Covariant Energy Density Functionals Research Articles

Universal reduced basis for the calibration of covariant energy density functionals

Universal reduced basis for the calibration of covariant energy density functionals

Impact of tensor couplings with scalar mixing on covariant energy density functionals

Impact of tensor couplings with scalar mixing on covariant energy density functionals

Bayesian refinement of covariant energy density functionals

The last five years have seen remarkable progress in our quest to determine the equation of state of neutron rich matter. Recent advances across the theoretical, experimental, and observational landscape have been incorporated in a Bayesian framework to refine existing covariant energy density functionals previously calibrated by the properties of finite nuclei. In particular, constraints on the maximum neutron star mass from pulsar timing, on stellar radii from the NICER mission, on tidal deformabilities from the LIGO-Virgo collaboration, and on the dynamics of pure neutron matter as predicted from chiral effective field theories have resulted in significant refinements to the models, particularly to those predicting a stiff symmetry energy. Still, even after these improvements, we find it challenging to reproduce simultaneously the neutron skin thickness of both $^{208}\mathrm{Pb}$ and $^{48}\mathrm{Ca}$ recently reported by the PREX/CREX collaboration.

Covariant energy density functionals with and without tensor couplings at the Hartree-Bogoliubov level

Background: The study of additional terms in functionals is relevant to better describe nuclear structure phenomenology. Among these terms, the tensor one is known to impact nuclear structure properties, especially in neutron-rich nuclei. However, its effect has not been studied on the whole nuclear chart yet.Purpose: The impact of terms corresponding to the tensor at the Hartree level is studied for infinite nuclear matter as well as deformed nuclei, by developing new density-dependent functionals including these terms. In particular, we study in details the improvement such a term can bring to the description of specific nuclear observables.Methods: The framework of covariant energy density functional is used at the Hartree-Bogoliubov level. The free parameters of covariant functionals are optimized by combining Markov-chain Monte Carlo and simplex algorithms. A sensitivity analysis on these parameters is undertaken.Results: An improvement of the rms binding energies, spin-orbit splittings, and gaps is obtained over the nuclear chart, including axially deformed ones, when including tensors terms. Small modifications of the potential energy surface and densities are also found. In infinite matter, the Dirac mass is shifted to a larger value, in better agreement with experiments.Conclusions: Taking into account additional terms corresponding to the tensor terms in the vector-isoscalar channel at the Hartree level improves the description of nuclear properties, both in nuclei and in nuclear matter.

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.

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.

Hyperheavy spherical and toroidal nuclei: The role of shell structure

The properties of toroidal hyperheavy even-even nuclei and the role of toroidal shell structure are extensively studied within covariant density functional theory. The general trends in the evolution of toroidal shapes in the $Z\approx 130-180$ region of nuclear chart are established for the first time. These nuclei are stable with respect of breathing deformations. The most compact fat toroidal nuclei are located in the $Z\approx 136, N\approx 206$ region of nuclear chart, but thin toroidal nuclei become dominant with increasing proton number and on moving towards proton and neutron drip lines. The role of toroidal shell structure, its regularity, supershell structure, shell gaps as well as the role of different groups of the pairs of the orbitals in its formation are investigated in detail. The lowest in energy solutions at axial symmetry are characterized either by large shell gaps or low density of the single-particle states in the vicinity of the Fermi level in at least one of the subsystems (proton or neutron). Related quantum shell effects are expected to act against the instabilities in breathing and sausage deformations for these subsystems. The investigation with large set of covariant energy density functionals reveals that substantial proton $Z=154$ and 186 and neutron $N=228$, 308 and 406 spherical shell gaps exist in all functionals. The nuclei in the vicinity of the combination of these particle numbers form the islands of stability of spherical hyperheavy nuclei. The study suggests that the $N=210$ toroidal shell gap plays a substantial role in the stabilization of fat toroidal nuclei.

Global study of separable pairing interaction in covariant density functional theory

A systematic global investigation of pairing properties based on all available experimental data on pairing indicators has been performed for the first time in the framework of covariant density functional theory. It is based on separable pairing interaction of Ref. [1]. The optimization of the scaling factors of this interaction to experimental data clearly reveals its isospin dependence in neutron subsystem. However, the situation is less certain in proton subsystem since similar accuracy of the description of pairing indicators can be achieved both with isospin-dependent and mass-dependent scaling factors. The differences in the functional dependencies of scaling factors lead to the uncertainties in the prediction of proton and neutron pairing properties which are especially pronounced at high isospin and could have a significant impact on some physical observables. For a given part of nuclear chart the scaling factors for spherical nuclei are smaller than those for deformed ones; this feature exists also in non-relativistic density functional theories. Its origin is traced back to particle-vibration coupling in odd-$A$ nuclei which is missing in all existing global studies of pairing. Although the present investigation is based on the NL5(E) covariant energy density functional (CEDF), its general conclusions are expected to be valid also for other CEDFs built at the Hartree level.

