Order-Disorder Behavior in the Phase Transition of PbTiO3B. Ravel, N. Sicron�, Y. Yacoby�, E.A. Stern, F. Dogan�� and J.J. RehrPhysics Department FM-15 University of Washington, Seattle WA 98195�Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, Israel 91904��Department of Materials Science and Engineering, University of Washington, Seattle WA98195 y(May 11, 1995)AbstractWe have quantitatively measured the temperature dependence of the localdistortions of PbTiO3 crystals below and above the structural ferroelectricphase transition using X-ray Absorption Fine Structure (XAFS) measure-ments. Both Pb and Ti edges were measured, providing quantitative de-terminations of the displacements of the atoms within the unit cell. Thesedistortions vary little with temperature below the transition and decrease onlyslightly above the transition temperature. In the paraelectric phase, the Pband Ti distortions are about 50% and 70% of the corresponding low tem-perature values respectively. These results show that an essential elementof order-disorder is present even in this ferroelectric crystal which displaysthe soft mode behavior and a dielectric constant typical of displacive ferro-electrics. The presence of the local distortions suggests that the displacementsshould have at least two correlation length scales, one associated with the thelocal distortions, the other with the order parameter.1

I. INTRODUCTION.In a material undergoing a structural phase transition, the average distortion producingthe lower symmetry structure is represented by an order parameter. However, the distor-tions manifested on a local scale can be quite di�erent from the average structure. There aretwo paradigmatic models for these phase transitions. If the order parameter shows the sametemperature behavior as the local distortions in a range including the transition tempera-ture, the transition is called `displacive'. If the magnitude of the local distortions remainunchanged while the order parameter shows the behavior of the phase transition, the tran-sition is called `order-disorder'. In an order-disorder transition only the relative orientationsof the local distortions change at the transition temperature. The distinction between thetwo is not absolute. Due to critical uctuations, local distortions will be �nite around thephase transition of a crystal undergoing a displacive transition. Similarly, the magnitude ofthe local distortions in an order-disorder crystal is never precisely constant because of elasticinteractions. Di�erent mechanisms drive these two types of transitions. In the displacivecase, an instability resulting from long range cooperative interactions produces the localdistortions. In the order-disorder case the local distortions are due to local instabilities.Understanding whether the distortions are driven by long-range or by local interactions isessential for understanding a phase transition.Perovskite crystals with an ABO3 type molecular structure have long been considereddisplacive ferroelectrics.1;2 The main evidence for this type of behavior has been the exis-tence of a Brillouin zone center soft mode3 both below and above the transition temperatureand the large size of the dielectric constant. Soft modes have been observed in many per-ovskite ferroelectrics.4;5 In some the soft mode is underdamped even within a few degreesof the transition temperature.6;7 Excluding critical phenomena, the dielectric constant ofcrystals undergoing second order or weakly �rst order transition follows a Curie-Weiss law,� / CjT�Tcj where Tc is the actual or extrapolated second order phase transition tempera-ture. It turns out that the proportionality constant is much larger in displacive than in2

order-disorder type ferroelectrics.1 The coe�cient, C, has values typical of displacive ferro-electrics for perovskite crystals. Consequently perovskite ferroelectrics have been describedas displacive,3;8;9 wherein the atoms vibrate about centrosymmetric positions in the cubicphase. Below the transition temperature the structure distorts and the distortion increaseswith decreasing temperature.During the last two decades much evidence accumulated indicating that local distortionsin perovskite crystals also exist above the phase transition temperature.10;11 The �rst reportwas by Comes et al.12 They found di�use X-ray lines above the transition temperature tothe cubic phase indicating the existence of some form of structural disorder. Many typesof optical experiments followed, including Raman scattering13{16, hyper-Raman17, Infra-redre ectivity,18 and optical refractive index.19 These experiments indicate the possibility of lo-cal distortions at temperatures far above the phase transition temperature. However, noneof these experiments provide direct quantitative information on the magnitude and temper-ature dependence of these distortions. The �rst experiments to provide direct quantitativeinformation on the temperature dependence of the distortions below and above the transi-tion in perovskite crystals were X-ray Absorption Fine Structure (XAFS) measurements20on KTa1�xNbxO3 crystals21{23 with x = 0:09. In these experiments, the positions of theniobium atoms were measured relative to the oxygen octahedra, to the potassium atomsat the corners of the cube and to the �rst tantalum neighbors. The results show that theniobium atoms are displaced by about 0:145�A in a h111i direction relative to the surround-ing atoms. The magnitude of the displacement changes only 12% from about 16K belowthe transition (Tc = 85:6K) to 215K above it. These results show conclusively that thetransition has an order-disorder character. A pure order-disorder type transition involvingthe niobium atoms alone will not account for some of the experimentally observed featuresin this material such as the soft phonon24 and a large inverse temperature coe�cient of thedielectric constant.25 It seems that, in this material the host KTaO3 provides a displacivecomponent which, combined with the Nb ions, may account for the properties of this system.PbTiO3 is a pure perovskite which has a well behaved soft mode and a large dielec-3

