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This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the crystallography of periodic crystals. A complete list of symmetry restrictions on modulation wave vectors is derived from this property. The symmetry of incommensurate crystals with an one-dimensional modulation is given by (3+1)-dimensional superspace groups. The latter are defined as a subset of the space groups in four-dimensional space. A thorough discussion is given of the notation of superspace groups, of equivalence relations between them, and of their various settings. Symmetry properties of modulation functions and other structural parameters are presented. An expression is derived for the structure factor of Bragg reflections that incorporates the full superspace symmetry of the incommensurately modulated structure.

This chapter gives an introduction to the distinctive structural features of aperiodic crystals, including incommensurately modulated crystals, composite crystals, and quasicrystals. Atomic structures are discussed in relation to the structures of periodic crystals, while it is shown that translational symmetry is lost. Diffraction by incommensurately modulated structures is shown to give rise to Bragg reflections that can be indexed by four or more integers. The modulation wave vector is introduced as the vector defining the periodicity of the modulation functions, as well as being the reciprocal vector employed in the indexing of Bragg reflections.

This chapter introduces the superspace theory for the description of the crystal structures of incommensurately modulated crystals. Two alternative constructions of superspace are presented. In reciprocal space the observed scattering vectors of Bragg reflections are considered to be the projections of reciprocal lattice points in (3+1)-dimensional superspace. Lattice translations in superspace applied to the atomic positions in three-dimensional, physical space generate the structure model in superspace. Thus, the latter is a periodic structure in superspace by definition. Structure factors of Bragg reflections are shown to be the Fourier transform of the electron density on one unit cell of the superspace lattice. t-Plots are defined, and their use in structural chemistry is demonstrated by the application to the incommensurately modulated structure of Sr₂Nb₂O₇.