Planar libration-regular precession approximation

Planar Libration-Regular Precession Approximation Hybrid Model ... 

All the dipoles could be distributed over three groups comprising librators, precessors, and rotators. Our calculations are made in the planar libration— regular precession approximation. For the set of parameters relevant to water (with ( 1 and u > 1) the contribution of the precessors to the spectral... 

Assumptions 1 and 2 constitute the planar libration-regular precession approximation. In Gaiduk et al. [56] and in GT2 the corresponding spectral functions L(z) and L(z) are found in analytical form, as well as the SF L(z) for the rotators. These SFs are expressed as simple integrals from elementary functions over the full energy of a dipole. The total spectral function is thus represented as... 

Using the so-called planar libration-regular precession (PL-RP) approximation, it is possible to reduce the double integral for the spectral function to a simple integral. The interval of integration is divided in the latter by two intervals, and in each one the integrands are substantially simplified. This simplification is shown to hold, if a qualitative absorption frequency dependence should be obtained. Useful simple formulas are derived for a few statistical parameters of the model expressed in terms of the cone angle (5 and of the lifetime x. A small (3 approximation is also considered, which presents a basis for the hybrid model. The latter is employed in Sections IV and VIII, as well as in other publications (VIG). 

In this section we have to calculate the complex permittivity s (v) and the absorption coefficient a(v) of ordinary (H2O) water over a wide range of frequencies. It is rather difficult to apply rigorous formulas because the fluctuations of the calculated characteristics occur at a small reduced collision frequency y typical for water (in this work we employ for calculations the standard MathCAD program). Such fluctuations are seen in Fig. 13b (solid curve). Therefore the calculations will be undertaken for two simplified variants of the hat model. Namely, we shall employ the planar libration-regular precession (PL-RP) approximation and the hybrid model.26... 

These assumptions constitute the planar libration—regular precession (PL-RP) approximation. Note that in Section IV the like one concerns the trajectories, which obey the same equation of motion. Here this approximation... 

The integrals (197b) will be calculated in Section V.E.7 in the planar libration-regular precession (PL-RP) approximation as functions of the parameters u, (5, / of the hat-curved model. For further consideration it is convenient to introduce the coefficient... 

In accord with the planar libration (PL)-regular precession (RP) approximation, in the subensemble of the librators we set l = 0,... 

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