Complex permittivity approximation

Employing the additivity approximation, we find dielectric response of a reorienting single dipole (of a water molecule) in an intermolecular potential well. The corresponding complex permittivity jip is found in terms of the hybrid model described in Section IV. The ionic complex permittivity A on is calculated for the above-mentioned types of one-dimensional and spatial motions of the charged particles. The effect of ions is found for low concentrated NaCl and KC1 aqueous solutions in terms of the resulting complex permittivity e p + Ae on. The calculations are made for long (Tjon x) and rather short (xion = x) ionic lifetimes. 

In this section we have to calculate the complex permittivity s (v) and the absorption coefficient a(v) of ordinary (H2O) water over a wide range of frequencies. It is rather difficult to apply rigorous formulas because the fluctuations of the calculated characteristics occur at a small reduced collision frequency y typical for water (in this work we employ for calculations the standard MathCAD program). Such fluctuations are seen in Fig. 13b (solid curve). Therefore the calculations will be undertaken for two simplified variants of the hat model. Namely, we shall employ the planar libration-regular precession (PL-RP) approximation and the hybrid model.26... 

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)... 

Double Debye Approximation for Complex Permittivity of Heavy Water... 

Figure 30. Imaginary (a) and real (b) parts of the complex permittivity of liquid water H20 at 22.2°C. Ordinary water is represented by solid lines, heavy water is represented by dashed lines. To the left from vertical lines (for v < 20 cm 1). calculation is performed using approximation [17] modified as described in Appendix 3.2 in the rest region, it is performed using the data 51 given in Table XII.

Now we apply the additivity approximation corresponding to Eq. (387). Namely, we sum up the contributions of three sources of dielectric loss due to (a) reorienting dipoles, (b) oscillating anions, and (c) oscillating cations. Combining Eqs. (387), (388), and (399a), we write the formula for the total complex permittivity ... 

Figure 2 Imaginary part of the complex permittivity for PETN, using the Hartree-Fock approximation, as a function of energy.

As far as measurements are concerned the perturbation method of Buravov and Shchegolev provides reliable results not only in the quasi-static region but practically at least one order of magnitude above the kb =>f limit. Moreover, for the spheroid of arbitrary complex permittivity we have found the universal perturbation relation valid for arbitrary wavelength inside the sample. This generalized approach not only covers various approximations used until now, but also strictly determines the limits of their applicability. ... 

We use the relationships (4)—(7) to calculate the water spectra. In such a calculation for ice the static permittivity, s(ice) is not involved. For molecules reorienting in the hat well the high-frequency approximation is employed. The complex permittivity of the LIB state is represented, instead of Eq. (4), as... 

We do not consider the low-frequency spectra for ice, since the contribution to complex permittivity of rigid reorienting dipoles is calculated from the simplified expression (A29), which is applicable only in the high-frequency approximation. Indeed, the ice permittivity is found for v > 0.1 cm-1 (see Figs. 20a,b and 24a), while for liquid water Eq. (4) is used, applicable also in the relaxation region. 

Michaels et al. [2, 3] found that the stoichiometric NaPSSA BTAC complex, when completely free of extraneous electrolytes, exhibits a high dc resistivity (approximately 10 ° Q. cm). The value of e measured at 100 Hz changes from 50 to 5 (for water-saturated PEC) and from about 5 to 3 (for dried PEC) at 0.1 MHz. When doped with simple electrolytes like NaBr, the absolute values of the complex permittivity as well as the dependence of s and e" on frequency change significantly. Eigure 2 shows the influence of the dopant salt [2]. 

DRS provides a particularly usdul means of monitoring the nature and extent of macroscopic alignment in SCLC samples that have been subjected to E fields, B fields, surface forces or are aligning/disaligning after electrical and/or thermal treatments. As we have shown [5Q, the complex permittivity of a uniaxial sample of intermediate alignment is given, to a good approximation, by the linear-addition relationship... 

Determination of all the tensor elements in an arbitrary coordinate system is a difficult task. Therefore, the optic axis is usually assumed to be perpendicular to the surface of the thin film. This makes the complex permittivity tensor 2 (suffix 2 refers to the thin film) a diagonal matrix in a imiaxial approximation (e — Then, 62 is given as... 

Since the reactants are molecules, the extent of solvation will be small and approximately the same for all solvents. But the extent of charge separation in the activated complex is very dependent on the solvent, by virtue of the magnitude of the relative permittivity, and a large variation in the extent of solvation of the activated complex is expected. The entropy of activation and the A factor are expected to vary with change of solvent. 

Wagner (1914) gave an approximate treatment of the important practical case where a very highly insulating dielectric suffers from inclusions of conductive impurities. Taking the model where the impurity (relative permittivity e2, conductivity a2) exists as a sparse distribution of small spheres (volume fraction f) in the dielectric matrix (relative permittivity e, negligible conductivity), he derived equations for the components of the complex relative permittivity of the composite ... 

Unfortunately, the Debye model provides only an approximate description of aprotic solvents. It has been applied extensively to determine their relaxation properties quite successfully, mainly because permittivity data are available over a limited frequency range. As a result, the high-frequency parameter is usually obtained by a long extrapolation. As experimental methods have become available at frequencies above 50 GHz, it has been found that the behavior of aprotic solvents is more complex [9]. 

Other modifications of the original Marcus model have been suggested [27]. Many reactants are not spherical in shape and are better approximated as ellipsoids. In this case a much more complex expression for the effective distance R is obtained which depends on the length of the two axes which describe the shape of the ellipsoid. Another improvement in the model is to describe each reactant as a dielectric cavity with fixed charges located within it. In this case, the calculation of Gxo requires a description of the charge distribution within the reactants and an estimate of the local permittivity in the dielectric cavity. 

We consider collective motion of pairs of water molecules. Let the unit volume of the medium comprise Avlb/2 of such pairs, with Ny being the concentration of molecules suffering elastic vibration. Using the high-frequency approximation, we calculate the complex susceptibility /vib — Xvib + Yvib °f the medium pertinent to harmonic vibration of the HB particles (we omit the complex-conjugation symbol). We assume that for an instant just after a strong collision, the velocities and position coordinates of the particles have Boltzmann distributions. Then the elastic-vibration complex susceptibility /vib and permittivity Asvib in view of TGN are determined by the formulas... 

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