PENELOPE 2003 - OECD Nuclear Energy Agency

ρ
ρ
ρ
λ
λ
λ
ρ
ρ
ρ
λ
λ
λ
3.1. Elastic collisions 75
paths. When E increases, the DCS becomes strongly peaked in the forward direction
and 〈µ 2 〉 becomes much smaller than 〈µ〉. In the high-energy limit, σ el,2 ≃ 3σ el,1 (λ el,2 ≃
λ el,1 /3). The total cross section, ∝ 1/(ρλ el ), decreases monotonously with E to reach a
constant value at high energies. This saturation is a relativistic effect: the total cross
section measures the interaction probability, which is proportional to the time spent
by the projectile within the region where the scattering field is appreciable. This time
is determined by the speed of the projectile, which approaches c from below when the
projectile energy increases. In the non-relativistic theory, the speed v n.r. = (2E/m e ) 1/2
increases without limit with E and the calculated non-relativistic total cross section
tends to zero at high energies.
1Ε+7
1Ε+7
1Ε+6
Al
el, 1
1Ε+6
Au
ρλ el
, ρλ el,1
, ρλ el,2
(µg /cm 2 )
1Ε+5
1Ε+4
1Ε+3
1Ε+2
1Ε+1
el, 2
el
ρλ el
, ρλ el,1
, ρλ el,2
(µg /cm 2 )
1Ε+5
1Ε+4
1Ε+3
1Ε+2
1Ε+1
el, 1
el, 2
el
1Ε+0
1Ε+0
1Ε− 1
1Ε− 1
1Ε+2 1Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7
1Ε+2 1Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7
E (eV)
E (eV)
Figure 3.3: Elastic mean free path, λ el , and first and second transport mean free paths, λ el,1
and λ el,2 , for electrons scattered in aluminium and gold as functions of the kinetic energy of
the projectile.
3.1.1 The modified Wentzel (MW) model
Although it is possible to do Monte Carlo simulation of electron and positron transport
using numerical partial-wave DCSs (Benedito et al., 2001), this procedure is too
laborious to be adopted as the basis of a simulation code for general purposes (mostly
because of the large volume of required numerical information). It is more convenient to
use suitable analytical approximate DCSs that may differ in detail from the partial-wave