Lattices and Space-Group Theory

Abstract

We continue our study of crystals by investigating the internal arrangements of crystalline materials. Crystals are characterized by periodicities in three dimensions. An atomic grouping, or pattern motif which, itself, may or may not be symmetrical, is repeated again and again by a symmetry mechanism, namely the space group of the crystal. There are 230 space groups, and each crystal substance belongs to one or other of them. In its simplest form, a space group may be derived from the repetition of a pattern motif by the translations of a lattice, as discussed below. It can be developed further by incorporating additional symmetry elements, as demonstrated through the following text and Problem 2.1. We now enlarge on these ideas, starting with an examination of lattices.