Kinematics of Elastic Collisions

The assumption of one collision partner being at rest initially, assumption (d), has been made in all previous work except molecular-dynamics computations. It is not fulfilled in very dense collision cascades, especially when the process of energy dissipation has proceeded to the point where most of the atoms in the cascade are in motion. 

Conservation of energy and conservation of momentum parallel and perpendicular to the direction of incidence are expressed by the equations 

These three equations, (3.1)—(3.3), can be solved in various forms (Nastasi et al. 1996). 

Ec Total kinetic energy in the center-of-mass system 

Velocity of the incident projectile in laboratory coordinates Velocity of the scattered projectile in laboratory coordinates Velocity of the recoiling atom in laboratory coordinates Velocity of the reduced mass in center-of-mass coordinates Velocity of the incident projectile (ion) in center-of-mass coordinates Velocity of the target atom in center-of-mass coordinates Laboratory angle of the scattered projectile Center-of-mass angle of the scattered projectile Maximum laboratory angle forATi scattering Mi M2) 

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Moreover, calculations are usually done in the center of mass frame, whereas the measurements are done in the laboratory frame [35]. Therefore, to calculate the kinematics of elastic collisions, it is often convenient to utilize both frames sequentially determining the scattering angles. That is, it is normally a better strategy to transform the problem to the center of mass frame, examine the kinetics of the collision and then transfer the result back to the laboratory frame, than to work directly in the laboratory frame. In particular, this procedure enables us to link the scattering angles in the laboratory frame 0L and the center of mass frame 0. 

Figure 18. Energy-transfer spectrum for Na +N2. Energy transfer A vlb ro, is measured in units of vibrational quanta v after collision. Shaded area indicates strong superposition of elastic scattering processes. Horizontal bars illustrate experimental resolution. Kinematic deconvolution is indicated 0.

Because all phases of the interaction of the incident energetic ion beam with materials, including kinematics and cross section of the elastic collision and the energy losses by means of inelastic interaction with the electrons are readily calculable, the analysis lends itself to computer simluation. One of the first such programs, developed at IBM (4), is used at NRL, while other programs have also been developed at a number of other laboratories. 

Molecular beam experiments are performed in a laboratory frame of reference but the chemically interesting events take place with respect to the center of mass of the colliding species. In order to interpret the data, differential cross sections measured in the laboratory (LAB) coordinate system must be transformed to reflect events which took place in the center-of-mass (CM) coordinate system. To effect this transformation the invariant motion of the center of mass must be subtracted from the scattering data obtained in the LAB system. A simple example which illustrates the difference between LAB and CM kinematics is shown, for an elastic collision, in Fig. 8.6. In CM the particles always move directly toward one another before interaction and directly apart afterwards. This condition is a consequence of momentum conservation in a system with a stationary center of mass. The interaction causes each particle to be deflected through... 

Energy transfer from a projectile to a target nucleus in an elastic two-body collision - concept of kinematic factor K = Ei/Eq i.e., ratio of energy of the scattered particle to the energy of the incident particle). 

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