We study the bulk deformation properties of the Skyrme nuclear energy density functionals. Following simple arguments based on the leptodermous expansion and liquid drop model, we apply the nuclear density functional theory to assess the role of the surface symmetry energy in nuclei. To this end, we validate the commonly used functional parametrizations against the data on excitation energies of superdeformed band-heads in Hg and Pb isotopes, and fission isomers in actinide nuclei. After subtracting shell effects, the results of our self-consistent calculations are consistent with macroscopic arguments and indicate that experimental data on strongly deformed configurations in neutron-rich nuclei are essential for optimizing future nuclear energy density functionals. The resulting survey provides a useful benchmark for further theoretical improvements. Unlike in nuclei close to the stability valley, whose macroscopic deformability hangs on the balance of surface and Coulomb terms, the deformability of neutron-rich nuclei strongly depends on the surface-symmetry energy; hence, its proper determination is crucial for the stability of deformed phases of the neutron- rich matter and description of fission rates for r-process nucleosynthesis.
We develop an energy functional with shell-model occupations as the relevant degrees of freedom and compute nuclear masses across the nuclear chart. The functional is based on Hohenberg-Kohn theory with phenomenologically motivated terms. A global fit of the 17-parameter functional to nuclear masses yields a root-mean-square deviation of χ= 1.31 MeV. Nuclear radii are computed within a model that employs the resulting occupation numbers.
The symmetry energy coefficients for nuclei with mass number A=20~250 are extracted from more than 2000 measured nuclear masses. With the semi-empirical connection between the symmetry energy coefficients of finite nuclei and the nuclear symmetry energy at reference densities, we investigate the density dependence of symmetry energy of nuclear matter at subnormal densities. The obtained results are compared with those extracted from other methods.
Because of the rotational invariance of the nuclear Hamiltonian, there exists a density functional for nuclei that depends only on two scalar densities. Practical calculations boil down to radial, one-dimensional ones.
The spin density matrix (SDM) used in atomic and molecular physics is revisited for nuclear physics, in the context of the radial density functional theory. The vector part of the SDM defines a "hedgehog" situation, which exists only if nuclear states contain some amount of parity violation.
Calculations of nuclear masses, using nuclear density functional theory, are presented for even-even nuclei spanning the nuclear chart. The resulting binding energy differences can be interpreted in terms of valence proton-neutron interactions. These are compared globally, regionally, and locally with empirical values. Overall, excellent agreement is obtained. Discrepancies highlight neglected degrees of freedom and can point to improved density functionals.
By using the functional path integral method, we obtain a model-independent formula for nuclear symmetry energy, which explicitly shows the relation between nuclear symmetry energy and isospin susceptibility. The latter one is found to be a probe to the QCD chiral phase transition. We further found that, the nuclear symmetry energy has an abrupt change at the critical nuclear density where the chiral symmetry restores partially, which could be detected from the experiments.
This note describes five subjects of some interest for the density functional theory in nuclear physics. These are, respectively, i) the need for concave functionals, ii) the nature of the Kohn-Sham potential for the radial density theory, iii) a proper implementation of a density functional for an "intrinsic" rotational density, iv) the possible existence of a potential driving the square root of the density, and v) the existence of many models where a density functional can be explicitly constructed.
A nuclear density functional can be used to find the binding energy and shell structure of nuclei and the energy gap in superconducting nuclear matter. In this paper, we study the possible application of a nuclear density functional theory to nuclear astrophysics. From energy density functional theory, we can deduce the interaction between nucleons to find a rough estimate of the charge radius of the specific nuclei. Compared to the Finite-Range Thomas Fermi model, we include three-body forces, which might be important at densities several times that of nuclear matter density. We also add the …
Extrapolations from nuclei to neutron stars hinge on the symmetry term in nuclear binding formulas. The term describes reduction in the binding associated with neutron-proton (np) imbalance. Regrettably, binding formulas in the literature commonly lack an intrinsic consistency with regard to the symmetry term. Our elementary considerations determine the universal macroscopic limit for the term and predict its weakening in light nuclei due to emergence of the nuclear asymmetry-skin. Experimental systematic of isobaric-analogue states allows for a determination of the volume and surface coeffici…
In the field of Energy Density Functionals (EDF) used in nuclear structure and dynamics, one of the unsolved issues is the stability of the functional. Numerical issues aside, some EDFs are unstable with respect to particular perturbations of the nuclear ground-state density. The aim of this contribution is to raise questions about the origin and nature of these instabilities, the techniques used to diagnose and prevent them, and the domain of density functions in which one should expect a nuclear EDF to be stable.
