Method

For the annihilation into two photons the cross-section formula of Heitler is used. The total cross-section is given by [,]

$\sigma$2γ(Z,E)	$=$	$\{Z\; \pi r$02γ+1}[{γ2+ 4 γ+1γ2-1}ln(γ+γ2-1) -{γ+3γ2-1}]
$r$0	$=$	$classical\; electron\; radius$

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

$\sigma$1γ(Z,E)	$=$	$4\; \pi r$02α4{Z5γβ(γ+ 1)2}[ γ2+ {23}γ+ {43}+ {γ+ 2γβ}ln(γ+ γβ) ]
$\alpha$	$=$	$fine\; structure\; constant$

In the derivation of this formula, only the interactions with the K-shell electrons are taken into account. As the cross-section depends on $Z5$ , a special value of $Z$_eff

is computed in GPROBI: $Z$_eff= ∑_i{p_iA_i}Z⁵

The notation of [PHYS010] is used.

The total cross-section for the positron annihilation is $\sigma =\; \sigma$_2γ+ σ_1γ . The value of $\sigma$_1γ is at it largest for heavy materials, for example, it is $\sim$ 20 % of $\sigma$ for a positron of 440 keV of kinetic energy in lead. For lower and higher energies the probability is lower.

PHYS351