A general introduction to space groups

Abstract

This chapter provides an introduction to the structure and classification of crystallographic space groups. Viewing space groups as groups of affine mappings which leave a crystal pattern invariant as a whole suggests a natural decomposition of a space group into its translation subgroup and its point group, since an affine mapping is composed from a linear part and a translation part. Starting with the translation part, one observes that the vectors by which the translations in the space group move the crystal pattern form a lattice, and we discuss fundamental concepts concerned with lattices, such as the metric tensor, the unit cell and the distinction into primitive and centred lattices. We then proceed to the point groups, which are formed by the linear parts of the affine mappings contained in a space group. A crucial point is that the point group acts on the translation lattice and the interplay between point groups and lattices is discussed in detail. In particular, the distinction between symmorphic and non-symmorphic groups is explained. The final part of this chapter deals with various schemes in which crystallographic space groups are classified. The most important of these are the classification into space-group types, geometric crystal classes and Bravais types of lattices.