Symmetry of composite crystals

Composite crystals are crystals that consist of two or more subsystems, in first approximation each one having its own three-dimensional periodicity. The symmetry of these subsystems is then characterized by an ordinary space group. Due to their mutual interaction the true structure consists of a collection of incommensurately modulated subsystems. In this paper we derive some general properties for intergrowth structures, using the superspace-group theory as developed by Janner and Janssen [Acta Crystallogr. A36, 408 (1980)]. In particular, the pseudoinverse is defined of the matrices relating the subsystem periodicities to the translation vectors in superspace. This pseudoinverse is then used to reformulate the relations between the structure and symmetry in three-dimensional space and in (3+d)-dimensional superspace. As an extension of the theory, subsystem superspace groups are defined, that characterize the symmetry of the individual, incommen- surately modulated subsystems. The relation between a unified description of the symmetry and an independent description of the subsystems is analyzed in detail, both on the level of the basic structure (translational symmetric subsystems) and on the level of the modulated structure (incommensurately modulated subsystems). The concepts are illustrated by the analysis of the diffraction symmetry of the intergrowth compound ${\mathrm{Hg}}_{3\mathrm{-}\mathrm{\delta }}$ ${\mathrm{AsF}}_6$.