Recent applications of the Simple Effective Interaction: The N=50 and Z=28 shell closure and charge radii differences of mirror nuclei

We review the recent application of the so-called finite range simple effective interaction (SEI) to study the Ni and Cu isotopes with neutron numbers $N=40$ to $N=50$. Recent experiments performed in neutron-rich Cu isotopes have revealed a crossing between the $3/2^-$ and $5/2^-$ spin-parity of the ground state and first excited state in $^{75}$Cu. In contrast to the common situation with Skyrme and other effective forces, where a tensor interaction needs to be added to account for this crossing and the crossing of the 2$p_{3/2}$ and 1$f_{5/2}$ proton single-particle levels in Ni isotopes, we find that SEI can reproduce the experimental crossing without the necessity of adding a tensor interaction. We also study with SEI the correlation of the charge radii differences in mirror nuclei pairs with the neutron skin thickness and with the slope $L$ of the symmetry energy. By comparing the SEI predictions with the measured charge radii difference in four mirror pairs and with the NICER astrophysical constraint for the radius of 1.4\,M$_{\odot}$ neutron stars, we obtain the constraint $L\approx70$--100 MeV. This range for $L$ is discussed in light of the recent PREX-2 and CREX data on the neutron skin thickness.