There is an ongoing quest to improve on the spectroscopic quality of nuclear energy density functionals (EDFs) of the Skyrme type through extensions of its traditional form. One direction for such activities is the inclusion of terms of higher order in gradients in the EDF. We report on exploratory symmetry-breaking calculations performed for an extension of the Skyrme EDF that includes central terms with four gradients at next-to-next-to-leading order (N2LO) and for which the high-quality parametrization SN2LO1 has been constructed recently [P. Becker et al, Phys. Rev. C 96, 044330 (2017)]. Up to now, the investigation of such functionals with higher-order terms was limited to infinite matter and spherically symmetric configurations of singly- and doubly-magic nuclei. We address here nuclei and phenomena that require us to consider axial and non-axial deformation, both for reflection-symmetric and also reflection-asymmetric shapes, as well as the breaking of time-reversal invariance. Achieving these calculations demanded a number of formal developments. These all resulted from the formulation of the N2LO EDF requiring the introduction of new local densities with additional gradients that are not present in the EDF at NLO. Their choice is not unique, but can differ in the way the gradients are coupled. While designing a numerical implementation of N2LO EDFs in Cartesian 3d coordinate-space representation, we have developed a novel definition and a new unifying notation for normal and pair densities that contain gradients at arbitrary order. The resulting scheme resolves several issues with some of the choices that have been made for local densities in the past, in particular when breaking time-reversal symmetry. Guided by general practical considerations, we propose an alternative form of the N2LO contribution to the Skyrme EDF that is built from a different set of densities.
The explicit form of the next-to-next-to-leading order (N2LO) of the Skyrme effective pseudopotential compatible with all required symmetries and especially with gauge invariance is presented in Cartesian basis. It is shown in particular that for such a particular pseudopotential there is no spin-orbit contribution and that the D-wave term suggested in the original Skyrme's formulation does not satisfy the invariance properties. The six new N2LO terms contribute to both the equation of state and Landau parameters. These contributions to symmetric nuclear matter are explicitly given and discuss…
I have investigated the $N$-dependence of the charge radii $r_c$ across $N=126$ in the Pb isotopes and the surface densities of $^{208}$Pb using new Skyrme interactions that contain a density-dependent spin-orbit term, which I have named the Skyrme-ddso interactions. I have compared the results obtained using Skyrme-Hartree-Fock-Bogoliubov calculations that employ the Skyrme-ddso interactions with the original Skyrme results and have discussed the effects of including the density-dependent spin-orbit term. The results for the kink behavior of $r_c$ at $N=126$ in the Pb isotopes were improved b…
Proton and neutron densities from Skyrme-Hartree-Fock (SHF) calculations are used to generate non-local (g-folding) proton-nucleus optical potentials. They are formed by folding the densities with realistic nucleon-nucleon interactions. The potentials are then used to calculate differential cross sections and spin observables for proton scattering. Good agreement with data has been found, supporting those found previously when using SHF charge densities in analyses of electron scattering data. That agreement was improved by use of (shell model) occupation numbers to constrain the HF iterations…
Time-odd densities and their effect on electric giant resonances are investigated within the self-consistent separable random-phase-approximation (SRPA) model for various Skyrme forces (SkT6, SkO, SkM*, SIII, SGII, SLy4, SLy6, SkI3). Time-odd densities restore Galilean invariance of the Skyrme functional, violated by the effective-mass and spin-orbital terms. In even-even nuclei these densities do not contribute to the ground state but can affect the dynamics. As a particular case, we explore the role of the current density in description of isovector E1 and isoscalar E2 giant resonances in a …
I discuss the inadequacy of the "projected density" prescription to be used in density dependent forces/functionals when calculations beyond mean field are pursued. The case of calculations aimed at the symmetry restoration of mean fields obtained with effective realistic forces of the Skyrme or Gogny type is considered in detail. It is shown that at least for the restoration of spatial symmetries like rotations, translations or parity the above prescription yields catastrophic results for the energy that drive the intrinsic wave function to configurations with infinite deformation, preventing…
We use the stability conditions of the Landau parameters for the symmetric nuclear matter and pure neutron matter to calculate the critical densities for the Skyrme type effective nucleon-nucleon interactions. We find that the critical density can be maximized by adjusting appropriately the values of the enhancement factor $κ$ associated with isovector giant dipole resonance, the quantity $L$ which is directly related to the slope of the symmetry energy and the Landau parameter $G_0^\prime$. However, restricting $κ$, $L$ and $G_0^\prime$ to vary within acceptable limits reduces the maximum val…
Nuclear dynamics of giant resonances are investigated with the real-time Skyrme TDHF method. The TDHF equation is explicitly linearized with respect to variation of single-particle wave functions. The time evolution of transition densities are calculated for giant dipole resonances. The time-dependent densities of protons and neutrons suggest that the dynamics of giant dipole resonance in neutron-rich nuclei are significantly different from that in stable nuclei with N=Z.
