Homogenization of spherical inclusions

In English The homogenization of cubically arranged, homogeneous spherical inclusions
in a background material is addressed. This is accomplished by the solution of
a local problem in the unit cell. An exact series representation of the effective
relative permittivity of the heterogeneous material is derived, and the functional
behavior for small radii of the spheres is given. The solution is utilizing
the translation properties of the solutions to the Laplace equation in spherical
coordinates. A comparison with the classical mixture formulas, e.g., the
Maxwell Garnett formula, the Bruggeman formula, and the Rayleigh formula,
shows that all classical mixture formulas are correct to the first (dipole) order,
and, moreover, that the Maxwell Garnett formula predicts several higher order
terms correctly. The solution is in agreement with the Hashin-Shtrikman
limits.