Relativistic Hartree-Bogoliubov Model Research Articles

Anatomy of octupole correlations in 96Zr with a symmetry-restored multidimensionally-constrained covariant density functional theory

A recent analysis based on the STAR measurement in relativistic heavy-ion collision experiments provides evidence of octupole correlation in the ground state of 96Zr, the description of which presents a challenge to nuclear structure models. In this work, we perform projection-after-variation calculations for 96Zr based on a multidimensionally-constrained relativistic Hartree-Bogoliubov model. Our results show that an octupole deformed shape is favored in energy after symmetry restoration, and this phenomenon cannot be reproduced in the pure mean-field calculations. These complex structures originate from the competition among various shell structures in this mass region. Our results suggest that the allowance of symmetry breaking at the mean-field level and the restoration of broken symmetries are essential elements for understanding the structure of 96Zr in density functional theories.

Magnetic quadrupole transitions in the relativistic energy density functional theory

Magnetic quadrupole (M2) excitation represents a fundamental feature in atomic nucleus associated to nuclear magnetism induced by spin and orbital transition operator. Since it has only been investigated within the non-relativistic theoretical approaches, and available experimental data are rather limited, it is interesting to study the properties of M2 transitions using the framework of relativistic nuclear energy density functional. In this work the nuclear ground state is calculated with relativistic Hartree-Bogoliubov model, while the M2 excitations are described using the relativistic quasiparticle random phase approximation with the residual interaction extended with the isovector-pseudovector term. The M2 transition strength distributions are described and analyzed for closed shell nuclei $$^{16}$$ O, $$^{48}$$ Ca, $$^{208}$$ Pb, open-shell $$^{18} \mathrm O$$ , $$^{42}$$ Ca, $$^{56}$$ Fe, and semi-magic $$^{90}$$ Zr. The evolution of M2 transition properties has been investigated within the $$^{36-64} \mathrm Ca$$ isotope chain. The main M2 transitions have rather rich underlying structure and detailed analysis shows that collectivity increases with the mass number due to larger number of contributing particle-hole configurations. Pairing correlations in open shell nuclei have strong effect, causing the reduction of M2 strength and shifts of the centroid energies to higher values. The analysis of M2 transition strengths indicate that considerable amount of experimental strength may be missing, mainly due to limitations to rather restricted energy ranges. The calculated M2 strengths for Ca isotopes, together with the future experimental data will allow constraining the quenching of the g factors in nuclear medium.

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.

Nuclear charge radii of Cr46–62 isotopes and reaction cross sections for p−Cr

The nuclear charge radii and density distributions for $^{46--62}\mathrm{Cr}$ have been calculated in the frameworks of both relativistic and nonrelativistic self-consistent mean-field models. The relativistic Hartree-Bogoliubov model with density-dependent meson-exchange functional DD-ME2 and the nonrelativistic Hartree-Fock-Bogoliubov model with the Gogny D1S interaction, hereafter referred to as DIRHB and HFB, respectively, are used to compute ground-state properties of even $^{46--62}\mathrm{Cr}$ isotopes. Both calculations for the root-mean-squared charge radii reveal the characteristic kink at the $N$ = 28 shell closure in accordance with the corresponding experimental radii. The point proton and neutron density distributions calculated from both DIRHB and HFB are used to obtain the optical potentials for $p$-Cr at an incident proton energy of 65 MeV. The elastic scattering differential and total reaction cross sections, computed from the semimicroscopic proton optical potentials, have been derived by folding each of the target matter densities with the Jeukenne-Lejeune-Mahaux-Bruye\ifmmode \acute{r}\else \'{r}\fi{}es (JLMB) energy- and density-dependent internucleon interaction. The calculated elastic scattering differential cross sections for stable even isotopes, $^{50,52,54}\mathrm{Cr}$, using the respective DIRHB and HFB densities in the folding model reproduce the corresponding cross-section data well. The correlation between root-mean-squared charge radii and nuclear reaction observables obtained from the DIRHB and HFB calculations have been used in the folding model approach to predict the reaction cross sections for $p\text{\ensuremath{-}}^{46--62}\mathrm{Cr}$ at an incident proton energy of 65 MeV.

Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei

Background: At the forefront of nuclear science, unstable nuclei are of special significance, in which the deformation and pairing correlations play an essential role in determining the nuclear structure. Under the mean field approach, the pairing and deformation effects have been uniformly considered within the deformed nonrelativistic Hartree-Fock-Bogoliubov model and the deformed relativistic Hartree-Bogoliubov (D-RHB) model. However, due to the limitation of the Hartree approach, an important ingredient of the nuclear force---the tensor force that contributes via the Fock diagram---is missing in the D-RHB model. Recently, the axially deformed relativistic Hartree-Fock (D-RHF) model was established for deformed nuclei, in which the $\ensuremath{\pi}$-pseudovector ($\ensuremath{\pi}$-PV) coupling can take into account the tensor force effects naturally. While limited by the BCS pairing, it cannot be safely applied to describe unstable nuclei.Purpose: The aim of this work is to develop the axially deformed relativistic Hartree-Fock-Bogoliubov (D-RHFB) model for the reliable description of a wide range of unstable nuclei, by utilizing the spherical Dirac Woods-Saxon (DWS) base.Method: Staring from the Lagrangian density that is based on the meson-propagated picture of nuclear force, the full Hamiltonian, that contains both mean field and pairing contributions, is derived by quantizing the Dirac spinor field in the Bogoliubov quasiparticle space, and the expectation with respect to the Bogoliubov ground state gives the full energy functional. As an extension of the D-RHF model, the degree of freedom associated with the $\ensuremath{\rho}$-tensor ($\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$) coupling is implemented, and incorporating with the Bogoliubov scheme the finite-range Gogny force D1S is utilized as the pairing force. Moreover, qualitative analyses on the nature of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings are presented to better understand their enhancements on the deformation effects.Results: Space convergence related to the spherical DWS base is confirmed for the D-RHFB model by taking light nucleus $^{24}\mathrm{Mg}$ and mid-heavy one $^{156}\mathrm{Sm}$ as candidates. Compared to light nuclei, extraordinarily more negative energy states are necessitated to keep the expansion completeness on the spherical DWS base for mid-heavy and heavy nuclei, due to the enhanced correlations between the expansion components with large $\ensuremath{\kappa}$ quantity, as indicated by the nature of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings. Furthermore, because of the enhanced deformation effects by the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings, the RHF Lagrangian PKA1 presents a deeper bound ground state for $^{24}\mathrm{Mg}$ than the other selected Lagrangians, in addition to predicting a fairly deep bound local minimum with large oblate deformation.Conclusions: The axially deformed relativistic Hartree-Fock-Bogoliubov model has been established with confirmed convergence checks. The effects of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings, coupled with nuclear deformation, are analyzed in both light and mid-heavy nuclear systems, and are expected to be manifested in a wide range of unstable nuclei.

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.

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.

Microscopic study of structure of light- and medium-mass even-even cadmium isotopes

The potential-energy surfaces (PESs) for ground states of even-even $^{102--112}\mathrm{Cd}$ isotopes are obtained by using the three-dimensional relativistic Hartree-Bogoliubov model with the density-dependent meson exchange interaction. The quadrupole deformation parameters are extracted from the PES calculations and employed in the beyond mean-field projected shell model to study the yrast and excited states of these isotopes. Besides, the $E2$ transition probabilities and $g$ factors are obtained and compared with the experimental data. The present calculations support the experimental finding that yrast ${8}^{+}$ states of light Cd isotopes are of ${(\ensuremath{\pi}{g}_{9/2})}^{2}$ character whereas the yrast ${10}^{+}$ states of medium-mass Cd isotopes are found to possess ${(\ensuremath{\nu}{h}_{11/2})}^{2}$ character.

Discerning nuclear pairing properties from magnetic dipole excitation

Pairing correlation of Cooper pair is a fundamental property of multi-fermion interacting systems. For nucleons, two modes of the Cooper-pair coupling may exist, namely of $$S_{12}=0$$ with $$L_{12}=0$$ (spin-singlet s-wave) and $$S_{12}=1$$ with $$L_{12}=1$$ (spin-triplet p-wave). In nuclear physics, it has been an open question whether the spin-singlet or spin-triplet coupling is dominant, as well as how to measure their role. We investigate a relation between the magnetic-dipole (M1) excitation of nuclei and the pairing modes within the framework of relativistic nuclear energy-density functional (RNEDF). The pairing correlations are taken into account by the relativistic Hartree-Bogoliubov (RHB) model in the ground state, and the relativistic quasi-particle random-phase approximation (RQRPA) is employed to describe M1 transitions. We have shown that M1 excitation properties display a sensitivity on the pairing model involved in the calculations. The systematic evaluation of M1 transitions together with the accurate experimental data enables us to discern the pairing properties in finite nuclei.

