Relativistic Continuum Hartree-Bogoliubov Research Articles

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

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.

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.

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.

Nuclear Bubble Configuration in Heavy-Ion Collisions

We study the effects of a bubble configuration in a nucleus on heavy-ion collisions at a few tens and hundreds A MeV. We first investigate the bubble structure of 206Hg using the relativistic continuum Hartree–Bogoliubov theory and then study the 206Hg + 208Pb and 206Hg+206Hg reactions using the DaeJeon–Boltzmann–Uehling–Uhlenbeck (DJBUU) transport model. To see the role of the bubble structure, we consider the maximum density of the produced nuclear matter, directed flow of neutrons and protons, and π−/π+ ratio. We observe that the maximum density is smaller with a bubble nucleus, and the directed flow of nucleons and π−/π+ ratio may depend on the bubble structure.

Systematic study of α decay half-lives for even–even nuclei within a deformed two-potential approach

In this work, we systematically study the α decay half-lives of 196 even–even nuclei using a two-potential approach improved by considering nuclear deformation. The results show that the accuracy of this model has been improved after considering nuclear deformation. In addition, we extend this model to predict the α decay half-lives of Z = 118 and 120 isotopes by inputting the α decay energies extracted from the Weizsacker–Skyrme-type (WS-type) mass model, a simple nuclear mass formula, relativistic continuum Hartree–Bogoliubov theory and Duflo-Zuker-19 (DZ19) mass model. It is useful for identifying the new superheavy elements or isotopes for future experiments. Finally, the predicted α decay energies and half-lives of Z = 118 and 120 isotopes are analyzed, and the shell structure of superheavy nuclei is discussed. It shows that the shell effect is obvious at N = 184, while the shell effect at N = 178 depends on the nuclear mass model.

Bubble nuclei with shape coexistence in even-even isotopes of Hf to Hg

The shape of a nucleus is one of the fundamental nuclear properties. We perform a systematic investigation of bubble nuclei that also exhibit shape coexistence in Hf, W, Os, Pt, and Hg even-even isotopes using the deformed relativistic Hartree-Bogoliubov theory in continuum. For a systematic study, we first consider nuclear bubble structures and shape coexistence separately. We confirm that deformations and pairing correlations hinder bubble structures by comparing our results with those from relativistic continuum Hartree-Bogoliubov theory that assumes spherical symmetry in nuclei. We then predict candidate isotopes with both bubble structure and shape coexistence. We observe that the depletion fraction factor that characterizes bubble structure is mostly smaller in oblate deformation than in prolate, while some isotopes such as $^{206}\mathrm{Os}$ have bubble structures both in oblate and prolate deformations. We compare the proton single-particle energy levels for the candidates of shape coexistence both with only prolate bubble structure and with prolate and oblate bubble structures.

Nuclear mass table in deformed relativistic Hartree–Bogoliubov theory in continuum, I: Even–even nuclei

Ground-state properties of even–even nuclei with 8≤Z≤120 from the proton drip line to the neutron drip line have been investigated using the deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) with the density functional PC-PK1. With the effects of deformation and continuum included simultaneously, 2583 even–even nuclei are predicted to be bound. The calculated binding energies, two-nucleon separation energies, root-mean-square (rms) radii of neutron, proton, matter, and charge distributions, quadrupole deformations, and neutron and proton Fermi surfaces are tabulated and compared with available experimental data. The rms deviation from the 637 mass data is 1.518 MeV, providing one of the best microscopic descriptions for nuclear masses. The drip lines obtained from DRHBc calculations are compared with other calculations, including the spherical relativistic continuum Hartree–Bogoliubov (RCHB) and triaxial relativistic Hartree–Bogoliubov (TRHB) calculations with PC-PK1. The deformation and continuum effects on the limits of the nuclear landscape are discussed. Possible peninsulas consisting of bound nuclei beyond the two-neutron drip line are predicted. The systematics of the two-nucleon separation energies, two-nucleon gaps, rms radii, quadrupole deformations, potential energy curves, neutron densities, neutron mean-field potentials, and pairing energies in the DRHBc calculations are also discussed. In addition, the α decay energies extracted are in good agreement with available data.

Canonical states in relativistic continuum theory with the Green's function method: Neutrons in continuum of zirconium giant-halo nuclei

The canonical states in the relativistic continuum Hartree-Bogoliubov theory with Green's function method are obtained by diagonalizing the density matrix on a spatial mesh. Taking the giant halo nucleus $^{128}\mathrm{Zr}$ as an example, the obtained canonical single-particle energies and wave functions are compared in detail with the corresponding calculations with the box-discretized method. The occupation number ${v}_{i}^{2}$ in the canonical basis and the neutrons in continuum ${N}_{c}$ are investigated for the Zr giant halo nuclei. The dependencies on the pairing strength ${V}_{0}$ and on the different density functionals PK1 and NL-SH are also studied. It is found that the calculations with Green's function and box-discretized methods are almost consistent in describing nuclear global properties and the neutrons in continuum, meanwhile the ${N}_{c}$ of giant halo nuclei may heavily depend on the adopted pairing strength as well as the density functional.

Mass relations of mirror nuclei in terms of Coulomb energies based on relativistic continuum Hartree-Bogoliubov calculations

In this paper we study mass relations of mirror nuclei in terms of Coulomb energies calculated by the microscopic relativistic continuum Hartree-Bogoliubov theory. An additional term corresponding to the Nolen-Schiffer anomaly is assumed to take the same form as the well-known empirical formula. The resultant root-mean-square deviations of these mass relations are typically less than 100 keV. Based on these relations, 61 unknown proton-rich nuclear masses are predicted for $18\ensuremath{\leqslant}A\ensuremath{\leqslant}87$ with positive two-proton separation energies and are tabulated in the Supplemental Material of this paper.

Neutron drip line in the deformed relativistic Hartree–Bogoliubov theory in continuum: Oxygen to Calcium

The location of the neutron drip line, currently known for only the lightest elements, remains a fundamental question in nuclear physics. Its description is a challenge for microscopic nuclear energy density functionals, as it must take into account in a realistic way not only the nuclear potential, but also pairing correlations, deformation effects and coupling to the continuum. The recently developed deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) includes all three aforementioned effects in the description of nuclei throughout the nuclear chart. Here, the DRHBc with the successful density functional PC-PK1 is used to investigate whether and how deformation influences the prediction for the neutron drip-line location for even–even nuclei with [Formula: see text], where many isotopes are predicted deformed. The results are compared with those based on the spherical relativistic continuum Hartree–Bogoliubov (RCHB) theory and discussed in terms of shape evolution and the variational principle. It is found that the Ne and Ar drip-line nuclei are different after the deformation effect is included. The direction of the change is not necessarily towards an extended drip line, but rather depends on the evolution of the degree of deformation towards the drip line. Deformation effects as well as pairing and continuum effects treated in a consistent way can affect critically the theoretical description of the neutron drip-line location.

Towards a high-precision nuclear mass table with deformed relativistic Hartree-Bogoliubov theory in continuum

<p indent=0mm>As one of the most fundamental quantities for the atomic nucleus, nuclear mass is very important for various fields such as nuclear physics, particle physics, astrophysics, and cosmology. It is the key to understand the interaction between nucleons, the evolution of shell structures, and the location of drip lines. It also provides the basic information for nuclear reactions and the necessary inputs to study the origin of elements. Although dramatic progress for the mass measurement of short-lived nuclei has been made, the masses of most exotic nuclei far away from the stability valley still rely on the predictions by nuclear models in the foreseeable future. The popular nuclear mass models are briefly reviewed, including the finite-range droplet model (FRDM), Weizsäcker-Skyrme 4 (WS4) model, and HFB-27* model, etc. The efforts in applying the machine learning to improve the nuclear mass predictions are introduced. The relativistic density functional theory has been proved to be a powerful theory in nuclear physics by its successful description of many nuclear phenomena. The relativistic density functional theory has many attractive advantages, such as the automatic inclusion of nucleonic spin degree of freedom, explaining naturally the pseudospin symmetry in the nucleon spectrum and the spin symmetry in antinucleon spectrum, and the natural inclusion of nuclear magnetism, which plays an important role in nuclear magnetic moments and nuclear rotations. The relativistic density functional PC-PK1 has turned out to be very successful in providing good descriptions of the isospin dependence of the binding energy along either the isotopic or the isotonic chain. The relativistic continuum Hartree-Bogoliubov (RCHB) theory, which takes into account self-consistently the pairing correlations and continuum effects, is very suitable to describe the exotic nuclei. The first nuclear mass table including the continuum effects based on the RCHB theory with PC-PK1 has been constructed, predicting 9035 bound nuclei with 8≤<italic>Z</italic>≤120, which remarkably extends the existing nuclear landscape. The deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) was developed for exotic nuclei including deformation and continuum effects, with the deformed relativistic Hartree-Bogoliubov equations solved in a Dirac Woods-Saxon basis. The success of the DRHBc theory has been demonstrated in predicting an interesting shape decoupling between the core and the halo in ⁴⁴Mg and ⁴²Mg, resolving the puzzles concerning the radius and configuration of valence neutrons in ²²C, and studying particles in the classically forbidden regions for magnesium isotopes. A DRHBc Mass Table Collaboration, presently consisting of sixteen institutions from China and South Korea, which aims to construct the first relativistic nuclear mass table including simultaneously the deformation and continuum effects based on the DRHBc theory and density functional PC-PK1, has been founded. The recent progress in the construction of the DRHBc mass table for even-even nuclei is introduced. By taking even-even Nd isotopes as examples, the improvement for the description of the nuclear masses by deformation effects is shown in comparison with the RCHB calculations. The investigation of the possible exotic structure on the neutron-rich side and the possible proton radioactivity beyond the proton drip line for Nd isotopes is discussed. Possible topics to be investigated based on the DRHBc mass table are proposed.

