PENELOPE 2003 - OECD Nuclear Energy Agency

136 Chapter 4. Electron/positron transport mechanics
It can be easily verified that these distributions have the required mean and variance.
It is also worth noticing that they yield ω values that are less than
⎧
〈ω〉 + 3σ in case I,
⎪⎨
ω max = ω 2 in case II,
(4.64)
⎪⎩
ω 0 in case III.
ω max is normally much less than the kinetic energy E of the transported particle. Energy
losses larger than E might be generated only when the step length s has a value of the
order of the Bethe range, but this never happens in practical simulation (see below). It
is worth noticing that, after a moderately large number of steps, this simple simulation
scheme effectively yields an energy loss distribution that has the correct first and second
moments and is similar in shape to the “true” distribution. Further improvements of
the distribution of soft energy losses would mean considering higher order moments of
the single scattering inelastic DCS given by eq. (4.49).
In spatial-dose calculations, the energy loss ω due to soft stopping interactions can be
considered to be locally deposited at a random position uniformly distributed along the
step. This procedure yields dose distributions identical to those obtained by assuming
that the energy loss is deposited at a constant rate along the step, but is computationally
simpler. According to this, penelope simulates the combined effect of all soft elastic
collisions and soft stopping interactions that occur between a pair of successive hard
events, separated a distance s, as a single event (a hinge) in which the particle changes
its direction of movement according to the distribution F a (s; µ), eqs. (4.30)-(4.32), and
loses energy ω that is generated from the distribution G a (s; ω), eqs. (4.59)-(4.63). The
position of the hinge is sampled uniformly along the step, as in the case of purely elastic
scattering (section 4.1.2). When the step crosses an interface (see fig. 4.2), the artificial
event is simulated only when its position lies in the initial material; otherwise the track
is stopped at the interface and restarted in the new material. It can be easily verified
that the particle reaches the interface not only with the correct average direction of
movement, but also with the correct average energy, E − S s t.
4.2.1 Energy dependence of the soft DCS
The simulation model for soft energy losses described above is based on the assumption
that the associated energy-loss DCS does not vary with the energy of the transported
particle. To account for the energy dependence of the DCS in a rigorous way, we have
to start from the transport equation [cf. eq. (4.48)]
∫
∂G(s; ω)
∞
= N G(s; ω − W ) σ s (E 0 − ω + W ; W ) dW
∂s
0
− N
∫ ∞
0
G(s; ω) σ s (E 0 − ω; W ) dW, (4.65)
where E 0 denotes the kinetic energy of the particle at the beginning of the step. We desire
to obtain expressions for the first and second moments, 〈ω〉 and 〈ω 2 〉, of the multiple