Imaginary permittivity

Principles in Processing Materials. In most practical apphcations of microwave power, the material to be processed is adequately specified in terms of its dielectric permittivity and conductivity. The permittivity is generally taken as complex to reflect loss mechanisms of the dielectric polarization process the conductivity may be specified separately to designate free carriers. Eor simplicity, it is common to lump ah. loss or absorption processes under one constitutive parameter (20) which can be alternatively labeled a conductivity, <7, or an imaginary part of the complex dielectric constant, S, as expressed in the foUowing equations for complex permittivity ... 

From Eq. (3) derive the relations for the real and imaginary parts of the refractive index as Auctions of the permittivity and the electrical conductivity of a given medium. Note drat both n and k are defined as real quantities. 

Fig. 1.4 Dependence of the complex dielectric permittivity on frequency (s is the real part and e is the imaginary part, or the dielectric loss).

In contrast with Eq. (5), Eq. (11) gives the frequency behavior in relation to the microscopic properties of the studied medium (polarizability, dipole moment, temperature, frequency of the field, etc). Thus for a given change of relaxation time with temperature we can determine the change with frequency and temperature of the dielectric properties - the real and imaginary parts of the dielectric permittivity. 

Both the conductivity and the susceptibility contribute to the imaginary part of the permittivity Im e) = Im x) + Re a/to. A nonzero value for Im e) manifests itself physically by absorption of electromagnetic energy in the medium. We may associate Im(x) with the bound charge current density and Re charge current density. Absorption is determined by the sum of these two quantities, however, and it is not possible to determine by absorption measurements their relative contributions. This underscores our assertion that there is no clearly defined distinction between free and bound charges. 

Let us consider a sphere composed of a material described by the constitutive relation (5.46). We assume that the principal axes of the real and imaginary parts of the permittivity tensor coincide this condition is not necessarily satisfied except for crystals with at least orthorhombic symmetry (Born and Wolf, 1965, p. 708). If we take as coordinate axes the principal axes of the permittivity tensor, the constitutive relation (5.46) in the sphere is... 

More general ellipsoidal particles in an anisotropic medium, where there is no restriction on the principal axes of either the real or imaginary parts of the permittivity tensors, have been treated by Jones (1945). 

There are two sets of quantities that are often used to describe optical properties the real and imaginary parts of the complex refractive index N = n + ik and the real and imaginary parts of the complex dielectric function (or relative permittivity) e = c + ie". These two sets of quantities are not independent either may be thought of as describing the intrinsic optical properties of matter. The relations between the two are, from (2.47) and (2.48),... 

The dielectric measurements were carried out in a plate capacitor and frequency dependences of complex permittivity e = e — is (e and e" being its real and imaginary part, respectively) were determined [33] in the range /= 20 Hz- 200 kHz. 

Fig. 19. Dependence of real s and imaginary s" permittivity component on acetone concentration a for two polyacrylamide networks (O, ) - nonionized network, ( ) - ionized network numbers denote frequency of measurement. From Liptak et al. [33]...

In particular, VF2/F3E copolymers have also been the subject of extensive research [6,17,96]. As an example to illustrate the dielectric behavior of these copolymers, the temperature dependence of the real and the imaginary part of the complex permittivity at two different frequencies (1 and 100 kHz) are shown in Figs. 23a and 23b respectively. The measurements correspond to the 60/40 copolymer. The data have been collected by using a sandwich geometry with gold evaporated electrodes [95]. Frequencies of 103 and 106 Hz have been used by employing a 4192 A HP Impedance Analyzer. From inspection of Fig. 23b... 

Fig. 25a, b. Temperature dependence of a. the real part, e and of b. the imaginary part, e" of the complex permittivity for copolymers with different compositions... 

The analysis of the real and imaginary part of the complex dielectric permittivity allows one to distinguish between the two main relaxation processes (a and P). The a-process is correlated to the transition from the ferro to the paraelectric phase and the p-process is attributed to segmental motions in the amorphous phase. 

The first term, which contains the the static dielectric permittivities of the three media , 2, and 3, represents the Keesom plus the Debye energy. It plays an important role for forces in water since water molecules have a strong dipole moment. Usually, however, the second term dominates in Eq. (6.23). The dielectric permittivity is not a constant but it depends on the frequency of the electric field. The static dielectric permittivities are the values of this dielectric function at zero frequency. 1 iv), 2 iv), and 3(iv) are the dielectric permittivities at imaginary frequencies iv, and v = 2 KksT/h = 3.9 x 1013 Hz at 25°C. This corresponds to a wavelength of 760 nm, which is the optical regime of the spectrum. The energy is in the order of electronic states of the outer electrons. 

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.

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.

Figure 35. Frequency dependence in the submillimeter wavelength region of the real (a, b) and imaginary (c, d) parts of the complex permittivity. Solid lines Calculation for the composite HC-HO model. Dashed lines Experimental data [51]. Dashed-and-dotted lines show the contributions to the calculated quantities due to stretching vibrations of an effective non-rigid dipole. The vertical lines are pertinent to the estimated frequency v b of the second stochastic process. Parts (a) and (c) refer to ordinary water, and parts (b) and (d) refer to heavy water. Temperature 22.2°C.

Figure 50. Calculated contributions of the ionic permittivity for the imaginary (a, c) and real (b, d) parts of the total complex permittivity (solid lines) dashed lines refer to the calculation, neglecting the ionic dispersion, (a, b) For NaCl-water solution (c, d) for KCl-water solution. Cm = 0.5 mol/liter, Tion/r = 10.

Figure 4.1 Variation of physical properties vs temperature, used to determine the glass transition (a) volume (V) or enthalpy (H) (b) expansion coefficient (a) or heat capacity (cp) (c) storage modulus (E ) (d) dissipation modulus (E") and dumping factor (tan 8) (e) real part of the complex dielectric permittivity (s ) (f) imaginary part of the complex dielectric permittivity (e").

Cole-Cole plot in the complex plane of the loss factor (the imaginary part) e" against the real part s in Eq. (3.30), for a solvent with the high frequency relative permittivity... 

Figure 1. Debye-type relaxation spectra for liquid water, adsorbed water, ice, and hydrate. Solid lines correspond to real relative permittivity n and dotted lines represent imaginary permittivity k [3, 5, 11, 10].

Notation. relative permittivity, K -imaginary relative permittivity, A-surface conduction, subscript "elT" denotes an effective quantity, f frequency, vv vvater content... 

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