J . Phys. Chem. 1993,97, 2368-2377
2368
Structures of the Ferroelectric Phases of Barium Titanate G. H. Kwei,' A. C. Lawson, and S. J. L. Billinge Los Alamos National Laboratory, Los Alamos, New Mexico 87545
S.-W. Cbeong AT& T Bell Laboratories, Murray Hill, New Jersey 07974 Received: November 17, 1992
The structures of the three ferroelectric phases of BaTi03 have been determined by Rietveld refinement using powder diffraction data collected at a spallation neutron source. The correlation between refined atomic displacements and thermal parameters, which has hampered previous structure determinations, has been partially alleviated by using data which extend over a wide range of d spacings. Data collected at a large number of sample temperatures provide information about the temperature dependence of the ferroelectric displacements and changes in the oxygen octahedra which surround the Ti ions. The temperature dependence of the thermal parameters gives atomic Debye-Waller temperatures that are remarkably similar to those for the cations in the high- Tcsuperconductors. Our results are insensitive to predictions of a soft-mode displacive model; however, values of the anisotropic thermal parameters do not support the order-disorder model suggested by Comes et al. Powder extinction and profile coefficients from the structural refinements show pronounced minima and maxima, respectively, near the phase transitions and provide information about the temperature dependence of the mosaic structure and the strain in the different phases.
I. Introduction Since the discovery of ferroelectricity in BaTi03 in 1945 by Wul and Goldman,' it has been one of the most exhaustively studied materials.* At high temperatures, it has the classic AB03 (with Ba2+ as A and Ti4+as B) perovskite structure. This is a centrosymmetric cubic structure with A at the corners, B at the center, and the oxygens at the face centers. However, as the temperatureis lowered, it goes through successivephase transitions to three different ferroelectric phases, each involving small distortions from the cubic symmetry (shown in Figure 1). At 393 K, it undergoes a paraelectric to ferroelectric transition to a tetragonal structure, it is orthorhombic between 278 and 183 K and, finally, it is rhombohedral below 183 K. Each of these distortions can be thought of as elongations of the cubic unit cell along an edge ([OOl]or tetragonal), along a face diagonal ([Ol 11 or orthorhombic), or along a body diagonal ([ 1111 or rhombohedral). These distortions result in a net displacement of the cations with respect to the oxygen octahedra along these directions. It is primarilythese displacementsthat give rise to the spontaneous polarization in the ferroelectric phases. Earlier theory suggested that the structural instabilities arise from the small Ti4+ ion 'rattling" around in the octahedral oxygen cage;3however, this is difficult to reconcile with the T i 0 bond lengths which do not differ much from the sum of the Ti4+and 02-ionic radii.4 Instead, recent electronic structure calculations by CohenS suggest that it is the competition between covalent and ionic forces involving Ti and 0 that leads to these instabilities. Since the structures of the ferroelectric phases of BaTi03 are closely tied to their ferroelectric properties, these structures also have been the subject of extensive study via X-ray and neutron diffraction techniques. Generally accepted structures are now available for all three phases at selected temperatures. However, no information is available on the temperature dependenceof the Ti displacements with respect to the oxygen octahedra or on the axial ratio of the octahedra. In addition, there has always been some question as to whether the atomic displacementsused in the structural models are correct. The problem is that the small displacements in the atomic positions are closely correlated with the thermal parameters in the structure refinement, and different
Cubic
Tetragonal
Orthorhombic
Rhombohedral
Figure 1. Schematic view of the structure of the different phases of BaTiO3. The Ba ions are located at the corners, the Ti ions at the ccll center, and the 0 ions at the face centers. The arrows show the direction of polarization along the cubic [OOl]direction for the tetragonal phase, the [ O l l ] direction for the orthorhombic phase (the A-centered orthorhombic cell used in the refinements is superimposed), and the [ 111) direction for the rhombohedral phase. The deformations in the ferroelectric structures are exaggerated for the sake of clarity.
sets of displacements and thermal parameters provide stable refinements of the structure.2a~d~cEven if the model for the displacements is *correct", there is the further question of the relevanceof the 'static" structure obtained from the Bragg peaks in a system that is known to display so much structural instability. An alternative approach has been used by Comes et a1.,6 who analyzed the diffuse scattering of X-rays from BaTiO3. They suggested that the Ti ion displacements are disordered along the eight possible directions corresponding to the eight faces of the oxygen octahedra. For the rhombohedral phase, all the displacements within a domain are ordered along one direction, giving rise to a polarization along the [ 11 11 direction. For the orthorhombic phase, these displacements become ordered along two adjacent directions, giving rise to a polarization along the [Ol 1J direction, and for the tetragonal phase, along four adjacent
0022-3654/93/2097-2368S04.00/0 0 1993 American Chemical Society
Ferroelectric Phases of Barium Titanate I
1
11
4 4
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The Journal of Physical Chemistry, Vo1. 97, No. 10, 1993 2369
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,
.
,
,
,
,
,
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
d-SPACING (A)
Figure 2. Part of the neutron diffraction data for BaTi03 at several different temperatures. Panel a represents data collected for the rhombohedral phase at 130 K, panel b represents data collected for the orthorhombic phase at 270 K, and panel c represents data collected for the tetragonal phase at 300 K. Points shown by plus (+) marks represent datacollectedon the+153O detectorbankofHIPD. Thecontinuousline through the data is the calculated profile from Rietveld refinement. Tick marks below the data indicate the positions for the allowed reflections. The lower curve in each panel represents the difference between the observed and calculated profiles. No Bragg peaks from impurities are observable for this sample.
directions togivea polarizationalongthe [001] direction. Finally, at high temperatures, the displacements are disordered along all eight directions, resulting in no net polarization in the cubic paraelectric phase. In this article, we report structures for all three ferroelectric phases of BaTiO3 at a series of temperatures in order to study the evolution of the structure through the different phases and phase boundaries. The structures are obtained from Rietveld refinement’ using powder neutron diffraction data collected at a spallation source. The wide range of momentum transfer available from such an epithermal source provides data over a wide range of d spacingswhich reduces the problem of parameter correlation in the structure refinement. Nevertheless, the parameter interaction problem still persists, and for all phases, a similar, but incorrect, structure with smaller Ti displacements and compenstaing larger thermal parameters provides stable refmementswith nearly as low an agreement factor. We conclude by analysis the changes in bond lengths, structural distortions, strain, and domain size across the different phases and how ferroelectricity in the different phases is affected. In a future article, we will report on short-range structural correlations that can be obtained from a pair distribution function analysis.
