Relativistic Hartree-Fock-Bogoliubov Research Articles

Study of the nuclear structure of some exotic nuclei using nonrelativistic and relativistic mean-field methods

The nuclear shell model with the Skyrme–Hartree–Fock (SHF), as a nonrelativistic approach, and the Relativistic Hartree–Fock–Bogoliubov (RHFB) methods have been used to study the nuclear structure of some exotic nuclei at the proton and neutron drip lines. Different Skyrme parametrizations, in particular SkM*, SkX, SkO, SLy4, Skxs25 and Z, have been used in the nonrelativistic region. In the relativistic region, the density-dependence meson-exchange models and density-dependence point-coupling models are used. Both methods are used to study ground state properties such as binding energy, mass radial density distribution and the corresponding root mean square (rms) mass radii. The fragmentation reaction cross-section is used as an important property to investigate the halo structure. Strong evidence for existence of a neutron halo in [Formula: see text]Li and [Formula: see text]Be and proton halo in [Formula: see text]Ne, [Formula: see text]Al and [Formula: see text]P are found using the SHF model and reaction cross-section. The ability of the SHF model to study the exotic structure with shell model occupation probability is confirmed.

Pseudospin-orbit splitting and its consequences for the central depression in nuclear density

The occurrence of the bubble-like structure has been studied, in the light of pseudospin degeneracy, within the relativistic Hartree-Fock-Bogoliubov (RHFB) theory. It is concluded that the charge/neutron bubble-like structure is predicted to occur in the mirror system of {$^{34}\mathrm{Si},\phantom{\rule{0.16em}{0ex}}^{34}\mathrm{Ca}$} commonly by the selected Lagrangians, due to the persistence of $Z(N)=14$ subshell gaps above which the $\ensuremath{\pi}(\ensuremath{\nu})2{s}_{1/2}$ states are not occupied. However, for the popular candidate $^{46}\mathrm{Ar}$, the RHFB Lagrangian PKA1 does not support the occurrence of the bubble-like structure in the charge (proton) density profiles, due to the almost degenerate pseudospin doublet ${\ensuremath{\pi}2{s}_{1/2},\ensuremath{\pi}1{d}_{3/2}}$ and coherent pairing effects. The formation of a semibubble in heavy nuclei is less possible as a result of small pseudospin-orbit (PSO) splitting, while it tends to appear at $Z=120$ superheavy systems which coincides with large PSO splitting of the doublet ${\ensuremath{\pi}3{p}_{3/2},\ensuremath{\pi}2{f}_{5/2}}$ and couples with significant shell effects. Pairing correlations, which can work against bubble formation, significantly affect the PSO splitting. Furthermore, we found that the influence on semibubble formation due to different types of pairing interactions is negligible. The quenching of the spin-orbit splitting in the $p$ orbit has been also stressed, and it may be considered the hallmark for semibubble nuclei.

Magicity of neutron-rich nuclei within relativistic self-consistent approaches

The formation of new shell gaps in intermediate mass neutron-rich nuclei is investigated within the relativistic Hartree–Fock–Bogoliubov theory, and the role of the Lorentz pseudo-vector and tensor interactions is analyzed. Based on the Foldy–Wouthuysen transformation, we discuss in detail the role played by the different terms of the Lorentz pseudo-vector and tensor interactions in the appearing of the N=16, 32 and 34 shell gaps. The nuclei 24O, 48Si and 52,54Ca are predicted with a large shell gap and zero (24O, 52Ca) or almost zero (48Si, 54Ca) pairing gap, making them candidates for new magic numbers in exotic nuclei. We find from our analysis that the Lorentz pseudo-vector and tensor interactions induce very specific evolutions of single-particle energies, which could clearly sign their presence and reveal the need for relativistic approaches with exchange interactions.

