PENELOPE 2003 - OECD Nuclear Energy Agency

4.2. Soft energy losses 135
deviation [Ω 2 s(E)s] 1/2 is much smaller than its mean S s (E)s (otherwise there would be
a finite probability of negative energy losses), i.e.
[
Ω
2
s (E)s ] 1/2
≪ Ss (E)s ≪ E. (4.58)
Requirement 1) implies that the cutoff energies W cc and W cr for delta ray production
and photon emission have to be relatively small. The second requirement holds for path
lengths larger than s crit = Ω 2 s/S 2 s .
Now, we address ourselves to the problem of simulating the energy losses due to soft
stopping interactions between two consecutive hard events. The distribution (4.57) gives
the desired result when conditions (4.58) are satisfied. In fact, the use of a Gaussian
distribution to simulate the effect of soft stopping interactions was previously proposed
by Andreo and Brahme (1984). Unfortunately, the step lengths found in our simulations
are frequently too short for conditions (4.58) to hold (i.e. s is usually less than s crit ). To
get over this problem, we replace the actual energy loss distribution G(s; ω) by a simpler
“equivalent” distribution G a (s; ω) with the same mean and variance, given by eqs. (4.54)
and (4.56). Other details of the adopted distribution have no effect on the simulation
results, provided that the number of steps along each track is statistically sufficient (say,
larger than ∼ 20). penelope generates ω from the following distributions
• Case I. If 〈ω〉 2 > 9 var(ω), we use a truncated Gaussian distribution,
⎧ [
]
⎪⎨
(ω − 〈ω〉)2
exp −
if |ω − 〈ω〉| < 3 σ.
G a,I (s; ω) = 2(1.015387σ) 2
⎪⎩ 0 otherwise.
(4.59)
where σ = [var(ω)] 1/2 is the standard deviation and the numerical factor 1.015387
corrects for the effect of the truncation. Notice that the shape of this distribution is
very similar to that of the “true” energy-loss distribution, eq. (4.57). Random sampling
from (4.59) is performed by means of the Box-Müller method, eq. (1.54), rejecting the
generated ω’s that are outside the interval 〈ω〉 ± 3σ.
• Case II. When 3 var(ω) < 〈ω〉 2 < 9 var(ω), the energy loss is sampled from the uniform
distribution
G a,II (s; ω) = U ω1 ,ω 2
(ω) (4.60)
with
ω 1 = 〈ω〉 − √ 3 σ, ω 2 = 〈ω〉 + √ 3 σ. (4.61)
• Case III. Finally, when 〈ω〉 2 < 3 var(ω), the adopted distribution is an admixture of a
delta and a uniform distribution,
G a,III (s; ω) = aδ(ω) + (1 − a)U 0,ω0 (ω) (4.62)
with
a =
3var(ω) − 〈ω〉2
3var(ω) + 3〈ω〉 2 and ω 0 =
3var(ω) + 3〈ω〉2
. (4.63)
2〈ω〉