Covariant density functional theory input for r -process simulations in actinides and superheavy nuclei: The ground state and fission properties

The systematic investigation of the ground state and fission properties of even-even actinides and superheavy nuclei with $Z=90-120$ from the two-proton up to two-neutron drip lines with proper assessment of systematic theoretical uncertainties has been performed for the first time in the framework of covariant density functional theory (CDFT). These results provide a necessary theoretical input for the r-process modeling in heavy nuclei and, in particular, for the study of fission recycling. Four state-of-the-art globally tested covariant energy density functionals (CEDFs), namely, DD-PC1, DD-ME2, NL3* and PC-PK1, representing the major classes of the CDFT models are employed in the present study. Ground state deformations, binding energies, two neutron separation energies, $\alpha$-decay $Q_{\alpha}$ values and half-lives and the heights of fission barriers have been calculated for all these nuclei. Theoretical uncertainties in these physical observables and their evolution as a function of proton and neutron numbers have been quantified and their major sources have been identified. Spherical shell closures at $Z=120$, $N=184$ and $N=258$ and the structure of the single-particle (especially, high-$j$) states in their vicinities as well as nuclear matter properties of employed CEDFs are two major factors contributing into theoretical uncertainties. However, different physical observables are affected in a different way by these two factors. For example, theoretical uncertainties in calculated ground state deformations are affected mostly by former factor, while theoretical uncertainties in fission barriers depend on both of these factors.

Landscape of pear-shaped even-even nuclei

The phenomenon of reflection-asymmetric nuclear shapes is relevant to nuclear stability, nuclear spectroscopy, nuclear decays and fission, and the search for new physics beyond the standard model. Global surveys of ground-state octupole deformation, performed with a limited number of models, suggest that the number of pear-shaped isotopes is fairly limited across the nuclear landscape. We carry out global analysis of ground-state octupole deformations for particle-bound even-even nuclei with $Z \leq 110$ and $N \leq 210$ using nuclear density functional theory (DFT) with several non-relativistic and covariant energy density functionals. In this way, we can identify the best candidates for reflection-asymmetric shapes. The calculations are performed in the frameworks of axial reflection-asymmetric Hartree-Fock-Bogoliubov theory and relativistic Hartree-Bogoliubov theory using DFT solvers employing harmonic oscillator basis expansion. We consider five Skyrme and four covariant energy density functionals. We predict several regions of ground-state octupole deformation. In addition to the "traditional" regions of neutron-deficient actinide nuclei around $^{224}$Ra and neutron-rich lanthanides around $^{146}$Ba, we identified vast regions of reflecion-asymmetric shapes in very neutron-rich nuclei around $^{200}$Gd and $^{288}$Pu, as well as in several nuclei around $^{112}$Ba. Our analysis suggests several promising candidates with stable ground-state octupole deformation, primarily in the neutron-deficient actinide region, that can be reached experimentally. Detailed comparison between Skyrme and covariant models is performed.

Propagation of statistical uncertainties in covariant density functional theory: Ground state observables and single-particle properties

Statistical errors in ground state observables and single-particle properties of spherical even-even nuclei and their propagation to the limits of nuclear landscape have been investigated in covariant density functional theory (CDFT) for the first time. In this study we consider only covariant energy density functionals with non-linear density dependency. Statistical errors for binding energies and neutron skins significantly increase on approaching two-neutron drip line. On the contrary, such a trend does not exist for statistical errors in charge radii and two-neutron separation energies. The absolute and relative energies of the single-particle states in the vicinity of the Fermi level are characterized by low statistical errors ($\sigma(e_i)\sim 0.1$ MeV). Statistical errors in the predictions of spin-orbit splittings are rather small. Statistical errors in physical observables are substantially smaller than related systematic uncertainties. Thus, at the present level of the development of theory, theoretical uncertainties at nuclear limits are dominated by systematic ones. Statistical errors in the description of physical observables related to the ground state and single-particle degrees of freedom are typically substantially lower in CDFT as compared with Skyrme density functional theory. The correlations between the model parameters are studied in detail. The parametric correlations are especially pronounced for the $g_2$ and $g_3$ parameters which are responsible for the density dependence of the model. The accounting of this fact potentially allows to reduce the number of free parameters of non-linear meson coupling model from six to five.

