Relative rates for reaction $\gamma -\gamma$ counts and positron annihilation

In the TaLL experiment, 68% of all the gamma-gamma events on tape were due to 511 keV coincidences in the single pair of opposite detectors at 90$^\circ$. This rate corresponds to an equivalent cross section of around 200 mbarns, which greatly exceeds the amount of beam stopped in the target or scattered at 90$^\circ$, and must be due to positrons from beam stopped elsewhere in the chamber but annihilating in view of this pair of detectors. It is useful to consider how this problem is changed when using a bigger array such as Exogam, which has many more opposite pairs of detectors.

If A is the rate of annihilation in view of each pair of opposite detectors, and there are $N_{{\rm opp}}$ such pairs, then the rate of back-to-back coincidences is $[(2\epsilon ) \times \epsilon _{{\rm int}} \times N_{{\rm opp}} \times A]$, where $\epsilon$ is the total absolute efficiency and $\epsilon _{{\rm int}}$ is the intrinsic efficiency, equal to $\epsilon$ divided by the geometrical factor $\epsilon _{{\rm geom}}$. Similarly, if R is the rate of reactions in the target, with average multiplicity $M _{\gamma }$, then with $N_{{\rm det}}$ detectors the coincidence rate is $[N_{{\rm det}} (N_{{\rm det}} -1) \times \epsilon ^2 \times (M_{\gamma } -1) \times R]$. The actual rate of reaction coincidences relative to annihilation coincidences, R/A, is then modified in the data by an enhancement factor given by

$$E=\frac{N_{{\rm det}} (N_{{\rm det}} -1)\ \epsilon ^2 \ (M_{\... ...}-1)}{2 \times N_{{\rm opp}} } \times \epsilon _{{\rm geom}}.$$

For the TaLL experiment $N_{{\rm det}}=7$, $N_{{\rm opp}}=1$ and $\epsilon _{{\rm geom}}=0.25$%, so for $M_{\gamma }=2$ this gives E=0.05. For Exogam $N_{{\rm det}}=16$, $N_{{\rm opp}}=8$ and $\epsilon _{{\rm geom}}=4.8$% and the corresponding figure is E=0.72. (For the Exogam-12 configuration $N_{{\rm det}}=12$, $N_{{\rm opp}}=4$ and E=0.79). Thus, the increased total detection efficiency outweighs the increase in opposite pairs and the relative rate of annihilation data is reduced by an order of magnitude compared to TaLL.