PENELOPE 2003 - OECD Nuclear Energy Agency

98 Chapter 3. Electron and positron interactions
and
[
f (+) (γ) = 2 ln 2 − β2
23 + 14
]
12 γ + 1 + 10
(γ + 1) + 4
2 (γ + 1) 3
(3.105)
for electrons and positrons, respectively. This formula can be derived from very general
arguments that do not require knowing the fine details of the GOS; the only information
needed is contained in the Bethe sum rule (3.45) and in the definition (3.46) of the mean
excitation energy (see e.g. Fano, 1963). Since our approximate analytical GOS model is
physically motivated, it satisfies the sum rule and reproduces the adopted value of the
mean ionization energy, it yields (at high energies) the exact Bethe formula.
It is striking that the “asymptotic” Bethe formula is in fact valid down to fairly small
energies, of the order of 10 keV for high-Z materials (see fig. 3.10). It also accounts
for the differences between the stopping powers of electrons and positrons (to the same
degree as our GOS model approximation).
For ultrarelativistic projectiles, for which the approximation (3.64) holds, the Bethe
formula simplifies to
{ ( )
}
S in ≃ N 2πe4 E
2
γ + 1
m e v 2Z ln
+ f (±) (γ) + 1 . (3.106)
2γ 2
Ω 2 p
The mean excitation energy I has disappeared from this formula, showing that at
very high energies the stopping power depends only on the electron density N Z of
the medium.
3.2.5 Simulation of hard inelastic collisions
The DCSs given by expressions (3.76)-(3.79) permit the random sampling of the energy
loss W and the angular deflection θ by using purely analytical methods. In the following
we consider the case of mixed (class II) simulation, in which only hard collisions, with
energy loss larger than a specified cutoff value W cc , are simulated (see chapter 4). As
the value of the cutoff energy loss can be selected arbitrarily, the sampling algorithm
can also be used in detailed (interaction-by-interaction) simulations (W cc = 0).
The first stage of the simulation is the selection of the active oscillator, for which we
need to know the restricted total cross section,
σ(W cc ) ≡
= ∑ k
∫ Wmax
W cc
dσ in
dW dW = σ dis,l(W cc ) + σ dis,t (W cc ) + σ clo (W cc )
σ k (W cc ), (3.107)
as well as the contribution of each oscillator, σ k (W cc ). The active oscillator is sampled
from the point probabilities p k = σ k (W cc )/σ(W cc ). Since these probabilities are calculated
analytically, the sampling algorithm is relatively slow. In mixed simulations, the
algorithm can be sped up by using a larger cutoff energy loss W cc , which eliminates all
the oscillators with W k < W cc from the sum.