Vibration Spectrum of a Simple Cubic Lattice

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.