Ion Implantation and Synthesis of Materials - Studium

44 4 Cross-Sectionsimilar to (4.11) can also be constructed for the differential angular scatteringcross-section as a function of the impact parameter.Following an analogous route to the development of the differential energycross-section given in (4.11), the probability function for a particle with energy Ebeing scattered into a solid angle dΩ in the angular region between θ c and θ c + dθ cis given byd PE ( ) d σ ( E)PE ( , Ω )dΩ≡ dΩ= N∆xdΩ.dΩdΩ(4.13)Equation (4.13) can be rewritten in terms of θ c by applying (4.6),d Ω = 2π sin θ c dθ c , which allows us to writed σ ( E)d σθ (c)P( E, Ω )dΩ = N∆x dΩ= 2π sinθcN∆xd θc.dΩdΩ(4.14)The relationship between energy-transfer, T, and the scattering angle, θ c , or thesolid angle, Ω, can be found by setting the probability functions given by (4.11)and (4.14) equal to each other, so thatPET ( , )d T=PE ( , Ω )d Ω,which is equivalent tod σ ( E) d σ ( θc)dT= 2 π sin θcd θcdTdΩ(4.15)ord σ ( E)dθcd σ( θc)= 2πsin θc.dTdTdΩ(4.16)The transferred energy, T, is given in (3.26) as2 1T = TM sin ( θc/ 2) = TM(1 −cos θc),2and the differential angular cross-section for scattering into a solid angle dΩ isgiven by (4.7) as