Cross-section prediction for isotopes near neutron drip line in 70, 80Zn projectile fragmentation reactions * * Supported by the National Natural Science Foundation of China (U1732135) and Natural Science Foundation of Henan Province (162300410179)

The cross sections for 59, 60Ca, recently measured in the 345 A MeV reaction, were estimated using the FRACS parametrization and an empirical formula, which are in good agreement. The FRACS parametrization and the empirical formula are combined to predict the cross sections for extreme calcium isotopes in the reactions at the incident energies of 60, 80, and 345 A MeV. The dependence of emperical formula parameters on the reaction system, as well as the incident energy, are discussed. The results indicate that can be discovered in reactions of 60, 80 A MeV . The predicted binding energy for extreme neutron-rich isotopes by the spherical relativistic continuum Hartree-Bogoliubov theory was adopted in the calculation. Hence, the planned Beijing Isotope-Separation-On Line Neutron-Rich Beam Facility (BISOL), which is a third generation radioactive ion beam facility, could provide the opportunity to discover and neighboring neutron-drip line nuclei.

Mass predictions of the relativistic continuum Hartree-Bogoliubov model with radial basis function approach * * Supported by the National Natural Science Foundation of China(11805004, 11875070 and 11711540016), the Natural Science Foundation of Anhui Province (1708085QA10), the Key Research Foundation of Education Ministry of Anhui Province (KJ2016A026 and SK2018A0577), the Doctor Foundation of Anhui Jianzhu University 2017 (2017QD18), and the Open

The radial basis function (RBF) approach is a powerful tool to improve nuclear mass predictions. By combining the RBF approach with the latest relativistic continuum Hartree-Bogoliubov (RCHB) model, the local systematic deviations between the RCHB mass predictions and the experimental data are eliminated, and the root-mean-square (rms) mass deviation is significantly reduced from 7.923 MeV to 0.386 MeV. However, systematic deviations between the RBF improved mass predictions and the experimental data remain for nuclei with four different odd-even parities, i.e. (even Z, even N), (even Z, odd N), (odd Z, even N), and (odd Z, odd N). They can be reduced by separately training RBF for the four groups of nuclei, and the resulting rms deviation decreases to 0.229 MeV. It is found that the RBF approach can describe the deformation effects neglected in the present RCHB mass calculations, and also improves the description of the shell effect and the pairing effect.

Predictions for cross sections of light proton-rich isotopes in the 40Ca + 9Be reaction**Supported by National Natural Science Foundation of China (U1732135), the Key Research Program of Frontier Sciences of CAS (QYZDJSSW-SLH002) and Natural and Science Foundation in Henan Province (162300410179)

The cross sections for Z = 10 – 19 with isotopes Tz = −3/2 to −5 in the 140A MeV 40Ca + 9Be projectile fragmentation reaction have been predicted. An empirical formula based on the correlation between the cross section and average binding energy of an isotope has been adopted to predict the cross section. The binding energies in the AME16, WS4, and the theoretical prediction by the spherical relativistic continuum Hartree–Bogoliubov theory have been used. Meanwhile, the fracs parametrization and the modified statistical abrasion-ablation model are also used to predict the cross sections for the proton-rich isotopes. The predicted cross sections for the Tz = −3 isotopes are close to 10−10 mb, which hopefully can be studied experimentally. In addition, based on the predicted cross sections, Z = 14 is suggested to be a new magic number in the light proton-rich nuclei with Tz ⩽ −3/2, for which the phenomenon is much more evident than it is from the average binding energy per nucleon.