Anisotropic flows in Au+Au collisions at $\sqrt{s_{\rm{NN}}} = 2.4\,\text{GeV}$ with a Skyrme pseudopotential
Henan Normal University（河南师范大学）；Shanghai Jiao Tong University（上海交通大学）；Istituto Nazionale di Fisica Nucleare (INFN), Laboratori Nazionali del Sud（意大利国家核物理研究所（INFN）南方实验室）；Sun Yat-sen University（中山大学）；Henan Academy of Science（河南省科学院）；Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University（上海理论核物理研究中心，国家自然科学基金委与复旦大学）
In the present paper we explore the neutron-drip region of cold non-rotating isolated neutron stars. We have performed extended nuclear-structure calculations for nuclei embedded in the electron gas. For modeling the outer crust we use a set of Wigner-Seitz cells, where every cell contains one nucleus surrounded by a cloud of relativistic electrons. Above the drip point a non-relativistic neutron gas occurs in the cell. These calculations are carried out within the Hartree-Fock approach in combination with Skyrme effective interactions. For every baryon density we have determined the configura…
The temperature dependence of the nuclear matter isovector symmetry energy coefficient ($\cA_{0,1}$) is investigated in the framework of the generalized nuclear polarizability with Skyrme interactions, as worked out in Refs. \cite{npa,prc}. The variation of $\cA_{0,1}(T)$ is very small (of the order of 1 MeV) for temperatures (T) in the range of 0 and 18 MeV. Different behaviors with temperature are found strongly depending on the Skyrme parameterization, in particular at densities lower than the saturation density $ρ_0$.
We develop the isospin-invariant Skyrme-EDF method by considering local densities in all possible isospin channels and proton-neutron (p-n) mixing terms as mandated by the isospin symmetry. The EDF employed has the most general form that depends quadratically on the isoscalar and isovector densities. We test and benchmark the resulting p-n EDF approach, and study the general properties of the new scheme by means of the cranking in the isospin space. We extend the existing axial DFT solver HFBTHO to the case of isospin-invariant EDF approach with all possible p-n mixing terms. Explicit expressi…
Based on the semi-classical extended Thomas-Fermi approach, we study the mass dependence of the symmetry energy coefficients of finite nuclei for 36 different Skyrme forces. The reference densities of both light and heavy nuclei are obtained. Eight models based on nuclear liquid drop concept and the Skyrme force SkM* suggest the symmetry energy coefficient $a_{\rm sym}=22.90 \pm 0.15 $ MeV at $A=260$, and the corresponding reference density is $ρ_A\simeq 0.1$ fm$^{-3}$ at this mass region. The standard Skyrme energy density functionals give negative values for the coefficient of the $I^4$ term…
The neutron-proton Fermi-energy difference and the correlation to nucleon separation energies for some magic nuclei are investigated with the Skyrme energy density functionals and nuclear masses, with which the nuclear symmetry energy at sub-saturation densities is constrained from 54 Skyrme parameter sets. The extracted nuclear symmetry energy at sub-saturation density of 0.11 fm$^{-3}$ is 26.2 $\pm$ 1.0 MeV with 1.5 $σ$ uncertainty. By further combining the neutron-skin thickness of 208Pb, ten Skyrme forces with slope parameter of 28<L<65 are selected for the description of the symmetry ener…
We develop a new formulation of the continuum quasiparticle random phase approximation (QRPA) in which the residual interaction is derived directly from the Skyrme energy functional, keeping all the velocity dependent terms of the Skyrme effective interaction. Numerical analysis using the SkM$^*$ parameter set is performed for the isovector dipole and the isovector/isoscalar quadrupole responses in $^{20}$O and $^{54}$Ca. It is shown that the energy-weighted sum rule including the enhancement factors for the isovector responses is satisfied with good accuracy. We investigate also how the veloc…
We generalize to N2LO Skyrme functionals the semi-classical approach of Grammaticos and Voros in order to calculate the Extended Thomas Fermi expressions of the new densities and currents appearing at the N2LO level. Within a one dimensional symmetric semi infinite nuclear matter model and using a simple Fermi-like density profile, we obtain an easy-to-use formula for the surface energy including the contributions of the central, density-dependent and spin-orbit terms up to $\hbar^2$. Such a formula can be easily used as a first attempt to constrain the surface properties of new N2LO Skyrme pa…
The magnetic dipole (M1) and the Gamow-Teller (GT) excitations of finite nuclei have been studied in a fully self-consistent Hartree-Fock (HF) plus random phase approximation (RPA) approach by using a Skyrme energy density functional with spin and spin-isospin densities. To this end, we adopt the extended SLy5st interaction which includes spin-density dependent terms and stabilize nuclear matter with respect to spin instabilities. The effect of the spin-density dependent terms is examined in both the mean field and the spin-flip excited state calculations. The numerical results show that those…
