PENELOPE 2003 - OECD Nuclear Energy Agency

120 Chapter 3. Electron and positron interactions
The annihilation mean free path is given by
λ −1
an = N Zσ an, (3.166)
where N Z is the density of electrons in the medium. The annihilation cross section is
displayed in fig. 3.17. The cross section decreases with the kinetic energy and, therefore,
high-energy positrons can travel path lengths of the order of the CSDA range before
annihilating.
3.4.1 Generation of emitted photons
The PDF of ζ is given by (normalization is irrelevant here)
p an (ζ) = S(ζ) + S(1 − ζ), ζ min ≤ ζ ≤ 1/2. (3.167)
To sample ζ, we may take advantage of the symmetry of this expression under the
exchange of the two photons, which corresponds to exchanging ζ and 1 − ζ. We first
consider the distribution
P (υ) ≡ S(υ), ζ min ≤ υ ≤ 1 − ζ min (3.168)
and write it in the form
P (υ) = π(υ)g(υ) (3.169)
with
and
g(υ) =
π(υ) =
[
ln
( )] −1
1 − ζmin 1
ζ min υ
(3.170)
[
−(γ + 1) 2 υ + (γ 2 + 4γ + 1) − 1 ]
. (3.171)
υ
π(υ) is a proper PDF (i.e. it is definite positive and normalized to unity) and g(υ) is a
monotonically decreasing function. Random values of υ from the distribution P (υ) can
be generated by using the following algorithm (rejection method):
(i) Sample a value υ from the distribution π(υ). This is easily done with the inverse
transform method, which yields the following sampling equation
υ = ζ min
(
1 − ζmin
ζ min
) ξ
. (3.172)
(ii) Generate a new random number ξ.
(iii) If ξg(ζ min ) > g(υ), go to step (i).
(iv) Deliver υ.