Equivalence among isotropy subgroups of space groups.

The Landau theory' of continuous phase transitions provides a powerful tool for understanding transitions between solid phases whose symmetries have a group-subgroup relationship. In the Landau theory, the thermodynamic free energy F of the crystal is written as a function of an order parameter Q. In the high-symmetry phase, the minimum of F is at Q = 0. In the low-symmetry phase, the minimum of I' occurs for some nonzero value of P. Let Go be the spacegroup symmetry of the high-symmetry phase. The spacegroup symmetry of the low-symmetry phase must be an isotropy subgroup of Go. Such subgroups can be obtained by group-theoretical methods. 2 Thus, from a complete list of isotropy subgroups one can obtain all possible values of Q for which the free energy may be at a minimum. Isotropy subgroups of a space group may be obtained by group-theoretical methods. We have implemented these methods on computer and obtained for all k points of symmetry a list of all isotropy subgroups of each of the 230 three-dimensional space groups, ' as well as each of the 17 two-dimensional space groups and each of the 80 diperiodic space groups. 5 In generating these lists, we have encountered a problem concerning "equivalent" isotropy subgroups. This question of equivalence has been treated ambiguously in the literature, as we will show below. In this paper, we briefly discuss the physical basis for defining equivalence and then through some specific examples demonstrate an appropriate application of this physical basis. Consider an isotropy subgroup G of Go. We can decompose Go into left cosets of G,