Skyrme Hartree-Fock Calculations Research Articles

Relativistic correction of the Coulomb interaction in the local density approximation for energies and radii in doubly-magic nuclei

Abstract Effects of the relativistic correction of the Coulomb interaction on doubly-magic nuclei are discussed with Skyrme Hartree–Fock calculations. The relativistic correction is treated by using the local density approximation. It is found that the correction to the total energy is about $2.4 \, \mathrm{MeV}$ for ${}^{208}_{}\mathrm{Pb}$, while the proton and neutron radii do not change significantly. This difference is larger than the difference of the Coulomb exchange (Fock) energy calculated with the local density (Hartree–Fock–Slater) approximation and that with the exact treatment and the neutron finite-size effect. Effects of the correction are also compared to the correction due to the vacuum polarization. It is shown that the two contributions to the total energy are comparable in light nuclei, but the latter dominates in heavy nuclei, while the contribution of the relativistic correction to the total energy is non-negligible compared to the target accuracy of the DFT calculation.

Low-energy Li7(n,γ)Li8 and Be7(p,γ)B8 radiative capture reactions within the Skyrme Hartree-Fock approach

The electromagnetic dipole transitions in $^{7}\mathrm{Be}(p,\ensuremath{\gamma})^{8}\mathrm{B}$ and $^{7}\mathrm{Li}(n,\ensuremath{\gamma})^{8}\mathrm{Li}$ reactions at the keV-energy region were analyzed simultaneously within the Skyrme Hartree-Fock potential model. The Skyrme Hartree-Fock calculation is adopted as a microscopic approach to obtain consistently the single-particle bound and scattering states in the calculation of the radial overlap function within the potential model. All nonresonant and resonant electromagnetic dipole transitions are taken into account. The electric dipole transitions are successfully described with the slightest adjustment. The resonant magnetic dipole transitions at 633 keV and 2184 keV of the $^{7}\mathrm{Be}(p,\ensuremath{\gamma})^{8}\mathrm{B}$ reaction, and the one at 222 keV of $^{7}\mathrm{Li}(n,\ensuremath{\gamma})^{8}\mathrm{Li}$ are also analyzed. The astrophysical ${\mathcal{S}}_{17}(0)$ factor of the $^{7}\mathrm{Be}(p,\ensuremath{\gamma})^{8}\mathrm{B}$ reaction is found to be 22.3 eV b.

Deformation effect on nuclear density profile and radius enhancement in light- and medium-mass neutron-rich nuclei

Abstract The mass number dependence of nuclear radii is closely related to the properties of nuclear matter. It is known that most nuclei exhibit some deformation. We discuss how the nuclear density profile is modified by nuclear deformation to elucidate the enhancement mechanism of nuclear radii through a systematic investigation of neutron-rich Ne, Mg, Si, S, Ar, Ti, Cr, and Fe isotopes. Skyrme–Hartree–Fock calculations are performed in a 3D Cartesian grid to describe the nuclear deformation in a non-empirical way. The role of nuclear deformation in nuclear density profiles is explored in comparison to calculations with a spherical limit. We find correlations between nuclear deformation and the internal nuclear density. The evolution of nuclear radii appears to follow the core swelling mechanism recently proposed in spherical nuclei [W. Horiuchi and T. Inakura, Phys. Rev. C 101, 061301(R) (2020)], and the radius is further enhanced by nuclear deformation. This study demands further theoretical and experimental investigations for the internal density.

Study of the Static and Dynamic Nuclear Properties and Form Factors for Some Magnesium Isotopes 29-34 Mg

Nuclear structure of 29-34Mg isotopes toward neutron dripline have been investigated using shell model with Skyrme-Hartree–Fock calculations. In particular nuclear densities for proton, neutron, mass and charge densities with their corresponding rms radii, neutron skin thicknesses and inelastic electron scattering form factors are calculated for positive low-lying states. The deduced results are discussed for the transverse form factor and compared with the available experimental data. It has been confirmed that the combining shell model with Hartree-Fock mean field method with Skyrme interaction can accommodate very well the nuclear excitation properties and can reach a highly descriptive and predictive power when investigating different nuclear configurations of stable and unstable nuclei.

