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ABSTRACT: Despite their crystallographic differences, the mechanisms of the ( F[`4]3m )\left( {F\bar 43m} \right)
) to orthorhombic (C2221), whereas that in SiO2 cristobalite is from cubic (
( Fd[`3]m )\left( {Fd\bar 3m} \right)
) to tetragonal (P43212 or P41212). These crystallographic differences stem from the fact that there are two distinct cation positions in AlPO4 cristobalite as opposed to one in SiO2 cristobalite and the ordered (Al,P) distribution is retained through the phase transition. As a result, there are significant differences in their crystal structures, domain configurations resulting from the phase transition and Landau free energy expressions. A symmetry analysis of the F[`4]3m ® C2221F\bar 43m \to C222_1
in AlPO4 cristobalite has been carried out based on the Landau formalism and the projection operator methods. The six-component order parameter, F[`4]3mF\bar 43m
and corresponds to the simultaneous translation and rotation of the [AlO4] and [PO4] tetrahedra coupled along 110. The Landau free energy expression contains a third order invariant, the minimization of which requires a first-order transition, consistent with experimental results. The tetrahedral configurations of twelve phase domains resulting from the transition in AlPO4 cristobalite are of two types: (1) transformation twins from a loss of the 3-fold axis, and (2) antiphase domains from the loss of the translation vectors 1/2[101] and 1/2[011] (FC). In contrast to -SiO2 cristobalite, the -AlPO4 cristobalite (C2221) does not have chiral elements (43, 41) and hence, enantiomorphous domains are absent. These transformation domains are essentially macroscopic and static in the phase and microscopic and dynamic in the phase. The order parameter, couples with the strain components, which initiates the structural fluctuations causing the domain configurations to dynamically interchange in the phase. An analysis of the MAS NMR data (29Si, 17O, 27Al) on the - transitions in SiO2 and AlPO4 cristobalites (Spearing et al. 1992, Phillips et al. 1993) essentially confirms the dynamical model proposed earlier for SiO2 cristobalite (Hatch and Ghose 1991) and yields a detailed picture of the transition dynamics. In both cases, small atomic clusters with the configuration of the low temperature phase persist considerably above the transition temperature, T0. The NMR data on the phases above T0 cannot be explained by a softening of the tetrahedral rotational and translational modes alone, but require the onset of an order-disorder mechanism resulting in a dynamic averaging due to rapidly changing domain configurations considerably below T0.
Physics and Chemistry of Minerals 01/1994; 21(1):67-77. · 1.73 Impact Factor
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ABSTRACT: Cryolite, Na3AlF6[ = 2Na+(Na0.5
+Al0.5
3+)F3] is a mixed fluoride perovskite, in which the corner-sharing octahedral framework is formed by alternating [NaF6] and [AlF6] octahedra and the cavities are occupied by Na+ ions. At 295 K, it is monoclinic ( phase), space group P2
1/n with a = 5.4139 (7), b = 5.6012 (5) and c = 7.7769 (8) and = 90.183 (3), Z = 2. A high temperature single crystal X-ray diffraction study in the range 295–900 K indicates a fluctuation-induced first-order phase transition from monoclinic to orthorhombic symmetry at T
0 885 K, in contrast to a previous report that it becomes cubic at 823 K. The space group of the high temperature phase is Immm with a = 5.632 (4), b = 5.627 (3) and c = 7.958 (4) , Z = 2 at 890 K. Above T
0, the coordination number of the Na+ ion in the cavity increases from eight to twelve and the zigzag Na1 — Al octahedral chains parallel to c become straight with the Na1-F-Al angle = 180 . The phase transition is driven by two coupled primary order parameters. The first corresponds to the rotation of the nearly rigid [AlF6] group and transforms according to the
4
+ irreducible representation of Immm. Coupled to the [AlF6] rotation is a second primary order parameter corresponding to the displacement of the Na2+ ion in the cavity from its equilibrium position. This order parameter transforms according to the X
3
+ irreducible representation of Immm. Following Immm P2
1
/n phase transition, four equivalent domains of P2
1/n are determined relative to Immm, which are in an antiphase and/or twin relationship. The abrupt shortening of the octahedral Al-F and Na-F bonds and a sudden change in orientations of the atomic thermal vibration ellipsoids above T
0 indicate a crossover from displacive to an order-disorder mechanism near the transition temperature. The phase is interpreted as a dynamic average of four micro-twin and -antiphase domains of the a phase. This view is consistent with the entropy of phase transition, Strans (11.43 JK–1 mol–1) calculated from heat capacity measurements (Anovitz et al. 1987), which corresponds closely to R ln4 (11.53 JK–1 mol–1), where 4 is the number of domains formed during the phase transition. The dynamic nature of the phase is independently confirmed from a considerable narrowing of the 27Al nuclear magnetic resonance (NMR) line-shape above T
0 (Stebbins et al. 1992).
