Space Groups in Reciprocal Space and Representations

Abstract

When moving from molecules to crystals, the physical properties will be described by dispersion relations in reciprocal space, rather than by energy levels. One of the most important applications of group theory to solid state physics relates to the symmetries and degeneracies of the dispersion relations, especially at high symmetry points in the Brillouin zone. As discussed for the Bravais lattices in Sect. 9.2, the number of possible types of Brillouin zones is limited. The reciprocal space for Bravais lattices is discussed in Sect. 10.1 and this topic is also discussed in solid state physics courses [6, 45]. The classification of the symmetry properties in reciprocal space involves the group of the wave vector, which is the subject of this chapter. The group of the wave vector is important because it is the way in which both the point group symmetry and the translational symmetry of the crystal lattice are incorporated into the formalism that describes the dispersion relations of elementary excitations in a solid. Suppose that we have a symmetry operatorˆP operatorˆ operatorˆP {Rα|τ } based on the space group element {R α |τ } that leaves the periodic potential V (r) invariant, ˆ P {Rα|τ } V (r) = V (r). (10.1) The invariance relation of (10.1) has important implications on the form of the wave function ψ(r). In particular if we consider only the translation operatorˆP operatorˆ operatorˆP {ε|τ } based on the translation group elements {ε|τ }, we have the result ˆ P {ε|τ } ψ(r) = ψ(r + τ). (10.2) Within this framework, we can prove Bloch's theorem in Sect. 10.2.2, and then we go on in Sect. 10.3 to determine the symmetry of the wave vector. We then discuss representations for symmorphic and nonsymmorphic space groups and illustrate the group of the wave vector. In Sect. 10.6 we consider the group of the wave vector in some detail for the simple cubic lattice and then we make a few comments to extend these results for the simple cubic lattice to the face centered and body centered cubic structures. The compatibility relations