Frequency dependence model

The theoretical result obtained for a continuous medium sphere confirms the computational results Eloc(x) x/R and is linearly proportional to the applied field Eo. The dispersion of values for Eioc(x) around the x/R line in Fig. 1 is due to the presence of Stone-Wales defects that induce some dispersion of the direction of Eloc which is perpendicular to the surface on the average. Furthermore, the fullerenes studied here are far from being a continuous surface due to their small size, which explains the deviation with the continuous model. The perspective of this approach is to extend these calculations with a frequency-dependent model by including dynamical polarizabilities and kinetic energy for dipoles and charges. 

Huheey, J. E. (1980b) Studies in warning coloration and mimicry. VIII. Further evidence for a frequency-dependent model of predation. J. Herp., 14, 223-30. 

The measurement of effective conductivities is complicated by the traditionally used electrode geometry. Typically, one uses a four-electrode technique [ Steendijk et al., 1993], in which two electrodes inject current and two others measure the potential (Figure 21.5). Gielen et al. [1984] used this method to measure the electrical properties of skeletal muscle and found that the effective conductivity depended on the interelectrode distance. Roth [1989] reanalyzed Gielen et al. s data using the spatial frequency dependent model and found agreement with some of the more unexpected features or their data (Figure 21.6). Table 21.1 contains typical values of skeletal muscle effective conductivities and microscopic tissue parameters. Table 21.2 Hsts nerve effective conductivities. 

Table 2.6 shows the maximum voltages calculated by the frequency-dependent Semiyen model and the frequency-independent distributed parameter line model of the EMTP. It is clear from Figure 2.45 and Table 2.6 that the results neglecting the frequency-dependent effect show a mi-nor difference from the results including the effect. Thus, it can be con-cluded that the frequency-dependent model does not have a significant effect on a lightning surge. 

Effect of frequency dependence on lightning surge for the no-flashover case, (a) Frequency-dependent model, (b) Frequency-independent model. 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

Figure 7-27 shows the frequency dependency of the in-phase PA bands in a-6T, measured at the maxima of the various PA bands. As expected, the two polaron bands are correlated with one another, having virtually the same dynamics. In contrast, the bipolaron PA band at 1.1 eV is virtually flat from 10 to 1000 Hz, indicating that trapped pol is are longer lived than trapped bipolarons in -6T. The excitation lifetimes modeled by comparing the data in Figure 7-27 to... 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

However, these parameters are temperature and Larmor frequency dependent. Such a model can be conveniently parameterized in terms of collision frequency (/), so that the rate of decorrelation is given by... 

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

To use this equation in evaluating I, one needs a model for e(t ) that is consistent with available experiments on the frequency-dependent dielectric constant. 

Some experiments outlined the frequency dependence of phonon scattering on surfaces [74]. Thus, Swartz made the hypothesis that a similar phenomenon could take place at the interface between solids and proposed the diffuse mismatch model [72]. The latter model represents the theoretic limit in which all phonons are heavily scattered at the interface, whereas the basic assumption in the acoustic mismatch model is that no scattering phenomenon takes place at the interface of the two materials. In the reality, phonons may be scattered at the interface with a clear reduction of the contact resistance value as calculated by the acoustic model. 

Chain stretching is governed by the covalent bonds in the chain and is therefore considered a purely elastic deformation, whereas the intermolecular secondary bonds govern the shear deformation. Hence, the time or frequency dependency of the tensile properties of a polymer fibre can be represented by introducing the time- or frequency-dependent internal shear modulus g(t) or g(v). According to the continuous chain model the fibre modulus is given by the formula... 

The relaxation spectrum H is independent of the experimental time t and is a fundamental description of the system. The exponential function depends upon both the experimental time and the relaxation time. Such a function in the context of this integral is called the kernel. In order to describe different experiments in terms of a relaxation spectrum H or retardation spectrum L it is the kernel that changes. The integral can be formed in time or frequency depending upon the experiment being modelled. The inclusion of elastic properties at all frequencies and times can be achieved by including an additional process in the relaxation... 

The sum over weighted relaxation times is heavily dominated by the longest time (the reptation time) r gp=L /7T Dp. Because of this the frequency-dependent dissipative modulus, G"(cd) is expected to show a sharp maximum The higher modes do modify the prediction from that of a single-mode Maxwell model, but only to the extent of reducing the form of G"(a>) to the right of the maximum from ccr to In fact, experiments on monodisperse linear polymers... 

In the next section, we will develop a simple model to predict the frequency dependence of the relative dielectric constants si and 2 of a given material. At that point, we will be able to determine the measurable optical magnitudes defined in Chapter 1 at any particular wavelength (or frequency) if the relative dielectric constants (and thus n and k) are known at that wavelength. 

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