PENELOPE 2003 - OECD Nuclear Energy Agency

146 Chapter 4. Electron/positron transport mechanics
where the summations extend over the oscillators with resonance energy less than W cc
and greater than W max , and each term contributes only for the µ-intervals indicated
above. The mean free path and the first and second transport mean free paths for soft
inelastic scattering are
[ ] ∫
(s) −1 µ2
λ in = N
0
dσ (s)
in
dµ
dµ, (4.114)
and
[ ] ∫
(s) −1 µ2
λ in,1 = N
0
[ ] ∫
(s) −1 µ2
λ in,2 = N
0
2µ dσ(s) in
dµ
6(µ − µ 2 ) dσ(s) in
dµ
dµ (4.115)
dµ. (4.116)
In penelope, soft electronic scattering is simulated together with soft elastic scattering,
by means of the artificial distribution (4.30). The combined process is described
by the transport mean free paths
and
[ ]
(s) −1 [ ]
(s) −1 [
(s) −1
λ comb,1 = λ el,1 + λ in,1]
(4.117)
[ ]
(s) −1 [ ]
(s) −1 [
(s) −1
λ comb,2 = λ el,2 + λ in,2]
. (4.118)
Thus, to account for soft electronic scattering we only have to replace the soft elastic
transport mean free paths by those of the combined process.
4.3.3 Bielajew’s alternate random hinge
Angular deflections due to soft interactions along a step of length s are generated from
the artificial distribution (4.30) with first and second moments given by eqs. (4.28)
and (4.29), which are determined by the transport mean free paths λ (s)
comb,1 and λ(s) comb,2 .
To account (at least partially) for the energy dependence of these quantities we use a
trick due to Alex Bielajew. The soft energy loss and angular deflection (which occur
at the hinge) are considered as independent processes and are simulated in random
order. That is, the soft angular deflection is evaluated for the energy at either the
beginning or the end of the step, with equal probabilities. This is equivalent to assuming
that the transport mean free paths λ (s)
comb,1(E) and λ (s)
comb,2(E) vary linearly with energy.
The method is fairly accurate and computationally inexpensive provided only that the
fractional energy loss along each step (which is of the order of C 2 ) is sufficiently small.
4.4 Generation of random tracks
Each simulated electron or positron history consists of a chronological succession of
events. These can be either hard events, artificial soft events (hinges) or other relevant