Bound-to-continuum potential model for the ( p,γ ) reactions of the CNO nucleosynthesis cycle

The study of CNO cycle involves the examination of the proton radiative capture, or the $(p,\gamma)$ reactions below 2 MeV. The astrophysical $\mathcal{S}$ factor characterizing the $(p,\gamma)$ reaction is usually reduced to the electric dipole transition $E1$ from the scattering state to the bound state. In this work, the partial scattering and the single-particle bound wave functions in the reduced matrix element of the transition are obtained from the single self-consistent mean-field potential deduced from the Skyrme Hartree-Fock calculation. The astrophysical $\mathcal{S}$ factors of the $(p,\gamma)$ reactions in the CNO cycle were successfully reproduced. The self-consistent Hartree-Fock calculation from the discrete to the continuum is a promising approach for the microscopic analysis of the nucleon-induced reactions in nuclear astrophysics.

Microscopic description of the fusion excitation function of 32,36S+90,94,96Zr

The fusion excitation function for the systems [Formula: see text]S+[Formula: see text]Zr is investigated using a microscopic internuclear potential derived from Skyrme energy density functional. The inputs in this approach are the proton and neutron density distributions of the interacting nuclei, which are derived from Skyrme–Hartree–Fock calculations. The SkM[Formula: see text] interaction is used in the calculation of the nuclear densities as well as the internuclear potential. The coupling to low lying inelastic excited states of target and projectile is considered. The role of the neutron transfer is discussed, where it is considered through the CCFULL model calculation. A good agreement with the experimental data is obtained without adjustable parameters.

Effects of finite nucleon size, vacuum polarization, and electromagnetic spin-orbit interaction on nuclear binding energies and radii in spherical nuclei

The electromagnetic effects of the finite size of the nucleon are implemented self-consistently on top of the Skyrme Hartree-Fock calculation, where the electric form factors of both protons and neutrons are considered. Furthermore, the vacuum polarization and the electromagnetic spin-orbit interaction are taken into account. The self-consistent finite-size effects give a different Coulomb potential from the conventional one and affect the neutrons as well. The contribution of the finite-size effects to the total energy reaches $ 7 \, \mathrm{MeV} $ in $ {}^{208} \mathrm{Pb} $. The vacuum polarization and the electromagnetic spin-orbit interaction are also non-negligible, especially in the heavy nuclei. These effects provide a comparable contribution of the total energy to that of the isospin symmetry-breaking terms of the nuclear interaction. The mirror nuclei mass difference in $ {}^{48} \mathrm{Ca} $--$ {}^{48} \mathrm{Ni} $ is also studied, and its value is improved by approximately one order of magnitude.

Self-consistent random-phase approximation based on the relativistic Hartree-Fock theory: Role of ρ -tensor coupling

The framework of the random-phase approximation (RPA) based on the relativistic Hartree-Fock (RHF) theory is extended to achieve a self-consistent calculation with the $\ensuremath{\rho}$-meson tensor coupling. The model self-consistency is verified by the check of the isobaric analog state, and it is found that the $\ensuremath{\rho}$-tensor and $\ensuremath{\rho}$-vector-tensor couplings play significant roles in maintaining the self-consistency. Using the RHF Lagrangian PKA1, the properties of the Gamow-Teller resonances (GTR) are investigated, in which the roles played by the particle-hole residual interaction of various meson-nucleon couplings are clarified in details. Furthermore, the effects of the tensor force, which is introduced naturally via the Fock terms, are analyzed by comparing the calculations with full Lagrangians and the ones artificially dropping the tensor force components. It is found that for the RHF Lagrangians $\mathrm{PKO}i$ $(i=1,2,3)$ and PKA1, the tensor forces play the role mainly via the RHF mean field rather than via the RPA residual interaction in determining the GTR. Moreover, the tensor-force effects are not as strong as those indicated by the Skyrme Hartree-Fock calculations.

