The Strong Collision Model

In the strong collision limit, the time taken for the transition from Cl to Cl is negligible in comparison with the residence time, r, spent by the molecule 

It can easily be shown that the symmetrized transition operator Q) 

Introducing K m n), the mean square average of the Wigner rotation matrices, 

The kappa values are readily evaluated using the Clebsch-Gordon expansion [7.14] and are given below in terms of D q = (P2) and D q = (P4), the ensemble average of the fourth-rank Legendre polynomial. 

Because of the simplicity of the model, it has been employed to describe [7.15] molecular reorientation of the nematogen 5CB. 

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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. 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).

An interesting question is whether the large fluctuations in the quantum mechanical decay rates have an influence on the temperature and pressure dependent unimolecular rate constant P) defined within the strong collision model, in Eq. (2). In the state-specific quantum mechanical approach the integral over the smooth temperature dependent rate k E) is replaced by a sum over the state-specific rates fc,-. Applications have been done for HCO [93], HO2 [94-96], and HOCl [97]. The effect of a broad distribution of widths is to decrease the observed pressure dependent rate constant as compared to the delta function-like distribution, assumed by statistical theories [98,99]. The reason is that broad distributions favor small decay rates and the overall dissociation slows down. This trend, pronounced in the fall-of region, was clearly seen in a study of thermal rate constants in the unimolecular dissociation of HOCl [97]. The extremely... 

Fig. 21. Zero-field dynamic Gaussian Kubo-Toyabe functions for different field fluctuation times. The curves are labeled by the value of t A (in rad). The calculation used the strong-collision model. In the Gaussian-Markovian approach G t) decays minimally slower.

Third, we must know something about the collisional transition rates between the grains, and many models have been explored in the past, particularly those known as the step-ladder and the exponential models [72.R 73.F 77.T1] I will confine myself almost exclusively in this discourse to the strong collision model, equation (2.26), and for the time being, at least, the apparent rate constant p for the relaxation of the total internal energy can be regarded as an adjustable parameter. 

Thus, unlike all model calculations dted in Section 3.3, the strong collision model gives the result that all states below threshold are in... 

Thus, the Debye model for isotropic rotation behaves like the strong collision model. 

Now the result given in Eq. (7.54) is very similar to the corresponding expression in the strong collision model given by Eq. (7.30) except the correlation times r rnM( = different for different... 

The small step rotational diffusion model has been extensively applied to interpret ESR linewidth [7.4, 7.9], dielectric relaxation [7.2], fluorescence depolarization [7.19], infrared and Raman band shapes [7.24], as well as NMR relaxation in liquid crystals [7.14, 7.25]. When dealing with internal rotations in flexible mesogens, they are often assumed to be uncoupled from reorientation to give the so-called superimposed rotations model. Either the strong collision model or the small step rotational diffusion model may be used to describe [7.26, 7.27] molecular reorientation. 

The strong collision model has been used in several studies to interpret spectral densities of motion 5CB-di5 [7.15], 50.7-d4 [7.7], and discotic hexa-hexyloxytriphenylene (THE6) [7.46]. The spectral densities can be obtained from Eq. (7.30). In the fast motion limit,... 

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