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:

$\gamma$	$=$	$\{Em\}$	$a$	$=$	$\gamma +1$
	$=$	$\{kE+m\}$	$S(x)$	$=$	$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 are:

₀= {1a+ γ²-1}≤&.gifi;≤{12}

Due to the symmetry of the formula () in , the range of can be expanded from (₀,1/2 ) to (₀,1-&.gifi;₀ ) 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: ₀&.gifi;₀}}{1&.gifi;}_f(&.gifi;){(a²+2a-2)-a²&.gifi;-{1&.gifi;}a²-2ln{1 - &.gifi;₀&.gifi;₀}}_g(&.gifi;)

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 st.gif:

1. sample from : ₀exp[ ln({1- &.gifi;₀&.gifi;₀}) r₁]

2. compute the rejection function and
   1. if $r$₂≤g(&.gifi;) accept

   2. if $r$₁> g(&.gifi;) 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 .