On Houston's Method of Evaluating Integrals Over the Brillouin Zone: Normalization Factor in a Simple Cubic Lattice

In evaluating integrals over the Brillouin zone by Houston's method, a normalization factor has to be used. A wave-vector-dependent normalization factor nj(q + Δq/2) is calculated by equating the volume common to a spherical shell of radius q and thickness Δq and the simple cubic Brillouin zone to the expression given by a j-direction Houston method. Some sample integrals are evaluated using this (GS) normalization procedure and Horton and Schiff's (HS) normalization procedure, originally developed for face-centered cubic lattices. The superiority of the GS method, especially with highly anisotropic functions, is demonstrated. From an evaluation of the moments of the frequency distribution function in a simple cubic lattice for various values of the anisotropy parameter, it is concluded that, when high accuracy is desired, the GS procedure is applicable over a wider range of anisotropy parameter than the HS procedure. The θ–T curve calculated by using the correct normalization procedure in a simple cubic lattice is in good agreement with that calculated by Blackman's method.