Transformations of coordinate systems

When dealing with descriptions of structures, it is not seldom that we are faced with the problem of comparing identical crystal structures described with respect to different coordinate systems. For example, two descriptions of the same structure can differ by an origin shift or by a different choice of the basis. Different phases of the same compound often differ in their symmetry at various temperatures or pressures. Any detailed comparison of their structures requires the selection of a common basis and consequently the transformation of the original data to a different coordinate system. The purpose of this chapter is to provide the mathematical tools to accomplish these transformations. The method for transforming the crystallographic data following a change of origin or a change of the basis is given and illustrated with some examples. The transformation rules of the metric tensor characterizing both the direct and reciprocal space and of the space-group symmetry operations under coordinate transformations are further derived and discussed. More than 40 different types of coordinate-system transformations representing the most frequently encountered cases are listed and illustrated. Finally, synoptic tables of space (plane) groups show different types of symmetry operations belonging to the same coset with respect to the translation subgroup and a large selection of alternative settings of space (plane) groups and their Hermann–Mauguin symbols covering most practical cases.