SYMMODES: a software package for group-theoretical analysis of structural phase transitions

Abstract

Group-theoretical methods are widely used in the investigation of phase transitions in crystals. On the one hand, important structural aspects concerning the relation between the lowand the highsymmetry structures follows from the the group±subgroup relations of the space groups G > H describing the symmetry of the two phases. For example, the number of low-symmetry domain states equals the index of H in the parent group G, while their symmetry groups are conjugate subgroups Hj (isomorphic to H) of G. The splitting schemes of the occupied atomic orbits for the symmetry break G > H (known also as splitting of Wyckoff positions) determine the structural relation between the two phases. Further, an essential feature of a continuous or quasicontinuous displacive phase transition is the appearance of a symmetry-breaking distortion in the low-symmetry phase (with respect to the highsymmetry one) which is mainly caused by the freezing of some mode, the so-called primary mode. In addition, secondary modes are also triggered by the phase transition and can also have non-zero spontaneous amplitudes in the distorted structure. The so-called symmetry-mode analysis allows the `separation' of the frozen primary mode from the contributions of modes of different symmetry present as a secondary effect, but for its application it is necessary ®rst to calculate the symmetry modes of the parent phase, compatible with the low-symmetry phase, and then using their orthonormality properties to decompose the global distortion as a sum of the contributions of each of them. Such a symmetry analysis of phase-transition phenomena is rather complex as it requires full use of group-theoretical methods, including detailed knowledge of the group±subgroup relations of space groups and their irreducible representations (`irreps'). The goal of the developed software package SYMMODES is to provide the necessary tools for a group-theoretical analysis of a structural phase transition, including (i) a detailed group±subgroup description of the chain G > H and (ii) a calculation of the primary and secondary modes for the symmetry break G ! H.