PENELOPE 2003 - OECD Nuclear Energy Agency

44 Chapter 2. Photon interactions
hydrogenic electron wave functions. The Sauter DCS (per electron) can be written as
( )
dσ ph Z 5
= α 4 re 2 β 3 sin 2 [
θ e
1 + 1 ]
dΩ e κ γ (1 − β cos θ e ) 4 2 γ(γ − 1)(γ − 2)(1 − β cos θ e) , (2.24)
where α is the fine-structure constant, r e is the classical electron radius, and
√
E
γ = 1 + E e /(m e c 2 e (E e + 2m e c 2 )
), β =
. (2.25)
E e + m e c 2
Strictly speaking, the DCS (2.24) is adequate only for ionization of the K-shell by highenergy
photons. Nevertheless, in many practical simulations no appreciable errors are
introduced when Sauter’s distribution is used to describe any photoionization event,
irrespective of the atomic shell and the photon energy. The main reason is that the
emitted photoelectron immediately starts to interact with the medium, and its direction
of movement is strongly altered after travelling a path length much shorter than the
photon mean free path. On the other hand, when the photon energy exceeds the K-
edge, most of the ionizations occur in the K-shell and then the Sauter distribution
represents a good approximation.
Introducing the variable ν = 1 − cos θ e , the angular distribution of photoelectrons
can be expressed in the form
[
1
p(ν) = (2 − ν)
A + ν + 1 ]
2 βγ(γ − 1)(γ − 2) ν
(A + ν) , A = 1 − 1, (2.26)
3 β
apart from a normalization constant. Random sampling of ν from this distribution can
be performed analytically. To this end, p(ν) can be factorized in the form
with
and
p(ν) = g(ν)π(ν) (2.27)
[
1
g(ν) = (2 − ν)
A + ν + 1 ]
2 βγ(γ − 1)(γ − 2)
π(ν) =
A(A + 2)2
2
(2.28)
ν
(A + ν) 3 . (2.29)
The variable ν takes values in the interval (0,2), where the function g(ν) is definite
positive and attains its maximum value at ν = 0, while the function π(ν) is positive
and normalized to unity. Random values from the probability distribution π(ν) are
generated by means of the sampling formula (inverse transform method, see section
1.2.2) ∫ ν
π(ν ′ ) dν ′ = ξ, (2.30)
which can be solved analytically to give
ν =
0
2A
(A + 2) 2 − 4ξ
[
2ξ + (A + 2)ξ
1/2 ] . (2.31)
Therefore, random sampling from Sauter’s distribution can be performed by the rejection
method (see section 1.2.4) as follows: