Carlo simulation {#sec:3}
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A Monte Carlo simulation of the experiment was performed using the GEANT code [@geant]. This simulation has produced four main conclusions about:

-   The proton detection efficiency:\
    High-energy protons can escape from the E4 implantation detector. The simulation allowed us to extract the proton detection efficiency of the E4 detector as a function of proton energy assuming an isotropic proton emission distribution. For energies from $2$ MeV to $8$ MeV, the efficiency decreases from 100% to 52% for an implantation depth of $420 \pm 10~\mu m$. The relative uncertainty of the proton detection efficiency depends on the uncertainty of the implantation depth in the silicon detector and on the uncertainty of the detector thickness. After a simulation using the extreme values of the implantation profile, the relative uncertainty of the proton detection efficiency was found to be less than 6%.\

-   The proton energy resolution:\
    This simulation and previous experiments have shown that a condition on the last silicon detector (E5 &gt;0) leads to a better energy resolution in the decay-energy spectrum. In figure \[fig:pro\_bet\], we have simulated the $\beta$ decay of $^{21}Mg$ to one state at 4.468 MeV in $^{21}Na$ followed by the emission of 