Global analysis of nuclear cluster structure from the elastic and inclusive electron scattering

Background: Recently, the researches of cluster structure have attracted considerable attention. Many microscopic structure models have been applied to describe the cluster configurations.Purpose: To better analyze the cluster structure and associate with the experimental observations, comparative studies are carried out in this paper by combining the nuclear structure model with the electron scattering theory.Method: The density distributions for candidate nuclei of normal and cluster states are obtained from the deformed relativistic Hartree-Bogoliubov (RHB) model. Using some moderate approximations, the corresponding Coulomb form factors $|{F}_{C}{(q)|}^{2}$ and inclusive cross sections for different configurations are calculated by the distorted wave Born approximation (DWBA) and coherent density fluctuation model (CDFM), respectively.Results: Comparing the $|{F}_{C}{(q)|}^{2}$ and inclusive cross sections of different configurations, the effects of nuclear cluster structure can be revealed, due to the differences of nuclear charge radii ${R}_{C}$ in coordinate space and the discrepancies of Fermi momentum ${k}_{F}$ in momentum space.Conclusions: Results illustrate that the cluster structure can be reflected from the elastic and inclusive electron scattering. The studies conducted in this paper provide a new approach to analyze the cluster configurations, and are also helpful to guide future electron scattering experiments.

Role of residual interaction in the relativistic description of M1 excitation

Magnetic dipole (M1) excitation is the leading mode of multi-nucleon excitations induced by the magnetic field, and is a phenomenon of the spin–orbit splitting and residual interactions involved. In this work, we investigate the effects of the residual interactions on the M1 excitation from a novel perspective, the framework of relativistic nuclear energy-density functional. The relativistic Hartree–Bogoliubov model is utilized to determine the nuclear ground state properties, while the relativistic quasi-particle random-phase approximation is employed for the description of M1-excitation properties. From the analysis of M1 mode in the Ca isotope chain, role of the isovector–pseudovector residual interaction is discussed. For open-shell nuclei, the pairing correlation also plays a noticeable role in the M1 mode. The experimental data on M1 mode is expected to provide a suitable reference to improve and optimize the theoretical models to describe the residual interactions.

Shape phase transitions in odd−A Zr isotopes

Spectroscopic properties that characterize shape phase transitions in neutron-rich odd-A Zr isotopes are investigated using the framework of nuclear density functional theory and particle-core coupling. The interacting-boson Hamiltonian of the even-even core nuclei, and the single-particle energies and occupation probabilities of the unpaired neutron are completely determined by deformation constrained self-consistent mean-field calculations based on the relativistic Hartree-Bogoliubov model with a choice of a universal energy density functional and pairing interaction. The triaxial $(\beta,\gamma)$ deformation energy surfaces for even-even $^{94-102}$Zr indicates transition from triaxial or $\gamma$-soft ($^{94,96}$Zr) to prolate ($^{98}$Zr), and triaxial ($^{100,102}$Zr) shapes. The corresponding low-energy excitation spectra of the odd-A Zr isotopes are in very good agreement with recent experimental results. Consistent with the structural evolution of the neighboring even-even Zr nuclei, the state-dependent effective deformations and their fluctuations in the odd-A isotopes indicate a pronounced discontinuity around the transitional nucleus $^{99}$Zr.

Possible dual bubble-like structure predicted by the relativistic Hartree–Bogoliubov model

Experimental observation of [Formula: see text]Si as a proton bubble nuclei has heated up the interest in the study of exotic bubble shaped nuclei. In this work, some of the potential doubly bubble-like (for proton as well as neutron both simultaneously) cases have been explored using relativistic Hartree–Bogoliubov (RHB) in light mass region, specially around N or [Formula: see text]. Further, the role of pairing and the evolution of new shell gaps around [Formula: see text] or [Formula: see text] and 34 have been investigated, as one moves toward drip lines. This study suggests that the occupancies/vacancies of neutron/proton orbitals for lower angular momentum state plays major role in nuclear structure to create bubble-like structure and [Formula: see text]O[Formula: see text], [Formula: see text]Mg8 and [Formula: see text]Si[Formula: see text] may have dual bubble-like structures.

Relativistic Hartree-Bogoliubov description of deformed light nuclei

The Relativistic Hartree-Bogoliubov model is applied in the analysis of ground- state properties of light nuclei with 4 < Ζ < 11. The model uses the NL3 effective interaction in the mean-field Lagrangian, and describes pairing correlations by the pairing part of the finite range Gogny interaction DIS. Neutron separation energies, quadrupole deformations, nuclear matter radii, and differences in radii of proton and neutron distributions are compared with recent experimental data.

