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We perform a systematic study of the impact of the J 2 tensor term in the Skyrme energy functional on properties of spherical nuclei. In the Skyrme energy functional, the tensor terms originate from both zero-range central and tensor forces. We build a set of 36 parametrizations, covering a wide range of the parameter space of the isoscalar and isovector tensor term coupling constants with a fit protocol very similar to that of the successful SLy parametrizations. We analyze the impact of the tensor terms on a large variety of observables in spherical mean-field calculations, such as the spin-orbit splittings and single-particle spectra of doubly-magic nuclei, the evolution of spin-orbit splittings along chains of semi-magic nuclei, mass residuals of spherical nuclei, and known anomalies of radii. The major findings of our study are as follows: (i) Tensor terms should not be added perturbatively to existing parametrizations; a complete refit of the entire parameter set is imperative. (ii) The free variation of the tensor terms does not lower the χ 2 within a standard Skyrme energy functional. (iii) For certain regions of the parameter space of their coupling constants, the tensor terms lead to instabilities of the spherical shell structure, or even to the coexistence of two configurations with different spherical shell structures. (iv) The standard spin-orbit interaction does not scale properly with the principal quantum number, such that single-particle states with one or several nodes have too large spin-orbit splittings, whereas those of nodeless intruder levels are tentatively too small. Tensor terms with realistic coupling constants cannot cure this problem.(v) Positive values of the coupling constants of proton-neutron and like-particle tensor terms allow for a qualitative description of the evolution of spin-orbit splittings in chains of Ca, Ni, and Sn isotopes. (vi) For the same values of the tensor term coupling constants, however, the overall agreement of the single-particle spectra in doubly-magic nuclei is deteriorated, which can be traced back to features of the single-particle spectra that are not related to the tensor terms. We conclude that the currently used central and spin-orbit parts of the Skyrme energy density functional are not flexible enough to allow for the presence of large tensor terms.well-defined transformation properties under rotations: (7) where δ µν is the Kronecker symbol and µνκ is the Levi-Civita tensor. The pseudoscalar, vector, and pseudotensor parts expressed in terms of the Cartesian tensor are given byν=x κµν J µν (r), 014312-6 TENSOR PART OF THE SKYRME ENERGY DENSITY . . . PHYSICAL REVIEW C 76, 014312 (2007) some authors [59]: z µ,ν=x J t,µν J t,µν = 1 3 J (0) t 2 + 1 2 J 2 t + z µ,ν=x J (2) t,µν J (2) t,µν , (24) 1 2 z µ=x J t,µµ 2 + z µ,ν=x J t,µν J t,νµ = 2 3 J (0) t 2 − 1 4 J 2 t + 1 2 z µ,ν=x J (2) t,µν J (2) t,µν . (25)

In order to extract informations on pairing correlations in nuclei from experimental masses, the different contributions to odd-even mass differences are investigated within the Skyrme Hartree-Fock-Bogoliubov ͑HFB͒ method. In this part of the paper, the description of odd nuclei within HFB is discussed since it is the key point for the understanding of the above mentioned contributions. To go from an even nucleus to an odd one, the advantage of a two steps process is demonstrated and its physical content is discussed. New results concerning time-reversal symmetry breaking in odd nuclei are also reported.

The nuclear incompressibility K ∞ is deduced from measurements of the Isoscalar Giant Monopole Resonance (ISGMR) in medium-heavy nuclei, and the resulting value turns out to be model dependent. Since the considered nuclei have neutron excess, it has been suggested that the model dependence is due to the different behaviour of the symmetry energy in different models. To clarify this issue, we make a systematic and careful analysis based on new Skyrme forces which span a wide range of values for K ∞ , for the value of the symmetry energy at saturation and for its density dependence. By calculating, in a fully self-consistent fashion, the ISGMR centroid energy in 208 Pb we reach, for the first time within the non-relativistic framework, three important conclusions:(i) the monopole energy, and consequently the deduced value of K ∞ , depend on a well defined parameter related to the shape of the symmetry energy curve and called K sym ; (ii) Skyrme forces of the type of SLy4 predict K ∞ around 230 MeV, in agreement with the Gogny force (previous estimates using Skyrme interactions having been plagued by lack of full self-consistency); (iii) it is possible to build forces which predict K ∞ around 250 MeV, although part of this increase is due to our poor knowledge of the density dependence and effective mass.

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We study the effect of the splitting of neutron and proton effective masses with isospin asymmetry on the properties of the Skyrme energy density functional. We discuss the ability of the latter to predict observable of infinite matter and finite nuclei, paying particular attention to controlling the agreement with ab-initio predictions of the spin-isospin content of the nuclear equation of state, as well as diagnosing the onset of finite size instabilities, which we find to be of critical importance. We show that these various constraints cannot be simultaneously fulfilled by the standard Skyrme force, calling at least for an extension of its P-wave part.