PENELOPE 2003 - OECD Nuclear Energy Agency

3.4. Positron annihilation 119
(1985) transformed this DCS to the laboratory system (where the electron is at rest),
their result can be written as
dσ an
dζ
=
πre
2 [S(ζ) + S(1 − ζ)] , (3.163)
(γ + 1)(γ 2 − 1)
where
S(ζ) = −(γ + 1) 2 + (γ 2 + 4γ + 1) 1 ζ − 1 ζ2. (3.164)
Owing to the axial symmetry of the process, the DCS is independent of the azimuthal
angle φ − , which is uniformly distributed on the interval (0, 2π). For fast positrons,
annihilation photons are emitted preferentially at forward directions. When the kinetic
energy of the positron decreases, the angular distribution of the generated photons
becomes more isotropical (see fig. 3.17).
0. 9
0. 8
1 MeV
0. 7
1
p (θ) (rad −1 )
0. 6
0. 5
0. 4
10 0 k eV
10 k eV
σ an
(barn)
0 . 1
0. 3
0. 2
0. 1
0 . 0 1
0. 0
0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0
θ (deg)
1Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7 1Ε+8
E (eV)
Figure 3.17: Left: angular distributions of photons produced by annihilation in flight of
positrons with the indicated kinetic energies. The dashed line represents the isotropic distribution.
Right: Annihilation cross section per target electron as a function of the kinetic
energy of the positron.
The cross section (per target electron) for two-photon annihilation is
σ an =
∫ 1/2
×
dσ an
dζ dζ = πre
2
(γ + 1)(γ 2 − 1)
{
[
(γ 2 + 4γ + 1) ln γ + ( γ 2 − 1 ) 1/2 ] − (3 + γ) ( γ 2 − 1 ) 1/2 } . (3.165)
ζ min