First the vibrational frequency distribution is derived from Appendix VI of Terrell L. Hill's An Introduction to Statistical Mechanics. Consider a three-dimensional isotropic elastic solid with N atoms in the shape of a rectangular parallelepiped with side-lengths L x , L y , L z . The elastic wave will obey the wave equation and will be plane waves ; consider the wave vector k = ( k x , k y , k z ) and define l x = k x | k | , l y = k y | k | , l z = k z | k | , such that