Crystallography of Quasicrystals

Abstract

Quasiperiodic structures (QSs), incommensurately modulated structures (IMSs) and composite structures (CSs) are the main types of aperiodic crystal structures (Cummins 1990,van Smaalen 1995,Axel and Gratias 1995,Yamamoto 1996,Steurer and Haibach 1998). Diffraction patterns of aperiodic crystals consist of sharp Bragg reflections like those of regular periodic crystals. In contrast, aperiodic crystal structures lack lattice periodicity (Burzlaff et al. 1992, Wondratschek 1992) in d-dimensional (dD) physical space. It was first shown by de Wolff (1974) that diffraction patterns of IMSs can be described as projections of appropriate n-dimensional (nD) reciprocal lattices onto physical space. Consequently, the direct space IMSs result from cutting nD periodic hypercrystals with physical space. All types of aperiodic crystals (excluding almost periodic structures) can be described entirely by the same nD approach. Crystallography of quasicrystals (QCs), therefore, corresponds mainly to crystallography extended to n dimensions. Hermann (1949) was the first to study the number of dimensions necessary for the existence of m-fold rotational symmetry preserving translation symmetry. For instance, a lattice being invariant under five-, eight-or twelvefold rotational symmetry has to be at least four-dimensional. In general, any aperiodic structure with a Fourier module corresponding to a ℤ-module of rank n, can be embedded in n dimensions to achieve a lattice periodic structure. Symmetry restrictions of the dD physical space drastically limit the number of symmetry groups needed in the corresponding nD symmetry analysis. A comprehensive description of the symmetry operations for nD periodic and quasiperiodic structures has been given, e.g., by Janssen (1992a).