II. Experimental Procedure Polycrystallinesamples of BaTi03were prepared by standard solid-statereaction in air. Stoichiometricmixtures of high-purity BaCO3 and Ti02 powder were thoroughly ground, pressed into a pellet, and calcined for 3 h at 1 100 OC. The pellet was reground,
repelletized,and then sintered for 20 hat 1270 OC. The sintering was repeated for 40 hat 1320 OC. Finally, thesinteredpellet was cooled to room temperature at the rate of 60 OC/h. Neutron diffraction data for a 4.3-g powder sample were collected over a series of temperatures using f 1 5 3 O and f 9 0 ° detector banks of the high-intensity powder diffractometer (HIPD) at the Manuel Lujan Jr. Neutron Scattering Center (LANSCE) at the Los Alamos National Laboratory. The diffractometer constants were calibrated to the lattice constant of CaFz [5.463 85(2) A at 302 K],which was in turn obtained from X-ray powder diffraction measurements using the Cu Ka radiation wavelengths given by Deslattes and Henins,B and the incident spectrum was measured using the scattering from a vanadium rod. Samples were cooled with an Applied Physics Laboratory helium refrigerator, and the temperature was controlled with a Lakeshore Cryogenics DRC-93 temperature controller. Three separate sets of runs were made: the first consisted of relatively long data collections (approximately 110 pA/h) with eventual pair density function analysis in mind; these were taken in the order 300, 270, 190, 180, 170, 130, 15, 230, and 320 K. These were followed 6 weeks later by shorter runs (approximately 35 pA/h) in theorder 250,210,150,100,70,40, and 20 K and several days after that in the order 290,280, and 350 K. Sample data and subsequent fits are shown in Figure 2. Structural models correspondingto the appropriate space groups (P4/mmm for tetragonal, Amm2 for orthorhombic, and R3m for rhombohedral) were refined using the Generalized Structure Analysis System (GSAS)Rietveld code.9 The use of Rietveld profile analysis of time-of-flight neutron diffraction data obtained at a spallation source, using data over the large d-spacing range of 0.35-6.6 A,provides the best opportunity to reduce the problems encountered with the correlation of various structural parameters in structure refinement. In addition to the lattice constants and the atomic parameters, we also refined sample absorption, powder extinction, and the isotropic strain term in the profile coefficients. The absorption for a cylindrical sample is calculated according to an empirical formula given by Rouse et a1.I0 and by Hewat.” The extinction is calculated according to a formalism developed by Sabine et a1.12 and accounts for the primary extinction effect within the crystal grains. The profile functions approximate line shapes in time-of-flight diffraction data and include an isotropic strain broadening term which can be refined to give the percent strain.13 The backgrounds were modeled with a nine-term cosine Fourier series with refinable coefficients. The latest recommended values of neutron scattering lengths of 0.507, 4.3438, and 0.5803 X 10-12 cm for Ba, Ti, and 0, respectively, were used.14 The value for Ba is somewhat smaller than the previously accepted value of 0.525 X cm. All differ considerably from those refined and used by Harada et a1.15
III. Results from Structural Refinement a. Rhombohedral Phase. Several structural studies of the rhombohedral phase of BaTiO3 have been reported. These used both neutron powder diffraction at 77 K16 and neutron singlecrystal diffraction at 132 and 196 K17 but were limited to data over relatively small ranges of d spacings. In our structure refinements, we took as initial values atomic coordinates given by Schildkamp and Fischer” and then refined lattice constants, atomic positions, anisotropic thermal parameters, absorption (constrained to be equal in oppositebanks), extinction (constrained to be equal for all banks), and isotropic strain in the profile coefficients, in addition to the usual background terms and scale factors. Initially, when only isotropic thermal parameters were varied, the refinements converged to a solution characterized by small Ti displacements(hTi a -0.005) with largeTi thermal parameters [Le., &,(Ti) > Ui,,(O) > Q,,(Ba)]. However, when anisotropic
2370 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
Kwei et al.
TABLE I: Resulting Structural Parameters for the Rhombohedral Phase of BaTi03 (Space Group R3m). sample temp, K lattice constants a, A a,deg
cell volume V,A3 atomic positions and thermal parameters Ba(O,O,O) UII
UI,
uwv
15
20
40
70
100
130
150
170
180
4.00360(25) 4.0036(4) 4.0035(4) 4.0042(4) 4.0042(4) 4.00385(25) 4.0057(4) 4.0041(3) 4.0043(3) 89.839( I ) 89.840(2) 89.843(2) 89.837(2) 89.836(2) 89.843(1) 89.846(2) 89.852(1) 89.855( 1) 64.172( 12) 64.170( 18) 64.168( 19) 64.200( 19) 64.201(20) 64.184( 12) 64.272(21) 64.196( 12) 64.206( 12)
-1.4(2) -3.9(2) -1.4(2)
-1.0(2) 4.2(2) -1.0(2)
-1.5(1) 4.1(1) -1.5(1)
-1.1(2) -4.3(2) -1.1(2)
-1.2(2) 4.3(2) -1.2(2)
-0.01 28(4) 2.8(3) 0.6(3) 2.8(3)
-0.0120(7) -0.01 53(6) -0.01 55(6) 2.5(4) 1.9(2) 0.14(4) 0.1(4) -0.5(4) -0.4(4) 2.5(5) 1.9(2) 1.4(5)
0.0109(4) 0.01930(23) 2.3(2) 1.4(3) 0.4(4) 0.7(2) 2.0(3)
0.01 16(5) 0.0195(3) 1.7(3) 1.9(4) 0.1(7) 0.4(4) 1.8(4)
0.0088(6) 0.0183(4) 3.9(3) 0.8(4) 0.2(6) 1.6(3) 2.8(4)
5.3113.92 3.619
8.0615.66 1.214
8.2215.73 1.165
-0.5(1) 4.7(1) -0.5(1)
1.2(2) -5.7(2) 1.2(2)
-0.1(1) 4.9(1) -0.2(1)
0.0(2)
-0.0107(5) -0.0147(4) 4.7(4) 1.9(3) 0.8(5) -0.9(3) 4.7(5) 1.9(3)
-0.01 36(6) 3.5(4) -2.0(5) 3.4(5)
-0.0140(4) 3.0(3) -0.5(3) 3.0(3)
-0.01 30(5) 3.6(3) 0.4(3) 3.6(3)
0.0086(6) 0.0175(4) 3.6(3) 1.6(5) 1.0(6) 1.4(4) 2.9(4)
0.01 13(7) 0.0200(3) 2.8(3) 2.4(5) 0.8(7) 0.8(4) 2.5(4)
0.0088(4) 0.0185(2) 3.8(2) 0.6(3) 0.4(4) 1.3(2) 2.7(3)
0.0093(6) 0.0181(4) 4.1(3) 0.6(4) 0.4(7) 1.0(4) 2.8(4)
0.0084(4) 0.0185(3) 4.1(2) 0.7(3) 0.1(4) 1.3(3) 3.0(3)
0.0092(5) 0.0189(3) 4.0(2) 1.2(3) 0.4(4) 1.0(3) 3.0(3)
8.5215.87 1.180
8.4815.91 1.161
5.0613.66 3.294
8.4715.87 1.176
5.0613.63 3.349
5.0313.61 3.248
-5.0(1) 0.0(2)
T ~ ( ' / ~ + ~ / T2 +I k, ~I t r I /2+ k T t ) k T l
UI I
ut,
Uw,
0('/2+~o,'l2+Axo,Azo) AX0 h 0
UII = u22
u33
UI2 u13
= u22
uwv RwplRcxpl% Xwd2
Units for thermal parameters are 1000 X
A2. Numbers in parentheses following refined parameters represent 1 standard deviation.
