Houston's Method and Its Application to the Calculation of Characteristic Temperatures of Cubic Crystals

Abstract

Houston's formula first derived for an approximate determination of the frequency spectrum of cubic crystals, essentially approximates the integral over the surface of a unit sphere of any function $I(\ensuremath{\theta}, \ensuremath{\phi})$ which is invariant under the operations of the cubic symmetry group in terms of the values of $I(\ensuremath{\theta}, \ensuremath{\phi})$ along the three directions (100), (110), and (111). In this paper approximate formulas are given for the integral in terms of the values of $I(\ensuremath{\theta}, \ensuremath{\phi})$ along the above three and any or all of the (210), (211), and (221) directions. As an application, Debye temperatures $\ensuremath{\Theta}$ are calculated for nine cubic crystals. It appears that the $\ensuremath{\Theta}$ values calculated from the formula containing the values of $I$ along all the above six directions may be expected to be correct to about 1% for crystals for which $0.25<\ensuremath{\eta}<4.0$ [where $\ensuremath{\eta}\ensuremath{\equiv}\frac{2{c}_{44}}{({c}_{11}\ensuremath{-}{c}_{12})}$]. For the alkali metals ($\ensuremath{\eta}\ensuremath{\sim}8$), the error can be as large as 10%.Houston's method involves the expansion of $I(\ensuremath{\theta}, \ensuremath{\phi})$ in terms of those Kubic harmonics which are invariant under all the operations of the cubic symmetry group. In the Appendix, formulas are derived from which the Kubic harmonics of this type of any degree may be written down readily.