Susceptibility complex

The appropriate (complex) susceptibility tensor for this generator is = co. + 01 + co ). 

Togni s [38] approach was therefore to test the ability of sparteine to act as an ancillary ligand in Pd(II)-allyl complexes—susceptible to nucleophilic attack by stabihzed anions such as Na[CH(COOMe)2]—which could be employed as catalyst precursors. In addition he speculated that the rather rigid and bulky sparteine would be able to induce significant differentiation between the two diastereotopic sites of 1,3-disubstituted allyl hgand, thus leading to enantioselection upon nucleophilic attack. 

Becker K.G., et al., Clustering of non-major histocompatibility complex susceptibility candidate loci in human autoimmune diseases, Proc. Natl. Acad. Sci., 95, 9979, 1998. 

In Refs. 80 and 81 it is shown that the Mittag-Leffier function is the exact relaxation function for an underlying fractal time random walk process, and that this function directly leads to the Cole-Cole behavior [82] for the complex susceptibility, which is broadly used to describe experimental results. Furthermore, the Mittag-Leffier function can be decomposed into single Debye processes, the relaxation time distribution of which is given by a mod-... 

It should be emphasized that these later expressions are valid for nonabsorbing media with a real susceptibility and refractive indices close to glass. A more general treatment of absorbing media and complex susceptibility is summarized below. The more complete expressions given by Kajzar and Messier, including the various transmission factors, should be used for accurate determinations. 

A. Spectral Function and Complex Susceptibility (General Expressions)... 

Expression of Complex Susceptibility Through Spectral Function... 

X = X + i%" I co = 2jicv cop = 2 e JnN/m 9 Complex susceptibility Current precessional shift of a dipole moment Angular frequency of radiation Plasma frequency Period of q(t) function (dimensional quantity)... 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

Third, the expression for the spectral function pertinent to the HO model is derived in detail using the ACF method. Some general results given in GT and VIG (and also in Section II) are confirmed by calculations, in which an undamped harmonic law of motion of the bounded charged particles is used explicitly. The complex susceptibility, depending on a type of a collision model,... 

In the first period, which ended with a review [18], the complex susceptibility x (0)) was expressed through the law of motion of the particles perturbed by a.c. external field E(t). The results of these calculations rigorously coincide with those obtained, for example, in Refs. 22 and 23, respectively, for the planar and spatial extended diffusion model (compare with our Ref. 18, pp. 65 and 68). The most important results of this period are (i) the planar confined rotator model [ 17, p. 70 20], which has found a number of applications in our and other [24—31] works (ii) the composite so-called confined rotator-extended diffusion model. However, this approach had no perspectives because of troublesome calculations of the susceptibility x ( )-... 

We have introduced the effective complex susceptibility x ( ) = X,( )+ X ) stipulated by reorienting dipoles. This scalar quantity plays a fundamental role in subsequent description, since it connects the properties and parameters of our molecular models with the frequency dependences of the complex permittivity s (v) and the absorption coefficient ot (v) calculated for these models. 

We shall show further that at F = FB the complex susceptibility is given by... 

Substituting (28c) into (14b) with account of (27b), we have the following relation between two components of the complex susceptibility ... 

Substituting this relation into Eq. (31), we finally derive unambiguous relation between the complex susceptibility and spectral function relevant to the Gross collision model12 ... 

To relate the complex permittivity s of a polar medium with the complex susceptibility % provided by motions of the dipoles, we suggest that a polar medium under study is influenced by the external macroscopic time-varying electric field Ee(f) = Re[Em exp(imf)], where Em is the complex amplitude. This field induces some local field EM(f) = Re[ ) exp(icof)] in a cavity surrounding each polar molecule. A given molecule directly experiences the latter field. 

If we neglect the difference between two complex amplitudes, Em and E , then the complex permittivity s of a polar medium and the complex susceptibility x provided by motions of the dipoles would be related as follows ... 

We employ the following equations Eq. (142) for the complex susceptibility X, Eq. (141) for the complex permittivity , and Eq. (136) for the absorption coefficient a. In (142) we substitute the spectral functions (132) for the PL-RP approximation and (133) for the hybrid model, respectively. In Table IIIB and IIIC the following fitted parameters and estimated quantities are listed the proportion r of rotators, Eqs. (112) and (127) the mean number m of reflections of a dipole from the walls of the rectangular well during its lifetime x, Eqs. (118)... 

The spectral function L(z) determines the complex susceptibility % of the medium,35... 

A theory, accounting for an internal field correction [40, 41], gives the relationship % ( ), Eq. (139), between the complex susceptibility and permittivity. For calculation of the wideband spectra it is more convenient to employ the reverse dependence (% ), Eq. (141). 

C. An analogous model was considered in Ref. 12b, but an important new step was made. Now it was assumed that the stochastic processes with two different relaxation times correspond to types of motion described by two wells. Two different complex susceptibilities were calculated, which have split Eq. (235) by two similar expressions for reorientation and vibration processes ... 

It is shown in Section VI.E that if the time-varying part of a dipole moment is small (i [Pg.208]

Let us calculate the broadband spectra of liquid water H20 and D20. The adopted experimental data are presented in Table XII. In accord with the scheme (238), we use Eq. (249) for the complex susceptibility x and use Eqs. (242) and (243) for the modified spectral function R(z). All other expressions used in these calculations are the same as were employed in Section V. 

If the time-varying moment jl is neglected, the complex susceptibility could be represented as a sum of two terms stipulated, respectively, by the a.c. perturbations of a dipole s trajectory and of the steady-state distribution ... 

Taking into account the internal-field correction, replacing % and s by x r and E r, we use the same relationships connecting the complex susceptibility and permittivity as were employed in Sections IV-VI ... 

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