thermal parameters were allowed, we discovered that the most stable solutions all resulted in a larger Ti displacement (-0.015) and equivalent 'isotropic" thermal parameters, defined as onethird the trace of the diagonalized thermal parameter tensor, U,,(O) > U,,(Ti) > U.,(Ba). The correlations between the various atomic parameters are all small enough that they do not present a problem in the structure refinement;however, it appears that there is a local minimum in which the smaller Ti displacements are compensated for by larger Ti thermal parameters, with correspondingchanges in the oxygen displacements and thermal parameters. Allowing anisotropic thermal parameters appears to alleviate this problem, giving solutions with larger Ti displacements, smaller equivalent thermal parameters, and better agreement. This is a problem that exists for all the other ferroelectric phases as well. Thus, all subsequent refinements were carried out with anisotropic models; when desired, values of the equivalent isotropic thermal parameters were then calculated by taking one-third the trace of the diagonalized anisotropic thermal parameter tensor. All refinements, with both isotropic and anisotropic thermal parameters, converged rapidly, and those with anisotropicthermal parameters gave the structural parameters shown in Table I. These parameters show little change over the temperature range 15-1 80 K, and the results for a sample temperature of 130 K are in moderate agreement with the 132 K data from Schildkamp and Fischer." For example,they find a = 4.004(3) A (assuming a = 89.87O), while we find a = 4.0034(3) A and a = 89.839( 1)". For the atomic displacements,they find A x ~= i -0.01 11(3), Ax0 = 0.0110(2), and AZO = 0.0180(2), while our results yield -0.0153(3), 0.0080(4),andO.0183(2),respectively. Asignificant discrepancy appears to exist for both the Ti and 0 ion displacements; our values give a larger value of the lattice polarization. Table I1 lists all elements of the correlation matrix for atomic parameters which have absolute magnitudes greater than 0.5. As expected, the Ti and 0 displacementsand the diagonal Ti thermal parameters are most strongly correlated. While correlation is severe, none of the elements exceeds 0.9, a value that has been found still to give reasonable refinements.Is Matrix elements from the single-crystal diffraction refinement from Schildkamp and Fischerl' are included for comparison. In addition to these correlations, there are also correlations between elements of the reciprocal metric tensor and the diffractometerconstants,between
TABLE II: Selected Terms in the Correlation Matrix from the Least-Squares Structural Refiaement of the Rhombohedral Phase of BaTi03 at 130 K Compared with Those from Ref 17. ~
parameters UII (Ba)-k(O)
UII ( B ~ ) - U I ~ ( B ~ ) UII(B~-UII(O) UdBa)-k(O) UdBa)-Ul1(0) UdBa)-U13(0) k(Ti)-Ul [(Ti) Ax(Ti)-Az( 0) UIl(Ti)-Az(O) UdTi)-k(O) Ax(O)-WO) k(O)-U1 I(0) 4 1(0)-U33(0)
~~
correlation matrix element this work ref 17 0.582 -0.877 -0,532 -0.536 0.531
0.61
-0.54 0.788 0.796 0.769 0.594 -0.776 -0,774
0.54 0.68 -0.52
Blanks in thesecondcolumn indicatematrixelementswithanabsolute value less than 0.5. Blanks in the third column indicate matrix elements that were not reported in ref 17. (I
scale factors and sample absorption, between the diffractometer constants, and, more severely, the usual correlations between the coefficients to the Fourier background terms. These should not significantly bias the atomic parameters. The variation of the lattice parameter a is shown in Figure 3, and the changes in the T i 4 and 0-0 distances are shown in Figures 4 and 5 . The temperature dependenceof the equivalent isotropic thermal parameters is shown in Figures 6 and 7. b. Orthorhombic Phase. The structure of the orthorhombic phase of BaTiOS at 263 K,determined by single-crystal neutron diffraction, has been reported by Shirane et al.,I9 and we used this structure as a starting point for our refinements. With isotropic thermal parameters, refinements of the structures at most temperatures resulted in a Ti displacement of about 0.015 with thermal parameters in the order Vi,(Ol) > Ui,(Ti) > vi,(02) = Vi,( Ba) . However, one of the refinements gave a structure with a smaller Ti displacement and a largeTi thermal parameter. On further exploration, we discovered the existence of an entire set of solutions with smaller Ti displacementsranging from -0.003 to 0.008 and thermal parameters in the order &(Ti) > Vi,(02)
Ferroelectric Phases of Barium Titanate 4.04 I
I
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2371
I
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bm=cm
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Rhombohedral
1
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Rhombohedral
I
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Tetragonal
I
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I,
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1
300
Temperature
I
2 o, 2.90
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400
(K)
I"
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2 2.10
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I-
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300
$00
100
1
200
1.80
1.70
0
100
200
300
400
Temperature (K) Figure 4. Variation of T i 4 bond lengths with temperature for BaTiO3. The numbers abovethe points indicate the number of bonds of that length. For the tetragonal phase, there is one long and one short axial bond and four intermediate equitorial bonds; for the orthorhombic phase, there are three sets of two bonds each, while for the rhombohedral phase there are two sets of three short and three long bonds.
-
I
I I -
I
I
I _ I I
n
PI
5-
1 -
&(01) > Ui,(Ba). These solutions all hadvery slightly lower
agreement factors. This ambiguitywas resolved when anisotropic thermal parameters were refined. These refinements resulted in a single set of stable solutions with Ti displacements ranging from 0.009 to 0.017A and equivalent thermal parameters in the order U,,(Ol) &(Ti) = U,,(02) > U,,(Ba). The results from these refinements are given in Table 111. The Ti displacements are considerably larger at the temperature extremes of this phase than they are for the structures at 230 and 250 K. However, it is not clear whether much significance can be attributed to this. The least-squares procedure just is not very sensitive to values for AzTi in this phase. For example, in refinements where A z ~ iwas set at 0.017 and not refined, we obtained values of xldZof 3.205 and 1.202 for 230 and 250 K, respectively, values that differed insigificantlyfrom thosein which A z ~ iwas varied to give the low values of 0.0079(8) and 0.0124(13), respectively.
