THE EGS5 CODE SYSTEM

where P Eh (t) is the probability that there is a hard collision of any kind over t, p(h) is uniform and
given by 1/s, with the probability for a given s that we have yet to encounter the hinge given by
(t − s)/t and the probability that we are past the hinge given by s/t. P Eh (t) is given by
∫ t
∫ s
P Eh (t) = ds dh p(s:h) p(h) (2.384)
0 0
∫ t ∫ s [ Σ0 (t − s)
= ds dh e −sΣ 0
+ Σ ]
1
0 0 st
t e−hΣ 0
e −(s−h)Σ 1
⎡
∫ t
= ds ⎣ Σ 0(t − s)
e −sΣ 0
+ Σ 1e −sΣ 1
(1 )
− e −s(Σ ⎤
0−Σ 1 )
⎦
0 t
t(Σ 0 − Σ 1 )
= 1 − e−tΣ 1
(1 )
− e −t(Σ 0−Σ 1 )
t(Σ 0 − Σ 1 )
Note that the distribution of collision distances, p(s), for the random energy hinge can be seen from
the integrand in the above expressions to be
p(s) = Σ 0(t − s)
e −sΣ 0
+ Σ 1e −sΣ 1
(1 )
− e −s(Σ 0−Σ 1 )
(2.385)
t
t(Σ 0 − Σ 1 )
In the exact case for electrons passing through media with varying cross sections, we have, of
course,
λ(t) = 1 ∫ t
{ ∫ s }
ds s Σ(s) exp − ds ′ Σ(s ′ )
(2.386)
P (t) 0
0
with the expression for P (t), the probability of any scatter,
∫ t
{ ∫ s }
P (t) = ds Σ(s) exp − ds ′ Σ(s ′ )
0
0
Expressed in terms of energy loss steps rather than distance these become
∫ E1
{
λ(t) = dE (R C (E 0 ) − R C (E))
dE
−1
∫ E
∣ ∣∣∣ ∣ dx ∣ Σ(E) exp − dE ′ dE ′
E 0
dx ∣
E 0
−1
Σ(E ′ )
}
(2.387)
(2.388)
and
P (t) =
∫ E1
E 0
dE
dE
∣ dx ∣
−1
Σ(E) exp
{
−
∫ E
∣ ∣∣∣
dE ′ dE ′
E 0
dx
}
−1
∣ Σ(E ′ )
(2.389)
Note that in the above, we have described energy hinge steps in both terms of the change in
energy loss (from E 0 to E 1 ) and also in terms of distance traveled t, as convenient. In EGS5 we
use a simple prescription for relating the two and for switching back and forth. For a given initial
energy E 0 and a pathlength t, E 1 is given as E 0 − ∆E(t) with ∆E(t) computed as follows. A table
of electron CSDA ranges R C (E) is constructed as a function of energy as
R C (E) =
∫ E
0
110
dE ′ ∣ ∣∣∣ dE
dx
−1
∣ . (2.390)