Binary collision approximation

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]. 

Moreover, since the mean free path is of the order of 100 times the molecular diameter, i.e., the range of force for a collision, collisions involving three or more particles are sufficiently rare to be neglected. This binary collision assumption (as well as the molecular chaos assumption) becomes better as the number density of the gas is decreased. Since these assumptions are increasingly valid as the particles spend a larger percentage of time out of the influence of another particle, one may expect that ideal gas behavior may be closely related to the consequences of the Boltzmann equation. This will be seen to be correct in the results of the approximation schemes used to solve the equation. 

This binary collision approximation thus gives rise to a two-particle distribution function whose velocities change, due to the two-body force F12 in the time interval s, according to Newton s law, and whose positions change by the appropriate increments due to the particles velocities. 

In impact theory the gas density is restricted by the criterion for the validity of the binary collision approximation. Roughly speaking, the time between collisions must be greater than their duration... 

Fig. 5.19. Experimental line width and calculated line widths predicted by the fitting laws in binary collision approximation [251] (o) experimental (+) PEG (A) ECS-EP ( ) MEG ( ) ECS-P.

Bessel function 40, 99, 201, 264 binary approximation 7, 41 binary collisions adiabatic/non-adiabatic 4 angular momentum, computer simulations 40... 

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,... 

Lulli, G., et al., The Monte Carlo Binary Collision Approximation Applied to the Simulation of the Ion Implantation Process in Single-Crystal SiC High-Dose Effects, Mater. Sci. Forum, Vol. 335-356, 2001, pp. 599-602. 

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. 

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 . 

It should be noted that although in Eq. (90) only the connected motion of the solute and the solvent is retained, in the argument presented on the time scale it is the disconnected parts which have been considered. This is because in the latter part, for the derivation of the expression of Ci. the solute and the solvent motions are assumed to be disconnected. This assumption is the same as those made in the density functional theory and also in mode coupling theories where a four-point correlation function is approximated as the product of two two-point correlation functions. This approximation when incorporated in Ci. means that after the binary collision takes place, the disturbances in the medium will propagate independently. A more exact calculation would be to consider the whole four-point correlation function, thus considering the dynamics of the solute and the solvent to be correlated even after the binary collision is over. Such a calculation is quite cumbersome and has not been performed yet. 

As mentioned before, the recollision term begins as I4, and thus the binary collision term contains all the contributions to order f. Also only even powers in t appear. Thus (B can be assumed to be given by Gaussian approximation and can be written as... 

B. Binary Collision Approximation for the Two-Particle Density Operator— Kinetic Equations for Free Particles and Atoms... 

The second problem is the decoupling of the chain of equations for the formal solutions. It is necessary to make approximations in order to truncate this chain. The simplest approximation is the binary collision approximation. That means, we neglect three-particle collisions. Then, we obtain... 

Let us discuss two approximations for the integral term. The first approximation is a more exact form of the binary collision approximation. Up to now this approximation has the form (2.11). However, this approximation is not quite consistent. Indeed we can show that the integral term of (2.34) contains terms that contribute to the binary collisions. Since V13 is different from zero only if the particles 1 and 3 are in finite distance, in binary collision approximations for the second term in (2.34) can be found ... 

This generalized binary collision approximation is discussed more in detail by Klimontovich.12... 

In order to analyze the retardation problem we consider the formal solution (1.30) for Fx. Using the binary collision approximation for F12, as before, the t integration can be carried out. The result is... 

Equation (2.57) represents the negative time derivative of the mean value of the potential energy in the approximation of the second virial coefficient (binary collision approximation). Therefore, we have from (2.44) and (2.57)... 

Let us first consider the effect of bound states in the binary collision approximation. In this approximation Fn satisfies the equation... 

Now we will introduce the distribution function of the atoms. Obviously, this function is related to the average of pafl(PP). In the spatially homogeneous system this quantity is diagonal (in the binary collision approximation)... 

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... 

More recently Morse produced a complete microscopic tube theory for stiff polymers that successfully interpolates between the rigid-rod and flexible chain limits. This theory explains many features of semiflexible polymer rheology, including the two mechanisms for plateau moduli described above (which depend on a comparison of timescales), with the tube diameter being the sole fitting parameter as in the Doi-Edwards theory. More recently, Morse successfully computed a tube diameter from two different approaches (self-consistent binary collision and continuum effective medium) that give similar results, e.g. modulus G p and respectively). An elastic network approximation... 

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