tric coe�cient characteristic of an ideal displacive ferroelectric. In the low temperature,ferroelectric phase the crystal is tetragonal with a maximum c/a ratio of 1.075. At roomtemperature the lead and titanium atoms are displaced from their respective oxygen planesin a h001i direction by 0:49�A and 0:32�A, respectively. Above 763K PbTiO3 is cubic.26 Thetransition at this temperature is weakly �rst order.The soft modes of PbTiO3 have been investigated as a function of temperature6;27 andpressure.7 The soft mode frequency decreases as the transition temperature is approached.It is underdamped at all temperatures and its frequency is about 55 cm�1 at the �rst ordertransition temperature.6 Near the transition temperature a central peak is observed.28 Thearea of this peak seems to grow as the transition temperature is approached. This result isinterpreted as a manifestation of some disorder on both sides of the transition temperature.Furthermore, the dielectric constant in this material is found to be even larger than thatexpected from the Lydane-Sachs-Teller relation, suggesting that even the ordinary displacivetype model would not be su�cient to account for the dielectric constant. Perturbed AngularCorrelation Spectroscopy29 and high-resolution refractive index measurements30 have alsofound evidence for disorder in the paraelectric phase of PbTiO3. Recent neutron di�rac-tion experiments32 further suggest the presence of disordered distortions above the phasetransition in PbTiO3.PbTiO3 has long been o�ered as a classic example of the displacive type of phasetransition.6 The discovery of order-disorder behavior in other perovskite ferroelectrics andrecent experimental evidence of disorder in the paraelectric phase of PbTiO3 suggests thatthe displacive model may not be wholly appropriate. We investigated the temperature de-pendence of the local structure of PbTiO3 in a temperature range including Tc. Here wediscuss the utility of the X-ray Absorption Fine Structure technique in resolving structurale�ects in phase transitions. We then present direct evidence for an order-disorder compo-nent to the phase transition in PbTiO3 and discuss our results in the context of other dataon this material. 4

II. XAFS AS A TOOL FOR STUDYING PHASE TRANSITIONSXAFS is ideally suited for investigating microscopic e�ects in phase transitions. XAFSis not dependent on long-range order, thus structurally disordered crystals as well as amor-phous and liquid materials can be studied using the same analytical techniques as are usedwith well-ordered materials. It is sensitive to order on an �angstrom scale, thus directlymeasures local e�ects that are inaccessible to many other experimental techniques. XAFSis sensitive to atomic species, thus local order can be measured about di�erent componentsof a material. We exploited this local sensitivity for investigating order-disorder behavior inthe paraelectric phase of PbTiO3.XAFS measures the energy dependence of the absorption coe�cient for the excitationof a deep core electron into the continuum. The X-ray absorption coe�cient is measuredas a function of energy in a range below and above the excitation energy of some deep corestate of an atom in the material. At energies below the excitation energy, the absorptioncoe�cient is small. At the excitation energy, a large jump in the absorption coe�cient ismeasured. This coe�cient then diminishes at higher energies. If the resonant atom were infree space, the measurement would be of this decaying step function. In condensed matter, inthe presence of surrounding atoms, the absorption coe�cient has an oscillatory �ne structurein the region around and above the edge.20 The frequency spectrum of these oscillations isdependent upon the distances to the surrounding atoms and is caused by the photoelectronwave function scattering o� the surrounding atoms, producing interference patterns withthe outgoing wave.Since the information about the local structure is contained in the oscillations and notin the smooth atomic background, we must isolate the oscillatory part of the spectrum byseparating it from the smooth part of the absorption coe�cient.36 We then �t parameterizedtheoretical standards37 to the oscillatory structure. From this �tting procedure we extractstructural information.The interference phenomenon measured by XAFS can be expressed as a sum over all5

