PENELOPE 2003 - OECD Nuclear Energy Agency

4.3. Combined scattering and energy loss 141
events, determined from the prescription (4.85) with C 1 = C 2 = 0.05 is also plotted. For
low energies, λ (h)
el = λ el and the simulation is purely detailed (µ c = 0). For intermediate
energies, λ (h)
el = C 1 λ el,1 , whereas λ (h)
el = C 2 E/S(E) in the high-energy domain. From fig.
4.3 it is clear that increasing the value of C 2 does not have any effect on the simulation
of electron tracks with initial energies that are less than ∼ 10 MeV.
4.3.1 Variation of λ (h)
T
with energy
With the definition (4.85) of the hard elastic mean free path, we only set a limit on
the average step length. However, since s is sampled from the exponential distribution,
its realizations fluctuate amply about the average value. On the other hand, the soft
energy loss ω along a step of given length s also fluctuates about the mean value 〈ω〉
given by eq. (4.78). This means that the inverse mean free path Σ h (E) varies along the
step in an essentially unpredictable way.
Let us consider for a moment that the CSDA is applicable (i.e. that the effect of
soft energy straggling is negligible). In this case, there is a one-by-one correspondence
between the kinetic energy E of the electron and the travelled path length s,
s =
∫ E0
E
dE ′
S s (E ′ ) , (4.86)
where E 0 is the initial energy (at s = 0) and S s (E) is the soft stopping power, eq. (4.46)
[we consider that no hard interactions occur along the step]. Equivalently,
ds
dE = − 1
S s (E) . (4.87)
Thus, the inverse mean free path Σ h can be formally considered as a function of the path
length s. The probability p(s) ds of having the first hard interaction when the particle
has travelled a length in the interval (s, s + ds) is determined by the equation [cf. eq.
(1.87)]
∫ ∞
p(s) = Σ h (s) p(s ′ ) ds ′ , (4.88)
s
with the normalization condition,
∫ ∞
0
p(s) ds = 1. (4.89)
Instead of the path length s, it is convenient to consider the dimensionless variable
q ≡
∫ E0
E
Σ h (E ′ ∫
)
s
S s (E ′ ) dE′ = Σ h (s ′ ) ds ′ , (4.90)
0
which varies with energy and
dq
dE = −Σ h(E)
S s (E) . (4.91)