PENELOPE 2003 - OECD Nuclear Energy Agency

3.2. Inelastic collisions 93
where N is the number of scattering centres (atoms or molecules) per unit volume. The
stopping power S in and the energy straggling parameter Ω 2 in are defined by
and
S in = N σ (1)
in = 〈W 〉
λ in
(3.84)
Ω 2 in = N σ (2)
in = 〈W 2 〉
λ in
. (3.85)
Notice that the stopping power gives the average energy loss per unit path length 4 . The
physical meaning of the straggling parameter is less direct. Consider a monoenergetic
electron (or positron) beam of energy E that impinges normally on a foil of material
of (small) thickness ds, and assume that the electrons do not scatter (i.e. they are
not deflected) in the foil. The product Ω 2 in ds then gives the variance of the energy
distribution of the beam after traversing the foil (see also section 4.2).
The integrated cross sections σ (n)
in
σ (n)
in
can be calculated as
= σ (n)
dis,l + σ (n)
dis,t + σ (n)
clo . (3.86)
The contributions from distant longitudinal and transverse interactions are
σ (n)
dis,l = 2πe4
m e v 2 ∑
k
f k W n−1
k
ln
(
Wk Q − + 2m e c 2 )
Θ(W
Q − W k + 2m e c 2 max − W k ) (3.87)
and
σ (n)
dis,t = 2πe4
m e v 2 ∑
k
f k W n−1
k
{
ln
( 1
1 − β 2 )
− β 2 − δ F
}
Θ(W max − W k ), (3.88)
respectively. Notice that for distant interactions W max
positrons.
The integrated cross sections for close collisions are
= E, for both electrons and
σ (n)
clo = 2πe4 ∑
m e v 2
k
f k
∫ Wmax
W k
W n−2 F (±) (E, W ) dW. (3.89)
In the case of electrons, the integrals in this formula are of the form
∫
J n (−) =
W n−2 [1 +
( ) W 2
(1 − a)W
−
E − W E − W + aW 2 ]
dW (3.90)
E 2
and can be calculated analytically. For the orders 0, 1 and 2 we have
J (−)
0 = − 1 W + 1
E − W + 1 − a ( ) E − W
E
ln + aW W E , (3.91)
2
4 The term “stopping power” is somewhat misleading; in fact, S in has the dimensions of force.