scatterings from the atoms surrounding the excited atom. Considered as a sum of scatteringpaths, the theoretical XAFS spectrum is expressed20 in the following form:�(k) = ImXj NjS20Fj(k) exp[i(2kRj +�j(k))] exp(�2k2�2j � �(k)2Rj ) (1)The subscript j denotes the various scattering con�gurations in the problem. Thephotoelectron wave number, k, is related to the energy of the incoming photon byk = q2me(E � E0)=�h2 . Here E is the energy of the incoming photon and E0 is the ex-citation energy. This equation contains both atomic and condensed matter contributions.The functions Fj(k), and �j(k) are respectively the scattering amplitudes and scatteringphase shifts for each scattering con�guration. �(k), the photoelectron mean free path, isrelated to the lifetime of the excited state. These functions depend on the characteristicsof the atomic species in the material and are obtained from theoretical calculations.37 Thecondensed matter contributions to the equation are Nj, the coordination of the surroundingatoms, Rj, the mean inter-atomic distances, and �j, the mean square displacements aboutthe Rj . We determine these condensed matter terms in our �tting procedure to extract thelocal structural information.This formalism can be used directly to reveal information on the nature of a phasetransition. Consider the eight site model proposed by Comes, et al.12 for the phase transitionsof BaTiO3 and KNbO3. In this model the B cation, Ti or Nb, is displaced in a h111idirection at all temperatures. In the rhombohedral phase, these displacements are correlatedsuch that all B cations are displaced in the same direction. As the temperature is raisedinto the orthorhombic phase, the B cation is allowed to occupy positions in two adjacenth111i directions, but it may not occupy the other six. Bragg di�raction would observe theaverage of these two allowed h111i displacements as a h110i displacement. In the tetragonalphase, the B cation is allowed to occupy four adjacent positions which average to a meandisplacement in a h100i direction. Finally in the paraelectric phase, all eight sites are allowed.This eight-site disorder averages to no net displacement, thus an observed cubic structure.With no sensitivity to long-range order, XAFS does not perform the averaging that yields6

this progression of structural phase transitions. In each phase the local structure about theB cation is essentially identical. The h111i displacement of the B cation will split the sixequidistant oxygen backscattering paths of the ideal perovskite structure into two groupsof three identical backscattering paths. Further coordination shells will be similarly split.Because this structure of the near coordination shells about the B cation remains essentiallythe same in all phases, the XAFS spectrum measured in the various phases would showlittle or no change in the measured values of Nj and Rj in Eq. 1. Thus XAFS would clearlyreveal the order-disorder nature of this phase transition. XAFS measurements38 on KNbO3do indicate distortions above Tc, although those results were not analyzed in detail.The deep core excitation measured by XAFS is extremely short lived. This feature is alsoexploited when using XAFS to study local structure. The excitations of the K electron intitanium and the LIII electron in lead both have mean lifetimes shorter than a femtosecond.41Consequently the XAFS measurement is a series of snap shots of the instantaneous structureof the material. The femtosecond lifetime of the core hole excitation is very short on thetime scales of ionic motion and is too short for relaxation of the lattice after the ionizationof the excited atom.The local structure about the titanium and lead sites in PbTiO3 in the tetragonal con-�guration is shown in Fig. VI. We shall refer to the oxygen backscatterers which are nearlycoplanar to the lead atom as O1. The oxygens which are nearly coplanar to the titaniumatom are O2. We use + and � to designate relative path lengths to atoms in the samecrystallographic site. At the titanium site, the oxygen shell is split into three paths. Thefour O2 paths are of equal length, 1:97�A. Along the c-axis is one O1� at 1:76�A and one O1+at 2:38�A. In our titanium K edge experiments, we measured the temperature dependenceof these bond lengths. At the lead site, the twelve oxygen near neigbors are split into threedistances. Four O1 atoms are at 2:79�A, four O2� are at 2:51�A, and four O2+ are at 3:21�A.Similarly the titanium and lead shells were each split into two distances. These variousdistances were measured as functions of temperature.Our data was taken in uorescence20 on a sintered pellet of pure PbTiO3, thus avoid-7

