Spin-orbit splittings

Before proceeding to readjustments of coupling constants so as to improve the agreement of the SO splittings with data, we analyze the influence of time-even (mass and shape) and time-odd (spin) polarization effects on the neutron SO splittings. Based on results presented in the preceding Section, we calculate the SO splittings as

$$\Delta\epsilon_{\text{SO}}^{n{\ell}} = \epsilon_{n{\ell}j_<} - \epsilon_{n{\ell}j_>}.$$ (26)

Figure 2 shows the SO splittings calculated using SLy4$_{L}$ -- the functional based on the original SLy4 [19] functional with spin fields readjusted to reproduce empirical Landau parameters according to the prescription given in Refs. [21,22]. Plotted values correspond to results presented in Tables 1 and 2. The results are labeled according to the following convention: open symbols mark results computed directly from the s.p. spectra in doubly-magic nuclei (bare s.p. energies). These bare values contain no polarization effect. Gray symbols label the SO splittings involving polarization due to the time-even mass- and shape-driving effects, i.e., those obtained with all time-odd components in the functional set equal to zero. Black symbols illustrate fully self-consistent results obtained for the complete SLy4$_{L}$ functional. Gray and black symbols are shifted slightly to the left-hand (right-hand) side with respect to the doubly-magic core in order to indicate the hole (particle) character of the SO partners. Mixed cases involving the particle-hole SO partners are also shifted to the right.

Figure 2: Neutron SO splittings (26) calculated using the SLy4$_{L}$ functional. White, gray, and black symbols mark bare, mass and shape polarized (time-even), and mass, shape, and spin polarized (time-even and time-odd) results, respectively. Results for hole (particle and particle-hole) orbitals are shifted to the left (right) with respect to the core (open symbols). A typical discrepancy with experiment is shown by the arrow in $^{40}$Ca.

The impact of polarization effects on the SO splittings is indeed very small, particularly for the cases where both SO partners are of particle or hole type. Indeed, for these cases, the effect only exceptionally exceeds 200keV, reflecting a cancellation of polarization effects exerted on the $j=\ell\pm 1/2$ partners. The smallness of polarization effects hardly allows for any systematic trends to be pinned down. Nevertheless, the self-consistent results show a weak but relatively clear tendency to slightly enlarge or diminish the splitting for hole or particle states, respectively.

The situation is clearer when the SO partners are of mixed particle ( $j=\ell-1/2$) and hole ( $j=\ell + 1/2$) character. In these cases, the shape polarization tends to diminish the splitting quite systematically by about 400-500keV. This behavior follows from naive deformed Nilsson model picture where the highest-$K$ members of the $j=\ell-1/2$ ( $j=\ell + 1/2$) multiplet slopes down (up) as a function of the oblate (prolate) deformation parameter. As discussed in the previous Section, the time-odd fields act in the opposite way, tending to slightly enlarge the gap. The net polarization effect does not seem to exceed about 300keV. In these cases, however, we deal with large $\ell$ orbitals having also quite large SO splittings of the order of $\sim$8MeV. Hence, the relative corrections due to polarization effects do not exceed about 4%, i.e., they are relatively small - much smaller than the effects of tensor terms discussed below and the discrepancy with data, which in Fig. 2 is indicated for the neutron 1f SO splitting in $^{40}$Ca. These results legitimate the direct use of bare s.p. spectra in magic cores for further studies of the SO splittings, which considerably facilitates the calculations.

As already mentioned, empirical s.p. energies are essentially deduced from differences between binding energies of doubly-magic core and their odd-$A$ neighbors. Different authors, however, use also one-particle transfer data, apply phenomenological particle-vibration corrections and/or treat slightly differently fragmented levels. Hence, published compilations of the s.p. energies, and in turn the SO splittings, differ slightly from one another depending on the assumed strategy. The typical uncertainties in the empirical SO splittings can be inferred from Table 3, which summarizes the available data on the SO splittings based on three recent s.p. level compilations published in Refs. [7,8,9].

