Klassengleiche and translationengleiche subgroups

Subgroups of space groups are often used to describe the structures of a family of compounds which are closely related. They can also describe a sequence of different phases of a single compound resulting from temperature or pressure changes.

Definition

Subgroups of space groups occur in two categories namely klassengleiche subgroups or translationengleiche subgroups. (The terms originate from the german language.) In the first case, the crystal class of the space group is maintained during the transition from the space group to the subgroup. In the second case, the lattice translations are maintained during the transition.

Usually only maximal subgroups of the space groups are considered. A subgroup of a space group is maximal is there are no other space group which is simultaneously a subgroup of the space group and a supergroup of the subgroup.

Maximal subgroups of the space groups are tabulated in IT vol A1

Each maximal subgroup is characterised by its index i.e. the inverse of the fraction of the symmetry operations which remained during the transition to the subgroup.

Only the maximal subgroups of the space groups are tabulated in IT vol A1

Examples

A subgroup of index k2 means that the crystal class is maintained during the transition and half of the translation symmetry operations have been lost during the transition. A subgroup of index t4 means that all the translations are maintained during the transition but only one quarter of the operations of the crystal class remained during the transition.

See also

International Tables Vol. A1 [1]

Symmetry Relationships between Crystal Structures. By Ulrich Müller. IUCr/Oxford Science Publications, 2013.