Superspace-group description of short-period commensurately modulated crystals

The symmetry of superstructures is described by an ordinary three-dimensional space group, based on a unit cell which is a multiple of the basic structure unit cell. Alternatively, superstructures can be looked upon as commensurately modulated structures. Then, the structure is described by its basic structure plus some distortion wave. In this paper the application to superstructures of superspace groups, originally devised to describe the symmetry of incommensurately modulated structures, is discussed. The Bravais classes of superspace groups for commensurately modulated structures are derived. A comparison is made between the superspace group and the ordinary three-dimensional space-group description for commensurately modulated structures. With the help of the structure of Ag_0.35TiS₂, the consequences for the interpretation of the diffraction pattern are discussed. Also, the nature and number of the independent parameters in the superspace-group approach are compared with the description of the structure by an ordinary space group.