Absorption coefficient complex permittivity

We have introduced the effective complex susceptibility x ( ) = X,( )+ X ) stipulated by reorienting dipoles. This scalar quantity plays a fundamental role in subsequent description, since it connects the properties and parameters of our molecular models with the frequency dependences of the complex permittivity s (v) and the absorption coefficient ot (v) calculated for these models. 

In this section we calculate the complex permittivity (v) and the absorption coefficient ot(v) of ordinary (H20) water and of fluoromethane CH3F over a wide range of frequencies. We shall first write down the list of the formulas useful for further calculations. 

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)... 

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... 

Figure 28. Experimental frequency dependences of dielectric parameters recorded for liquid water (a) Real (curve 1) and imaginary (curve 2) parts of the complex permittivity at 27°C. The data are from Refs. 42 (solid lines) and 17 (circles), (b) Absorption coefficient. Solid line and crosses 1 refer to 1°C filled circles 2 refer to 27°C dashed line and squares 3 refer to 50°C. For lines the data from Ref. 17 were employed, for circles the data are from Ref. 42, for crosses and squares the data are from Ref. 53.

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. 

We shall combine the (A) and (B) mechanisms within a composite HC-HO model capable of describing the complex permittivity e (v) and absorption coefficient a(v) of liquid H20 and D20. The theory will be given in a simple analytical form. We shall see that such a modeling could give an agreement with... 

Using Eqs. (278), (281b), (282), and (301), we can calculate the total complex permittivity s (v) from the above theory. After that we can find the absorption coefficient a(v), its components aor(v) and avib(v) due to the reorientation and vibration processes, and the refraction index n ... 

The permittivity of a vacuum Eq has SI units of (C /J m). The specific conductivity (Tc (l/( 2-m)) couples the electric field to the electric current density by J= OcE. From the relations described in (6b), it becomes evident that optically generated gratings correspond to spatial modulations of n, , or Xg. The parameters AA, , and Xg are tensorial. This means that the value of Xg depends on the material orientation to the electric field (anisotropic interactions). In general, P and E can be related by higher-rank susceptibility tensors, which describe anisotropic mediums. The refractive index n, and absorption coefficient K, can be joined to specify the complex susceptibility when K (Xp) 471/Xp such that... 

The second part (Sections VI through X) is devoted to the derivations of the formulas used in calculations of the complex permittivity and absorption coefficient. Here the basics of the employed analytical approach are also briefly discussed. 

In Sections III-VI of this chapter we shall demonstrate that the main features of the complex permittivity e(v) and the absorption coefficient a(v), pertinent to water and ice, may be properly described if these four mechanisms are used. 

We calculate the complex permittivity s and absorption coefficient a using the general formulas given in Section II.E. A good agreement between the wideband theoretical and experimental water spectra will be demonstrated. 

Preliminary analysis indicates a layer thickness of 3.09 nm and a complex permittivity of (2.34 + 0.25 i), corresponding to a refractive index, n, of 1.53 and absorption coefficient, k, of 0.08. These values are in agreement with figures produced by ellipsometric analysis of multilayer films (7 to 29 layers) of the same material [9]. The film thickness also coincides with the molecular length of 3.0 nm, measured from a precision space-filled model, suggesting vertical alignment of the molecules within the layers. 

In these equations, eq and a are respectively the vacuum permittivity and the linear absorption coefficient. E a>) represents the apphed electric field at the frequeney co [ (complex conjugate of ( )], whilea>, —m) and/( >,< , —< ), which are fourth-order tensors, are respectively the third-order susceptibUity and hyperpolarisabUity. The real part of these tensors represents the induced refractive index change and is a function of the laser intensity (m) oc (imaginary part describes the 2PA process [40]. 

The absorptive losses are referred to as the dielectric loss or the loss factor, and can also be expressed as the complex coefficient of the relative permittivity, i.e. 

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