab and dT; which measure the displacements of thePb and Ti atoms, respectively, along the (001) direction.The first two parameters were not used as variable pa-rameters in fitting theory to experiment. The Ti XAFSresults discussed further down show that these values donot change significantly with temperature. Furthermore,the tetragonal distortion hardly affects the Pb-0 and Pb-Ti distances. Thus, these values were taken as constantsequal to their values at 150 K, namely, c=4.17 A anda=3.90 A. The other two parameters were the main ob-jective of this work.

(b) The energy threshold Eo and the overall ampli-tude So2. Eo is not determined by theory at this stage tobetter than a few eV; similarly So2 can be off by +15%.Therefore, both parameters are treated as variables and

13 172 SICRON, RAVEL, YACOBY, STERN, DOGAN, AND REHR

since they are temperature independent their values wereadjusted so as to optiinize the fits at all temperatures.

(c) Mean squared relative displacements (ir~). Thenumber of 0 ~ parameters used at low and high tempera-tures was rliiFerent. At the lowest temperature and up toroom temperature four shells were fit and so we used six0~ parameters. One for the four nearest-neighbor oxy-gen bonds. Another for the other eight first shell oxy-gens. At high temperatures the total contribution of themore distant oxygens becomes small and the correspond-ing Debye-Wailer factor is ill defined. Thus, we imposedthe requirement that the ratio between the o~'s of theshort distance oxygens and the other oxygen subshells beconstant and equal to the low temperatures (15—300 K)value, namely 0.78. This ratio is expected to be inde-pendent of temperature significantly above the Einsteintemperature of the Pb-O bond. The Einstein temper-ature was found to be 252 K. Thus, the ratio obtainedat room temperature is expected to remain constant athigher temperatures. Two additional rr~ parameters wereused for the two Ti subshells. Since the XAFS contribu-tions of these shells extend over a relatively long range ink, the values of these parameters are well defined at alltemperatures. Finally, for the third and fourth shells weused just one o~ parameter each. The third and fourthshells were fitted only at low temperatures. At high tem-peratures their contributions were too small to be usedin the fitting process. The 0 ~ parameters of the multiplescattering paths were estimated in terms of the previousparameters. In summary, when four shells were fit sixa ~'s were used as variable parameters. At higher temper-atures when only two shells were fit, three o ~ parameterswere used.

(d) Mean cubic relative displacements (Cs). This pa-rameter refiects the anharmonicity in the relative vibra-tions between the probe and the neighboring atoms. It isexpected to become non-negligible only at high temper-atures. Cs was used only for the closest oxygens. Whenused for Ti shells its value was negligible even at thehighest temperatures.

The procedure for subtracting the background hasbeen described in detail elsewhere4s and will only be sum-marized here. In subtracting the background we used aspline with seven sections equally spaced in k space andwith eight corresponding parameters. The spline was ad-justed so as to minimize the difference between theoryand experiment at the r & 1.0 A region of the Fouriertransform. The theoretical and experimental spectrawere fit in the Fourier transform space. At low tem-peratures all four shells were used and the 6t extendedover 1.0 & r & 3.9 A. range. At high temperatures onlythe first two shells were used and the fit extended overthe range 1.0 & r & 3.6 k For completeness the two-shell fit was also performed at low temperatures. Theparameters obtained from the two types of fits were ingood agreement with each other and varied much lessthan the experimental error.

The number of relevant independent experimentalpoints N " ", where 6k and b,r are the k rangewith valid data and the fitting r range, respectively. Asseen in Fig. 2, Lk decreases with rising temperature. For

12K

L

12K— 300

523848

0

0000

0

i

0 2

r()FIG. 4. The Fourier transforms of the Pb XAFS spectra

X(r ) as a function of the distance to the probe r, for severalrepresentative temperatures. Dashed and solid lines repre-sent the experimental results and the two-shell theoreticalfits, respectively. (a) Absolute value, (b) real part, and (c)imaginary part. The temperatures are indicated in the fig-

ure. The vertical lines indicate the fitting ranges. Note thatat 12 K both four- and two-she11 fits are shown. The smalldifFerence between the two experimental curves is a result ofa leV difFerence in Eo.

the four-shell model N —24 and 20 at 12 and 300 K, re-spectively. For the two-shell model N —20 and 10 at12 and 848 K, respectively. The number of parametersused in the four-shell fit was nine and in the two-shell fitwas six with two additional constant parameters whichwere the same in all fits. Thus, in both types of fit andat aO temperatures the number of parameters was smallcompared to the number of relevant independent experi-mental points.

