Relativistic Mean-field Lagrangian Research Articles

Fusion cross section of the superheavy Z=120 nuclei within the relativistic mean-field formalism

Background: Isotopes of $Z=107\text{--}118$ have been synthesized using cold fusion at GSI, Darmstadt, and hot fusion reactions at JINR, Dubna. Recently theoretical models have predicted $Z=120$ with $N=184$ as an island of stability in the superheavy valley. Hence it is crucial and exciting to predict theoretically the possible combination of projectiles and targets for the synthesis of $Z=120$, which can be informative for upcoming experiments.Purpose: Present theoretical investigations aim to explore the fusion characteristics of various isotopes of $Z=120$ within the relativistic mean-field formalism. We predict the most suitable projectile-target combination for the synthesis of element $Z=120$. The increase in fusion cross section of nuclei in the superheavy island directly signals the nuclear shell effects. Besides these, the analysis will be crucial and relevant for future experiments to synthesize superheavy nuclei.Methods: The microscopic nucleon-nucleon R3Y interaction and the density distributions for targets and projectiles are calculated using a relativistic mean-field formalism with the ${\mathrm{NL3}}^{*}$ parameter set. These densities and R3Y nucleon-nucleon (NN) interaction are then used to calculate the nuclear interaction potential using the double folding approach. Seventeen different projectile-target combinations that allow a high $N$/$Z$ ratio are considered in the present analysis to calculate the capture and/or fusion cross sections of various isotopes of $Z=120$ within the $\ensuremath{\ell}$-summed Wong formula.Results: The nuclear density distributions for the interacting projectile and target nuclei are obtained from relativistic mean-field Lagrangian for the ${\mathrm{NL3}}^{*}$ parameter set. The nucleus-nucleus interaction potential is estimated for seventeen possible projectile-target combinations using the mean-field density and the R3Y NN potential via a double folding approach. The fusion barriers are obtained by adding the Coulomb potential to the nucleus-nucleus interaction potential. Finally, the fusion and/or capture cross section is calculated for all the systems within the $\ensuremath{\ell}$-summed Wong formula. Further, the equivalent surface diffusion parameter is estimated to correlate the surface properties of interacting nuclei with the fusion cross section.Conclusions: The four Ti-based reactions with the heaviest available target $^{\mathrm{x}}\mathrm{Cf}$, namely, $^{46}\mathrm{Ti}+^{248}\mathrm{Cf}$, $^{46}\mathrm{Ti}+^{249}\mathrm{Cf}$, $^{50}\mathrm{Ti}+^{249}\mathrm{Cf}$, and $^{50}\mathrm{Ti}+^{252}\mathrm{Cf}$, and also $^{54}\mathrm{Cr}+^{250}\mathrm{Cm}$ are found to have the most suitable target-projectile combinations for the synthesis of various isotopes $Z=120$. We also notice that $^{48}\mathrm{Ca}$ beams merely provide the required number of protons to synthesize the element with $Z=120$. We established a correlation among the surface properties of interacting nuclei with the fusion characteristics in terms of the equivalent surface diffusion parameter.

Novel relativistic mean field Lagrangian guided by pseudo-spin symmetry restoration * * Supported by National Natural Science Foundation of China (11675065, 11875152, 11905088), Fundamental Research Funds for the central universities (lzujbky-2019-11), and the Supercompuer Center of HIRFL

The relativistic mean field (RMF) model has achieved great success in describing various nuclear phenomena. However, several serious defects are common. For instance, the pseudo-spin symmetry of high-l orbits is distinctly violated in general, leading to spurious shell closures and . This leads to problems in describing structure properties, including shell structures, nuclear masses, etc. Guided by the pseudo-spin symmetry restoration [Geng et al., Phys. Rev. C, 100: 051301 (2019)], a new RMF Lagrangian DD-LZ1 is developed by considering the density-dependent meson-nucleon coupling strengths. With the newly obtained RMF Lagrangian DD-LZ1, satisfactory descriptions can be obtained for the bulk properties of nuclear matter and finite nuclei. In particular, significant improvements on describing the single-particle spectra are achieved by DD-LZ1. In particular, the spurious shell closures and , commonly found in previous RMF calculations, are eliminated by the new effective interaction DD-LZ1, and consistently the pseudo-spin symmetry (PSS) around the Fermi levels is reasonably restored for both low-l and high-l orbits. Moreover, the description of nuclear masses is also notably improved by DD-LZ1, as compared to the other RMF Lagrangians.

