THE EGS5 CODE SYSTEM

= ∆E
( ∣ ∣∣∣ dE
−1 ∣ )
∣∣∣
f(E 0 )
dE
−1
2 dx ∣ + f(E 1 )
E 0
dx ∣
E 1
Thus, the random energy hinge step distances are limited by the accuracy which can be achieved
in numerically integrating energy dependent quantities of interest using the trapezoid rule. This
limit suggests a prescription for determining the energy hinge step-sizes in EGS5: we take t as
the longest step-size which assures that Equation 2.379 is accurate to within a given tolerance ɛ fE
when applied to the integration of the following: the stopping power to compute the energy loss;
the scattering power to compute the scattering strength; and the hard collision cross section to
compute the hard collision total scattering probability and mean free path. Thus, in general, if
∆F | anal
is the analytic integral of one of our functions f over t, we wish to satisfy
or
or
∫ E1
E 0
dEf(E)
∣ ∣∣∣ dE
dx
∫ t
0
∆F | anal
− F (t) ≤ ɛ fE (2.379)
[ ]
ds f(t)
t
∣ −
anal 2 (f(E 0) + f(E 1 )) ≤ ɛ fE (2.380)
∣
∣−1 ∣ [
∣∣∣anal RC (E 0 ) − R C (E 1 )
−
2
where R C (E) is the CSDA range for an electron with energy E.
]
(f(E 0 ) + f(E 1 )) ≤ ɛ fE (2.381)
For the scattering strength, the function f is the scattering power G 1 , and for energy loss
f is the stopping power, in which case the analytical expression reduces simply to ∆E. Note
that in the case of the electron mean free path and total scattering probability, the expressions
for both the analytical function and the random hinge results are somewhat different from the
results described above, as the integrands for those quantities contain the spatial distribution of
the collision distances. For the random energy hinge methodology, the probability per unit path of
an interaction taking place over a step of length t is given by
p(s:h) = Σ 0 e −sΣ 0
s ≤ h, (2.382)
Σ 1 e −hΣ 0
e −(s−h)Σ 1
s > h
where h is the hinge distance, Σ 0 the cross section at the initial energy, and Σ 1 the cross section
after the energy hinge. The random hinge mean free path over t is then given as
λ Eh (t) =
=
=
=
=
∫ t
1
ds s p(s) (2.383)
P Eh (t) 0
∫
1 t ∫ s
ds s dh p(s:h) p(h)
P Eh (t) 0 0
∫
1 t ∫ s [ Σ0 (t − s)
ds s dh e −sΣ 0
+ Σ ]
1
P Eh (t) 0 0 st
t e−hΣ 0
e −(s−h)Σ 1
⎡
∫
1 t
ds s ⎣ Σ 0(t − s)
e −sΣ 0
+ Σ 1e −sΣ 1
(1 )
− e −s(Σ ⎤
0−Σ 1 )
⎦
P Eh (t) 0 t
t(Σ 0 − Σ 1 )
⎡
1
⎣ 1 − e−tΣ 1
(1 )
( )
− e −t(Σ 0−Σ 1 ) (
1 + 1 )
+ (Σ ⎤
0 − Σ 1 ) 1 − e −tΣ 0
P Eh (t) Σ 0 (Σ 0 − Σ 1 )
tΣ 1 tΣ 2 0 Σ ⎦
1
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