Section 2.2.14.1. Characterization by group theory

The energy bands are primarily characterized by the wavevector in the first BZ that is associated with the translational symmetry according to (2.2.4.23). The star of determines an irreducible basis provided that the functions of the star are symmetrized with respect to the small representations, as discussed in Section 2.2.6. Along symmetry lines in the BZ (e.g. from along towards X in the BZ shown in Fig. 2.2.7.1), the corresponding group of the vector may show a group–subgroup relation, as for example for and . The corresponding irreducible representations can then be found by deduction (or by induction in the case of a group–supergroup relation). These concepts define the compatibility relations (Bouckaert et al., 1930; Bradley & Cracknell, 1972), which tell us how to connect energy bands. For example, the twofold degenerate representation (the symmetry in a cubic system) splits into the and manifold in the direction, both of which are one-dimensional. The compatibility relations tell us how to connect bands. In addition, one can also find an orbital representation and thus knows from the group-theoretical analysis which orbitals belong to a certain energy band. This is very useful for interpretations.