In Fig. 4 we show one example of the four-shell fit at12 K and four examples of the two-shell fit. The contri-butions of certain shells and subshells to the theoreticalspectrum at 12 K are shown in Fig. 5. All the subshellcontributions of the first two shells are shown explicitly

50 NATURE OF THE FERROELECTRIC PHASE TRANSITION IN PbTi03 13 173

LO

0)+

FIG. 5. The absolute values of the Fourier transforms ofthe 12 K Pb XAFS experimental spectrum and the contri-butions of various subshells and shells as a function of thedistance to the probe t.. The various contributions are in-dicated in the Sgure. Oq and Oq+ represent the short andlong distance Srst-shell oxygens and Oq represents the oxy-gens close to the Pb plane. Similarly Ti and Ti+ representthe short and long distance titanium atoms. Pb and 0 rep-resent the total single scattering contributions of the Pb andfourth-neighbor oxygens, respectively. (Notice that the theo-retical contributions do not add up to the experiment resultbecause these are absolute values. )

are in a plane perpendicular to the c axis. This plane isslightly offset from the midpoint between the Oz's in thec direction. The distance of these oxygens from the Tiprobe is 1.97 A..

The amplitude and phase functions E(k) and e(k)were calculated using the ub initio FEFF5 code4 andthe theoretical XAFS spectr»m was expressed in terms ofsix parameters. A single-energy threshold Eo, an overallamplitude factor So2, a single 02 for all three oxygen sub-shells, and three additional structural parameters wereused. These structural parameters were the distances ofthe Ti probe to the three oxygen subshells. From thesedistances one can calculate the a and c nuit cell dimen-sions and the displacement of the Ti atom from the centerof the octahedron. An attempt to use two o2's for theTi-0 bonds was made. The difference between the two

300K

730K

780K

together with the contributions of the third and fourthshells. Note that the contributions are shown in absolutevalue. Thus, they do not add up to the absolute valueof the experiment. It is clear from this figure, that thepeaks, seen in the absolute value of the Fourier trans-form of the experimental spectr»m, cannot be assignedto specific shells or subshells. However, it is seen thatthe contributions of the third and fourth shells are quitesmall below ~3.6 A. even at 12 K. At higher tempera-tures the relative contributions of these shells are evensmaller. This fact enabled us to confine our analysis athigh temperatures to the first two shells.

Finally, the energy threshold Ee was found to be atthe midpoint of the absorption step and So2=0.93. Thestructural and 02 parameters are discussed in Sec. IV C.

a

- {b)Q-

I

a ~ I a

la

Q =-

&x

0 =-IX

ri~'

a a a I a ~ ~ a I a a I

~

Ia

950K~ t ~ l a r

300Krr

730K

780K

950K

300K

730K

B.Ti edge&x

780K

Because the Ti-edge data were measured in the fiuores-cence mode, its quality was not as good as the Pb-edgedata. However, the first shell is well separated from themore distant shells. These data were therefore analyzedonly for the Srst shell and a simpler analysis procedurewas adopted.

As a starting point for the theoretical calculation of theXAFS spectr»m we used the room temperature structureof PbTiOs. 42 Seen from the Ti probe (see Fig. 1), thefirst shell splits into three subshells. Two of the pathsare to Oq atoms lying along the c axis at distances of2.39 and 1.77 A. from the Ti atom. The other four 02

950K

I. . . . I

2 0 3r (A)

I

4

FIG. 6. The Fourier transforms of the Ti-edge XAFS spec-tra X(r) as a function of the distance to the probe r, for severalrepresentative temperatures. Dashed and solid lines representthe experimental results and the theoretical Sts, respectively.(a) Absolute value, (b) real part, and (c) imaginary part. Thetemperatures are indicated in the Sgure. The vertical lines in-dicate the Stting range.

13 174 SICRON, RAVEL, YACOBY, STERN, DOGAN, AND REHR

was within the error brackets. An attempt to add a C3as in Eq. (1) did not yield any significant improvementin the fit nor did it significantly change the values of theother parameters.