The liquid-gas phase transition in hot asymmetric nuclear matter is investigated within relativistic mean-field model using the density dependence of nuclear symmetry energy constrained from the measured neutron skin thickness of finite nuclei. We find symmetry energy has a significant influence on several features of liquid-gas phase transition. The boundary and area of the liquid-gas coexistence region, the maximal isospin asymmetry and the critical values of pressure and isospin asymmetry all of which systematically increase with increasing softness in the density dependence of symmetry ene…
Based on the general analysis of the grand canonical partition function in the S-matrix framework, the calculated results on symmetry energy, free energy and entropy of dilute warm nuclear matter are presented. At a given temperature and density, the symmetry energy or symmetry free energy of the clusterized nuclear matter in the S-matrix formulation deviates, particularly at low temperature and relatively higher density, in a subtle way, from the linear dependence on the square of the isospin asymmetry parameter $X=(ρ_n-ρ_p)/(ρ_n+ρ_p)$, contrary to those obtained for homogeneous nucleonic mat…
A model is developed, based on the density functional perturbation theory and the inverse Kohn-Sham method, that can be used to improve relativistic nuclear energy density functionals towards an exact but unknown Kohn-Sham exchange-correlation functional. The improved functional is determined by empirical exact ground-state densities of finite systems. A test of the model and an illustrative calculation are performed, starting from two different approximate functionals, to reproduce the parameters and density dependence of a target functional, using exact ground-state densities of symmetric N=…
We carried out a simulation of heavy ion collision using a time-dependent density functional theory. We call it the canonical-basis time-dependent Hartree-Fock-Bogoliubov theory (Cb-TDHFB) which can describe nuclear dynamics in three-dimensional coordinate space, treating nuclear pairing correlation. We simulate 20O+20O collision using the Cb-TDHFB with a contact-type pairing functional, and show the behavior of gap energy which is decreasing and vibrating while colliding.
We discuss some of the challenges that the DFT community faces in its quest for the truly universal energy density functional applicable over the entire nuclear chart.
Using the relativistic impulse approximation with empirical NN scattering amplitude and the nuclear scalar and vector densities from the relativistic mean-field theory, we evaluate the Dirac optical potential for neutrons and protons in asymmetric nuclear matter. From the resulting Schrö% dinger-equivalent potential, the high energy behavior of the nuclear symmetry potential is studied. We find that the symmetry potential at fixed baryon density is essentially constant once the nucleon kinetic energy is greater than about 500 MeV. Moreover, for such high energy nucleon, the symmetry potential …
When applied to a single nucleon, nuclear energy density functionals may yield a non-vanishing internal energy thus implying that the nucleon is interacting with itself. It is shown how to avoid this unphysical feature for semi-local phenomenological functionals containing all possible bilinear combinations of local densities and currents up to second order in the derivatives. The method outlined in this Rapid Communication could be easily extended to functionals containing higher order terms, and could serve as a guide for constraining the time-odd part of the functional.
The effect of density dependent symmetry energy on nuclear-stopping is studied using isospin-dependent quantum molecular dynamics model(IQMD). We have used the reduced isospin-dependent cross-section with soft(S) equation of state for the systems having different isostopic content, to explore the various aspects of nuclear stopping. The aim is to pin down the nature of the nuclear stopping with density dependent symmetry energy. Nuclear stopping is found to be sensitive towards the various forms of the density dependent symmetry energy. The nuclear stopping tends to decrease for the stiffer eq…
An electron density functional approach for the calculation of the nuclear multipole moments is presented. The electronic matrix elements entering the experimentally observed hyperfine electron-nucleus interaction constants in atoms are expressed in terms of the electron density functionals of the charge or spin distributions. In principle, the construction of the charge or spin distribution density functionals can be obtained by means of every relativistic or non-relativistic quantum-mechanical or DFT method. The electronic matrix elements for all the electronic operators of the existing Hami…
The nuclear magnetic moment is an important physical observable and serves as a useful tool for the stringent test of nuclear models. For the past decades, the covariant density functional theory and its extension have been proved to be successful in describing the nuclear ground-states and excited states properties. However, a long-standing problem is its failure to predict magnetic moments. This article reviews the recent progress in the description of the nuclear magnetic moments within the covariant density functional theory. In particular, the magnetic moments of spherical odd-A nuclei wi…
We examine how nuclear masses are related to the density dependence of the symmetry energy. Using a macroscopic nuclear model we calculate nuclear masses in a way dependent on the equation of state of asymmetric nuclear matter. We find by comparison with empirical two-proton separation energies that a smaller symmetry energy at subnuclear densities, corresponding to a larger density symmetry coefficient L, is favored. This tendency, which is clearly seen for nuclei that are neutron-rich, nondeformed, and light, can be understood from the property of the surface symmetry energy in a compressibl…
We present background concepts of the nuclear density functional theory (DFT) and applications of the time-dependent DFT with the Skyrme energy functional for nuclear response functions. Practical methods for numerical applications of the time-dependent Hartree-Fock-Bogoliubov theory (TDHFB) are proposed; finite amplitude method and canonical-basis TDHFB. These approaches are briefly reviewed and some numerical applications are shown to demonstrate their feasibility.
