The determination of the relationship between derivative lattices

Derivative lattices are related to one another by transformation matrices having rational elements. A simple algorithm for finding these matrices consists in testing if the scalar products of the vectors defining two arbitrary primitive cells of two lattices can be exactly or approximately related by equations with rational coefficients. A rational relationship indicates that two or more lattices have a number of geometrical features in common such as common superlattices, sublattices, etc. The algorithm can, therefore, be applied to a variety of crystallographic problems such as the study of twinning, the indexing of powder patterns, single-crystal diffractometry and the critical evaluation of crystal data. Five examples are discussed in detail.