Physics and Chemistry of Minerals 01/1993; 19(8):528-544. · 1.73 Impact Factor
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ABSTRACT: The paraelectric to antiferroelectric phase transition in titanite at 500 K involves a displacement of the titanium atom from the center of the [TiO6] octahedron in the paraelectric phase (A2/a) to an off-center position in the antiferroelectric (P2
1/a) phase. We have carried out a detailed single crystal high temperature x-ray diffraction study of the phase transition including structure refinements at 294, 350, 400, 430, 440, 450, 500, 600, and 700 K. The unit cell dimensions show a pronounced hysteresis effect in the 450–500 K range on heating and cooling during the first cycle along with a reduction of the transition temperature, T
c
from 495 5 K on heating to 445 5 K on cooling. The hysteresis effect disappears on further heating and the superstructure reflections show residual intensities above T
c
(445 K). An order parameter treatment of the phase transition is presented in terms of Landau theory and induced representation theory. The Ti-displacements parallel and antiparallel to a are taken as the primary order parameter , which transforms as the Y
2
+
representation. A coupling of Y
2
+
with T
1
+
results in the linear-quadratic coupling of the spontaneous strain components,
ij with . The Ti-displacements are coupled linearly to the Cadisplacements. Both sets of displacements predicted from induced representation theory are observed experimentally. The phase transition is initially driven by the soft mode at the zone boundary point Y
2
+
; near T
c
critical fluctuations set in and an order-disorder mechanism finally drives the phase transition, whereby parallel and antiparallel Ti-displacements related by [0, 1/2, 1/2] in adjacent domains are dynamically interchanged. Immediately above T
c
, the high temperature (A2/a) phase is a statistical average of small dynamic antiphase domains of the low temperature (P2
1/a) phase. Vacancies and defects pinning the domain boundaries may drastically alter the transition behavior and affect the domain mobility.
Physics and Chemistry of Minerals 02/1991; 17(7):591-603. · 1.73 Impact Factor
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ABSTRACT: Cristobalite, a high temperature phase of silica, SiO2, undergoes a (metastable) first-order phase transition from a cubic,
Fd[`3]mFd\bar 3m
, to a tetragonal, P43212 (or P41212), structure at around 220 C. The cubic C9-type structure for Fd[`3]m ® P43 21 2Fd\bar 3m \to P4_3 2_1 2
(or P41212) transition in cristobalite has been carried out based on the Landau formalism and projection operator methods. The starting point is the ideal cubic (
Fd[`3]mFd\bar 3m
) C9-type structure with the unit cell dimension a (7.432 ) slightly larger than the known a dimension (7.195 at 205 C) of -cristobalite, such that the Si-O-Si angle is still 180, but the Si-O bond length is 1.609 . The six-component order parameter driving the phase transition transforms according to the X4 representation. The transition mechanism essentially involves a simultaneous translation and rotation of the silicate tetrahedra coupled along 110. A Landau free-energy expression is given as well as a listing of the three types of domains expected in -cristobalite from the transition. These domains are: (i) transformation twins from a loss of 3-fold axes, (ii) enantiomorphous twins from a loss of the inversion center, and (iii) antiphase domains from a loss of translation vectors 1/2 110 (FP). These domains are macroscopic and static in -cristobalite, and microscopic and dynamic in -cristobalite. The order parameter , couples with the strain components as 2, which initiates the structural fluctuations, thereby causing the domain configurations to dynamically interchange in the -phase. Hence, the - cristobalite transition is a fluctuation-induced first-order transition and the -phase is a dynamic average of -type domains.
