PENELOPE 2003 - OECD Nuclear Energy Agency

4.2. Soft energy losses 137
scattering energy-loss distribution, which define the artificial distribution G a (s; ω) as described
above. Unfortunately, for a realistic DCS, these moments can only be obtained
after arduous numerical calculations and we have to rely on simple approximations that
can be easily implemented in the simulation code.
Let us consider that, at least for relatively small fractional energy losses, the DCS
varies linearly with the kinetic energy of the particle,
σ s (E 0 − ω; W ) ≃ σ s (E 0 ; W ) −
[ ]
∂σs (E; W )
∂E
E=E 0
ω. (4.66)
We recall that we are considering only soft energy-loss interactions (inelastic collisions
and bremsstrahlung emission) for which the cutoff energies, W cc and W cr , do not vary
with E. Therefore, the upper limit of the integrals in the right hand side of eq. (4.65)
is finite and independent of the energy of the particle. The stopping power S s (E 0 − ω)
can then be approximated as
∫
S s (E 0 − ω) ≡ N W σ s (E 0 − ω; W ) dW ≃ S s (E 0 ) − S s ′ (E 0) ω, (4.67)
where the prime denotes the derivative with respect to E. Similarly, for the straggling
parameter Ω 2 s(E) we have
∫
Ω 2 s(E 0 − ω) ≡ N
W 2 σ s (E 0 − ω; W ) dW ≃ Ω 2 s(E 0 ) − Ω 2′ s (E 0 ) ω. (4.68)
From eq. (4.65) it follows that the moments of the multiple scattering distribution,
∫
〈ω n 〉 = ω n G(s; ω) dω,
satisfy the equations
∫
d
ds 〈ωn 〉 = N
∫
− N
∫
dω
∫
dω
dW [(ω + W ) n G(s; ω)σ s (E 0 − ω; W )]
dW ω n G(s; ω)σ s (E 0 − ω; W )
= N
n∑
k=1
∫
n!
k!(n − k)!
∫
dω
dW ω n−k W k G(s; ω)σ s (E 0 − ω; W ). (4.69)
By inserting the approximation (4.66), we obtain
d
ds 〈ωn 〉 =
n∑
k=1
n! (〈 〉 ω
n−k
M k − 〈 ω n−k+1〉 )
M k
′ , (4.70)
k!(n − k)!
where
∫
M k ≡ N
W k σ s (E 0 ; W ) dW (4.71)