CERN Program Library Long Writeup W5013 - CERNLIB ...

Geant 3.16 GEANT User’s Guide PHYS350
Origin : L.Urbán Submitted: 26.10.84
Revision : G.Azuelos Revised: 16.12.93
Documentation :
1 Subroutines
Total cross-section for e+e- annihilation
CALL GANNII
GANNII tabulates the mean free path at initialisation as a function of the medium and of the energy (see
JMATE data structure). The energy binning is set within the array ELOW (common block /GCMULO/) in the
routine GPHYSI. GANNII is called from GPHYSI.
2 Method
For the annihilation into two photons the cross-section formula of Heitler is used. The total cross-section is
given by [12, 79]
[
γ 2 +4γ +1
γ 2 − 1
r 0 = classical electron radius
σ 2γ (Z, E) = Zπr2 0
γ +1
( √ )
ln γ + γ 2 − 1 − √ γ +3
]
γ 2 − 1
(1)
For compounds, the cross-section is calculated using an effective atomic number as explained in [PHYS010].
Positrons can annihilate in a single photon if the electron with which it interacts is bound to a nucleus. The
total cross-section for such a process is
σ 1γ (Z, E) = 4πr0α 2 4 Z 5 [
γβ(γ +1) 2 γ 2 + 2 3 γ + 4 3 + γ +2
]
γβ ln(γ + γβ)
α = fine structure constant
(2)
In the derivation of this formula, only the interactions with the K-shell electrons are taken into account. As
the cross-section depends on Z 5 , a special value of Z eff is computed in GPROBI:
Z eff = ∑ i
p i
A i
Z 5 (3)
The notation of [PHYS010] is used.
The total cross-section for the positron annihilation is σ = σ 2γ + σ 1γ . The value of σ 1γ is at it largest for
heavy materials, for example, it is ∼ 20 % of σ for a positron of 440 keV of kinetic energy in lead. For
lower and higher energies the probability is lower.
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