A point group approach to selection rules in crystals

Abstract

The problem of generation of the selection rules for a transition between Bloch states at any point of the Brillouin zone in crystals is equivalent to the problem of the decomposition of Kronecker products of two representations (reps) of a space group into irreducible components (the full group method). This problem can be solved also by the subgroup method where small reps of little groups are used. In this article, we propose a third method of generation of the selection rules, which is formulated in terms of projective reps of crystal point groups. It is based on a well-known relation between small irreducible reps (irreps) of little space groups and projective irreps of the corresponding little cogroups. The proposed procedure is illustrated by calculations of the Kronecker products for different irreps at the W point of the Brillouin zone for the nonsymmorphic space group O h 7 which is one of the most complicated space groups for the generation of selection rules. As an example, the general procedure suggested is applied to obtain the selection rules for direct and phonon-assisted electrical dipole transitions between certain states in crystals with the space group O h 7 .