Physics and Chemistry of Minerals 01/1991; 17(6):554-562. · 1.73 Impact Factor
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ABSTRACT: The transition from P213(T
4) to P212121(D
2
4
) in the langbeinite K2Cd2(SO4)3 has been analyzed using group theoretical methods and previously published structural data above and below the transition. We find that because the transition is strongly first-order, the primary-order parameter has relatively large values at the transition temperature, and higher order terms which involve the order parameter, the strain, and the coupling of the two must be included in the Landau expansion for the free energy. Complex displacements occur at the transition for all atoms of the unit cell, but these displacements can be resolved into contributions which can be shown from symmetry considerations to transform as the
2
3 irrep of P2
1 3(T
4) as well as contributions from symmetry-preserving displacements which transform under the irrep
1. Therefore, the transition is not a simple one and involves sulfate rotations and cadmium and potassium ion displacements.
Physics and Chemistry of Minerals 08/1990; 17(4):334-343. · 1.73 Impact Factor
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ABSTRACT: The non-ferroic triclinic to triclinic
I[`1] - P[`1]I\bar 1 - P\bar 1
phase transition in anorthite is described in terms of the spontaneous onset of an order parameter η. A triclinic to triclinic
phase transition can be driven by order parameters (representations) arising from the Γ, Z, X, U, V, R, Y, and T points of
symmetry of the Brillouin zone. Each point leads to a set of two inequivalent representations and thus there is a total of
sixteen inequivalent order parameters. However, only the R
1
+
representation is consistent with the change from the body-centered to primitive cell (increase of primitive cell size of
two) and also with the origin of the two space groups (inversion center) being at the same position. The R
1
+
order parameter of the high symmetry triclinic phase
P[`1]0P\bar 1_0
(or equivalently
I[`1]I\bar 1
) causes a reciprocal lattice change and, in terms of the lower symmetry reciprocal lattice, the order parameter corresponds
to the b* point. This is consistent with experimentally observed x-ray diffuse scattering. Using induced representation theory, microscopic
distortions compatible with the R
1
+
order parameter are obtained. Assuming a distortion in an arbitrary direction at the general 2(i) Wyckoff position (x0,y0,z0) of
P[`1]0P\bar 1_0
(the higher symmetry phase) induced representation theory demands an opposite displacement at the position (x0, y0, z0), an opposite displacement at (x0+1,y0+1,z0+1), and the same displacement at (
[`(x)]\bar x
0+1,
[`(y)]\bar y
0+1,
[`(z)]\bar z
0+1) of
P[`1]0P\bar 1_0
. This is also consistent with experiment. The presence of the weak c-type reflections above the transition is attributed
to the fluctuating lower symmetry antiphase domains related by the translation (1/2, 1/2, 1/2).
Physics and Chemistry of Minerals 06/1989; 16(6):614-620. · 1.73 Impact Factor
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ABSTRACT: Ilvaite, Ca(Fe2+, Fe3+) Fe2+Si2O7O(OH), a mixed-valence iron silicate shows an insulator-semimetal transition with a band gap of 0.13 eV due to thermally induced charge delocalization between Fe2+ and Fe3+ ions (A sites) in double octahedral chains. The charge delocalization induces a second order crystallographic phase transition on heating from monoclinic (P21/a) to orthorhombic (Pnam) symmetry at 346 K. The unit cell dimensions within the 295–420 K range and the crystal structures at 295, 320, 340, 360, 380 and 400 K have been determined by high temperature single crystal X-ray diffraction. The degree of charge delocalization determined from the sizes of the Fe(Ao) and Fe(Am) octahedra is the primary order parameter, Q which couples linearly with the spontaneous strain component, 13. The order parameter coupling and the associated free energy expression is given. The calculated normal modes of the space group symmetry change are consistent with the experimentally observed atomic displacements, which are parallel and antiparallel to c. Formation of antiphase lamellar twin domains parallel to (001) in the monoclinic phase is predicted to occur as a result of the phase transition. Above Tc (= 346 K), the slow asymptotic decrease of 13 attaining a zero value at 380 K indicates the presence of fluctuating precursor clusters with considerable short-range order above Tc. A peak in the specific heat (Cp) measurements coincides with the onset of longrange order at 380 K, whereas 57Fe Mssbauer measurements indicate the onset of charge localization at a considerbly higher temperature (470 K). The coupling of the d6 electron of the Fe2+ (A) ion with a longitudinal optic phonon with the polarization vector along c
* is the likely mechanism to drive the phase transition. The electronphonon coupling also provides a charge conduction mechanism through electron hopping, whereby the short-bonded Fe2+-Fe3+ pair containing the d6 electron (intermediate polaron) will break up and re-form, thereby propagating the electron one step along the c axis.
Physics and Chemistry of Minerals 03/1989; 16(5):483-496. · 1.73 Impact Factor
