The $N=20$ and $N=28$ “islands of inversion” are described by large scale shell model calculations with an extension of the interaction SDPF-U that makes it possible to mix configurations with different $\mathrm{N}\hslash \omega$ or equivalently with different numbers of particles promoted from the $sd$ shell to the $pf$ shell. It allows to connect the classical $sd$-shell calculations below $N=18$ with the $sd$ (protons)-$pf$ (neutrons) calculations beyond $N=24–26$, for all the isotopes from oxygen to sulfur, using the same interaction. For some isotopes this range contains all the nuclei between the proton and the neutron drip lines and includes the $N=20$ and $N=28$ islands of inversion. We pay particular attention to the properties of the states at fixed $\mathrm{N}\hslash \omega$ which turn out to be the real protagonists of the physics at $N=20$. The existence of islands of inversion or deformation are explained as the result of the competition between the spherical mean field which favors the $0\hslash \omega$ configurations and the nuclear correlations which favor the deformed $N\hslash \omega$ configurations. The magnesium chain is exceptional because in it the $N=20$ and $N=28$ islands of inversion merge, enclosing all the isotopes between $N=19$ and $N=30$. Indeed, this would be also the case for the neon and sodium chains if their drip lines would reach $N=28$.

• Received 1 April 2014
• Revised 3 June 2014

DOI:https://doi.org/10.1103/PhysRevC.90.014302