Ion Implantation and Synthesis of Materials - Studium

3.4 Center-of-Mass Coordinates 29and are independent of the final scattering angle between the two particles(Fig. 3.3b). Thus, regardless of whether the collision is elastic or inelastic, the totalmomentum is unchanged in a collision. In addition, from (3.9) and (3.10), we seethat the ratio of the ion-to-atom velocities is inversely proportional to the ratio oftheir massesvion v0 − vc M2= = =v v Matom c 11 .µ(3.11)Another advantage to the CM reference frame is that the CM total energy, E c , isequal to the CM initial kinetic energyE12c= Mcv02,(3.12)1 MM M ME v E E ,1 2 2 2cc=0=0=02 M1 + M2 M1 + M2 M1(3.13)2where M1v0/2 = E 0.In converting the scattering angles from the laboratory system to the CMsystem, it can be shown thatφc = 2 φ.(3.14)From the CM diagram, Fig. 3.3, we have θ c + φ c = π, which allows us to rewrite(3.14) in the formπ −θφ =2c .(3.15)Equation (3.15) relates the target atom scattering angle in the laboratory to theCM ion scattering angle. In addition:1. The CM angle of the scattered projectile is θ c = π − 2φ = π − φ c . WhenM 1 ≤ M 2 ⇒ µ ≤ 1, θ c is defined for all θ ≤ π and θ c = θ + sin −1 (µ sin θ ).2. When M 1 > M 2 ⇒ µ > 1, θ c is double valued and the laboratoryscattering angle is limited to the range θ > sin −1 (1/µ). In this case,θ c = θ + sin −1 (µ sin θ ), or θ c = π + θ − sin −1 (µ sin θ ).The transferred energy, T, can be related to the ion scattering angle, θ c , by (3.15)to yield