-
I
7
-t J
I I I 1 I
Barium
Rhombohedral 6
100
I I
I
I _
I
I
I
]Orthorhombic I
,
200
Tetragonal I
,
I,
300
400
Temperature (K) Figure 7. Equivalent isotropic thermal parameters, Uoqv2, for the oxygen ions of BaTiO3 as a function of temperature.
Table IV lists all elements of the correlation matrix for atomic parameters which have absolute magnitudes greater than 0.5. While correlation is extensive, none of the elements of the correlation matrix is large enough to cause problems in the
Kwei et al.
2372 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 TABLE III: Resulting StrucMl Parameters for the Orthorhombic Phase of BaTIOs (Space Group A D " ) * sample temperature, K parameter
190
210
230
250
270
lattice constants a, A
b, A c, A
3.9828(3) 5.6745(5) 5.6916(5)
3.9806(5) 5.67 1O(8) 5.6904(8)
3.9841 (3) 5.6741(5) 5.69 16(5 )
3.9855(5) 5.6738(8) 5.6903(8)
3.9874(3) 5.6751(5) 5.6901 (5)
4.0 185(4) 89.828(7) 128.63(3)
4.0169(6) 89.804(11) 128.46(5)
4.01 84(4) 89.824(7) 128.66(3)
4.0178(6) 89.834( 11) 128.67(5)
4.0182(4) 89.849(7) 128.76(3)
4.2(5) 8.4(7) -9.8(2) 0.9(6)
3.2(7) 10.4(12) -10.1(2) 1.2(9)
4.2(4) 5.7(8) -9.9(1) 0.0(6)
437) 8.2(13) -10.0(2) 0.9(10)
4.9(5) 10.6(8) -1O.O( 1) 1.8(6)
0.0170(5) 1.9(5) 3.0( 11) 2.8(8) 2.6(10)
0.0143(10) 3.1(7) 4.9( 16) 7.3(13) 5.1 (15)
0.0079(8) 0.5(5) 8.9( 14) 12.8(10) 7.4(12)
0.0124(13) 3.4(8) 8.8(22) 7.5(16) 6.6( 19)
0.0169(6) 2.6(5) 4.2(12) 3.5(9) 3.4(11)
-0.01 lO(6) 6.0(5) 4.1 (14) 4.3(11) 4.8(12)
-0.01 10( 13) 6.5(8) 3.2(20) 6.8(19) 5.5(19)
-0.0146(6) 7.5(5) 3.1(22) 3.9(16) 4.8(19)
-0,009 1( 12) 6.8(9) 4.3(27) 6.6(16) 5.9(22)
-0).0090(8) 4.7(5) 6.0( 18) 5.5( 10) 5.4(14)
0.0061(3) -0.01 57(4) 1.9(3) 0.9(7) 4.6(8) 3.2(5) 2.5(7)
0.0061(6) -0.0167(7) 2.2(4) 0.6(10) 6.4( 12) 2.3(9) 3.1(11)
0.0044(3) -0.0189(4) 1.2(2) 4.2(11) 4.2(9) 1.0(6) 3.2(10)
0.0067(6) -0).0146(8) 2.2(4) 3.1 (12) 7.4(14) 3.8(9) 4.2(13)
0.0060(4) -0.0140(5) 2.7(3) 2.5(8) 6.3(9) 3.1(6) 3.8(8)
4,9913.59 3.007
7.9015.48 1.192
5.0013.57 3.204
8.1815.71 1.201
4.83 / 3.46 2.975
equiv pseudomonoclinic cell constantsb Cm.
A
a,deg
cell volume V,A3 atomic positions and thermal parameters Ba(O,O,O) 111I u22 u33
Ti(1/210,1/2+&~i) &Ti
UII
u22 u33 UWV
01(O,O,l / ~ + A ~ O I ) Azo I
UI I
u22
0 Units for thermal parameters are 1000 X A2. Numbers in parentheses following refined parameters represent 1 standard deviation. The pseudomonoclinic lattice constant umequals the orthorhombic lattice constant u and is not shown.
TABLE I V Terms in the Correlation Matrix with an Absolute Value Greater Than 0.5 from the Least-Squarer, Structural Refinement of the Orthorhombic Phase of BaTiOJ at 270 K
constants, and, more severely, the usual correlations between the coefficients to the Fourier background terms. The results for a sample temperature of 270 K agree reasonably well with the 263 K data from Shirane et al.I9 for the lattice constants and most atomic displacements but not so well for the Ti displacement, A z ~ i .For example, they find a = 3.990 A, 6 = 5.669 A, and c = 5.682 A, while we find a = 3.9873(3) A, b = 5.6750(5) A, and c = 5.6900(5) A. For the atomic displacements, they find k ~ = 0.010, i Azol = -0.010, Ay02 = 0.003, and Azo2 = -0.013, while our results yield 0.0169(6), -0.0100(7), 0.0061(4), and -0.0143(4), respectively. It is not clear which value for AZT~is more correct, but our values should again result in a larger value for the spontaneous polarization. The orthorhombic lattice parameters, a, 6, and c are related to the equivalent pseudomonoclinicparameters a,, c,, and a by a = a,
b = 2c, sin(a/2) refinement. This is very surprising since the orthorhombic structures are the most difficult of all three phases to refine and the resulting structural parameters have the largest uncertainty. For example, given the difficulties described in the previous paragraph, it is very surprising that there is little correlation between the Ti displacements and the Ti anisotropic thermal parameters. At present, we have no explanation for the difficulties encountered in the refinements. Again, in addition to these correlations, there are also correlations between elements of the reciprocal metric tensor and the diffractometerconstants,between scale factors and sample absorption, between the diffractometer
c = 2c, cos(a/2)
