Space Groups and Crystalline Structures

Abstract

The classification of the crystalline electron and phonon states requires the knowledge of the full symmetry group of a crystal (space group G) and its irreducible representations. The group G includes both translations, operations from the point groups of symmetry and combined operations. The application of symmetry transformations means splitting all space into systems of equivalent points known also as Wyckoff positions in crystals, irrespective of whether there are atoms in these points or not. The crystal-structure typeis specified when one states which sets of the Wyckoff positions for the corresponding space group are occupied by atoms. To distinguish between different structures of the same type, one needs the numerical values of lattice parameters and additional data if there exist occupied Wyckoff positions with free parameters in the coordinates. We briefly discuss the 15 crystal-structure types ordered by the space-group index.