The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal

TLDR

For crystals with harmonic interatomic forces and periodic structure, the elastic vibration frequency distribution exhibits analytic singularities, a consequence of Morse’s theorem on saddle points on a torus. The singularities depend only on crystal dimensionality: in two dimensions they are logarithmically infinite peaks, while in three dimensions the distribution is continuous but its first derivative has infinite discontinuities.

Abstract

It is shown that for a crystal, under the assumption of harmonicity for the interatomic forces and as a consequence of the periodic structure, the frequency distribution function of elastic vibrations has analytic singularities. In the general case, the nature of the singularities depends only on the number of dimensions of the crystal. For a two-dimensional crystal, the distribution function has logarithmically infinite peaks. In the three-dimensional case, the distribution function itself is continuous whereas its first derivative exhibits infinite discontinuities. These results are elementary consequences of a theorem of Morse on the existence of saddle points for functions defined on a torus.