The dielectric constant of a simple cubic array of identical spheres

New methods are described for calculating the dielectric constant epsilon e or the conductivity sigma e of a two-phase composite and these are applied to a simple cubic array of identical spherical inclusions embedded in a homogeneous host. A spectral representation is derived for epsilon e, and numerical results are presented for the poles and the residues. Analytical and numerical methods are used to discuss the conductivity threshold of the cubic array, which occurs when the host is an insulator and the conducting spheres begin to touch each other. It is argued that sigma e as a function of sigma 1/ sigma 2 has an essential singularity at the conductivity threshold.