Mean field theory and scaling laws for the optical properties of inhomogeneous media

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Abstract
The Bruggeman theory for the optical effective dielectric function of inhomogeneous media is a mean field theory. However, the characteristic behaviour of such theories, namely the power law relation between the effective dielectric quantities and the filling factor q, of the form (q — q c)-δ, where δ is a critical exponent depending of the physical quantity under consideration (s = 1 for polarization and t = 1 for conductivity), was never analitically demonstrated. In fact, these values are wrong, as confirmed by Monte-Carlo calculations predicting the values s = 0.73 and t = 1.94 and by our experimental results on Au-MgO, Pt-Al2O3, Fe-Al2O3, as well as by many other measurements on Co-Al 2O3 and Ag-KCl, for example. We confirm that these values are frequency independent far from the dielectric anomaly and we present a new formulation of the Bruggeman theory in which the different contributions to the dielectric function appear explicitly with their own exponents. A new formulation is then proposed by replacing the exponents by their proper values. One shows, on comparing to experimental results, that, in contrast to the BR theory, our approach is valid close to the percolation critical fraction. The agreement between theory and experiments is rather satisfactory, except for some discrepancies on the imaginary part of the dielectric function. This problem can be solved using experimental arguments but the validity of any approach based on an effective medium theory has to be reconsidered near the percolation. We also propose an original but simple way to determine the experimental percolation threshold which is the most important parameter in this approach