Computers and group-theoretical methods for studying structural phase transitions
The paper analyses various approaches based on Landau's theory of continuous structural phase transitions in crystals, characterized by a group/subgroup relation of the symmetry of the crystal above and below a transition point, having space groups G and \(G_ D\), respectively. The crystal density of the low symmetry phase can be expanded in basis functions of irreducible representations (IR) of the space-group G of the high symmetry phase. For thermodynamical reasons, Landau concluded that in fact only one IR does occur in the expansion, denoted as critical IR, the corresponding coefficient vector being called the order parameter responsible for the transition. There are two methods for obtaining a classification of the continuous phase transitions by means of the IR's of the 3-dim. space groups. The first one by Birman is based on the number of times the trivial IR occurs in the representation of G restricted to the subgroup \(G_ D\). The second one, adopted by the Rostov Group as basis for a set of group theoretical computer programs, involves the idea of a stationary vector (SV) which expresses the invariance of the order parameter vector with respect to the action of the elements of the subgroup \(G_ D\). The basic property exploited is the 1-to-1 correspondence between these stationary vectors (for given IR of G) and the admissible subgroups \(G_ D.\) In the paper the various programs are discussed, the most important being: REPS: for deriving the IR of a space group, VECTOR: for finding a set of non-equivalent stationary vectors, GROUP: for identifying the corresponding space groups \(G_ D\) and FONON: for obtaining basis functions of the IR of a number of space groups. The procedure is illustrated on the basis of a concrete example, and an extensive list of references (71) is given.
continuous structural phase transitions in crystals
scientific article; zbMATH DE number 4177357
scientific article