PENELOPE 2003 - OECD Nuclear Energy Agency

140 Chapter 4. Electron/positron transport mechanics
where
S(E) = S in (E) + S br (E) (4.84)
is the total stopping power. Since the mean free path between consecutive hard events of
any kind is shorter than the mean free path between hard elastic events, the energy loss
per step can be limited by re-defining the hard mean free path. If we wish to tolerate
average fractional energy losses ∆E/E along a step of the order of C 2 (a small value,
say, 0.05), we simply take
{
[
]}
λ (h)
E
el (E) = max λ el (E), min C 1 λ el,1 (E), C 2 . (4.85)
S(E)
This effectively limits the average energy loss per step at the expense of increasing the
frequency of hard elastic events. The parameters C 1 and C 2 in eq. (4.85), to be selected
by the user, determine the computer time needed to simulate each track. Ideally, they
should not have any influence on the accuracy of the simulation results. This happens
only when their values are sufficiently small (see below).
1Ε+1
1Ε+1
Al, electrons
Pb, electrons
1Ε+0
1Ε+0
ρ×length (g /cm 2 )
1Ε− 1
1Ε− 2
1Ε− 3
1Ε− 4
λ el,1
E / S
➤
(h)
λ el
ρ×length (g /cm 2 )
1Ε− 1
1Ε− 2
1Ε− 3
1Ε− 4
λ el,1
E / S
➤
(h)
λ el
λ el
λ el
1Ε− 5
1Ε− 5
1Ε− 6
1Ε− 6
1Ε+2 1Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7 1Ε+8
E (eV)
1Ε+2 1Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7 1Ε+8
E (eV)
Figure 4.3: Elastic mean free path λ el , first transport mean free path λ el,1 and E/S(E) for
electrons in aluminium and lead. The solid line represents the mean free path between hard
elastic events λ (h)
el
obtained from eq. (4.85) with C 1 = C 2 = 0.05.
It should be noted that C 1 and C 2 act on different energy domains. This is illustrated
in fig. 4.3, where the lengths λ el , λ el,1 and E/S for electrons in aluminium and lead are
represented as functions of the kinetic energy. The mean free path λ (h)
el for hard elastic