Ion Implantation and Synthesis of Materials - Studium

3.4 Center-of-Mass Coordinates 27Fig. 3.3. Elastic collision diagrams between two unequal masses as seen in the CMreference frame3.4 Center-of-Mass CoordinatesThe collision and scattering problem defined by Fig. 3.2 now will be restated interms of CM coordinates. The motivation for this transformation will be obviouswhen we discuss scattering in a central force field later in this chapter. Throughthe use of CM coordinates it will be shown that, no matter how complex the forceis between the two particles, so long as it acts only along the line joining them (notransverse forces), the relative motion of the two particles can be reduced to thatof a single particle moving in an interatomic potential centered at the origin of theCM coordinates. By introducing the CM system, the mutual interaction of the twocolliding particles can be described by a force field, V(r), which depends only onthe absolute value of the interatomic separation, r. The motion of both particles isgiven by one equation of motion. This equation has r as the independent variableand describes a particle moving in the central force field, V(r).The CM coordinates for a two-particle system are defined in a zero-momentumreference frame in which the total force on two particles that interact only witheach other is zero. We can define the total force of two interacting particles asdFT = F T1+ F2= p,dt(3.4)where F T is the total force, F 1 and F 2 are the individual forces on particles 1 and 2,respectively, and p T is the total linear momentum of the two-particle system. ForF T = 0, dp T = 0, indicating that the total momentum is unchanged or conservedduring the interaction process.One consequence associated with observing elastic collisions in the CMcoordinates is that the individual particle kinetic energies are unchanged by the