Compulsory degeneracy

Application of group theory can also explain the important problem of degeneracy of excitonic states. This degeneracy can be conditioned by invariance of the crystal Hamiltonian upon elements of its space group (in this case it is sometimes called compulsory degeneracy, see (21), 40) or, as shown by Herring (23), it can result from the crystal invariance upon the time-reversal operation. 

Let us first consider the case of compulsory degeneracy. To this end we assume that an exciton state (u(k) is degenerated for k = ko, which means there are p excitonic states associated with the wavevector ko, t = 0,1,..., p,... 

As an example we consider the case of the naphthalene crystal, G ft being its space group. The point group C-2h (see Table 2.1) has only one-dimensional representations. Since any subgroup of the group C h can also have only onedimensional representations, it is clear that in crystals of naphthalene type the compulsory degeneracy for excitonic states inside the first Brillouin zone is not possible. 

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