Wreath groups symmetry of crystals with structural distortions

A new class of groups is defined for describing the symmetry of atom arrangements in crystals with structural distortions. The general structure of these new groups, which we name wreath groups, is determined in a mathematically rigorous manner. Wreath groups are particularly suitable for describing atom arrangements in crystals with periodic structural distortions. The effect of such wreath-group symmetry on x-ray-diffraction patterns is briefly discussed. A new Bloch theorem, based on wreath-group symmetry, is formulated for crystals with structural distortions. This new Bloch theorem is applied to determine the form and corresponding charge density of one-electron eigenfunctions in the nearly-free-electron approximation for crystals with periodic structural distortions.