The useful range in the photoelectron wave number kwas 8.0 A. i. The Hanning window sills in k space were2 A. i wide and ki weighting was used. The fitting rangein r space was from 1.0 to 2.2 A. Using these ranges thenumber of relevant independent experimental points isabout 7. Thus, using all six parameters as &ee variableswould be a serious strain on the information content. Infact the model was not so seriously information limited.Two of the variables Eo and S02 are independent of tem-perature. Thus, at the end, a fixed value was used foreach of these two parameters producing optimal fits tothe spectra at all 13 temperatures. In the initial fits thecr2 parameter was allowed to vary and the results werefitted to an Einstein model. In the final fits, the valuesobtained from the Einstein model were used as fixed 02parameters. Thus in these fits we used three variableparameters at each temperature, namely the three addi-tional structural parameters, and three parameters whichwere corinnon and fixed at all temperatures.

The background from the experimental data was re-moved using the method described elsewhere. Thebackground spline function was characterized by sevenparameters giving the spline values at seven equallyspaced points in k space. The theoretical and experi-mental spectra were fit in r space using a Levenberg-Marquardt nonlinear least squares minimization. TheFourier transforms of the experimental and theoreticaldata are shown for several temperatures in Fig. 6. Onesees that in this case the quality of background removaland the quality of the fits are not as good as in the Pb-edge data. We attribute this to the fact that the fiuores-cence data in this case are not as good as the transmis-sion Pb-edge data. The error analysis includes the un-certainties introduced by the poorer quality of the data.The two nonstructural parameters found in this analy-sis are the energy threshold at 3.5 eV below the mid-point of the absorption step and the amplitude factor of0.72+0.04. Because the the fiuorescence data were takenon a concentrated sample, this value must be correctedfor self-absorption in the sample. This is a 15'% effect,thus S02 ——0.83 6 0.05. The structural parameters arediscussed in the next subsection.

C. The mean squared relative displacements and thestructural parameters

0.0l 2

O.OI 0—

0.008

0%0.006

b0.004—

I I II

I I I II

l I 1 1I

i I I II

f I I

~CI 2pQ

suits reported below the systematic errors are dominantin determining the error brackets. Thus the scatter of theparameter values is small compared to the error brackets.

The temperature dependence of the Ti-0 cr is pre-sented in Fig. 7. The experimental points were obtainedwhen this parameter was allowed to vary. As mentionedbefore the final fits were done using the values on the Ein-stein curve. The o 2's of the near oxygens and the two Tisubshells relative to the Pb probe are shown in Fig. 8. Inall four examples the results are well fit with an Einsteinmodel. The Pb-02 0 is particularly large. This may beconsistent with large Pb vibration amplitudes observedin neutron difFraction. 35

The temperature dependence'of the third cumulant,Cs, of the Pb-02 bond is presented in Fig. 9. Up to300 K this parameter is negligible, but as the temperatureincreases further, it increases (within the error brackets)as T Thi. s parameter refiects the anharmonicity of thePb-02 vibrations and therefore the asyminetric charac-ter of the Pb-02 distance distribution function. Thisfunction was calculated using the three lowest cumulantsobtained in the fit: the mean Pb-02 distance, 02 andC3. The results for various temperatures are presented inFig. 10. We define the distortion distribution function asthe distance distribution function with the abscissa equalto half of the diagonal distance between 02 and 02+minus the Pb-O2 distance (see upper scale in Fig. 10).At low temperatures the function is synUnetric. As thetemperature increases the function becomes asymmetric.The function is much steeper at the short distance sideand as the temperature rises the peak moves towardssmaller distortions.

The temperature dependence of the local unit cell di-mensions are shown in Fig. 11. The c dimension is equalto the sum of the Ti to oxygen distances along the c di-rection and the a dimension can be calculated &om theTi to 02 distance taking into account the displacementof the Ti in the c direction. Also shown by circles arethe geometric average of the local cell dimensions, given

Before discussing the structural parameters obtained,we discuss error estimation. The errors have been care-fully estimated using two difFerent methods with compa-rable results. In these estimates, the effects of random ex-perimental noise, and systematic experimental and the-oretical errors have been included. Results of two dif-ferent samples were analyzed and the parameter valuesagree better than the error brackets. A detailed discus-sion of the error estimation technique is available andin U%'XAFS 2.0 software documentation. In all the re-

0.002—

,F,Einstein temperature: 695 + 9OK

I I I I I i j I I i I I I I I I I

200 400 600temperature ( K )

I

800 I000

I'IG. 7. The temperature dependence of the o of the Tito oxygen bond. They were obtained when this parameterwas allowed to vary. The solid line is an Einstein St to theexperimental points. The transition temperature is indicatedby the arrow.