Effects of symmetry energy on the radius and tidal deformability of neutron stars in the relativistic mean-field model

Abstract The radii and tidal deformabilities of neutron stars are investigated in the framework of the relativistic mean-field (RMF) model with different density-dependent behaviors of symmetry energy. To study the effects of symmetry energy on the properties of neutron stars, $\omega$ meson and $\rho$ meson coupling terms are included in a popular RMF Lagrangian, i.e., the TM1 parameter set, which is adopted for the widely used supernova equation of state (EoS) table. The coupling constants relevant to the vector–isovector meson, $\rho$, are refitted by a fixed symmetry energy at subsaturation density and its slope at saturation density, while other coupling constants remain the same as the original ones in TM1 so as to update the supernova EoS table. The radius and mass of maximum neutron stars are not so sensitive to the symmetry energy in these family TM1 parameterizations. However, the radii in the intermediate-mass region are strongly correlated with the slope of symmetry energy. Furthermore, the dimensionless tidal deformabilities of neutron stars are also calculated within the associated Love number, which is related to the quadrupole deformation of the star in a static external tidal field and can be extracted from the observation of a gravitational wave generated by a binary star merger. We find that its value at $1.4 \mathrm{M}_\odot$ has a linear correlation to the slope of symmetry energy, unlike that previously studied. With the latest constraints of tidal deformabilities from the GW170817 event, the slope of symmetry energy at nuclear saturation density should be smaller than $60$ MeV in the family TM1 parameterizations. This fact supports the usage of a lower symmetry energy slope for the updated supernova EoS, which is applicable to simulations of neutron star mergers. Furthermore, an analogous analysis is also done within the family IUFSU parameter sets. It is found that the correlations between the symmetry energy slope with the radius and tidal deformability at $1.4 \mathrm{M}_\odot$ have very similar linear relations in these RMF models.

Analysis of the reaction cross-sections of 22−36Na+12C and 19−27F+12C at 240MeV/u

The nucleus–nucleus reaction cross-sections of [Formula: see text]C and [Formula: see text]C at 240[Formula: see text]MeV/u are calculated using the optical limit of the Glauber model. The deformation and radii of a deformed Fermi density are calculated from the relativistic mean field (RMF). The results are compared with the recent experimental data. It is found that the Fermi density whose quadrupole deformation parameters and radii are derived from RMF, using TMA effective interaction in the RMF Lagrangian, provide a satisfactory explanation of the experimental data of Na isotopes, except for [Formula: see text]Na, which are expected to be strongly deformed. For F isotopes, the deformed Fermi density adjusted to the radii derived from RMF, using TMA interaction, presents a lower reaction cross-section. The Lagrangian parameters set NL3* gives a good description of the data which is better than that predicted by TMA. On the other hand, the two forces cannot describe the reaction cross-section of [Formula: see text]F since it is expected to have a deformed halo structure. The radius deduced from the data is found to be of the order 3.5[Formula: see text]fm.

Calculation of a complete set of spin observables for proton elastic scattering from stable and unstable nuclei

A microscopic study of proton elastic scattering from unstable nuclei at intermediate energies using a relativistic formalism is presented. We have employed both the original relativistic impulse approximation (IA1) and the generalized impulse approximation (IA2) formalisms to calculate the relativistic optical potentials, with target densities derived from relativistic mean field (RMF) theory using the NL3 and FSUGold parameter sets. Comparisons between the optical potentials computed using both IA1 and IA2 formalisms and the different RMF Lagrangians are presented for both stable and unstable targets. The comparisons are required to study the effect of using IA1 versus IA2 optical potentials, with different RMF parameter sets, on elastic scattering observables for unstable targets at intermediate energies. We also study the effect of full-folding form versus the factorized form of the optical potentials on elastic scattering observables. As with the case for stable nuclei, we found that the use of the full-folding optical potential improves the scattering observables (especially spin observables) at low intermediate energy (e.g., $200\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$). No discernible difference is found at a projectile incident energy of $500\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$. To check the validity of using localized optical potential, we calculate the scattering observables using nonlocal potentials by solving the momentum space Dirac equation. The Dirac equation is transformed to two coupled Lippmann-Schwinger equations, which are then numerically solved to obtain elastic scattering observables. The results are discussed and compared to calculations involving local coordinate-space optical potentials.

