Frequency Spectrum of Crystalline Solids

It is generally accepted that a normal crystalline solid can be pictured at absolute zero as an assembly of molecules arranged at periodically placed lattice points. Since at higher temperatures each molecule becomes a harmonic oscillator about its lattice point, in order to calculate thermodynamic properties of the crystal it is necessary to know the distribution of its internal normal modes of vibration. On the basis of the Born-von Kármán model these normal modes of vibration are roots of a secular determinant. In this paper it is shown that the 2nth moment of the distribution function of normal modes is proportional to the trace of the nth power of the matrix of the Born-von Kármán determinant. By expressing the distribution function as a linear combination of Legendre polynomials it is shown that the coefficient of each polynomial is a linear combination of the moments. The frequency distribution function of a two-dimensional simple cubic lattice is calculated by the above method and turns out to have two maxima. Usually the equation for a thermodynamic function F(T) involves the integral of the product of the frequency distribution function g(v) and a known function K(T, v). We show here that when F(T) is a known function of T an integral equation results with g(v) under the integral sign. This integral equation can be solved for g(v) by use of Fourier transforms.