THE EGS5 CODE SYSTEM

That is, using Equations 2.122 and 2.123 in Equation 2.121, we obtain
( )/ ∣ f(E) = g h −1 (E) ∣∣
dh
dy (h−1 (x))
∣ (2.125)
= g(E/p)/p
1
=
ln 2
1
=
ln 2
( 1
p if E p ∈ (
0, 1 2
)
,
1 − E/p
E/p
( 1
p if E ∈ (0, p/2), 1 − E/p
E
· 1
( )) 1
p if E/p ∈ 2 , 1 )
if E ∈ (p/2, p) Q.E.D.
Thus we sample f 1j (E) by first sampling E ′ from Equation 2.121 and then letting E = E ′ p.
The g(E ′ ) in Equation 2.122 may be decomposed according to
g(E ′ ) =
2∑
i=1
α ′ i f ′ i (E′ ) , (2.126)
with
α ′ 1 = 1
2 ln 2 , f 1 ′ (E′ ) =
(
α ′ 2 = 1 − 1 )
(
, f ′
2 ln 2
2(E ′ 1
) =
(ln 2) − 1 2
( (
2 if E ′ ∈ 0, 1 ) )
, 0
2
· 1 − E′
E ′
if E ′ ∈
, (2.127)
( ) )
1
2 , 1 , 0
. (2.128)
We sample f 1 ′(E′ ) by letting E ′ = ζ/2, where ζ is a random number uniformly drawn on the interval
(0, 1). To sample f 2 ′(E′ ) we let E ′ = 1− 1 2x, x ∈ (0, 1), and sample x from the frequency distribution
function
h(x) = α ′′ f ′′ (x) g ′′ (x) (2.129)
where
α ′′ =
1
(4 ln 2) − 2 , f ′′ (x) = 2x, g ′′ (x) = 1
2 − x = 0.5
E ′ (2.130)
We already know how to sample f ′′ (x) (e.g., see Equation 2.111), so this completes the details of
sampling the bremsstrahlung spectrum.
Let us now consider the pair production interaction. The general developments are quite analogous
to the bremsstrahlung case and we obtain the following formulas:
E ≡ Ĕ
˘k , (2.131)
where Ĕ is the energy of one of the secondary electrons, ˘k is the incident photon energy,
∆ E =
1
˘kE(1 − E) , (2.132)
δ ′ = ∆ C ∆ E , (2.133)
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