Note on the Theory of the Frequency Spectrum of Crystalline Solid

Abstract

ReSDme. In order to examine the accurancy of Houston's method of approximation for calculating the frequency spectrum of crystalline solid, the approximate frequency spectrum of square lattice obtained from this method is compared with the exact one obtained from Montroll's· method. In the neighborhood of the origin the curve obtained from Houston's approximation agrees rather exactly with the accurate curve, while its slop!! becomes too steep as the frequency approaches the first peak. The portion of this approximate curve to the right of the first peak has no physical significance. The frequency distribution function consists of two or more analytically different parts. If we try to apply Houston's method to such parts separately, the frequency distribution function without spurious peaks will be obtained. And it is of importance to investigate the topological properties of the curve of equal frequency in llrillouin zone for the purpose of carrying out the calculation according to this method. §l. In recent years, there appeared three illuminating theories of the frequency spectrum of crystalline solid based on the Born-von Karman model, two by MontroU and one by Houston. First, MontrolPl reduced the calculation of the frequency distribution function to the so called moment problem in statistics, where moments are given by the trace of the products of the matrix in the eigen­ wert problem as H. Thirring2) already found. This method of approximation for the frequency distribution function, hereinafter referred to as Thirring-Montroll's method or tracemoment method, gives the mean approximation in the sense of the least square method, but not. the local approximation. And calculations of moments of higher order necessary to obtain the more accurate distribution funntion are tedious. Secondly, Montrol!') gave the exact solution for the square lattice, according to which the distribution function has two points of loganthmic infinity. But, it seems to be difficult to extend Montroll's exact solution to three dimensional lattices, although it is desirable. Hereinafter MontroU's paper concerning his exact solution is referred to as M2. Finally, Houston4) developed one approximate method, where unknown solutions of the secular equation are interpolated from known'ones of the particular directions in Brillouin zone along which we easily obtain solution of the secular equation. This method of appro­ ximation, hereinafter referred to as Houston's method, is easy in calculation, but has unfortunately spurious peaks. In spite of its defects, it seems to have some convenience because of its simplicity. Therefore, we consider it is worth while to inquire the nature of Houston's method of approximation.