Tensor-force effects on single-particle levels and proton bubble structure around the $Z$ or $N=20$ magic number

TL;DR

This study uses semi-realistic tensor-inclusive interactions in Hartree-Fock calculations to explore single-particle levels and potential proton bubble structures around the $Z$ or $N=20$ magic number, emphasizing tensor force effects.

Contribution

It demonstrates the importance of tensor forces in reproducing experimental single-particle energy differences and predicts the delayed inversion of levels in calcium isotopes, also assessing bubble structures in argon and silicon isotopes.

Findings

Tensor force reproduces experimental energy differences in Ca isotopes.

Level inversion in Ca occurs only at N≥46 due to tensor effects.

Proton bubble structure is unlikely in Ar but possible in $^{34}$Si.

Abstract

Applying the semi-realistic $NN$ interactions that include realistic tensor force to the Hartree-Fock calculations, we investigate tensor-force effects on the single-particle levels in the Ca isotopes. The semi-realistic interaction successfully describes the experimental difference between $\varepsilon(p1s_{1/2})$ and $\varepsilon(p0d_{3/2})$ (denoted by $\mathit{\Delta}\varepsilon_{13}$) both at $^{40}$Ca and $^{48}$Ca, confirming importance of the tensor force. The tensor force plays a role in the $N$-dependence of $\mathit{\Delta}\varepsilon_{13}$ also in neutron-rich Ca nuclei. While the $p1s_{1/2}$-$p0d_{3/2}$ inversion is predicted in heavier Ca nuclei as in $^{48}$Ca, it takes place only in $N\geq 46$, delayed by the tensor force. We further investigate possibility of proton bubble structure in Ar, which is suggested by the $p1s_{1/2}$-$p0d_{3/2}$ inversion in $^{48}$Ca and more neutron-rich Ca nuclei, by the spherical Hartree-Fock-Bogolyubov calculations. Even with the inversion at $^{48}$Ca the pair correlation prohibits prominent bubble distribution in $^{46}$Ar. Bubble in Ar is unlikely also near neutron drip line because either of unboundness or of deformation. However, $^{34}$Si remains a candidate for proton bubble structure, owing to large shell gap between $p1s_{1/2}$ and $p0d_{5/2}$.