PENELOPE 2003 - OECD Nuclear Energy Agency

142 Chapter 4. Electron/positron transport mechanics
The PDF of q is
π(q) = p(s) ds
dq
= p(s)
ds
dE
From eq. (4.88) it follows that π(q) satisfies the equation
π(q) =
Therefore, q is distributed exponentially,
∫ ∞
q
dE
dq = p(s) 1
Σ h (s) . (4.92)
π(q ′ ) dq ′ . (4.93)
π(q) = exp(−q). (4.94)
The PDF of the step length s is obtained by inverting the transformation (4.90),
( ∫ s
)
p(s) = Σ h (s) exp − Σ h (s ′ ) ds ′ . (4.95)
0
It is not practical to sample s from this complicated PDF. It is much more convenient
to sample q [as − ln ξ, cf. eq. (1.36)] and then determine s from (4.90), which can be
inverted numerically (for practical details, see Berger, 1998). Although this sampling
method effectively accounts for the energy dependence of Σ s (E), it is applicable only to
simulations in the CSDA.
A more versatile algorithm for sampling the position of hard events, still within the
CSDA, is the following. We let the electron move in steps of maximum length s max , a
value specified by the user. This determines the maximum energy loss along the step,
ω max =
∫ smax
0
S s (s) ds. (4.96)
Let Σ h,max denote an upper bound for the inverse mean free path of hard events in the
swept energy interval, i.e.
Σ h,max > max {Σ h (E), E ∈ (E 0 − ω max , E 0 )} (4.97)
We now assume that the electron may undergo fictitious events in which the energy and
direction remain unaltered (delta interactions). The inverse mean free path of these
interactions is defined as
Σ δ (E) = Σ h,max − Σ h (E), (4.98)
so that the inverse mean free path of the combined process (delta interactions + hard
events) equals Σ h,max , a constant. Owing to the Markovian character of the processes,
the introduction of delta interactions does not influence the path-length distribution
between hard events. Therefore, the occurrence of hard events can be sampled by
means of the following simple algorithm,
(i) Sample a distance s from the exponential distribution with inverse mean free path
Σ h,max , i.e. s = (− ln ξ)/Σ h,max .