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.
Recently one has seen a growing interest in systems, like modulated crystals and crystals with charge or spin density waves, which can be considered as crystals with a distortion which is periodic in space or in space time. The Euclidean symmetry of these systems is, in general, not a three-dimensional space group and is fairly low. It is shown that enlarging the admitted group of transformations the symmetry group is a space group with dimension higher than three. For static systems the additional dimensions are related to internal degrees of freedom associated with relative Euclidean motions of the distortion with respect to an average crystal structure and for time-dependent ones to the time. A discussion of the symmetry is given, both for point particle systems and for continuous density distributions. These higher-dimensional space groups are relevant for the physical properties of such crystals, as shown here in particular for systematic extinctions occurring in their diffraction patterns.
In this second part [part !: Acta Cryst. (1980), A36, 399-408] the superspace-group approach is formulated for a class of crystals (called composite crystals) which involve a basic structure composed of subsystems, each one having three-dimensional space-group symmetry, but being mutually incommensurate. By taking into account the interaction among these subsystems, or other second-order effects, one is led to the actual structure, which very often is modulated, and in any case incommensurate. Neither the basic structure nor the actual one has a three-dimensional space-group symmetry but both allow a superspace-group characterization of their symmetry properties. The aim of the present paper is to show how these concepts apply in practice. Accordingly, two composite crystals, extensively studied in the literature, are considered from the present point of view: the organic compound (TTF)7Is_ x, i.e. C42H2sS2s.Is_x, and the polymercury cation compound Hg3_ ~ AsF 6. The regularities found in these two compounds are interpreted and fit naturally with the corresponding superspace-symmetry groups.
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