Hartree Bogoliubov Research Articles

Nuclear Matter and Finite Nuclei: Recent Studies Based on Parity Doublet Model

In this review, we summarize recent studies on nuclear matter and finite nuclei based on parity doublet models. We first construct a parity doublet model (PDM), which includes the chiral invariant mass m0 of nucleons together with the mass generated by the spontaneous chiral symmetry breaking. We then study the density dependence of the symmetry energy in the PDM, which shows that the symmetry energy is larger for smaller chiral inavariant mass. Then, we investigate some finite nuclei by applying the Relativistic Continuum Hartree–Bogoliubov (RCHB) theory to the PDM. We present the root-mean-square deviation (RMSD) of the binding energies and charge radii, and show that m0=700 MeV is preferred by the nuclear properties. Finally, we modify the PDM by adding the isovector scalar meson a0(980), and show that the inclusion of the a0(980) enlarges the symmetry energy of the infinite nuclear matter.

Study of Shape Evolution and Ground State Properties of Even-Even Tellurium Isotopic Mass Chain by using Relativistic Hartree Bogoliubov framework

Study of Shape Evolution and Ground State Properties of Even-Even Tellurium Isotopic Mass Chain by using Relativistic Hartree Bogoliubov framework

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.

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.

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.

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.

Octupole shape phase transitions and critical points in neutron rich actinides

The evolution of octupole shapes and shape phase transitions in neutron rich actinides is studied within the covariant density functional framework. Octupole constrained energy surfaces, and spectroscopic observables of four isotopic chains of: Cm, Cf, Fm and No with neutron numbers 186le N le 200 are analysed using a collective quadrupole–octupole Hamiltonian (QOCH). The parameters of the Hamiltonian are determined by axially reflection-asymmetric relativistic Hartree–Bogoliubov calculations based on the energy density functional DD-PC1, and a finite-range pairing interaction. The results suggest quantum phase transitions from non-octupole to octupole deformed shapes and to octupole vibrations with increasing neutron number. ^{288}Cm is possibly close to the critical point of a simultaneous phase transition from spherical to prolate deformed and from non-octupole to stable octupole deformed configurations.

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.

Structure and decay chain of fermium isotopes using relativistic mean-field approach

In this paper, we have analyzed the ground state structural properties of fermium-isotopes in the mass range [Formula: see text] within the relativistic mean-field with NL3[Formula: see text] and Relativistic Hartree–Bogoliubov approach with DD-ME2 parameter sets. The bulk properties, such as binding energy, root mean-square charge radius, quadruple deformation parameter, chemical potentials, two-neutron separation energy and differential two- neutron separation energy, are estimated for the above Fm- isotopic chain. All the calculated observables are compared with finite range droplet model (FRDM) predictions and experimental data wherever available. The analysis predicts the existence of shell closure in the neutron-rich region of the isotopic chain at [Formula: see text]. The decay properties such as the [Formula: see text]-decay energy and corresponding half-life for three decay chains i.e. [Formula: see text]Fm, [Formula: see text]Fm and [Formula: see text]Fm are studied. Six different empirical formulae, namely; Viola-Seaborg, Royer, modified B. Alex Brown, Parkhomenko–Sobiczewski, Universal decay law and modified universal decay law are employed to ascertain the numerical dependency of the half-life for each decay energy. A comparative study of [Formula: see text]-values and half-lives for these decay chains gives a good signature of the calculated results. The half-life calculations of three decay chains suggest that [Formula: see text]Pu, [Formula: see text]Pu, [Formula: see text]Pu are shell closure nuclei having maximum half-lives, whereas [Formula: see text]Fm, [Formula: see text]Fm and [Formula: see text]Fm are shell stabilized with lower half-lives.

Semi-bubble structure of superheavy nuclei

The nuclear structure properties of heavy and superheavy nuclei lying at the extremes of charge, especially [Formula: see text], and their density profiles have been explored using the relativistic Hartree–Bogoliubov in axial symmetry and the Skyrme–Hartree–Fock–Bogoliubov framework in transformed harmonic oscillator basis. In this study, we found that the semi-bubble (reduced central density) structure is mainly driven by Coulomb repulsion which stabilizes the nuclei either by pushing the nuclear matter in the surface region or by shape deformation. The shell effect is found maximum in spherical symmetric nuclei as the vanishing fission barrier in Liquid Drop formula indicates that the shell correction energy is responsible for the stabilization of these nuclei.

