Introduction

In this work we propose and explore two new ideas pertaining to the energy density functional (EDF) methods. First, we suggest a necessity of shifting attention and focus of these methods from ground-state bulk properties (e.g. total nuclear masses) to single-particle (s.p.) properties, and to look for a spectroscopic-quality EDFs that would correctly describe nuclear shell structure. Proper positions of s.p. levels are instrumental for good description of deformation, pairing, particle-core coupling, and rotational effects, and many other phenomena.

On the one hand, careful adjustment of these positions were at the heart of tremendous success of phenomenological mean-field (MF) models, like those of Nilsson, Woods-Saxon, or folded Yukawa [1]. On the other hand, similarly successful phenomenological description of nuclear masses, within the so-called microscopic-macroscopic method [2], relies on the liquid-drop mass surface, which is entirely decoupled from the s.p. structure.

Up to now, methods based on using EDFs, in any of its variants like local Skyrme, non-local Gogny, or relativistic-mean-field (RMF) [3] approach, were mostly using adjustments to bulk nuclear properties. As a result, shell properties were described poorly. After so many years of investigations, a further increase in precision and predictability of all methods based on the EDFs may require extensions beyond forms currently in use. Before this can be fully achieved, we propose to first take care of the s.p. properties, and come back to precise adjustment of bulk properties once these extensions are implemented.

Second, we propose to look at the s.p. properties of nuclei through the magnifying glass of odd-even mass differences. This idea has already been put forward in a seminal paper by Rutz et al. [4], where calculations performed within the RMF approach were presented. In this paper we perform analogous analysis for the EDFs based on the Skyrme interactions.

On the one hand, it was recognized long time ago, cf., e.g., Refs. [5,6], that the theoretical s.p. energies, defined as eigenvalues of the MF Hamiltonian, cannot be directly compared to experiment, because they are strongly renormalized by the particle-core coupling. On the other hand, procedures used to deduce the s.p. energies from experiment [7,8,9] require various theoretical assumptions, by which these quantities cease to result from direct experimental observation. In the past, these theoretical and experimental caveats hampered the use of s.p. energies for proper adjustments of EDFs. However, the odd-even mass differences carry very similar physical information to that given by s.p. energies, and have advantage of being clearly defined, both experimentally and theoretically.

Indeed, experimental difference in mass between an odd nucleus and its lighter even-even neighbor, i.e., the particle separation energy, is an easily available and unambiguous piece of data, which reflects the physical role of the s.p. energy, with all polarizations and couplings taken into account. Similarly, differences of masses between the low-lying excited states in an odd system and lighter even-even neighbor may illustrate effective positions of higher s.p. states. Physical connections between these mass differences and s.p. energies are closest in semi-magic nuclei, which will be studied in the present paper.

In theory, the primary goal of the EDF methods is to describe ground-state energies of fermion systems, i.e., in nuclear-physics applications - masses of nuclei. For odd systems, the EDF methods should give masses of ground states and of several low-lying excited states; the latter being obtained by blocking specific s.p. orbitals. We stress here that in an odd system, separate self-consistent calculations have to be performed for each of the blocked states, so as to allow the system for exploring all possible polarizations exerted by the odd particle on the even-even core. Again, this procedure is clearly defined and entirely within the scope and spirit of the EDF method, which is supposed to provide exact energies of correlated states. Note, that although one here calculates total masses of odd and even systems, the comparison with experiment involves the differences of masses, for which many effects cancel out. Therefore, one can confidently attribute calculated differences of masses to properties of effective s.p. energies, with all polarization effects taken into account, like in the experiment. Again, in semi-magic nuclei such a connection is most reliable.

Of course, there remains a general question of whether one should adjust theoretical parameters at the mean-field level or including explicitly beyond-mean-field effects. This question is particularly important when we discuss the s.p. spectra, where large effects beyond the mean-field may, in principle, be important. However, within our approach we propose a thorough adjustment at the mean-field level first. This strategy may clearly identify the experimental data that cannot at all be explained within such a restricted methodology.

Following the above line of reasoning, in the present paper we study experimental and theoretical aspects of the spin-orbit (SO) splitting between the s.p. states. In particular, we analyze the role of the SO and tensor MFs in providing the correct values of the SO splittings across the nuclear chart. The paper is organized as follows: In Sect. 2 we briefly recall basic theoretical building blocks related to the SO and tensor terms of the EDFs and interactions. In Sect. 3 we discuss in details three major sources of the core polarization: the mass, shape, and spin polarizabilities. In Sect. 4 we present a novel method that allows for a firm adjustment of the SO and tensor coupling constants arguing that currently used functionals require major revisions concerning strengths of both these terms. The analysis is based on the $f_{7/2} - f_{5/2}$ SO splittings in three key nuclei, including spin-saturated isoscalar $^{40}$Ca, spin-unsaturated isoscalar $^{56}$Ni, and spin-unsaturated isovector $^{48}$Ca systems, and is subsequently verified by systematic calculations of the SO splittings in magic nuclei. In Sect. 5 we discuss an impact of these changes on binding energies in magic nuclei. Conclusions of the paper are presented in Sect. 6.