THE EGS5 CODE SYSTEM

The general formula for the f (i) is (Bethe, Equation 26)
f (n) (θ) = (n!) −1
For n = 0 this reduces to
∫ ∞
0
[
n
udu J 0 (θu) × exp(−u 2 /4) 1/4 u 2 ln (u /4)] 2 . (2.282)
f (0) (θ) = 2e −θ2 . (2.283)
Instead of using the somewhat complicated expressions when n = 1 and 2, we have elected to use
a) the numerical values presented in Bethe’s paper (for 29 selected values of θ from 0 to 10), b) the
fact that f (i) (θ) behaves as θ −2i−2 for large θ, and c) the fact that f (1) (θ) goes over into the single
scattering law at large θ. That is,
lim f (1) (θ)θ 4 = 2. (2.284)
θ→∞
This also implies that
lim f (2) (θ)θ 4 = 0. (2.285)
θ→∞
The f (i) (θ) functions are not needed in EGS directly, but rather PEGS needs the f (i) (θ) to
create data that EGS does use. Let
η = 1/θ (2.286)
and
f (i)
η (η) = f (i) (1/η)η −4 = f (i) (θ(η))θ(η) 4 . (2.287)
As a result of Equations 2.285 and 2.286 we see that f η (1) (0) = 2 and f η
(2) (0) = 0. We now do a
cubic spline fit to f (i) (θ) for θɛ(0, 10) and f η
(i) (η) for ηɛ(0, 5). If we use ˆf (i) (i)
(θ) and ˆf η (η) to denote
these fits, then we evaluate the f (i) (θ) as
(
)
f (i) (θ) = ˆf (i) 1 (i)
(θ) if θ < 10, ˆf
θ 4 η (1/θ) . (2.288)
Similarly if we want f η
(i) (η) for arbitrary η we use
f (i)
η (η) = (
ˆf (i)
η (η) if η < 5, 1
η 4
)
ˆf (i) (1/η)
. (2.289)
To complete the mathematical definition of f(Θ) we now give additional formulas for the evaluation
of χ c and B. We have
B − ln B = b, (2.290)
b = ln Ω 0 , (2.291)
Ω 0 = b c t/β 2 , (2.292)
b c = ‘6680′ ρZ S e Z E/Z S
Me Z X/Z S
(2.293)
(Note, PEGS computes ˘b c = X 0 b c ),
( ) [ ]
¯h
2
(0.885)
‘6680 ′ 2
= 4πN a = 6702.33, (2.294)
m e c 1.167 × 1.13
84