These are alsogiven in Table 111. Thevariation with tempertaure of the equivalent pseudomonocliniclattice parameters a,,,and c, is shown in Figure 3, and the changes in the T i 4 and 0-0 distances are shown in Figures 4 and 5. The temperature dependence of the equivalent isotropic thermal parameters is shown in Figures 6 and 7. Since the refinements only converged when heavily damped, we thought that a different structural model might actually be more appropriate. Since the ferroelectric distortion along the
Ferroelectric Phases of Barium Titanate
TABLE V
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2373
R e ~ u l t h pStructural Parameters for the Tetragonal Phase of BaTiOj (Space Croup P4/mmm)* sample temperature, K Darameter
280
290
300
320
350
3.9970(5) 4.03 14(6) 64.406( 26)
3.9925(5) 4.0365(5) 64.341(24)
3.99095(29) 4.0352(3) 64.271 (14)
3.9938(3) 4.0361(3) 64.378( 16)
3.9956(5) 4.0354(5) 64.426(24)
5.0(4) -0.9(9) 3.0(7)
4.7(3) 1.4(8) 3.6(6)
2.9(2) 3.5(5) 3.1(4)
3.3(2) 3.5(6) 3.4(4)
4.5(3) 2.6(9) 3.9(7)
0.0203( 10) 6.0(4) 5.2(9) 5.7(7)
0.021 5 ( 10) 7.0(4) 4.1(11) 6.0(8)
0.0224(6) 5.4(2) 5.1(7) 5.3(5)
0.0215(7) 6.5(3) 4.9(8) 6.0(6)
0.0195(11) 6.6(4) 6.0( 12) 6.4(9)
-0.0258(6) 1.0(3) -6.2(3) -1.4(3)
-0.0253(6) -2.1(2) -3.4(4) -2.5(3)
-0.0244(4) -0.3(1) -4.8(2) -1.8(2)
-0.0233(5) 0.4(2) -5.0(2) -1.4(2)
-0.0251 (7) -1.0(2) -2.7(4) -1.6(3)
-0.0123(10) 10.5(6) 8.7(5) 6.0(4) 8.4(6)
-0.0105(9) 10.7(5) 6.9(4) 9.0(5) 8.9(5)
-0.0105(5) 9.8(3) 5.4(2) 10.5(3) 8.6(3)
-0.0095(6) 9.4(4) 6.9(3) 9.6(3) 8.6(4)
-0.01 lO(10) 9.9(5) 734) 9.3(4) 8.9(5)
7.3915.05 1.665
7.7915.42 1.275
4.5513.30 2.839
5.0513.61 2.624
6.68/4.63 1.408
~
lattice constants a, A c, A
cell volume V, A3 atomic positions and thermal parameters Ba(O,O,O) UII = u22 u 3 3
uw
Ti(1/2.1/2,'/~+k~i) &TI
UII = u22
u33
Numbers in parentheses following refined parameters represent 1 standard deviation.
face diagonal may be construed as a monoclinic distortion, we attempted to refine the structure in the monoclinic Pml 1 space group. However, we found that this model converged more slowly, if at all, and with higher agreement factors, despite the larger number of degrees of freedom. c. TetragonalPhase. The structure of tetragonal BaTiOS has been studied using single-crystal X-ray diffraction?" neutron diffraction,Z' and a combination of X-ray and neutron diffraction.15 In the latter case, the serious parameter interaction problems encountered in the earlier workz2were 'avoided" by alternately varying selected parameters in independent refinements using X-ray and neutron data. Finally, all parameters were varied in these refinements to obtain structural parameters and correlation matrices. Both X-ray and neutron refinement agreement factorsvaried slowly withvalues of the Ti displacement, but agreement between anisotropic thermal parameters along the c axis for the various atoms in the two refinements was used to support the previous structural conclusions. Our structural refinements using powder neutron diffraction data, with anisotropic thermal parameters, gave two stable solutions. As before, one set had small Ti displacements(ranging from -0.006 to 0.004 A) with values of the equivalent isotropic thermal parameter in the order &,(Ti) > U,,(Ol) U,,(02) > U,,(Ba), while the other set had much larger Ti displacements with equivalentisotropicthermal parameters in theorder U,,(02) > U,,(Ti) > U,,(Ba) > Uqv(Ol). For the temperatures 280350 Kin ascending order, the values of the minimization function xrdZ were 1.676, 1.277, 2.851, 2.419, and 1.406 for the former and 1.665,1.275,2.839,2.624, and 1.408 for the latter. A choice between these two models based on the values of xrd2would be difficult, but the former set of refinements can be ruled out because they would give too small a value of the spontaneous polarization and because our experience from refinement of the structures for the other phases has shown that the solution in which small, or negligible, Ti displacementsare compensated by very large values of the Ti thermal parameter is incorrect. Values of the refined parameters based on the latter set of refinements are shown in Table V. Table VI lists all elements of the correlation matrix for atomic
-
TABLE VI: Selected Terms in the Correlation Matrix from the Least-SquPres Structural Refiaement of the Tetragod Phase of BaTiOJ at 300 K Compared with Those from Ref 15. correlation matrix element parameters
this work
ref 15 (neutron)
ref 15 (X-ray)
