THE EGS5 CODE SYSTEM

then we see that for k ≫ k c , F P goes to one; for k ≈ k c , it is about 1/2; and for k ≪ k c it goes as
k 2 /k 2 c . In the latter case, the k2 factor multiplied by the usual 1/k dependence results in an overall
k dependence as k → 0. Thus we have finite differential and total cross sections and the infrared
catastrophe is averted. It can be seen that the ratio of the cutoff energy to the incident electron
energy is independent of energy, and depends on the medium only through its electron density. For
lead, the ratio is
k c /E 0 =
√
nr 0 λ 2 0 /π (2.39)
= 1.195 × 10 −4
The natural log of the inverse of this ratio (≃ 9) is then approximately equal to the total bremsstrahlung
cross section in units of inverse radiation lengths (if one takes dσ/dk [X0 −1 ] = 1/k for
k > k c , = 0 for k < k c ).
As the corrections presented above have not been implemented in EGS5, it is instructive to
investigate the magnitude of the error expected this introduces. First, it is clear that at energies
above 100 GeV, ignoring the LPM suppression effect will have a significant impact on the predicted
gross behavior of a shower because of the over-estimation of the bremsstrahlung and pair production
cross sections. We therefore set 100 GeV as a safe upper limit to the present EGS5 version. Next,
we see that by ignoring the bremsstrahlung cutoff k c , EGS5 simulations of high energy electron
transport produce too many low energy secondary photons. For example, a 10 GeV electron should
not emit many photons below 1 MeV, whereas EGS5 would continue production down possibly as
low as 1 keV. This should not disturb the general shower behavior much, as there will be many
more low energy electrons than high energy electrons, so that the few extra low energy photons
produced by the high energy electrons should be insignificant compared to the number of low energy
photons produced by the lower energy electrons. It should be clear, however, that if the user were
using EGS5 to determine thin target bremsstrahlung spectra from high energy electrons, the results
would be in error below the cutoff k c .
Neglecting possible crystal diffraction effects[55, 170], the macroscopic cross section for bremsstrahlung
or pair production is given in terms of the microscopic cross sections, σ i , for the atoms
of type i by
Σ = N ∑
aρ ∑
i
p i σ i = N a ρ
p iσ i
M
∑i p . (2.40)
iA i
i
We see that the macroscopic cross sections do not depend on the absolute normalization of the p i ’s,
only the ratios. With the exception of ionization losses (where polarization effects are important),
Equation 2.40 is also valid for the other interactions that are considered (e.g., Møller, Compton,
etc.).
For conciseness in what follows we shall use the notation
(E 1 if B 1 , . . . , E n if B n , E n+1 ) (2.41)
to denote the conditional expression which takes E i for its value if B i is the first true expression,
and takes E n+1 for its value if no B i is true. We will also make use of the Kronecker delta function
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