Dielectric function for a material containing hyp er spherical inclusions to O(c²) I. Multipole expansions

By averaging over pairs of hyperspheres, we have obtained the dielectric function for a binary mixture containing hyperspherical inclusions up to order c2, where c is the volume fraction of inclusions. The method used is based on multipole expansions for the potential of two spheres in a uniform field and is a generalization of the method of Jeffrey to d-dimensional space. Numerical results are presented for the second-order coefficient K in the low-c expansion of the dielectric constant for arbitrary d; these verify earlier known results, as well as showing the dependence of K on dimensionality, which is particularly simple as d -»1 and as d -> oo.