A systematic analysis is made on the total reaction cross sections for Ne, Mg, Si, and S isotopes. The high-energy nucleus-nucleus collision is described based on the Glauber model. Using the Skyrme-Hartree-Fock method in the three-dimensional grid-space representation, we determine the nuclear density distribution for a wide range of nuclei self-consistently without assuming any spatial symmetry. The calculated total reaction cross sections consistently agree with the recent cross section data on Ne$+^{12}$C collision at 240$A$\,MeV, which makes it possible to discuss the radius and deformati…
We present the temperature and density dependence of symmetry energy for nuclear matter at finite temperature based on the approach of the thermodynamics with Skyrme energy density functional. We first classify the Skyrme interactions into 7 groups according to the range of neutron and proton effective mass in neutron matter limit(99.99 per cent neutron in the matter). We find that there is obvious correlation between the temperature dependence of the symmetry energy and the splitting of the neutron and proton effective mass. For some Skyrme interactions with $m^{*}_{n}>m^{*}_{p}$ and strong s…
Based on an extended Skyrme interaction that includes the terms in relative momenta up to sixth order, corresponding to the so-called Skyrme pseudopotential up to next-to-next-to-next-to leading order (N3LO), we derive the expressions of Hamiltonian density and single nucleon potential under general non-equilibrium conditions which can be applied in transport model simulations of heavy-ion collisions induced by neutron-rich nuclei. While the conventional Skyrme interactions, which include the terms in relative momenta up to second order, predict an incorrect behavior as a function of energy fo…
A study of the charge and matter densities and the corresponding rms radii for even-even isotopes of Ni, Kr, and Sn has been performed in the framework of deformed self-consistent mean field Skyrme HF+BCS method. The resulting charge radii and neutron skin thicknesses of these nuclei are compared with available experimental data, as well as with other theoretical predictions. The formation of a neutron skin, which manifests itself in an excess of neutrons at distances greater than the radius of the proton distribution, is analyzed in terms of various definitions. Formation of a proton skin is …
Energy for a nucleus is considered in macroscopic limit, in terms of nucleon numbers. Further considered for a nuclear system is the Hohenberg-Kohn energy functional, in terms of proton and neutron densities. Finally, Skyrme-Hartree-Fock calculations are carried out for a half-infinite particle-stable nuclear-matter. In each case, the attention is focused on the role of neutron-proton asymmetry and on the nuclear symmetry energy. We extend the considerations on the symmetry term from an energy formula to the respective term in the Hohenberg-Kohn functional. We show, in particular, that in the …
The first results of a new three-dimensional, finite temperature Skyrme-Hartree-Fock+BCS study of the properties of inhomogeneous nuclear matter at densities and temperatures leading to the transition to uniform nuclear matter are presented. A constraint is placed on the two independent components of the quadrupole moment in order to self-consistently explore the shape phase space of nuclear configurations. The scheme employed naturally allows effects such as (i) neutron drip, which results in an external neutron gas, (ii) the variety of exotic nuclear shapes expected for extremely neutron hea…
Within the framework of BCS theory and Skyrme-Hartree-Fock model, we employ various microscopic pairing gaps and effective pairing interactions to study pairing effects on the equation of state (EOS) of neutron matter and the symmetry energy at subsaturation densities. We find pairing effects may have considerable contributions to the EOS of neutron matter at very low densities ($\lesssim 0.02~\rm{fm}^{-3}$), while only have a small impact on the symmetry energy at subsatruation densities. In addition, the reliability of the parabolic approximation for the isospin asymmetry dependence of nucle…
The time-dependent Hartree-Fock calculation with a full Skyrme energy functional has been carried out on the three-dimensional Cartesian lattice space to study E1 giant dipole resonances (GDR) in light nuclei. The outgoing boundary condition for the continuum states is treated by the absorbing complex potential. The calculation for GDR in O-16 suggests a significant influence of the residual interaction associated with time-odd densities in the Skyrme functional. We also predict a large damping for superdeformed Be-14 at the neutron drip line.
