Weak collision

Infrared Spectroscopy. The infrared spectroscopy of adsorbates has been studied for many years, especially for chemisorbed species (see Section XVIII-2C). In the case of physisorption, where the molecule remains intact, one is interested in how the molecular symmetry is altered on adsorption. Perhaps the conceptually simplest case is that of H2 on NaCl(lOO). Being homo-polar, Ha by itself has no allowed vibrational absorption (except for some weak collision-induced transitions) but when adsorbed, the reduced symmetry allows a vibrational spectrum to be observed. Fig. XVII-16 shows the infrared spectrum at 30 K for various degrees of monolayer coverage [96] (the adsorption is Langmuirian with half-coverage at about 10 atm). The bands labeled sf are for transitions of H2 on a smooth face and are from the 7 = 0 and J = 1 rotational states Q /fR) is assigned as a combination band. The bands labeled... 

An important example for the application of general first-order kinetics in gas-phase reactions is the master equation treatment of the fall-off range of themial unimolecular reactions to describe non-equilibrium effects in the weak collision limit when activation and deactivation cross sections (equation (A3.4.125)) are to be retained in detail [ ]. 

Gilbert R G, Luther K and Troe J 1983 Theory of thermal unimolecular reactions in the fall-off range. II. Weak collision rate constants Ber. Bunsenges. Phys. Chem. 87 169-77... 

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

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],... 

In the case of weak collisions, the moment changes in small steps AJ (1 — y)J < J, and the process is considered as diffusion in J-space. Formally, this means that the function /(z) of width [(1 — y2)d]i is narrow relative to P(J,J, x). At t To the latter may be expanded at the point J up to terms of second-order with respect to (/ — /). Then at the limit y -> 1, to — 0 with tj finite, the Feller equations turn into a Fokker-Planck equation... 

This shows the time evolution of the distribution over J when it was a (5-function at t = 0. The distribution spreads and shifts to J = 0, until it finally takes the equilibrium shape (pe- Gradual transformation of the distribution is typical for correlated change in J t) caused by weak collisions. 

By substituting Eq. (1.25) into the above equation and taking into account Eq. (1.27) one can easily see that Eq. (1.29) holds for weak collisions. 

In summary, the Langevin model which addresses not Eq. (1.26) but rather solution (1.25) fits solely the limiting case of weak collisions. Only in this limit does rotational friction acquire the meaning of a mobility in J-space, i.e. 

This conclusion is not unexpected. The molecule s response to weak collisions considered as a random process may be described by perturbation theory (with respect to interaction) if [53]... 

Relations (2.46) and (2.47) are equivalent formulations of the fact that, in a dense medium, increase in frequency of collisions retards molecular reorientation. As this fact was established by Hubbard within Langevin phenomenology [30] it is compatible with any sort of molecule-neighbourhood interaction (binary or collective) that results in diffusion of angular momentum. In the gas phase it is related to weak collisions only. On the other hand, the perturbation theory derivation of the Hubbard relation shows that it is valid for dense media but only for collisions of arbitrary strength. Hence the Hubbard relation has a more general and universal character than that originally accredited to it. 

For one-dimensional rotation (r = 1), orientational correlation functions were rigorously calculated in the impact theory for both strong and weak collisions [98, 99]. It turns out in the case of weak collisions that the exact solution, which holds for any happens to coincide with what is obtained in Eq. (2.50). Consequently, the accuracy of the perturbation theory is characterized by the difference between Eq. (2.49) and Eq. (2.50), at least in this particular case. The degree of agreement between approximate and exact solutions is readily determined by representing them as a time expansion... 

Fig. 2.9. Comparison of the experiment for CIO3F [94] with theoretical predictions of Te,2( j) dependence. The Hubbard relation is presented by horizontal line (4), the lines with non-zero slopes obtained correspond to the strong collision case y = 0 (1), the weak collision case y = 1 (3) and the intermediate case y = 0.6 (2).

In the case of weak collisions the change in J is so slight that one may proceed from an integral description of the process to a differential one, just as in Eq. (1.23). However, the kernel of the integral equation (3.26) specified in Eq. (3.28) is different from that in the Feller equation. Thus, the standard procedure described in [20] is more complicated and gives different results (see Appendix 3). The final form of the equation obtained in the limit y — 1, to —> 0 with... 

Fig. 3.2. Q-branch transformation with increase of density in strong collision (a) and weak collision (b) approximation at T = 0.1 (I) T = 0.3 (II) T = 10 (III). All spectra are normalized to 1 at their maxima.

