Maxwell-Garnett formula

The main issue in the effective-medium approximations is to relate the polarizability a. to the relative permittivity r, where = i/ s, and E[ is the relative permittivity of the inclusion. For spherical particles and at frequencies for which the inclusions can be considered very small, the Maxwell-Garnett formula uses the relation... 

Parenthetically we note that the discrete dipole approximation can be used to derive the size-dependent Maxwell-Garnett formula. For a collection of identical spheres of volume V and relative refractive index the polarizability dyad is... 

The size-dependent Maxwell-Garnett formula can now be obtained by combining (3.11) and (2.168) [129]. 

A. Lakhtakia, Size-dependent Maxwell-Garnett formula from an integral equation formalism. Optik 91, 134 (1992)... 

In the statistic route, an effechve dielectric function e(v) is calculated from the dielectric funchon of the metal Enie(v) and the of polymer material po(v) by using a formula, the effective medium theory. The most general effective medium theory is the Bergman theory in which the nanostructure of the composite material can be considered by a spectral density function. The Bergman theory includes the soluhons from the Bruggeman theory and the Maxwell Garnett theory for spherical, parallel-oriented, and random-oriented ellipsoidal parhcles. 

For the following calculation, experimentally determined dielectric functions for silver [30] and for a plasma polymer [31] were taken. The effective dielectric functions e(v) were calculated with the Maxwell Garnett theory for parallel-oriented particles, equation (13). From the effective dielectric function, transmission or extinction spectra can be calculated by using the Fresnel formulas [10] for the optical system air-composite media-quartz substrate. As a further parameter, the thickness of the film with embedded particles and the thickness of other present layers that do not contain metal nanoparticles have to be included. The calculated extinction spectra can be compared with the experimental spectra. 

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