PENELOPE 2003 - OECD Nuclear Energy Agency

92 Chapter 3. Electron and positron interactions
where Q − and Q + are the minimum and maximum kinematically allowed recoil energies
given by eq. (A.31). The contributions from distant longitudinal and transverse
interactions are
dσ dis,l
dW
( = 2πe4 ∑ 1 Wk Q − + 2m e c 2 )
f
m e v 2 k ln
δ(W − W
W k Q − W k + 2m e c 2 k ) Θ(W k − Q − ) (3.77)
k
and
dσ dis,t
dW
{ ( )
} = 2πe4 ∑ 1 1
f
m e v 2 k ln − β 2 − δ
W k 1 − β 2 F δ(W − W k ) Θ(W k − Q − ), (3.78)
k
respectively. The energy-loss DCS for close collisions is
dσ (±)
clo
dW
= 2πe4 ∑ 1
f
m e v 2 k
W F (±) (E, W ) Θ(W − W 2 k ). (3.79)
k
Our analytical GOS model provides quite an accurate average description of inelastic
collisions (see below). However, the continuous energy loss spectrum associated with
single distant excitations of a given atomic electron shell is approximated here as a single
resonance (a δ-distribution). As a consequence, the simulated energy loss spectra show
unphysically narrow peaks at energy losses that are multiples of the resonance energies.
These spurious peaks are automatically smoothed out after multiple inelastic collisions
and also when the bin width used to tally the energy loss distributions is larger than
the difference between resonance energies of neighbouring oscillators.
where
The PDF of the energy loss in a single inelastic collision is given by
p in (W ) = 1
σ in
dσ in
dW , (3.80)
σ in =
∫ Wmax
0
dσ in
dW
is the total cross section for inelastic interactions.
quantities
σ (n)
in
≡
∫ Wmax
0
dW (3.81)
It is convenient to introduce the
W n dσ ∫
in
dW dW = σ Wmax
in W n p in (W ) dW = σ in 〈W n 〉, (3.82)
0
where 〈W n 〉 denotes the n-th moment of the energy loss in a single collision (notice
that σ (0)
in = σ in ). σ (1)
in and σ (2)
in are known as the stopping cross section and the energy
straggling cross section (for inelastic collisions), respectively.
The mean free path λ in for inelastic collisions is
λ −1
in = N σ in, (3.83)