ing concerns about small particles. The complete details of our sample preparation andexperimental procedures are presented elsewhere.40 A thorough explanation of the analysisof our data is presented there as well. We present details and examples of each stage of ouranalysis. III. RESULTSOur results show that large distortions persist in the local structure at a temperature190K above the ferroelectric to paraelectric phase transition. We show that the tempera-ture dependence of the local structures about the titanium and lead sites is indicative of asigni�cant order-disorder contribution in the phase transition on PbTiO3. In this sectionwe present results of our measurements of the local structures about the titanium and leadsites as a function of temperature and through the phase transition.The length of the axes of the unit cell are measured from the Ti-O bond lengths of thetetragonal con�guration in Fig. VI. The c axis is the sum of the Ti-O1� and the Ti-O1+bond lengths. The a axis length is twice the length of the Ti-O2 bond length minus a smallcorrection for the distortion that raises the titanium atom out of the plane of the O2's. Thetemperature dependence of the local cell dimensions is shown in Fig. VI. The local celldimensions are plotted with the average cell dimensions as measured by X-ray di�raction.31We see that the local and average structures are the same only at very low temperature.As the temperature is raised, the two measurements deviate signi�cantly. Above Tc, thetetragonal distortion to the perovskite structure persists. The c axis length shows no changewith temperature outside of the error bars. The a axis is about 14 �A shorter than the c axiseven at 950K, � 190K above Tc. From this measurement alone it is clear that the localstructure is signi�cantly di�erent from the average structure, indicating that the crystal iscomposed of small regions of correlated, distorted cells. These distorted regions must thenbe uncorrelated in orientation above Tc.Fig. VI shows the displacement of the titanium atom from the midpoint between the two8

O1 atoms as measured by XAFS and by X-ray31 and neutron32 di�raction. Again the localand average distortions agree at low temperature and the two diverge at higher temperature.The two di�raction measurements disagree slightly in the ferroelectric phase, but both showa distortion which vanishes at Tc as expected of a displacive transition. The local distortionmeasured by XAFS is constant throughout the ferroelectric phase. This local distortionpersists into the high temperature phase, diminishing only � 30% above Tc. Again wesee a discrepancy between the local and average structures, suggesting an order-disordercomponent to this transition.The displacement of the lead from the midpoint of the planes de�ned by the O2 atomsis plotted in Fig. VI. Here the result of the lead edge measurement is compared with theX-ray and neutron data. The displacement of the lead atom vanishes at Tc in the x-raydata. The temperature behavior of the neutron data shows the disorder about the leadsite as modeled in that analysis.32 The XAFS measurement of this displacement is againdi�erent from the di�raction measurements. Like the displacement of the titanium atom,the lead atom displacement is constant throughout the ferroelectric phase and diminishesonly �50% in the paraelectric phase.Fig. VI shows the temperature dependence of the shorter of the two Pb-Ti bond lengthsas measured in the lead edge XAFS data. This bond length is constant at all temperatureswithin the error bars. The longer Pb-Ti bond length, which is not plotted, is likewiseconstant. This measurement again shows the discrepancy between the local and averagestructures. The average structure is that of a displacive transition wherein centrosymmetryis restored above Tc. The local structure is distorted at all temperatures. Figures VI throughVI taken together suggest a robust model for the temperature behavior of the local structure.The tetragonal framework of the metal atom is a rigid structure that shows little temperaturedependence. The a axis has only a small thermal expansion as shown in Fig. VI. That thePb-Ti bond is constant shows that the titanium atom is not moving within the tetragonalbox of lead atoms. At Tc, the displacements of the lead and titanium atoms relative totheir surrounding oxygen atoms decrease slightly. We conclude that the displacement of the9

