Prediction of two-neutron halos in the $N=28$ isotones ${}^{40}$Mg and ${}^{39}$Na

Abstract

The ground states of the nuclei ${}^{40}$Mg and ${}^{39}$Na are investigated using the hyperspherical formalism. Since they are located at the edge of the "big island of inversion", we concentrate on whether we are likely to find two-neutron Borromean halos in these nuclei. A three-body model with effective $n$-$n$ and ${}^{38}$Mg$+n$ interactions is built for ${}^{40}$Mg based on the available data. We also give predictions for the low-lying spectrum of ${}^{38}$Na${=}^{37}$Na$+n$ and two-neutron separation energy of the ${}^{39}$Na nucleus. Depending on parameter choice, we report an increase in the matter radii in the range $0.1$-$0.5$ fm relative to those of the core nuclei. The results suggest a two-neutron halo structure in ${}^{40}$Mg for a subset of parameters, reinforcing the prediction of a Borromean halo nucleus. The calculations indicate that a two-neutron halo is even more likely for ${}^{39}$Na. As expected, the halo is linked to the disappearance of the shell gap in these nuclei due to the inversion of the $2p_{3/2}$ and $1f_{7/2}$ orbitals. We study the total cross section for scattering of these nuclei from a carbon target using a Glauber model and show that these provide a clear signal to assess the halo structure.

9 pages, 5 figures, 1 table (Final published version)

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