Structure of krypton isotopes within the interacting boson model derived from the Gogny energy density functional

The evolution and coexistence of the nuclear shapes as well as the corresponding low-lying collective states and electromagnetic transition rates are investigated along the Krypton isotopic chain within the framework of the interacting boson model (IBM). The IBM Hamiltonian is determined through mean-field calculations based on the several parametrizations of the Gogny energy density functional and the relativistic mean-field Lagrangian. The mean-field energy surfaces, as functions of the axial $\beta$ and triaxial $\gamma$ quadrupole deformations, are mapped onto the expectation value of the interacting-boson Hamiltonian that explicitly includes the particle-hole excitations. The resulting boson Hamiltonian is then used to compute low-energy excitation spectra as well as E2 and E0 transition probabilities for $^{70-100}$Kr. Our results point to a number of examples of the prolate-oblate shape transitions and coexistence both on the neutron-deficient and neutron-rich sides. A reasonable agreement with the available experimental data is obtained for the considered nuclear properties.

Relativistic self-consistent mean-field description of Sm isotopes

The evolution of the shape from the spherical to the axially deformed shapes of the neutron-rich, even-even $^{144\text{--}164}\mathrm{Sm}$ transitional nuclei is investigated. The investigations are performed with explicit density-dependent meson-nucleon and point-coupling models within the framework of the covariant density functional theory. A nonlinear meson-nucleon coupling model represented by the NL3* parametrization of the relativistic mean-field Lagrangian has also been used. The bulk and the microscopic properties of these nuclei have been investigated to analyze the phase-transition region and the critical-point behavior. The microscopic and self-consistent quadrupole deformation-constrained calculations show a clear shape change for even-even Sm isotopes with $N=82\ensuremath{-}102$. The potential energy surfaces for $^{148}\mathrm{Sm},^{150}\mathrm{Sm}$, and $^{152}\mathrm{Sm}$ obtained using different interactions are found to be relatively flat, which may be the possible critical-point nuclei. By examining the single-particle spectra, it is found that these nuclei distribute more uniformly as compared to other isotopes. Investigations also support the proposed shell-closure properties of $^{162}\mathrm{Sm}$. Overall good agreement is found within the different models used and between the calculated and experimental results wherever available.

Microscopic particle-rotor model for the low-lying spectrum ofΛhypernuclei

We propose a novel method for low-lying states of hypernuclei based on the particle-rotor model, in which hypernuclear states are constructed by coupling the hyperon to low-lying states of the core nucleus. In contrast to the conventional particle-rotor model, we employ a microscopic approach for the core states, that is, the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this microscopic particle-rotor model to $^9_\Lambda $Be employing a point-coupling version of the relativistic mean-field Lagrangian. A reasonable agreement with the experimental data for the low-spin spectrum is achieved using the $\Lambda N$ coupling strengths determined to reproduce the binding energy of the $\Lambda$ particle.

Deformed one-quasiparticle states in covariant density functional theory

Systematic investigation of the accuracy of the description of the energies of deformed one-quasiparticle states has been performed in covariant density functional theory in actinide and rare-earth mass regions. The sources of the discrepancies between theory and experiment are analyzed. Although some improvements in the description of ground state configurations and one-quasiparticle spectra can be achieved by better parametrization of the relativistic mean field Lagrangian, the analysis suggests that spectroscopic quality of their description can be achieved only in theoretical framework which takes into account particle–vibration coupling.

NUCLEAR EXCITATIONS AND WEAK INTERACTION RATES AT FINITE TEMPERATURE

Self-consistent theory frameworks have been introduced for description of excitations in nuclei at finite temperature, based on energy density functionals formulated using i) relativistic mean field Lagrangian with density dependent meson-nucleon vertex functions, and ii) Skyrme-type functionals. Finite temperature random phase approximation (FTRPA) has been employed in description of multipole excitations and charge-exchange modes. It is shown that in the temperature range T = 1–2 MeV additional transition strength appears at low energies due to thermal unblocking of single-particle orbitals close to the Fermi level. Within the finite temperature Hartree-Fock+RPA based on Skyrme functionals, the electron capture cross sections have been studied for target nuclei at temperatures relevant in modeling the supernova evolution.

