Chapter 7 - Crystallographic Space Groups

Abstract

This chapter describes various aspects of crystallographic space groups. An infinite three-dimensional lattice may be defined in terms of three linearly independent real basic lattice vectors. The set of all lattice vectors of the lattice is provided. The cubic system is probably the most significant body-centered and face-centered lattices occurring for a large number of important solids. For the body-centered cubic lattice, the basic lattice vectors join a point at the center of a cube to three of the vertices of the cube, so that the lattice points form a repeated cubic array with lattice points also occurring at every cube center. It is found that for a perfect crystalline solid, the group of the Schrodinger equation is a crystallographic space group, which contains rotations as well as pure primitive translations. It is suggested that the energy eigenfunctions must transform according to the irreducible representations of this subgroup. It is found that the infinite crystal is considered to consist of a set of basic blocks in the form of parallelepipeds having edges and that the physical situation is identical in corresponding points of different blocks. These boundary conditions cannot affect the behavior of electrons inside each basic block to any significant extent, so the bulk properties are again unchanged. The irreducible representations of the group of pure primitive translations and Bloch's theorem are also elaborated.