PENELOPE 2003 - OECD Nuclear Energy Agency

74 Chapter 3. Electron and positron interactions
Notice that we can write
dσ el
dµ = σ el p el (µ), (3.11)
where p el (µ) is the normalized PDF of µ in a single collision. The mean free path
between consecutive elastic events in a homogeneous single-element medium is
where N is the number of atoms per unit volume.
λ el = 1/(N σ el ), (3.12)
Other important quantities (see section 4.1) are the transport cross sections 1
∫
σ el,l ≡
[1 − P l (cos θ)] dσ el
dΩ
The l-th transport mean free path is defined by
dΩ. (3.13)
λ el,l ≡ 1/(N σ el,l ). (3.14)
The first and second transport cross sections, σ el,1 and σ el,2 , are given by
and
σ el,2 =
∫
σ el,1 =
(1 − cos θ) dσ ∫ 1
el
dΩ dΩ = 2σ el µp el (µ) dµ = 2σ el 〈µ〉 (3.15)
0
∫ 3
2 (1−cos2 θ) dσ ∫ 1
el
dΩ dΩ = 6σ (
el (µ−µ 2 )p el (µ) dµ = 6σ el 〈µ〉 − 〈µ 2 〉 ) , (3.16)
0
where 〈· · ·〉 indicates the average value in a single collision. The quantities λ el,1 and
λ el,2 , eq. (3.14), determine the first and second moments of the multiple scattering
distributions (see section 4.1). The inverse of the first transport mean free path,
λ −1
el,1 = N σ el,1 = 2
λ el
〈µ〉, (3.17)
gives a measure of the average angular deflection per unit path length. By analogy with
the “stopping power”, which is defined as the mean energy loss per unit path length
(see section 3.2.3), the quantity 2λ −1
el,1 is sometimes called the “scattering power”2 .
Fig. 3.3 shows elastic mean free paths and transport mean free paths for electrons
in aluminium and gold. At low energies, the differences between the DCS of the two
elements (see fig. 3.2) produce very visible differences between the transport mean free
1 The Legendre polynomials of lowest orders are
P 0 (x) = 1, P 1 (x) = x, P 2 (x) = 1 2 (3x2 − 1).
2 At high energies, where the scattering is concentrated at very small angles, 〈µ〉 ≃ 〈θ 2 〉/4 and
λ −1
el,1 ≃ 〈θ2 〉/(2λ el ).