Dorian M. Hatch

ISODISPLACE: a web-based tool for exploring structural distortions

ISODISPLACE is a new internet-server tool for exploring structural phase transitions. Given parent-phase structural information, it generates atomic displacement patterns induced by irreducible representations of the parent space-group symmetry and allows a user to visualize and manipulate the amplitude of each distortion mode interactively. ISODISPLACE is freely accessible at http://stokes.byu.edu/isodisplace.html via common internet browsers.

Isotropy subgroups of the 230 crystallographic space groups

This book gives a rather exhaustive list of isotropy subgroups of the 230 crystallographic space groups. The symmetry changes for the vast majority of observed phase transitions in crystalline solids can be found in the list. With each entry, information is given concerning both physical and abstract characteristics of the phase transitions.

FINDSYM: program for identifying the space-group symmetry of a crystal

Given the lattice parameters of a crystal and the positions of atoms in the unit cell, it is often useful to identify the space-group symmetry of the crystal in a standard setting along with the Wyckoff positions to which the atoms belong. This can be a very difficult problem to do by hand, especially if the unit cell is given in a non-standard setting or if there are many atoms in the unit cell. The aim of this program is to identify the space-group symmetry and give the lattice parameters and Wyckoff positions of the atoms in a standard setting, no matter what setting the original information is given in. We have compared FINDSYM with other programs that address the same general problem. MISSYM (Le Page, 1987) and its implementation as ADDSYM in the program PLATON (Spek, 2003) searches for possibly missed additional symmetry in a given coordinate set. Two examples of its use are given at the internet site for ADDSYM (1998). FINDSYM failed in both cases to find the additonal symmetry found by ADDSYM. FINDSYM requires that all atomic positions be reasonably close to their idealized positions in the space-group symmetry we are seeking. TRS, RGS and SFND implemented in the program KPLOT (Hannemann et al., 1998; Hundt et al., 1999) use algorithms very similar to those in FINDSYM. Two examples of its function are given by Hannemann et al. (1998) and two more by Hundt et al. (1999). Each of these four examples consists of structures found during the investigation of the energy landscape of a system using global optimization techniques. In each of the four examples, FINDSYM successfully found the same space-group symmetry as that found by the KPLOT routines (this required using a tolerance of 0.03–0.06 Å). FINDSYM was not developed for the purpose of extracting a space-group symmetry from noisey data, although it can accomplish this if the data are not too noisey, as illustrated by its success with the KPLOT examples. Its limitations in this regard are illustrated by its failure with the ADDSYM examples. The main strength of FINDSYM is its ease of use. It can be run over the internet by filling out a simple and user-friendly html form. 2. Method of solution

Group-theoretical analysis of octahedral tilting in ferroelectric perovskites

Group-theoretical methods are used to analyze perovskite structures where both ferroelectric cation displacements and simple tilting of octahedral units are present. This results in a list of 40 different structures, each with a unique space-group symmetry. The list is compared with that of Aleksandrov & Bartolomé [Phase Transit. (2001), 74, 255-335] and a number of differences are found. The group-subgroup relationships between the structures are also determined, along with an indication of those phase transitions that must be first order by Landau theory.

New mechanism for the alpha to omega martensitic transformation in pure titanium.

We propose a new direct mechanism for the pressure driven alpha-->omega martensitic transformation in pure titanium. A systematic algorithm enumerates all possible pathways whose energy barriers are evaluated. A new, homogeneous pathway emerges with a barrier at least 4 times lower than other pathways. The pathway is shown to be favorable in any nucleation model.

Ferroelastic phase transition in cryolite, Na3AlF6, a mixed fluoride perovskite: High temperature single crystal X-ray diffraction study and symmetry analysis of the transition mechanism

Cryolite, Na3AlF6[ = 2Na+(Na0.5+Al0.53+)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 (α phase), space group P21/n with a = 5.4139 (7), b = 5.6012 (5) and c = 7.7769 (8) Å and β = 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 T0 ∼ 885 K, in contrast to a previous report that it becomes cubic at ∼823 K. The space group of the high temperature β phase is Immm with a = 5.632 (4), b = 5.627 (3) and c = 7.958 (4) Å, Z = 2 at 890 K. Above T0, 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 Γ4+ 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 X3+ irreducible representation of Immm. Following Immm → P21/n phase transition, four equivalent domains of P21/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 T0 indicate a crossover from displacive to an order-disorder mechanism near the transition temperature. The β 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, ΔStrans (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 β phase is independently confirmed from a considerable narrowing of the 27Al nuclear magnetic resonance (NMR) line-shape above T0 (Stebbins et al. 1992).

Non-linear normal modes for systems with discrete symmetry

Abstract Normal modes in linear mechanical systems with a discrete symmetry group in their equilibrium state can be classified by irreducible representations (irreps) of this group. In non-linear dynamical systems, excitation of a given mode spreads to a number of other modes associated with different irreps, and this collection of modes was called a “bush” of modes in previous papers. There are some special cases where, because of symmetry restrictions , a bush is “irreducible” — it contains modes associated with a single irrep only. We looked for all irreducible bushes of vibrational modes for N -particle mechanical systems with the symmetry of any of the 230 space groups and, for the case of analytical potentials, found that there exist only 19 classes of such bushes. As a result, all modal subspaces to which symmetry determined similar non-linear normal modes (introduced by Rosenberg) belong, were found, as well as all analytical mechanical systems whose dynamics, with a certain mode being initially excited, strictly reduces to only one resonance subspace corresponding to a single irrep. We found that the dimensionality of such resonance subspaces does not exceed four.

Coupled order parameters in the Landau theory of phase transitions in solids

Abstract We discuss the role of both primary and secondary order parameters in a transition between two solid phases. We show how microscopic distortions which appear at the transition can be calculated using group-theoretical methods. We also show how the form of the coupling term in the Landau free energy influences the behavior of secondary order parameters.

Phase transitions and renormalization group hamiltonian densities in the 80 diperiodic space groups

Abstract Phase transitions for systems with diperiodic symmetry are discussed. Direct group-theoretical methods are employed to obtain a list of possible commensurate lower-symmetry phases (subgroups) which are induced by a single order parameter. The lower-symmetry phases for all 80 diperiodic space groups are given, along with specific details of the group-subgroup relationships. Results for the 17 two-dimensional space groups are also contained in our list. The renormalization-group Hamiltonian densities for the diperiodics are calculated. The 12 densities listed constitute the complete set of densities which may arise in the diperiodic space groups. Critical properties for the diperiodics can thus be obtained from analysis of these densities.

The α-β phase transition in AlPO4 cristobalite: Symmetry analysis, domain structure and transition dynamics

AbstractDespite their crystallographic differences, the mechanisms of the α-β phase transitions in the cristobalite phases of SiO2 and AlPO4 are very similar. The β→α transition in AlPO4 cristobalite is from cubic (