Table 3: Empirical SO splittings. Compilation is based on the empirical s.p. levels taken from Refs. [7,8,9].

Nucleus	orbitals	Ref. [7]	Ref. [8]	Ref. [9]
$^{ 16}$O	$\nu 1p_{3/2}^{-1}-\nu 1p_{1/2}^{-1}$	6.17	6.18	$--$
	$\nu 1d_{5/2}- \nu 1d_{3/2}$	5.08	5.72	5.08
	$\pi 1p_{3/2}^{-1}-\pi 1p_{1/2}^{-1}$	6.32	6.32	$--$
	$\pi 1d_{5/2}- \pi 1d_{3/2}$	5.00	4.97	5.00
$^{40}$Ca	$\nu 2p_{3/2}- \nu 2p_{1/2}$	2.00	2.00	1.54
	$\nu 1f_{7/2}- \nu 1f_{5/2}$	4.88	5.24	5.64
	$\nu 1d_{5/2}^{-1}-\nu 1d_{3/2}^{-1}$	6.00	6.75	6.75
	$\pi 2p_{3/2}- \pi 2p_{1/2}$	2.01	1.72	1.69
	$\pi 1f_{7/2}- \pi 1f_{5/2}$	4.95	5.41	6.05
	$\pi 1d_{5/2}^{-1}-\pi 1d_{3/2}^{-1}$	6.00	5.94	6.74
$^{48}$Ca	$\nu 2p_{3/2}- \nu 2p_{1/2}$	$--$	1.67	1.77
	$\nu 1f_{7/2}^{-1}- \nu 1f_{5/2}$	$--$	8.01	8.80
	$\nu 1d_{5/2}^{-1}-\nu 1d_{3/2}^{-1}$	$--$	5.30	3.08
	$\pi 2p_{3/2}- \pi 2p_{1/2}$	$--$	2.14	1.77
	$\pi 1f_{7/2}- \pi 1f_{5/2}$	$--$	4.92	$--$
	$\pi 1d_{5/2}^{-1}-\pi 1d_{3/2}^{-1}$	$--$	5.01	5.29
$^{56}$Ni	$\nu 2p_{3/2}- \nu 2p_{1/2}$	$--$	1.88	1.12
	$\nu 1f_{7/2}^{-1}- \nu 1f_{5/2}$	$--$	6.82	7.16
	$\pi 2p_{3/2}- \pi 2p_{1/2}$	$--$	1.83	1.11
	$\pi 1f_{7/2}^{-1}- \pi 1f_{5/2}$	$--$	7.01	7.50
$^{90}$Zr	$\nu 2d_{5/2}- \nu 2d_{3/2}$	$--$	2.43	$--$
	$\nu 1g_{9/2}^{-1}- \nu 1g_{7/2}$	$--$	7.07	$--$
	$\nu 2p_{3/2}^{-1}- \nu 2p_{1/2}^{-1}$	$--$	0.37	$--$
	$\nu 1f_{7/2}^{-1}- \nu 1f_{5/2}^{-1}$	$--$	1.71	$--$
	$\pi 2d_{5/2}- \pi 2d_{3/2}$	$--$	2.03	$--$
	$\pi 1g_{9/2}- \pi 1g_{7/2}$	$--$	5.56	$--$
	$\pi 2p_{3/2}^{-1}- \pi 2p_{1/2}^{-1}$	$--$	1.50	$--$
	$\pi 1f_{7/2}^{-1}- \pi 1f_{5/2}^{-1}$	$--$	4.56	$--$
$^{100}$Sn	$\nu 2d_{5/2}- \nu 2d_{3/2}$	1.93	$--$	1.93
	$\nu 1g_{9/2}^{-1}- \nu 1g_{7/2}$	7.00	$--$	7.00
	$\pi 1g_{9/2}^{-1}- \pi 1g_{7/2}$	6.82	$--$	6.82
	$\pi 2p_{3/2}^{-1}- \pi 2p_{3/2}^{-1}$	2.85	$--$	2.85
$^{132}$Sn	$\nu 2f_{7/2}- \nu 2f_{5/2}$	2.00	1.94	$--$
	$\nu 3p_{3/2}- \nu 3p_{1/2}$	0.81	0.59	1.15
	$\nu 1h_{11/2}^{-1}- \nu 1h_{9/2}$	6.53	6.51	6.68
	$\nu 2d_{5/2}^{-1}- \nu 2d_{3/2}^{-1}$	1.65	1.93	1.66
	$\pi 2d_{5/2}- \pi 2d_{3/2}$	1.48	1.83	1.75
	$\pi 1g_{9/2}^{-1}- \pi 1g_{7/2}$	6.08	5.33	6.08
$^{208}$Pb	$\nu 3d_{5/2}- \nu 3d_{3/2}$	0.97	0.89	0.97
	$\nu 2g_{9/2}- \nu 2g_{7/2}$	2.50	2.38	2.50
	$\nu 1i_{13/2}^{-1}- \nu 11_{11/2}$	5.84	5.81	6.08
	$\nu 3p_{3/2}^{-1}- \nu 3p_{1/2}^{-1}$	0.90	0.90	0.89
	$\nu 2f_{7/2}^{-1}- \nu 2f_{5/2}^{-1}$	2.13	2.18	1.87
	$\pi 2f_{7/2}- \pi 2f_{5/2}$	1.93	2.02	1.93
	$\pi 3p_{3/2}- \pi 3p_{1/2}$	0.85	0.45	0.52
	$\pi 1h_{11/2}^{-1}- \pi 1h_{9/2}$	5.56	5.03	5.56
	$\pi 2d_{5/2}^{-1}- \pi 2d_{3/2}^{-1}$	1.68	1.62	1.46

