Relativistic Hartree-Bogoliubov Theory Research Articles

Inner fission barriers of uranium isotopes in the deformed relativistic Hartree-Bogoliubov theory in continuum

Abstract The inner fission barriers of the even-even uranium isotopes from the proton to the neutron drip line are studied with the deformed relativistic Hartree-Bogoliubov theory in continuum. A periodic evolution for the ground state shapes is shown with the neutron number, i.e., spherical shapes at shell closures N=126, 184, 258, and prolate dominated shapes between them. In analogy to the shape evolution, the inner fission barriers also exhibit a periodic behavior: peaks at the shell closures and valleys in the mid-shells. The triaxial effect to the inner fission barrier is evaluated using the triaxial relativistic mean field calculations plus a simple BCS method for pairing. With the triaxial correction included, good consistency in the inner barrier heights is found with the available empirical data.
Besides, the evolution from the proton to the neutron drip line is in accord with the results by the multi-dimensionally constrained relativistic mean field theory. A flat valley in the fission barrier height is predicted around the neutron-rich nucleus 318U which may play a role of fission recycling in the astrophysical r-process nucleosynthesis.

Nuclear magnetism in the deformed halo nucleus 31Ne

Based on the point-coupling density functional, the time-odd deformed relativistic Hartree-Bogoliubov theory in continuum (TODRHBc) is developed. The effects of nuclear magnetism on halo phenomenon are explored by taking the experimentally suggested deformed halo nucleus 31Ne as an example. For 31Ne, nuclear magnetism contributes 0.09 MeV to total binding energy, and the breaking of Kramers degeneracy results in a 0 – 0.2 MeV splitting in the canonical single particle spectra. The blocked neutron level has a dominant component of the p wave and is marginally bound. However, if we ignore nuclear magnetism, the level becomes unbound. This shows a subtle mechanism that nuclear magnetism changes the single particle energies, causing a nucleus to become bound. Based on the TODRHBc results, a prolate one-neutron halo is formed around the near-spherical core in 31Ne. The nucleon current is mostly contributed by the halo rather than the core, except near the center of the nucleus. A layered feature in the neutron current distribution is observed and studied in detail.

Nuclear mass predictions of the relativistic continuum Hartree-Bogoliubov theory with the kernel ridge regression

Nuclear mass predictions of the relativistic continuum Hartree-Bogoliubov theory with the kernel ridge regression

A unified description of the halo nucleus 37Mg from microscopic structure to reaction observables

Based on the structure input directly from the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), we fully study reaction observables to search for halo evidence of 37Mg with the Glauber model. In this scheme, we evaluate the reaction cross sections of 20–37Mg bombarding a C target at 240 MeV/nucleon and find that most of the results agree well with the experimental data within 1σ uncertainty, especially good for the case of 37Mg. Our predicted doubling of the cross section change from 36Mg to 37Mg over that from 34Mg to 35Mg is consistent with a halo structure in 37Mg. Additionally, our calculated narrow longitudinal momentum distribution of 36Mg residues from 37Mg + 12C breakup agrees well with data and imply a nuclear size much larger than 35Mg. Furthermore, our study shows a dominant p-wave contribution to the longitudinal momentum distribution, supporting the conclusion that 37Mg is a p-wave halo nucleus. Our study provides the first unified description of the halo characteristics of 37Mg from nuclear structure to reaction dynamics.

Evolution of N = 20, 28, 50 shell closures in the 20 ≤ Z ≤ 30 region in deformed relativistic Hartree-Bogoliubov theory in continuum* *Supported in part by the National Natural Science Foundation of China (NSFC) (11975041, 11961141004). Xiang-Xiang Sun is supported in part by NSFC (12205308), and the Deutsche Forschungsgemeinschaft (DFG) and NSFC through the funds provided to the

