Application of unitary groups for determining tensor components in crystals and selection principles for phase transitions

Abstract

The paper is concerned with compound series of unitary groups, of rotation groups and point groups for classification of various tensor components i n crystals, which have various components in relation to a rotation group and a permutation group. For this purpose the following series Ulvly 1 R[,ly 3 ... 3 G is being introduced, where y represents a vector (a pseudovector); then the number of various branches which end with a unit group G determines the number of various tensor components, which have obvious properties for reduction symmetry. When classifying in this way, a new interpretation of selection principles for phase transitions in crystals can be introduced.Fur verbundene Serien von unitiiren Gruppen, Rotationsgruppen und Punktgruppen werden verschiedene Tensorkomponenten in Kristallen klassifiziert, die verschiedene Komponenteii bezuglich einer Rotationsgruppe und einer Permutationsgruppe besitzen. Dazu werden die folgenden Serien U [ y l y 3 Rl,,lv 3 ... 3 G eingefiihrt, wobei y einen Vektor (einen Pseudovektor) darstellt; dann bestimmt die Zahl der verschiedenen Zweige, die mit einer Einheitsgruppe G enden, die Zahl der verschiedenen Tensorkomponenten, die deutlich Eigenschaften der Reduktionssymmetrie besitzen. Durch eine derartige Klassifizierung kann eine neue Interpretation von Aaswahlprinzipien fiir Phasenubergange in Kristallen eingefuhrt werden.

Allowed symmetries of soft-modes in cubic crystals

“…Group theory and Landau theory of phase transition have been reviewed in several papers [14,15]. The application of unitary groups for determining tensor components in crystals and selection principles for phase transitions and the application of tensor field in crystals and colour groups have been considered in papers [16,17]. Assuming the Landau theory, the theory of colour permutation groups and the representations associated with them have been applied to the group-theoretical analysis and the classification of continuous phase transitions has been discussed in paper [ 18].…”