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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The superspace-group approach [Janner & Janssen (1977), Phys. Rev. B, 15, 643-658] is used to solve the symmetry problem of incommensurate crystal phases in the case of displacive-and occupation-wave modulation. Generalization is given to cover magnetic modulation as well. The symmetry conditions imposed by the superspace group on the crystal structure are derived and applied to the following incommensurate crystals, whose structures have been discussed in the literature independently from the present point of view: K2SeO 4, 2H-TaSe2, NaNO 2 and Cr. The superspace groups describing the symmetry of these compounds are indicated and the structural implications of the corresponding symmetry elements discussed.

A characterization of the dielectric properties in incommensurately modulated phases is presented. First a microscopic description is adopted. It takes into account the structural restrictions imposed on a microscopic scale by the superspace group. The macroscopic optical properties are derived from it. The optical activity as observed in the incommensurate phases of several compounds cannot be explained within a plain microscopic approximation. Therefore, a mesoscopic level is considered, implying averaging distances large with respect to the basic unit cell parameters or even the modulation wavelength, but small with respect to the size of the finite crystal. The phenomenological approach presented is compatible both with the microscopic description based on a superspace group symmetry, as well as with the macroscopic one involving the point group of the average structure. In the case considered here (one-dimensional incommensurate modulation) the optical properties of the dielectric medium are analysed in terms of subperiodic groups, periodic along the direction of the modulation wave and homogeneous along the other two directions. It is shown that the optical properties are affected by the incommensurability and that small changes in symmetry related to boundary conditions can lead to macroscopic effects. To interpret the experimental data more quantitatively a Jones model is applied, which describes the mesoscopic spatial dispersion of the modulated dielectric tensor to vary periodically along the modulation and being constant in the plane perpendicular to it. This model is compared with the experimental results for the ellipticity angle chi of (N(CH3)4)2ZnCl4 obtained for light propagating along the modulation.