Introduction

Density functional theory (DFT) is a universal ab initio approach designed and used to calculate properties of electronic systems entrapped in the external Coulomb field of nuclei. It has been successfully applied to atoms, molecules, or condensed media. Universality of the DFT means independence of a functional form of the shape of external one-body potential holding the electronic system together. The existence of such a universal and, in principle, exact density functional describing ground-states of externally bound fermionic systems is guaranteed by the Hohenberg-Kohn [1] and Kohn-Sham [2] (HKS) theorems.

Generalization of the DFT theory to self-bound systems like atomic nuclei encounters problems associated with intrinsic rather than laboratory density which characterizes the atomic nuclei, see Ref. [3,4,5]. In spite of that, the HKS theorem has strongly influenced the way of thinking in the field of nuclear structure. Nowadays, the nuclear structure theorists employ the functionals that are treated as separate entities, independently, to a large extent, of the underlying effective nucleon-nucleon (NN) interactions like, for example, the local Skyrme interaction [6]. Free parameters of these functionals are directly adjusted to fit empirical data. There are also attempts to enrich their functional form as compared to the form resulting from the mean-field (MF) averaging of the effective Skyrme interaction [7], which are motivated by a rather mediocre performance of the conventional Skyrme-type functionals, see for example Refs. [8,9].

Adequateness of the fitting strategy, that is, the choice of the data set, is a key factor determining performance of the nuclear energy density functional (EDF) method. In this work we explore the Skyrme-type local EDF approach to nuclear structure and focus on the spin-orbit (SO) and tensor parts of the functional. Throughout the years, not much attention was paid to, in particular, the tensor part, mainly due to the lack of clear experimental data constraining the strength of this part of the EDF. Hence, the tensor term in the existing Skyrme functionals is either trivially set to zero by hand, see the review in Ref. [10], or is a result of a global fit to, predominantly, bulk nuclear properties [11].

Recent revival of interest in the tensor term was triggered by systematic observation of non-conventional shell evolution in isotopic chains of light nuclei far from stability, including new magic shell-gap opening at $N$=32, see for example Refs. [12,13]. Otsuka and collaborators associated these effects with the two-body tensor interaction [14,15]. This interpretation was soon confirmed within the local Skyrme-type EDF models [16,17,18,19,20,21]. Inclusion of single-particle (s.p.) energies in the fitting data sets appears to lead to the tensor coupling constants [17,20,21,22,23,24] which are at variance with bulk fits, see Fig. 1 and extensive discussion in Ref. [11].

The aim of this work is to look into consequences of strong attractive isoscalar and isovector tensor fields resulting from the fitting method proposed by our group [22]. In this work we concentrate mostly on the tensor fields, studying their impact on the nuclear binding energies. The paper is organized as follows. In Sec. 2, theoretical framework is briefly outlined. In Sec. 3, the procedure used to fit the tensor and SO coupling constants is discussed. In Sec. 4, numerical results showing tensor energy contribution to the total binding energy, followed by a discussion of tensorial magic structure, is presented. The paper is summarized in Sec. 5.

Figure 1: Unlike-particle $C_u^J = (C_0^J -C_1^J)/2$ versus like-particle $C_l^J = (C_0^J +C_1^J)/2$ tensor coupling constants resulting form fits to: bulk nuclear properties (black dots) and the s.p. levels and the SO splittings (triangles). Open triangles represent fits of Ref. [17,20,21]. Black triangles mark our results [22,23,24] from the fit to the $f_{7/2}-f_{5/2}$ SO splittings. Shaded area represents the parameters established by Brink, Stancu and Flocard (BSF) in their seminal paper [25]