Librational band model, narrowing

A. Previous models of water (see 1-6 in Section V.A.l) and also the hat-curved model itself cannot describe properly the R-band arising in water and therefore cannot explain a small isotope shift of the center frequency vR. Indeed, in these models the R-band arises due to free rotors. Since the moment of inertia I of D20 molecule is about twice that of H20, the estimated center of the R-band for D20 would be placed at y/2 lower frequency than for H20. This result would contradict the recorded experimental data, since vR(D20) vR(H20) 200 cm-1. The first attempt to overcome this difficulty was made in GT, p. 549, where the cosine-squared (CS) potential model was formally (i.e., irrespective of a physical origin of such potential) applied for description of dielectric response of rotators moving above the CS well (in this work the librators were assumed to move in the rectangular well). The nonuniform CS potential yields a rather narrow absorption band this property agrees with the experimental data [17, 42, 54]. The absorption-peak position Vcs depends on the field parameter p of the model given by... 

The solid curve in Fig. 31 shows the frequency dependence of absorption [Eq. (255)]. We see the shoulder in the R-band region (at x 2) and the main (librational) absorption peak at x 5.5. This dependence (solid curve) agrees with the experimental data [42, 51] (a more detailed analysis is given in the next section). If the vibration lifetime r b were twice as large, we would get an unreasonably large and narrow R-peak (dashed curve). On the other hand, if the parameter, v were twice as large, the intensity of the R-band would increase unreasonably (dash-and-dotted curve). This example shows convincingly that the form of the FIR water spectra sharply determines the parameters (25 lb) of our polarization model. 

The main advantage of the hat-curved potential is that it is possible to narrow the width Avor of the librational absorption band by decreasing the form factor /. Indeed, Avor attains its maximum value when/ = 1. Note that / = 1 is just the case of the hat flat or its simplified variant, the hybrid model, both of which were described in Section IV. The latter was often applied before (VIG) and is characterized by a rather wide absorption band, especially in the case of heavy water. In another extreme case, / — 0, the linewidth Avor becomes very low. When / = 0, we have the case of the parabolic potential well, whose dielectric response was described, for example, in GT and VIG. Thus, when the form factor/of the hat-curved well decreases from 1 to 0, the width Avor decreases from its maximum to some minimum value. 

Other more unusual pictures that have been proposed include the libron model [45], where librations (molecular rotations) are responsible for ph T). It was suggested that the T2 behavior arose from the dominance of two-libron processes, and that the first-order, single-libron contribution is small by symmetry. The T dependence of the spin susceptibility was discussed within the same framework, in terms of a band-narrowing effect [46]. However, this approach has been criticized [44], and furthermore, no evidence has been found that librations play any role in metal-semiconductor transitions [9]. 

In Fig. 2a we depict the wideband loss spectrum of water calculated in terms of the hat model for room temperature (27°C). For ice at —7°C a similar calculation (but in a much narrower band) is presented in Fig. 2b. The fitted molecular parameters are presented in Table I (some of these parameters will be determined below). During the lifetime Tor a dipole performs in water and ice about two librations and about six librations in supercooled water (mor = 5.6). 

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