CERN Program Library Long Writeup W5013 - CERNLIB ...

where
α 1 = (0.5 − ɛ 0) 2
F 10 α 2 = 1 3
2 F 20
3
f 1 (ɛ) =
(0.5 − ɛ 0 ) 3 (ɛ − 1
0.5)2 f 2 (ɛ) =
0.5 − ɛ 0
g 1 (ɛ) = F 1 /F 10 g 2 (ɛ) = F 2 /F 20
and
F 1 = F 1 (δ) =3Φ 1 (δ) − Φ 2 (δ) − F (Z) F 10 = F 1 (δ min )
F 2 = F 2 (δ) = 3 2 Φ 1(δ)+ 1 2 Φ 2(δ) − F (Z) F 20 = F 2 (δ min )
δ min = 4 136m
Z 1 3 3E
δ min is the minimal value of the screening variable δ. It can be seen that the functions f i (ɛ) are normalised
and that the functions g i (ɛ) are “valid” rejection functions.
Therefore, if r i are uniformly distributed random numbers (0 ≤ r i ≤ 1), we can sample the ɛ (x in the
program) in the following way:
1. select i to be 1 or 2 according to the following ratio:
BPAR = α 1
α 1 + α 2
If r 0 < BPAR then i =1, otherwise if r 0 ≥ BPAR i =2;
2. sample ɛ from f 1 (ɛ). This can be performed by the following expressions:
⎧
1
⎨ 0.5 − (0.5 − ɛ 0 )r 1 3
( )
when i =1
ɛ = 1
⎩ ɛ 0 +
2 − ɛ 0 r 1 when i =2
3. calculate the rejection function g i (ɛ). Ifr 2 ≤ g i (ɛ), accept ɛ, otherwise return to step 1.
It should be mentioned that we need a step just after sampling ɛ in the step 2, because the cross-section
formula becomes negative at large δ and this imposes an upper limit for δ;
[ ]
42.24 − F (Z)
δ max =exp
− 0.952
8.368
If we get a δ value using the sampled ɛ such that δ>δ max , we have to start again from the step 1. After the
successful sampling of ɛ, GPAIRG generates the polar angles of the electron with respect to an axis defined
along the direction of the parent photon. The electron and the positron are assumed to have a symmetric
angular distribution. The energy-angle distribution is given by Tsai [13, 14] as following:
dσ
dpdΩ = 2α2 e 2 {[ 2x(1 − x)
πkm 4
+
]
12lx(1 − x)
(1 + l) 4 (Z 2 + Z)
(1 + l) 2 −
[ ]
2x 2 − 2x +1 4lx(1 − x) [
(1 + l) 2 +
(1 + l) 4 X − 2Z 2 f((αZ) 2 )
where k is the photon energy, p and E are the momentum and the energy of the electron of the e + e − -pair,
x = E/k and l = E 2 θ 2 /m 2 . This distribution is quite complicated to sample and, furthermore, considered
214 PHYS211 – 3
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