In the framework of nuclear energy density functional (EDF) methods, many nuclear phenomena can be related to the deformation of intrinsic states. Their accurate modeling relies on the correct description of the change of nuclear binding energy with deformation. The two most important contributions to the deformation energy have their origin in shell effects and the surface energy coefficient of nuclear matter. In a first step, we build nine series of parametrizations with a systematically varied surface-energy coefficient a_surf for three frequently-used options for the CM correction (none, one-body term only, full one-body and two-body contributions) combined with three values for the isoscalar effective mass m^*_0/m (0.7, 0.8, 0.85) and analyse how well each of these parametrizations can be adjusted to the properties of spherical nuclei and infinite nuclear matter. In a second step, we performed additional fits without the constraint on surface energy, adding one ``best-fit" parametrization to each of the nine series. We then benchmark these parametrizations to the deformation properties of heavy nuclei by means of three-dimensional Hartree-Fock-Bogoliubov calculations that allow for non-axial and/or non-reflection symmetric configurations. We perform a detailed correlation analysis between surface and volume properties of nuclear matter using the nine series of parametrizations. The best fits out of each series are then benchmarked on the fission barriers of Pu240 and Hg180, as well as on the properties of deformed states at normal and superdeformation for actinides and nuclei in the neutron-deficient Hg region. (see paper for full abstract)

In the framework of nuclear energy density functional (EDF) methods, many nuclear phenomena are related to the deformation of intrinsic states. Their accurate modeling relies on the correct description of the change of nuclear binding energy with deformation. The two most important contributions to the deformation energy have their origin in shell effects that are correlated to the spectrum of single-particle states, and the deformability of nuclear matter, that can be characterized by a model-dependent surface energy coefficient a_{surf}. With the goal of improving the global performance of nuclear EDFs through fine-tuning of their deformation properties, the purpose of this study is threefold. First, to analyze the impact of systematic variations of a_{surf} on properties of nuclei; second, to identify observables that can be safely used to narrow down the range of appropriate values of a_{surf} to be targeted in future parameter fits; third, to analyze the interdependence of a_{surf} with other properties of a nuclear EDF. Results for a large variety of relevant observables of deformed nuclei obtained from self-consistent mean-field calculations with a set of purpose-built SLy5sX parameterizations of the Skyrme EDF are correlated with the value of a_{surf}. The three main conclusions are that there is an evident preference for a comparatively low value of a_{surf}, as expected from the performance of existing parameterizations; that the isospin dependence of the surface energy also needs further finetuning in order to describe trends across the chart of nuclei; and that a satisfying simultaneous description of fission barriers and superdeformed states requires a better description of the single-particle spectra. [Note: The abstract has been abbreviated because of length restrictions imposed by the arXiv. See the paper for the full abstract.]

Starting from general expressions of well-chosen symmetric nuclear matter quantities derived for both zero-and finite-range effective theories, we derive some universal relations between them. We first showthat, independently of the range, the two-body contribution is enough to describe correctly the saturation mechanism but gives an effective mass value around m(*)/m similar or equal to 0.4 when the other properties of the saturation point are set near their generally accepted values. Then, we show that a more elaborated interaction (for instance, an effective two-body density-dependent term on top of the pure two-body term) is needed to reach the accepted value m(*)/m similar or equal to 0.7-0.8.

We present recent developments obtained in the so-called N2LO extension of the usual Skyrme pseudo-potential.In particular, we discuss the isovector splitting mass in infinite nuclear matter and the pairing gaps of selected semi-magic even-even nuclei.

We present the development of the extended Skyrme N2LO pseudopotential in the case of spherical even-even nuclei calculations. The energy density functional is first presented. Then we derive the mean-field equations and discuss the numerical method used to solve the resulting fourth-order differential equation together with the behavior of the solutions at the origin. Finally, a fitting procedure for such an N2LO interaction is discussed and we provide a first parametrization. Typical ground-state observables are calculated and compared against experimental data.

