Orientation relaxation

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

The orientational relaxation, considered in Chapters 6 and 7, is a more complex problem. The impact theory is the only model capable of tracing the transition from quasi-free rotation in the rare gas to... 

From a mathematical perspective either of the two cases (correlated or non-correlated) considerably simplifies the situation [26]. Thus, it is not surprising that all non-adiabatic theories of rotational and orientational relaxation in gases are subdivided into two classes according to the type of collisions. Sack s model A [26], referred to as Langevin model in subsequent papers, falls into the first class (correlated or weak collisions process) [29, 30, 12]. The second class includes Gordon s extended diffusion model [8], [22] and Sack s model B [26], later considered as a non-correlated or strong collision process [29, 31, 32],... 

If the resolving capacity of the instruments is ideal then vibrational-rotational absorption and Raman spectra make it possible in principle to divide and study separately vibrational and orientational relaxation of molecules in gases and liquids. First one transforms the observed spectrum of infrared absorption FIR and that of Raman scattering FR into spectral functions... 

Although Ki and Ki are defined by physical quantities of different nature, their time evolution is universally determined by orientational relaxation. This discussion is restricted to linear molecules and vibrations of spherical rotators for which / is a symmetric tensor / = fiki- In this case the following relation holds... 

As far as indirect methods are concerned (for instance, that of magnetic resonance), they measure solely the correlation times of orientational relaxation, which are integral characteristics of the process ... 

Of course, knowledge of the entire spectrum does provide more information. If the shape of the wings of G (co) is established correctly, then not only the value of tj but also angular momentum correlation function Kj(t) may be determined. Thus, in order to obtain full information from the optical spectra of liquids, it is necessary to use their periphery as well as the central Lorentzian part of the spectrum. In terms of correlation functions this means that the initial non-exponential relaxation, which characterizes the system s behaviour during free rotation, is of no less importance than its long-time exponential behaviour. Therefore, we pay special attention to how dynamic effects may be taken into account in the theory of orientational relaxation. 

This means that the theory may be applied only to dense media where rotational relaxation proceeds at a higher rate than does orientational relaxation. 

This ratio of orientational relaxation times is sometimes used to identify the situation corresponding to perturbation theory [85]. 

This is nothing other than a long-time asymptotics of orientational relaxation valid at t > xj. Sometimes deviations from Debye s relaxation were also found at t > xg/. This is an effect of long-range Coulomb and... 

Markovian theory of orientational relaxation implies that it is exponential from the very beginning but actually Eq. (2.26) holds for t zj only. If any non-Markovian equations, either (2.24) or (2.25), are used instead, then the exponential asymptotic behaviour is preceded by a short dynamic stage which accounts for the inertial effects (at t < zj) and collisions (at t < Tc). 

Comparison of formulae (2.51) and (2.64) allows one to understand the limits and advantages of the impact approximation in the theory of orientational relaxation. The results agree solely in second order with respect to time. Everything else is different. In the impact theory the expansion involves odd powers of time, though, strictly speaking, the latter should not appear. Furthermore the coefficient /4/Tj defined in (2.61) differs from the fourth spectral moment I4 both in value and in sign. Moreover, in the impact approximation all spectral moments higher than the second one are infinite. This is due to the non-analytical nature of Kj and Kf in the impact approximation. In reality, of course, all of them exist and the lowest two are usually utilized to find from Eq. (2.66) either the dispersion of the torque (M2) or related Rq defined in Eq. (1.82) ... 

The mutual correspondence of non-Markovian and Markovian (impact) approximations becomes clear, if the second derivative of K/(t) is considered. It varies differently within three time intervals with the following bounds xc < xj < Tj1 (Fig. 2.5). Orientational relaxation occurs in times Fj1. The gap near zero has a scale of xj. A parabolic vertex of extent xc and curvature I4 > 0 is inscribed into its acute end. The narrower the vertex, the larger is its curvature, thus, in the impact approximation (tc = 0) it is equal to 00. In reality xc =j= 0, and the... 

The same was done recently for their spectral properties [111, 112] using Eq. (2.68) to express the spectrum of orientational relaxation (2.13)... 

To prove this let us make more precise the short-time behaviour of the orientational relaxation, estimating it in the next order of tfg. The estimate of U given in (2.65b) involves terms of first and second order in Jtfg but the accuracy of the latter was not guaranteed by the simplest perturbation theory. The exact value of I4 presented in Eq. (2.66) involves numerical coefficient which is correct only in the next level of approximation. The latter keeps in Eq. (2.86) the terms quadratic to emerging from the expansion of M(Jf ). Taking into account this correction calculated in Appendix 2, one may readily reproduce the exact... 

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