50 NATURE OF THE FERROELECTRIC PHASE TRANSITION IN PbTi03 13 175

0.05(a)

Bond - Length Distortion (A)

0.04—0.6 0.4 0.2 0.0

515K

-0.2 -0.4

0.03—

0.02—

0.01—

~150K!if

!' ', ', 473K

724K.'I

0.00

0.02

(b) Tc

2.2 2.4I i . I

2.6 2.8Distance (A)

3.0 3.2 3.4

0.01

0.000.03

(c)

0.02—

0.01—

0.000 200

I I I I

400 600Temperature (K]

I

800

FIG. 8. The temperature dependence of Pb edge cr 's. Thesolid D's represent the experimental results. The solid linesare Einstein model fits to the data. The Einstein tempera-tures obtained are shown in the figures. (a) o 's of the Pb tonear oxygen bond. (b) o 's of the Pb to near Ti bond. (c)n 's of the Pb to far Ti bond.

FIG. 10. The calculated Pb to nearest oxygen distance dis-tribution function (DDF) at various temperatures. The ver-tical line indicates half the diagonal distance between Oqand Oq+. The upper scale is the distortion. Notice that theshortest Pb-Oq distance reaches a minimum value at 150 Kand remains almost constant all the way to 848 K.

by (aac)!!. The results are quite surprising because theyshow that the cells are strongly distorted even far abovethe transition temperature. In fact the local tetragonaldistortion changes by only ~15% from 12 to 950 K. Thisin itself shows that the local structure above the tran-sition differs significantly from the cubic structure. Thefact that in the cubic phase the cells are locally tetragonaland the structure on the average is cubic suggests thatthe crystal consists of correlated nanoregions of distortedcells with different distortion orientations.

The error bars shown are much larger than the randomfiuctuations of the data points, refiecting the fact that theerrors are domir!ated by systematic effects. The relativevariation of the data is more reliable than indicated bythe error bars. In Fig. 11 the average of the local cell di-

0.012 II

! I

Q Q1Q — (C3/T ) = 1.4X10 (A /K )

0.008—

0.006—c

O 0.004—

0.002—

0.000 = 0

—.0020

I I I I

400 600Temperature (K)

I

200I

800

FIG. 9. The temperature dependence of the Pb to nearestoxygen third cumulant, C3. The solid line represents a T fit.

4,2—U)C(D

co 4.0 —, 6'0

)t X—X

0 ()

3.80 200

0I ! I

400 600Temperature (K)

I

800 1000

FIG. 11. Temperature dependence of the a and c unit celldimensions. Solid lines represent the average dimensions ob-tained from x-ray difiraction (Ref. 30). The dashed line rep-resents the geometric average of the lattice constants. The x'sand diamonds represent the c and a axes obtained from theTi-edge XAFS. The circles are the average lattice constantsabove 7, as obtained from the XAFS results by (a c) & .

SICRON, RAVEL, YACOBY, STERN, DOGAN, AND REHR

mensions is consistent, within the uncertainties, with theaverage structure measured by dHFraction. These agree-ments verify the accuracy of the XAFS results

The average Pb distortion and the distortion at thepeak of the distortion distribution function, DDF, areshown in Fig. 12. Notice that at low temperatures theXAFS results are slightly larger than the x-rays2 so andneutron diffractionM results. However at higher temper-atures the distortions obtained Rom x-ray diffraction fallofF roughly as (T—T,)~~2. The results obtained from neu-tron difFraction are systematically larger and are nonzeroeven above the phase transition. These results were ana-lyzed allowing for disorder in the Pb positions similar tothe distortions present below the transition temperature.The average distortions obtained from XAFS follow theneutron difFraction results below T,' but above T, they areslightly larger and are about 50% of the value at 12 K.The distortions at the peaks of the DDF remain constantalmost all the way to T, . Above T, they are larger thanthe mean distortions by 0.08 A and are more than 7070of the peak distortion at 12 K.

