THE EGS5 CODE SYSTEM

Now suppose that the f(x) in Equation 2.62 is decomposed in a manner that allows for a
combination of “composition” and “rejection” techniques (e.g., see Equation 2.10; that is
or
where
/
dΣ(x 0 , x)
Σ(x 0 ) =
dx
dΣ(x 0 , x)
dx
=
N e ∑
i=1
N e ∑
i=1
α i = a ′ i Σ(x 0) .
α ′ i f i(x) g i (x) (2.63)
α i f i (x) g i (x) , (2.64)
But it can be shown (e.g., see Equation 2.7) that if all α i are multiplied by the same factor, it
will not change the function being sampled. We conclude that to properly sample from the p.d.f.
(Equation 2.62), it is not necessary to obtain Σ(x 0 ), but it is sufficient to decompose dΣ(x 0 , x)/dx
as shown in Equation 2.64 without bothering to normalize. We seek decompositions of the form
n∑
j=1
α j f j (x)g j (x) =
N e ∑
i=1
p i
dσ uncorrected (Z i , x 0 , x)
dx
. (2.65)
Let us now do this for the bremsstrahlung process (in a manner similar to that of Butcher and
Messel[39]) using Equation 2.43 for dσ i /dx with A ′ (Z i , Ĕ0) = 1. The first point to be noted[114] is
that φ 1 (δ) and φ 2 (δ) → 0 as δ → ∞, so that the expressions in square brackets in Equation 2.43
go to zero, and in fact go negative, at sufficiently large δ. As can be seen from Equation 2.51,
φ 2 (δ) = φ 1 (δ) for δ > 1. By checking numerical values it is seen that the value of δ for which the
expressions go to zero is greater than 1, so that both [ ] expressions in Equation 2.43 go to zero
simultaneously. Clearly the differential cross section must not be allowed to go negative, so this
imposes an upper kinematic limit δ max (Z i , Ĕ0) resulting in the condition (from Equation 2.50)
21.12 − 4.184 ln(δ max (Z, Ĕ0) + 0.952) − 4 3 ln Z − (4f c(Z) if Ĕ 0 > 50, 0) = 0. (2.66)
Solving for δ max (Z, Ĕ0) we obtain
[
δ max (Z, Ĕ0) = exp (21.12 − 4 ]
3 ln Z − (4f c(Z) if Ĕ 0 > 50, 0))/4.184 − 0.952 . (2.67)
From PEGS routines BREMDZ, BRMSDZ, and BRMSFZ, which compute dσ Brem (Z, Ĕ0, ˘k)/d˘k as given
by Equation 2.43, it may be seen that the result is set to zero if δ > δ max (Z, Ĕ0). Another way of
looking at this is to define
(
)
φ ′ i(Z, Ĕ0, δ) = φ i (δ) if δ ≤ δ max (Z, Ĕ0), φ i (δ max (Z, Ĕ0)) . (2.68)
We now define
E = ˘k/Ĕ0 (2.69)
44