Space Lattice Symmetry

Abstract

Some properties of crystalline solids, such as the directions and symmetry of diffraction (X-ray, neutron, electron), the anisotropy of transport phenomena (electrical, thermal conduction, matter diffusion), and the anisotropy of thermal expansion and interactions of crystals with optical radiation are directly related to the underlying three-dimensional periodicities that characterize many crystalline materials. It is therefore important to understand such periodicity in order to better understand the phenomenology of the interactions of crystalline solids. It is also necessary to understand three-dimensional periodicity as a basis for development of the abstract theory of symmetry in crystalline solids (the theory of space groups and their representations). The three-dimensional periodicities are conveniently discussed in terms of space lattices, which are three-dimensional spatial arrays of descrete points which correspond to the translational symmetries of solids. The theory of space lattices, developed below, is the theory of the allowed symmetries of the periodicity of three-dimensionally crystalline solids.