Magicity of the ${}^{52}\mathrm{Ca}$ and ${}^{54}\mathrm{Ca}$ isotopes and tensor contribution within a mean-field approach

I investigate the magicity of the isotopes ${}^{52}\mathrm{Ca}$ and ${}^{54}\mathrm{Ca}$, which was recently confirmed by two experimental measurements, and relate it to like-particle and neutron-proton tensor effects within a mean-field description. By analyzing Ca isotopes, it is shown that the like-particle tensor contribution induces shell effects that render these nuclei more magic than would be predicted by neglecting it. In particular, such induced shell effects are stronger in the ${}^{52}\mathrm{Ca}$ nucleus, and the single-particle gaps are increased in both isotopes due to the tensor force. By studying $N=32$ and $N=34$ isotones, neutron-proton tensor effects may be isolated and their role analyzed. It is shown that neutron-proton tensor effects lead to increasing $N=32$ and $N=34$ gaps, when going along isotonic chains, from ${}^{58}\mathrm{Fe}$ to ${}^{52}\mathrm{Ca}$ and from ${}^{60}\mathrm{Fe}$ to ${}^{54}\mathrm{Ca}$, respectively. Mean-field calculations are perfomed by employing one Skyrme parameter set, which was introduced in a previous work by fitting the tensor parameters together with the spin-orbit strength. The signs and values of the tensor strengths are thus checked within this specific application. The obtained results indicate that the employed parameter set, even if generated with a partial adjustment of the parameters of the force, leads to the correct shell behavior and provides, in particular, a description of the magicity of ${}^{52}\mathrm{Ca}$ and ${}^{54}\mathrm{Ca}$ within a pure mean-field picture with the effective two-body Skyrme interaction.

Figure 1

Single-particle neutron gaps for ${}^{40}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}$, ${}^{52}\mathrm{Ca}$, and ${}^{54}\mathrm{Ca}$. The solid black line corresponds to the results obtained with the new Skyrme force; the dashed black line corresponds to the results obtained with the new Skyrme force by neglecting the n-n tensor contribution. Results obtained using the original SLy5 force are also reported [dotted (blue) line]. Experimental values for ${}^{40}\mathrm{Ca}$ and ${}^{48}\mathrm{Ca}$ are represented by (red) triangles. (Grasso 2013): Ref. [1].

Figure 2

Neutron single-particle states for ${}^{48}\mathrm{Ca}$ (a), ${}^{52}\mathrm{Ca}$ (b), and ${}^{54}\mathrm{Ca}$ (c) with and without the like-particle tensor contribution.

Figure 3

Neutron gap $N=32$ calculated for the isotones ${}^{52}\mathrm{Ca}$, ${}^{54}\mathrm{Ti}$, ${}^{56}\mathrm{Cr}$, and ${}^{58}\mathrm{Fe}$. (Grasso 2013): Ref. [1].

Figure 4

Single-particle energies of the proton state $1f_{7/2}$ and of the neutron states $2p_{3/2}$ and $2p_{1/2}$ for the isotones ${}^{52}\mathrm{Ca}$, ${}^{54}\mathrm{Ti}$, ${}^{56}\mathrm{Cr}$, and ${}^{58}\mathrm{Fe}$.

Figure 5

Neutron gap $N=34$ calculated for the isotones ${}^{54}\mathrm{Ca}$, ${}^{56}\mathrm{Ti}$, ${}^{58}\mathrm{Cr}$, and ${}^{60}\mathrm{Fe}$, (Grasso 2013): Ref. [1].

Figure 6

Single-particle energies of the proton state $1f_{7/2}$ and of the neutron states $2p_{1/2}$ and $1f_{5/2}$ for the istones ${}^{54}\mathrm{Ca}$, ${}^{56}\mathrm{Ti}$, ${}^{58}\mathrm{Cr}$, and ${}^{60}\mathrm{Fe}$.

Figure 7

Two-neutron separation energies for the isotopes ${}^{50}\mathrm{Ca}$, ${}^{52}\mathrm{Ca}$, and ${}^{54}\mathrm{Ca}$ calculated with the new Skyrme set. Experimental values are reported [6, 16]. The theoretical value for the nucleus ${}^{56}\mathrm{Ca}$ is also shown. Inset: Difference between theoretical and experimental values. (Grasso 2013): Ref. [1].