Librational band

The latter is determined by the oscillation frequency, decaying coefficient, and vibration lifetime. This nonrigid dipole moment stipulates a Lorentz-like addition to the correlation function. As a result, the form of the calculated R-band substantially changes, if to compare it with this band described in terms of the pure hat-curved model. Application to ordinary and heavy water of the so-corrected hat-curved model is shown to improve description (given in terms of a simple analytical theory) of the far-infra red spectrum comprising superposition of the R- and librational bands. 

Furthermore, we consider the moments of inertia I and I2 and thus I in Eq. (149) to be known, as well as the experimental position lh. We remark that actually there is no need to know exactly the value of moment of inertia I. Indeed, a possible uncertainty in the employed I value given by Eq. (149), and therefore in determination of the center librational-band frequency Vub from Eqs. (136), (137), (141), and (142), could easily be compensated by fitting the half-width (3 of the potential well. 

The calculated width of the librational band is wider than the recorded one, cf. solid and dashed lines in Figs. 15a and 15b. This disagreement is... 

In spite of a considerable success and simplicity, the models 1 and 2 fail to explain the FIR absorption band in water, where beside the librational band (which is placed in the vicinity of 700 cm 3) there arises also a less intensive R-band with its peak-absorption frequency at 200 cm . 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

Figure 24. Absorption coefficient (a, c) and wideband diecltric loss (b, d) calculated for liquid H20 water at 22.2°C (a, b) and 27°C (c, d) for the hat-curved model (solid lines). The experimental a(v) dependencies [17, 42, 56] are shown by dashed lines. The horizontal lines in Figs, (a) and (c) denote the maximum absorption recorded in the librational band.

In Table IX we present the list of optical constant of ordinary (H20) water [42] at 27°C covering very wide range of frequencies (from 10 cm-1 until 1000 cm-1). For two other temperatures (1°C and 50°C) we present in Table X such constants recorded in Ref. 53 for a narrower region from 400 cm-1 to 820 cm-1. Both tables comprise the absorption maximum of the librational band, and the first one includes also the maximum in the R-band. For lower frequencies we can use the empirical formulas of Liebe et al. [17], They are represented in Section G.2.a. Note that the absorption coefficient a is determined by the imaginary component... 

With respect to water we shall conditionally extend the SWR from 10 to 300 cm this frequency region falls between the Debye relaxation range and the librational band. 

Here g is the Kirkwood correlation factor (146), which is determined by the static permittivity ss, permittivity irx at the HF edge of the librational band and by the reduced concentration of the dipoles G. The total number of free parameters of our model is now six ... 

In the librational band we have attained now a satisfactory agreement of the theoretical and experimental absorption frequency dependences. Comparing Figs. 32a and 32b with Figs. 26a and 26c calculated in Section V for a pure ... 

Figures 32d-f, placed on the right-hand side of Fig. 32, demonstrate a wideband dielectric-loss frequency dependence. This loss is calculated (solid lines) or measured [17, 42, 51, 54] (dashed lines) for water H20 and D20 at the same temperatures, as correspond to the absorption curves shown on the left-hand side of Fig. 32. Our theory gives a satisfactory agreement with the experimental data, obtained for the Debye region, R- and librational bands, to which three peaks (from left to right) correspond. However, in the submillimeter wavelength region (namely, from 10 to 100 cm ) the calculated loss is less than the recorded one. The fundamental reason for this difference will be discussed at the end of the next section.

The dash-and-dotted curves mentioned here are obtained from the empirical description [17] of the FIR spectra. However, approximately the same contour of the libration band is relevant [9] for the rectangular potential well. 

The first group of the parameters determine the frequency dependences in the Debye and librational bands and the second group—in the submillimeter wavelength range and in the R-band. A few statistical parameters of the composite model are determined by the same formulas as were given in Sections V and VI. 

Thus, the parameter (308) is for heavy water about twice that estimated for ordinary water. Our result for water corresponds to a known [59, 60] statement that in heavy water the H-bond is stronger than in ordinary water. Moreover, one may suggest that in ice Ih the form factor / should be lower than in D20, since the librational band in ice is narrower than in water (the comparison of these bands are given in VIG, in Ref. 61, and in Section X. On the other hand, in a nonassociated liquid the so found spread is about 4 times lower than in the case of H20 this result also looks reasonable. 

In terms of the four parameters in Eq. (316a) the latter alone gives a satisfactory description of the Debye-relaxation and librational bands of various liquids. One can control the width of the librational band of liquid H2O by changing the form factor/. 

Returning to the rotational dynamics, we remark that the hat-curved model was applied in Sections V-VII for description of the librational band,... 

The width of librational band and the left and right cut-off values reflect the orientational restrictions of the potentials. These restrictions usually involve the charge-charge interaction of the pair-wise potential (for details see Table 3). 

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