Spin-orbit interaction near the neutron drip line and shell effects at N = 82 near the r-process path

We have studied the behavior of charge and neutron radii across the shell-closure at N = 82 near the neutron dripline in a comparative approach which includes Relativistic Mean Field (RMF) Theory and the nonrelativistic Skyrme-Hartree-Fock (SHF) Theory. We have investigated the extent and the effects of the spin-orbit interaction on the ground state properties of neutron-rich nuclei in the extreme region. RMF and SHF calculations have been performed for nuclei in the isotopic chains from Kr (Z = 36) to Sn (Z = 50) near the neutron dripline across the magic number N = 82. We present the results of shell-effects on charge radii, neutron radii, and the neutron skin thickness. We have examined the single particle energies (SPE) in the context of the kinks in the radii. It is shown that RMF theory produces a kink in charge radii, similar to what was seen experimentally for Pb (Z = 82) isotopic chains. This is in contrast to the Skyrme forces, which do not produce such a kink.

Macroscopic and Skyrme–Hartree–Fock test of E(5) symmetry in xenon isotopes

Algebraic collective model calculations for even–even xenon isotopes 128,130Xe have been performed to describe collective properties of these two isotopes, considered to be candidates for an E(5) transition from spherical to γ-soft, O(6)-like nuclei. The shape transitional point and the E(5) symmetry are characterised by large fluctuations in the triaxiality parameter γ reflected by a very flat potential in the γ direction resulting in two symmetric prolate and oblate minima in the potential energy curves. This has been studied using the Skyrme–Hartree–Fock calculations with different parametrisations of the Skyrme interaction.

Continuum damping effects in nuclear collisions associated with twisted boundary conditions

The time-dependent Skyrme Hartree-Fock calculations have been performed to study $^{24}$Mg +$^{24}$Mg collisions. The twisted boundary conditions, which can avoid finite box-size effects of the employed 3D coordinate space, have been implemented. The prolate deformed $^{24}$Mg has been set to different orientations to study vibrations and rotations of the compound nucleus $^{48}$Cr. Our time evolution results show continuum damping effects associated with the twist-averaged boundary condition play a persistent role after the fusion stage. In particular, a rotational damping in continuum is presented in calculations of both twist-averaged and absorbing boundary conditions, in which damping widths can be clearly extracted. It is unusual that the rotating compound nucleus in continuum evolves towards spherical but still has a considerable angular momentum.

Coulomb exchange functional with generalized gradient approximation for self-consistent Skyrme Hartree-Fock calculations

We perform the self-consistent Skyrme Hartree-Fock calculation with the Coulomb exchange functional in the form of generalized gradient approximation (GGA). It is found that the Perdew-Burke-Ernzerhof GGA (PBE-GGA) Coulomb exchange functional is able to reproduce the exact-Fock energy for the nuclei in a large region of the nuclear chart with one adjustable parameter. The remaining error of Coulomb exchange energy by the GGA with respect to the exact-Fock energy dominantly comes from the functional-driven error.

β -decay properties of neutron-rich Ca, Ti, and Cr isotopes

Beta-decay properties of neutron-rich Ca, Ti, and Cr isotopes are studied within a deformed proton-neutron quasiparticle random-phase approximation. The underlying mean field is described self-consistently from deformed Skyrme Hartree-Fock calculations with pairing correlations. Residual spin-isospin interactions in the particle-hole and particle-particle channels are also included in the formalism. The energy distributions of the Gamow-Teller strength, the beta-decay feedings, the beta-decay half-lives, and the beta-delayed neutron emission probabilities are discussed and compared with other theoretical results, as well as with the available experimental information. The evolution of these nuclear beta-decay properties is investigated in isotopic chains in a search for structural changes. A reliable estimate of the beta-decay properties in this mass region is a valuable information for evaluating decay rates in astrophysical scenarios.

