PENELOPE 2003 - OECD Nuclear Energy Agency

4.3. Combined scattering and energy loss 139
where the logarithmic derivatives have been introduced for numerical convenience. The
factors in curly brackets account for the global effect of the energy dependence of the soft
energy-loss DCS (within the linear approximation). To simulate soft energy losses, we
sample ω from the artificial distribution G a (ω; s), eqs. (4.59) to (4.63), with the “correct”
first moment and variance, given by expressions (4.78) and (4.79). In penelope, we
use step lengths s such that the fractional energy loss along each step is relatively small
(see below) and, consequently, the energy-dependence correction is also small (i.e. the
correcting factors are close to unity).
4.3 Combined scattering and energy loss
Up to this point, soft scattering and energy loss have been regarded as essentially independent
processes, while in reality they coexist. In this section, we consider their
interplay and set the basis of an algorithm that simulates their combined effect.
Ours is a mixed algorithm, where hard interactions are described individually from
the associated DCSs (see chapter 3). These interactions are 1) hard elastic collisions,
“el”, 2) hard inelastic collisions, “in”, 3) hard bremsstrahlung photon emission, “br”,
4) ionization of inner shells, “si”, and, in the case of positrons, 5) positron annihilation,
“an”. The mean free path between consecutive hard events, λ (h)
T , is given by
where σ (h)
T
[ ]
(h) −1 (h)
λ T = N σ T = N [ σ (h)
el + σ (h)
in + σ (h)
br + σ si (+σ an ) ] ≡ Σ h , (4.80)
is the total atomic cross section for hard interactions. We recall that the
inverse mean free path, Σ h , gives the interaction probability per unit path length. In the
absence of soft energy-loss events, the PDF of the step length s between two successive
hard events (or from a given point in the track to the next hard event) is
p(s) = Σ h exp (−Σ h s) . (4.81)
In each hard event, one and only one interaction (i=“el”, “in”, “br”, “si” or “an”) occurs
with probability
p i = σ (h)
i /σ (h)
T . (4.82)
When soft energy-losses are considered, the PDF of the distance s travelled by the
particle to the following hard interaction is not given by eq. (4.81), because the mean
free path λ (h)
T varies with energy and may change appreciably along a single step. The
simplest way to cope with this problem is to limit the length of the step to make sure
that the average energy loss is much smaller than the kinetic energy E at the beginning
of the step, and consider that λ (h)
T (E) remains essentially constant along the step. Then,
the mean energy loss in a step is given by
〈∆E〉 = λ (h)
T S(E), (4.83)