The nuclear $α$ decay of heavy nuclei is investigated based on the nuclear energy density functional, which leads to the $α$ potential inside the parent nucleus in terms of the proton and neutron density profiles of the daughter nucleus. We use the Skyrme force model, Gogny force model, and relativistic mean field model to get the nucleon density profiles inside heavy nuclei. Once the nucleon density profiles are determined, the parameters of the nuclear $α$ potential are fitted to the observed $α$ decay half-lives of heavy nuclei. This approach is then applied to predict unknown $α$ decay hal…
These are personal recollections of how Gerry Brown's work and thinking influenced the development of the chiral effective Lagrangian based theory of nuclear forces.
A microscopic framework of nuclear energy density functionals is reviewed, which establishes a direct relation between low-energy QCD and nuclear structure, synthesizing effective field theory methods and principles of density functional theory. Guided by two closely related features of QCD in the low-energy limit: a) in-medium changes of vacuum condensates, and b) spontaneous breaking of chiral symmetry; a relativistic energy density functional is developed and applied in studies of ground-state properties of spherical and deformed nuclei.
A study of the correlation between nuclear temperatures and symmetry energy is presented for heavy-ion collisions at intermediate energies via the isospin-dependent quantum molecular-dynamics model. It is found that different symmetry energy parameters will change the density and kinetic energy distribution of the hot nuclei. More importantly, nuclear temperatures that are based on kinetic energy properties can be used to study symmetry energy information.
We propose a method for microscopic calculations of nuclear ground-state properties in the framework of density functional theory. We discuss how the density functional is equivalent to the effective action for the density, thereby establishing a constructive framework for density functional calculations from nuclear forces. The presented approach starts from non-interacting nucleons in a background potential (a simple approximation for the mean field). The nuclear forces are then gradually turned on, while the background potential is removed. The evolution equation yields the ground-state ene…
Interplay between the dependence of symmetry energy on density and the variation of nucleonic densities across nuclear surface is discussed. That interplay gives rise to the mass dependence of the symmetry coefficient in an energy formula. Charge symmetry of the nuclear interactions allows to introduce isoscalar and isovector densities that are approximately independent of the magnitude of neutron-proton asymmetry.
We present the basic concepts and our recent developments in the density functional approaches with the Skyrme functionals for describing nuclear dynamics at low energy. The time-dependent density-functional theory (TDDFT) is utilized for the exact linear response with an external perturbation. For description of collective dynamics beyond the perturbative regime, we present a theory of a decoupled collective submanifold to describe for a slow motion based on the TDDFT. Selected applications are shown to demonstrate the quality of their performance and feasibility. Advantages and disadvantages…
A chiral model based on nucleons interacting via boson exchange is investigated. Fluctuation effects are included consistently beyond the mean-field approximation in the framework of the functional renormalization group. The liquid-gas phase transition of symmetric nuclear matter is studied in detail. No sign of a chiral restoration transition is found up to temperatures of about 100 MeV and densities of at least three times the density of normal nuclear matter. Moreover, the model is extended to asymmetric nuclear matter and the constraints from neutron star observations are discussed.
We review the main achievements of the research programme for the study of nuclear forces in the framework of chiral symmetry and discuss some problems which are still open.
The pervasive role of the nuclear symmetry energy in establishing some nuclear static and dynamical properties, and in governing some attributes of neutron star properties is highlighted.
Two recent experiments have demonstrated that the effective $ρ$-mass in nuclear medium, as extracted from the $^3He(γ, π^+ π^-)$ reaction, is substantially reduced. This has been advocated as an indication of partial restoration of chiral symmetry in nuclear matter. We show that even in the absence of chiral symmetry, effective mean field nuclear matter models can explain these findings quantitatively.
