Single-particle levels

Figure 3: Single-particle properties of neutron levels in $N$=32 isotones, calculated within the HFB method with the SLy4 Skyrme interaction. Left panels correspond to the standard SLy4 parametrization with no tensor terms ($t_e$=0) while the right panels correspond to the tensor-even interaction included ($t_e$=200MeVfm$^5$). Single-particle energies in the $sdpf$ shell (top panels) are shown along with their spin-orbit-splitting energies (middle panels) and centroid energies (bottom panels).

In Fig. 3 are shown properties of neutron single-particle levels calculated for the chain of the $N$=32 isotones (these levels are relevant for the changes of the shell structure discussed in Sec. 2). Single-particle energies $\varepsilon_N^{n{\ell}j}$ (top panels) were calculated as the canonical energies[21] of the HFB method. In order to better visualize the dependence of the single-particle energies on the proton number, in the middle and bottom panels are shown the SO splittings and centroids, respectively, of the SO partners, defined as

$$\begin{array}{rcl} \varepsilon_{\mbox{\scriptsize\rm {SO}}}^... ...ell}j_<} + \varepsilon_N^{n{\ell}j_>}\right) . \\ \end{array}$$ (11)

Without the tensor terms (left panels of Fig. 3), the neutron single-particle energies vary smoothly with the proton numbers. Apart from a gradual increase with decreasing $Z$, one observes two clear type of changes in the shell structure. First, in each shell the centroids of levels with different values of $\ell$ become degenerate towards the neutron drip line. This effect is related to the increase of the surface diffuseness of particle distributions, which renders the shell structure of very neutron-rich nuclei similar to that of a harmonic oscillator.[22] Second, near the neutron drip line the SO splitting of the weakly bound $p$ orbitals becomes smaller, because such orbitals start to decouple from the SO potential due to their increasing spatial dimensions.[23]

In the right panels of Fig. 3 are shown the analogous results obtained with the tensor-even interaction $\hat{V}_{Te}$, Eq. (1), taken into account for $t_e$=200MeVfm$^5$. (This particular value of the coupling constant was not optimized in any sense, and it is used here only to illustrate some general trends.) The use of the tensor-even interaction ($T$=0, $S$=1 neutron-proton channel) corresponds to the energy density that depends on the product of neutron and proton SO densities, cf. Eq. (3). Therefore, the effect of the tensor term vanishes at the closed proton shell $Z$=20 (SS system). For $Z$ higher (lower) than 20, the effect of the tensor term increases as a result of increasing contributions to the proton SO density coming from the $\pi$f$_{7/2}$ ($\pi$d$_{3/2}$) orbitals. This is clearly visible in the middle right panel of Fig. 3, where the SO splitting decreases on both sides of $^{52}$Ca. This is so because, for positive values of the coupling constant $t_e$, the effect of the tensor force partly cancels that of the standard SO force. As a result, the $\pi$f$_{5/2}$ orbital is in $^{60}$Ni much closer to the $p$ orbitals than it is in $^{54}$Ti; the shift which is compatible with the changes of the shell structure discussed in Sec. 2.

A detailed reproduction of the level positions is not the goal of the present study. The coupling constants of the tensor terms have to be adjusted together with other parameters of the EDF, by considering not only this particular region of nuclei, and not only this particular set of observables. Indeed, the tensor terms included in the EDF will influence many different global nuclear properties throughout the mass chart, and a global analysis is therefore necessary. Before this is done, in the next section, the impact of the tensor interaction on nuclear binding energies is studied in a preliminary way.