THE EGS5 CODE SYSTEM

2.11 Bhabha ScatteringThe differential Bhabha[25], cross section, as formulated in PEGS, iswhered˘Σ Bhabha (Ĕ0)dĔ−= X 0n2πr 2 0 m˘T 2 0[ 1E( )]1Eβ 2 − B 1 + B 2 + E (EB 4 − B 3 )(2.208)Ĕ 0 = energy of incident positron (MeV),˘T 0 = kinetic energy of incident positron (MeV),β = v/c for incident positron,γ = Ĕ0/m,Ĕ − = energy of secondary electron (MeV),E = (Ĕ− − m)/ ˘T 0 = ˘T − / ˘T 0 ,y = 1/(γ + 1),B 1 = 2 − y 2 ,B 2 = (1 − 2y)(3 + y 2 ),B 3 = B 4 + (1 − 2y) 2 ,B 4 = (1 − 2y) 3 .If Equation 2.208 is integrated between Ĕ1 and Ĕ2, we obtain∫ Ĕ2Ĕ 1d˘Σ Bhabha (Ĕ0)dĔ−dĔ− = X 0n2πr 2 0 m˘T 2 0[ 1β 2 ( 1E 1− 1 E 2)− B 1 ln E 2+B 2 (E 2 − E 1 ) + E 2 2 (E 2B 4 /3 − B 3 /2) − E 2 1 (E 1B 4 /3 − B 3 /2)E 1](2.209)whereand other symbols are the same as in Equation 2.208.E i = (Ĕi − m)/ ˘T 0 , i = 1, 2 (2.210)Unlike in Møller scattering, in Bhabha scattering, the final state particles are distinguishable,so the upper limit for E is 1. Note that E is the fraction of the kinetic energy that the negativeatomic electron gets. There is still a singularity at E = 0 which is circumvented in the same way asfor Møller by requiring that the energy transfered to the atomic electron be at least T E = A E − m.It should be noted that there is no singularity at E = 1 as there was for Møller, and in fact, thefinal positron energy may be less than A E (down to m). Thus, the threshold for a discrete Bhabhainteraction is A E , and as long as the positron is above the cutoff energy, it will have some non-zeroBhabha cross section. Using the minimum and maximum Ĕ− for Ĕ1 and Ĕ2 in Equation 2.209, weobtain the total cross section as˘Σ Bhabha (Ĕ0) = (Equation 2.209 with Ĕ1 = A E & Ĕ2 = Ĕ0 if Ĕ 0 > A E , 0). (2.211)68