Ion Implantation and Synthesis of Materials - Studium

4.3 Energy-Transfer Cross-Section 43From the above analysis of Fig. 4.5, we can define the probability of a projectilewith energy E undergoing a scattering event or a collision with a target nucleuswhile traversing a thickness ∆x asPE ( ) = Nσ( E) ∆ x.(4.10)Equation (4.10) defines the total collision cross-section, σ (E), between an energeticparticle of energy E and the target atoms. The total cross-section gives ameasure of the probability for any type of collision to occur where energytransfersare possible, for energies up to and including the maximum valueT M = 4M 1 M 2 E 0 /(M 1 + M 2 ) 2 .In addition to the total cross-section, we also wish to consider the more restrictivetypes of interactions that can occur between target nuclei and particles withenergy E. Consider the condition where we wish to know the probability that aprojectile with energy E will transfer an amount of energy between T and T + dTto a target atom. Such a probability function defines the differential energytransfercross-section, dσ (E)/dT, and is obtained by differentiating (4.10)d PE ( ) d σ( E) 1 d σ( E)PET ( , )dT≡ dT= N∆xdT=d T,dT d T σ ( E) dT(4.11)where P(E, T) is the probability that an ion with energy E will undergo a collisionproducing an energy-transfer in the range T and T + dT while traversing a distance∆x, and is simply defined as the ratio of the differential cross-section to the totalenergy-transfer cross-section.Probability functions can be constructed based on the scattering processes describedby Figs. 4.2 and 4.3. The probability of a collision producing a deflectionθ c in the incident projectile’s trajectory is given byP( θ ) = σ( θ ) Nd x,cc(4.12a)and the probability of scattering the projectile into the angular range between θ cand θ c + dθ c while it travels a distance dx is given byd P( θc) d σ( θc) 1 d σ( θc)P( θc, b)db ≡ = N dbdx =d b,db d b σθ ( ) dbc(4.12b)where σ (θ c ) is the total angular scattering cross-section given in (4.3), and dσ (θ c )is the differential angular scattering cross-section given in (4.4). An expression