Figure 3: Figure illustrates the three-step procedure used to fit the isoscalar SO coupling constant $C^{\nabla J}_0$ in $^{40}$Ca (upper panel), the isoscalar tensor strength $C_0^J$ in $^{56}$Ni (middle panel), and the isovector tensor strength $C_1^J$ in $^{48}$Ca (lowest panel). These particular calculations have been done for the SkP functional, but the pattern is common for all the analyzed parameterizations including SLy4 and SkO. See text for further details.

Instead of large-scale fit to the data (see, e.g., Ref. [16]), we propose a simple three-step method to adjust three coupling constants $C_0^{\nabla J}$, $C_0^J$, and $C_1^J$. The entire idea of this procedure is based on the observation that the empirical $1f_{7/2}-1f_{5/2}$ SO splittings in $^{40}$Ca, $^{56}$Ni, and $^{48}$Ca form very distinct pattern, which cannot be reproduced by using solely the conventional SO interaction.

The readjustment is done in the following way. First, experimental data in the spin-saturated (SS) nucleus $^{40}$Ca are used in order to fit the isoscalar SO coupling constant $C^{\nabla J}_0$. One should note that in this nucleus, the SO splitting depends only on $C^{\nabla J}_0$, and not on $C^J_0$ (because of the spin saturation), nor on $C^{\nabla J}_1$ (because of the isospin invariance at $N=Z$), nor on $C^J_1$ (because of both reasons above). Therefore, here one experimental number determines one particular coupling constants.

Second, once $C^{\nabla J}_0$ is fixed, the spin-unsaturated (SUS) $N=Z$ nucleus $^{56}$Ni is used to establish the isoscalar tensor coupling constant $C_0^J$. Again here, because of the isospin invariance, the SO splitting is independent of either of the two isovector coupling constants, $C^{\nabla J}_1$ or $C^J_1$. Finally, in the third step, $^{48}$Ca is used to adjust the isovector tensor coupling constant $C_1^J$. Such a procedure exemplifies the focus of fit on the s.p. properties, as discussed in the Introduction.