Nuclear phase transitions near the critical points: a study with the Relativistic Hartree-Bogoliubov model, the Interacting Boson Model and the Boson Coherent-State Framework

We present an analysis of the intensity of 2-particle transfer reactions in the Interacting Boson Model (IBM), and in the Boson Coherent-State framework, as a tool to study nuclear phase transitions. We study transfer reactions between two ground states, and between the ground state and the band head of the beta-vibrational band. We suggest characteristic fingerprints that should allow experimentalists to identify the critical points of the nuclear phase transition. Two analytical solutions, X(5) and E(5), have been proposed recently for two of the critical points. We present a study within the Relativistic Hartree-Bogoliubov model (RHB), using Potential-Energy Surfaces (PES), to test whether the initial approximations made in deriving the analytical solutions are valid.

Proton drip line nuclei in the Relativistic Hartree-Bogoliubov model

The Relativistic Hartree Bogoliubov (RHB) theory is applied to the description of ground-state properties of proton-rich odd-Z nuclei. The model predicts the location of the proton drip-line, the ground-state quadrupole deformations and one-proton separation energies at and beyond the drip-line, the deformed single-particle Orbitals occupied by the odd valence proton, and the corresponding spectroscopic factors. The results of fully self-consistent RHB calculations are compared with experimental data on deformed and transitional proton emitters and the predictions of various mass models.

Microscopic core-quasiparticle coupling model for spectroscopy of odd-mass nuclei with octupole correlations

[Background] Predictions of spectroscopic properties of low-lying states are critical for nuclear structure studies. Theoretical methods can be particularly involved for odd-mass nuclei because of the interplay between the unpaired nucleon and collective degrees of freedom. Only a few models have been developed for systems in which octupole collective degrees of freedom play a role. [Purpose] We aim to predict spectroscopic properties of odd-mass nuclei characterized by octupole shape deformation, employing a model that describes single-particle and collective degrees of freedom within the same microscopic framework. [Method] A microscopic core-quasiparticle coupling (CQC) model based on the covariant density functional theory is developed, which includes collective excitations of even-mass core nuclei and single-particle states of the odd nucleon, calculated using a quadrupole-octupole collective Hamiltonian combined with a constrained reflection-asymmetric relativistic Hartree-Bogoliubov model. [Results] Model predictions for low-energy excitation spectra and transition rates of odd-mass radium isotopes $^{223, 225, 227}$Ra are shown to be in good agreement with available data. [Conclusions] A microscopic CQC model based on covariant density functional theory has been developed for odd-mass nuclei characterized by both quadrupole and octupole shape deformations. Theoretical results reproduce data in odd-mass Ra isotopes and provide useful predictions for future studies of octupole correlations in nuclei and related phenomena.

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.

Microscopic study of evolution of shape change across even-even mass chain of tellurium isotopes using relativistic Hartree-Bogoliubov model

The density dependent point coupling and meson exchange effective interactions are employed in the relativistic Hartree-Bogoliubov (RHB) model to study the development of ground state deformation and occurrence of shape transitions with change of neutron number in 104−144Te isotopes. Besides this, the systematics of other nuclear structure properties with increasing neutron number are also obtained. The RHB model reproduces well the available experimental data on binding energies, quadrupole deformation parameters, two neutron separation energies and root mean square charge radii. The results of three-dimensional potential energy surfaces predict prolate ground state minima for proton rich and neutron rich tellurium isotopes, oblate minima for 114−124Te, triaxial minima for 126,128Te and spherical minima for 130−134Te. Besides, oblate prolate shape coexistence is predicted in 116−120Te isotopes.

Optimizing the relativistic energy density functional with nuclear ground state and collective excitation properties

We introduce a new relativistic energy density functional constrained by the ground state properties of atomic nuclei along with the isoscalar giant monopole resonance energy and dipole polarizability in $^{208}$Pb. A unified framework of the relativistic Hartree-Bogoliubov model and random phase approximation based on the relativistic density-dependent point coupling interaction is established in order to determine the DD-PCX parameterization by $\chi^2$ minimization. This procedure is supplemented with the co-variance analysis in order to estimate statistical uncertainties in the model parameters and observables. The effective interaction DD-PCX accurately describes the nuclear ground state properties including the neutron-skin thickness, as well as the isoscalar giant monopole resonance excitation energies and dipole polarizabilities. The implementation of the experimental data on nuclear excitations allows constraining the symmetry energy close to the saturation density, and the incompressibility of nuclear matter by using genuine observables on finite nuclei in the $\chi^2$ minimization protocol, rather than using pseudo-observables on the nuclear matter, or by relying on the ground state properties only, as it has been customary in the previous studies.