Symmetry of periodically distorted crystals

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.