Full-Group and Subgroup Methods in Crystal Physics

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Abstract
Two group-theoretical methods are developed and contrasted. These enable reduction of space-group representations to be effected. Such reduction is essential in order to obtain selection rules for a variety of processes in crystal physics. In the full-group method an irreducible representation of the entire space group is induced from that of a particular subgroup; the reduction of the resulting representation is then treated by methods recently developed. In the subgroup method the analysis is carried out in terms only of a variety of subgroups of the full space group. The subgroup procedure is economical in that relatively few group elements are used. However, owing to the need to obtain a "complete" character system for the relevant subgroups, it is more cumbersome and more liable to errors. The precise relation between these methods is developed, free from ambiguities. As an illustration, selection rules for intervalley scattering in zinc blende are obtained by both methods. The analysis is carried out for the vector representations of the single space groups in order to clarify the contrast between the two methods; extension to include spin and time-reversal effects is straightforward. To date, full-group methods have been successfully applied to more complex cases (e.g. reduction of symmetrized cubes) than subgroup methods.