Note on the Theory of the Frequency Spectrum of Crystalline Solid

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 slope 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 Brillouin zone for the purpose of carrying out the calculation according to this method.