Stuart A. Hayward

Cubic-tetragonal phase transition in SrTiO3 revisited: Landau theory and transition mechanism

Abstract Existing experimental data for the antiferroelastic phase transition in strontium titanate are reviewed and analysed using a Landau free energy of the form ΔG = 1/2Aθs (cothθs/ Tc-colb.θ/T)Q2 + 1/4BQ 4 + 1/6CQ 6, with A = 0·6472 J K−1mol−1, B = 29·12 Jmol−1, C = 39·27 Jmol, T c= 105·6 K, θ S = 60·8 K. The temperature dependence of the critical exponent is found to be due to the delicate balance between the Q 4 and Q 6 terms in the free energy expansion, and the saturation of the order parameter at low temperatures. The spontaneous strains observed in this phase transition are not consistent with simple rotation of the TiO6 octahedra around [001], An alternative model is proposed, where these octahedra expand in order to preserve the volume of the twelve-fold co-ordinated Sr site and the spacing between SrO3 pseudo-closepacked layers.

Low-temperature phase diagrams: non-linearities due to quantum mechanical saturation of order parameters

At low temperatures, the behaviour of structural phase transitions is modified by the influence of quantum fluctuations. Such fluctuations enhance the stability of the high-symmetry phase, reducing the observed transition temperature. The effect on phase diagrams of temperature versus some other control parameter (e.g. pressure or chemical composition) is described. A simple low-temperature extension of Landau theory is used, where the extent of quantum mechanical effects is characterized by a saturation temperature . The theory is used to model the phase diagrams of the mineral anorthoclase , the ferroelectric materials ( for the 20 K transition), and SbSI . The saturation temperature is related to the extent to which changes in the hard phonon modes influence the transition mechanism.

Order parameter saturation in LaAlO3

High-resolution x-ray rocking diffraction has been used to measure the spontaneous strain associated with the cubic–rhombohedral phase transition in LaAlO3 in the range 10K ≤ T ≤ 750 K. The data are consistent with a second-order Landau-like model at high temperatures, with TC = 834(2) K. At lower temperatures, the strain data display order parameter saturation, related to the quantum saturation of the phonon modes. Comparison of the saturation temperature for the spontaneous strain (θS = 95 K) with the saturation temperatures for independent measurements of the rotation (θS = 260 K) and distortion (θS = 150 K) of the AlO6 octahedra indicates that the phase transition consists of two coupled processes, and that the coupling does not have the same effect in the classical and quantum saturation limits.

Thermodynamic nature of and spontaneous strain below the cubic–monoclinic phase transition in La2Mo2O9

The pseudocubic lattice parameter of lanthanum molybdate, La2Mo2O9 ,h as been measured as a function of temperature between 13 and 1100 K. These data show an anomaly at 842 K, associated with the alpha (pseudocubic monoclinic, non-conducting) to beta (cubic, conducting) phase transition. For as ample w hich has been quenched from high temperatures, the a(T ) data are continuous across this transition, indicating that the transition is smeared or thermodynamically second order. When the same material is cooled slowly, subsequent measurements show the transition to be first order, with as imilar transition temperature. The difference between these two behaviours is attributed to changes in oxygen ordering resulting from different thermal histories.

Displacive phase transition in anorthoclase: The “plateau effect” and the effect of T1-T2 ordering on the transition temperature

Abstract The effects of compositional changes and nonsymmetry-breaking Al-Si ordering on the displacive phase transition in anorthoclase have been studied using X-ray powder diffraction. The results were analyzed using a model based on Landau theory. The observed transition temperature, T∗ c, was found to be independent of orthoclase content in the range 0 < XOr < 0.02. For samples with XOr > 0.02, Tf decreases linearly with increasing XOr. This plateau behavior is explained by the finite volume of the strain field around each K+ ion: Chemical mixing behavior is observed only for average K -K+ distances of <20 Å. Increasing Al-Si order between the T1 and T2 sites causes Tc to increase; this is unlike T1o-T1m ordering, which prevents the transition to monoclinic symmetry altogether. The range of T∗ c values reported in the literature is shown to be consistent with different degrees of T1-T2 order.

