Section 4.6.2.1. Basic concepts

An incommensurate modulation of a lattice-periodic structure destroys its translational symmetry in direct and reciprocal space. In the early seventies, a method was suggested by de Wolff (1974) for restoring the lost lattice symmetry by considering the diffraction pattern of an incommensurately modulated structure (IMS) as a projection of an nD reciprocal lattice upon the physical space. n, the dimension of the superspace, is always larger than or equal to d, the dimension of the physical space. This leads to a simple method for the description and characterization of IMSs as well as a variety of new possibilities in their structure analysis. The nD embedding method is well established today and can be applied to all aperiodic crystals with reciprocal-space structure equivalent to a module with finite rank n (Janssen, 1988). The dimension of the embedding space is determined by the rank of the module, i.e. by the number of reciprocal-basis vectors necessary to allow for indexing all Bragg reflections with integer numbers. The point symmetry of the 3D reciprocal space (Fourier spectrum) constrains the point symmetry of the nD reciprocal lattice and restricts the number of possible nD symmetry groups.

In the following sections, the nD descriptions of the four main classes of aperiodic crystals are demonstrated on simple 1D examples of incommensurately modulated phases, composite crystals, quasicrystals and structures with fractally shaped atomic surfaces. The main emphasis is placed on quasicrystals that show scaling symmetry, a new and unusual property in structural crystallography. A detailed discussion of the different types of 3D aperiodic crystals follows in Section 4.6.3.