PENELOPE 2003 - OECD Nuclear Energy Agency

128 Chapter 4. Electron/positron transport mechanics
⎡
= 2 s 3
⎣1 − 1 λ (s) ⎛ ⎞ ⎤
el,1 ⎝
s ⎠ + . . . ⎦ , (4.21)
9 λ (s) 2
el,2 λ (s)
el,2 λ (s)
el,1
which does not differ much from the exact value given by eq. (4.19). From these facts,
we may conclude that the random hinge method provides a faithful description of the
transport when the step length s is much shorter than the first transport mean free path
λ el,1 , so that the global angular deflection and lateral displacement are small. Surprisingly,
it does work well also in condensed (class I) simulations, where this requirement
is not met. In spite of its simplicity, the random hinge method competes in accuracy
and speed with other, much more sophisticated transport algorithms (see Bielajew and
Salvat, 2001, and references therein). It seems that the randomness of the hinge position
τ leads to correlations between the angular deflection and the displacement that
are close to the actual correlations.
1
2
➤
r
➤
^
r+τ d
χ
^
r+s d
➤
t
➤
Figure 4.2: Simulation of a track near the crossing of an interface.
The random hinge algorithm can be readily adapted to simulate multiple elastic
scattering processes in limited material structures, which may consist of several regions
of different compositions separated by well-defined surfaces (interfaces). In these geometries,
when the track crosses an interface, we simply stop it at the crossing point,
and resume the simulation in the new material. In spite of its simplicity, this recipe
gives a fairly accurate description of interface crossing. To see this, consider that a hard
collision has occurred at the position r in region “1” and assume that the following hard
collision occurs in region “2”. The step length s between these two hard collisions is
larger than the distance t from r to the interface (see fig. 4.2). If the artificial soft elastic
collision occurs in region “1”, the angular deflection in this collision is sampled from the
distribution F (s) (s; χ). Otherwise, the electron reaches the interface without changing
its direction of movement. Assuming s ≪ λ (s)
el,1 , the mean angular deflection due to soft
collisions is
1 − 〈cos χ〉 (s) = 1 − exp(−s/λ (s)
el,1 ) ≃
s
λ (s)
el,1
. (4.22)
Moreover, when this assumption is valid, lateral displacements due to soft collisions are
small and can be neglected to a first approximation. As the probability for the soft
collision to occur within region “1” equals t/s, the average angular deflection of the