oxygen octahedron relative to the rigid tetragonal frame of the metal atoms relaxes slightlyat Tc. Because the local distortions persist into the paraelectric phase, we must consideran order-disorder component to the phase transition. We will discuss this in greater detailin section IV. Because the displacement of the oxygen octahedron relative to the metalatoms relaxes somewhat at Tc, a purely order-disorder model for this transition would beinappropriate.The error bars shown in �gures VI through VI are much larger than the random devia-tions of the data points from the straight lines that describe the temperature behavior of thelocal structure. This re ects the fact that systematic e�ects dominate the determinationof the error bars in the analysis of the data. The relative variation of the XAFS resultsshows that it is more reliable than the systematic error bars. The accuracy of the XAFSmeasurement is demonstrated, for instance, in Fig. VI. The circles are the average of thecell lengths, given by (a2c) 13 , in the high temperature phase. The cube of this is the volumeof the unit cell. Our measurement of the unit cell volume is consistent with the average unitcell volume with an accuracy better than that suggested by the size of the error bars.Considerable care was taken to verify our model for the behavior of the local structure.40We performed �ts to the data at both edges assuming a local cubic structure in the hightemperature phase. At both edges the quality of the �t was two times worse by a �2 test whena locally centrosymmetric and cubic con�guration was assumed than when local distortionswere introduced. We also considered the possibility of anharmonicity in the variation of thebond lengths. Signi�cant anharmonicity was measured only in the Pb-O2� bond. This wasintroduced as a higher moment in the distribution of that bond length and was included inthe model that produced the results presented here. Anharmonicity was also considered inthe tests of a cubic con�guration. Thermal anharmonicty without local structural disordercannot account for the temperature behavior of the XAFS data.We also considered the possibility of an eight-site disorder model for the titanium atomin the high temperature phase. Recent self-consistent total energy calculations within thelocal density approximation indicate that the large tetragonal strain in the ferroelectric10

phase of PbTiO3 stabilizes the distorted tetragonal structure.39 Above Tc, in the absence ofthis large tetragonal strain, the calculations predict that the titanium atom will randomizeamong the eight h111i directions as in the Comes model for BaTiO3. The titanium edgedata would be sensitive to the di�erence between the h111i and the h100i con�gurations42.The �t assuming the eight-site model was 1.5 times worse than the tetragonal model. Dueto this signi�cant di�erence, the XAFS results favor a tetragonal distortion for the hightemperature local structure. IV. DISCUSSION.It is important to understand the di�erences among the results obtained from X-raydi�raction, neutron di�raction and XAFS. Given any long range order, di�raction spectraare essentially composed of delta functions, even when structural disorder is present. Ina disordered case there is also a weak, di�use background which is usually ignored. Thedisorder does a�ect the relative integrated intensities of the various delta functions andthe di�raction spectrum does contain information on the disordered structural distortions.To obtain this information the disordered distortions must be introduced to the �t modeland the data must have su�cient range in k-space to resolve these disordered distortions.Since disordered distortions were not considered in the X-ray di�raction models for PbTiO3discussed in the literature, these experiments provided information only about the averagestructure. In this case local distortions show up only as in ated Debye-Waller factors. Anal-ysis of the neutron di�raction found local distortions in the paraelectric phase.32 However,the quantitative accuracy of these results is inferior to that of XAFS. In the neutron di�rac-tion data the range of momentum transfer was limited and, as stated by the authors, itis di�cult to distinguish between a large anharmonic mean thermal amplitude and disor-dered displacements. XAFS is more capable of distinguishing between these as the range ink-space in our experiments is considerably larger.40The local distortions cannot be a result of defect or surface strains. Such e�ects would11

