THE EGS5 CODE SYSTEM

2.11 Bhabha Scattering
The differential Bhabha[25], cross section, as formulated in PEGS, is
where
d˘Σ Bhabha (Ĕ0)
dĔ−
= X 0n2πr 2 0 m
˘T 2 0
[ 1
E
( )
]
1
Eβ 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
∫ Ĕ2
Ĕ 1
d˘Σ Bhabha (Ĕ0)
dĔ−
dĔ− = X 0n2πr 2 0 m
˘T 2 0
[ 1
β 2 ( 1
E 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)
where
and 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 negative
atomic electron gets. There is still a singularity at E = 0 which is circumvented in the same way as
for 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, the
final positron energy may be less than A E (down to m). Thus, the threshold for a discrete Bhabha
interaction is A E , and as long as the positron is above the cutoff energy, it will have some non-zero
Bhabha cross section. Using the minimum and maximum Ĕ− for Ĕ1 and Ĕ2 in Equation 2.209, we
obtain 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)
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