An elementary introduction to superspace crystallography

Abstract

Aperiodic crystals are defined as a crystalline state of matter, that has atomic structures with long-range order but without translational symmetry. Experimentally, they are characterized by sharp Bragg reflections in the X-ray diffraction, that can be indexed by integers, if four or more reciprocal basis vectors are used. An introduction is given to the basic concepts of the superspace theory for structural analysis of incommensurately modulated crystals and incommensurate composite crystals [De Wolff, Janner and Janssen, Acta Crystallogr. 37 (1981) 625-636]. It is concluded that major challenges of the crystallography of incommensurate phases lie in the determination of the precise shapes of modulation functions, and in finding the relations between physical properties of incommensurate crystals and their atomic structures.