A product representation for cubic harmonics and special directions for the determination of the Fermi surface and related properties

A product representation of the fully symmetric cubic harmonics has been developed which greatly simplifies the analysis of problems involving these functions. An important example is the determination of special directions which optimize the description of various physical quantities with cubic symmetry. For example, one can obtain for a metal the Fermi energy, density of states, and details of the Fermi surface by calculation or measurement along a few special directions in the irreducible segment of the Brillouin zone. These special directions are also the basis for high-precision formulae for integration over the unit sphere of functions with full cubic symmetry. We obtain such integration formulae with a generalization to two dimensions of the one-dimensional Gauss technique. These new formulae represent a major advance in precision for with them one can determine accurately nearly as many coefficients in a cubic harmonic expansion of a physical variable as there are sample values of the variable. The integration formulae presented here should be adequate for most situations of current physical interest; if not, the method described in this paper can be used to obtain formulae with even higher precision.