The displacement of Pb along the c axis can be cal-culated from the Pb distortions by taking its compo-nent along the c direction. The average displacementscalculated are 0.53, 0.48, and 0.27 jt at 12, 473, and848 K, respectively. The calculated displacements above

500 K are only estimates because the Pb and Oz vi-brations may be correlated producing a larger displace-ment than the one calculated here. For this reason wedisplay the distortions as defined above rather than thedisplacements along the c direction.

The temperature dependence of the lead to the neartitanium distance is shown in Fig. 13. Within the exper-imental error this distance does not change with temper-ature. It corresponds to a 0.16(.03) A. vertical displace-ment of the Ti atoms from the center of the Pb cube. Thedisplacement of the Ti atoms from the center point be-tween the Oq oxygens as measured from Ti-edge XAFSis presented in Fig. 14. This displacement can also be

c

CD

C3

3.3H '-03

3.34 I-

3.30 '

200 400 600Temperature (K)

800

FIG. 13. The temperature dependence of the Pb to nearTi distance. +'s represent x-ray data (Refs. 32 and 42) audSolid A' s, the results obtained from Pb-edge XAFS.

calculated from the Pb-edge XAFS using the Pb-Ti andPb-02 distances. At low temperatures the two valuesmatch but as temperature increases the Pb edge value isconsiderably smaller than the Ti-edge result. The reasonfor this mismatch is that at high temperatures the vibra-tions are large as can be seen in Fig. 8. If correlations areimportant the Ti displacement can not be reliably calcu-lated from the Pb-edge XAFS. One possible scenario isthe following: Suppose at high temperature the Pb vi-bration in the z direction is large and correlated withthe 02 displacement in the (zy) plane. Then when thePb moves closer the oxygens move away to make room.In this case the actual average Pb displacement in the zdirection will be larger than the one we calculate. Thiswill account for the smaller calculated Ti displacement.The results in Fig. 14 show that the displacement of theTi atoms &om the center point between the Oq's changesonly little below the transition temperature. It then de-creases faster in the vicinity of the transition and remainsclearly nonzero above the transition temperature. The

0o 0.2--CO

Cl

7 II

I

400200 600 800

0 00-+------~O ~ p

+

k '~k

++

\tt

I t

0.4

c0.3

C2

—0.2Ci

Il

T

( )

+.'+

Temperature {K)

FIG. 12. The temperature dependence of the Pb distortion.The open circles indicate the distortion calculated from thepeak of the DDF. The solid A's are the average distortionsobtained from the Pb-edge XAFS data, the +'s are from x-raydiffraction (Refs. 32 aud 42), aud the x's are from neutrondifFraction (Ref. 36).

200 400 600Temperature (K)

800 1000

FIG. 14. The displacement of Ti from the midpoint be-tvreen the two O~ planes. +'s and x's represent the x-ray(Refs. 32 aud 42) aud neutron diffraction (Ref. 36) results,respectively. Open Q's represent the results obtained &omTi XAFS.

50 NATURE OF THE FERROELECTRIC PHASE TRANSITION IN PbTi03 13 177

displacement at 950 K is still about 70%%up of the displace-ment at 12 K. The results of the x-ray42 s2 and neutronmeasurements are also displayed in Fig. 14. The neu-tron diffraction and Ti XAFS results agree fairly wellbelow the transition temperature. However above thetransition the neutron results suggest that the Ti goesback on center in contrast to the results of both the Pband Ti XAFS.

The fact that the local distortions are nonzero abovethe transition while the order parameter is zero showsthat the distortions must have orientational disorder.Partial disorder must be present also below the transi-tion because the local distortions are larger than the av-erage. However since the Pb data can be fitted with justtwo Ti subshells at all temperatures, the displacementsof neighboring Ti and Pb atoms must remain correlatedover at least several »»it cells. If the displacements of Pband Ti were not correlated at least four Pb-Ti distanceswould be needed to fit the data.

