levels up to a $Q_{\beta^+}$ = 17.56 MeV. Thus, we are able to compare directly the experimental spectrum with the predicted one using a comparison between the branching ratios measured and those calculated (see figure \[fig:sum\_br\]).

We performed $sd$ shell-model calculations using the OXBASH code [@oxbash]. Two different interactions were used, the USD interaction [@wild] and the isospin non-conserving Ormand-Brown OB interaction [@ormand]. Only the OB interaction allowed us to calculate the particle decay from the IAS in $^{22}Mg$, because it is isospin forbidden. Since the ground state spin of $^{22}Al$ is not known, we performed two calculations, one for a spin-parity of $3^+$ and one for a spin-parity of $4^+$. For the $\beta$ transitions, we used effective single-particle matrix elements -an overall quenching factor of $0.76 \pm 0.03$ is included- for the Gamow-Teller operator [@brown85].

The phase-space factor $f$ for $\beta ^+$-decay was calculated by means of a program available at [@nndc]. For the particle transitions, it is necessary to calculate the spectroscopic factors $\theta ^2$ for all possible transitions from states in $^{22}Mg$ to states in the daughter nuclei. The partial widths were estimated in the standard way from the expression $\Gamma = 2 \theta ^2 \gamma ^2 