Microscopic description of even–even medium-mass tellurium isotopes within the projected shell model

The low-lying energy levels, B(E2) transition probabilities and g-factors of even–even medium-mass tellurium isotopes have been studied by employing the projected shell model (PSM) approach. The oblate ground state deformations are taken in this study that were predicted in previous relativistic Hartree Bogoliubov mean field study. The low-lying yrast and yrare states are reproduced well in these calculations. The energy and configuration of yre states are also predicted. The experimental B(E2) values and their systematics with neutron number are reproduced well in the medium-mass Te isotopes. The experimental data are available for [Formula: see text]-factors of 2[Formula: see text] states of [Formula: see text]Te isotopes which is in line with these calculations. The enhancement of g-factors at spin I = 6[Formula: see text] in all the Te isotopes considered may be attributable to the alignment of protons in [Formula: see text] subshell. Furthermore, the reduction of g-factors at spin [Formula: see text] in [Formula: see text]Te and [Formula: see text] in [Formula: see text]Te may be attributable to the alignment of neutrons in the [Formula: see text] subshell. The 8[Formula: see text] states of [Formula: see text]Te are envisaged to be of [Formula: see text] nature and in [Formula: see text]Te these states are found to be of [Formula: see text] nature.

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.

Relativistic and nonrelativistic mean field analysis of the shell evolution in 44−52Ar isotopes

The systematic search of axial ground state and the emergence of new gaps in neutron-rich [Formula: see text]Ar isotopes are the challenging of current calculations of relativistic and nonrelativistic mean field models. For this purpose, we have employed the model based on Relativistic Hartree–Bogoliubov (RHB) with the density-dependent effective interaction of point coupling DD-PCX type. However, the nonrelativistic mean field calculations based on Hartree-Fock model were carried out with Skyrme SV-min parametrization. All the present calculations were performed within axial symmetries. The results provide that most of isotopes show an oblate-deformed ground state. In particular, [Formula: see text]Ar and [Formula: see text]Ar display a spherical shape in their ground states with SV-Min calculations. There is a notable signature of the sub-shell closure at neutron number [Formula: see text], whereas [Formula: see text] is expected to be nearly quenched in the spherical ground. Furthermore, coexistence features are predicted in the neutron-rich [Formula: see text]Ar isotope. We have also studied the neutron axial densities and its central depletion in the well-deformed grounds for the first time. In addition to these, the proton single-particle evolution and bubble-like structure at [Formula: see text]Ar are very well reproduced with DD-PCX calculations.

Impact of the New 65As(p,γ)66Se Reaction Rate on the Two-proton Sequential Capture of 64Ge, Weak GeAs Cycles, and Type I X-Ray Bursts Such as the Clocked Burster GS 1826−24

We reassess the 65As(p,γ)66Se reaction rates based on a set of proton thresholds of 66Se, S p(66Se), estimated from the experimental mirror nuclear masses, theoretical mirror displacement energies, and full p f-model space shell-model calculation. The self-consistent relativistic Hartree–Bogoliubov theory is employed to obtain the mirror displacement energies with much reduced uncertainty, and thus reducing the proton-threshold uncertainty up to 161 keV compared to the AME2020 evaluation. Using the simulation instantiated by the one-dimensional multi-zone hydrodynamic code, Kepler, which closely reproduces the observed GS 1826−24 clocked bursts, the present forward and reverse 65As(p,γ)66Se reaction rates based on a selected S p(66Se) = 2.469 ± 0.054 MeV, and the latest 22Mg(α,p)25Al, 56Ni(p,γ)57Cu, 57Cu(p,γ)58Zn, 55Ni(p,γ)56Cu, and 64Ge(p,γ)65As reaction rates, we find that though the GeAs cycles are weakly established in the rapid-proton capture process path, the 65As(p,γ)66Se reaction still strongly characterizes the burst tail end due to the two-proton sequential capture on 64Ge, not found by the Cyburt et al. sensitivity study. The 65As(p,γ)66Se reaction influences the abundances of nuclei A = 64, 68, 72, 76, and 80 up to a factor of 1.4. The new S p(66Se) and the inclusion of the updated 22Mg(α,p)25Al reaction rate increases the production of 12C up to a factor of 4.5, which is not observable and could be the main fuel for a superburst. The enhancement of the 12C mass fraction alleviates the discrepancy in explaining the origin of the superburst. The waiting point status of and two-proton sequential capture on 64Ge, the weak-cycle feature of GeAs at a region heavier than 64Ge, and the impact of other possible S p(66Se) are also discussed.

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.

