Critical Points and Lattice Vibration Spectra

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Abstract
A method is developed for obtaining directly from group-theoretical and topological considerations a consistent set of critical points belonging to a given secular equation. The treatment includes a detailed discussion of nonanalytic behavior near degeneracies. Comparison with many specific calculations shows that the minimal set so obtained is often the set actually present in the case of short-range forces. The contributions of nonanalytic critical points to the frequency distribution are analyzed. Behavior near critical points is made the basis of a scheme for calculating the main features of frequency distributions from the corresponding minimal sets. An approximate frequency distribution is calculated for aluminum. Possible applications to energy bands are discussed.

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