Magicity, or shell closure, plays an important role in our understanding of complex nuclear phenomena. In this work, we employ one of the state-of-the-art density functional theories, the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) with the density functional PC-PK1, to investigate the evolution of the shell closures in the region. We show how these three conventional shell closures evolve from the proton drip line to the neutron drip line by studying the charge radii, two-neutron separation energies, two-neutron gaps, quadrupole deformations, and single-particle levels. In particular, we find that in the region, the shell closure disappears or becomes quenched, mainly due to the deformation effects. Similarly, both experimental data and theoretical predictions indicate that the shell closure disappears in the Mn isotopic chain, mainly due to the deformation effects. The DRHBc theory predicts the existence of the shell closure in the Ca, Sc, and Ti isotopic chains, but the existing data for the Ti isotopes suggest the contrary, and therefore further research is needed.

Impact of nuclear rotation corrections on alpha decay half-lives of superheavy nuclei within 98 ≤ Z ≤ 120

The influence of nuclear rotation on the decay half-lives of superheavy nuclei within the range 98≤Z≤120 is investigated using the axially deformed relativistic Hartree-Bogoliubov theory in the continuum (DRHBc) with the PC-PK1 parameter set. The deduced DRHBc decay energies (with and without the rotation effect) are compared with those calculated from the macroscopic-microscopic WS4 and the available experimental binding energies. Six semi-empirical formulae, such as the Viola-Seaborg formula (VSS), the modified Brown formula (mB1), the semi-empirical relationship based on fission theory (SemFIS2), the Royer formula (R), the Wang formula (Wang) and the modified YQZR formula (MYQZR) are employed to estimate the half-lives of α decay. Among these formulae, the half-live predictions of SemFIS2 are found to gradually deviate from the systematic trend beyond Nd=184, showing that the probability of undergoing spontaneous fission is less feasible beyond this point. A minimum is observed at Nd=184, reflecting its neutron shell closure for the mass region considered on the nuclear chart. The results further indicate that, for 98≤Z≤104, the predictions of all semi-empirical formulae are more closely matched to the available experimental data when the rotation effect is taken into account. However, we have demonstrated that the effect of nuclear rotation gradually reduces as the atomic nucleus becomes heavier.

Odd-even shape staggering and kink structure of charge radii of Hg isotopes by the deformed relativistic Hartree–Bogoliubov theory in continuum

We examined the shape staggering of relative charge radii in 180−186Hg isotopes, which was first measured in 1977 and recently confirmed using advanced spectroscopy techniques. To understand the nuclear structure underlying this phenomenon, we employed the deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc). Our analysis revealed that the shape staggering can be attributed to nuclear shape transition in the Hg isotopes. Specifically, we demonstrated that prolate shapes of 181,183,185Hg lead to an increase in the charge radii compared to oblate shapes of 180,182,184,186Hg isotopes. We explained the nuclear shape staggering in terms of the evolution of occupation probability (OP) of ν1i13/2, ν1h9/2, π1h9/2, and π3s1/2 states. Additionally, we clarified the kink structure of the charge radii in the Hg isotopes near N=126 magic shell does not come from the change of the OP of π1h9/2 state, but mainly by the increase of the OPs of ν1i11/2 and ν2g9/2 states.

Isospin-dependence of the charge-changing cross-section shaped by the charged-particle evaporation process

We present the charge-changing cross sections (CCCS) of 11−15C, 13−17N, and 15,17−18O at around 300 MeV/nucleon on a carbon target, which extends to p-shell isotopes with N<Z for the first time. The Glauber model, which considers only the proton distribution of projectile nuclei, underestimates the cross sections by more than 10%. We show that this discrepancy can be resolved by considering the contribution from the charged-particle evaporation process (CPEP) following projectile neutron removal. Using nucleon densities from the deformed relativistic Hartree-Bogoliubov theory in continuum, we investigate the isospin-dependent CPEP contribution to the CCCS for a wide range of neutron-to-proton separation energy asymmetry. Our calculations, which include the CPEP contribution, agree well with existing systematic data and reveal an “evaporation peak” at the isospin symmetric region where the neutron-to-proton separation energy is close to zero. These results suggest that analysis beyond the Glauber model is crucial for accurately determining nuclear charge radii from CCCSs.

