on the $\beta$ particle energy, which is most often not known.

    So, to extract the peak area we have used the method described here. First of all, we performed a Gaussian fit of the low energy part of the simulated proton spectrum, conditioned by the E5 detector (E5 &gt;0) as shown by the dashed line in fig. \[fig:pro\_bet\]. This fit allowed us to calculate the corresponding area. By normalising this area with the total peak area without any condition (the total number of simulated particles for this proton energy), we obtain the ratio of events selected by such a procedure. We have introduced two ratios:

    1.  $R_1 = \frac{N_{p (E5 \textgreater 0)}}{N_{p total}}$\
        This ratio is determined by dividing the total number of counts in the conditioned proton spectrum (E5 &gt;0) by the total number of counts in the proton spectrum without any condition. However, the charged-particle spectrum contains a $\beta$ contribution from $^{22}Al$ $\beta$-$\gamma$ transitions. To get the total number of protons, one has to subtract this contribution from the spectrum. In fact, simulations have shown that the $\beta$ contribution can be fitted by the sum of two exponential functions and subtracted from the unconditioned proton spectrum.\

    2.  $R_2 