It turns out that current experimental data, and in particular lack of information in $^{48}$Ni, do not allow for adjusting the fourth coupling constant, $C^{\nabla J}_1$. For this reason, in the present study we fix it by keeping the ratio of $C^{\nabla J}_0/C^{\nabla J}_1$ equal to that of the given standard Skyrme force. In the process of fitting, all the remaining time-even coupling constants $C_t$ are kept unchanged. Variants of the standard functionals obtained in this way are below denoted by SkP$_{T}$, SLy4$_{T}$, and SkO$_{T}$. When the time-odd channels, modified so as to reproduce the Landau parameters, are active, we also use notation SkP$_{LT}$, SLy4$_{LT}$, and SkO$_{LT}$.

For the SkP functional, the procedure is illustrated in Fig. 3. We start with the isoscalar $N=Z$ nucleus $^{40}$Ca. The evolution of the SO splittings in function of $f^{\nabla J}$, which is the factor scaling the original SkP coupling constant $C_0^{\nabla J}$, is shown in the upper panel of Fig. 3. As it is clearly seen from the Figure, fair agreement with data requires about 20% reduction in the conventional SO interaction strength $C_0^{\nabla J}$, cf. results of the recent study in Ref. [38]. It should also be noted that the reduction in the SO interaction considerably improves the $1f_{7/2}-1d_{3/2}$ and $1f_{7/2}-2p_{3/2}$ splittings but slightly spoils the $2p_{3/2}-2p_{1/2}$ SO splitting. Qualitatively, similar results were obtained for the SLy4 and SkO interactions. Reasonable agreement to the data requires $\sim$20% reduction of the original $C_0^{\nabla J}$ in case of the SkO interaction and quite drastic $\sim$35% reduction of the original $C_0^{\nabla J}$ in case of the SLy4 force.

Having fixed $C_0^{\nabla J}$ in $^{40}$Ca we move to the isoscalar nucleus $^{56}$Ni. This nucleus is spin-unsaturated and therefore is very sensitive to the isoscalar $C_0^J$ tensor coupling constant. The evolution of theoretical s.p. levels versus $C_0^J$ is illustrated in the middle panel of Fig. 3. As shown in the Figure, reasonable agreement between the empirical and theoretical $1f_{7/2}-1f_{5/2}$ SO splitting is achieved for $C_0^J\sim -40$MeVfm$^5$, which by a factor of about five exceeds the original SkP value for this coupling constant. It is striking that a similar value of $C_0^J$ is obtained in the analogical analysis performed for the SLy4 interaction.

Finally, the isovector tensor coupling constant $C_1^{\nabla J}$ is established in $N$$\ne$$Z$ nucleus $^{48}$Ca. The evolution of theoretical neutron s.p. levels versus $C_1^J$ is illustrated in the lowest panel of Fig. 3. As shown in the Figure, the value of $C_1^{\nabla J}\sim -70$MeVfm$^5$ is needed to reach reasonable agreement for the $1\nu f_{7/2}-1\nu f_{5/2}$ SO splitting in this case. For this value of the $C_1^J$ strength one obtains also good agreement for the proton $1\pi f_{7/2}-1\pi f_{5/2}$ SO splitting (see Figs. 4 and 5 below), without any further readjustment of the $C_1^{\nabla J}$ strength. Again, very similar value for the $C_1^J$ strength is deduced for the SLy4 force. Note also the improvement in the $1f_{7/2}-1d_{3/2}$ splitting caused by the isovector tensor interaction. Dotted lines show results obtained from the mass differences, i.e., with all the polarization effects included.

During the fitting procedure all the remaining functional coupling constants were kept fixed at their Skyrme values. The ratio of the isoscalar to the isovector coupling constant in the SO interaction channel was locked to its standard Skyrme value of $C_0^{\nabla J}/C_1^{\nabla J}=3$. Since no clear indication for relaxing this condition is seen (see also Figs. 4 and 5 below), we have decided to investigate the isovector degree of freedom in the SO interaction (see [17]) by performing our three-step fitting process also for the generalized Skyrme interaction SkO [20], for which $C_0^{\nabla J}/C_1^{\nabla J}\sim -0.78$.

Table 4: Spin-orbit $C^{\nabla J}$ and tensor isoscalar $C_{0}^J$ and isovector $C_{1}^J$ functional coupling constants adopted in this work and subsequently used in Figs. 4, 5, and 6, where global calculations of the SO splittings are presented.