One-proton emission from 148−151Lu in the DRHBc+WKB approach

One-proton radioactivity in 149Lu, the latest identified proton emitter, is studied in the Wentzel-Kramers-Brillouin (WKB) approach with the proton-nucleus potential extracted from the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) for the first time. The predicted half-life turns out to be consistent with the experimental measurement within uncertainties and (almost) independent of the density functionals in the DRHBc theory. Such a microscopic self-consistent calculation reveals that 149Lu is oblately deformed with a quadrupole deformation β2≈−0.18, and rules out the possibility of a prolate quadrupole deformation suggested in the nonadiabatic quasiparticle model. We also check the validity of this approach in the description of 150,151Lu and their isomeric states. The deviations of the predicted half-lives from their experimental counterparts are mostly smaller than those of the theoretical studies without considering deformation effects. Furthermore, we predict 148Lu to be a more oblately deformed proton-emitter with a longer half-life than that of 149Lu, which can be checked in the future. Our studies show that the DRHBc + WKB approach provides a new alternative method to evaluate the half-lives of well-deformed proton emitters.

Nuclear charge radii and shape evolution of Kr and Sr isotopes with the deformed relativistic Hartree-Bogoliubov theory in continuum

Nuclear charge radii and shape evolution of Kr and Sr isotopes with the deformed relativistic Hartree-Bogoliubov theory in continuum

Missed prediction of the neutron halo in 37Mg

Halo phenomena have long been an important frontier in both experimental and theoretical nuclear physics. 37Mg was identified as a halo nucleus in 2014 and remains the heaviest nuclear halo system to date. While the halo phenomenon in 37Mg was not predicted before the discovery, its description has been still challenging afterwards. In this Letter, we report a microscopic and self-consistent description of the neutron halo in 37Mg using the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) that was developed in 2010. The experimental neutron separation energies and empirical matter radii of neutron-rich magnesium isotopes as well as the deformed p-wave halo characteristics of 37Mg are well reproduced without any free parameters. In particular, the orbital occupied by the halo neutron in 37Mg, exhibiting p-wave components comparable to those suggested in experiments, remains consistent across various employed density functionals including PC-F1, PC-PK1, NL3*, and PK1. The DRHBc theory investigated only even-even magnesium isotopes in previous works and for that reason missed predicting 37Mg as a halo nucleus before 2014. Although the core and the halo of 37Mg are both prolate, higher-order shape decoupling on the hexadecapole and hexacontatetrapole levels is predicted.

Prolate-shape dominance in atomic nuclei within the deformed relativistic Hartree-Bogoliubov theory in continuum

Prolate-shape dominance in atomic nuclei within the deformed relativistic Hartree-Bogoliubov theory in continuum

Spherical, Axial, and Triaxial Symmetries in the Study of Halo Nuclei with Covariant Density Functional Theory

The halo phenomenon in exotic nuclei has long been an important frontier in nuclear physics research since its discovery in 1985. In parallel with the experimental progress in exploring halo nuclei, the covariant density functional theory has become one of the most successful tools for the microscopic study of halo nuclei. Based on spherical symmetry, the relativistic continuum Hartree–Bogoliubov theory describes the first halo nucleus 11Li self-consistently and predicts the giant halo phenomenon. Based on axial symmetry, the deformed relativistic Hartree–Bogoliubov theory in continuum has predicted axially deformed halo nuclei 42,44Mg and the shape decoupling effects therein. Based on triaxial symmetry, recently the triaxial relativistic Hartree–Bogoliubov theory in continuum has been developed and applied to explore halos in triaxially deformed nuclei. The theoretical frameworks of these models are presented, with the efficacy of exploiting symmetries highlighted. Selected applications to spherical, axially deformed, and triaxially deformed halo nuclei are introduced.

