Dielectric relaxation spectral function

Figure 2. Dielectric spectral responses of clay-water mixtures at 298 K. Parameters used in the Debye-type relaxation spectral functions are complied from Ishida et al. (2000).

Using the Onsager model, the function Av-l(t) can be calculated for all time domains of dielectric relaxation of solvents measured experimentally for commonly used liquids (see, for example, [39]). Such simulations, for example, give for alcohols, at least, three different time components of spectral shift during relaxation, which are due to appropriate time domains of solvents relaxation. 

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... 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

With local information given by INM analysis in mind, we next see the character of rotational relaxation in liquid water. The most familiar way to see this, not only for numerical simulations [76-78] but for laboratory experiments, is to measure dielectric relaxation, by means of which total or individual dipole moments can be probed [79,80]. Figure 10 gives power spectra of the total dipole moment fluctuation of liquid water, together with the case of water cluster, (H20)io8- The spectral profile for liquid water is nearly fitted to the Lorentzian, which is consistent with a direct calculation of the correlation function of rotational motions. The exponential decaying behavior of dielectric relaxation was actually verified in laboratory experiments [79,80]. On the other hand, the profile for water cluster deviates from the Lorentzian function. As stated in Section III, the dynamics of finite systems may be more difficult to be understood. 

The spectroscopic-active medium is regarded as that comprising pairs of the HB molecules. Hence, we consider a model of their collective motions and the relevant dielectric relaxation. The point is that for the dimer scheme, shown in Fig. 42a, it would be incorrect to represent the spectral function (SF) as a sum of functions corresponding, respectively, to the positively and negatively charged molecules suffering vibrations (a similar reasoning was used in Section VI). 

Where M2 is the second moment of the NMR lineshape, J the spectral density function, with (Dq the Larmor frequency, and (0i the frequency of the spin-locking field. The spectral density can be written in terms of the molecular correlation time, x, and the overall shape of the Tjp - temperature dispersion and the relatively shallow minima arc due to the correlation time distribution, although the location of the minimum is unaffected by this distribution. We have examined several models for the distribution, all of which give essentially the same results. One of the more simple is the Cole-Davidson function (75), which has also been applied to the analysis of dielectric relaxations. The relevant expression for the spectral density in this case is given by Equation 4. 

Figure 6.23 Relaxation times for (A) shift of fluorescence spectral maximum and (B) dielectric relaxation, in n-propanol at low temperatures. Relaxation times as a function of temperature Arrhenius plots, logr against r. (o) for shift of fluorescence spectral maximum for 4-aminophthalimide in n-propanol. ( ) fOT dielectric relaxation of liquid and n-propanol. From W.R. Ware et al. Ref. [55,a].

Simple Spectral Method [23] In the simple spectral method, a model dielectric response function is used. It combines a Debye relaxation term to describe the response at microwave frequencies with a sum of terms of classical form of Lorentz electron dispersion (corresponding to a damped harmonic oscillator model) for the frequencies from IR to UV ... 

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