Spectral function complex susceptibility

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

Expression of Complex Susceptibility Through Spectral Function... 

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,... 

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

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)... 

In Table V the fitted free and estimated statistical parameters are presented. For calculation of the spectral function we use rigorous formulas (130) and Eqs. (132) for the hybrid model. For calculation of the susceptibility %, complex permittivity , and absorption coefficient a we use the same formulas as those employed in Section IV.G.2 for water.29... 

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

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. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

The analytical formulas (138) and (139) or equivalent formulas (141) describe the harmonic-vibration contribution % to the total complex susceptibility. A simpler variant of this model, in which the partial spectral functions are represented by the formulas (148), gives for water a graphical coincidence with the results of the above-mentioned rigorous theory. For ice, both approaches agree well (cf. the solid and dashed curves in Fig. 41). 

Let a unit volume of an isotropic medium comprise Vvib/2 of such pairs (nonrigid dipoles). We shall calculate the generated complex susceptibility x by using the high-frequency approximation for which it is assumed that at the instant just after a strong collision the velocities and position coordinates are given by the Boltzmann distribution (marked by the subscript B). Then, in view of Eq. (3.5) in GT1, the complex susceptibility x is proportional to the spectral function L ... 

There are three reasonable combinations of metal oxidation states for oxidized Type 3 copper that are consistent with spectral and redox data (1) Cu(I) Cu(I) with some other group, e.g., disulfide, functioning as a two-electron acceptor (2) Cu(I)-Cu(III) where Cu(III) is low spin and (3) an antiferromagnetically coupled Cu(II)-Cu(II) dimer. Magnetic susceptibility studies on Rhus vernicifera laccase have established that the two Type 3 copper atoms in this enzyme are present as an antiferromagnetically coupled Cu(II) dimer (4). The Type 3 copper atoms of hemocyanin and tyrosinase appear to be similarly coupled and separated by 3-5 A (5,6,7). Further structural information on the Type 3 copper chromophore is scanty neither the identity of the ligands nor the geometry of the site has been ascertained. There is likewise a paucity of literature on binuclear copper complexes that exhibit structural features expected for Type 3 copper. 

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