Close-collision model

Unlike V(R the effective potential shows a maximum at R for each 6, the so-called centrifugal barrier Vz = Eb /Rl,. With the assumption that reaction occurs if E exceeds the centrifugal barrier, the maximum impact parameter b ax ch energy E and hence the reaction cross section (j(E) = nb j, may be calculated. One obtains for this close-collision model (Levine and Bernstein, 1974)... 

Figure 5. (a) Variation of potential energy in collisions between hard spheres with attraction, (b) Corresponding cross section for the close-collision model. (- - -) indicates the interaction appropriate for the hard-sphere, energy invariant cross section nd. ... 

I like to emphasize that Fig. 1 is not meant to indicate any fundamental limitation of quantum mechanics both Bohr s and Bethe s formulae invoke mathematical approximations to the underlying physical models, and Bethe s formula in particular relies on first-order perturbation theory for both distant and close collisions. 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

Owing to the presence of the pair correlation function, the collision model in Eq. (6.2) is unclosed. Thus, in order to close the kinetic equation (Eq. 6.1), we must provide a closure for written in terms of /. The simplest closure is the Boltzmann Stofizahlansatz (Boltzmann, 1872) ... 

Second, the SF (254) is used in the case of the Gross collision model as a constitutive element of the formula for the complex susceptibility Xg I 1 this case the orientational distribution Fq, differing from FB, changes radically the calculated low-frequency spectrum, while the far-IR spectrum is very close to that given by the Boltzmann susceptibility Xb (252). We shall return to this point in Section IX.D. 

Before closing this section, we should remark that although this analysis of velocity relaxation effects has focused on a simple collision model, we expect that the detailed structure of the rate kernel for short times will depend on the precise form of the chemical interactions in the system under consideration. It is clear, however, that a number of fundamental questions need to be answered before more specific calculations can be undertaken form the kinetic theory point of view. 

We consider first the question of collision mechanism first, that the calculation of a close-collision cross section places no mechanistic constraints on the ensuing reaction and second, the important corollary that mechanistic statements may not be deduced from the shape of excitation functions. Next, the validity of the potential is examined,f this necessarily setting an upper bound to the energy at which the model may be applied. Various applications of the model are then considered and its success evaluated. Finally, the various explanations which have been advanced to rationalize its failure are discussed. 

The extensive use by Dugan and Magee of trajectory calculations to compute close-collision cross sections for the collision of ions with polar molecules has been reviewed in Section 4.2.2d. For such calculations, a form for only the attractive part of the potential need be assumed and, in this case, a particular value of the ion-molecule separation was used to define a close collision. The form chosen for the potential was the simple, anisotropic, electrostatic ion-dipole potential plus the ion-induced-dipole potential and, for reasons discussed in that section, such a model may only be applied to ion-molecule collisions at thermal energies. 

While the potential selected for this very simple model is appropriately the simplest, it should be remembered, from the discussion in Section 4.2.1, that it is quite unrealistic, as evidenced by the discontinuity discussed above. Moreover, it should be remembered that, even at those low collision energies when the electrostatic potential is a reasonable approximation for the calculation of close-collision cross sections, the ion-quadrupole potential plays a significant role, the resultant potential exhibiting dramatic differences for the various M,J states of the deuterium molecule rotor. While it would be interesting to explore the effect of employing a more realistic potential in this calculation, such refinements are contrary to the spirit of the model, which enquires how adequate the simplest possible model may be. 

The potential is constructed using a model which considers the H2 as an atomic ion and the H2 as an atom, to give two potential curves, one attractive and one repulsive (corresponding to the and states of H2 for example). The treatment amounts to approximating H4 hypersurfaces by effective two-body potentials. It is by no means clear to the present author that there can be a repulsive hypersurface of H4 which correlates with the H2- and H2 reactants in their ground electronic states. If there is not, the contributions to chemical reaction from this repulsive state is a fiction. If there is, this has profound implications for the calculation of close-collision cross sections for such systems using models for the interparticle potential (see Section 4.2.1c). 

Since it is not possible in this simple model to calculate the relative phase of a, and a , the only information on the magnitude of the interference term that can be obtained is the largest possible value of Since only processes occurring at the same impact parameter may interfere, it is expected that the interference term stems from the two close-collision processes TS-1 and TS-2. Therefore,... 

The success of this semiempirical model further supports the validity of our physical picture of the double ionization of He for impact of fast projectiles of low charge. Double ionization stems from the SO process, which gives its main contribution in distant collisions, and from the TS-1 and TS-2 processes which are the important mechanisms for close collisions. Furthermore, the observed difference between for projectiles of opposite charge stems from an interference between the two close collision processes TS-1 and TS-2. 

For expUcit results we need, however, to adopt specific models. One such model is that of rebound reactions. Here one assumes tiiat reaction only occurs on close collisions when the reactants are subject to the short-range repulsive part of the intermolecular potential. The rearrangement thus takes place at close quarters and the newly formed products recede under the influence of the short-range repulsion. Hence, the net deflection is that typical of hard-sphere scattering. 

RBS is based on collisions between atomic nuclei and derives its name from Lord Ernest Rutherford who first presented the concept of atoms having nuclei. When a sample is bombarded with a beam of high-energy particles, the vast majority of particles are implanted into the material and do not escape. This is because the diameter of an atomic nucleus is on the order of 10 A while the spacing between nuclei is on the order of 1 A. A small fraction of the incident particles do undergo a direct collision with a nucleus of one of the atoms in the upper few pm of the sample. This collision actually is due to the Coulombic force present between two nuclei in close proximity to each other, but can be modeled as an elastic collision using classical physics. 

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. 

The closer two particles pass, the greater is their interaction. Still, AJ/J may turn out to be less than 1 even in the case of face to face collision. In this limit collisions are weak, y 1 and the model of the correlated process fits the situation well. If close impacts produce a strong effect, then the influence of more distant paths is negligible, and the process approaches the non-correlated limit y 0. 

In the gas phase, the reaction of O- with NH3 and hydrocarbons occurs with a collision frequency close to unity.43 Steady-state conditions for both NH3(s) and C5- ) were assumed and the transient electrophilic species O 5- the oxidant, the oxide 02 (a) species poisoning the reaction.44 The estimate of the surface lifetime of the 0 (s) species was 10 8 s under the reaction conditions of 298 K and low pressure ( 10 r Torr). The kinetic model used was subsequently examined more quantitatively by computer modelling the kinetics and solving the relevant differential equations describing the above... 

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