PENELOPE 2003 - OECD Nuclear Energy Agency

134 Chapter 4. Electron/positron transport mechanics
where Θ(x) is the step function. A closed formal solution of the integral equation (4.48)
may be obtained by considering its Fourier, or Laplace, transform with respect to ω (see
e.g. Landau, 1944, Blunck and Leisegang, 1950). For our purposes it is only necessary
to know the first moments of the energy loss distribution after the path length s,
From eq. (4.48) it follows that
∫
d
∞
ds 〈ωn 〉 = N dω
0
(∫ ∞
= N
=
n∑
k=1
0
∫ ∞
0
dω ′ ∫ ∞
0
〈ω n 〉 ≡
n!
k!(n − k)! 〈ωn−k 〉N
∫ ∞
0
ω n G(s; ω) dω. (4.50)
dW ω n [G(s; ω − W ) − G(s; ω)] σ s (E; W )
∫ ∞
dW (ω ′ + W ) n G(s; ω ′ )σ s (E; W ) − 〈ω n 〉
∫ ∞
0
0
)
σ s (E; W ) dW
W k σ s (E; W ) dW, (4.51)
where use has been made of the fact that σ s (E; W ) vanishes when W < 0. In particular,
we have
∫
d
∞
ds 〈ω〉 = N W σ s (E; W ) dW = S s , (4.52)
0
and, hence,
∫
d
∞
∫ ∞
ds 〈ω2 〉 = 2〈ω〉N W σ s (E; W ) dW + N W 2 σ s (E; W ) dW
0
0
= 2〈ω〉S s + Ω 2 s (4.53)
The variance of the energy loss distribution is
〈ω〉 = S s s, (4.54)
〈ω 2 〉 = (S s s) 2 + Ω 2 ss. (4.55)
var(ω) = 〈ω 2 〉 − 〈ω〉 2 = Ω 2 ss, (4.56)
i.e. the energy straggling parameter Ω 2 s equals the variance increase per unit path length.
The key point in our argument is that soft interactions involve only comparatively
small energy losses. If the number of soft interactions along the path length s is statistically
sufficient, it follows from the central limit theorem that the energy loss distribution
is Gaussian with mean S s s and variance Ω 2 ss, i.e.
[
1
G(s; ω) ≃
(2πΩ 2 s(E)s) exp − (ω − S s(E)s) 2 ]
1/2 2Ω 2 s (E)s . (4.57)
This result is accurate only if 1) the average energy loss S s (E)s is much smaller than
E (so that the DCS dσ s /dW is nearly constant along the step) and 2) its standard