Theoretical evidence for 'Lichtenecker's mixture formulae' based on the effective medium theory

In order to calculate the electrical permittivity property of a multiphase mixture, a beta function distribution of the geometrical shapes of inclusions is considered. By using the effective medium theory together with the assumption of self-consistency, it is shown that the beta function distribution leads to Lichtenecker's formulae. This gives a theoretical model for this mixing formula, which had hitherto been considered empirical. This result is important because the first Lichtenecker's is already used for time domain reflectometry (TDR) calibration purposes, namely conversion from permittivity measurement to water content of heterogeneous materials.