Relativistic Energy Density Functionals Research Articles

Microscopic description of α, 2α , and cluster decays of Rn216−220 and Ra220−224

$\ensuremath{\alpha}$ and cluster decays are analyzed for heavy nuclei located above $^{208}\mathrm{Pb}$ on the chart of nuclides: $^{216--220}\mathrm{Rn}$ and $^{220--224}\mathrm{Ra}$, which are also candidates for observing the $2\ensuremath{\alpha}$ decay mode. A microscopic theoretical approach based on relativistic energy density functionals (EDF), is used to compute axially symmetric deformation-energy surfaces as functions of quadrupole, octupole, and hexadecupole collective coordinates. Dynamical least-action paths for specific decay modes are calculated on the corresponding potential-energy surfaces. The effective collective inertia is determined using the perturbative cranking approximation, and zero-point and rotational energy corrections are included in the model. The predicted half-lives for $\ensuremath{\alpha}$ decay are within one order of magnitude of the experimental values. In the case of single-$\ensuremath{\alpha}$ emission, the nuclei considered in the present study exhibit least-action paths that differ significantly up to the scission point. The differences in $\ensuremath{\alpha}$-decay lifetimes are not only driven by $Q$ values, but also by variances of the least-action paths prior to scission. In contrast, the $2\ensuremath{\alpha}$ decay mode presents very similar paths from equilibrium to scission, and the differences in lifetimes are mainly driven by the corresponding $Q$ values. The predicted $^{14}\mathrm{C}$ cluster decay half-lives are within three orders of magnitudes of the empirical values, and point to a much more complex pattern compared with the $\ensuremath{\alpha}$-decay mode.

Time-dependent Dirac equation applied to one-proton radioactive emission

Relativistic energy-density functional (REDF) theory has been developed and utilized for self-consistent mean-field calculations of atomic nuclei. The proton-emitting radioactivity can provide a suitable reference to improve the predicting ability of REDF especially on the proton-drip line. One needs to consider the quantum tunneling effect, which plays an essential role in nucleon-emitting radioactive processes. However, the relativistic quantum tunneling has been less investigated compared with the nonrelativistic case. This work is devoted to a theoretical evaluation of one-proton ($1p$) radioactivity based on the relativistic Dirac formalism. For this purpose, I develop the time-dependent (TD) Dirac-spinor calculation to simulate the $1p$ emission. By utilizing the relativistic Hartree-Bogoliubov (RHB) calculation with the DD-PCX parameters, single-proton potentials for the time-dependent Dirac spinor are determined. The TD-Dirac calculation is applied to the $1p$ emissions from the $^{37}\mathrm{Sc}$ and $^{39}\mathrm{Sc}$ nuclei, which can be well approximated as the valence proton and the proton-close-shell cores. The sensitivity of $1p$-emission energy and decaying width to the mass number is demonstrated. Remarkable sensitivity exists due to the size of the system, which affects the nuclear part of potentials and energy levels, whereas the Coulomb barrier is common with the same atomic number. The calculated $1p$ energy and decaying lifetime are roughly consistent to the experimental limitation. The present TD-Dirac calculation is expected as applicable widely to proton-rich nuclides in order to improve the REDF by utilizing the $1p$-emission data.

Isotopic Shift in Hg-Isotopes within Brückner versus Relativistic Energy Density Functional

The present study is focused on revealing a characteristic kink of the neutron shell closure N = 126 across the Hg-isotopic chain within the relativistic mean-field (RMF) approach with the IOPB-I, DD-ME2, DD-PC1 and NL3 parameter sets. The RMF densities are converted to their spherical equivalence via the Wood–Saxon approximation and used as input within the parametrization procedure of the coherent density fluctuation model (CDFM). The nuclear matter symmetry energy is calculated using the Brückner energy density functional, and its surface, as well as volume components, are evaluated within Danielwicz’s liquid drop prescription. In addition, a comparison between Brückner and relativistic energy density functionals using the NL3 parameter set is shown as a representative case. The binding energy, charge distribution radius and symmetry energy are used as indicators of the isotopic shift in both ground and isomeric states. We have found the presence of a kink at the shell/sub-shell closure at N = 126 for neutron-rich 206Hg. The formation of the kink is traceable to the early filling of the 1i11/2 orbitals rather than 2g9/2, due to the large spin-orbit splitting. As such, the link between the occupational probability and the magicity of nuclei over the Hg-isotopic chain is established.

