Dielectric function average

We take as our model of an inhomogeneous medium a two-component mixture composed of inclusions embedded in an otherwise homogeneous matrix, where e and are their respective dielectric functions. The inclusions are identical in composition but may be different in volume, shape, and orientation we shall restrict ourselves, however, to ellipsoidal inclusions. The average electric field (E) over a volume V surrounding the point x is defined as... 

It is not difficult to extend (8.49) or (8.47) to multicomponent mixtures. If we make the same assumptions for each inclusion that were made preceding (8.48), then the average dielectric function is... 

The average dielectric function (8.50) was first derived by Maxwell Garnett (1904) subsequently, it has been rederived under various sets of assumptions (see, for example, Genzel and Martin, 1972, 1973 Barker, 1973 Bohren and Wickramasinghe, 1977). 

The following expression for an average dielectric function was first obtained by Bruggeman (1935) ... 

An approach somewhat similar to that in Section 8.5 was taken by O Neill and Ignatiev (1978), who obtained an expression for the average dielectric function of a mixture containing spheroidal inclusions with ratios of semi-minor to semimajor axes given by a probability distribution function. 

An elementary treatment of the free-electron motion (see, e.g., Kittel, 1962, pp. 107-109) shows that the damping constant is related to the average time t between collisions by y = 1 /t. Collision times may be determined by impurities and imperfections at low temperatures but at ordinary temperatures are usually dominated by interaction of the electrons with lattice vibrations electron-phonon scattering. For most metals at room temperature y is much less than oip. Plasma frequencies of metals are in the visible and ultraviolet hu>p ranges from about 3 to 20 eV. Therefore, a good approximation to the Drude dielectric functions at visible and ultraviolet frequencies is... 

It follows from (8.47) that the average dielectric function of a suspension of identical oblate spheroids is... 

Some applications of the method will be shown for the ternary salt trimethylammonium (TMA)-TCNQ-I. Tanner et al. [43] have obtained a best fit with a Drude-Lorentz dielectric function with a>p = 5290 cm-1, -y = 2560 cm"1, ec = 2.65, , is the frequency and T is the relaxation rate of the second oscillator. The average value of the frequency-dependent conductivity below 200 cm"1 should be 19 IS- cm"1. The energy gap of 0.10 to 0.14 eV, the effective quantity of electrons participating in the electric transport Ncfi = 0.67 per molecule, and the effective mass of the carriers m = 5.2 m were found from the plasma frequency. 

Consider a medium with a low volume fraction of nanoparticles and complex dielectric function c( co), each with radius R, such that 2iiRnjA< 1. An electric field in the medium polarizes the solvent at the frequency of the field. The key is to determine the average dielectric function of the combined system. The average electric field in the medium is... 

An ansatz for treating the resultant optical properties is to treat the shell using MG theory to yield an approximate average dielectric function for the shell, and then to use this function in the core-shell geometry. The predictions of this model are shown in Fig. 8. A fixed latex or silica core of 50 nm is shown surrounded by a 5 nm shell of Au Si02 where the Au volume fraction is varied. The medium is taken to be water for simplicity. 

One can see that the effective dielectric function of the effective medium differs from the simple average of the dielectric functions of the constituents. Moreover, Eq. (1.122) is inherently asymmetric in the treatment of the two constituents the transformation of e -o- egm and / 1 - / will result in different effective dielectric functions. 

Assuming that the individual grains, representing both the particles and the surroundings, exist in some effective medium, in 1935 Bmggeman [189] derived the expression for the average dielectric function, which can be generalized to ellipsoids as [190]... 

This possibility is extremely useful for the structural characterization of polymer blends. The optical properties of the polymer layer can be described by a so-caUed effective dielectric function, which is a suitable average of the dielectric functions of the two components. Three averaging effective medium approximation (EMA)s -the linear, Maxwell-Garnett and Bruggeman EMAs - are widely used for this purpose [9]. These approximations differ in their spectral densities for a given volume fraction. 

---