The limits of the nuclear landscape explored by the relativistic continuum Hartree–Bogoliubov theory

The ground-state properties of nuclei with 8⩽Z⩽120 from the proton drip line to the neutron drip line have been investigated using the spherical relativistic continuum Hartree–Bogoliubov (RCHB) theory with the relativistic density functional PC-PK1. With the effects of the continuum included, there are totally 9035 nuclei predicted to be bound, which largely extends the existing nuclear landscapes predicted with other methods. The calculated binding energies, separation energies, neutron and proton Fermi surfaces, root-mean-square (rms) radii of neutron, proton, matter, and charge distributions, ground-state spins and parities are tabulated. The extension of the nuclear landscape obtained with RCHB is discussed in detail, in particular for the neutron-rich side, in comparison with the relativistic mean field calculations without pairing correlations and also other predicted landscapes. It is found that the coupling between the bound states and the continuum due to the pairing correlations plays an essential role in extending the nuclear landscape. The systematics of the separation energies, radii, densities, potentials and pairing energies of the RCHB calculations are also discussed. In addition, the α-decay energies and proton emitters based on the RCHB calculations are investigated.

Evolution of N = 28 shell closure in relativistic continuum Hartree-Bogoliubov theory**Supported by Major State 973 Program of China (2013CB834400), National Natural Science Foundation of China (11175002, 11335002, 11375015, 11461141002) and Research Fund for Doctoral Program of Higher Education (20110001110087)

The N = 28 shell gap in sulfur, argon, calcium and titanium isotopes is investigated in the framework of relativistic continuum Hartree-Bogoliubov (RCHB) theory. The evolutions of neutron shell gap, separation energy, single particle energy and pairing energy are analyzed, and it is found that the N = 28 shell gap is quenched in sulfur isotopes but persists in argon, calcium and titanium isotopes. The evolution of the N = 28 shell gap in the N = 28 isotonic chain is discussed, and the erosion of the N = 28 shell gap is understood with the evolution of potential with proton number.

Global α-decay study based on the mass table of the relativistic continuum Hartree-Bogoliubov theory**Supported by Major State 973 Program of China (2013CB834400), National Natural Science Foundation of China (11175002, 11335002, 11375015, 11461141002), Research Fund for the Doctoral Program of Higher Education (20110001110087) and National Undergraduate Innovation Training Programs of Peking University.

The α-decay energies (Qα) are systematically investigated with the nuclear masses for 10 ⩽ Z ⩽ 120 isotopes obtained by the relativistic continuum Hartree-Bogoliubov (RCHB) theory with the covariant density functional PC-PK1, and compared with available experimental values. It is found that the α-decay energies deduced from the RCHB results present a similar pattern to those from available experiments. Owing to the large predicted Qα values (⩾ 4 MeV), many undiscovered heavy nuclei in the proton-rich side and super-heavy nuclei may have large possibilities for α-decay. The influence of nuclear shell structure on α-decay energies is also analysed.

Proton radioactivity in relativistic continuum Hartree-Bogoliubov theory

We investigate spherical proton emitters in the region from I $(Z=53)$ to Bi $(Z=83)$ isotopes as a crucial application of the relativistic energy density functional PC-PK1 beyond the drip line. The continuum effect is treated carefully by using the relativistic continuum Hartree-Bogoliubov theory. We calculate the half-lives of the proton emitters using the Wentzel-Kramers-Brillouin (WKB) method and observe that the calculated values are in reasonable agreement with experimental results for most of emitters considered in this study. We also discuss a clue to improve the functional PC-PK1 in future.

Nuclear mass table in relativistic continuum Hartree-Bogoliubov theory

We present nuclear mass table in the framework of relativistic continuum Hartree-Bogoliubov (RCHB) theory which properly treats pairing correlations in the presence of the continuum. We study the nuclear mass table mainly with spherical symmetry using the density functional PC-PK1. We then briefly discuss on-going calculations for deformed nuclei with axial symmetry in the framework of deformed RCHB theory.