PENELOPE 2003 - OECD Nuclear Energy Agency

118 Chapter 3. Electron and positron interactions
(iii) If ξ 1 ≥ A, then
4) Sample a random number ξ and set cos θ ′ = −1 + 2ξ.
5) Sample a random number ξ.
6) If ξ > 1 − cos 2 θ ′ , go to 4).
(iv) Deliver cos θ = cos θ′ + β ′
1 + β ′ cos θ ′ .
The efficiencies of the rejections in steps (ii) and (iii) are both equal to 0.66. That is,
on average, we need 4 random numbers to generate each value of cos θ.
3.4 Positron annihilation
Following Nelson et al. (1985), we consider that positrons penetrating a medium of
atomic number Z with kinetic energy E can annihilate with the electrons in the medium
by emission of two photons. We assume that the target electrons are free and at rest,
thus disregarding electron binding effects, which enable one-photon annihilation (Heitler,
1954). When annihilation occurs in flight, i.e. when the kinetic energy E of the positron
is larger than the “absorption” energy, the two photons may have different energies, say
E − and E + , which add to E +2m e c 2 . In what follows, quantities referring to the photon
with the lowest energy will be denoted by the subscript “−”. Each annihilation event
is then completely characterized by the quantity
ζ ≡
E −
E + 2m e c 2 . (3.159)
Assuming that the positron moves initially in the direction of the z-axis, from conservation
of energy and momentum it follows that the two photons are emitted in directions
with polar angles [see eqs. (A.21) and (A.22) in appendix A]
and
cos θ − = (γ 2 − 1) −1/2 (γ + 1 − 1/ζ) (3.160)
cos θ + = (γ 2 − 1) −1/2 [γ + 1 − 1/(1 − ζ)], (3.161)
and azimuthal angles φ − and φ + = φ − + π. The quantity γ = 1 + E/(m e c 2 ) is the total
energy of the positron in units of its rest energy.
The maximum value of ζ is 1/2, its minimum value is found when cos θ − = −1 and
is given by
1
ζ min =
γ + 1 + (γ 2 − 1)
1/2.
(3.162)
The DCS (per electron) for two-photon annihilation, as observed in the centre-ofmass
system of the positron and the electron, is given by Heitler (1954). Nelson et al.