Debye relaxation spectrum

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. 

In the low-frequency range (with x spectral function L(z) depends weakly on frequency x. Then Eq. (32) comes to the Debye-relaxation spectrum given by Eq. (33). Its main characteristics, such as the dielectric-loss maximum Xd and its frequency xD, are given by Eq. (34). A connection between these quantities and the model parameters becomes clear in an example of a very small collision frequency y. In this case, relations (34) come to... 

Mechanism (a) responsible for libration band near the edge of far IR spectrum and for nonresonance Debye relaxation spectrum... 

Thus, at low frequencies the Gross susceptibility (280) reduces to the Debye relaxation spectrum. 

Figure 9.15. Typical trajectories of a Gaussian stochastic process x(t) with zero mean and Gaussian (a) or exponential (i>) correlation function. Circles are crossing points of x = 0. Trajectories were generated by regular sampling in the frequency domain, (c) corresponds to the Debye relaxation spectrum with a cutoff frequency. Reorganization energy of the discarded part of the spectrum is 7% of the total. The sampling pattern was the same as in (b).

A principal drawback of the hat-curved model revealed here and also in Section V is that we cannot exactly describe the submillimeter (v) spectrum of water (cf. solid and dashed lines in Figs. 32d-f). It appears that a plausible reason for such a difference is rather fundamental, since in Sections V and VI a dipole is assumed to move in one (hat-curved) potential well, to which only one Debye relaxation process corresponds. We remark that the decaying oscillations of a nonrigid dipole are considered in this section in such a way that the law of these oscillations is taken a priori—that is, without consideration of any dynamical process. 

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 change in the emission spectrum with time after pulsed excitation (TRES) is a method for assessing the overall response of the solvent to a change in solute geometry or polarity [22]. The precise values of the relaxation times depend upon the method of measurement. At room temperature the TRES solvent correlation times are subnanosecond and, in some cases subpicosecond. The Debye relaxation time in water is 8 ps, while the TRES correlation time is shorter [22]. Although there is not, in general, a... 

The next issue to concern us will be anomalous relaxation in which the smearing out of a relaxation spectrum (i.e., the deviation of complex susceptibility from its Debye form) is associated with the concept of a relaxation time distribution. As is well known, this concept implies an assembly of dipoles with a continuous distribution of relaxation times of Eq. (379). 

In most liquids and to a good approximation, y0= (1-n). Therefore n can be evaluated by evaluating p. P itself is given by the ratio WJW, where and W are the Debye and actual widths of the relaxation spectrum. Even when loss curves (dielectric, mechanical or any other) are fitted to other standard analytical functions such as Cole-Cole (CC), Davidson-Cole (DC) or Haveriliac-Negami expressions, (see earlier section) one can determine p using the empirical relations... 

We assume that the above-indicated drawback of the present model can be avoided (or at least reduced) if a new paradigm [mentioned below in Section X.B.4(ii)] of the molecular model will be constructed. In our opinion, this drawback of the present model is stipulated by the following. In view of Eq. (11) the libration lifetime T0r is determined by the experimental Debye relaxation time td, so variation of Tor cannot be used for other corrections of the calculated spectra. In the proposed new paradigm it is desirable to use Tor for the latter purpose, while a correct describing of the low-frequency Debye spectrum is assumed to be reached by variation of additional parameter(s). 

The first process prevails at relatively low frequencies. The electric component E of radiation orients dipole moments p along the field direction, while chaotic molecular motions hinder this orientation p and E are the vectors, and the field E is assumed to vary harmonically with time t. Due to inertia of reorienting molecules the time dependence of the polarization lags behind the time dependence E(f), so that heating of the medium occurs (the heating effect is not considered in this work). The dielectric spectrum obeys the Debye relaxation, for which the absorption monotonically increases with frequency. 

The LIB fraction (the fraction of monomers) containing about 55-70% of all water molecules comprises a permanent dipole, namely, a neutral molecule librating in a tight surrounding of other molecules in a condensed medium. This motion, governed by mechanism a, is responsible for the librational band, placed near the boundary of the IR region, and for the low-frequency Debye relaxation band, placed at microwaves. The Debye spectrum is not considered in this work. 

Let us first discuss estimates fi om DR measurements that provide several important pieces of information. These experiments measure the frequency-dependent dielectric constant and provide a measure of a liquid s polarization response at different frequencies. In bulk water, we have two dominant regions. The low-frequency dispersion gives us the well-known Debye relaxation time, Tq, which is equal to 8.3 ps. There is a second prominent dispersion in the high-frequency side with relaxation time constant less than Ips which contains combined contributions from low-frequency intermolecular vibrations and libra-tion. Aqueous protein solutions exhibit at least two more dispersions, (i) A new dispersion at intermediate frequencies, called, d dispersion, which appears at a timescale of about 50 ps in the dielectric spectrum, seems to be present in most protein solutions. This additional dispersion is attributed to water in the hydration layer, (ii) Another dispersion is present at very low frequencies and is attributed to the rotation of the protein. 

In Figure 17 (curve 1), the dielectric loss spectrum for PS at room temperature as taken from Bur [18] is presented. There are no pronounced relaxation loss peaks due to a- and / -processes in this polymer which is considered to be nonpolar , although in fact it possesses a smaU dipole moment due to the asymmetry at the phenyl side group. The loss tangent is seen to be constant and relatively small over a very broad frequency range from subaudio to 10 Hz. A loss peak occurs at vs2 x 10 Hz, a very high frequency for Debye relaxation dispersion. It appears to be the 5-peak which has been n asured by McCammon et al. at 46 K (1 kHz) with an activation energy of 12 kJ mole" ... 

Several types of spin-lattice relaxation processes have been described in the literature [31]. Here a brief overview of some of the most important ones is given. The simplest spin-lattice process is the direct process in which a spin transition is accompanied by the creation or annihilation of a single phonon such that the electronic spin transition energy, A, is exchanged by the phonon energy, hcoq. Using the Debye model for the phonon spectrum, one finds for k T A that... 

At higher temperatures, the two-phonon (Raman) processes may be predominant. In such a process, a phonon with energy hcOq is annihilated and a phonon with energy HcOr is created. The energy difference TicOq — ha>r is taken up in a transition of the electronic spin. In the Debye approximation for the phonon spectrum, this gives rise to a relaxation rate given by... 

This treatment is oversimplified because, in addition to neglecting inner sphere contributions to the reorganization energy it approximates the dielectric frequency spectrum to a single frequency, 0jo — 1011 s 1, corresponding to the Debye dielectric relaxation which probably varies in the vicinity of the ions. The cathodic current is given by... 

Figure 15.8 A model of temperature-dependent relaxation time spectrum for PMN. G (r, T) is the number of polar regions having a relaxation time r, T/ is the freezing temperature, and td is the inverse Debye frequency (from [16]).

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