Complex permittivity response

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. 

We employ the linear response theory based on a phenomenological molecular model of water. In the proposed composite HC-HO model the complex permittivity is represented as the sum... 

The theory of wideband complex permittivity of water described in the review drastically differs from the empirical double Debye representation [17, 54] of the complex permittivity given for water by formula (280b). Evolution of the employed potential profiles, in which a dipole moves, explored by a dynamic linear-response method can be illustrated as follows ... 

As a result of these very general considerations, one expects the dielectric response function, as expressed by the complex permittivity, k (oj), or the attenuation function, a(oi), of ordinary molecular fluids to be characterized, from zero frequency to the extreme far-infrared region, by a relaxation spectrum. To first order, k (co) may be represented by a sum of terms for individual relaxation processes k, each given by a term of the form ... 

Finally, in this section, the possible role of counterion displacements in relaxation of globular proteins should be mentioned. These can result in fluctuating dipole moments in addition to permanent moments p. along principal molecular axes. If their relaxations are independent and separately exponential with rate constants k and k, response theory formulation gives the complex permittivity in the form... 

Dielectric relaxation — Dielectric materials have the ability to store energy when an external electric field is applied (see -> dielectric constant, dielectric - permittivity). Dielectric relaxation is the delayed response of a dielectric medium to an external field, e.g., AC sinusoidal voltage, usually at high frequencies. The resulting current is made up of a charging current and a loss current. The relaxation can be described as a frequency-dependent permittivity. The real part of the complex permittivity (e1) is a measure of how much energy from an external electric field is stored in a material, the imaginary part (e") is called the loss factor. The latter is the measure of how dissipative a material is to an exter-... 

Materials that exhibit a single relaxation time constant can be modeled by the Debye relation which appears as a characteristic response in the permittivity as a function of frequency. The complex permittivity diagram is called Cole-Cole diagram constructed by plotting e" vs. e with frequency as independent parameter. 

The complex permittivity is obtained as follows For nondisperse materials (frequency-independent permittivity), the reflected signal follows the exponential response of the RC line-cell arrangement for disperse materials, the signal follows a convolution of the line-cell response with the frequency response of the sample. The actual sample response is found by writing the total voltage across the sample as follows ... 

The response of a dipole, surrounded by neighbours, to an applied fidd can in fact be related in detail to the frequency variation of the complex permittivity. A theoretical treatment of the effect on the (fipole of the lag in the reaction field has been given by Scaife and by Fatuzzo and Mason. Equation (14) has been derived by Klug et al. and also by Nee and Zwanzig, in which y(t), defined in equation (IS), is the normalized correlation function of the permanent dipole moment /r. The function... 

Meanwhile, experimentalists have measured and analyzed, in particular, the dielectric response of liquid water in the frequency range from 10 up to 1000 cm 1 [344,358]. An analysis of the dielectric spectra of water shows the availability of complex permittivity in the microwave region and two absorption... 

Figure 6. Measured complex permittivity maps at 24°C for O, a segmented polyether-polyurethane (UET46-1) and , a segmented polyether-polyester (H49). Measurements were made with the steady-state response dielectric spectrometer.

When a field E is applied across a dielectric (a simple parallel plate condenser, for example), the resulting displacement current, D is related to E as D = eE. In glasses, which are dielectrically isotropic, the permittivity, 8 behaves as a scalar quantity and is equal to D E. While E is an experimentally controlled alternating field of arbitrary frequency, D and s are the material dependent responses and Z), which represents the polarization current is not always in phase with E. s is, therefore, a complex frequency dependent quantity. The complex permittivity, e, is defined as... 

This system of charges is spectroscopically active. It contributes to the complex permittivity and absorption in the THz region.4 In water the relevant absorption peak is located at 200 cm-1. Our estimate shows that in the VIB state the concentration Vvib of water molecules is commensurable5 with their total concentration N (estimation gives Avib = Afrvib with rvib 35-45%). The b mechanism is responsible for the translational band (T-band) located in the vicinity of 180 cm-1 and the c mechanism is responsible for the band that we term the V-band, with the center placed in the vicinity of 150 cm-1. 

The loss and absorption peaks at v 700 cm-1, located near the border of the IR region, arise due to mechanism a—that is, due to reorientation of a rigid (permanent) dipole in the hat well. This mechanism is also responsible for the microwave loss peak located between the frequencies 0.1 and 1cm-1. The complex permittivity s of the corresponding relaxation band is actually governed by Debye theory, which is involved formally in our calculation scheme. 

In the first part of this work (Sections II through V) we have combined the formula for x given there without derivation, with the formulas for xq, and Xor> accounting for dielectric response, arising, respectively, from elastic harmonic vibration of charged molecules along the H-bond (HB), from elastic reorientation of HB permanent dipoles about this bond, and from a rather free libration of a permanent dipole in a defect of water/ice structure modeled by the hat well. The set of four frequency dependences, namely of Xor(v)> (v), X (v), and X (v), allows us to describe the water/ice wideband spectra. For these dependences and those similar to them—namely 0r(v), Asq(v), Ae/1(v), and Ae (v) for the partial23 complex permittivity—we refer to mechanisms a, b, c, and d. 

D and s are the material dependent responses and D, which represents the polarization current is not always in phase with E. s is, therefore, a complex frequency dependent quantity. The complex permittivity, e, is defined as... 

Under the action of an alternating electric field, the electrical response of a system having dipolar interaction may be characterized by the complex permittivity e = s ((o) — ie"(ro) as discussed above. Methods to measure the frequency-dependent permittivity use coaxial lines. The cell of our... 

The phenomenological theory of the dielectric relaxation behaviour of linear systems is well-established [1-5]. The fundamental relationship joining the frequency-dependent complex permittivity c(cu) measured at frequency / = (ofln and the transient step-response function t) is the Fourier transform relationship... 

As.discussed in ibe previous section, the response of the system of charges rqnescoting a molecule, group, or even a crystallite to the action of an external force field may be treated as a perturbation, and quantum mechanical methods may be used, la this way the response of the dielectric material to a time-depeadent (e.g., periodic) force field can be delennined in terms of general complex susceptibilities, which are related directly to the complex permittivity tensor (Ret 12, pp. 178-201). 

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