The temperature dependence of the symmetry energy for isotopic chains of even-even Ni, Sn, and Pb nuclei is investigated in the framework of the local density approximation (LDA). The Skyrme energy density functional with two Skyrme-class effective interactions, SkM* and SLy4, is used in the calculations. The temperature-dependent proton and neutron densities are calculated through the HFBTHO code that solves the nuclear Skyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformed deformed harmonic-oscillator basis. In addition, two other density distributions of $^{208}$Pb, nam…
Models based on nuclear energy density functionals can provide access to a multitude of observables for thousands of nuclei in a single framework with microscopic foundations. Such models can rival the accuracy of more phenomenological approaches, but doing so requires adjusting parameters to thousands of nuclear masses. To keep such large-scale fits feasible, several symmetry restrictions are generally imposed on the nuclear configurations. One such example is time-reversal invariance, which is generally enforced via the Equal Filling Approximation (EFA). Here we lift this assumption, enablin…
We apply the direct variational method to derive the equation of state for finite nuclei within the stability valley. The extended Thomas-Fermi approximation for the energy functional with Skyrme forces is used. Using the leptodermous expansion for the profile nucleon densities, we have studied the neutron coat and the isospin symmetry energy for neutron-rich nuclei. Using equation of state for the pressure, we derive the region of spinodal instability of finite nuclei and its dependence on the mass number, the asymmetry parameter and the Skyrme force parameters. We suggest the procedure of de…
Within the newly updated version of the Ultra-relativistic quantum molecular dynamics (UrQMD) model in which the Skyrme potential energy-density functional is introduced, the yield ratio between $^3$H and $^3$He clusters emitted from central $^{40}$Ca+$^{40}$Ca, $^{96}$Zr+$^{96}$Zr, $^{96}$Ru+$^{96}$Ru, and $^{197}$Au+$^{197}$Au collisions in the beam energy range from 0.12 to 1 GeV$/$nucleon is studied. The recent FOPI data for the $^3$H$/$$^3$He ratio are compared with UrQMD calculations using 13 Skyrme interactions (all exhibiting similar values of iso-scalar incompressibility but very diff…
The first results of a new three-dimensional, finite temperature Skyrme-Hartree-Fock+BCS study of the properties of inhomogeneous nuclear matter at densities and temperatures leading to the transition to uniform nuclear matter are presented. Calculations are carried out in a cubic box representing a unit cell of the locally periodic structure of the matter. A constraint is placed on the two independent components of the quadrupole moment of the neutron density in order to investigate the dependence of the total energy-density of matter on the geometry of the nuclear structure in the unit cell.…
Spinodal instabilities and early growth of baryon density fluctuations in symmetric nuclear matter are investigated in the basis of stochastic extension of relativistic mean-field approach in the semi-classical approximation. Calculations are compared with the results of non-relativistic calculations based on Skyrme-type effective interactions under similar conditions. A qualitative difference appears in the unstable response of the system: the system exhibits most unstable behavior at higher baryon densities around $ρ_{b}=0.4 ~ρ_{0}$ in the relativistic approach while most unstable behavior o…
To extract structure information from proton elastic scattering off S isotopes at 320 MeV and Ni isotopes at $E_p=180$ MeV, this study proposes isotopic analyses combining nuclear structure and reaction calculations. The isotonic analysis was repeated on $^{48}$Ca and $^{50}$Ti. The structure calculations were performed by using the spherical and deformed Relativistic Hartree--Bogoliubov (RHB) calculations, and the spherical nonrelativistic Skyrme Hartree--Fock--Bogoliubov calculations. The $(p,p)$ reactions were calculated using the relativistic impulse approximation (RIA) assuming the theore…
This paper deals with the theoretical foundation of effective two-body forces for the Generator Coordinate Method (GCM) and the projected mean-field method. The first aim of this paper is to reduce into various local-densities the in-medium content of a generalized $G$ matrix removing the hard core problem in this extended context. Then, we consider the possible renormalization of multi-body forces through a density-dependent two-body interaction in the context of configuration mixing calculations. A density dependence of the form $ρ^σ$, as used in Skyrme and Gogny forces, is successfully inte…
The pasta phase in core-collapse supernova matter (finite temperatures and fixed proton fractions) is studied within relativistic mean field models. Three different calculations are used for comparison, the Thomas-Fermi (TF), the Coexisting Phases (CP) and the Compressible Liquid Drop (CLD) approximations. The effects of including light clusters in nuclear matter and the densities at which the transitions between pasta configurations and to uniform matter occur are also investigated. The free energy, pressure, entropy and chemical potentials in the range of particle number densities and temper…