Several steps have been taken to verify that a localcubic symmetric structure contradicts the XAFS dataabove T,. We calculated directly the theoretical PbXAFS spectra using the cubic structure with the x-raydetermined»»t cell (a = c = 3.97 A.) as a starting point.We then used four variable parameters: Eo, the 02's of

CO

C

Xl

Pb-0 and Pb-Ti, and a Ti phase shift correction. The fitresults at the highest temperature are shown in Fig. 15 incomparison with the fit of the distorted structure. The fitwas performed in the range 3 ( k ( 10.5 jt ~. As men-tioned above, there is no signal above k = 10A ~. The fitof the cubic model is clearly much worse. The least sumof squares in the cubic fit is five times worse than the cor-responding value for the tetragonal fit. When we allowedthe distortion parameters to vary, they converged withinthe experimental error to the values we found previously.A similar test was performed on the Ti-edge XAFS. Theresults are shown in Fig. 16. The fit to the cubic modelis two times worse than to the distorted model and thefact that the latter had one additional variable parametercannot by itself account for this improvement.

We also checked the eight-site model for the Ti dis-placements above T, and verified that it contradicts theXAFS data. Self-consistent total energy calculationswithin the local density approximation for PbTiO& in-dicate that the large tetragonal strain in the ferroelec-tric phase stabilizes the tetragonal structure. Above T,where no long range tetragonal strain exists, the calcu-lation suggests that the Ti atoms will randomly displacealong the eight (ill) directions rather than the six (001)directions. A fit to the Ti-edge data above T, was per-formed assuming the eight-site model. In this case sym-metry requires that the six oxygen backscattering pathssplit into two groups of three paths. These path lengthscan be parametrized by a single parameter, br, whichdefines the displacement from the cell center in a (111)direction. In this model the fit is 1.5 times worse thanthat of the tetragonal model. The result of this fit isshown in Fig. 16(c) and as can be seen the tetragonal fitis much better than the eight-site model.

a ~ s ~ I ~ I ~ ~ I ~ ~

~ ~ I l ~ ~ l % l ~ ~ ~ I/

~ ~ 0 ~

r ~a

CO

C

0 1 2 3 4

r(A)F&G. 15. The Bt of theory (solid line) and experiment

(dashed line) of the absolute, real, and imaginary spectra ofthe Fourier transforms of the Pb-edge XAFS results at 848 K.(a) The distorted model. (b) The cubic model. The verticallines represent the fitting range. The two experimental curvesare a little difFerent because Eo eras allured to vary in the cu-bic fit in order to optimize it. Notice that the distorted modelfits the experimental results much better than the cubic one.

FIG. 16. The 6t of theory (solid line) and experiment(dashed line) of the absolute value of the Fourier transformsof the Ti-edge XAFS results at 950 K. (a) The tetragonallydistorted model, (b) the cubic model, and (c) the eight-sitemodel. The vertical lines represent the fitting range. Noticethat the tetragonally distorted model fits the experimentalresults much better than the others.

13 178 SICRON, RAVEL, YACOBY, STERN, DOGAN, AND REHR 50

V. DISCUSSXGN

It is important to understand the differences among theresults obtained &om x-ray diffraction, neutron diffrac-tion, and XAFS. As long as some long range order ispresent, diffraction spectra are essentially composed ofb functions, even if structural disorder is present. Inthis case there is also a weak broad background whichis usually ignored. The disorder does however affect therelative integrated intensities of the various b functions.Thus in principle, the spectra do contain information onthe disordered structural distortions. To obtain this in-formation it is necessary to introduce the disordered dis-tortions to the fit model and to have s»fRcient range ink space in order to resolve the disordered distortions.Since disordered distortions were not taken into accountin most x-ray models, these experiments provided infor-mation only about the average structure. In this caselocal distortions show up only as infiated Debye-Wailerfactors. In analyzing the neutron diffraction results localdistortions were taken into account and such distortionswere detected in the paraelectric phase. M However, thequantitative accuracy of the results is inferior to thatof XAFS. In neutron diffractionM the range of momen-t»m transfer used in this case was about 9 A. i and, aspointed by the authors, it is difficult to distinguish be-tween a large anharmonic mean thermal amplitude anddisordered displacements. XAFS is in a better state todistinguish the two apart because the corresponding mo-ment»m range is between 32 and 21 A. i, two times themaxim»m photoelectron wave number at 12 and 950 K,respectively.

