Frequency Spectrum and Momentum Autocorrelation Function for a Simple Lattice

Abstract

Calculations are made on the dynamics of a familiar lattice model: the simple cubic lattice with central and noncentral harmonic forces between nearest neighbors only. For the case of equal force constants, natural extensions are made of existing analytical approximations for the spectrum of squared frequency. The behavior of the spectrum near its singular points is described more accurately than before and expressions are derived which make it easy to obtain the density of modes at any frequency to about one part in a thousand. The new description of the spectrum is used to improve existing approximations for the classical momentum autocorrelation function for the infinite lattice ρ(τ), and for the function X₃(τ′) used by Goodman in calculations on the response of surface atoms in a semi‐infinite lattice. Good agreement with numerical results of Goodman for τ′ = 20, 25, and 30 is obtained. The results for the spectrum also apply to the density of states of electrons in a simple cubic lattice in the tight‐binding approximation.