tend to smear the position distribution functions measured in our experiments40. Also, thedefect and surface e�ects observed have been strongly temperature dependent and showedup only close to Tc. Our results do not show this temperature dependence. It appears thatthe local distortions result from local instabilities rather than long range interactions. Thisbehavior has been observed explicitly in KTa1�xNbxO3. In that case, the niobium displace-ments were almost independent of temperature and of niobium concentration indicating thatthe range of the interaction that causes the displacement is of the order of the length of theunit cell axis.The temperature dependence of the local distortions suggests the following qualitativepicture. At very low temperature, the crystal is uniformly distorted and the local and averagedistortions equal one another. As the temperature rises clusters with di�erent polarizationorientations form. The clusters of di�erent polarization grow while the polarization per unitcell may diminish. In this way, the local distortions decrease much less than the averagepolarization. Above the transition, the local distortions are smaller but well above zero.The contributions from the di�erent types of clusters average to zero. The polarizationautocorrelation function is thus expected to have at least two length scales. One is relatedto the cluster sizes, the other to the correlation of the order parameter. The latter scalewould be large compared to the cluster size below and, at least, in the vicinity above Tc,and would grow critically as the transition temperature is approached. The XAFS resultsprovide direct information that the length scales of the cluster sizes extend to at least a fewunit cells at all temperatures measured.We �nd that little happens to the local structure when the long range order disappearsabove Tc. The local distortions change only slightly and the correlations between the dis-placements in near neighbor cells remain about the same. If the range of interactions betweenatoms that causes the local instability is smaller or of the order of the cluster size, then onecan understand why the local structure changes only slightly above Tc. These results agreewith the theoretical calculations for PbTiO3 that predict that the ferroelectric transitionhas an essential order-disorder element,11;39 as local displacements remain above Tc. We do12

not �nd a change39 in the displacements from tetragonal to rhombohedral above Tc. Ratherthe persistence of the tetragonal displacement indicates that the range of the interactioncausing the instability is smaller than the cluster size.11Soft modes and central peaks are measured at well de�ned points in the reciprocal spaceof the crystal. The linear dimension of reciprocal space involved is usually only about 10�4to 10�2 of the linear size of the Brillouin zone. This corresponds to a correlation length of102 to 104 unit cells in real space. We expect that the linear size of the clusters in real spaceis at least several unit cells. This corresponds to a much larger fraction of the Brillouin zone.Thus, soft modes can coexist with local distortions.The fact that the lattice is clearly distorted locally above Tc for at least as high as 190Kabove the transition temperature and the fact that the distortions there are 50-70% of thedistortions at 12K, clearly show that this system is not a classic displacive ferroelectric.Any microscopic theory for this ferroelectric phase transition must consider these facts.However, it should be equally emphasized that PbTiO3 cannot be described as a pure order-disorder ferroelectric either for several reasons. First, the local distortions do change nearTc. Second, PbTiO3 displays a clear underdamped zone center soft mode both below andabove the transition temperature. Third, as mentioned above the proportionality constantrelating the dielectric susceptibility and (T � Tc) of an order-disorder ferroelectric is aboutone to two orders of magnitude smaller than that of displacive ferroelectrics. In PbTiO3 theexperimental coe�cient is even larger than that calculated for a purely displacive transitionfrom the Lydane-Sachs-Teller relation28. Thus, neither a pure displacive nor a pure order-disorder theory is su�cient to describe the properties of PbTiO3.V. SUMMARY AND CONCLUSIONS.In this paper we presented XAFS measurements of PbTiO3 measured at various tem-peratures both below and above the ferroelectric phase transition. Both Ti and Pb X-rayabsorption edges were measured. Direct quantitative information on the local structure13

surrounding both probe atoms was obtained. We summarize:� The local structure of the unit cells remains tetragonal even in the paraelectric phase.Locally the a and c axes do not show any change of length associated with the ferro-electric phase transition. The a axis is 14 �A shorter than the c axis even at 950K.� The titanium atoms are displaced in a h001i direction relative to the oxygen octahedraboth below and above the transition temperature. The displacement is constant belowthe transition and relaxes to about 70% of the low temperature value in the paraelectricphase.� The lead atoms are displaced in a h001i direction both below and above the phasetransition temperature. The displacement remains about 50% of the low temperaturevalue at 850K.� The Ti atoms are displaced from the center of the lead tetragons. Within the experi-mental error these displacements are independent of temperature.� Above Tc the tetragonal distortions remain well correlated in neighboring cells andthis correlation extends a �nite length producing a cluster which is larger than therange of the interaction that causes the instability.We conclude that even a pure perovskite like PbTiO3, which was considered to be anexemplary displacive ferroelectric, is not purely displacive. That both Pb and Ti atomsare displaced from their corresponding centrosymmetric positions well above the transitiontemperature shows that the system is clearly disordered. That the magnitude of thesedisplacements is a�ected by the phase transition shows that the displacive nature of thetransition cannot be ignored.The observation of local distortions above Tc in PbTiO3 suggests that all structural phasetransitions might require a local instability. If so, then all such transitions have an essentialorder-disorder component. Long-range interactions might then enhance the local distortions.14