The purpose of this study is threefold: first, to identify a scheme for the determination of the surface energy coefficient a_surf that offers the best compromise between robustness, precision, and numerical efficiency; second, to analyze the correlation between values for a_surf and the characteristic energies of the fission barrier of Pu240; and third, to lay out a procedure how the deformation properties of the Skyrme energy density functional (EDF) can be constrained during the parameter fit. There are several frequently used possibilities to define and calculate the surface energy coefficient a_surf of effective interactions. The most direct access is provided by the model system of semi-infinite nuclear matter, but a_surf can also be extracted from the systematics of binding energies of finite nuclei. Calculations can be carried out either self-consistently (HF), which incorporates quantal shell effects, or in one of the semi-classical Extended Thomas-Fermi (ETF) or Modified Thomas-Fermi (MTF) approximations. The surface energy coefficient of 76 parameterizations of the Skyrme EDF have been calculated. Values obtained with the HF, ETF and MTF methods are not identical, but differ by fairly constant systematic offsets. Despite having some drawbacks compared to the other methods studied here, the MTF approach provides sufficiently precise values for a_surf such that it can be used as a very robust constraint on surface properties during a parameter fit at negligible additional cost. Through the construction of a series of eight parameterizations SLy5s1-SLy5s8 of the standard Skyrme EDF with systematically varied a_surf value it is shown how to arrive at a fit with realistic deformation properties. [Note: The abstract has been abbreviated because of length restrictions imposed by the arXiv. See the paper for the full abstract.]

We discuss the contributions to the equation of state for the NlLO Skyrme pseudopotential (l = 2,3). We show that by adding fourth- and sixth-order gradient terms, it is possible to fairly reproduce the spin/isospin decomposition of an equation of state obtained from ab initio methods. Moreover, by inspecting the partial-wave decomposition of the equation of state, we show for the first time a possible way to add explicit constraints on the sign of the tensor terms of the Skyrme interaction.

By means of a partial wave decomposition, we separate their contributions to the equation of state of symmetric nuclear matter for the N3LO pseudo-potential. In particular, we show that although both the tensor and the spin-orbit terms do not contribute to the equation of state, they give a non-vanishing contribution to the separate (JLS) channels.

The possibility of adding a D-wave term to the standard Skyrme effective interaction has been widely considered in the past. Such a term has been shown to appear in the next-to-next-to-leading order of the Skyrme pseudo-potential. The aim of the present article is to provide the necessary tools to incorporate this term in a fitting procedure: first, a mean-field equation written in spherical symmetry in order to describe spherical nuclei and second, the response function to detect unphysical instabilities. With these tools it will be possible to build a new fitting procedure to determine the coupling constants of the new functional.

It has recently been shown that the linear response theory in symmetric nuclear matter can be used as a tool for detecting finite-size instabilities for different Skyrme functionals. In particular, it has been shown that there is a correlation between the density at which instabilities occur in infinite matter and the instabilities in finite nuclei. In this paper, we present a new fitting protocol that uses this correlation to add a new additional constraint in symmetric infinite nuclear matter in order to ensure the stability of finite nuclei against matter fluctuation in all spin and isospin channels. As an application, we give the parameter set for a new Skyrme functional which includes central and spin–orbit parts and which is free from instabilities by construction.

The authors thank T. Duguet for the stimulating discussions that motivated the present article. We also acknowledge the fruitful discussions with K. Bennaceur, J. Dobaczewski, and M. Kortelainen. A.P. is also grateful to the University of Jyvaskyla and the FIDIPRO group for the warm hospitality during the time in which part of this work was realized. D.T. was supported in part by the ERANET-NuPNET grant SARFEN of the Polish National Centre for Research and Development (NCBiR) and by the Academy of Finland and the University of Jyvaskyla within the FIDIPRO program. J.N. was supported by Mineco (Spain), Grant No. FIS2014-51948-C2-1-P.

