Symmetry Relationships Between Crystal Structures and Their Practical Applications

Abstract

The space group of a structure describes the infinite set of symmetry transformations which leaves the structure unchanged. Subgroups of a particular space group are also of infinite order but consists of a subset of all symmetry transformations contained in the group. The group-subgroup relations can be used to study common features in parent crystal structures and establish a classitication scheme. The latest edition of the International Tables lists the most important subgroups for each space group. For the crystal chemist, this information is a convenient tool which can facilitate the establishment of yet undiscovered structural similarities. Some examples of structure relations are given for various families of crystal structures which are derived from the highly symmetrical compounds of rutile, α-Po and perowskite. It is often useful to consider additional parameters like e.g. rotation or distortion expressing the relative displacement of structural units to refine the classification scheme based on subgroup relations.