Ion Implantation and Synthesis of Materials - Studium

52 5 Ion Stoppingwhere E is the energy of the moving particle and T is the recoil or transfer energy.The average energy-loss by the moving particle in the distance dx is obtained bymultiplying (4.11) by the transfer energy T and integrating over all possible valuesof T.d PE ( ) TMd σ ( E)〈 dE〉 = ∫T dT = N∫T dTdx.TdTmin dT(5.4)For infinitesimal dx, omitting the averaging symbol on dE, we haveσdETMd ( E) = N∫T d T ,dxTminndT(5.5)where dE/dx| n is the nuclear stopping power. The lower limit in the integrationT min is the minimum energy-transferred and need not be zero. One value used forT min is the energy needed to displace an atom from its lattice site, approximately20–30 eV. Atomic displacement processes will be discussed in Chap. 7. The upperlimit, T M , is the maximum transfer energy given by T M = 4M 1 M 2 E/(M 1 + M 2 ) 2 .In (5.3) we defined the stopping cross-section. The nuclear stopping crosssectionfor an ion of energy E is given by1 dET d σ ( E)S E = = T T∫Mn( ) d ,N dx Tn min dT(5.6)where dσ (E)/dT is the energy-transfer differential cross-section. The nuclearstopping cross-section can be evaluated using the power representation of theenergy-transfer differential cross-section given in (4.19)d σ ( E)=dTCmm 1+mE T,where the constant C m is defined in (4.20). Taking T min = 0, the nuclear stoppingcross-section is now given byor−mC Tm M −mCmE TSn( E) = d =m ∫ T T0E1−m1−mTM0