Collision integral inelastic

Again using Eqs. (4.65) and (4.67) we may write the collision integral for the first process of (4.80), corresponding to an inelastic scattering between an atom and free particles. We then have... 

VI. Pair Recombination—A Stochastic Approach A. Generalized Encounter Theory An Inelastic Collision Integral Atomic and Molecular Recombination Dynamics... 

The results are plotted in Fig. 19. The values of Eqs. (6.9) and (6.8) differ by only 6% over the entire range that QJj is defined, so that the inelastic collision integral has an overall T dependence. A comparison in Fig. 19 shows that significant differences are observed between QJj and at low tempera-... 

Figure 19. Temperature dependence of inelastic collision integral for attractive limb of Lennard-...

For elastic collisions, several different kinetic models have been proposed in order to close the Boltzmann hard-sphere collision term (Eq. 6.9). For inelastic collisions (e < 1), one must correctly account for the dependence of the dissipation of granular energy on the value of e. One method for accomplishing this task is to start from the exact (unclosed) collision integral in Eq. (6.68). From the definition of if given in Eq. (6.60), it can be... 

Here -Bq denotes the charge of the electrons, E(x, t) is the electric field, and C (F) and C[ (F) are the collision integrals for elastic collisions and important conservative inelastic collisions, i.e., the /th excitation or dissociation process of the electrons in collisions with the ground-state atoms or molecules of the gas. For simplicity in the further representation, only the most essential electron... 

The techniques discussed in Sects. 8.2-S.4 allowed the measurement of absolute rate constants of selected collision-induced transitions, which represent integral inelastic cross sections integrated over the angular distribution and averaged over the thermal velocity distribution of the collision partners. Much more detailed information on the interaction potential can be extracted from measured differential cross sections, obtained in crossed molecular beam experiments [958, 1074, 1075]. 

Fxmctions Adj, Bdj, Fdj and Gdj satisfy the linear integral equations with the linearized operators of elastic collisions and inelastic ones with internal energy transitions. 

For inelastic collisions the Q integrals cannot yet be calculated from first principles. The procedure adopted with polyatomic gases is therefore to treat them as if the collisions were elastic, that is, to use tables of elastic-collision integrals and to derive effective values of eps)ij/kg and (si)y from the temperature variation of the transport properties. The first approximations to the binary diffusion coefficients l ij] 1, as well as and (1.2C — 1), may... 

For electronic transitions in electron-atom and heavy-particle collisions at high unpact energies, the major contribution to inelastic cross sections arises from scattering in the forward direction. The trajectories implicit in the action phases and set of coupled equations can be taken as rectilinear. The integral representation... 

The application of HRTEM and ED can be difficult, however, because of the tendency of the catalysts to become amorphous under the beam. Use of higher accelerating voltages (to minimize the production of inelastic collisions which are primarily responsible for the structural disintegration of the sample), better vacuum in the sample chamber and, when possible, dealumination of the aluminosilicate (by replacing AP+ with in the framework) without losing structural integrity avoids these problems. These studies have indicated that Al-O bonds are more susceptible to rupture under electron-beam irradiation than Si-O bonds. 

Eq. (11.30) is strictly applicable only to elastic collisions, in which a = a, and is thus of limited utility. However, it is physically appealing to assume that the cross section o(a, a, / ,/ ) for an inelastic process a = a and ft (i can be written as the integral of the electron scattering cross section oe(J3, f, q) over the velocity distribution of the Rydberg electron in the initial state a. Making this notion explicit, we write19... 

Here. .. depends on the W-kernels introduced above, but in general stands for terms of the form cross-section, appropriately weighted by factors [1 — cosl(0)], l = 1, 2,. .., if the efficiency of exchange of quantity A in a collision depends on the scattering angle 0 in this way (as, e.g., in case of elastic neutral particle - ion collisions, see Sect. 2.2.4). In case of inelastic collision processes a is simply the total cross-section, denoted weighting exponent l = 1, 2,... is possible) or a1. In principle the detector functions qa must be obtained by numerical integration and tabulated for the parameters of the relevant distribution functions fa. 

Figure 3. Integral cross-sections for rotationaUy inelastic collisions between two para-tl2 in their...

To calculate the integrals defining the source term and the flux term, appropriate expressions for Aip and ip[ — ipi have to be determined from an analysis of the inelastic binary particle collision dynamics. 

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