PENELOPE 2003 - OECD Nuclear Energy Agency

124 Chapter 4. Electron/positron transport mechanics
4.1 Elastic scattering
Let us start by considering electrons (or positrons) with kinetic energy E moving in a
hypothetical infinite homogeneous medium, with N scattering centres per unit volume,
in which they experience only pure elastic collisions (i.e. with no energy loss).
4.1.1 Multiple elastic scattering theory
Assume that an electron starts off from a certain position, which we select as the origin
of our reference frame, moving in the direction of the z-axis. Let f(s; r, ˆd) denote the
probability density of finding the electron at the position r = (x, y, z), moving in the
direction given by the unit vector ˆd after having travelled a path length s. The diffusion
equation for this problem is (Lewis, 1950)
∂f
∂s + ˆd
∫ [
· ∇f = N f(s; r, ˆd′ ) − f(s; r, ˆd) ] dσ el (θ)
dΩ, (4.1)
dΩ
where θ ≡ arccos(ˆd · ˆd ′ ) is the scattering angle corresponding to the angular deflection
ˆd ′ → ˆd. This equation has to be solved with the boundary condition f(0; r, ˆd) =
(1/π)δ(r)δ(1 − cos χ), where χ is the polar angle of the direction ˆd. By expanding
f(s; r, ˆd) in spherical harmonics, Lewis (1950) obtained exact expressions for the angular
distribution and for the first moments of the spatial distribution after a given path length
s. The probability density F (s; χ) of having a final direction in the solid angle element
dΩ around a direction defined by the polar angle χ is given by
∫
F (s; χ) =
f(s; r, ˆd) dr =
∞∑
l=0
2l + 1
4π exp(−s/λ el,l)P l (cos χ), (4.2)
where P l (cos χ) are Legendre polynomials and λ el,l = 1/(N σ el,l ) is the l-th transport
mean free path defined by eq. (3.14). The result given by eq. (4.2) coincides with the
multiple scattering distribution obtained by Goudsmit and Saunderson (1940a, 1940b).
Evidently, the distribution F (s; χ) is symmetric about the z-axis, i.e. independent of
the azimuthal angle of the final direction.
From the orthogonality of the Legendre polynomials, it follows that
∫ 1
〈P l (cos χ)〉 ≡ 2π P l (cos χ)F (s; χ) d(cos χ) = exp(−s/λ el,l ). (4.3)
−1
In particular, we have
and
〈cos χ〉 = exp(−s/λ el,1 ) (4.4)
〈cos 2 χ〉 = 1 3 [1 + 2 exp(−s/λ el,2)] . (4.5)