Pairing phase transition: A finite-temperature relativistic Hartree-Fock-Bogoliubov study

Background: The relativistic Hartree-Fock-Bogoliubov (RHFB) theory has recently been developed and it provides a unified and highly predictive description of both nuclear mean field and pairing correlations. Ground state properties of finite nuclei can accurately be reproduced without neglecting exchange (Fock) contributions. Purpose: Finite-temperature RHFB (FT-RHFB) theory has not yet been developed, leaving yet unknown its predictions for phase transitions and thermal excitations in both stable and weakly bound nuclei. Method: FT-RHFB equations are solved in a Dirac Woods-Saxon (DWS) basis considering two kinds of pairing interactions: finite or zero range. Such a model is appropriate for describing stable as well as loosely bound nuclei since the basis states have correct asymptotic behaviour for large spatial distributions. Results: Systematic FT-RH(F)B calculations are performed for several semi-magic isotopic/isotonic chains comparing the predictions of a large number of Lagrangians, among which are PKA1, PKO1 and DD-ME2. It is found that the critical temperature for a pairing transition generally follows the rule $T_c = 0.60\Delta(0)$ for a finite-range pairing force and $T_c = 0.57\Delta(0)$ for a contact pairing force, where $\Delta(0)$ is the pairing gap at zero temperature. Two types of pairing persistence are analysed: type I pairing persistence occurs in closed subshell nuclei while type II pairing persistence can occur in loosely bound nuclei strongly coupled to the continuum states. Conclusions: This first FT-RHFB calculation shows very interesting features of the pairing correlations at finite temperature and in finite systems such as pairing re-entrance and pairing persistence.

Superheavy magic nuclei in Relativistic Hartree-Fock-Bogoliubov theory

The occurrence of the spherical shell closures in superheavy nuclear systems is explored within the relativistic Hartree-Fock-Bogoliubov (RHFB) theory with density- dependent meson-nucleon couplings. We use the two-nucleon gaps δ2n(p) and pairing gaps Δv(π) to characterize the shell effects. The results depend slightly on the forces used, but the general set of magic numbers beyond 208Pb are Z = 120, 138 and N = 172, 184, 228 and 258. Our calculations are in favor of the nuclide 304120 as the next spherical doubly magic nucleus. Combined with the bulk properties of symmetric matter, we find that the shell effects are sensitive to the values of both scalar mass and effective mass. These two masses essentially determine the spin-orbit splittings and single-particle level density, respectively. Furthermore, the breaking of relativistic pseudo-spin symmetry influences the level structure and the occurrence of closed shells.

Superheavy magic structures in the relativistic Hartree–Fock–Bogoliubov approach

We have explored the occurrence of the spherical shell closures for superheavy nuclei in the framework of the relativistic Hartree–Fock–Bogoliubov (RHFB) theory. Shell effects are characterized in terms of two-nucleon gaps δ2n(p). Although the results depend slightly on the effective Lagrangians used, the general set of magic numbers beyond 208Pb are predicted to be Z=120, 138 for protons and N=172, 184, 228 and 258 for neutrons, respectively. Specifically the RHFB calculations favor the nuclide 120304 as the next spherical doubly magic one beyond 208Pb. Shell effects are sensitive to various terms of the mean-field, such as the spin-orbit coupling, the scalar and effective masses.

Odd-even staggering of the nuclear binding energy described by covariant density functional theory with calculations for spherical nuclei

The odd-even staggerings (OES) on nuclear binding energies are studied systematically within the covariant density functional (CDF) theories, specifically the relativistic Hartree-Fock-Bogoliubov (RHFB) and the relativistic Hartree-Bogoliubov (RHB) theories. Taking the finite-range Gogny force D1S as an effective pairing interaction, both CDF models can provide appropriate descriptions on the OESs of nuclear binding energies for C, O, Ca, Ni, Zr, Sn, Ce, Gd and Pb isotopes as well as for N=50 and 82 isotones. However, due to the inconsistence between the non-relativistic pairing interaction and the relativistic effective Lagrangians, there exist some systematical discrepancies from the data, i.e., the underestimated OESs in light C and O isotopes and the overestimated ones in heavy region, respectively. Such discrepancies can be eliminated partially by introducing a $Z$- or $N$-dependent strength factor into the pairing force Gogny D1S. In addition, successful descriptions of the occupation numbers of Sn isotopes are achieved with the optimized Gogny pairing force. Furthermore, the analysis of the systematics of both pairing effects and nuclear binding energy indicate the requirement of an unified relativistic mechanism in both p-p and p-h channels to improve the quantitative precision of the theory.

