THE EGS5 CODE SYSTEM

we have
Now define
δ ′ = ∆ C ∆ E . (2.88)
A(δ ′ ) = Â(δ′ /∆ C , Ĕ0)/Âmax(Ĕ0) , (2.89)
B(δ ′ ) = ˆB(δ ′ /∆ C , Ĕ0)/ ˆB max (Ĕ0) . (2.90)
Then A and B have maximum values of 1 at δ ′ = 0, and are thus candidate rejection functions.
We are now ready to explain the reason for introducing the parameter δ ′ and to introduce our
final approximation; namely, we assume that the ˆφ j can be obtained using
ˆφ j (δ ′ /∆ c ) =
N e ∑
i=1
p i Z i (Z i + ξ i )φ ′ j(Z i , Ĕ0, 136 Z −1/3
i mδ ′ /∆ c )
N e
≈ φ j (δ ′ ∑
) p i Z i (Z i + ξ i ) = φ j (δ ′ )Z T . (2.91)
i=1
In order to justify the reasonableness of this approximation, assume for all i that
Then using Equations 2.50 and 2.51 for the φ j we obtain
1 ≪ 136 Z −1/3
i mδ ′ /∆ c < δ max (Z i , Ĕ0) . (2.92)
N e ∑
i=1
p i Z i (Z i + ξ i )φ j (136 Z −1/3
i mδ ′ /∆ c )
=
≈
N e ∑
i=1
∑N e
i=1
[
((
p i Z i (Z i + ξ i ) 21.12 − 4.184 ln 136 Z −1/3 ) )]
i mδ ′ /∆ c + 0.952
[
p i Z i (Z i + ξ i ) 21.12 − 4.184 ln
(
136 Z −1/3
i mδ ′ /∆ c
)]
= [ 21.12 − 4.184 ln (136 mδ ′ /∆ c ) ] Z T − 4.184
= [ 21.12 − 4.184 (ln (136 mδ ′ /∆ c ) + Z G ) ] Z T
N e ∑
i=1
p i Z i (Z i + ξ i )ln Z −1/3
i
= ( 21.12 − 4.184 ln δ ′) Z T
≈
[ 21.12 − 4.184 ln (δ ′ + 0.952) ] Z T
= φ j (δ ′ )Z T . Q.E.D (2.93)
Butcher and Messel[39] make this approximation, although they don’t mention it explicitly. As
with our previous approximation, this approximation could be avoided (i.e., if we were willing to
have PEGS fit the A and B functions in some convenient way for EGS).
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