In the present case XAFS was able to provide both av-erage distortions and o2's. The average local distortionsof Pb and Ti at 800 K are 0.18 and 0.20 A. , respec-tively. The value of the cr of the Ti-0 bonds is 0.12 A.In the case of Pb the width of the DDF can be deter-mined visually &om Fig. 10. It is found that above T,the DDF falls to half of its maximum value at zero distor-tion. Thus, the DDF's of Ti and Pb are clearly difFerent&om a broad distribution with a maximum at zero whichwould be generally expected for a simple displacive tran-sit;ion.

This result has two important consequence. First, thelocal distortions cannot be a result of defect or surfacestrains. Such eff'ects would tend to smear the distribu-tion function. Also, the defect and surface effects ob-served so far have been strongly temperature dependentand showed up only close to T, in contrast to presentresults. Second, it seems that the local distortions area result of local instabilities rather than the ferroelec-tric phase transition. This behavior has been observedin a more pronounced way in KTaOs. Nb. In that casethe Nb displacements were almost independent of tem-perature and Nb concentration indicating that the Nbdisplacements takes place whether the system undergoesa phase transition or not.

The Pb-02 distribution function shown in Fig. 10 dis-plays one very interesting feature. In spite of the factthat the width of the distribution function grows strongly

with temperature the minimum Pb-Q2 distance doesnot change above 150 K. This means that at this dis-tance the repulsion between the two grows very steeply.This further suggests that the displacement of the Pbtowards the center may be due simply to the increasein the vibrational amplitude combined with the wall-likerepulsion &om the nearest oxygen.

The temperature dependence of the local distortionssuggests the following qualitative picture. At very lowtemperatures the crystal is»»formly distorted and thelocal and average distortions are equal to each other. Asthe temperature rises clusters with difFerent polarizationorientations form. Their number and volumes grow whilethe polarization per unit cell may somewhat decrease.Thus, the local distortions decrease much less than theaverage polarization. Above the transition, the local dis-tortion, at the peak of the distortion distribution func-tion, is smaller but well above zero and the contributionsof the difFerent types of clusters average out to zero. Thepolarization autocorrelation function is expected to haveat least two length scales. One is related to the clustersizes, the other to the correlation of the order param-eter. The latter scale would be large compared to thecluster size below and, at least, in the vicinity above T„and would grow critically as the transition temperatureis approached. The XAFS results provide direct infor-mation that the length scales of the cluster sizes extendto at least a few unit cells at all temperatures measured.

Our results indicate that locally little happens as thelong range order disappears above T, . The local distor-tions change only slightly and the correlations betweenthe displacements in neighboring cells remain about thesame. If the range of interactions between atoms thatcauses the local instability is smaller or of the order ofthe cluster size, then one can understand why the lo-cal structure changes only slightly above T, . These re-sults agree with the theoretical calculations for PbTiOsthat predict that the ferroelectric transition has an es-sential order-disorder element, 4s ii as local displacementsremain above T,. However we do not find the predicted4schange in the displacements &om tetragonal to rhombo-hedral above T,. The persistence of the tetragonal dis-placement is indicative that the range of the interactioncausing the instability is smaller than the cluster size.

Soft modes and central peaks are measured at a well-defined point in the reciprocal space of the crystal. Thelinear dimension of reciprocal space involved is usuallyonly about 1 x 10 4 to 1 x 10 2 of the linear size of theBrillouin zone. This corresponds to a correlation lengthof 10~—104»nit cells in real space. In contrast the linearsize of the clusters in real space is expected to be muchsmaller, corresponding to a much larger &action of theBrillouin zone. Thus, soft modes can coexist with localdistortions and the temperature dependencies and originof central peaks, observed by diff'raction and light scat-tering tec&»piques, can be difFerent &om those of the localdistortions measured by XAFS.

The fact that the lattice is clearly distorted locallyabove T, for at least as high as 190 K above the transi-tion temperature and the fact that the distortions thereare more than YOFFE& of the distortions at 12 K, clearly

50 NATURE OF THE FERROELECTRIC PHASE TRANSITION IN PbTi03 13 179

show that this system can by no means be described asa classic displacive ferroelectric. Any microscopic theorydiscussing the ferroelectric phase transition must eitherexplain or at least take these facts into account. However,it should be equally emphasized that PbTiOs cannot bedescribed as a pure order-disorder ferroelectric either forseveral reasons. First, the local distortions do changenear T', . Second, PbTiO& displays a clear underdampedzone center soft mode both beloved and above the tran-sition temperature. Third, as mentioned above the pro-portionality constant relating the dielectric susceptibilityand (T —T,) of an order-disorder ferroelectric is about 1to 2 orders of magnitude smaller than that of displaciveferroelectrics. In PbTiOs the experimental coefficient iseven larger than that calculated for a purely displacivetransition &om the Lydane-Sachs-Teller relation. s4 Thus,neither a pure displacive nor a pure order-disorder theoryis sufficient to describe the properties of PbTi03.