That even a classic example of a displacive transition shows a large local distortion in itsparaelectric phase lends much weight in favor of this speculation.VI. ACKNOWLEDGEMENTS.This work was supported in part by DOE Grant No. DE-FG06-90ER45425 and by NSFGrant No. 9200348. Our experiments were performed at NSLS Beamline X11-A, which issupported by DOE Grant No. DE-FG05-89ER45384. We wish to thank M. Newville andY. Zhang for their assistance in collecting data, M. Wilber for his help with the manuscript,and Dr. R. Cohen, and Profs. I. Bersuker, D. Thouless, and B. Spivak for stimulatingdiscussion.

15

REFERENCESyPresent Address: Princeton University, Dept.of Chemical Engineering and Princeton Ma-terials Institute, Princeton, NJ 085421M.E. Lines and A.M. Glass,\Principles and Applications of Ferroelectrics and RelatedMaterials", Clarendon Press, Oxford, 1977.2R. Blinc and B. Zeks, \Soft Modes in Ferroelectrics and Antiferroelectrics", Elsevir, New-York, 1974.3W. Cochran, Adv. Phys., 9, 387 (1960).4 J.F. Scott, Rev. of Mod. Phys., 46, 83 (1974).5 J.D. Axe, J. Harada and G. Shirane, Phys. Rev., B1, 1227 (1970).6G. Burns and B.A. Scott, Phys. Rev., B7, 3088 (1973).7 J.A. Sanjurjo, E. Lopez-Cruz and G. Burns, Phys. Rev., B28, 7260 (1983).8 E. Pytte and J. Feder, Phys. Rev., 187, 1077 (1969).9R. Migoni, H. Bilz and D. Bauerle, Phys. Rev. Lett., 37, 1155 (1976).10B.E. Vugmeister and M.D. Glinchuk, Rev. of Mod. Phys., 62, 933 (1990).11 I.B. Bersuker and V.Z. Polinger, \Vibronic Interactions in Molecules and Crystals",Springer Series in Chemical Physics 49, Edited by V.I. Goldanski, (Springer-Verlag, New-York, 1989).12R. Comes, M. Lambert, A. Guinier Solid State Comm., 6, 715 (1968)13A.M. Guitet, M. Lambert and A. Guinier, Solid State Comm., 12, 1053 (1973).14Y. Yacoby and S. Just, Solid State Comm., 15, 715 (1974).15Y. Yacoby, Z. Physik, B Condensed Matter, 31, 275 (1978).16

16Y. Yacoby, Z. Physik, B Condensed Matter, 41, 269 (1981).17H. Vogt and H. Uwe, Phys. Rev., B29, 1030 (1984).18 F. Gervais, Ferroelectrics, 53, 91 (1984).19G. Burns and F. Dacol, Ferroelectrics, 37, 661 (1981).20 E.A. Stern and S.M. Heald, Chapter 10 in Handbook of Synchrotron Radiation, edited byE.E. Koch (North-Holland, Amsterdam, 1983) Vol. 1, p.995.21O. Hanske-Petitpierre, E.A. Stern and Y. Yacoby, J. of Physics C, 8, 675 (1986).22 J. Mustre, Y. Yacoby, E.A. Stern, J.J. Rehr and M. Dell'ariccia, Physica, B158, 263(1989).23O. Hanske-Petitpierre, Y. Yacoby, J. Mustre-deLeon, E.A. Stern and J.J. Rehr, Phys.Rev., B44, 6700 (1991).24 S.K. Manlief and H.Y. Fan, Phys. Rev., 5, 4046 (1972).25D. Rytz, U.T. H�ochli and H. Bilz, Phys. Rev., B22, 359 (1980).26G. Shirane and S. Hoshino, J. of Physical Society of Japan, 6, 265 (1951).27G. Shirane, J.D. Axe, J. Harada and J.P. Remeika, Phys. Rev., B2, 155 (1970).28M.D. Fontana, K. Wojcik, H. Idrissi and G.E. Kugel, Ferroelectrics, 107, 91 (1990).29G.L. Catchen, S.J. Wukitch, D.M. Spaar and M. Blaskiewicz, Phys. Rev., B42, 1885(1990)30W. Kleeman, F.J. Sch�afer and D. Rytz, Phys. Rev., B34, 7873 (1986)31A.M. Glazer and S.A. Mabud, J. Appl. Cryst. 12, 49 (1979)32R.J. Nelmes and W.F. Kuhs, Solid State Comm., 54, 721 (1985); R.J. Nelmes, R.O. Piltz,W.F. Kuhs, Z. Tun and R. Restori, Ferroelectrics, 108, 165 (1990).17

