PENELOPE 2003 - OECD Nuclear Energy Agency

76 Chapter 3. Electron and positron interactions
DCSs but lead to nearly the same multiple scattering distributions. In penelope we
use a model in which the DCS is expressed as
dσ (MW)
el
dµ
= σ el p MW (µ). (3.18)
The single scattering distribution p MW (µ) is defined by a simple analytical expression,
with a physically plausible form, depending on two adjustable parameters. These parameters
are determined in such a way that the values of 〈µ〉 and 〈µ 2 〉 obtained from
p MW (µ) are equal to those of the actual (partial-wave) DCS:
and
〈µ 2 〉 MW ≡
〈µ〉 MW ≡
∫ 1
0
∫ 1
0
µp MW (µ) dµ = 〈µ〉 = 1 2
µ 2 p MW (µ) dµ = 〈µ 2 〉 = 1 2
σ el,1
σ el
(3.19)
σ el,1
− 1 σ el,2
. (3.20)
σ el 6 σ el
Thus, the MW model will give the same mean free path and the same first and second
transport mean free paths as the partial-wave DCS. As a consequence (see chapter
4), detailed simulations using this model will yield multiple scattering distributions
that do not differ significantly from those obtained from the partial wave DCS, quite
irrespectively of other details of the “artificial” distribution p MW (µ).
To set the distribution p MW (µ), we start from the Wentzel (1927) angular distribution,
p W,A0 (µ) ≡ A 0(1 + A 0 )
(µ + A 0 ) 2 , (A 0 > 0) (3.21)
which describes the scattering by an exponentially screened Coulomb field within the
Born approximation (see e.g. Mott and Massey, 1965), that is, it provides a physically
plausible angular distribution, at least for light elements or high-energy projectiles. It
is also worth mentioning that the multiple scattering theory of Molière (1947, 1948) can
be derived by assuming that electrons scatter according to the Wentzel distribution (see
Fernández-Varea et al., 1993b). The first moments of the Wentzel distribution are
〈µ〉 W,A0 =
∫ 1
0
µ A [
( ) ]
0(1 + A 0 )
1 +
(µ + A 0 ) dµ = A A0
2 0 (1 + A 0 ) ln − 1
A 0
(3.22)
and
〈µ 2 〉 W,A0 =
∫ 1
0
µ 2 A 0(1 + A 0 )
(µ + A 0 ) 2 dµ = A 0 [1 − 2〈µ〉 W,A0 ] . (3.23)
Let us define the value of the screening constant A 0 so that 〈µ〉 W,A0 = 〈µ〉. The value
of A 0 can be easily calculated by solving eq. (3.22) numerically, e.g. by the Newton-
Raphson method. Usually, we shall have 〈µ 2 〉 W,A0 ≠ 〈µ 2 〉. At low energies, the Wentzel
distribution that gives the correct average deflection is too “narrow” [〈µ 2 〉 W,A0 < 〈µ 2 〉
for both electrons and positrons and for all the elements]. At high energies, the angular
distribution is strongly peaked in the forward direction and the Wentzel distribution