Skyrme	$C_{0}^{\nabla J}$	$C_{0}^{\nabla J}/C_{1}^{\nabla J}$	$C_{0}^J$	$C_{1}^J$
force	[MeVfm$^5$]		[MeVfm$^5$]	[MeVfm$^5$]
SkP$_{T}$	$-$60.0	3	$-$38.6	$-$61.7
SLy4$_{T}$	$-$60.0	3	$-$45.0	$-$60.0
SkO$_{T}$	$-$61.8	$-$0.78	$-$33.1	$-$91.6

All the adopted functional coupling constants resulting from our calculations are collected in Table 4. Note, that the procedure leads to essentially identical SO interaction strengths $C_0^{\nabla J}$ for all three forces irrespective of their intrinsic differences, for example in effective masses. The tensor coupling constants in both the SkP and the SLy4 functionals are also very similar. In the SkO case, one observes rather clear enhancement in the isovector tensor coupling constant which becomes more negative to, most likely, counterbalance the non-standard positive strength in the isovector SO channel.

Figure 4: Experimental [8] (black symbols) and theoretical SO splittings calculated using the original SkP functional (gray symbols) and our modified SkP$_{LT}$ functional (white symbols) with the SO and tensor coupling constants given in Table 4. Upper left and right panels show neutron SO splittings for low- $\ell = 1,2$ ($p$ and $d$) and high- $\ell \geq 3$ orbitals, respectively. Analogical information but for proton SO splittings is depicted in the lower panels.

Figure 5: Same as in Fig. 4 but for the SLy4 functional.

Figure 6: Same as in Fig. 4 but for the SkO functional.

The functionals were modified using only three specific pieces of data on the neutron $1f_{7/2}-1f_{5/2}$ SO splittings. In order to verify the reliability of the modifications, we have performed systematic calculations of the experimentally accessible SO splittings. The results are depicted in Figs. 4, 5, and 6 for the SkP, SLy4, and SkO functionals, respectively. Additionally, Fig. 7 shows neutron and proton magic gaps (24) calculated using the SkP functional. In all these Figures, estimates taken from Ref. [8] are used as reference empirical data.

Figure 7: Experimental [8] (dots) and theoretical values of magic gaps (24), calculated using the original SkP functional (open triangles) and the SkP$_{T}$ functional (full triangles) with the SO and the tensor coupling constants from Table. 4. The gaps were computed using the bare unpolarized s.p. spectra.

This global set of the results can be summarized as follows:

• The $\ell=1$, $p_{3/2}-p_{1/2}$, SO splittings are slightly better reproduced with original rather than modified functionals.
• The $1d$ SO splittings ($^{ 16}$O, $^{40,48}$Ca) are rather poorly reproduced by both the original and modified functionals. These splitting are also subject to relatively big empirical uncertainties as shown in Table 3 and, therefore, need not be very conclusive. In particular, the $1d$ SO splittings in $N$=$Z$ $^{ 16}$O and $^{40}$Ca nuclei deduced from Ref. [8] and depicted in the figures show surprisingly large isospin dependence.
• The $2d$ and $3d$ splittings are quite well reproduced by both the original and modified functionals with slight preference for the modified functional, in particular for the SLy4 interaction.
• All the $\ell \geq 3$ SO splittings are reproduced considerably better by the modified functionals.
• Magic gaps are also better (although not fully satisfactorily) reproduced by the modified functionals.

Without any doubt the SO splittings are better described by the modified functionals. It should be stressed that the improvements were reached using only three additional data points without any further optimization. The tensor coupling constants deduced in this work and collected in table 4 should be therefore considered as reference values. Indeed, direct calculations show that variations in $C_t^J$ within $\pm$10% affect the calculated SO splittings only very weakly. The price paid for the improvements concerns mostly the binding energies, which for the nuclei $^{56}$Ni, $^{132}$Sn, and $^{208}$Pb become worse as compared to the original values. This issue is addressed in the next section.