THE EGS5 CODE SYSTEM

over the range of energy transfers which still give rise to soft final state secondary electrons, as in:
−
( ) dE±
dx Sub-Cutoff
Atomic Electrons
=
∫ Tmax
0
T dΣ ±
dT
dT . (2.254)
For values of T on the order of the atomic excitation levels, the frequencies and strengths
of the atomic oscillators must be taken into account and the evaluation of this integral is quite
complicated. For values of T large enough that the atomic electrons may be considered free, the
Møller or Bhabha cross sections can be used to describe the scattering. If we let T med be a value
of T which is sufficiently above the atomic excitation level, but is still small compared to T max , we
have ( )
− dE −
dx
=
∫ Tmed
0
T dΣ −
dT
dT + ∫ Tmax
T med
T dΣ Møller
dE
dT (2.255)
and (
− dE +
dx
)
=
∫ Tmed
0
T dΣ +
dT
dT + ∫ Tmax
T med
T dΣ Bhabha
dE
dE (2.256)
When appropriate approximations are made, Equations 2.224 and 2.225 are independent of
T med . We use the formulas recommended by Berger and Seltzer [17] for restricted stopping power 7
which are based on the Bethe-Bloch formula [21, 22, 36]. Note that we have corrected typographic
errors in Berger and Seltzer’s Equations 22-24 for F + (τ, ∆)). Other useful references regarding this
topic are Rohrlich and Carlson[140], Jauch and Rohrlich[83], Turner[173], Sternheimer[159, 160,
161, 162, 163, 165], Evans[53] (for Bhabha and Møller cross sections), Armstrong and Alsmiller[10],
and Messel and Crawford[103]. The formula used in PEGS for the restricted stopping power (i.e.,
due to sub-cutoff electrons) is
(
−X 0
dĔ±
dx
)
Sub-Cutoff
Atomic Electrons
= X 0n2πr0 2m
[
β 2 ln
]
2(τ + 2)
(Īadj/m) + F ± (τ, ∆) − δ
(2.257)
where γ = Ĕ 0 /m = usual relativistic factor,
√
(2.258)
η = γ 2 − 1 = βγ = ˘p 0 c/m,
√
(2.259)
β = 1 − γ −2 = v/c for incident particle, (2.260)
T ′ E = T E /m = K.E. cutoff in electron mass units, (2.261)
τ = γ − 1, (2.262)
y = (γ + 1) −1 (See Bhabha formula), (2.263)
T ′ max = maximum energy transfer
= (τ if positron, τ/2 if electron), (2.264)
∆ = restricted maximum energy transfer
= min(T ′ E, T ′ max), (2.265)
7 The concept of “restricted stopping power” is discussed in detail in the book by Kase and Nelson[87].
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