Abstract
The simplest crystallographic color groups are the permutational color groups. Elements of these groups combine two types of transformations: One is a rotation and/or translation of physical space and the other is a permutation. The groups considered here are subgroups of direct products and abstractly isomorphic to crystallographic groups, hence their relative simplicity. Despite this simplicity, there is a richness of information contained in each such group. The group symbol conveys the following: the isomorphic crystallographic group , a subgroup of , the largest normal subgroup of , contained in , and a transitive group of permutations isomorphic to the factor group . We derive and tabulate here all classes of equivalent permutational color point groups using a definition of equivalence classes which we physically motivate. For applications we require and report here the permutation representation of associated with each and we reduce into irreducible components. The major application given here is to the Landau theory of symmetry change in continuous phase transitions. A complete set of tables is presented for all allowed equitranslational ("Zellengleich" or ) phase transitions in crystals based on group-theoretical criteria, including a new "kernel-core" criterion. The tables may be used to determine all active representations for transitions between two specific groups or alternatively, all possible subgroups which can be obtained from a specific group and irreducible representation. We also relate two classifications schemes for phase transitions to the structure of permutational color groups.
- Received 2 June 1982
DOI:https://doi.org/10.1103/PhysRevB.26.6947
©1982 American Physical Society