Deformation of Λ hypernuclei

We study the deformation property of {lambda} hypernuclei using the relativistic mean-field (RMF) method. We find that {sub {lambda}}{sup 29}Si and {sub {lambda}}{sup 13}C hypernuclei have a spherical shape as a consequence of the additional {lambda} particle, whereas the corresponding core nuclei, {sup 28}Si and {sup 12}C, are oblately deformed. Most other hypernuclei have a similar deformation parameter to the core nucleus, in accordance with the previous study with the nonrelativistic Skyrme-Hartree-Fock method. We discuss the sensitivity of our results to the choice of pairing interaction and to the parameter set of the RMF Lagrangian.

Scalar–vector Lagrangian without nonlinear self-interactions of bosonic fields in the relativistic mean-field theory

A new Lagrangian model without nonlinear scalar self-interactions in the relativistic mean-field (RMF) theory is proposed. Introducing terms for scalar–vector interactions (SVI), we have developed a RMF Lagrangian model for finite nuclei and nuclear matter. It is shown that by inclusion of SVI in the basic RMF Lagrangian, the nonlinear σ3 and σ4 terms can be dispensed with. The SVI Lagrangian thus obtained provides a good description of ground-state properties of nuclei along the stability line as well as far away from it. This Lagrangian model is also able to describe experimental data on the breathing-mode giant monopole resonance energies well.

Erratum: Isospin dependence of incompressibility in relativistic and nonrelativistic mean field calculations [Phys. Rev. C 76, 034327 (2007)

The isospin dependence of incompressibility is investigated in the Skyrme Hartree-Fock (SHF) and relativistic mean field (RMF) models. The correlations between the nuclear matter incompressibility and the isospin dependent term of the finite nucleus incompressibility is elucidated by using the Thomas-Fermi approximation. The Coulomb term is also studied by using various different Skyrme Hamiltonians and RMF Lagrangians. The symmetry energy coefficient of incompressibility is extracted to be K_{\tau}=-(500\pm50) MeV from the recent experimental data of isoscalar giant monopole resonances (ISGMR) in Sn isotopes. Microscopic HF+random phase approximation (RPA) calculations are also performed with Skyrme interactions for ^{208}Pb and Sn isotopes to study the strength distributions of ISGMR. .

Neutron-skin thickness of finite nuclei in relativistic mean-field models with chiral limits

We study several structure properties of finite nuclei using relativistic mean-field Lagrangians constructed according to the Brown-Rho scaling due to the chiral symmetry restoration at high densities. The models are consistent with current experimental constraints for the equations of state of symmetric matter at both normal and supranormal densities and of asymmetric matter at subsaturation densities. It is shown that these models can successfully describe the binding energies and charge radii of finite nuclei. Compared to calculations with usual relativistic mean-field models, these models give a reduced thickness of neutron skin in $^{208}\mathrm{Pb}$ between 0.17 fm and 0.21 fm. The reduction of the predicted neutron skin thickness is found to be due to not only the softening of the symmetry energy but also the scaling property of $\ensuremath{\rho}$ meson required by the partial restoration of chiral symmetry.

Isospin dependence of incompressibility in relativistic and nonrelativistic mean field calculations

The isospin dependence of incompressibility is investigated in Skyrme Hartree-Fock (SHF) and relativistic mean field (RMF) models. The correlations between the nuclear matter incompressibility and the isospin-dependent term of the finite nucleus incompressibility is elucidated using the Thomas-Fermi approximation. The Coulomb term is also studied by using various different Skyrme Hamiltonians and RMF Lagrangians. Microscopic $\mathrm{HF}+\mathrm{random}$ phase approximation (RPA) calculations are performed with Skyrme interactions for $^{208}\mathrm{Pb}$ and Sn isotopes to study the strength distributions of isoscalar giant monopole resonances. The symmetry term of nuclear incompressibility is extracted to be ${K}_{\ensuremath{\tau}}=\ensuremath{-}(500\ifmmode\pm\else\textpm\fi{}50)$ MeV from the recent experimental data of isoscalar giant monopole resonances (ISGMR) in Sn isotopes.

