Permeability and permittivity spectra of granular materials

We present an extended analytic technique for calculating the permeability and permittivity spectra of granular materials. It addresses the relationship between grain properties, their size relative to the wavelength, and the permeability and permittivity spectra of polycrystalline material. The scattered multipolar fields about a single sphere are related to the polarizability of an ordered congregation of such spheres. The product of the external wave vector k and sphere radius a is small and the product of the internal wave vector ${\mathit{k}}_{\mathit{i}}$ and a unrestricted. The Clausius-Mossotti relation is used in combination with the scattering results to yield an equation that permits us to calculate the effective permeability and permittivity spectra of a cubic array. The result is a variety of possible spectral types, including complex permittivity and permeability spectra that have been measured and explained either by including multiple atomic-level sources or by statistical weighting over an ensemble of grains. This theory predicts both ‘‘classical’’ and ‘‘anomalous’’ spectra from a single source, as well as certain conglomerate permeability spectra that have been measured and for which no direct explanation is available. Our results show that an understanding of the permeability and permittivity spectra of composite materials cannot be complete without inclusion of a dependence on the ratio of the grain size to the wavelength.