Symmetry properties of Coulomb excitons

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. 

As a second example consider the case of a quartz crystal, with the space group Ds. One of the representations of the point group 3 has dimension two. Thus, if the vector ko is parallel to the three-fold symmetry axis when the point group Gk0 coincides with the point group 3, a double degeneracy of excitonic terms is, in general, possible. 

There is another reason for degeneracy of excitonic states, being a consequence of the structure of the Schrodinger equation. Indeed, since the Hamiltonian is a self-conjugated operator, wavefunctions l ko (( = 1,2. p), where the star means complex conjugate, as well as wavefunctions kotJ ( = 1,2. p), 

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