Collapse of the N=28 shell closure in the newly discovered Na39 nucleus and the development of deformed halos towards the neutron dripline

Halos and changes of nuclear magicities have been extensively investigated in exotic nuclei during past decades. The newly discovered $^{39}\mathrm{Na}$ with the neutron number $N=28$ provides a new platform to explore such novel phenomena near the neutron dripline of the sodium isotopic chain. We study the shell property and the possible halo structure in $^{39}\mathrm{Na}$ within the deformed relativistic Hartree-Bogoliubov theory in continuum. It is found that the lowering of $2p$ orbitals in the spherical limit results in the collapse of the $N=28$ shell closure in $^{39}\mathrm{Na}$, and a well-deformed ground state is established. The pairing correlations and the mixing of $pf$ components driven by deformation lead to the occupation of weakly bound or continuum $p$-wave neutron orbitals. An oblate halo is therefore formed around the prolate core in $^{39,41}\mathrm{Na}$, making $^{39}\mathrm{Na}$ a single nucleus with the coexistence of several exotic structures, including the quenched $N=28$ shell closure, Borromean structure, deformed halo, and shape decoupling. The microscopic mechanisms behind the shape decoupling phenomenon and the development of halos towards dripline are revealed.

Finite-nuclear-size effect in hydrogenlike ions with relativistic nuclear structure

The finite-nuclear-size (FNS) effect has a large contribution to atomic spectral properties, especially for heavy nuclei. By adopting the microscopic nuclear charge-density distributions obtained from the relativistic continuum Hartree-Bogoliubov (RCHB) theory, we systematically investigate the FNS corrections to atomic energy levels and bound-electron $g$ factors of hydrogenlike ions with nuclear charge up to 118. The comparison of the present numerical calculations with the predictions from empirical nuclear charge models, the nonrelativistic Skyrme-Hartree-Fock calculations, and the results based on experimental charge densities indicate that both the nuclear charge radius and the detailed shape of the charge-density distribution play important roles in determining the FNS corrections. The variation of FNS corrections to energy levels and $g$ factors with respect to the nuclear charge are investigated for the several lowest bound states of hydrogenlike ions. It is shown that they both increase by orders of magnitude with increasing nuclear charge, while the ratio between them has a relatively weak dependence on the nuclear charge. The FNS corrections to the ${s}_{1/2}$ and ${p}_{1/2}$ bound-state energies from the RCHB calculations are generally in good agreement with the analytical estimations by Shabaev [V. M. Shabaev, J. Phys. B 26, 1103 (1993)] based on the homogeneously charged sphere nuclear model, with the discrepancy indicating the distinct contribution of microscopic nuclear structure to the FNS effects.

Triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis

A triaxially deformed relativistic Hartree–Bogoliubov theory in the Woods–Saxon basis is developed with the aim of treating the triaxial deformation, pairing correlations and continuum in a unified way. In order to consider the triaxial deformation, the deformed potentials are expanded in terms of spherical harmonic functions in the coordinate space. In order to take the pairing correlations into account and treat the continuum properly, by using the Dirac Woods–Saxon basis, which has correct asymptotic behavior, the relativistic Hartree–Bogoliubov equation with triaxial deformation is solved. The formalism of triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis is presented. Taking an axially deformed nucleus 24Ne and a triaxially deformed nucleus 76Ge as examples, the numerical checks are performed. A weakly bound nucleus 112Ge is taken as an example to carry out the necessary converge checks for the numerical parameters. In addition, the ground-state properties of even–even germanium isotopes are investigated. The evolutions of two-neutron separation energy, deformation, root-mean-square radii and density distribution with mass number are analyzed. The comparison between the calculations from the relativistic Hartree–Bogoliubov theory based on harmonic-oscillator basis and the triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis is performed. It is found that the neutron drip line is extended from 114Ge to 118Ge in the triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis.

