Hat-flat model

E. Summation of Series in Expression for the Spectral Function IV. Hat-Flat Model... 

For a reasonable set of the parameters the calculated far-infrared absorption frequency dependence presents a two-humped curve. The absorption peaks due to the librators and the rotators are situated at higher and lower frequencies with respect to each other. The absorption dependences obtained rigorously and in the above-mentioned approximations agree reasonably. An important result concerns the low-frequency (Debye) relaxation spectrum. The hat-flat model gives, unlike the protomodel, a reasonable estimation of the Debye relaxation time td. The negative result for xD obtained in the protomodel is explained as follows. The subensemble of the rotators vanishes, if u —> oo. 

A brief list of basic assumptions used in the ACF method precedes the detailed analysis of the results of calculations. The derivation of the formula for the spectral function is given at the end of the section. The calculations demonstrate a substantial progress as compared with the hat-flat model but also reveal two drawbacks related to disagreement with experiment of (i) the form of the FIR absorption spectrum and (ii) the complex-permittivity spectrum in the submillimeter wavelength region. We try to overcome these drawbacks in the next two sections, to which Fig. 2c refers. 

It will be calculated in Section IV.F for an example of a finite-depth rectangular well (viz., for the hat-flat model), where also a more general definition of this quantity will be given. 

In the case of water we actually shall employ two approximations of the hat-flat model. 

We remind the Reader that we use the hat-flat model in order to describe a two-humped absorption spectmm characteristic for water. The larger peak frequency (x) we refer to the librators and the smaller one (x) to the rotators of both types. We estimate x by the same way as in Eq. (85) ... 

The comparison of the spectra calculated for rigorous hat-flat model with various simplified variants will be given below. 

The hat-flat model cannot be applied to heavy (D20) water. Namely, for this liquid the theoretical center frequency of the R-absorption band is lower than for H20 due to larger moment of inertia I, while the... 

Figure 18. The absorption (a) and loss (b) spectrum of CH3F at T = 133 K. Solid and dashed lines refer, respectively, to rigorous and the hybrid-model versions of the hat-flat model, circles do to experiment. The vertical line marks the frequency of relaxational loss peak.

Rigorous and simplified variants of the hat-flat model agree satisfactorily with the experiment at low temperature. The hybrid model yields worse result for the temperature T = 293 K near the critical one The theoretical loss-peak position is noticeably shifted to lower frequencies in comparison with the experimental position. This important result shows that rigorous30 consideration is preferable in the case of a nonassociated liquid. 

The hat (more precisely, the hat-flat) model is described in Section IV of this chapter. 

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