The mass-dependent symmetry energy coefficients $a_{sym}(A)$ has been extracted by analysing the heavy nuclear mass differences reducing the uncertainties as far as possible in our previous work. Taking advantage of the obtained symmetry energy coefficient $a_{sym}(A)$ and the density profiles obtained by switching off the Coulomb interaction in $^{208}\text{Pb}$, we calculated the slope parameter $L_{0.11}$ of the symmetry energy at the density of $0.11\text{fm}^{-3}$. The calculated $L_{0.11}$ ranges from 40.5 MeV to 60.3 MeV. The slope parameter $L_{0.11}$ of the symmetry energy at the dens…
We construct the first analytic self-gravitating Skyrmions with higher Baryon charge in four dimensions for the $SU(3)$-Skyrme-Einstein-$Λ$ theory by combining the generalized hedgehog ansatz with the approach developed by Balachandran et al. to describe the first (numerical) example of a non-embedded solution. These are genuine $SU(3)$ analytic solutions instead of trivial embeddings of $SU(2)$ into $SU(3)$ and its geometry is that of a Bianchi IX Universe. The Skyrme ansatz is chosen in such a way that the Skyrme field equations are identically satisfied in the sector with Baryon charge 4. T…
Results of charge form factors calculations for several unstable neutron-rich isotopes of light, medium and heavy nuclei (He, Li, Ni, Kr, Sn) are presented and compared to those of stable isotopes in the same isotopic chain. For the lighter isotopes (He and Li) the proton and neutron densities are obtained within a microscopic large-scale shell-model, while for heavier ones Ni, Kr and Sn the densities are calculated in deformed self-consistent mean-field Skyrme HF+BCS method. We also compare proton densities to matter densities together with their rms radii and diffuseness parameter values. Wh…
The connection from the structure and dynamics of atomic nuclei (finite nuclear system) to the nuclear equation of state (thermodynamic limit) is primarily made through nuclear energy-density functional (EDF) theory. Failure to describe both entities simultaneously within existing EDF frameworks means that we have either seriously misjudged the scope of EDF or not fully taken advantage of it. Enter the versatile KIDS Ansatz, which is based on controlled, order-by-order extensions of the nuclear EDF with respect to the Fermi momentum and allows a direct mapping from a given, immutable equation …
Constraining Energy Density Functionals via Bayesian Analysis of Nuclear Densities
The University of Osaka（大阪大学）；RIKEN Nishina Center for Accelerator-Based Science（理化学研究所仁科加速器科学研究中心）；Center for Computational Sciences, University of Tsukuba（筑波大学计算科学中心）
We study isoscalar (IS) and isovector (IV) densities in ${}^{40} \mathrm{Ca}$ in comparison with theoretical densities calculated by Skyrme Hartree-Fock (HF) models. The charge symmetry breaking and the charge independence breaking forces are introduced to study the effect on the IV density. The effect of isospin mixing in the ground-state density is examined by using the particle-vibration coupling model taking into account the collective IV giant monopole excitation. We show a clear correlations in the IV density and isospin impurity of ${}^{40} \mathrm{Ca}$ within the HF and the particle-vi…
We study the nuclear symmetry energy S(rho) and related quantities of nuclear physics and nuclear astrophysics predicted generically by relativistic mean-field (RMF) and Skyrme-Hartree-Fock (SHF) models. We establish a simple prescription for preparing equivalent RMF and SHF parametrizations starting from a minimal set of empirical constraints on symmetric nuclear matter, nuclear binding energy and charge radii, enforcing equivalence of their Lorenz effective masses, and then using the pure neutron matter (PNM) equation of state (EoS) obtained from ab-initio calculations to optimize the pure i…
We investigate the high-temperature effect on the nuclear matter that consists of mixture of nucleons and all nuclei in the dense and hot stellar environment. The individual nuclei are described within the compressible liquid-drop model that is based on Skyrme interactions for bulk energies and that takes into account modifications of the surface and Coulomb energies at finite temperatures and densities. The free-energy density is minimized with respect to the individual equilibrium densities of all heavy nuclei and the nuclear composition. We find that their optimized equilibrium densities be…
Based on systematic analyses of several popular non-relativistic energy density functionals with mean-field approximation, we estimate the value of the fourth-order symmetry energy $E_{\text{sym,4}}(ρ)$ at nuclear normal density $ρ_0$ and its density dependence, and explore the correlation between $E_{\text{sym,4}}(ρ_0)$ and other macroscopic quantities of nuclear matter properties. We use the empirical values of some nuclear macroscopic quantities to construct model parameter sets by Monte Carlo method for the conventional Skyrme-Hartree-Fock (SHF) model, the extended Skyrme-Hartree-Fock (eSH…
