Method

The type of annihilation is sampled from the total cross-sections for the annihilation into two photons and into one photon (see section [PHYS350]).

Annihilation into two photons

The differential cross-section of the two-photon positron-electron annihilation can be written as [,]:

where m is the electron mass Z is the atomic number of the material. If we indicate with E the initial energy of the positron, with $r$₀ the classical electron radius and with k the energy of the less energetic photon generated, we have:

$\gamma$	$=$	$\{Em\}$	$a$	$=$	$\gamma +1$
$\epsilon$	$=$	$\{kE+m\}$	$S(x)$	$=$	$C$1[ -1 + {C2x}-{1x2}]
$C$1	$=$	$\{Z\; \pi r$02a(E-m)}	$C$2	$=$	$a\; +\; \{e\; \gamma a\}$

The kinematical limits for the variable $\epsilon$ are:

$\epsilon$₀= {1a+ γ²-1}≤ε≤{12}

Due to the symmetry of the formula () in $\epsilon$ , the range of $\epsilon$ can be expanded from ($\epsilon$₀,1/2 ) to ($\epsilon$₀,1-ε₀ ) and the second function S can be eliminated from the formula. Having done this, the differential cross-section can be decomposed (apart from the normalisation) as: $\{d\; \sigma d\; \epsilon \}=\{1ln\{1\; -\; \epsilon$₀ε₀}}{1ε}_f(ε){(a²+2a-2)-a²ε-{1ε}a²-2ln{1 - ε₀ε₀}}_g(ε)

Figure: Comparison between the K-shell binding energies given by the expression in the text and the tabulated values.

Using the expression () with random numbers $r$_i∈]0,1[, i=1,2 , the secondary photon energy is sampled by the following steps:

1. sample $\epsilon$ from $f(\epsilon )$ : $\epsilon =\epsilon$₀exp[ ln({1- ε₀ε₀}) r₁]

2. compute the rejection function $g(\epsilon )$ and
   1. if $r$₂≤g(ε) accept $\epsilon$

   2. if $r$₁> g(ε) go back to 1.

and then GANNI generates the polar angles of the photon with respect to an axis defined by the momentum of the positron. The azimuthal angle $\Phi$ is generated isotropically and $\Theta$ is computed from the energy-momentum conservation. With this information, the momentum vector of both photons can be calculated and transformed into the GEANT coordinate system.

The routine GANNIR treates the special case when a positron falls below the cut-off energy ( CUTELE in common block /GCCUTS/) before annihilating. In this case, it is assumed that the positron comes to rest before annihilating. GANNIR generates two photons with energy k=m. The angular distribution is isotropic.

Annihilation into one photon

The generated photon is assumed to be collinear with the positron. Its energy will be $k\; =\; E\; +\; m$_e- E_bind

where $E$_bind is the binding energy of the K-shell electron. It can be estimated as follows $E$_bind= 0.5 (Z  α)² m_e

where $\alpha$ is the fine stucture constant. The comparison of this expression with the experimental data from [] is shown in figure .