Information about the nuclear matter under the extreme conditions of temperature and density and the role of symmetry energy under these conditions is still a topic of crucial importance in the present day nuclear physics research. The multifragmentation, collective flow and the nuclear stopping is among the various rare phenomenon which can be observed in heavy-ion collisions at intermediate energies. The nuclear stopping, which is sensitive towards the symmetry energy has gained a lot of interest because it provides the possibility to examine the degree of thermalization or equilibration in …
We review the notion of symmetry breaking and restoration within the frame of nuclear energy density functional methods. We focus on key differences between wave-function- and energy-functional-based methods. In particular, we point to difficulties encountered within the energy functional framework and discuss new potential constraints on the underlying energy density functional that could make the restoration of broken symmetries better formulated within such a formalism. We refer to Ref.~\cite{duguet10a} for details.
We present the first application of a new approach, proposed in [Journal of Physics G: Nuclear and Particle Physics, 43, 04LT01 (2016)] to derive coupling constants of the Skyrme energy density functional (EDF) from ab initio Hamiltonian. By perturbing the ab initio Hamiltonian with several functional generators defining the Skyrme EDF, we create a set of metadata that is then used to constrain the coupling constants of the functional. We use statistical analysis to obtain such an ab initio-equivalent Skyrme EDF. We find that the resulting functional describes properties of atomic nuclei and i…
We recall the main features of the Td (tetrahedral) symmetry in atomic nuclei and present realistic mean-field calculations supporting the existence such a symmetry all over the nuclear chart. A few potential candidate-nuclei are investigated and the possible experimental signatures of the tetrahedral symmetry are also briefly discussed.
Using the relativistic mean-field theory, we adopt two different methods, namely, the coexisting phase method and the self-consistent Thomas-Fermi approximation, to study the impacts of the nuclear symmetry energy on properties of neutron star crusts within a wide range of densities. It is found that the nuclear symmetry energy and its density slope play an important role in determining the pasta phases and the crust-core transition.
We study a particular class of relativistic nuclear energy density functionals in which only nucleon degrees of freedom are explicitly used in the construction of effective interaction terms. Short-distance (high-momentum) correlations, as well as intermediate and long-range dynamics, are encoded in the medium (nucleon density) dependence of the strength functionals of an effective interaction Lagrangian. Guided by the density dependence of microscopic nucleon self-energies in nuclear matter, a phenomenological ansatz for the density-dependent coupling functionals is accurately determined in s…
Within the nuclear density functional theory (DFT) we study the effect of reflection-asymmetric shapes on ground-state binding energies and binding energy differences. To this end, we developed the new DFT solver AxialHFB that uses an approximate second-order gradient to solve the Hartree-Fock-Bogoliubov equations of superconducting DFT with the quasi-local Skyrme energy density functionals. Illustrative calculations are carried out for even-even isotopes of radium and thorium.
Response functions of infinite nuclear matter with arbitrary isospin asymmetry are studied in the framework of the random phase approximation. The residual interaction is derived from a general nuclear Skyrme energy density functional. Besides the usual central, spin-orbit and tensor terms it could also include other components as new density-dependent terms or three-body terms. Algebraic expressions for the response functions are obtained from the Bethe-Salpeter equation for the particle-hole propagator. Applications to symmetric nuclear matter, pure neutron matter and asymmetric nuclear matt…
A study of parameter sensitivity of nuclear energy density functionals, initiated in the first part of this work \cite{NV.16}, is extended by the inclusion of data on ground-state properties of finite nuclei in the application of the manifold boundary approximation method (MBAM). Density functionals used in self-consistent mean-field calculations, and nuclear structure models based on them, are generally "sloppy" and exhibit an exponential range of sensitivity to parameter variations. Concepts of information geometry are used to identify the presence of effective functionals of lower dimension…
With the global color symmetry model being extended to finite chemical potential, we study the density dependence of the local and nonlocal scalar quark condensates in nuclear matter. The calculated results indicate that the quark condensates increase smoothly with the increasing of nuclear matter density before the critical value (about 12$ρ_0$) is reached. It also manifests that the chiral symmetry is restored suddenly as the density of nuclear matter reaches its critical value. Meanwhile, the nonlocal quark condensate in nuclear matter changes nonmonotonously against the space-time distance…
A non-relativisitic nuclear density functional theory is constructed, not as usual, from an effective density dependent nucleon-nucleon force but directly introducing in the functional results from microscopic nuclear and neutron matter Bruckner G-matrix calculations at various densities. A purely phenomenological finite range part to account for surface properties is added. The striking result is that only four to five adjustable parameters, spin-orbit included, suffice to reproduce nuclear binding energies and radii with the same quality as obtained with the most performant effective forces,…
The origin of the nuclear tensor interaction in the covariant energy density functional (EDF) is presented in this work, associated with the Fock diagrams of Lorentz scalar and vector couplings. With this newly obtained relativistic formalism of the nuclear tensor interaction, more distinct tensor effects are found in the Fock diagrams of the Lorentz scalar and vector couplings, as compared to the Lorentz pseudo-vector and tensor channels. A unified and self-consistent treatment on both the nuclear tensor and spin-orbit interactions, which dominate the spin-dependent features of the nuclear fo…
The Interacting Boson Model of nuclear structure is introduced to the unitarity limit. Non relativistic conformal symmetry creates a scale invariant state from which the stationary states of atomic nuclei are obtained. A brief discussion of the nature of the conformal field theory is included along with the BCS-BEC crossover and hydrodynamic properties.