33B. Rechav, N. Sicron, Y. Yacoby, B. Ravel, M. Newville and E.A. Stern, Physica C 209,55 (1993).34B. Ravel, E.A. Stern, Y. Yacoby and F. Dogan, Jpn. J. of Applied Physics, 32 Suppl.,782 (1992).35A.M. Glazer and S.A. Mabud, Acta Cryslalogr., B34, 1065 (1978).36M. Newville, P. Livins, Y. Yacoby, J.J. Rehr and E.A. Stern, Phys. Rev., B47, 14126(1993).37 J.J. Rehr, R.C. Albers and S.I. Zabinsky, Phys. Rev. Lett., 69, 3397 (1992); J.J. Rehr,S.I. Zabinski and R.C. Albers, Phys. Rev., B41, 8139 (1990).38K.H. Kim, W.T. Elam and E.F. Skelton, \Optical Fiber Materials and Processing", Editedby J.F. Fleming et al. M.R.S. Proceedings, 172, 291 (1990).39R.E. Cohen and H. Krakauer, Ferroelectrics, 136, 65 (1992).40N. Sicron, B. Ravel, Y. Yacoby, E.A. Stern, F. Dogan and J.J. Rehr, To be published inPhys. Rev. B41O. Keske-Rakonnen and M.O. Krause, Atom. Data and Nucl. Tables, 14, 139, (1974)42Y. Yacoby and E.A. Stern, Ferroelectrics, 125, 689 (1992).43O. Hanske-Petitpierre, PhD. Thesis, University of Washington, (1986)18

FIGURESFIG. 1. Schematic structure of PbTiO3 around the titanium (a) and lead (b) sites. The whitecircle represents a lead atom, the shaded circle represent titanium atoms, and the solid circlesrepresent oxygen atoms.

catemperature (K)axislengths(Angstroms)

100080060040020004.44.34.24.143.93.8FIG. 2. Temperature dependence of the a and c unit cell dimensions. Solid lines represent theaverage dimensions obtained from X-ray di�raction.35 The 4's and 's represent the c and a axisobtain from the Ti edge XAFS. The �'s are the average lattice constants above Tc as obtained fromthe XAFS results by (a2c) 13 19

temperature (K)Tidisplacement(Angstroms)

100080060040020000.50.40.30.20.10FIG. 3. The displacement of titainum atom from the mid point between the two O1 planes.'s and 4's represent the X-ray31 and neutron di�raction32 results respectively. �'s represent theresults obtained from the titanium K edge XAFS.

Temperature (K)Pbdisplacement(Anstroms)

80060040020000.50.40.30.20.10FIG. 4. The temperature dependence of the h001i displacement of the lead atoms with respectto the mid point between the two adjacent O2 planes. The 's are from X-ray di�raction31, The+'s are from neutron di�raction32 and the �'s are the results obtained from the lead LIII edgeXAFS.

20

temperature (K)Pb-Tibondlength(Angstroms)

100080060040020003.53.453.43.353.3FIG. 5. The temperature dependence of the Pb to near Ti distance. 4's represent X-ray data31and �'s represent the results obtained from the lead LIII edge XAFS.

21