Ion Implantation and Synthesis of Materials - Studium

24 3 Dynamics of Binary Elastic CollisionsThe simplest collision event is the collision between a charged particle and theatomic nucleus. This can be treated as a two-body collision provided that the meanfree path between collisions is much greater than the interatomic spacing. Thechance of correlated effects, due to neighboring atoms recoiling simultaneously, isthen very small. The momentum of the recoiling target atoms is the parameter thatdetermines the amount of damage that occurs in the solid target. The momentumtransferred to the recoiling atom also is responsible for a large portion of theenergy-loss process of the ion.In developing our understanding of ion–solid interactions for the purposes ofion beam modification of materials, we will first derive some general relationsgoverning two-body collisions, considering only the asymptotic values of mom-entum at great distances from the collision. The principles of conservation ofmomentum and energy are all that are required to obtain the recoil energy as afunction of recoil angle. We shall assume that collisions are elastic and, further,that velocities are small enough for nonrelativistic mechanics to apply.3.2 Classical Scattering TheoryThe following assumptions are usually made in the description of the scatteringprocesses between energetic particles in solids:(a) Only two-atom collisions are considered;(b) Classical dynamics is applied;(c) Excitation or ionization of electrons only enters as a source of energy-loss,but does not influence the collision dynamics;(d) One of the two colliding atoms initially is at rest.Assumption (a) is appropriate for violent collisions. Violent collisions betweenatoms of reasonably high energy range (keV) require the collision partners toapproach very closely, so that the probability of a collision between three or moreatoms is small. Soft collisions can take place at large distances and therefore caninvolve more than two atoms simultaneously. However, soft collisions usually canbe treated by perturbation theory (the momentum or impulse approximation), inwhich case no restriction to binary collisions is necessary. At lower energies(below 1 keV), collective effects become increasingly important and assumption(a) starts to break down. However, the problems associated with many-bodycollisions in this low-energy regime can be overcome by molecular-dynamicsimulations, where assumption (a) is not required.In the limit of assumption (b), the applicability of classical mechanics isnormally limited to specific quantities, one of which is the differential scatteringcross-section, dσ (θ c ), where θ c is the center-of-mass (CM) scattering angle.Neglecting the effect of electronic excitation on the collision dynamics,assumption (c), is justified if either the energy transferred to electrons is smallcompared with the exchange of kinetic energy between the atoms (so that the