Octupole deformation in the ground states of even-even Z∼96, N∼196 actinides and superheavy nuclei

A systematic search for axial octupole deformation in the actinides and superheavy nuclei with proton numbers $Z=88-126$ and neutron numbers from two-proton drip line up to $N=210$ has been performed in covariant density functional theory (DFT) using four state-of-the-art covariant energy density functionals representing different model classes. The nuclei in the $Z\sim 96, N\sim 196$ region of octupole deformation have been investigated in detail and the systematic uncertainties in the description of their observables have been quantified. Similar region of octupole deformation exists also in Skyrme DFT and microscopic+macroscopic approach but it is centered at somewhat different particle numbers. Theoretical uncertainties in the predictions of the regions of octupole deformation are increasing on going to superheavy nuclei with $Z\sim 120, N\sim 190$. There are no octupole deformed nuclei for $Z=112-126$ in covariant DFT calculations. This agrees with Skyrme DFT calculations, but disagrees with Gogny DFT and microscopic+macroscopic calculations which predict extended $Z\sim 120, N\sim 190$ region of octupole deformation.

Assessing theoretical uncertainties in fission barriers of superheavy nuclei

Theoretical uncertainties in the predictions of inner fission barrier heights in superheavy elements have been investigated in a systematic way for a set of state-of-the-art covariant energy density functionals which represent major classes of the functionals used in covariant density functional theory. They differ in basic model assumptions and fitting protocols. Both systematic and statistical uncertainties have been quantified where the former turn out to be larger. Systematic uncertainties are substantial in superheavy elements and their behavior as a function of proton and neutron numbers contains a large random component. The benchmarking of the functionals to the experimental data on fission barriers in the actinides allows to reduce the systematic theoretical uncertainties for the inner fission barriers of unknown superheavy elements. However, even then they on average increase on moving away from the region where benchmarking has been performed. In addition, a comparison with the results of non-relativistic approaches is performed in order to define full systematic theoretical uncertainties over the state-of-the-art models. Even for the models benchmarked in the actinides, the difference in the inner fission barrier height of some superheavy elements reaches $5-6$ MeV. This uncertainty in the fission barrier heights will translate into huge (many tens of the orders of magnitude) uncertainties in the spontaneous fission half-lives.

Global analysis of quadrupole shape invariants based on covariant energy density functionals

Coexistence of different geometric shapes at low energies presents a universal structure phenomenon that occurs over the entire chart of nuclides. Studies of the shape coexistence are important for understanding the microscopic origin of collectivity and modifications of shell structure in exotic nuclei far from stability. The aim of this work is to provide a systematic analysis of characteristic signatures of coexisting nuclear shapes in different mass regions, using a global self-consistent theoretical method based on universal energy density functionals and the quadrupole collective model. The low-energy excitation spectrum and quadrupole shape invariants of the two lowest $0^{+}$ states of even-even nuclei are obtained as solutions of a five-dimensional collective Hamiltonian (5DCH) model, with parameters determined by constrained self-consistent mean-field calculations based on the relativistic energy density functional PC-PK1, and a finite-range pairing interaction. The theoretical excitation energies of the states: $2^+_1$, $4^+_1$, $0^+_2$, $2^+_2$, $2^+_3$, as well as the $B(E2; 0^+_1\to 2^+_1)$ values, are in very good agreement with the corresponding experimental values for 621 even-even nuclei. Quadrupole shape invariants have been implemented to investigate shape coexistence, and the distribution of possible shape-coexisting nuclei is consistent with results obtained in recent theoretical studies and available data. The present analysis has shown that, when based on a universal and consistent microscopic framework of nuclear density functionals, shape invariants provide distinct indicators and reliable predictions for the occurrence of low-energy coexisting shapes. This method is particularly useful for studies of shape coexistence in regions far from stability where few data are available.

Covariant energy density functionals: Nuclear matter constraints and global ground state properties

The correlations between global description of the ground state properties (binding energies, charge radii) and nuclear matter properties of the state-of-the-art covariant energy density functionals have been studied. It was concluded that the strict enforcement of the constraints on the nuclear matter properties (NMP) defined in Ref.\ \cite{RMF-nm} will not necessary lead to the functionals with good description of the binding energies and other ground and excited state properties. In addition, it will not substantially reduce the uncertainties in the predictions of the binding energies in neutron-rich systems. It turns out that the functionals, which come close to satisfying these NMP constraints, have some problems in the description of existing data. On the other hand, these problems are either absent or much smaller in the functionals which are carefully fitted to finite nuclei but which violate some NMP constraints. This is a consequence of the fact that the properties of finite nuclei are defined not only by nuclear matter properties but also by underlying shell effects. The mismatch of phenomenological content, existing in all modern functionals, related to nuclear matter physics and the physics of finite nuclei could also be responsible.

