PENELOPE 2003 - OECD Nuclear Energy Agency

4.1. Elastic scattering 125
Lewis (1950) also derived analytical formulae for the first moments of the spatial
distribution and the correlation function of z and cos χ. Neglecting energy losses, the
results explicitly given in Lewis’ paper simplify to
∫
〈z〉 ≡ 2π zf(s; r, ˆd) d(cos χ) dr = λ el,1 [1 − exp(−s/λ el,1 )] , (4.6)
〈x 2 + y 2 〉 ≡ 2π
∫ (
x 2 + y 2) f(s; r, ˆd) d(cos χ) dr
= 4 3
〈z cos χ〉 ≡ 2π
∫ s
0
∫
∫ t
dt exp(−t/λ el,1 ) [1 − exp(−u/λ el,2 )] exp(u/λ el,1 ) du, (4.7)
0
z cos χf(s; r, ˆd) d(cos χ) dr
∫ s
= exp(−s/λ el,1 ) [1 + 2 exp(−t/λ el,2 )] exp(t/λ el,1 ) dt. (4.8)
0
It is worth observing that the quantities (4.4)–(4.8) are completely determined by the
values of the transport mean free paths λ el,1 and λ el,2 ; they are independent of the elastic
mean free path λ el .
4.1.2 Mixed simulation of elastic scattering
At high energies, where detailed simulation becomes impractical, λ el,1 ≫ λ el (see fig.
3.3) so that the average angular deflection in each collision is small. In other words,
the great majority of elastic collisions of fast electrons are soft collisions with very small
deflections. We shall consider mixed simulation procedures (see Fernández-Varea et al.,
1993b; Baró et al., 1994b) in which hard collisions, with scattering angle θ larger than a
certain value θ c , are individually simulated and soft collisions (with θ < θ c ) are described
by means of a multiple scattering approach.
In practice, the mixed algorithm will be defined by specifying the mean free path
λ (h)
el between hard elastic events, defined by [see eqs. (3.10) and (3.12)]
1
λ (h)
el
∫ π dσ el (θ)
= N 2π sin θ dθ. (4.9)
θ c dΩ
This equation determines the cutoff angle θ c as a function of λ (h)
el . A convenient recipe
to set the mean free path λ (h)
el is
λ (h)
el (E) = max {λ el (E), C 1 λ el,1 (E)} , (4.10)
where C 1 is a pre-selected small constant (say, less than ∼ 0.1). For increasing energies,
λ el attains a constant value and λ el,1 increases steadily (see fig. 3.3) so that the formula
(4.10) gives a mean free path for hard collisions that increases with energy, i.e. hard
collisions are less frequent when the scattering effect is weaker. The recipe (4.10) also
ensures that λ (h)
el will reduce to the actual mean free path λ el for low energies. In this case,