Collision density distribution function

If three-body collisions are neglected, which is permitted at sufficiently low densities, all the interactions take place between pairs of particles the two-particle distribution function will, therefore, satisfy Liouville s equation for two interacting particles. For /<2)(f + s) we may write Eq. (1-121) ... 

The polymerization sites at the molecule s exterior are likely to experience more fruitful collisions than sites on the interior. To account for such complex phenomena, an additional separable function was introduced to simulate steric hindrance factors at the molecular level. An arbitrary, but continuous, exponential function was selected. The resulting rate constant that yields a good fit of experimental population density distributions is... 

For high densities, g cannot be set equal to one, and the collision function becomes much more complex and so is not given here. It turns out, however, that instead of using the full radial distribution function, it is sufficient to use the value at contact r — R, so that we define a new function ... 

Results. The theory of ternary processes in collision-induced absorption was pioneered by van Kranendonk [402, 400]. He has pointed out the strong cancellations of the contributions arising from the density-dependent part of the pair distribution function (the intermolecular force effect ) and the destructive interference effect of three-body complexes ( cancellation effect ) that leads to a certain feebleness of the theoretical estimates of ternary effects. 

We should hasten to note that these fundamental difficulties do not mean that this theory does not often work. The most common application of IBC theory points to its particularly simple prediction for the dependence of relaxation rates on the thermodynamic state of the solvent with the Enskog estimate of collision rates, the ratio of vibrational relaxation rates at two different liquid densities p and p2 is just the ratio of the local solvent densities [pigi(R)//02g2(R)], where g(r) is the solute-solvent radial distribution function and R defines the solute-solvent distance at... 

For some variables, for example, the relative collision velocity, the cumulative distribution function does not have closed form, and then a third Monte Carlo method must be adopted. Here, another random number R is used to provide a value of v, but a decision on whether to accept this value is made on the outcome of a game of chance against a second random number. The probability that a value is accepted is proportional to the probability density in the statistical distribution at that value. The procedure is repeated until the game of chance is won, and the successful value of v is then incorporated into the set of starting parameters. 

The kd (v) rate coefficients have been obtained by using the cross sections of Fig. 12 and the non-Maxwellian electron distribution functions of Fig. 13. The edfs have been obtained by a numerical solution of the Boltzmann equation (BE) which includes the superelastic vibrational collisions involving the first three vibrational levels, and the dissociation process from all vibrational levels (see Ref.9) for details). The vibrational population densities inserted in the BE are self-consistent with the quasi-stationary values reported in Figs. 8 and 10. It should be noted that the DEM rates (Fig. 14) depend on E/N as well as on the vibrational non equilibrium present in the discharge, which affects the electron distribution functions, as discussed in Sect. 2.1. 

The radial distribution function implicitly takes account of the enhancement of the number density in the neighborhood of the collision sphere due to the excluded volume of the other molecules making up the system. 

Specified electron energy distribution function The EEDF is specified, normally assumed Maxwellian (Eq. 9). The electron energy balance (Eq. 31) is solved assuming an adiabatic condition for electron temperature at the wall. The Maxwellian assumption is very common in the literature [100, 125, 126, 130, 133, 135-137]. Measured EEDFs in ICPs, however, have a Maxwellian bulk (due to electron-electron collisions), and a depleted tail due to inelastic losses and escape of fast electrons to the walls. Thus a bi-Maxwellian distribution may be more appropriate [154]. A Maxwellian distribution is not expected to have a great effect on ion densities since the ionization rate is self-adjusted to balance the loss rate of ions to the walls and the latter depends only very weakly on the EEDF. The good agreement with experimental data [101, 130, 148, 152] is an indirect evidence that the Maxwellian EEDF is reasonable for obtaining species densities and their distributions. Other forms of... 

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