Derivation of Lichtenecker's Logarithmic Mixture Formula From Maxwell's Equations

Lichtenecker's logarithmic mixture formula for dielectrics has proven to be a useful practical formulation for determining the effective permittivity of homogenized dielectric mixtures. However, the formula itself has long been regarded as semiempirical in nature without firm theoretical justification. This paper shows that Lichtenecker's formula can be derived by applying Maxwell's equations and the principle of charge conservation to a mixture for which the spatial distribution of shapes and orientations of the components is randomly distributed. Furthermore, the relationship between the Lichtenecker formula and the well-known mixture formulas due to Maxwell-Garnett is demonstrated. Similarly, it is shown that the symmetric mixture formula of Bruggeman can also be obtained from the Lichtenecker formula.