Spectral functions permittivity

A brief list of basic assumptions used in the ACF method precedes the detailed analysis of the results of calculations. The derivation of the formula for the spectral function is given at the end of the section. The calculations demonstrate a substantial progress as compared with the hat-flat model but also reveal two drawbacks related to disagreement with experiment of (i) the form of the FIR absorption spectrum and (ii) the complex-permittivity spectrum in the submillimeter wavelength region. We try to overcome these drawbacks in the next two sections, to which Fig. 2c refers. 

We employ the following equations Eq. (142) for the complex susceptibility X, Eq. (141) for the complex permittivity , and Eq. (136) for the absorption coefficient a. In (142) we substitute the spectral functions (132) for the PL-RP approximation and (133) for the hybrid model, respectively. In Table IIIB and IIIC the following fitted parameters and estimated quantities are listed the proportion r of rotators, Eqs. (112) and (127) the mean number m of reflections of a dipole from the walls of the rectangular well during its lifetime x, Eqs. (118)... 

In Table V the fitted free and estimated statistical parameters are presented. For calculation of the spectral function we use rigorous formulas (130) and Eqs. (132) for the hybrid model. For calculation of the susceptibility %, complex permittivity , and absorption coefficient a we use the same formulas as those employed in Section IV.G.2 for water.29... 

The hat model is parameterized to obtain an agreement between the complex permittivity s and absorption a with experiment in the FIR range 400-1000 cm-1 and in the low-frequency region. To achieve correct low-frequency spectra, we set xor = 0 in the argument of the spectral function Lor and vary the lifetime Tor (and thus the collision frequency yor = f]or/Tor) in order to satisfy the equation ... 

The representation (6.69) of e s indicates that this quantity includes contributions from all the resonance frequencies (C0 =P ) of the frequency dependent permittivity (6.68). Moreover it also is determined by some dissipative properties of the material reflected in the spectral function (,(p)) and also in the optical permittivity =n p,), [6.6, 6.29]. 

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]. 

To estimate necessary k values. Equations (9.16a) and (9.16b) can be employed, but it is not possible to estimate the n values in a similar way. Refractive indices should be taken from appropriate literature, or should be determined by spectral simulations using permittivity functions [15]. 

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