A theory of sublattices and superlattices

An algorithm is presented by which the smallest common supercell of two non-equivalent unit cells of two lattices can be calculated with a minimum of trial and error. Necessary and sufficient conditions for the existence of such a common supercell are given. The existence of a common supercell implies the existence of a common subcell. The common supercell as well as the common subcell are of practical value as an appropriate reference frame for the comparison of polymorphous or related structures. By means of this theory, possible epitaxic relationships between two different lattices can be predicted. The algorithm is illustrated by an example.