Inhomogeneous particles, dielectric functions

The notion of homogeneity is not absolute all substances are inhomogeneous upon sufficiently close inspection. Thus, the description of the interaction of an electromagnetic wave with any medium by means of a spatially uniform dielectric function is ultimately statistical, and its validity requires that the constituents—whatever their nature—be small compared with the wavelength. It is for this reason that the optical properties of media usually considered to be homogeneous—pure liquids, for example—are adequately described to first approximation by a dielectric function. There is no sharp distinction between such molecular media and those composed of small particles each of which contains sufficiently many molecules that they can be individually assigned a bulk dielectric function we may consider the particles to be giant molecules with polarizabilities determined by their composition and shape. 

Effective medium theories characterize the frequency-dependent transport in systems with large-scale inhomogeneities such as metal particles dispersed in an insulating matrix [118,119]. An IMT in the effective medium model represents a percolation problem where a finite a c as T 0 is not achieved until metallic grains in contact span the sample. To understand the frequency dependence of the macroscopic material, an effective medium is built up from a composite of volume fraction /of metallic grains and volume fraction 1 — / of insulator grains. The effective dielectric function semaCw) and conductivity function (Tema(w) are solved self-consistently. 

According to Ekj. (7), it is the dielectric dynamics of the homogeneous solvent, as expressed in C (fc, ), that is the source of the time dependence of the estimate Z t) of the solvation tcf. In the RDT approximation the effect of the solute-solvent interactions is carried by the static coupling function B (fc). This factorization (to a function of the homogeneous solvent dynamics times a function of the static solute-solvent structure) is a characteristic feature of the RDT theory. The renonnalized character of the coupling function allows us to bypass the two-time many-point correlation functions that would necessarily appear in a dynamical theory that explicitly addressed the inhomogeneity of the solvent in the neighborhood of the solute particle. 

The above consideration of nanoparticles has been carried out in a supposition that they have more or less the same size. To be more precise, we assumed that the width of the nanoparticles sizes distribution function is smaller then its mean value. The mean value R is usually extracted from, e.g., X-Ray diffraction measurements [91] and it is supposed, that the size of all the particles corresponds to R. In this part we will show, that the neglection of sizes distribution can lead to incorrect results, when measurements are performed on the samples with essential scattering of sizes. Besides that, actually the size distribution defines the spectral lines inhomogeneous broadening. Moreover, it essentially influences the observed anomalies of many physical properties (like specific heat and dielectric or magnetic permittivity) of nanomaterials. Note that in real nanomaterials, like nanoparticles powders and/or nanogranular ceramics there is unavoidable size distribution which in general case should be taken into account. However, we will show below, that in perfect samples, where the width of size distribution is small, it is possible to suppose safely that all particles have the same size. In this part we primarily follow the approaches from the paper [92]. 

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