Section 9.2.2.2.2. Polytype families and OD groupoid families

All polytypes of a substance built on the same structural principle are said to belong to the same family. All polytypic structures, even of different substances, built according to the same symmetry principle also belong to a family, but different from the previous one since it includes structures of various polytype families, e.g. SiC, ZnS, AgI, which differ in their composition, lattice dimensions, etc. Such a family has been called an OD groupoid family; its members differ only in the relative distribution of coincidence operations³ describing the respective symmetries, irrespective of the crystallochemical content. These coincidence operations can be total or partial (local) and their set constitutes a groupoid (Dornberger-Schiff, 1964, pp. 16 ff.; Fichtner, 1965, 1977). Any polytype (abstract) belonging to such a family has its own stacking of layers, and its symmetry can be described by the appropriate individual groupoid. Strictly speaking, these groupoids are the members of an OD groupoid family. Let us recall that any space group consists of total coincidence operations only, which therefore become the symmetry operations for the entire structure.