$\gamma$ ray at 1633.8 keV, corresponding to the transition from the first excited state to the ground state in $^{20}Ne$. This is a firm confirmation of the $\beta$-2p decay to the $^{20}Ne$ first excited state. The branching ratio of this $\beta$-2p decay is calculated from the number of counts in the 2p peak in the charged-particle spectrum conditioned by the $\gamma$ ray at 1633.8 keV. After correction for the $\gamma$ and proton detection efficiencies and normalisation to the total number of $^{22}Al$ implanted, we find a branching ratio of $0.69 \pm 0.08 \%$.

        The energy of the charged-particle peak $15^*$ ($E=4.017 \pm 0.008$ MeV) is compatible with the $\beta$-$\alpha$ transition measured by Blank [*et al.*]{} [@blank] at $E = 4.01 \pm 0.05$ MeV. The $\gamma$ spectrum conditioned by this peak shows a clear $\gamma$ ray at 1887 keV (see figure \[fig:coinc\_alpha\]). This $\gamma$-ray energy corresponds to the $\gamma$ transition from the first excited state to the $^{18}Ne$ ground state. Again, this observation is the first firm confirmation of this $\beta$-$\alpha$ transition. The branching ratio of the $\beta$-$\alpha$ decay to the first excited state of $^{18}Ne$ is $0.038 \pm 0.017~\%$.

Shell model calculations {#sec:5}
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Shell model calculations were performed to 