The pressure-temperature phase diagram of BaTiO3: a macroscopic description of the low-temperature behaviour

The pressure–temperature phase diagram of BaTiO3 has been investigated using a modification of the standard Landau potential to take account of quantum saturation of the order parameter at low temperatures. The calculated phase diagram agrees well with experiment for the cubic–tetragonal and tetragonal–orthorhombic transitions, but underestimates the orthorhombic–rhombohedral transition temperature somewhat. The saturation temperature (θS = 160 K) is sufficiently high that the expected critical point is not observed experimentally. Instead, each phase boundary bends sharply down, so each of the four crystalline structures of BaTiO3 has a stability field with increasing pressure at 0 K.

Antiferroelectric phase transition in titanite: Excess entropy and short range order

Abstract The antiferroelectric A2/a↔P21/a phase transition in titanite may be described using a thermodynamic model where the principal contribution to the excess entropy is assumed to be configurational rather than vibrational; G = A/2 Q2 + B/4 Q4 + λT [(1 + Q) ln (1 + Q) + (1 - Q) ln (1 - Q)]. Such a model is likely to be a valid description of a phase transition where strain effects are large enough to maintain mean field behavior, but not so large that vibrational effects dominate in the excess entropy. Best-fit parameters for A/λ and B/λ are determined from X-ray measurements of the order parameter. The magnitudes of the three parameters are then determined using calorimetric data, with the results A = -337.6 J/mol, B = -112.5 J/mol, λ = 0.34 J/mol-K. The model is compared with measurements of dielectric susceptibility, and is found to give good, but not perfect, agreement with experiment. The excess entropy associated with the transition is far smaller than expected for a simple order-disorder model. This is interpreted as evidence for significant short-range order above TC.

Twin walls and hierarchical mesoscopic structures

Abstract Networks of interacting twin walls form hierarchical, mesoscopic structures in minerals. The typical thickness of transformation twins at T << TC is ~3 nm and increases at T → TC as |T– TC|-1/2. The internal structure of twin walls is derived to be chiral in systems with coupled order parameters. The effect of wall bending is discussed.

Phase transitions in lawsonite a calorimetric study

The specific heat of lawsonite, CaAl 2 Si 2 O 7 (OH) 2 .H 2 O, has been measured in the temperature range 125 K - 325 K. An anomaly is seen at 273 K, which is related to the Cmcm — Pmcn phase transition. The magnitude of the total excess entropy associated with this transition is not reproducible, varying in the range 5.93 — 6.24 J K −1 mol −1 . On heating, the specific heat anomaly is consistent with a tricritical phase transition. However, on cooling, significant hysteresis is observed, and the form of the C P anomaly is quite different. In all measurements extensive pre-transitional effects are observed above T C . Analysis of existing specific heat data in the temperature range 75 K — 175 K shows an anomaly associated with the Pmcn — P2 1 cn phase transition. The excess entropy associated with this transition is 6 (1) J K −1 mol −1 . These data are interpreted as showing that both transitions are caused by the interaction of proton ordering and displacive changes in the aluminosilicate framework. The standard entropy of lawsonite at 298 K is recalculated, incorporating the effects of the two transitions. Two methods are used for this recalculation, giving values of S 0 298 = 233.27 and 234.96 JK −1 mol −1 respectively.

Twin memory and twin amnesia in anorthoclase

Abstract Many natural minerals and synthetic materials display twin microstructures resulting from displacive phase transitions. These microstructures may be removed temporarily from the sample by heating above the relevant transition temperature, though the twinning generally returns on subsequent cooling. In anorthoclase, the spatial distributions of twins before and after brief annealing above TC are often identical. This property appears to be a common feature in many materials which undergo ferroelastic phase transitions, and is known as ‘twin memory’. The atomic mechanisms responsible for this twin memory may be investigated by studying the annealing regimes required to remove the memory effect; how long must a sample be annealed, and at what temperature, to induce ‘twin amnesia’. High-resolution X-ray diffraction (XRD) has been used to investigate twin memory and twin amnesia in anorthoclase. In anorthoclase, the primary constraint on twin amnesia is thermodynamic, rather than kinetic. The critical temperature to induce amnesia correlates well with the top of the (Na, K) solvus in disordered alkali feldspar. For this reason, the proposed mechanism for twin memory involves the segregation of alkali cations in thin lamellae at the twin boundaries.