Signatures of shape phase transitions in krypton isotopes based on relativistic energy density functionals

Spectroscopic properties that characterize the shape phase transitions in krypton isotopes with the mass $A\approx80$ region are investigated within the framework of the nuclear density functional theory. Triaxial quadrupole constrained self-consistent mean-field calculations that employ relativistic energy density functionals and a pairing interaction are carried out for the even-even nuclei $^{76-86}$Kr. The spectroscopic properties are computed by solving the triaxial quadrupole collective Hamiltonian, with the ingredients, i.e., the deformation-dependent moments of inertia and mass parameters, and the collective potential, determined by using the SCMF solutions as microscopic inputs. Systematic behaviors of the SCMF potential energy surfaces, the corresponding low-energy spectra, electric quadrupole and monopole transition probabilities, and the fluctuations in the triaxial quadrupole deformations indicate evolution of the underlying nuclear structure as functions of the neutron number, that is characterized by a considerable degree of shape mixing. A special attention is paid to the transitional nucleus $^{82}$Kr, which has been recently identified experimentally as an empirical realization of the E(5) critical-point symmetry.

Symmetry breaking of Gamow-Teller and magnetic-dipole transitions and its restoration in calcium isotopes

Nuclear magnetic-dipole (M1) and Gamow-Teller (GT) transitions provide insight into the spin-isospin properties of atomic nuclei. By considering them as unified spin-isospin transitions, the M1/GT transition strengths and excitation energies are subject to isospin symmetry. The excitation properties associated to the M1/GT symmetry need to be clarified within consistent theoretical approach. In this work, the relationship between the M1 and GT transitions in Ca isotopes is investigated in a unified framework based on the relativistic energy-density functional (REDF) with point-coupling interactions, using the relativistic quasi-particle random-phase approximation (RQRPA). It is shown that the isovector-pseudovector (IV-PV) residual interaction affects both transitions, and the symmetry of M1 and giant-GT transitions is disrupted by this interaction in closed-shell nuclei. In open-shell Ca isotopes, the proton-neutron pairing in the residual RQRPA interaction also plays a role in GT transitions. Due to the interplay between these interactions, the M1/GT symmetry can be restored especially in the $^{42}$Ca nucleus, i.e., the giant-GT strength can become comparable to that of the M1 mode in terms of the unified spin-isospin transitions by adjusting the PN-pairing strength to reproduce the experimental low-lying GT-excitation energies. The mirror symmetry of both M1 and GT transitions is also demonstrated for open-shell mirror partners, $^{42}$Ca and $^{42}$Ti. Further improvements are required to achieve simultaneous reproduction of M1 and GT-transition energies in the REDF framework.

Linking Partial Dynamical Symmetry to Nuclear Energy Density Functional

We use self-consistent mean-field methods in combination with the interacting boson model (IBM) of nuclei, to establish a linkage between universal energy density functionals (EDFs) and partial dynamical symmetry (PDS). An application to $^{168}$Er shows that IBM Hamiltonians derived microscopically from known nonrelativistic and relativistic EDFs in this region, conform with SU(3)-PDS.

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.

Nuclear energy density functionals from empirical ground-state densities

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=Z systems.

Proton density polarization of the doubly magic 40Ca core in 48Ca and EoS parameters

Abstract The proton density polarization in ${\rm ^{48}Ca}$ by the eight neutrons added on the ${\rm ^{40}Ca}$ core is discussed. We adopt the proton density distributions determined by electron scattering and the neutron density distributions determined by proton elastic scattering in ${\rm ^{40}Ca}$ and ${\rm ^{48}Ca}$. As a measure of the proton density polarization effect due to the neutrons, a ratio $R$ of the peak values of the neutron density difference $\Delta\rho_{n,{\rm peak}}$ between ${\rm ^{48}Ca}$ and ${\rm ^{40}Ca}$ to the proton one $\Delta\rho_{p,{\rm peak}}$ is introduced and compared with the Hartree–Fock and Hartree calculations by non-relativistic and relativistic energy density functionals (EDFs), respectively. The value $R$ shows clear correlations with the symmetry energy parameters $J$ and $L$, but not with the isoscalar incompressibility $K_\infty$. The experimental value $R$ can be reproduced by theoretical calculations using the EDFs with $L=25$–50 MeV.