VI. SUMMARY AND CONCLUSIONS

In this paper we presented the XAFS spectra ofPbTiOs measured at various temperatures both belowand above the ferroelectric phase transition. Both Tiand Pb x-ray absorption edges were used. The spectrawere analyzed in detail by fitting theoretical calculationsto the experimentally measured XAFS spectra. The low-temperature Pb-edge XAFS were fit up to a distance of3.9 A. using four shells. The theoretical model includedcontributions &om single and multiple scattering config-urations. At high temperatures only two shells were in-cluded because the contributions of the other two shellswere too small and too noisy to be useful. The Ti-edgefiuorescence XAFS spectra were noisier than the Pb spec-tra and were therefore analyzed for the first shell only.Yet, the Ti analysis provided important information onthe temperature dependence of the local unit cell struc-ture and the Ti off-center displacement.

Direct quantitative information on the local structuresurrounding both probes was obtained and it can be sum-marized as follows.

(1) Locally i»it cells remain tetragonal even above thetransition temperature with the cell distortions &om thecubic structure changing by less than 1Vp over the entiretemperature range.

(2) The Pb atoms are displaced &om the correspond-ing oxygen planes of the ideal perovskite structure in the[001] direction both below and above the phase transi-tion temperature. The distortion distribution function(DDF) is symmetric and off center at low temperatures.As the temperature rises the distribution function broad-ens asynunetrically leaving the shortest Pb oxygen dis-tance constant. The peak of the distribution moves tosmaller values but remain at 75'%%uo of the value at low

temperatures even at the end of the measureinent range,85 K above the transition temperature.

(3) The Ti atoms are displaced from the center of thePb cubes. Within the experimental error these displace-ments are independent of temperature.

(4) The Ti-edge XAFS show that the Ti atoms aredisplaced relative to the oxygen octahedra both belowand above the transition temperature. The displacementvaries little below the transition, more rapidly close tothe transition and saturates at a value of about 70%%uo

&om the low-temperature value at least up to the endof the experimental temperature range, 190 K above thetransition.

(5) Above T, the tetragonal distortions remain wellcorrelated in neighboring cells and this correlation ex-tends a finite length producing a cluster which is largerthan the range of the interaction that causes the insta-bility.

The results show that the DDF's of both Pb and Tihave well-defined peaks at off zero displacements evenabove T, The f. act that the peaks are well defined im-plies that the local distortions are not a result of defectsor strained surfaces. Furthermore, the microscopic mech-anism creating the distortions is local instabilities ratherthan long range cooperative interactions.

In sunimary we are led to the conclusion that evena pure perovskite like PbTiOs which was considered tobe an exemplary displacive ferroelectric is by no meanspurely displacive. The fact that both Pb and Ti atomsare displaced &om their corresponding center of inversionpositions well above the transition temperature showsthat the system is clearly disordered but the fact thatthe magnitude of these displacements is affected by thephase transition shows that the displacive nature of thetransition cannot be ignored either.

The observation of local distortions above T, poses thequestion whether all structural phase transitions have anecessary requirement of a local instability. If this is thecase then all such transitions have an essential element oforder-disorder with a possible enhancement of the localdistortions by long range interactions. The fact that evena classic example of a displacive transition still shows alarge remnant local distortion above the transition lendsinuch weight in favor of this speculation.

ACKNOWLEDGMENTS

This work was supported in part by DOE Grant No.DE-FG06-90ER45425 and BSF Grant No. 92-00348.Beamline Xll-A is supported by DOE Grant No. DE-FG05-89ER45384. We wish to thank M. Newville andY. Zhang for their assistance in collecting data and Dr.R. Cohen, Professor. I. Bersuker, Professor D. Thouless,and Professor B. Spivak for stimulating discussion.

13 180 SICRON, RAVEL, YACOSY, STERN, DOGAN, AND REHR 50

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