Within a dynamical description of nuclear fragmentation, based on the liquid-gas phase transition scenario, we explore the relation between neutron-proton density fluctuations and nuclear symmetry energy. We show that, along the fragmentation path, isovector fluctuations follow the evolution of the local density and approach an equilibrium value connected to the local symmetry energy. Higher density regions are characterized by smaller average asymmetry and narrower isotopic distributions. This dynamical analysis points out that fragment final state isospin fluctuations can probe the symmetry …
The symmetry energy coefficients of dilute clusterized nuclear matter are evaluated in the $S$-matrix framework. Employing a few different definitions commonly used in the literature for uniform nuclear matter, it is seen that the different definitions lead to perceptibly different results for the symmetry coefficients for dilute nuclear matter. They are found to be higher compared to those obtained for uniform matter in the low density domain. The calculated results are in reasonable consonance with those extracted recently from experimental data.
Reliable predictions of nuclear properties are needed as much to answer fundamental science questions as in applications such as reactor physics or data evaluation. Nuclear density functional theory is currently the only microscopic, global approach to nuclear structure that is applicable throughout the nuclear chart. In the past few years, a lot of effort has been devoted to setting up a general methodology to assess theoretical uncertainties in nuclear DFT calculations. In this paper, we summarize some of the recent progress in this direction. Most of the new material discussed here will be …
The formalism of linear response theory for a Skyrme functional including spin-orbit and tensor terms is generalized to the case of infinite nuclear matter with arbitrary isospin asymmetry. Response functions are obtained by solving an algebraic system of equations, which is explicitly given. Spin-isospin strength functions are analyzed varying the conditions of density, momentum transfer, asymmetry and temperature. The presence of instabilities, including the spinodal one, is studied by means of the static susceptibility.
The Skyrme nuclear energy density functional theory (DFT) is used to model neutron-induced fission in actinides. This paper focuses on the numerical implementation of the theory. In particular, it reports recent advances in DFT code development on leadership class computers, and presents a detailed analysis of the numerical accuracy of DFT solvers for near-scission calculations.
Relativistic energy density functionals (EDF) have become a standard tool for nuclear structure calculations, providing a complete and accurate, global description of nuclear ground states and collective excitations. Guided by the medium dependence of the microscopic nucleon self-energies in nuclear matter, semi-empirical functionals have been adjusted to the nuclear matter equation of state and to bulk properties of finite nuclei, and applied to studies of arbitrarily heavy nuclei, exotic nuclei far from stability, and even systems at the nucleon drip-lines. REDF-based structure models have a…
We suggest a particular procedure of derivation of the beta-stability line and isotopic symmetry energy. The behavior of the symmetry energy coefficient $b(A,N-Z)$ is analyzed. We redefine the surface tension coefficient and the surface symmetry energy for an asymmetric nuclear Fermi-liquid drop with a finite diffuse layer. Following Gibbs-Tolman concept, we introduce the equimolar radius at which the surface tension is applied. The relation of the nuclear macroscopic characteristics like surface and symmetry energies, Tolman length, etc. to the bulk properties of nuclear matter is considered.…
The nuclear symmetry energy represents a response to the neutron-proton asymmetry. In this survey we discuss various aspects of symmetry energy in the framework of nuclear density functional theory, considering both non-relativistic and relativistic self-consistent mean-field realizations side-by-side. Key observables pertaining to bulk nucleonic matter and finite nuclei are reviewed. Constraints on the symmetry energy and correlations between observables and symmetry-energy parameters, using statistical covariance analysis, are investigated. Perspectives for future work are outlined in the co…
Basic issues of the time-dependent density-functional theory are discussed, especially on the real-time calculation of the linear response functions. Some remarks on the derivation of the time-dependent Kohn-Sham equations and on the numerical methods are given.
