Debye model, frequency dependence

Even though the Einstein and Debye models are not exact, these simple one-parameter models illustrate the properties of crystals and should give reliable estimates of the volume dependence of the vibrational entropy [15]. The entropy is given by the characteristic vibrational frequency and is thus related to some kind of mean interatomic distance or simpler, the volume of a compound. If the unit cell volume is expanded, the average interatomic distance becomes larger and the... 

Let us summarize by modeling the velocity autocorrelation function using Debye-Huckel type interactions between charged point defects in ionic crystals, one can evaluate the frequency-dependent conductivity and give an interpretation of the universal dielectric response. 

The temperature and frequency dependence of the dielectric properties of polar molecules such as water was first modeled by Debye (1929). Early work on dielectric properties has been described by von Hippel (1954a, 1954b). Excellent recent reviews have been published by Ohlsson and Bengtsson (1975), Mudgett (1985 1990) and Geyer (1990). 

For a reasonable set of the parameters the calculated far-infrared absorption frequency dependence presents a two-humped curve. The absorption peaks due to the librators and the rotators are situated at higher and lower frequencies with respect to each other. The absorption dependences obtained rigorously and in the above-mentioned approximations agree reasonably. An important result concerns the low-frequency (Debye) relaxation spectrum. The hat-flat model gives, unlike the protomodel, a reasonable estimation of the Debye relaxation time td. The negative result for xD obtained in the protomodel is explained as follows. The subensemble of the rotators vanishes, if u —> oo. 

The calculated spectra are illustrated by Fig. 25. In Fig. 25a we see a quasiresonance FIR absorption band, which, unlike water, exhibits only one maximum. Figure 25b demonstrates the calculated and experimental Debye-relaxation loss band situated at microwaves. Our theory satisfactorily agrees with the recorded a(v) and e"(v) frequency dependencies. Although the fitted form factor/is very close to 1 (/ 0.96), the hat-curved model gives better agreement with the experiment than does a model based on the rectangular potential well, where / = 1 (see Section IV.G.3). 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

The first group of the parameters determine the frequency dependences in the Debye and librational bands and the second group—in the submillimeter wavelength range and in the R-band. A few statistical parameters of the composite model are determined by the same formulas as were given in Sections V and VI. 

In Figs. 66 and 68 the calculated absorption and loss spectra are depicted for ordinary water at the temperatures 22.2°C and 27°C and for heavy water at 27°C. The solid curves refer to the composite model, and the dashed curves refer to the experimental spectra [42, 51]. For comparison of our theory with experiment at low frequencies, in the case of H20 we use the empirical formula [17] comprising double Debye-double Lorentz frequency dependences. In the case of D20 we use empirical relationship [54] aided by approximate formulae given in Appendix 3 of Section V. The employed molecular constants were presented in previous sections, and the fitted/estimated parameters are given in Table XXIV. The parameters of the composite model are chosen so that the calculated absorption-peak frequencies ilb and vR come close to the... 

In Fig. 11.1 the time dependent dielectric constant (left) and both dynamic dielectric constants (right) are plotted vs. reduced time, t/x, and reduced angular frequency, cox, respectively, for amorphous polyethylene terephthalate at 81° C. It illustrates a continuous increase of s(t) from 3.8 to 6. It also shows that the e (co) decreases with frequency in a similar way as e(t) increases with time. The maximum in s", situated at cox = 1, is equal to (fis— oo)/2 = 1.1. A Cole-Cole plot where s" is plotted vs. s is shown in Fig. 11.2. For a Debye model with only one relaxation time this should be a semi-circle. In reality the decrease of s from s to 00 is not so fast and the maximum in s" not so sharp as the Debye... 

Dielectric relaxation of complex materials over wide frequency and temperature ranges in general may be described in terms of several non-Debye relaxation processes. A quantitative analysis of the dielectric spectra begins with the construction of a fitting function in selected frequency and temperature intervals, which corresponds to the relaxation processes in the spectra. This fitting function is a linear superposition of the model functions (such as HN, Jonscher, dc-conductivity terms see Section II.B.l) that describes the frequency dependence of the isothermal data of the complex dielectric permittivity. The temperature behavior of the fitting parameters reflects the structural and dynamic properties of the material. 

The theory starts from description of the dielectric loss spectra, frequency-dependent permittivity of the solvent e uj), in the framework of the Debye model [86], in which the reorientation of the solvent dipoles gives the main contribution to the relaxation of solvent polarization ... 

As for the previous example described in Sect. 3.2.1, the relaxation of the magnetization has been studied using combined ac (Fig. 6a) and dc (Fig. 6b) measurements. In order to extract the relaxation time of the system (r), the obtained frequency dependence of the in-phase x and out-of-phase x" susceptibilities and furthermore the Cole-Cole plots (x" vs. x plot) were fitted simultaneously to a generalized Debye model (solid lines in Figs. 6a and 6b). The fact that the found a parameters of this model are less than 0.06, indicates that the system is close to a pure Debye model with hence a single relaxation time. This indication is confirmed by the quasi-exponential decay of the magnetization observed between 1.8 and 0.8 K (Fig. 6b). 

The role of vibrational relaxation and solvation dynamics can be probed most effectively by fluorescence experiments, which are both time- and frequency-resolved,66-68 as indicated at the end of Sec. V. We have recently developed a theory for fluorescence of polar molecules in polar solvents.68 The solvaion dynamics is related to the solvent dielectric function e(co) by introducing a solvation coordinate. When (ai) has a Lorentzian dependence on frequency (the Debye model), the broadening is described by the stochastic model [Eqs. (113)], where the parameters A and A may be related to molecular... 

Theoretical Estimates The use of the Debye model (Figure 3.2), which assumes that a solid behaves as a three-dimensional elastic continuum with a frequency distribution/(j ) = allows accurate prediction of the temperature dependence of the vibrational heat capacity C / of solids at low temperatures Cy oc r ), as well as at high temperatures (Cy = Wks). One may also use the same model with confidence to evaluate the temperature dependence of the surface heat capacity due to vibrations of atoms in the surface. 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

Electrical circuits with lumped components are often used as models mimicking the electrical properties of tissue. The simplest models are with three components. If all components are ideal (not frequency-dependent values), the model is a Debye model. 

The main assumption in all these approaches is that the characteristic sizes of the single-phase regions are much larger than the Debye screening length (26). Provided that the dielectric permittivity and electric conductivity of the individual phases are known, the MW models enable us to calculate the total frequency-dependent permittivity of the system. 

In order to analyze carefully the frequency-dependent ellipsometric measurements described in the previous section, a precise determination of the frequency dependence of the dielectric constant e is needed. While, the dielectric constant of nonpolar polymers is nearly constant over a wide range of frequencies, that of polar materials decreases with increasing frequency (50), In the optical range, e generally increases with the frequency and this behavior is known as normal dispersion. At these high frequencies, the origin of the polarizability is mainly electronic. However, at moderate and low frequencies the dielectric constant is enhanced compared with its optical frequency value due to the motion of the molecular dipoles. This regime is called anomalous dispersion. The orientational and electronic contributions are found in the well-known Clausius-Mossotti formula for instance. In the simplest model, the frequency dependence of the dielectric constant can be described by the Debye formula (50) ... 

This has been used for two-level tunnelling systems in insulating glasses. The coupling coefficient Fip from the phonon deformation potential should be independent of T and A, because the density of phonon modes in the Debye model is proportional to co up to the maximum frequency cod and this co-dependence counteracts the smaller overlap for larger A. The electron rate Re may therefore dominate the total rate at small values of A, while Rip may be faster for large A up to the Debye energy k T. ... 

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