Ion Implantation and Synthesis of Materials - Studium

40 4 Cross-Sectionimpact parameters between b and b + db will be scattered through angles betweenθ c and θ c + dθ c . The differential cross-section for this process is found by takingthe differential of (4.3) with respect to the impact parameter.2d σθ (c) = d( πb) = 2πbd b.(4.4)From the description given in (4.4) and the schematic presented in Fig. 4.3, thedifferential cross-section of each target nucleus is presented as a ring of radius b, acircumference 2πb, and a width db. Any incident particles with an impact parameterwithin db will be scattered into angles between θ c and θ c + dθ c .From the examples presented in Figs. 4.2 and 4.3, we see that there is a uniqueconnection between the value of b and the scattering angle, θ c . To find the dependenceof dσ (θ c ) on the scattering angle, we rewrite (4.4) in the formd b( θ )d σ ( θ ) 2 π ( θ ) dθcc= bc c.dθc(4.5)We use the absolute value of db(θ c )/dθ c to maintain dσ (θ c ) as a positive value; θ cincreases as b decreases, indicating that db(θ c )/dθ c is negative.To determine an expression for the differential scattering cross-section per unitsolid angle, (4.1), we note that scattering experiments are performed by observingthe number of incident particles that are scattered into a solid angle located at θ c .Measurements give information in units of the number of scattering particles perelement of solid angle. A schematic of this process is presented in Fig. 4.4. Theannular region represents the solid angle dΩ subtended between the scattering anglesθ c and θ c + dθ c . The entire area of the sphere of radius R is 4πR 2 , and the totaldθ cθ cbdbCentral Force(Target Atom)Fig. 4.3. Nuclear target area for the differential cross-section dσ = 2πb db