Fitting procedure

Fitting procedure used to constrain the coupling constants $C^J_t$ and $C^{\nabla J}_t$ was described in detail in Ref. [22] and we only recall it here very briefly. The idea is to reproduce experimental $f_{7/2}-f_{5/2}$ SO splittings in three key nuclei: $^{40}$Ca, $^{48}$Ca, and $^{56}$Ni. Since $^{40}$Ca is, as discussed above, a SS system, the conventional SO term of Eq. (5) is the only source of the SO splitting. Hence, this nucleus is used to set the isoscalar strength of the SO term $C^{\nabla J}_0$. Having set $C^{\nabla J}_0$, we next constrain the $C^J_0$ strength by using the $f_{7/2}-f_{5/2}$ SO splitting in the isoscalar ($N=Z$) SUS nucleus $^{56}$Ni. Finally, we move to $^{48}$Ca, where protons and neutrons constitute a SS and SUS system, respectively. This nucleus is used to fit the isovector coupling constants or, more precisely, to fit $C^{J}_1$, because the ratio $C^{\nabla J}_0/C^{\nabla J}_1$ is kept equal to the value characteristic for the given Skyrme parametrization. There is one piece of data on the $f_{7/2}-f_{5/2}$ SO splittings, preferably in $^{48}$Ni or $^{78}$Ni, which is badly needed to fit the tensor and SO terms uniquely.

Figure 2: Empirical (horizontal lines) and theoretical (inclined lines) $f_{7/2}-f_{5/2}$ splittings in $^{40}$Ca, $^{56}$Ni, and $^{48}$Ca nuclei. The splittings in $^{40}$Ca (upper panel) are drown as a function of $C_0^{\nabla J} / C_0^{\nabla J}($SLy4$)$ isoscalar spin-orbit coupling constants ratio where $C_0^{\nabla J}($SLy4$)$ denotes the original SLy4 value. The splittings in $^{56}$Ni (middle panel) are plotted versus $C_0^J$ tensor coupling constant. Finally, the splittings in $^{48}$Ca (lower panel) are plotted versus $C_1^J$ tensor coupling constant. Solid and dashed lines represent neutron an proton splittings, respectively. The theoretical results are obtained by modifying the SO and tensor strengths in the SLy4 functional. Empirical data are taken from [27]. See text for further details.

The procedure outlined above is illustrated in Fig. 2 for the case of the SLy4 functional [28] but it is qualitatively independent of the initial parameterization. As shown in the figure, a good agreement with empirical data requires, for this low-effective-mass force, circa 35% reduction of $C_0^{\nabla J}$ as compared to the original value, a large attractive isoscalar tensor coupling constant $C_0^{J}$ of about $-45$MeVfm$^5$, and $C_1^{J}$ of about $-70$MeVfm$^5$. It appears that the resulting tensor coupling constants $C_t^{J}$ (as well as the SO strengths $C_0^{\nabla J}$) are, to large extent, independent on the initial parameterization. This is illustrated in Fig. 1 where different functionals modified according to our prescription, see Refs. [22,23,29,24], are collected. They are labeled by a subscript $T$ following the force acronym and marked by black triangles. Note, that the SkO$_T$ and SkO $_{T^\prime}$ parameterizations have slightly larger (smaller) values of the $C_u^J$ ($C_l^J$) coupling constants as compared to the other parameterizations, respectively. These values are needed in order to compensate for non-standard, very strong isovector spin-orbit strength characterizing the SkO based functionals. The SkO $_{T^\prime}$ functional has an interesting property. It appears to reproduce reasonably well both the SO splittings and masses of spherical nuclei [29,24]. This result seems to be at variance not only with our SLy4 $_{T\mbox{min}}$ functional fitted in Ref. [22], which reproduces masses of spherical nuclei much worse than the original SLy4 functional, but also with the results of systematic study in Ref. [11].