J. M. Perez-Mato

Mode crystallography of distorted structures

The description of displacive distorted structures in terms of symmetry-adapted modes is reviewed. A specific parameterization of the symmetry-mode decomposition of these pseudosymmetric structures defined on the setting of the experimental space group is proposed. This approach closely follows crystallographic conventions and permits a straightforward transformation between symmetry-mode and conventional descriptions of the structures. Multiple examples are presented showing the insight provided by the symmetry-mode approach. The methodology is shown at work, illustrating its various possibilities for improving the characterization of distorted structures, for example: detection of hidden structural correlations, identification of fundamental and marginal degrees of freedom, reduction of the effective number of atomic positional parameters, quantitative comparison of structures with the same or different space group, detection of false refinement minima, systematic characterization of thermal behavior, rationalization of phase diagrams and various symmetries in families of compounds etc. The close relation of the symmetry-mode description with the superspace formalism applied to commensurate superstructures is also discussed. Finally, the application of this methodology in the field of ab initio or first-principles calculations is outlined. At present, there are several freely available user-friendly computer tools for performing automatic symmetry-mode analyses. The use of these programs does not require a deep knowledge of group theory and can be applied either a posteriori to analyze a given distorted structure or a priori to parameterize the structure to be determined. It is hoped that this article will encourage the use of these tools. All the examples presented here have been worked out using the program AMPLIMODES [Orobengoa et al. (2009). J. Appl. Cryst. 42, 820-833].

Structural phase transitions in crystals with P63/mmc symmetry

A study of the possible phase symmetries derived from a prototype phase with space group P63/mmc (D6h4) has been performed. Only those phases that can be explained as the result of the onset of an order parameter transforming according to one irreducible representation of the prototype space group are considered. The different space groups for the possible distorted phases have been investigated. All the irreducible representations which belong to symmetry points in the hexagonal Brillouin zone are analysed. The Landau and Lifshitz criteria have been considered to determine those transitions which should necessarily be first order. For every representation the authors have calculated the subspace in the representation space along which the displacement of the order parameter must take place to describe the different permitted symmetries for the distorted phase. Every considered irreducible representation D(k, j) and every relevant subgroup G of P63/mmc, of the number of times that the identity representation is contained in the representation subduced in G by D(k,j). This information together with the chain subduction criterion permits a complete investigation of the symmetry in the representation space.

Diffraction symmetry of incommensurate structures

The symmetry of the diffusion pattern for an incommensurate displacive structure is analysed in a general context. The rotational symmetry and systematic extinctions in the diffraction diagram of this type of structure are shown to be directly based on a particular transformation property of the amplitudes of the atomic modulation waves intervening in the distortion for a subgroup of the basic structure space group. Hence a general three-dimensional framework can be established, where diffraction symmetry can straight-forwardly be used to restrict and relate the amplitudes corresponding to the atomic modulation waves of the distortion. It is shown how a full direct exploitation of diffraction symmetry can therefore be accomplished to determine the characteristics of the incommensurate distortion, without referring to the superspace group formalism. The equivalence of the present simpler approach with the latter is then demonstrated. Finally a concept of atomic scattering modulation factors is proposed to reduce the expression of the structure factor for these phases to a very simple form analogous to those used in commensurate cases.

Competing Instabilities in Ferroelectric Aurivillius Compounds

The stability of SrBi2Ta2O9 has been studied by means of ab-initio calculations. The main instability of the tetragonal configuration concerns a non-polar mode at the Brillouin zone border. It can be related through Brillouin zone folding to the well-known soft rigid-unit mode R25 in perovskites. This implies a complex scenario for the whole Aurivillius family with two competing order parameters, which can result in a sequence of two continuous phase transitions or a single discontinuous one. A third hard-mode is shown to play an essential role in the stabilization of the ferroelectric phase.

Group theoretical analysis of the structural phase transitions sequence in the quasi-one dimensional compound ((CCH3)2NH)CdCl3

The two recently reported structural phase transitions in (CH3)3NHCdCl3 are investigated from a group theoretical point of view and the symmetries of their corresponding order parameters have been determined. The authors also present the faint variables and modes for every phase and their faintness indices. From a study of the distortion which relates the prototype phase with the room temperature structure, they show how the orthorhombic distortion can be understood as mainly the result of two normal modes, one corresponding to antiphase rotations of successive octahedra columns, and the other related to y displacements with Eg symmetry of the nitrogen atoms. The two motions are clearly correlated.