Shell structure and ρ-tensor correlations in density dependent relativistic Hartree-Fock theory

A new effective interaction PKA1 with \ensuremath{\rho}-tensor couplings for the density dependent relativistic Hartree-Fock (DDRHF) theory is presented. It is obtained by fitting selected empirical ground state and shell structure properties. It provides satisfactory descriptions of nuclear matter and the ground state properties of finite nuclei at the same quantitative level as recent DDRHF and relativistic mean field (RMF) models. Significant improvement in the single-particle spectra is also found due to the inclusion of \ensuremath{\rho}-tensor couplings. As a result, PKA1 cures a common disease of the existing DDRHF and RMF Lagrangians, namely, the artificial shells at 58 and 92, and recovers the realistic subshell closure at 64. Moreover, the proper spin-orbit splittings and well-conserved pseudospin symmetry are obtained with the new effective interaction PKA1. Due to the extra binding introduced by the \ensuremath{\rho}-tensor correlations, the balance between the nuclear attractions and the repulsions is changed, and this constitutes the physical reason for the improvement of the nuclear shell structure.

Equation of state of isospin-asymmetric nuclear matter in relativistic mean-field models with chiral limits

Using in-medium hadron properties according to the Brown–Rho scaling due to the chiral symmetry restoration at high densities and considering naturalness of the coupling constants, we have newly constructed several relativistic mean-field Lagrangians with chiral limits. The model parameters are adjusted such that the symmetric part of the resulting equation of state at supra-normal densities is consistent with that required by the collective flow data from high energy heavy-ion reactions, while the resulting density dependence of the symmetry energy at sub-saturation densities agrees with that extracted from the recent isospin diffusion data from intermediate energy heavy-ion reactions. The resulting equations of state have the special feature of being soft at intermediate densities but stiff at high densities naturally. With these constrained equations of state, it is found that the radius of a 1.4 M ⊙ canonical neutron star is in the range of 11.9 km ⩽ R ⩽ 13.1 km , and the maximum neutron star mass is around 2.0 M ⊙ close to the recent observations.

Proton and neutron skins of light nuclei within the relativistic mean field theory

The relativistic mean field (RMF) theory is applied to the analysis of ground-state properties of Ne, Na, Cl and Ar isotopes. In particular, we study the recently established proton skin in Ar isotopes and neutron skin in Na isotopes as a function of the difference between the proton and the neutron separation energy. We use the TMA effective interaction in the RMF Lagrangian, and describe pairing correlation by the density-independent delta-function interaction. We calculate single neutron and proton separation energies, quadrupole deformations, nuclear matter radii and differences between proton radii and neutron radii, and compare these results with the recent experimental data.

Self-consistent mean-field models for nuclear structure

The authors review the present status of self-consistent mean-field (SCMF) models for describing nuclear structure and low-energy dynamics. These models are presented as effective energy-density functionals. The three most widely used variants of SCMF's based on a Skyrme energy functional, a Gogny force, and a relativistic mean-field Lagrangian are considered side by side. The crucial role of the treatment of pairing correlations is pointed out in each case. The authors discuss other related nuclear structure models and present several extensions beyond the mean-field model which are currently used. Phenomenological adjustment of the model parameters is discussed in detail. The performance quality of the SCMF model is demonstrated for a broad range of typical applications.

Relativistic mean field approach with density dependent couplings for finite nuclei

We study finite nuclei within the relativistic mean field approach with density dependent couplings (RMFD) based on the relativistic Brueckner-Hartree-Fock (RBHF) results on nuclear matter. We take the linear relativistic mean-field Lagrangian with mesons \ensuremath{\sigma}, \ensuremath{\omega}, \ensuremath{\delta}, and \ensuremath{\rho}, whose coupling constants with nucleons are determined so as to reproduce the RBHF results of nuclear matter at various densities and proton fractions. We apply the RMFD approach to various nuclei with spherical shape including unstable ones in the periodic table. We find satisfactory results on the nuclear properties. We emphasize here that the proton and neutron effective masses are largely different from each other as the proton fraction is decreased from ${Y}_{p}=0.5$, which forces us to include the isovector scalar meson \ensuremath{\delta} in our approach.