Binary collision approximation model

The dynamics of ion surface scattering at energies exceeding several hundred electronvolts can be described by a series of binary collision approximations (BCAs) in which only the interaction of one energetic particle with a solid atom is considered at a time [25]. This model is reasonable because the interaction time for the collision is short compared witii the period of phonon frequencies in solids, and the interaction distance is shorter tlian the interatomic distances in solids. The BCA simplifies the many-body interactions between a projectile and solid atoms to a series of two-body collisions of the projectile and individual solid atoms. This can be described with results from the well known two-body central force problem [26]. 

The essence of Monte-Carlo models is to calculate the path of an ion as it penetrates a crystal. Early versions of these models used the binary collision approximation, i.e., they only treated collisions with one atom at a time. Careful estimates have shown that this is an accurate procedure for collisions with a single row of atoms (Andersen and Feldman, 1970). However, when the rows are assembled into a crystal the combined potentials of many neighboring atomic rows affect ion trajectories near the center of a channel. For this reason, the more sophisticated models used currently (Barrett, 1971, 1990 Smulders and Boerma, 1987) handle collisions with far-away atoms using the continuum string approximation,... 

Robinson and Torrens have discussed the limitation of the binary-collision approximation. They concluded, for example, that it was likely to fail at energies below 9 eV for collisions between copper atoms in a copper lattice and below 33 eV for gold atoms in a gold lattice. Therefore, phenomena which depend sensitively upon motions of particles with very low energy are likely to be described only qualitatively by the binary collision model. 

The reflection of carbon atoms and ions at the limiter or divertor surfaces in the ERO modeling is determined by TRIM [29], However, the binary collision approximation used in the TRIM code is no longer valid at small energies of the incoming particles where chemical effects start to influence the interaction of the particles with the solid. To take this into account reflection coefficients calculated with a molecular dynamic code MolDyn [55] were implemented. 

The results of detailed gas/surface scattering experiments reveal the precise momentum transferred to a surface by an incident gas particle. Relatively simple models, such as the binary collision approximation, provide... 

It is possible to show for many collision models and even for the simple model of linear trajectories that Ivh(co) will have an exponential line shape (to within a factor that is some power of the frequency) that is, Ivh(co) — he mU with A oc Thus the observed exponential wing seems to be a consequence of the binary collision approximation. 

Since the introduction of the isolated binary collision model there has been considerable controversy over its applicability. A number of theoretical papers have challenged its basic assumptions and proposed corrections due to collective effects (usually within the weak coupling approximation of Section II.B), while others have supported and extended the model. In this section, we outline the development of the controversy over the binary collision approximation, which is not resolved even today. 

The binary collision approximation (BCA) model was the first to be used in computer simulations of ion-solid interactions (Bredov et al, 1958). The usefulness of computer simulations was further demonstrated by Robinson and Oen (1963) during their discovery of the channeling effect. Computer simulations based on the BCA model in essence fall into two categories, those that assume a crystalline structure of the solid and those that, as in calculations based on the TRIM code, assume a randomized or structureless target. 

Even though the binary collision model is very useful, it is still an approximation to the real situation and a more detailed understanding probably requires computer simulation studies such as those pioneered by Harrison . 

As discussed in Chapter 2, the one-particle NDF does not usually provide a complete description of the microscale system. For example, a microscale system containing N particles would be completely described by an A-particle NDF. This is because the mesoscale variable in any one particle can, in principle, be influenced by the mesoscale variables in all N particles. Or, in other words, the N sets of mesoscale variables can be correlated with each other. For example, a system of particles interacting through binary collisions exhibits correlations between the velocities of the two particles before and after a collision. Thus, the time evolution of the one-particle NDF for velocity will involve the two-particle NDF due to the collisions. In the mesoscale modeling approach, the primary physical modeling step involves the approximation of the A-particle NDF (i.e. the exact microscale model) by a functional of the one-particle NDF. A typical example is the closure of the colli-sionterm (see Chapter 6) by approximating the two-particle NDF by the product of two one-particle NDFs. 

The first pubUshed criticism of the binary collision model was due to Fixman he retained the approximation that the relaxation rate is the product of a collision rate and a transition probabihty, but argued that the transition probability should be density dependent due to the interactions of the colliding pair with surrounding molecules. He took the force on the relaxing molecule to be the sum of the force from the neighbor with which it is undergoing a hard binary collision, and a random force mA t). This latter force was taken to be the random force of Brownian motion theory, with a delta-function time correlation ... 

The steric factor, p, of equation (2-20) can be estimated directly from TST using either exact or approximate methods. This is satisfying, since in the manner in which it was introduced p seemed to be yet another adjustable parameter. In general the steric factor is given by the ratio of the pre-exponential factor of the reaction in question to that for the binary collision model ... 

To determine if there is a correlation between the observed local density enhancements in the model system and the computed relaxation lifetimes, we use isolated binary collision theory [81-83], which, although approximate, has some empirically established validity [75,82,84], to provide a framework for our analysis [12]. Specifically, the isolated binary collision theory yields an expression for the vibrational relaxation rate. 

Most theories of the polymeric nematic phase are based upon the model in which the liquid crystal is considered as a packed system interacting through its hard-core diameters. Historically, attention was first focused on lyotropic liquid crystals. The binary collision or second virial approximation is taken into account at low volume factions. A slightly different approach is to use a scaling law to describe the fluctuation of nematic order parameters. In this section we select three different methods to represent the three different treatments. 

Dilute gas approximation requires the average distance between the molecules to be an order of magnitude larger than flieir diameter, a, that is, - 1. This almost guarantees that all collisions between molecules are binary collisions, avoiding the complexity of modeling multiple encounters. Dissociation and ionization phenomena involve triple collisions. 

Storer model used in this theory enables us to describe classically the spectral collapse of the Q-branch for any strength of collisions. The theory generates the canonical relation between the width of the Raman spectrum and the rate of rotational relaxation measured by NMR or acoustic methods. At medium pressures the impact theory overlaps with the non-model perturbation theory which extends the relation to the region where the binary approximation is invalid. The employment of this relation has become a routine procedure which puts in order numerous experimental data from different methods. At low densities it permits us to estimate, roughly, the strength of collisions. 

Criteria for choosing specific models. We have seen above that spectral profiles of binary complexes can be computed from a rigorous quantum formalism. These describe the measurements well. Long profiles, i.e., profiles with a peak-to-wing intensity of several orders of magnitude, can be readily obtained. As an illustration of how well the various available model profiles approximate the exact computations, we show results for one basic profile type characteristic of absorption by H2-H2 pairs [69], a pure quadrupole-induced profile which accounts for roughly 90% of the total intensity of the rototranslational H2-H2 collision-induced absorption spectra at 77 K. 

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