Philippe Wolff

A modulated structure can be depicted as a section through a four-dimensional periodic structure. In the latter, each atom is represented by a string continuing endlessly in the overall direction (e4) of the normal to R3, R3 being the hyperplane of the section. The strings have periodic bends or densifications for displacive and substitutional modulation respectively. Formulae for structure factors can be derived from this picture with little effort. The pseudo-symmetry of modulated structures can be described conveniently in this picture. Each four-dimensional space group to which the four-dimensional structure can belong is a possible MS3 (modulated three-dimensional structure) group of pseudo-symmetry, and is called an MS3 space group. It is shown that MS3 point groups are reducible in the form Q⊕ɛ. 1, where 1 is the unit 2 × 2 matrix, and ɛ = ± 1. A list is presented of these 31 groups written as black-and-white or colourless groups of three-dimensional symmetry. The MS3 space groups are discussed briefly. As an example of the peculiar differentiations caused by e4 being a unique direction, the 23 MS2 space groups are listed explicitly. Finally, it is shown that MS groups are essential for the description of MS symmetry, because very often the latter cannot be represented completely and unambiguously by the normal space group of an approximate superstructure.

A complete list of (3 + 1)-dimensional superspace groups is presented. These groups describe the symmetry of incommensurate crystal structures with a one-dimensional modulation. A short discussion is given of applications. Extinction rules and Bravais types are tabulated in order to facilitate the determination of the superspace-group symmetry.

An earlier report [Acta Cryst. (1977), A33, 681-684] by a joint IUCr-IMA committee on the nomenclature of polytypism has been revised and extended. Two kinds of symbolism are recommended for use with either simple or complicated polytypic structures. The first consists of 'indicative symbols' in a modified Gard notation, the second of 'descriptive symbols' based on earlier proposals by Dornberger-Schiff, Ďurovič and Zvyagin. The polytypism of ZnS and SrGeO3 provides examples for the use of descriptive symbolism.

Displacive modulation is defined as periodic distortion with an incommensurable k vector. If the phase of the distortion is t, a symmetry operation can be expressed as a normal space-group operation combined with a sign reversal and/or a shift in the variable t. Earlier results obtained by de Wolff [Acta Cryst. (1974), A30, 777-785] are reformulated in terms of these operations. The periodic functions defining the displacements of two symmetry-related atoms are shown to be related by a simple equation. Applications to published structures are given. The validity of this equation depends on a symmetry-adapted choice of the vector k. It is shown by a two-dimensional example that there are cases for which that choice of k requires the introduction of 'improper symmetry translations' with an extra t shift of ½ or ½. The ensuing Bravais lattice types are similar to those used for the description of magnetic symmetry.