Splitting of Wyckoff positions (orbits). II. Group-subgroup chains of index 6.1

Abstract

Abstract A crystal structure is usually described as the union of sets of symmetrically equivalent atoms, i.e. as the union of orbits. During a continuous or nearly continuous phase transition, symmetrically equivalent atoms of a crystal structure may become inequivalent, their site symmetries may be reduced, or both may happen simultaneously. The same effect may occur as the result of an applied external mechanical, electric, magnetic, etc. homogeneous field. Using the orbit approach, this phenomenon may be described as ‘Splitting of orbits due to symmetry reduction’. The splitting behavior is the same for orbits of the same Wyckoff position. Therefore, one may speak loosely of ‘Splitting of Wyckoff positions’. In a preceding paper (Part I) the dependence of orbit splitting on the index and on the kind of group-subgroup relation [unk] – [unk] between groups has been considered. General laws have been derived and applied to the possible group-subgroup relations of index 4. In this paper (Part II) the splitting rules are formulated as two lemmata and applied to group-subgroup relations of index 6. There are eight different types of such relations, seven of which occur between space groups. Restrictions for the possible splittings are due to groups intermediate between [unk] and [unk] and due to the size of the normalizer [unk][unk]([unk]). The possible kinds of splittings are calculated. With two exceptions all of them are covered by theoretical examples. Furthermore, many of them describe relations between actual crystal structures.