The width of this Lorentzian line is half as large as that found in (3.37). This, however, is not a surprise because the perturbation theory equation (3.23) predicted exactly this difference in the width of the line narrowed by strong and weak collisions. This is the maximal difference expected within the framework of impact theory when the Keilson-Storer kernel is used and 0 < y < 1. 

Fig. 3.7. The CARS line width dependence on T = 1 /cqqte at different collision strengths [144] (1) y = 0 (strong collisions) (2) y = 0.4 (3) y =0.7 (4) y = 0.9 (5) y — 1 (weak collisions). The dots denote the perturbation theory result A < = 2coQ/r.

The other cause of trouble in interpretation is far less trivial. The weak collision limit of the Kielson-Storer model implies that y -> 1, to = (ncot )-1 0 but puts no restriction for oj = (1 /nv) lim (1 — y)/%o... 

It should be remembered, however, that a linear relationship between T and n is only valid within the limits of binary impact theory. Its restrictions have already been discussed in connection with Fig. 1.23, where the straight line drawn through zero corresponds to relation (3.46). The latter is acceptable within the whole region of the gas phase up to nearly the critical point. Therefore we used Eq. (3.46) to plot experimental data in Fig. 3.8. The coincidence of maxima in theoretical and experimental dependence Aa)i/2(r) is rather good, as it is achieved by choice of cross-section (3.44), which is the only fitting parameter of the theory. Moreover, within the whole range of the gas phase the experimental widths do not fall outside the narrow corridor of possible values established by the theory. The upper curve corresponds to strong collisions and the lower to the weak collision limit. As follows from (3.23), they differ by a factor... 

At higher pressures only Raman spectroscopy data are available. Because the rotational structure is smoothed, either quantum theory or classical theory may be used. At a mixture pressure above 10 atm the spectra of CO and N2 obtained in [230] were well described classically (Fig. 5.11). For the lowest densities (10-15 amagat) the band contours have a characteristic asymmetric shape. The asymmetry disappears at higher pressures when the contour is sufficiently narrowed. The decrease of width with 1/tj measured in [230] by NMR is closer to the strong collision model in the case of CO and to the weak collision model in the case of N2. This conclusion was confirmed in [215] by presenting the results in universal coordinates of Fig. 5.12. It is also seen that both systems are still far away from the fast modulation (perturbation theory) limit where the upper and lower borders established by alternative models merge into a universal curve independent of collision strength. 

Fig. 5.12. Q-branch narrowing in classical. /-diffusion theory in strong collision (1) and weak collision (2) models [215], The widths are taken from experimental spectra shown in Fig. 5.11 for systems CO-He ( ) and N2-Ar (o).

Fig. 5.15. Theoretical dependences of HWHM on the rate of rotational energy relaxation perturbation theory asymptotics (1), classical weak-collision. /-diffusion model (2), quantum theory without (3) and with (4) adiabatic correction.

Weak collisions are quite different. According to Appendix 4, the kinetic equation corresponding to such collisions has the form ... 

The lower boundary corresponds to strong collisions, and the upper one to weak collisions. This conclusion can be confirmed by experiment. According to [259], nitrogen dissolved in SF6 has a symmetrical spectrum of isotropic scattering, indicating that collapse of the spectrum has already occurred. At the same densities, the Q-branch of the anisotropic spectrum is still well separated from the side branches, and in [259] the lower bound for its half-width is estimated as 5 cm-1. So,... 

In fact, such a method was proposed by Sack in the classical work [99], which was far ahead of its time. This method provides the general solution of Eq. (6.4) in the form of a continuous fraction, which is, however, rather difficult to analyse. In the case of weak collisions, there is no good alternative to this method, but for strong collisions, the solution can be found analytically. Let us first consider this case. 

The impact theory defines uniquely the spectral transformation in the limit of weak collisions. Expanding in a series over J — J the integrand of Eq. (6.4) one can obtain at T = /coq%j 1... 

Kluk was the first to obtain these results [269]. They differ drastically from (6.25). When the medium becomes more rarefied, intensity in the central part of the IR spectrum decreases to zero. Hence, tJ j shortens, unlike t, which lengthens. The time t e i behaves in the same manner for the case of weak collisions, though a formula quantitatively analogous to (6.63) is not found for this case. One can refer only to numerical calculations based on the general formulae by Sack or by Fixman and Rider. These calculations provide identical results [85]. Fig. 6.5 shows that, in rarefied media, the difference between weak and strong collision... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

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