PENELOPE 2003 - OECD Nuclear Energy Agency

3.2. Inelastic collisions 101
Table 3.1: Efficiency (%) of the random sampling algorithm of the energy loss in close
collisions of electrons and positrons for different values of the energy E and the cutoff energy
loss κ c .
E (eV)
κ c
0.001 0.01 0.1 0.25 0.4
10 3 99.9 99.9 99.8 99.7 99.6
10 5 99.7 98 87 77 70
10 7 99 93 70 59 59
10 9 99 93 71 62 63
After sampling the energy loss W = κE, the polar scattering angle θ is obtained
from eq. (A.40) with Q = W . This yields
cos 2 θ = E − W
E
E + 2m e c 2
E − W + 2m e c2, (3.116)
which agrees with eq. (A.17). The azimuthal scattering angle φ is sampled uniformly in
the interval (0, 2π).
Hard close collisions of positrons
The PDF of the reduced energy loss κ ≡ W/E in positron close collisions with the k-th
oscillator is given by [see eqs. (3.74) and (3.75)]
P (+)
k (κ) = κ −2 F (+)
k (E, W ) Θ(κ − κ c ) Θ(1 − κ)
[ 1
=
κ − b ]
1
2 κ + b 2 − b 3 κ + b 4 κ 2 Θ(κ − κ c ) Θ(1 − κ) (3.117)
with κ c ≡ max(W k , W cc )/E. The maximum allowed reduced energy loss is 1. Again,
normalization is not important.
Consider the distribution
Φ (+) (κ) ≡ κ −2 Θ(κ − κ c ) Θ(1 − κ). (3.118)
It is easy to see that Φ (+) > P (+)
k in the interval (κ c , 1). Therefore, we can generate κ
from the PDF, eq. (3.117), by using the rejection method with trial values sampled from
the distribution of eq. (3.118) and acceptance probability P (+)
k /Φ (+) . Sampling from the
PDF Φ (+) can easily be performed with the inverse transform method.
The algorithm for random sampling from the PDF (3.117), is: