Energy Density Functionals Research Articles

Finite-temperature infinite matter with effective-field-theory-inspired energy-density functionals

Finite-temperature infinite matter is analyzed with the recently introduced effective-field-theory(EFT)-inspired YGLO (Yang–Grasso–Lacroix–Orsay) and ELYO (extended Lee–Yang, Orsay) functionals, which are designed to describe very low-density regimes in symmetric (YGLO) and in pure neutron (YGLO and ELYO) matter. The article deals with neutron matter and aims to verify whether the use of these functionals allows us to correctly incorporate finite-temperature effects. We compare our results for some relevant thermodynamical quantities with the corresponding ones computed with a chosen reference ab-initio model, namely the many-body-perturbation-theory scheme. We validate the reliability of both EFT-inspired functionals at least at rather low densities and not too high temperatures and we discuss the effects related to the effective mass. We conclude that, at the present stage, the ELYO functional, having a higher neutron effective mass around saturation (closer to ab-initio values), allows us to describe finite-temperature properties more satisfactorily, in better agreement with ab-initio predictions up to higher densities and temperatures, compared to YGLO.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Constraints on the curvature of nuclear symmetry energy from recent astronomical data within the KIDS framework

In this paper, we investigate the density dependence of the nuclear symmetry energy [Formula: see text] in the KIDS (Korea-IBS-Daegu-SKKU) framework for the nuclear equation of state (EoS) and energy-density functional (EDF). The aim is to constrain the value of the curvature parameter ([Formula: see text]) based on recent astronomical data. First, assuming a standard saturation point, we calculate bulk nuclear properties within KIDS-EDF for different values of the compression modulus of symmetric nuclear matter ([Formula: see text]) and of the leading-order symmetry energy parameters, i.e., the symmetry energy ([Formula: see text]) and slope ([Formula: see text]) at saturation density, each within a broadly accepted range, as well as [Formula: see text]. All of the above EoS parameters are varied independently of each other. The skewness parameter ([Formula: see text]) is presently kept fixed at 650[Formula: see text]MeV. For all EoS parameter sets which describe the selected nuclear data within better than [Formula: see text], we calculate the neutron star EoS and mass–radius relation and analyze the results in terms of Pearson correlation coefficients [Formula: see text]. We find that the value of [Formula: see text] is strongly correlated with the radius of both a canonical and a massive star [Formula: see text]. If we impose that all known astronomical constraints on the neutron star radii must be satisfied, we deduce [Formula: see text]. As a result, the symmetry energy as a function of the density is consistently found to have an inflection point at [Formula: see text]. We take the opportunity to report that the neutron skin thickness of [Formula: see text]Pb shows no correlation at all with the neutron star radii [Formula: see text], in contrast with studies which focus on the role of [Formula: see text] only.

Global Performance of Covariant Density Functional Theory in Description of Charge Radii and Related Indicators

A short review of existing efforts to understand charge radii and related indicators on a global scale within the covariant density functional theory (CDFT) is presented. Using major classes of covariant energy density functionals (CEDFs), the global accuracy of the description of experimental absolute and differential charge radii within the CDFT framework has been established. This assessment is supplemented by an evaluation of theoretical statistical and systematic uncertainties in the description of charge radii. New results on the accuracy of the description of differential charge radii in deformed actinides and light superheavy nuclei are presented and the role of octupole deformation in their reproduction is evaluated. Novel mechanisms leading to odd-even staggering in charge radii are discussed. Finally, we analyze the role of self-consistency effects in an accurate description of differential charge radii.

Viscoelastic up-scaling rank-one effects in in-silico modelling of electro-active polymers

This paper analyses the viscoelastic up-scaling effects in electro-active polymers endowed with a micro-structure architecture in the form of a rank-one laminate. The principles of rank-n homogenisation and thermodynamical consistency are combined in the context of extremely deformable dielectric elastomers actuated well beyond the onset of geometrical instabilities. To ensure the robustness of the resulting methodology, Convex Multi-Variable (CMV) energy density functionals enriched with a nonlinear continuum viscoelastic description are used to describe the physics of the individual microscopic constituents. The high nonlinearity of the visco-electro-mechanical problem is resolved via a monolithic multi-scale Newton–Raphson scheme with a Backward-Euler (implicit) time integration scheme. A tensor cross product operation between vectors and tensors and an additive decomposition of the micro-scale deformation gradient (in terms of macro-scale and fluctuation components) are used to considerably reduce the complexity of the algebra. The resulting computational framework permits to explore the time-dependent in-silico analysis of rank-one electro-active polymer composites exhibiting extremely complex deformation patterns, paying particular attention to viscoelastic up-scaling effects. A comprehensive series of numerical examples is presented, where specially revealing conclusions about the rate-dependency of the composite electro-active polymer are observed as a function of its microstructure orientation and viscoelastic content. In a rectangular film subjected to extreme bending deformation, two different deformation modes are observed with one prevailing mode depending on the laminate composition. For the case of a square membrane where extreme deformation induces buckling, it is shown that the viscoelastic contribution leads to larger values of (stable) deformation, due to the regularisation that viscoelasticity inherently provides.

Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: effect of triaxial shape

The modeling of nuclear reactions and radioactive decays in astrophysical or earth-based conditions requires detailed knowledge of the masses of essentially all nuclei. Microscopic mass models based on nuclear energy density functionals (EDFs) can be descriptive and used to provide this information. The concept of intrinsic symmetry breaking is central to the predictive power of EDF approaches, yet is generally not exploited to the utmost by mass models because of the computational demands of adjusting up to about two dozen parameters to thousands of nuclear masses. We report on a first step to bridge the gap between what is presently feasible for studies of individual nuclei and large-scale models: we present a new Skyrme-EDF-based model that was adjusted using a three-dimensional coordinate-space representation, for the first time allowing for both axial and triaxial deformations during the adjustment process. To compensate for the substantial increase in computational cost brought by the latter, we have employed a committee of multilayer neural networks to model the objective function in parameter space and guide us towards the overall best fit. The resulting mass model BSkG1 is computed with the EDF model independently of the neural network. It yields a root mean square (rms) deviation on the 2457 known masses of 741 keV and an rms deviation on the 884 measured charge radii of 0.024 fm.

Information Content of the Parity-Violating Asymmetry in ^{208}Pb.

The parity-violating asymmetry A_{PV} in ^{208}Pb, recently measured by the PREX-2 Collaboration, is studied using modern relativistic (covariant) and nonrelativistic energy density functionals. We first assess the theoretical uncertainty on A_{PV} which is intrinsic to the adopted approach. To this end, we use quantified functionals that are able to accommodate our previous knowledge on nuclear observables such as binding energies, charge radii, and the dipole polarizability α_{D} of ^{208}Pb. We then add the quantified value of A_{PV} together with α_{D} to our calibration dataset to optimize new functionals. Based on these results, we predict a neutron skin thickness in ^{208}Pb r_{skin}=0.19±0.02 fm and the symmetry-energy slope L=54±8 MeV. These values are consistent with other estimates based on astrophysical data and are significantly lower than those recently reported using a particular set of relativistic energy density functionals. We also make a prediction for the A_{PV} value in ^{48}Ca that will be soon available from the CREX measurement.

Primary rainbow and Airy minima in 12C + 12C elastic scattering with families of non-monotonic potentials*

The experimental differential cross section of the 12C + 12C elastic scattering over a wide laboratory energy scale 25 ⩽ E lab ⩽ 360 MeV are analyzed within the framework of the optical model (OM) using non-monotonic (NM) nucleus–nucleus potential. The real parts of the NM potentials are derived from the energy density functional (EDF) formalism with the sudden approximation embodying the Pauli exclusion principle and the imaginary parts are taken as phenomenology. The diffractive and the refractive scattering with Airy structures in the whole angular region of the 12C + 12C elastic scattering in the above mentioned energy range are found profoundly successful in OM with the NM nucleus–nucleus potentials in five of different potential families. All the families at lower energies converge at 240 MeV to the unique potential family starting with the EDF potential. This, in conjunction with our recent findings for the 16O + 16O system in Islam et al (2021 J. Phys. G 48 075109), not only verifies the Goldberg criterion for elimination of discrete potential ambiguities for NM potentials, but also establishes that this leads uniquely to the EDF family. The 90° excitation function is also successfully reproduced with extension using NM potentials. The near- and far-side (N/F) decomposition of total elastic scattering amplitude has also been studied using our NM potentials.

Properties of a quantum vortex in neutron matter at finite temperatures

We have studied systematically microscopic properties of a quantum vortex in neutron matter at finite temperatures and densities corresponding to different layers of the inner crust of a neutron star. To this end and in preparation of future simulations of the vortex dynamics, we have carried out fully self-consistent 3D Hartree-Fock-Bogoliubov calculations, using one of the latest nuclear energy-density functionals from the Brussels-Montreal family, which has been developed specifically for applications to neutron superfluidity in neutron-star crusts. By analyzing the flow around the vortex, we have determined the effective radius relevant for the vortex filament model. We have also calculated the specific heat in the presence of the quantum vortex and have shown that it is substantially larger than for a uniform system at low temperatures. The low temperature limit of the specific heat has been identified as being determined by Andreev states inside the vortex core. We have shown that the specific heat in this limit does not scale linearly with temperature. The typical energy scale associated with Andreev states is defined by the minigap, which we have extracted for various neutron-matter densities. Our results suggest that vortices may be spin-polarized in the crust of magnetars. Finally, we have obtained a lower bound for the specific heat of a collection of vortices with given surface density, taking into account both the contributions from the vortex core states and from the hydrodynamic flow.

Scaling analysis of a physics-guided kinetic energy density expansion

An approach guided by physical consistency in determining the general forms of D-dimensional kinetic energy density functionals (KEDF) has been demonstrated previously, producing an expansion which contains the majority of the known one-point KEDF forms. It is known that any noninteracting KEDF shall necessarily have a homogeneity degree of 2 in coordinate scaling. This paper demonstrates that this condition is already satisfied in the general expansion despite not being conceived with the scaling as a constraint.

Partial dynamical symmetry from energy density functionals

We show that the notion of partial dynamical symmetry is robust and founded on a microscopic many-body theory of nuclei. Based on the universal energy density functional framework, a general quantal boson Hamiltonian is derived and shown to have essentially the same spectroscopic character as that predicted by the partial SU(3) symmetry. The principal conclusion holds in two representative classes of energy density functionals: nonrelativistic and relativistic. The analysis is illustrated in application to the axially-deformed nucleus $^{168}$Er.

Microscopic self-consistent description of induced fission: Dynamical pairing degree of freedom

The role of dynamical pairing in induced fission dynamics is investigated using the time-dependent generator coordinate method in the Gaussian overlap approximation, based on the microscopic framework of nuclear energy density functionals. A calculation of fragment charge yields for induced fission of $^{228}$Th is performed in a three-dimensional space of collective coordinates that, in addition to the axial quadrupole and octupole intrinsic deformations of the nuclear density, also includes an isoscalar pairing degree of freedom. It is shown that the inclusion of dynamical pairing has a pronounced effect on the collective inertia, the collective flux through the scission hyper-surface, and the resulting fission yields, reducing the asymmetric peaks and enhancing the contribution of symmetric fission, in better agreement with the empirical trend.

Nuclear symmetry energy components and their ratio: A new approach within the coherent density fluctuation model

A different alternative approach to calculate the ratio of the surface to volume components of the nuclear symmetry energy is proposed in the framework of the coherent density fluctuation model (CDFM). An alternate expression (scheme II) for the ratio is derived consistently within the model. This expression appears in a form more direct and physically motivated than the expression (scheme I) that was used in our previous works within the CDFM and avoids preliminary assumptions and mathematical ambiguities in scheme I. The calculations are based on the Skyrme and Brueckner energy-density functionals for nuclear matter and on the nonrelativistic Brueckner-Hartree-Fock method with realistic Bonn B and Bonn CD nucleon-nucleon potentials. The approach is applied to isotopic chains of Ni, Sn, and Pb nuclei using nuclear densities obtained in self-consistent Hartree-Fock$+\mathrm{BCS}$ calculations with the SLy4 Skyrme effective interaction. The applicability of both schemes within the CDFM is demonstrated by a comparison of the results with the available empirical data and with results of other theoretical studies of the considered quantities. Although in some instances the results obtained for the studied ratio and the symmetry energy components are rather close in both schemes, the proposed scheme II leads to more realistic values that agree better with the empirical data and exhibits conceptual and operational advantages.

Skyrme pseudopotentials at next-to-next-to-leading order: Construction of local densities and first symmetry-breaking calculations

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.

Isovector giant monopole and quadrupole resonances in a Skyrme energy density functional approach with axial symmetry

[Background] Giant resonance (GR) is a typical collective mode of vibration. The deformation splitting of the isovector (IV) giant dipole resonance is well established. However, the splitting of GRs with other multipolarities is not well understood. [Purpose] I explore the IV monopole and quadrupole excitations and attempt to obtain the generic features of IV giant resonances in deformed nuclei by investigating the neutral and charge-exchange channels simultaneously. [Method] I employ a nuclear energy-density functional (EDF) method: the Skyrme-Kohn-Sham-Bogoliubov and the quasiparticle random-phase approximation are used to describe the ground state and the transition to excited states. [Results] I find the concentration of the monopole strengths in the energy region of the isobaric analog or Gamow-Teller resonance irrespective of nuclear deformation, and the appearance of a high-energy giant resonance composed of the particle-hole configurations of $2\hbar \omega_0$ excitation. Splitting of the distribution of the strength occurs in the giant monopole and quadrupole resonances due to deformation. The lower $K$ states of quadrupole resonances appear lower in energy and possess the enhanced strengths in the prolate configuration, and vice versa in the oblate configuration, while the energy ordering depending on $K$ is not clear for the $J=1$ and $J=2$ spin-quadrupole resonances. [Conclusions] The deformation splitting occurs generously in the giant monopole and quadrupole resonances. The $K$-dependence of the quadrupole transition strengths is largely understood by the anisotropy of density distribution.

BigApple force and its implications to finite nuclei and astrophysical objects

The secondary component of the GW190814 event left us with a question, “whether it is a supermassive neutron star or lightest black-hole?” Recently, Fattoyev et al. obtained an energy density functional (EDF) named as BigApple, which reproduces the mass of the neutron star is 2.60 [Formula: see text] which is well consistent with GW190814 data. This study explores the properties of finite nuclei, nuclear matter and neutron stars by using the BigApple EDF along with four well-known relativistic mean-field forces, namely NL3, G3, IOPB-I and FSUGarnet. The finite nuclei properties like binding energy per particle, skin thickness, charge radius, single-particle energy, and two-neutron separation energy are well predicted by the BigApple for a series of nuclei. The calculated nuclear matter quantities, such as incompressibility, symmetry energy and slope parameters at saturation density, are consistent with the empirical or experimental values where ever available. The predicted canonical tidal deformability by the BigApple parameter set is well-matched with the GW190814 data. Also, the dimensionless moment of inertia lies in the range given by the analysis of PSR J0737-3039A.

Implications of PREX-2 on the electric dipole polarizability of neutron-rich nuclei

The recent announcement by the PREX collaboration of an unanticipated thick neutron skin in 208Pb (Rskin208) has challenged our understanding of neutron rich matter in the vicinity of nuclear saturation density. Whereas earlier constraints indicate that the symmetry energy is relatively soft, the PREX-2 result seems to suggest the opposite. We confront constraints on the symmetry energy obtained from measurements of the electric dipole polarizability against those informed by the PREX-2 measurement of Rskin208 and by the correlations that it entails. Covariant energy density functionals informed by the properties of finite nuclei are used to compute the electric dipole response of 48Ca, 68Ni, 132Sn, and 208Pb. The set of functionals used in this work are consistent with experimental data, yet are flexible enough in that they span a wide range of values of Rskin208. It is found that theoretical predictions of the electric dipole polarizability that are consistent with the PREX-2 measurement systematically overestimate the corresponding values extracted from the direct measurements of the distribution of electric dipole strength. The neutron skin thickness of 208Pb extracted from parity violating electron scattering and the electric dipole polarizability measured in photoabsorption experiments are two of the cleanest experimental tools used to constrain the symmetry energy around nuclear saturation density. However, the recent value of Rskin208 that suggests a fairly stiff symmetry energy stands in stark contrast to the conclusions derived from the electric dipole polarizability. At present, we offer no solution to this dilemma.

Interplay between pairing and triaxial shape degrees of freedom in Os and Pt nuclei

The effect of coupling between pairing and quadrupole triaxial shape vibrations on the low-energy collective states of $\gamma$-soft nuclei is investigated using a model based on the framework of nuclear energy density functionals (EDFs). Employing a constrained self-consistent mean-field (SCMF) method that uses universal EDFs and pairing interactions, potential energy surfaces of characteristic $\gamma$-soft Os and Pt nuclei with $A\approx190$ are calculated as functions of the pairing and triaxial quadrupole deformations. Collective spectroscopic properties are computed using a number-nonconserving interacting boson model (IBM) Hamiltonian, with parameters determined by mapping the SCMF energy surface onto the expectation value of the Hamiltonian in the boson condensate state. It is shown that, by simultaneously considering both the shape and pairing collective degrees of freedom, the EDF-based IBM successfully reproduces data on collective structures based on low-energy $0^{+}$ states, as well as $\gamma$-vibrational bands.

Pairing and nonaxial-shape correlations in N=150 isotones

Background: Rotational bands have been measured around $^{250}\mathrm{Fm}$ associated with strong deformed-shell closures. The ${K}^{\ensuremath{\pi}}={2}^{\ensuremath{-}}$ excited band emerges systematically in $N=150$ isotones raging from plutonium to nobelium with even-$Z$ numbers, and a sharp drop in energies was observed in californium.Purpose: I attempt to uncover the microscopic mechanism for the appearance of such a low-energy ${2}^{\ensuremath{-}}$ state in $^{248}\mathrm{Cf}$. Furthermore, I investigate the possible occurrence of the low-energy ${K}^{\ensuremath{\pi}}={2}^{+}$ state, the $\ensuremath{\gamma}$ vibration, to elucidate the mechanism that prefers the simultaneous breaking of the reflection and axial symmetry to the breaking of the axial symmetry alone in this mass region.Method: I employ a nuclear energy-density functional (EDF) method: the Skyrme-Kohn-Sham-Bogoliubov and the quasiparticle random-phase approximation are used to describe the ground state and the transition to excited states.Results: The Skyrme-type SkM* and SLy4 functionals reproduce the fall in energy but not the absolute value of the ${K}^{\ensuremath{\pi}}={2}^{\ensuremath{-}}$ state at $Z=98$ where the proton two-quasiparticle excitation $[633]7/2\ensuremath{\bigotimes}[521]3/2$ plays a decisive role for the peculiar isotonic dependence. I find interweaving roles by the pairing correlation of protons and the deformed-shell closure at $Z=98$. The SkM* model predicts the ${K}^{\ensuremath{\pi}}={2}^{\ensuremath{-}}$ state appears lower in energy in $^{246}\mathrm{Cf}$ than in $^{248}\mathrm{Cf}$ as the Fermi level of neutrons is located in between the [622]5/2 and the [734]9/2 orbitals. Except for $^{250}\mathrm{Fm}$ in the SkM* calculation, the ${K}^{\ensuremath{\pi}}={2}^{+}$ state is predicted to appear higher in energy than the ${K}^{\ensuremath{\pi}}={2}^{\ensuremath{-}}$ state because the quasiproton [521]1/2 orbital is located above the [633]7/2 orbital.Conclusions: A systematic study of low-lying collective states in heavy actinide nuclei provides a rigorous testing ground for microscopic nuclear models. The present paper shows a need for improvements in the EDFs to describe pairing correlations and shell structures in heavy nuclei, that are indispensable in predicting the heaviest nuclei.

Highly accurate machine learning model for kinetic energy density functional

Inspired by the remarkable ongoing progress of data-driven approaches, a very accurate predictive model is developed to estimate one-dimensional kinetic energy density functionals (KEDF) using Machine Learning (ML). Starting from possible analytical forms of kinetic energy density and by utilizing a variety of solvable models, a simple – yet highly – accurate linear regression model is statistically trained to estimate the kinetic energy as functionals of the density. The mean relative accuracy for even a small number of randomly generated potentials is found to be better than the standard KEDF (Thomas-Fermi (TF) and von Weizsäcker (vW)) by several orders of magnitudes. As more different potentials of model problems are mixed, the coefficients of the linear model significantly approach the known values of Thomas-Fermi and von Weizsäcker, suggesting the reliability of the statistical training approach. This work can provide an important step toward more accurate large-scale orbital free density functional theory (OF-DFT) calculations.