Octupole deformation in the ground states of even-even nuclei: A global analysis within the covariant density functional theory

A systematic investigation of octupole deformed nuclei is presented for even-even systems with $Z\leq 106$ located between the two-proton and two-neutron drip lines. For this study we use five most up-to-date covariant energy density functionals of different types, with a non-linear meson coupling, with density dependent meson couplings, and with density-dependent zero-range interactions. Pairing correlations are treated within relativistic Hartree-Bogoliubov (RHB) theory based on an effective separable particle-particle interaction of finite range. This allows us to assess theoretical uncertainties within the present covariant models for the prediction of physical observables relevant for octupole deformed nuclei. In addition, a detailed comparison with the predictions of non-relativistic models is performed. A new region of octupole deformation, centered around $Z\sim 98, N\sim 196$ is predicted for the first time. In terms of its size in the $(Z,N)$ plane and the impact of octupole deformation on binding energies this region is similar to the best known region of octupole deformed nuclei centered at $Z\sim 90, N\sim 136$. For the later island of octupole deformed nuclei, the calculations suggest substantial increase of its size as compared with available experimental data.

Covariant density functional theory: Reexamining the structure of superheavy nuclei

A systematic investigation of even-even superheavy elements in the region of proton numbers $100 \leq Z \leq 130$ and in the region of neutron numbers from the proton-drip line up to neutron number $N=196$ is presented. For this study we use five most up-to-date covariant energy density functionals of different types, with a non-linear meson coupling, with density dependent meson couplings, and with density-dependent zero-range interactions. Pairing correlations are treated within relativistic Hartree-Bogoliubov (RHB) theory based on an effective separable particle-particle interaction of finite range and deformation effects are taken into account. This allows us to assess the spread of theoretical predictions within the present covariant models for the binding energies, deformation parameters, shell structures and $\alpha$-decay half-lives. Contrary to the previous studies in covariant density functional theory, it was found that the impact of $N=172$ spherical shell gap on the structure of superheavy elements is very limited. Similar to non-relativistic functionals some covariant functionals predict the important role played by the spherical $N=184$ gap. For these functionals (NL3*, DD-ME2 and PC-PK1), there is a band of spherical nuclei along and near the $Z=120$ and $N=184$ lines. However, for other functionals (DD-PC1 and DD-ME$\delta$) oblate shapes dominate at and in the vicinity of these lines. Available experimental data are in general described with comparable accuracy and do not allow to discriminate these predictions.

Properties of nuclei in the nobelium region studied within the covariant, Skyrme, and Gogny energy density functionals

We calculate properties of the ground and excited states of nuclei in the nobelium region for proton and neutron numbers of 92≤Z≤104 and 144≤N≤156, respectively. We use three different energy-density-functional (EDF) approaches, based on covariant, Skyrme, and Gogny functionals, each with two different parameter sets. A comparative analysis of the results obtained for quasiparticle spectra, odd–even and two-particle mass staggering, and moments of inertia allows us to identify single-particle and shell effects that are characteristic to these different models and to illustrate possible systematic uncertainties related to using the EDF modelling.

Nuclear tensor interaction in a covariant energy density functional

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 force, is then achieved by the relativistic models. Moreover, careful analysis on the tensor strengths indicates the reliability of the nuclear tensor interaction in the covariant EDF for exploring the nuclear structure, excitation and decay modes.

Global study of beyond-mean-field correlation energies in covariant energy density functional theory using a collective Hamiltonian method

We report the first global study of dynamic correlation energies (DCEs) associated with rotational motion and quadrupole shape vibrational motion in a covariant energy density functional (CEDF) for 575 even-even nuclei with proton numbers ranging from $Z=8$ to $Z=108$ by solving a five-dimensional collective Hamiltonian, the collective parameters of which are determined from triaxial relativistic mean field plus BCS calculation using the PC-PK1 force. After taking into account these beyond mean-field DCEs, the root-mean-square (rms) deviation with respect to nuclear masses is reduced significantly down to 1.14 MeV, which is smaller than those of other successful CEDFs: NL3* (2.96 MeV), DD-ME2 (2.39 MeV), $\mathrm{DD}\text{\ensuremath{-}}\mathrm{ME}\ensuremath{\delta}$ (2.29 MeV), and DD-PC1 (2.01 MeV). Moreover, the rms deviation for two-nucleon separation energies is reduced by $\ensuremath{\sim}34%$ in comparison with the cranking prescription.