CERN Program Library Long Writeup W5013 - CERNLIB ...

• the mean number of tries is not too large.
Thus the different possible decompositions of the distribution f(x) are not equivalent from the practical
point of view (e.g. they can be very different in computational speed) and it can be very useful to optimise the
decomposition. A remark of practical importance is that if our distribution is not normalised ( ∫ f(x)dx =
C>0; C ≈ 1), the method can be used in the same manner, the only difference is that the mean number of
tries in this case is given by ∑ i α i/C .
2.2 Differential cross-section for pair production
The Bethe-Heitler differential cross-section with the Coulomb correction for a photon of energy E to produce
a e − e + -pair with one of the particles having energy ɛE (ɛ is the fraction of the photon energy carried
by one particle of the pair) is given as in [12]:
dσ(Z, E, ɛ)
dɛ
{
= r2 0αZ[Z + ξ(Z)]
E 2
[ɛ 2 +(1− ɛ) 2 ][Φ 1 (δ) − F (Z)
2
]+ 2 3 ɛ(1 − ɛ)[Φ 2(δ) − F (Z) }
] (2)
2
where Φ i (δ) are the screening functions depending on the screening variable δ
δ = 136m 1
Z 1/3 E ɛ(1 − ɛ)
}
Φ 1 (δ) =20.867 − 3.242δ +0.625δ 2
Φ 2 (δ) =20.209 − 1.930δ − 0.086δ 2 δ ≤ 1
Φ 1 (δ) =Φ 2 (δ) =21.12 − 4.184 ln(δ +0.952) δ>1
m = electron mass
F (Z) =
{
8/3lnZ
8/3lnZ +8f c (Z)
E