Coupling of shape and pairing vibrations in a collective Hamiltonian based on nuclear energy density functionals

The quadrupole collective Hamiltonian, based on relativistic energy density functionals, is extended to include a pairing collective coordinate. In addition to quadrupole shape vibrations and rotations, the model describes pairing vibrations and the coupling between shape and pairing degrees of freedom. The parameters of the collective Hamiltonian are determined by constrained self-consistent relativistic mean-field plus Bardeen-Cooper-Schrieffer (RMF+BCS) calculations in the space of intrinsic shape and pairing deformations. The effect of coupling between shape and pairing degrees of freedom is analyzed in a study of low-energy spectra and transition rates of four axially symmetric $N=92$ rare-earth isotones. When compared to results obtained with the standard quadrupole collective Hamiltonian, the inclusion of dynamical pairing increases the moment of inertia, lowers the energies of excited $0^+$ states and reduces the E0-transition strengths, in better agreement with data.

Parametrisations of relativistic energy density functionals with tensor couplings

The relativistic density functional with minimal density dependent nucleon–meson couplings for nuclei and nuclear matter is extended to include tensor couplings of the nucleons to the vector mesons. The dependence of the minimal couplings on either vector or scalar densities is explored. New parametrisations are obtained by a fit to nuclear observables with uncertainties that are determined self-consistently. The corresponding nuclear matter parameters at saturation are determined including their uncertainties. An improvement in the description of nuclear observables, in particular for binding energies and diffraction radii, is found when tensor couplings are considered, accompanied by an increase of the Dirac effective mass. The equations of state for symmetric nuclear matter and pure neutron matter are studied for all models. The density dependence of the nuclear symmetry energy, the Dirac effective masses and scalar densities is explored. Problems at high densities for parametrisations using a scalar density dependence of the couplings are identified due to the rearrangement contributions in the scalar self-energies that lead to vanishing Dirac effective masses.

Empirical constraints on the high-density equation of state from multimessenger observables

We search for possible correlations between neutron star observables and thermodynamic quantities that characterize high density nuclear matter. We generate a set of model-independent equations of state describing stellar matter from a Taylor expansion around saturation density. Each equation of state which is a functional of the nuclear matter parameters is thermodynamically consistent, causal and compatible with astrophysical observations. We find that the neutron star tidal deformability and radius are strongly correlated with the pressure, the energy density and the sound velocity at different densities. Similar correlations are also exhibited by a large set of mean-field models based on non-relativistic and relativistic nuclear energy density functionals. These model independent correlations can be employed to constrain the equation of state at different densities above saturation from measurements of NS properties with multi-messenger observations. In particular, precise constraints on the radius of PSR J0030+0451 thanks to NICER observations would allow to better infer the properties of matter around two times the nuclear saturation density.

Nuclear energy density functionals constrained by low-energy QCD

Relativistic nuclear energy density functionals are formulated and developed, guided by two important features that establish connections with chiral dynamics and the sym- metry breaking pattern of low-energy QCD: a) strong scalar and vector fields related to in-medium changes of QCD vacuum condensates; b) the long- and intermediate-range interactions generated by one-and two-pion exchange, derived from in-medium chiral per- turbation theory.

Constraints on the inner edge of neutron star crusts from relativistic nuclear energy density functionals

The transition density nt and pressure Pt at the inner edge between the liquid core and the solid crust of a neutron star are analyzed using the thermodynami- cal method and the framework of relativistic nuclear energy density functionals. Starting from a functional that has been carefully adjusted to experimental binding energies of finite nuclei, and varying the density dependence of the cor- responding symmetry energy within the limits determined by isovector prop- erties of finite nuclei, we estimate the constraints on the core-crust transition density and pressure of neutron stars: 0.086 fm−3 ≤ nt < 0.090 fm−3 and 0.3 MeV fm−3 < Pt ≤ 0.76 MeV fm−3 [1].

Cluster structures in C12 from global energy density functionals

Spectroscopic properties of low-lying states and cluster structures in $^{12}$C are analyzed in a "beyond mean-field framework" based on global energy density functionals (EDFs). To build symmetry-conserving collective states, axially-symmetric and reflection-asymmetric solutions of the relativistic Hartree-Bogoliubov equations are first projected onto good values of angular momentum, particle number, and parity. Configuration mixing is implemented using the generator coordinate method formalism. It is shown that such a global microscopic approach, based on a relativistic EDF, can account for the main spectroscopic features of $^{12}$C, including the ground-state and linear-chain bands as well as, to a certain approximation, the excitation energy of the Hoyle state. The calculated form factors reproduce reasonably well the available experimental values, and display an accuracy comparable to that of dedicated microscopic cluster models.