The angular momentum and parity projected multidimensionally constrained relativistic Hartree–Bogoliubov model

Nuclear deformations are fundamentally important in nuclear physics. We recently developed a multidimensionally constrained relativistic Hartree–Bogoliubov (MDCRHB) model, in which all multipole deformations respecting the V 4 symmetry can be considered self-consistently. In this work we extend this model by incorporating the angular momentum projection and parity projection to restore the rotational and parity symmetries broken in the mean-field level. This projected MDCRHB (p-MDCRHB) model enables us to connect certain nuclear spectra to exotic intrinsic shapes such as triangles or tetrahedrons. We present the details of the method and an exemplary calculation for 12C. We develop a triangular moment constraint to generate the triangular configurations consisting of three α clusters arranged as an equilateral triangle. The resulting 12C spectra are consistent with that from a triangular rigid rotor for large separations between the α clusters. We also calculate the B(E2) and B(E3) values for low-lying states and find good agreement with the experiments.

A combined Glauber model plus relativistic Hartree–Bogoliubov theory analysis of nuclear reactions with light and medium mass nuclei

A combined Glauber model plus relativistic Hartree–Bogoliubov theory analysis of nuclear reactions with light and medium mass nuclei

Finite-temperature linear response theory based on relativistic Hartree Bogoliubov model with point-coupling interaction

The finite-temperature linear response theory based on the finite-temperature relativistic Hartree-Bogoliubov (FT-RHB) model is developed in the charge-exchange channel to study the temperature evolution of spin-isospin excitations. Calculations are performed self-consistently with relativistic point-coupling interactions DD-PC1 and DD-PCX. In the charge-exchange channel, the pairing interaction can be split into isovector ($T = 1$) and isoscalar ($T = 0$) parts. For the isovector component, the same separable form of the Gogny D1S pairing interaction is used both for the ground-state calculation as well as for the residual interaction, while the strength of the isoscalar pairing in the residual interaction is determined by comparison with experimental data on Gamow-Teller resonance (GTR) and Isobaric analog resonance (IAR) centroid energy differences in even-even tin isotopes. The temperature effects are introduced by treating Bogoliubov quasiparticles within a grand-canonical ensemble. Thus, unlike the conventional formulation of the quasiparticle random-phase approximation (QRPA) based on the Bardeen-Cooper-Schrieffer (BCS) basis, our model is formulated within the Hartree-Fock-Bogoliubov (HFB) quasiparticle basis. Implementing a relativistic point-coupling interaction and a separable pairing force allows for the reduction of complicated two-body residual interaction matrix elements, which considerably decreases the dimension of the problem in the coordinate space. The main advantage of this method is to avoid the diagonalization of a large QRPA matrix, especially at finite temperature where the size of configuration space is significantly increased. The implementation of the linear response code is used to study the temperature evolution of IAR, GTR, and spin-dipole resonance (SDR) in even-even tin isotopes in the temperature range $T = 0 - 1.5$ MeV.

Bubble structure in superheavy nuclei around neutron and proton shell closure

We have performed a systematic study for the nuclear structure of superheavy nuclei with a special emphasis on the nuclei with possible central depletion of proton and neutron density in the mass region [Formula: see text] using the Relativistic Hartree–Bogoliubov (RHB) framework. It has been observed that in the case of neutron density distribution, the occurrence of central depletion is related to the occupancy of 4s orbital and it is found to decrease with increasing occupancy of the 4s orbital. On the other hand, in the case of proton density distribution, the central density depletion is mainly due to the lowering of weakly bound p-orbital states close to the continuum as it is energetically favored to lower the Coulomb repulsion in the case of superheavy nuclei. Also, occupation probability of the lower angular momentum states (p-orbitals) lying near the Fermi level is strongly suppressed due to the weak centrifugal barrier and strong Coulomb repulsion in comparison to large angular momentum states (contributing to surface region mainly), resulting in central density depletion. Among the considered cases in the present work, the maximum depletion is observed for [Formula: see text] and for [Formula: see text]Og under spherically symmetric and axially deformed cases, respectively.

Shape isomerism and magicity in Ni isotopes

A systematic study of shapes in Ni [Formula: see text] isotopes has been made in the Relativistic–Hartree–Bogoliubov (RHB) formalism with two types of density-dependent NN interactions which are based on the range of meson-exchange. The constraint calculations assuming the axial and triaxial-symmetry predict the shape isomerism in the case of [Formula: see text] isotopes. Significant jumps at [Formula: see text] in the binding energy per nucleon (BE/A) and in the [Formula: see text] correspond to the neutron shell closure, and [Formula: see text] as doubly magic nuclei. The present calculation supports the recently reported calculations using the non-relativistic Hartree–Fock (HF) Skyrme SIII [1] interaction predicting the importance of tensor parameter in order to reproduce the experimental findings of the proton level crossing at [Formula: see text]. The results obtained are in agreement with experiment and with other theoretical studies.