RI 5995 The Frequency Spectrum Of A Square Lattice ? Introduction

In this paper, closed expressions have been obtained for the frequency spectrum of a Born-Karman square lattice in terms of elliptic and hyperelliptic integrals. This work is part of continuing research by the Bureau of Mines with the thermodynamic properties of metals and minerals. Although other workers have obtained various types of approximations for the general case of the frequency spectrum of a square lattice, no one previously has obtained closed analytic expressions for this spectrum. To understand the optical and thermodynamic properties of crystals, several investigators have studied the distribution of frequencies of normal modes of vibration of the crystal lattice. Born and Karman3 laid the foundation for the theory of lattice dynamics and obtained detailed results for the case of a one-dimensional lattice. Blackman4 used numerical techniques to estimate the frequency distribution of a two-dimensional square lattice. Mbntroll5 analytically determined the frequency distribution of a square lattice for a particular value of the force constants. He also studied the more general problem and obtained some analytic approximations, using the method of moments. However, the calculations by this method were so difficult that it was not always practicable to carry them out. Van Hove6 and Montroll7 studied the nature of the singularities in the frequency spectrum of both a two and