Computers and group-theoretical methods for studying structural phase transitions
type: Item
instance of: scholarly article
title: Computers and group-theoretical methods for studying structural phase transitions
zbMATH Open document ID: 0715.20003
DOI: 10.1016/0898-1221(89)90162-4
author: G. M. Chechin
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20C35
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 20G45
zbMATH DE Number: 4177357
zbMATH Keywords: Landau's theory
zbMATH Keywords: continuous structural phase transitions in crystals
zbMATH Keywords: space groups
zbMATH Keywords: irreducible representations
zbMATH Keywords: group theoretical computer programs
zbMATH Keywords: stationary vectors
reviewed by: A. Janner
describes a project that uses: ISOTROPY
MaRDI profile type: Publication
full work available at URL: 0898-1221(89)90162-4
OpenAlex ID: W2035772904
cites work: Q3247767
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 4177357