Vibration Spectrum of a Simple Cubic Lattice

Abstract

Analytic expressions are derived for the frequency distribution function g(f) of a simple cubic monoatomic lattice. Only nearest and next‐nearest neighbor interactions are considered, and the latter are assumed to be weak compared with the former. The procedure is based upon considering the next‐nearest neighbor interactions as a perturbation, and the results are correct to the first power of a parameter τ which is essentially a measure of strength of the next‐nearest neighbor forces as compared with the nearest neighbor forces. It has been known for some time that a simple cubic lattice with only nearest neighbor interactions degenerates into the equi alent of three independent one‐dimensional lattices giving a nonzero g(f) for f=0 and an infinite value at the maximum frequency. If, however, one includes even a small interaction between next‐nearest neighbors, the behavior of g(f) near both ends of the spectrum changes considerably; in fact it vanishes at both ends. For intermediate frequencies, g(f) is continuous but has four analytic singularities with vertical tangents of the type predicted by van Hove for a quite general type of crystal. Although the calculated g(f) is exact only in the limit τ→0, it does properly describe the exact location and type of singularities for 0≤τ≤1/10.