THE EGS5 CODE SYSTEM

Hence,
so that
ln [1.13 + 3.76(αZ i ) 2 ] = ln 1.13 + ln[1 + 3.34(αZ i ) 2 ] . (2.320)
ln χ 2 α =
[
Z S ln
(
m
2
e e 4 )
]
1.13
p 2¯h 2 + Z
(0.885) 2 X − Z E ZS −1 , (2.321)
χ 2 α = m2 ee 4 1.13 e Z X/Z S
p 2¯h 2 (0.885) 2 e Z E/Z S
, (2.322)
Now, recalling that Equations 2.297 through 2.299 are equivalent to
χ 2 c = N aρ
M
and using Equations 2.302, 2.303, 2.322 and 2.323, we obtain
Ω 0 =
N aρ
M
(
4πr2 0Z S t m 2 ec 2 /E 2 β 4) , (2.323)
4πr2 0 Z Stm 2 e c4 (E 2 β 2 /c 2 )¯h 2 (0.885) 2 e Z E/Z S
m 2 e (r2 0 m2 e c4 )e Z X /Z SE 2 β 4 (1.167)(1.13)
= ‘6680 ′ [
ρZS e Z E/Z S
Me Z X/Z S
]
t
β 2 (2.324)
= b c t/β 2 . Q.E.D.
Molière’s B parameter is related to b by the transcendental Equation 2.290. For a given value
of b, the corresponding value of B may be found using Newton’s iteration method. As a rough
estimate, B = b+ln b. It can be seen that b, and hence B, increases logarithmically with increasing
transport distance.
The intuitive meaning of Ω 0 is that it may be thought of as the number of scatterings that take
place in the slab. If this number is too small, then the scattering is not truly multiple scattering and
various steps in Molière’s derivation become invalid. In Molière’s original paper[107], he considered
his theory valid for
Ω 0 (his Ω b ) ≥ 20 , (2.325)
which corresponds to
From Equations 2.325 and 2.325 we arrive at the condition
B ≥ 4.5, and b ≥ 3 . (2.326)
t/β 2 ≥ 20/b c = (t eff ) 0 .
Another restriction on the validity of Equation 2.279 is mentioned by Bethe[23], namely,
χ 2 c B < 1 . (2.327)
Resuming our presentation of the method used to sample Θ, we return now to the problem of
sampling θ from f r (θ) given by Equation 2.281. It might at first appear that f r (θ) is already
decomposed into sub-distribution functions. However, f (1) (θ) and f (2) (θ) are not always positive,
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