Examination of machine learning for assessing physical effects: Learning the relativistic continuum mass table with kernel ridge regression* *Supported by the National Natural Science Foundation of China (11875075, 11935003, 11975031, 12141501, 12070131001), the China Postdoctoral Science Foundation under (2021M700256), the State Key Laboratory of Nuclear Physics and Technology, Peking University (NPT2023ZX01, NPT2023KFY02), and the President's Undergraduate Research Fellowship (PURF) of

The kernel ridge regression (KRR) method and its extension with odd-even effects (KRRoe) are used to learn the nuclear mass table obtained by the relativistic continuum Hartree-Bogoliubov theory. With respect to the binding energies of 9035 nuclei, the KRR method achieves a root-mean-square deviation of 0.96 MeV, and the KRRoe method remarkably reduces the deviation to 0.17 MeV. By investigating the shell effects, one-nucleon and two-nucleon separation energies, odd-even mass differences, and empirical proton-neutron interactions extracted from the learned binding energies, the ability of the machine learning tool to grasp the known physics is discussed. It is found that the shell effects, evolutions of nucleon separation energies, and empirical proton-neutron interactions are well reproduced by both the KRR and KRRoe methods, although the odd-even mass differences can only be reproduced by the KRRoe method.

Microscopic study of shape variation across even–even mass chain of cadmium isotopes

The ground state shapes and properties of even–even [Formula: see text]Cd isotopes have been studied in the framework of relativistic Hartree–Bogoliubov model with two effective interactions that are density-dependent meson exchange and point coupling. The theoretical results on the binding energies, charge radii and two neutron separation energies are presented in this paper in comparison to experimental data and a reasonable agreement is found between the two. Both the interactions predict spherical to axially prolate, prolate to triaxial, triaxial to prolate and prolate to spherical transitions in cadmium isotopic mass chain. In addition, the present calculations confirm the existence of shape coexistence observed in [Formula: see text]Cd. The comparison of calculated ground state properties with the results based on PC-PK1 functional shows that PC-PK1 employed in the deformed relativistic Hartree–Bogoliubov theory in continuum has the lowest root mean square deviation for average binding energies and two neutron separation energies.

Gross theory of β decay by considering the spin-orbit splitting from relativistic Hartree-Bogoliubov theory

Nuclear $\ensuremath{\beta}$-decay half-lives are predicted with the so-called gross theory, which is improved by including the spin-orbit splitting from relativistic Hartree-Bogoliubov theory. The calculated differences between the Gamow-Teller and Fermi transition energies are in excellent agreement with experimental data for Zr, Sn, and Pb isotopes. The influences of the $Q$ value and the integrated Fermi function on the calculations of $\ensuremath{\beta}$-decay half-lives are carefully studied. Based on the mass predictions of the latest Weizs\"acker-Skyrme model, the half-lives from Ca to Pb isotopes are systematically calculated. It is found that the Weizs\"acker-Skyrme model well reproduces the experimental data with accuracy better than that of quasiparticle random-phase approximation approaches. When extrapolated to the unknown region, our results are generally close to those from the Skyrme finite-amplitude method. This improved gross theory can be employed to calculate half-lives based on mass predictions of various models and hence can provide relatively consistent half-life inputs for the $r$-process studies.

Microscopic analysis of the ground state properties of the even-even Dy isotopes in the reflection-asymmetric relativistic mean-field theory

Reflection-asymmetric relativistic mean-field theory is used to explore the ground state properties for the even-even Dy isotopes. The results are compared with those from the finite-range droplet model (FRDM), the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), and the available data. The calculated binding energies and quadrupole deformations agree with the experimental data as well as the FRDM and DRHBc calculations. The obtained matter density distributions and potential energy surfaces clearly exhibit the ground state shapes and the presence of deformation in these nuclei. The experimental fingerprints of octupolarity in $^{148,154,156,160}\mathrm{Dy}$ around $N=88$ are convincing in the present calculations. The predictions on the octupole deformation in $^{190\text{--}194,200,202}\mathrm{Dy}$ around $N=134$ agree with the results from the FRDM calculations. In addition, we obtain more knowledge about the shape evolution and phase transition from oblate to pear shape and then to prolate shapes in the Dy isotopes.