-0.508 0.99 0.99 -0.65 -0.74 0.783 -0.56 0.96 -0.573 -0.79 -0.6 10
-0.732
4.88 -0.92 0.94 0.54 0.98 0.98 0.99
0.52 0.79 0.64 0.81 0.79 -0.98 0.64 0.84 4-62 -0.86 -0.84 0.60 -0.54
0.94 Blanks indicate matrix elements with an absolute value less than 0.5.
parameters which have absolute magnitudes greater than 0.5. Only five pairs of parameters have correlation matrix elements greater than 0.5: as expected, the Ti displacement and the anisotropic thermal parameter along the c axis are anticorrelatcd and the 0 1 displacement is correlated with both Ti displacement and the anisotropic thermal parameter. Matrix elements from the single-crystalX-ray and neutron diffraction refinements from Harada et al.' are included for comparison. Correlation appears much less severe in the powder neutron data than in either the single-crystal neutron or X-ray refinements. The X-ray refinement may help in the case of correlation between &(Ti) and U33(Ba) but cannot help in the cases where oxygen atom parameters are involved because of the weak scattering of X-rays
2374 The Journal of Physical Chemistry, Vol. 97, No. 10, 19'93
by oxygen, at least not in independent refinements. Again, in addition to these correlations, there are also correlations between elements of the reciprocal metric tensor and the diffractometer constants, between scale factors and sample absorption, between the diffractometer constants, and, more severely, the usual correlations between the coefficients to the Fourier background terms. The results for a sampletemperature of 300 K agree reasonably well with the previous determination23 of lattice constants which gave values of a = 3.9945 A and c = 4.0335 A. We find a = 3.99095(29) A and c = 4.0352(3) A. However, the atomic positions differ somewhat from those given by Harada et al.I5 at approximately 298 K (the X-ray data were collected at 301 K, while the neutron data were collected at 295 K) in that the displacements for Ti and 0 2 are both significantly larger, again resulting in a larger value for the spontaneous polarization. For example, they find A x ~ i= 0.0135(4), Azo1 = -0.0250(4), and Azo2 = -0.0150(3), whileour results yield0.0224(6),-0.0244(4), and -0.0105( 5 ) , respectively. The variation of the lattice parameters is shown in Figure 3, and the changes in the Ti-O and 0-0 distances are shown in Figures 4 and 5, respectively. The temperature dependence of the equivalent isotropic thermal parameters is shown in Figures 6 and 7. Because of the increasingacceptanceof the Comes et a1.6model for the ordered displacements of the Ti ions,5v24+25 we also examined two additionalstructural models early in the data analysisprocess. If the Comes et al. model gives a correct description of the tetragonal phase, then theTi ions shouldbelocatedat ('/~+Ax,l/ ~ + A x , ~ / 2 + A xand ) the threesymmetry equivalentpositions, each with an occupancy of instead of the (I/Z,I/~,I/~+AZ) position used earlier. Refinement of this model using the 300 K data gave a stable refinement with the parameters Ax = 0.0205(5) and U,,, = 0.7(4) X 10-3 A2, slightly different values for the other refined atomic parameters, with a higher value of xrd2= 2.857and with poorer agreement factors Rwp/Rexp = 4.57%/3.32%. The poor fit led us to examine another model, intermediate between the two we considered, in which the Ti displacements in the a-6 plane are uncoupled from the displacements along z,e.g., Ti ions in the positions ( ~ / ~ + A X , ~ / Z + A X , ' / ~ +This A ~ )refinement . resulted in Ax = 0.004(5), Az = 0.0224(6),and vi, = 5.1(7) X 10-3 A2, with other atomic parameters similar to those shown in Table V and with agreement factors identical to our original refinement. In this refinement, the value of Ax becomes indistinguishablefrom zero. Clearly what is happening is that, when given the freedom, the structure obtained from Rietveld refinement begins to deviate from the Comes et al. model and approaches the original model incorporating displacements only along the z axis. IV. Evolution of Structure with Temperature In this section, we examine the behavior of various structural parameters, as a function of temperature, across the three ferroelectric phases. Lattice parameters give changes in cell volume across the phase boundarieswhich are in better agreement than earlier X-ray results with dilatometry measurements and thermodynamic considerations. The discussion continues by examining bond lengths and relative atom positions in BaTiO3. Using a simple model, we can calculate the spontaneous polarization due to the relative displacement of the ionic sublattices. Finally, we examine the temperature dependence of the thermal parameters to estimate atomic DebyeWaller temperatures and to gain some insight into the various structural models that have been proposed. a. Volume Changes at the Phase Traositions. Changes in the lattice constants with temperature have been reported in the vicinity of the tetragonal-to-cubic phase transition by Megaw26 and later for all phases by Kay and V ~ u s d e n Lattice . ~ ~ parameters
Kwei et al. 4.02
I
I
t t
'
I I I I
I
I I I
I I
Rhombohedral
3.98
I
0
100
I
I I I I
:Orthorhombic I
1
200
I I
'
1'1
I1 1 I i
Tetragonal
i1 I
I
1
I,
300
400
Temperature (K) Figure& Temperature depndence of the pseudocubic lattice parameter, defined as 0 = p/3where Vis the volume of the pseudocubic cell. The lines are linear least-squares fits through the data in each phase.
are plotted in Figure 3 for all the phases considered. In a Rietveld refinement, these parameters are given with good precision and are rather insensitive to the structure model used. The volume parameter, expressed as the pseudocubic lattice parameter, it = PI3,is plotted as a function of temperature in Figure 8. The volume increases overall due to normal thermal expansion. In addition, there are volume anomalies associated with the tetragonal/orthorhombic(TO) and the orthorhombic/rhombohedral (OR) transitions. Anearlier study of the temperaturedependence of this parameter27 indicated that the volume anomaly was negative for both the TO and the OR transition. This disagreed with dilatometry measurements28which showed a volume expansion at the TO and a contraction at the OR transition. The expected sign of the volume anomaly can be predicted from the pressure dependence of the transition temperature, TT,for the transitions by using the Clausiudlapeyron equation. MeasurementsZBof dTT/dP suggested the results of the dilatometry experiments were correct, even though it was a less direct measurementof the volume change. Our results are in agreement with the dilatometry experiments and the predictions from the pressure dependence. From Figure 8, we can estimate the vdues of AV(T0) and AV(R0) as +0.11 and -0.038 AS,respectively. As is evident in Figures 3 and 8, significant fluctuations of the measured volume are evident in the premonitory region as the transition is approached. Since in our sample these fluctuations form a significant proportion of the data, the best line to fit to the data is somewhat subjective. Nonetheless, the sign of the volume anomalies is beyond question. The observation of the fluctuationeffects as the transition is approached is very dependent on the internal stresses felt by the crystal.2d They are observed when the crystal is unstressed but are quickly suppressed by internal (or external) stressessuchas thoseproducedby differently oriented domains impinging on one another. The fact that they are so prominent in our sample may be an indication of a good degree of crystal perfection within the grains. b. Displacements of the Ions. The displacementsof the Ti and oxygen ions from their symmetry sites in the cubic phase result in changes in the Ti-O bond lengths. As shown in Figure 4,these displacements produce one long bond, one short bond, and four normal bonds (according to ref 3, the sum of ionic radii for Ti4+ and 02-in octahedral coordination is 2.005 A) in the tetragonal phase. For the orthorhombic phase, this becomes two short, two long, and two normal length bonds, while for the rhombohedral phase, it becomes three short and three long bonds. The curved temperature dependenceof the bond lengthsfor the orthorhombic phase is probably spurious,resultingfrom thedifficulty in refining the structures at 230 and 250 K. The bond lengths for the short and long bonds seem to diverge very slightly as the temperature is lowered.
I I
I
I
I
(
I I
II
I
I
I
lOrthorhombic
I Tetragonal
II .
I I
I I
I
I _ I I . I I _ I I . I
Rhombohedral 0.14
I
0
100
I I
I
I
I
;Orthorhombic 1
1
200
I I I . I I _
I Tetragonal
I
,
300
I . I I t
-
a 1
-
0.04 0.02
-
Rhombohedral
I
I
I
I
-
I _ I . ,
II
400
Temperature (K) Figure 9. Spontaneous polarization of the sample as a function of temperature calculated from the relative displacements of ions within the unit cell.
