Spectral functions libration

Appendix 1. Calculation of Fourier Amplitudes b -i for Librators Appendix 2. Transformation of Integral for Spectral Function of Precessors Appendix 3. Optical Constants of Liquid Water... 

Spectral function pertinent to librating and rotating particles... 

In this chapter, dielectric response of only isotropic medium is considered. However, in a local-order scale, such a medium is actually anisotropic. The anisotropy is characterized by a local axially symmetric potential. Spatial motion of a dipole in such a potential can be represented as a superposition of oscillations (librations) in a symmetry-axis plane and of a dipole s precession about this axis. In our theory this anisotropy is revealed as follows. The spectral function presents a linear combination of the transverse (K ) and the longitudinal (K ) spectral functions, which are found, respectively, for the parallel and the transverse orientations of the potential symmetry axis with... 

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 the second period, which was ended by review GT after the average perturbation theorem was proved, it became possible to get the Kubo-like expression for the spectral function L(z) (GT, p. 150). This expression is applicable to any axially symmetric potential well. Several collision models were also considered, and the susceptibility was expressed through the same spectral function L(z) (GT, p. 188). The law of motion of the particles should now be determined only by the steady state. So, calculations became much simpler than in the period (1). The best achievements of the period (2) concern the cone-confined rotator model (GT, p. 231), in which the dipoles were assumed to librate in space in an infinitely deep rectangular well, and applications of the theory to nonassociated liquids (GT, p. 329). 

A rigorous analyses shows that at spectral function due to planar libration involves the averaging over time during a half-period. 

Thus, we should discriminate three subensembles. We mark the quantities referring to the librators by the superscript -, and we mark hindered and free rotators by the common superscript °. So, we represent the total spectral function (SF) as the sum... 

Comparing Eqs. (99) and (67b), we find that at the instant of reflection of the librator from the wall, when ii (5, its axial energy g d2 is restricted by the value u, while in the case of the protomodel such energy tends to oo. Therefore, in order to modify the spectral function (71a) of the protomodel with respect to librators, in the integral over q = fg we replace the infinite upper limit... 

Such a doubling will be employed further while calculating the other (besides the spectral function) additive quantities relating to the librators. 

We modify Eqs. (79) and (80) in the fashion described in Section IV.B.l. Then we have the following expression for the librational spectral function ... 

We shall write down the formulas for the above-mentioned spectral functions pertinent to different dipolar subensembles. The spectral function of the librators is given by... 

Using this result, we may simplify calculation of the spectral function Liz) by neglecting the precessional contribution to L. We shall estimate also in this approximation the peak frequencies X ib and xrot of the absorption bands determined by the librational and the rotational subensembles. 

The phase regions occupied by the librators and precessors are depicted in Fig. 27 in coordinates h, l2 for the parameters u = 5.9, p = %/9. We take two values of the form factor/ 0.65 in (a) and 0.85 in (b). When/increases, the / and 2P areas extend to the larger h and l values. The values of Vmin are shown as functions of l in Fig. 21c. In this example (and in the calculations described in Section V.C) the potential well depth U0 is much greater than kBT, that is, u> 1. We see in Fig. 27 that for the J and 2P areas the boundary values for h are still greater than it. This property is used to simplify analytical expressions for the spectral functions. 

In accord with the planar libration approximation, we first come from representation of the spectral function for motion of a dipole in a plane, where integration over l is lacking by definition, so only integration over energy h is employed. We shall find in this way the function (203). As a next step we carry out integration over l, so that a rather simple expression (171) for the spectral function L(z) will be obtained. 

On the other hand, in the high-frequency region, where R- and librational quasi-resonance bands arise, we have gx y L(z. It follows from Eq. (249) that in this region the susceptibility yf (x) becomes proportional to the spectral function R(z). 

The second group, with 0 = 0, and 0 = 0 for which the spectral function is denoted L(z), comprises all the V-particles. They precess with the constant azimuthal velocity at the frozen polar angle 0 = 0, in which the effective potential is minimal. Particles of this group, unlike the librators, do not occupy the flat part of the hat curved potential well. 

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... 

Now we shall modify the spectral function (224a) by taking into account a one-dimensional libration scheme proposed in Section VIII.B. 

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