Strong collisions

Troe J 1977 Theory of thermal unimolecular reactions at low pressures. II. Strong collision rate constants. Applications J. Chem. Phys. 66 4758... 

Troe J 1983 Theory of thermal unimolecular reactions in the fall-off range. I. Strong collision rate constants Ber. Bunsenges. Phys. Chem. 87 161-9... 

The density matrix p describes the pure state, as seen from the equality p = p, while p does not. The transition from (2.35a) to (2.35b) describes a strong collision , which fully localizes the particle, but in general the off-diagonal elements may not completely vanish. This however does not affect the qualitative picture. 

After each strong collision the system, having been localised in the left or right well, resumes free tunneling from the diagonal state. Thus, after N collisions the probability to survive in the left well is... 

The proposed specification of the kernel for m- and J-diffusion models is mathematically closed, physically clear and of quite general character. In particular, it takes into consideration that any collisions may be of arbitrary strength. The conventional m-diffusion model considers only strong collisions (0(a) = 1 /(27c)), while J-diffusion considers either strong (y = 0) or weak (y = 1) collisions. Of course, the particular type of kernel used in (1.6) restricts the problem somewhat, but it does allow us to consider kernels with arbitrary y < 1. 

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],... 

Though these are alternative models, they are both particular cases of the non-adiabatic impact theory of angular momentum relaxation in gases. Thus, we prefer to call them models of weak and strong collisions , as is usually done in analogous problems [13, 33],... 

Strictly speaking, the process of J-diffusion described by the above equation is not diffusion at all. The very first collision restores equilibrium in the whole J-space. In this sense, strong collisions represent the hopping mechanism of J-relaxation, which is the only alternative to the diffusion mechanism [20, 24, 25, 36]. Since the term J-diffusion is so pervasive, we do not like to reject it. However, it should be understood in a wider sense and used to denote J-migration . Then it remains valid for both weak and strong collisions. Still, it should be remembered that there is a considerable difference between these limits. For strong collision we obtain from Eq. (1.30)... 

For strong collisions (y = 0), is still equal to //to, but qj does not exist and the diffusion mechanism of rotational relaxation is replaced by a hopping mechanism. 

In NMR theory the analogue of the relation (1.57) connects the times of longitudinal (Ti) and transverse (T2) relaxation [39]. In the case of weak non-adiabatic interaction with a medium it turns out that T = Ti/2. This also happens in a harmonic oscillator [40, 41] and in any two-level system. However, if the system is perturbed by strong collisions then Ti = T2 as for y=0 [42], Thus in non-adiabatic theory these times differ by not more than a factor 2 regardless of the type of system, or the type of perturbation, which may be either impact or a continuous process. 

In the case of strong collisions, corresponding to y = 0, inequality (1.88) is reduced to a conventional validity criterion of the binary theory (1.59). However, if collisions are weak (y 1), the actual criterion given in Eq. (1.88) is considerably weakened and Langevin phenomenology is valid at larger densities. 

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).

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.

Although from a mathematical point of view formulae (3.34) and (3.40) have little in common, the spectral transformation described by them proceeds in a similar way (Fig. 3.2). Just as with strong collisions, the contour is gradually symmetrized and its centre is shifted to the average frequency coq with an increase in the density. When the spectrum is narrowed (at T 1), its central part ( Aco] < 1/tj) takes the following form ... 

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.

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).

The simple fitting procedure is especially useful in the case of sophisticated nonlinear spectroscopy such as time domain CARS [238]. The very rough though popular strong collision model is often used in an attempt to reproduce the shape of pulse response in CARS [239]. Even if it is successful, information obtained in this way is not useful. When the fitting law is used instead, both the finite strength of collisions and their adiabaticity are properly taken into account. A comparison of... 

Firstly, we are going to demonstrate how branch interference may be taken into account within the quasi-classical impact theory. Then we shall analyse a quasi-static case, when the exchange frequency between branches is relatively small. An alternative case, when exchange is intensive and the spectrum collapses, has been already considered in Chapter 2. Now it will be shown how the quasi-static spectrum narrows with intensification of exchange. The models of weak and strong collisions will be compared with each other and with experimental data. Finally, the mutual agreement of various theoretical approaches to the problem will be considered. 

In the case of strong collisions, the integral part of Eq. (6.14) becomes even simpler and has the form... 

Since xe,2 oc tj, it is analogous to the Hubbard relation however, it is not universal. The numerical coefficient provides information on the strength of collisions the value 1/3 is peculiar to strong collisions. 

As can be seen, the difference in behaviour of orientational relaxation times Te,2 in models of weak and strong collisions is manifested more strongly than in the case of isotropic scattering. Relation (6.26) is... 

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. 

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... 

The behaviour of te,2 (tj) is qualitatively different. In the dense media this dependence also satisfies the Hubbard relation (6.64), and in logarithmic coordinates of Fig. 6.6 it is rectilinear. As t increases, it passes through the minimum and becomes linear again when results (6.25) and (6.34) hold, correspondingly, for weak and strong collisions ... 

Fig. 6.7. The first-order (curve 1), second-order (curve 2) and third-order (curve 3) approximations to the exact dependence x x ) in the strong collision model (curve 4).

Even without using numerical methods, one can analyse some physically sound limiting cases of the exact solution for the case of strong collisions (7.64). First of all, (7.64) evidently reduces to the results of Robert and Galatry in the quasi-static case, i.e. when tc — oo. An opposite limiting case of fast fluctuations... 

Kolomoitsev D. V., Nikitin S. Yu. Analysis of experimental data on nonstationary active spectroscopy of molecular nitrogen in the strong-collision approximation, Opt. Spectr. 66, 165-8 (1989) [Optika i Spectr. 66, 286-93 (1989)]. 

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