THE EGS5 CODE SYSTEM

Actually, b = 0 does not correspond to t = 0, but rather to t ≈ 2 × 10 −6 X 0 . We nevertheless
set θ = 0 if b ≤ 0. The case where bɛ(0, 3) is not too likely either, since b = 3 roughly corresponds
to t ≈ 10 −4 X 0 , and is not very important since the scattering angles should be small. However,
Rogers[136] has found that in low energy applications, it is possible to take too small a step, thereby
running into this constraint and effectively turning off the multiple scattering.
To complete our discussion on sampling we note that f 1 (θ) is sampled directly by means of
θ = √ −ln ζ . (2.342)
The p.d.f. of f 2 (θ) is sampled by merely choosing a uniformly distributed random number. The
p.d.f. of ˆf3 (η) is sampled by taking the larger of two uniformly distributed random numbers.
Finally, g 2 (θ) and g η3 (η) are divided into “B-independent” parts
g 2 (θ) = g 21 (θ) + g 22 (θ)/B for θɛ(0, 1) , (2.343)
g η3 (θ) = g 31 (η) + g 32 (η)/B for ηɛ(0, 1) . (2.344)
The functions g 21 , g 22 , g 31 , and g 32 have been fit by PEGS over the interval (0,1) using a piecewise
quadratic fit. This completes our discussion of the method used to sample θ.
Note that the Bethe condition, χ 2 cB < 1, places a limit on the length of the electron step size
which can be accurately modeled using Molière’s multiple scattering p.d.f. We can determine the
maximum total step size consistent with this constraint, t B , by starting with
From Equation 2.290 we can write
χ 2 c (t B)B(t B ) = 1 . (2.345)
e b = e B /B , (2.346)
and using Equations 2.291, 2.292, 2.297 with Equations 2.345 and 2.346, we have
b c t B
β 2
= exp[(ĔMSβ 2 ) 2 /χ 2 cct B ]χ 2 cct B
(ĔMSβ 2 ) 2 . (2.347)
Solving for t B we obtain
t B = (ĔMSβ 2 /χ cc ) 2
ln[b c (ĔMSβ/χ cc ) 2 ] . (2.348)
2.14.2 The Goudsmit-Saunderson Multiple Scattering Distribution
It has long been acknowledged that Molière’s multiple scattering distribution breaks down under
certain conditions in addition to that given above in 2.348. In particular: the basic form of the
cross section assumed by Molière is in error in the MeV range, when spin and relativistic effects are
important; various approximations in Molière’s derivation lead to significant errors at pathlengths
less than 20 elastic scattering mean free paths (recall Equation 2.325); and the form of Molière’s
cross section is incapable of accurately modeling the structure in the elastic scattering cross section
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