Towards a more comprehensive crystallography

Abstract

Despite the intrinsic difference from the point of view of structure of various types of crystals, such as commensurate, incommensurately modulated, intergrowth crystals and quasicrystals, a common approach to their symmetry seems to be possible which eventually will lead to a more comprehensive crystallography. The unifying elements become apparent through treatments which, at first, seem to be contradictory with the geometry of the crystal structures involved. Examples are the description of aperiodic crystals in terms of lattice-periodic structures (going beyond three-dimensionality), the investigation of scaling symmetry in quasicrystals by means of a Z-module of translations generating a dense set of translationally equivalent atomic positions (going beyond discreteness) and finally the characterization of Euclidean properties of normal crystals through non-Euclidean symmetries (going beyond Euclidean metric). These changing approaches do not modify, however, the fundamental nature of crystals to be three-dimensional, discrete and Euclidean. They only allow implicit symmetry groups like the superspace groups (unifying the crystallography of incommensurate and commensurate crystals) and the multimetrical space groups (unifying the possible symmetries of quasicrystals and normal crystals) to be made explicit. Aspects of crystal diffraction, morphology and crystal structure are presented from this unifying point of view, without intending to cover the whole crystallography.