The shape of the oxygen octahedra is perhaps best shown by the changes of the 12 nearest-neighbor 0-0 distances as the octahedron deforms (Figure 5 ) . For the tetragonal phase, there are four 0-0 distances of each length as the octahedron is stretched along the z axis. These correspondto the four distances from each of the 0 1 apical oxygens to the 0 2 in-plane oxygens and to the four distances for the oxygens within the plane. For the orthorhombic phase, the four 0 2 oxygens, lying in the same plane as the polarization vector, form one short, one long, and two intermediate length 0-0 distances. In addition, there are two sets of four short and long distances between the 0 1 apical oxygensand the oxygens in the plane. In the rhombohedralphase, the distortion involves a stretch (with a slight twist) along the axis of polarization which joins two opposite faces of the octahedron. The intermediate distances are from the two octahedron faces and the sets of short and long distances alternately join the oxygens from the two faces. Both Ti-0 and 0-0distanceschange substantiallyat the phase transitions, but surprisingly, they do not change much with temperature within each phase. c. SpontraeolrpPolarization from Relative Ionic Displacements. The lattice polarizationcan be calculated from the atom positions. We used the simplest possible formula:
P = V'cQizi i
Here Qiis simply the ionic charge of each ion, and zi is its displacement with respect to the unpolarized cubic structure. The results are shown in Figure 9. The magnitude of the polarization vector is essentially continuous through the lowtemperature transitions. We have also plotted in Figure 10 the component of the polarization vector along the cube edge to allow comparison of our results with the direct polarization measurements of Merz29 and Kitnzig and Meier.30 In accordance with Figure 1, this means dividing the polarizationof the orthorhombic phase by 2112 and that of the rhombohedral phase by 3Il2. Our calculated values are consistently higher than Merz's; however, the measured values for the tetragonal phase were later revised" in measurementson a better sample and valuesof the spontaneous polarization increased from about 15 to about 26 C/m2. No allowancewas made in the calculationsfor apparent ionic charges, which take into account contributions from ionic polarizability;16 use of such apparent charges would have increased the calculated values by approximately 50%. d. Thermal Parameters nad DebyeWaUer Temperature. The temperature dependence of thermal parameters can reveal some information about the strength of the local atomic potential in a crystal in the form of a Debye-Waller temperature. In a crystallographic refinement, they may also suggest the presence
of local deviations from the average crystal structure, or indicate deficiencies in the structural model used for refinement. The equivalent isotropic thermal parameters, defined as one-third the trace of the diagonalized thermal parameter tensor, are shown in Figures 6 for Ba and Ti and in Figure 7 for the oxygens. Since the structural changes are small, the forces that the oxygen ions experience should be similar in all phases, and we would expect the thermal parameters in the different phases to remain similar. This appears to be the case for both Ba and Ti but not for the oxygens. As pointed out by Lawson et al.,32 the temperature dependence of the thermal parameters can be fit to a DebyeWaller model
where the temperature-dependent term ( u2)ideal can be related to the Debye-Waller temperature and a temperature-independent term or "offset", ( ~ ~ ) ~which f f ~ arises ~ , from certain inadequacies in our experimental technique, including the fitting procedure, as well as static displacements. For theoxygen ions, theequivalent isotropic thermal parameters show a relatively flat temperature dependence for all three phases, indicating a high atomic DebyeWaller temperature. However, they are discontinuous across the phase boundaries and indicate different offsets, ( u2)offse,,for the two oxygen sites in the orthorhombic and tetragonal phases. We believe that these shifts in the offsets reflect inadequacies of the structural models in treating the oxygen displacements and thermal vibrations. This does not appear to be the case for the Ba and Ti ions. In this case, the temperature dependence of the equivalent isotropic thermal parameters is continuous across the phase boundaries and shows the expected temperature dependence. Fits to the Debye-Waller model are shown in Figure 6 and yield the atomic Debye-Waller temperatures of 251(3) K for Ba and 434(10)KforTi,withoffsetsof0.3(2)X lt3and-2.6(1) X l t 3 A2, respectively. These values are strikingly similar to those for the cations in the closely related cuprate perovskite analogues which exhibit high-temperature superconductivity: for example, in the case of the La2Cu04 semicond~ctor~~ where La and Cu were found to have Debye-Waller temperatures of 268(3) and 430(7) K, respectively, or in the case of the YBa2Cu307 superconductor34where Y and Ba were found to have temperatures of 345(6) and 231(3) K, respectively, and the chain and plane Cu ions temperatures of 487(12) and 402(5) K, respectively. In both of these systems,the oxygen ions had average Debye-Waller temperatures of about 600 K. It might be hoped that the refined thermal parameters could help differentiate between soft-mode and orderdisorder behavior. An orderdisorder model such as that of Comes et aL6 implies that more than one minimum in the atom potential permanently
2316
Kwei et al.
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
exists at a given lattice site. For example, in the Comes et al. model, thereareeight minima associated with theTi sitedisplaced along each of the ( 1 1 1 ) directions from the cell center. When the Ti ions are all ordered along one unique direction, as happens in the rhombohedral phase, it will be very clear in the crystal structure refinement. However, when the displacements are randomly distributed among a number of allowed directions, the crystal structure refinement will yield the mean atom displacement along the polar direction. The local displacive disorder which is not described by this average displacement will appear as enlarged thermal parameters. If the Comes et al. model is correct, then in the tetragonal phase one would expect to see an average displacement of Ti refined along thez direction along with thermal parameters on this site enlarged in directions perpendicular to the z axis. However, this is not observed: if AZT~is constrained is to zero, then U33 is much larger than UII = U22. As allowed to give the preferred displacement of 0.085A, averaged over all five temperatures, the values of U33become nearly equal to U II = U22,with average values of 0.63 X 10-2 and 0.51 X 10-2 A2,respectively. This would imply thermal amplitudes of 0.079 A perpendicular to the c axis and 0.071 A, superposed along a displacement of 0.085A, along the c axis. This observation does not support the Comes et al. model. On the other hand, softmode behavior would be hard to detect in the thermal parameters. The thermal and zero-point vibrational amplitude of atoms has contributionsfrom phonon modes throughout the Brillouin zone. If one phonon branch softens in one region of the Brillouin zone, the population of these modes will increase and their vibrational amplitude will increase, but it should have a rather small effect on the total vibrational amplitude measured by thermal parameters. The refined thermal parameters appear tovary rather smoothly with temperature, and none of them have thermal ellipsoids which are excessively anisotropic, which would indicate some displacive disorder. Based on the thermal parameters, there does not appear to be good evidence for significant order4isorder behavior. This is difficult to reconcile with the observation of significant diffuse scattering in the X-ray experiments of Comes et a1.6 An analysis sensitive to the local structure is in progress to investigate this further. V. Changes in Macroscopic Properties with Temperature Finally, in this section, we consider changes of mosaic block size and inhomogeneous strain with temperature. Estimates of these parameters can be obtained from the powder extinction coefficient and the strain term in the profile coefficients, respectively, fitted in the structural refinements. a. Change in Mosaic Block Size. K t i n ~ i gnoticed ~ ~ J ~ a sharp peak in the extinction sensitive reflections near the cubic-totetragonal transition and suggested that fluctuations diminish the extinction near the phase transition. We have also noticed large changes in the average (over the four banks of data) powder extinction, obtained from the structure refinement, as the sample temperature is changed. In particular, as shown in Figure 11, extinction is largest away from the phase transitions and decreases as the phase transition is approached either from above or from below. These changes in the extinction reflect changes in the mosaic structure of the crystallites. An effective mosaic block size can be estimated from the powder extinction coefficient obtained from these refinements and is roughly equal to the square root of the extinction coefficient in micr0ns.~-'2For the tetragonal phase, the mosaic size decreases dramatically as the temperature is lowered from 300 K, where the extinction may have been unusually high because the sample had not yet undergone any phase transitions, to 280 K. As the temperature is increased from 300 toward 350 K, the mosaic size again decreases. This behavior clearly reflects the strains that build up as the temperature is lowered and the thermal defects that grow as the
""I 70
'
60
-
50
-
'
'
I I I I
I I I I I
.-
0
I
240.-
'I
I I I
.y
z30-
I ' I
I
I
1'1
'
I
I I
90
I
I
I
Rhombohedral I
I Orthorhombic1
Tetraaonal
I
I
I
I1
,
I
- ,
Figure 12. Strain broadening coefficient, in units of ps2/A2,from the data collected in the + 1 5 3 O detector bank as a function of sample temperature.
