Ion Implantation and Synthesis of Materials - Studium

4.2 Scattering Cross-Section 39unit time. Since the solid angle Ω unit (steradian) is dimensionless, the differentialscattering cross-section has units of area.The cross-section is simply the effective target area presented by each scatteringcenter (target nucleus) to the incident beam. At a more microscopic level, thescattering cross-section can be shown to be dependent on b, the impact parameter.In Fig. 4.2 we present the collision process in which the incident particle is scatteredby a target nucleus through an angle θ c . The projectile moves in a nearlystraight line until it gets fairly close to the target nucleus, at which point it is deflectedthrough an angle θ c . After being deflected, the trajectory of the particle isagain nearly a straight line. If there had been no interaction force between the projectileand the target nucleus, the projectile would have maintained a straight trajectoryand passed the target nucleus at a distance b.On examining Fig. 4.2, we see that all incident particles with impact parameterb are headed in a direction to strike the rim of the circle drawn around the targetnucleus and will be deflected by an angle θ c . The area of this circle is πb 2 , and anyparticle with a trajectory that strikes anywhere within this area will be deflected byan angle greater than θ c . The target area defined by the impact parameter is calledthe total cross-section, σ (θ c ):σθ (c)= πb2 .(4.3)For projectiles moving with small values of b, the cross-section defined by (4.3)will be small, but, due to the interaction forces, the scattering angle will be large.Thus, b is proportional to σ (θ c ), while b and σ (θ c ) are inversely related to θ c .From this discussion we see that b = b(θ c ).In addition to the total cross-section, there is the differential cross-section,dσ (θ c ), and its relationship to b. As shown in Fig. 4.3, particles incident withProjectileθ cbCentral Force(Target Atom)Fig. 4.2. Scattering of a particle that approaches a nucleus with an impact parameter b. Thetotal cross-section is σ = πb 2