Ion Implantation and Synthesis of Materials - Studium

283 Dynamics of Binary Elastic Collisionscollision process. Thus, the CM velocities of the two colliding particles are thesame before and after the collision process. In addition, the CM scattering angle ofparticle 1 will equal the scattering angle of particle 2. Finally, all scattering anglesin the CM system are allowed, unlike the scattering angles in the laboratoryreference frame, where the allowed scattering angles depend on the ratio M 1 /M 2 .For CM coordinates, Fig. 3.3, we define the system velocity, v c , such that inthis coordinate system there is no net momentum change, so thatM v = ( M + M ) v .1 0 1 2 c(3.5)We also define in CM coordinates a reduced mass, M c , given by the relation1 1 1= +M M Mc 1 2(3.6)orMMM 1 2c = M M 1+2.(3.7)From (3.7) we see that, for large mass difference between M 1 and M 2 , M capproaches the value of the lower mass. For example, if M 2 < < M 1 , M c ≅ M 2 .We can represent the CM velocity in terms of reduced mass asMv v vMc 1c=0=0M2 M1 + M2.(3.8)From the velocity vector diagram in Fig. 3.3 and (3.8), the ion and target atomvelocities in the CM coordinates are:Mv v v v v vc 21=ion=0−c=0=0M1 M1 + M2Mv v v v vMMc 12=atom=c=0=0M2M1 + M2.,(3.9)(3.10)Equation (3.10) shows that the target atom, which has zero velocity before thecollision in the laboratory reference frame, has the system velocity v c before andafter the collision in the CM reference frame.Equations (3.9) and (3.10) show the advantage of the CM reference frame. Thesystem velocity, v c , and the atom and ion velocities, v atom and v ion, remain constant