Background: In one way or another, all modern parametrizations of the nuclear energy density functional (EDF) do not respect the exchange symmetry associated with Pauli's principle. It has been recently shown that this practice jeopardizes multireference (MR) EDF calculations by contaminating the energy with spurious self-interactions that, for example, lead to finite steps or even divergences when plotting it as a function of collective coordinates [J. Dobaczewski et al., Phys. Rev. C 76, 054315 (2007); D. Lacroix et al., Phys. Rev. C 79, 044318 (2009)]. As of today, the only viable option to bypass these pathologies is to rely on EDF kernels that enforce Pauli's principle from the outset by strictly and exactly deriving from a genuine, i.e., density-independent, Hamilton operator.Purpose: The objective is to build cutting-edge parametrizations of the EDF kernel deriving from a pseudopotential that can be safely employed in symmetry restoration and configuration mixing calculations.Methods: We wish to develop the most general Skyrme-like EDF parametrization containing linear, bilinear, and trilinear terms in the density matrices with up to two gradients, under the key constraint that it derives strictly from an effective Hamilton operator. While linear and bilinear terms are obtained from a standard one-body kinetic energy operator and a (density-independent) two-body Skyrme pseudopotential, the most general three-body Skyrme-like pseudopotential containing up to two gradient operators is constructed to generate the trilinear part. The present study is limited to central terms. Spin orbit and tensor will be addressed in a forthcoming paper.Results: The most general central Skyrme-type zero-range three-body interaction is built up to second order in derivatives. The complete trilinear EDF, including time-odd and T = 1 pairing parts, is derived along with the corresponding normal and anomalous fields entering the Hartree-Fock-Bogoliubov equations of motion. Its building blocks are the same local densities that the standard Skyrme functional is constructed from. The central three-body pseudopotential is defined out of six independent parameters. Expressions for bulk properties of symmetric, isospin-asymmetric, and spin-polarized homogeneous nuclear matter, as well as associated Landau parameters, are given.Conclusions: This study establishes a first step towards a new generation of nuclear EDFs that respect Pauli's principle and that can be safely used in predictive and spuriosity-free SR and MR EDF calculations.

Nuclear effective interactions are often modelled by simple analytical expressions such as the Skyrme zero-range force. This effective interaction depends on a limited number of parameters that are usually fitted using experimental data obtained from doubly magic nuclei. It was recently shown that many Skyrme functionals lead to the appearance of instabilities, in particular when symmetries are broken, for example unphysical polarization of odd-even or rotating nuclei. In this article, we show how the formalism of the linear response in infinite nuclear matter can be used to predict and avoid the regions of parameters that are responsible for these unphysical instabilities.

The formalism of the linear response for the Skyrme energy density functional including tensor terms derived in Refs. [1,2] for nuclear matter is applied here to the case of pure neutron matter. As in Ref. [2] we present analytical results for the response function in all channels, the Landau parameters, and the odd-power sum rules. Special emphasis is given to the inverse energy weighted sum rule because it can be used to detect nonphysical instabilities. Typical examples are discussed and numerical results shown. Moreover, as a direct application, neutrino propagation in neutron matter is investigated through its neutrino mean-free path at zero temperature. This quantity turns out to be very sensitive to the tensor terms of the Skyrme energy density functional.

The formalism of linear response theory for Skyrme forces including tensor terms presented by Davesne et al. [Phys. Rev. C 80, 024314 (2009)] is generalized for the case of a Skyrme energy density functional in infinite matter. We also present analytical results for the odd-power sum rules, with particular attention to the inverse energy weighted sum rule, M-1, as a tool to detect instabilities in Skyrme functionals.

The effects of a zero-range tensor component of the effective interaction on nuclear response functions are determined in the random phase approximation approach. Explicit formulas in the case of symmetric homogeneous isotropic nuclear matter are given for each spin-isospin excitation channel. It is shown for a typical interaction with tensor couplings that the effects are quantitatively important, mainly in vector channels.

We perform systematic calculations of pairing gaps in semi-magic nuclei across the nuclear chart using the Energy Density Functional method and a non-empirical pairing functional derived, without further approximation, at lowest order in the two-nucleon vacuum interaction, including the Coulomb force. The correlated single-particle motion is accounted for by the SLy4 semi-empirical functional. Rather unexpectedly, both neutron and proton pairing gaps thus generated are systematically close to experimental data. Such a result further suggests that missing effects, i.e. higher partial waves of the NN interaction, the NNN interaction and the coupling to collective fluctuations, provide an overall contribution that is sub-leading as for generating pairing gaps in nuclei. We find that including the Coulomb interaction is essential as it reduces proton pairing gaps by up to 40%.

We perform systematic calculations of pairing gaps in semi-magic nuclei across the nuclear chart using the Energy Density Functional method and a non-empirical pairing functional derived, without further approximation, at lowest order in the two-nucleon vacuum interaction, including the Coulomb force. The correlated single-particle motion is accounted for by the SLy4 semi-empirical functional. Rather unexpectedly, both neutron and proton pairing gaps thus generated are systematically close to experimental data. Such a result further suggests that missing effects, i.e. higher partial waves of the NN interaction, the NNN interaction and the coupling to collective fluctuations, provide an overall contribution that is sub-leading as for generating pairing gaps in nuclei. We find that including the Coulomb interaction is essential as it reduces proton pairing gaps by up to 40%.