Beyond mean-field approach for pear-shaped hypernuclei

We develop both relativistic mean field and beyond approaches for hypernuclei with possible quadrupole-octupole deformation or pear-like shapes based on relativistic point-coupling energy density functionals. The symmetries broken in the mean-field states are recovered with parity, particle-number, and angular momentum projections. We take $^{21}_\Lambda$Ne as an example to illustrate the method, where the $\Lambda$ hyperon is put on one of the two lowest-energy orbits (labeled as $\Lambda_s, \Lambda_p$), respectively. We find that the $\Lambda$ hyperon in both cases disfavors the formation of a reflection-asymmetric molecular-like $^{16}$O$+\alpha$ structure in $^{20}$Ne, which is consistent with the Nilsson diagram for the hyperon in $(\beta_2, \beta_3)$ deformation plane. In particular, we show that the negative-parity states with the configuration $^{20}$Ne($K^\pi=0^-)\otimes \Lambda_s$ are close in energy to those with the configuration $^{20}$Ne($K^\pi=0^+)\otimes \Lambda_p$, even though they have very different structures. The $\Lambda_s$ ($\Lambda_p$) becomes more and more concentrated around the bottom (top) of the "pear" with the increase of octupole deformation.

Lagrange-Mesh Method for Deformed Nuclei With Relativistic Energy Density Functionals

The application of relativistic energy density functionals to the description of nuclei leads to the problem of solving self-consistently a coupled set of equations of motion to determine the nucleon wave functions and meson fields. In this work the Lagrange-mesh method in spherical coordinates is proposed for the numerical calculation. The essential field equations are derived from the relativistic energy density functional and the basic principles of the Lagrange-mesh method are delineated for this particular application. The numerical accuracy is studied for the case of a deformed relativistic harmonic oscillator potential with axial symmetry. Then the method is applied to determine the point matter distributions and deformation parameters of self-conjugate even-even nuclei from 4He to 40Ca.

Effects of tensor forces in nuclear spin–orbit splittings from ab initio calculations

A systematic and specific pattern due to the effects of the tensor forces is found in the evolution of spin–orbit splittings in neutron drops. This result is obtained from relativistic Brueckner–Hartree–Fock theory using the bare nucleon–nucleon interaction. It forms an important guide for future microscopic derivations of relativistic and nonrelativistic nuclear energy density functionals.

Difference in proton radii of mirror nuclei as a possible surrogate for the neutron skin

It has been recently suggested that differences in the charge radii of mirror nuclei are proportional to the neutron-skin thickness of neutron-rich nuclei and to the slope of the symmetry energy $L$ [B.A. Brown Phys. Rev. Lett. 119, 122502 (2017)]. The determination of the neutron skin has important implications for nuclear physics and astrophysics. Although the use of electroweak probes provides a largely model-independent determination of the neutron skin, the experimental challenges are enormous. Thus, the possibility that differences in the charge radii of mirror nuclei may be used as a surrogate for the neutron skin is a welcome alternative. To test the validity of this assumption we perform calculations based on a set of relativistic energy density functionals that span a wide region of values of $L$. Our results confirm that the difference in charge radii of various neutron-deficient nickel isotopes and their corresponding mirror nuclei is indeed strongly correlated to both the neutron-skin thickness and $L$. Moreover, given that various neutron-star properties are also sensitive to $L$, a data-to-data relation emerges between the difference in charge radii of mirror nuclei and the radius of low-mass neutron stars.

Spectroscopy of reflection-asymmetric nuclei with relativistic energy density functionals

Quadrupole and octupole deformation energy surfaces, low-energy excitation spectra and transition rates in fourteen isotopic chains: Xe, Ba, Ce, Nd, Sm, Gd, Rn, Ra, Th, U, Pu, Cm, Cf, and Fm, are systematically analyzed using a theoretical framework based on a quadrupole-octupole collective Hamiltonian (QOCH), with parameters determined by constrained reflection-asymmetric and axially-symmetric relativistic mean-field calculations. The microscopic QOCH model based on the PC-PK1 energy density functional and $\delta$-interaction pairing is shown to accurately describe the empirical trend of low-energy quadrupole and octupole collective states, and predicted spectroscopic properties are consistent with recent microscopic calculations based on both relativistic and non-relativistic energy density functionals. Low-energy negative-parity bands, average octupole deformations, and transition rates show evidence for octupole collectivity in both mass regions, for which a microscopic mechanism is discussed in terms of evolution of single-nucleon orbitals with deformation.