temperature is increased. The mosaic size is approximately 8.5 pm a t 300 K and decreases to about 4.5pm at 280 K and 7.1 pm at 350 K. Similar behavior is observed for the orthorhombic phase, where the mosaic size reaches a maximum of 4.8 pm at ca. 230 K but decreases to about 3.1 pm near the phase boundaries. The extinction behavior is again similar in the rhombohedral phase, although it is not clear why the domain size should decrease at low temperatures, or why there appears to be so much scatter in the point at 130 K (or perhaps 150 K). It is interesting that the maximum mosaic size is much larger for the tetragonal phase than for the other two phases. This may result from domain clamping effects which prevent domains in the low-temperature phases from reaching their equilibrium size. b. Inhomogeneous Strain at the Phase Transitions. Both the premonitory nature of the transitions and the decrease in mosaic block size, especially at the tetragonal-to-orthorhombic transition, suggest that there must be an increase in the inhomogeneous strain of the sample as the phase transition is approached. The strain broadening term in the profile functions used in the refinements provides an estimate of this strain. Figure 12 shows the dependence of this term, u12,for data from the + 1 5 3 O bank as a function of temperature. The strain broadening term shows a large increase from a value of 48-50 ps2/A2 to approximately 90 ps2/A2as the temperature is decreased to 280 K. Similarly, this term again shows signs of increasing at the orthorhombicto-rhombohedral transition, but this time the increaseis not nearly as large, most probably because the strain really is lower and partly because, in this case, no data are available very close to the transition. In any event, the microscopic strain appears to
Ferroelectric Phases of Barium Titanate be a fairly sharp function of how closely the phase transition is approached. A more quantitative estimate of the dimensionless strain S is given by the r e l a t i ~ n s h i p ? J ~ , ~ ~ S = 100%C1[(8In 2)(uI2- u l i2)I 1/2 where Cis the so-calleddiffractometer constant that relates TOF in microseconds to d spacing (in this case, for the + 153O detector bank of HIPD, C = 5034.00 p s / A and uli2= 47 ps2/A2). This would then give 0.31% strain at the peak corresponding to 280 K,0,14%strainat 19OK,andO.O7%strainat100K,a temperature where the strain appears small.
VI. Conclusions We have carried out a careful analysis of the crystal structure of BaTiO3 as a function of temperature between 15 and 350 K. This encompasses all three ferroelectric phases of the material. The technique used was the Rietveld structure refinement with neutron powder diffraction data collected from a pulsed source. Structure analysis of BaTiO3 is notoriously difficult because the small atom displacements associated with the ferroelectric distortions are highly correlated with each other and with the thermal parameters. As discussed, this can result in inaccurate refined values and a lack of uniqueness in the solution. We have attempted to minimize this problem by using data covering a wide range of d spacings, which reduces the correlation between parameters, by considering changes to parameter values as a function of temperature, and by relating structural parameters to the known propertiesof the material. The structure parameters refined in this study are in reasonable agreement with earlier models, although the Ti displacements are consistently slightly larger than in earlier reports. Using a simplified model, we calculated the spontaneous polarization which successfully reproduced the qualitative temperature dependence of the spontaneous polarization. Fits of a DebyeWaller model to the temperature dependence of the thermal parameters give atomic DebyeWaller temperatures for the cations which are strikingly similar to the analogous cations in the LazCu04 semiconductorand the YBa2Cu307 highTc superconductor. This suggests that these rather unusual perovskite-like systems share common elastic properties. For the tetragonal phase, it has not been possible to differentiate directly between models in which the titanium ion displaces directly in the polar direction and the Comes et aL6 model in which it displaces along the pseudocubic ( 111) directions but has, on the average, a net displacement along the polar direction. In the latter case, the displacements are not ordered over long distances and the long-range periodic structure reduces to that of the average displacement along the polar direction. The best evidence from this work to differentiate between these models is from anisotropic thermal factors. Based on the relative size of UII and U33 from our refinements for the Ti site in the tetragonal phase, it appears highly unlikely that the true displacement directionis along ( 111 ). An analysis of the local atomic structure, utilizing both Bragg and diffuse scattering, is in progress to try and address this issue in greater detail. Finally, we have presented an analysis of the temperature dependence of the extinction and microscopic strain parameters which provide information about the structural coherence of our sample. These indicate that the domain size in the rhombohedral and orthorhombic phases is considerably smaller than observed in the tetragonal phase. Fluctuationsat the phase transformations also reduce the structural coherence, and a marked increase in microscopic strain is observed in the sample as the transition is approached.
Acknowledgment. This article is dedicated, in celebration of his 60th birthday, to Dudley R. Herschbach, who taught one of us (G.H.K.) the importance of a knowledge of structure in
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2377 understanding dynamics. We thank Allen C. Larson and Robert B. Von Dreele for many helpful discussions. The neutron powder diffraction experiments were carried out at the Manuel Lujan Jr. Neutron Scattering Center which is partially funded as a national user facility by the Division of Material Sciences,Office of Basic Energy Sciences of the United States Department of Energy. This work was done under auspices of the United States Department of Energy.
References and Notes (1) Wul, W.; Goldman,I. M. C.R . Acad. Sci. Russ. 1946,46,139; 1946, 49, 177; 1947, 51, 21.
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