β-decay half-lives of neutron-rich nuclei and matter flow in the r-process

The β-decay half-lives of neutron-rich nuclei with 20≤Z≤50 are systematically investigated using the newly developed fully self-consistent proton–neutron quasiparticle random phase approximation (QRPA), based on the spherical relativistic Hartree–Fock–Bogoliubov (RHFB) framework. Available data are reproduced by including an isospin-dependent proton–neutron pairing interaction in the isoscalar channel of the RHFB+QRPA model. With the calculated β-decay half-lives of neutron-rich nuclei a remarkable speeding up of r-matter flow is predicted. This leads to enhanced r-process abundances of elements with A≳140, an important result for the understanding of the origin of heavy elements in the universe.

Tensor effects on the evolution of theN=40shell gap from nonrelativistic and relativistic mean-field theory

Tensor effects on the $N=40$ gap evolution of $N=40$ isotones are studied by employing the Skyrme-Hartree-Fock-Bogoliubov (SHFB) and relativistic Hartree-Fock-Bogoliubov (RHFB) theories. The results with and without the inclusion of the tensor components are compared with each other. It is found that when the tensor force is included, both approaches present the same trend and qualitatively agree with the experimental one indicated by the $\ensuremath{\beta}$-decay studies, which implies an important role played by the tensor force in the gap evolution of $N=40$. Furthermore, it is shown that the gap evolution is dominated by the corresponding tensor contributions from $\ensuremath{\pi}$ and $\ensuremath{\rho}$-tensor coupling in the relativistic framework.

Description of carbon isotopes within relativistic Hartree-Fock-Bogoliubov theory

Within the relativistic Hartree-Fock-Bogoliubov (RHFB) theory, the structure properties of Carbon isotopes are systematically studied. In order to reproduce the experiment data, we take the finite-range Gogny D1S with a strength factor $f$ as the pairing force. The self-consistent RHFB calculations with density-dependent meson-nucleon couplings indicate the single-neutron halo structures in both $^{17}$C and $^{19}$C, whereas the two-neutron halo in $^{22}$C is not well supported. It is also found that close to the neutron drip line there exists distinct odd-even staggering on neutron radii, which is tightly related with the blocking effects and correspondingly the blocking effect plays a significant role in halo formation.

Description of the ground state of axially deformed nuclei within the Relativistic Hartree–Fock–Bogoliubov model

The Relativistic Hartree–Fock–Bogoliubov model for axially deformed nuclei (RHFBz) is presented. The model involves a phenomenological Lagrangian with density-dependent meson–nucleon couplings in the particle-hole channel and the central part of the Gogny force in the particle–particle channel. The RHFBz equations are solved by expansion in the basis of a deformed harmonic oscillator. Illustrative RHFBz calculations are performed for Neon isotopes.

Relativistic Hartree-Fock-Bogoliubov theory with density dependent meson-nucleon couplings

Relativistic Hartree-Fock-Bogoliubov (RHFB) theory with density-dependent meson-nucleon couplings is presented. The integro-differential RHFB equations are solved by expanding the different components of the quasi-particle spinors in the complete set of eigen-solutions of the Dirac equations with Woods-Saxon potentials. Using the finite-range Gogny force D1S as an effective interaction in the pairing channel, systematic RHFB calculations are performed for Sn isotopes and N=82 isotones. It is demonstrated that an appropriate description of both mean field and pairing effects can be obtained within RHFB theory with finite range Gogny pairing forces. Better systematics are also found in the regions from the stable to the neutron-rich side with the inclusion of Fock terms, especially in the presence of $\rho$-tensor couplings.

Polarization effects in relativistic pairing in nuclear matter

We discuss the effects of medium polarization on nuclear pairing in the context of the relativistic Hartree–Fock–Bogoliubov (HFB) approximation in the σ – ω model. The medium polarization effects on the scalar and vector meson masses, and on the pairing gap are shown for several values of the coupling constants scaled by a parameter x. We find that the medium polarization increases the magnitude of the pairing gap at low density while the vacuum polarization decreases its magnitude at higher density.

Phenomenological construction of a relativistic nucleon–nucleon interaction for the superfluid gap equation in finite density systems

We construct phenomenologically a relativistic particle–particle channel interaction which suits the gap equation for nuclear matter. This is done by introducing a density-independent momentum-cutoff parameter to the relativistic mean field (Hartree and Hartree–Fock) models so as to reproduce the pairing properties obtained by the Bonn-B potential and not to change the saturation property. The interaction so obtained can be used for the relativistic Hartree–Bogoliubov calculation, but some reservation is necessary for the relativistic Hartree–Fock–Bogoliubov calculation.