a 2.036 MeV proton. In the E4 spectrum, we can see that the $\beta$ particle associated with the transition has a strong effect. The proton peak is shifted by 36 keV when the positrons escape on the downstream side of E4, and by 158 keV when they escape in the upstream direction. These results do not depend on the $Q_{\beta}$ value in the 4-14 MeV range of interest, however, they depend strongly on the implantation depth in the E4 detector. If we use a condition on E5, we can mainly select the $\beta$ particles which escape on the downstream side. Such a condition selects the first peak and gets rid of the second one as shown by the dotted line in figure \[fig:pro\_bet\]. This technique leads to a much better energy resolution.\

-   The branching ratio determination:\
    In order to determine the proton branching ratios, we have to determine the area of each proton peak without any condition. However, it is difficult to fit the area of the proton peaks with a complex function that takes into account the exact shape of the double-bump distribution (see figure \[fig:pro\_bet\]). Moreover, simulations have shown that the exact shape of the peak depends 