Symmetry in the solid state; working beyond the space group

Abstract

Central to the mechanics of modern crystallography is the notion of a unit cell. It has long been a dogma that crystals are made by repetition of these fundamental units in three-dimensional space. In the 1970s so called ‘aperiodic’ structures were found that could not be described by the framework of the crystallographic space groups. Rather, these structures were found to be periodic only in a space with greater than three dimensions and their description required crystallographers to employ coloured space groups or superspace groups. There are, however, fundamental limitations to a symmetry description that is based on a unit cell and simple crystallographic symmetry operations. A more general description of commensurate and incommensurate crystallographic symmetry can be constructed from representation theory, as has been developed to describe magnetic symmetry and ordering processes. This article reviews these different extensions to the crystallographic symmetry and details examples and limitations of space group theory.