Symmetry energy III: Isovector skins

Isoscalar density is a sum of neutron and proton densities and isovector is a normalized difference. Here, we report the experimental evidence for the displacement of the isovector and isoscalar surfaces in nuclei, by ∼0.9 fm from each other. We analyze data on quasielastic (QE) charge exchange (p,n) reactions, concurrently with proton and neutron elastic scattering data for the same target nuclei, following the concepts of the isoscalar and isovector potentials combined into Lane optical potential. The elastic data largely probe the geometry of the isoscalar potential and the (p,n) data largely probe a relation between the geometries of the isovector and isoscalar potentials. The targets include 48Ca, 90Zr, 120Sn and 208Pb and projectile incident energy values span the range of (10–50) MeV. In our fit to elastic and QE charge-exchange data, we allow the values of isoscalar and isovector radii, diffusivities and overall potential normalizations to float away from those in the popular Koning and Delaroche parametrization. We find that the best-fit isovector radii are consistently larger than isoscalar and the best-fit isovector surfaces are steeper. Upon identifying the displacement of the potential surfaces with the displacement of the surfaces for the densities in the Skyrme–Hartree–Fock calculations, and by supplementing the results with those from analyzing excitation energies to isobaric analog states in the past, we arrive at the slope and value of the symmetry energy at normal density of 70<L<101 MeV and 33.5<aaV<36.4 MeV, respectively.

β-decay properties of neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes from deformed quasiparticle random-phase approximation

Beta-decay properties of even and odd-A neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes involved in the astrophysical rapid neutron capture process are studied within a deformed proton-neutron quasiparticle random-phase approximation. The underlying mean field is described self-consistently from deformed Skyrme Hartree-Fock calculations with pairing correlations. Residual interactions in the particle-hole and particle-particle channels are also included in the formalism. The isotopic evolution of the various nuclear equilibrium shapes and the corresponding charge radii are investigated in all the isotopic chains. The energy distributions of the Gamow-Teller strength as well as the beta-decay half-lives are discussed and compared with the available experimental information. It is shown that nuclear deformation plays a significant role in the description of the decay properties in this mass region. Reliable predictions of the strength distributions are essential to evaluate decay rates in astrophysical scenarios.

Uses of tensor-induced spin–orbit splitting to locate spherical doubly-magic SHE

Using optimized iso-scalar and iso-vector tensor coupling constants in Skyrme Hartree–Fock calculation which reproduces the splitting of spin–orbit partners 1f7/2–1f5/2 in 40,48Ca and 56Ni, the evolution of shells in the nuclear chart have been studied through developments in spin–orbit splitting. Signatures for all the magic numbers have been generated without any exception. The procedure when extended to superheavy region indicates the element with Z = 114 and N = 184 to be a doubly-magic spherical nucleus.

Gamow-Teller response in deformed even and odd neutron-rich Zr and Mo isotopes

Beta-decay properties of neutron-rich Zr and Mo isotopes are investigated within a microscopic theoretical approach based on the proton-neutron quasiparticle random-phase approximation. The underlying mean field is described self-consistently from deformed Skyrme Hartree-Fock calculations with pairing correlations. Residual separable particle-hole and particle-particle forces are also included in the formalism. The structural evolution in these isotopic chains including both even and odd isotopes is analyzed in terms of the equilibrium deformed shapes. Gamow-Teller strength distributions, beta-decay half-lives, and beta-delayed neutron-emission probabilities are studied, stressing their relevance to describe the path of the nucleosynthesis rapid neutron capture process.

A study of Λ hypernuclei within the Skyrme–Hartree–Fock model

We investigate the properties of the single Lambda hypernuclei within a Skyrme-Hartree-Fock (SHF) model. The parameters of the Skyrme type effective lambda-nucleon (Lambda N) interaction are obtained by fitting to the experimental Lambda binding energies of hypernuclei with masses spanning a wide range of the periodic table. Alternative parameter sets are also obtained by omitting nuclei below mass number 16 from the fitting procedure. The SHF calculations are performed for the binding energies of the Lambda single-particle states over a full mass range using the best fit parameter sets obtained in these fitting procedures and the results are compared with the available experimental data. The data show some sensitivity to the parameter sets obtained with or without including the nuclei below mass 16. The radii of the Lambda orbits in the hypernuclear ground states and the Lambda effective mass in nuclear matter show some dependence on different parameter sets. We present results for the total binding energy per baryon of the hypernuclei over a large mass region to elucidate their stability as a function of the baryon number. We have also employed the our best fit Lambda N parameter sets to investigate the role of hyperons in some key properties of neutron stars.

Neutron-proton radii inN≈Znuclei

A simple formula is derived that describes how the Coulomb interaction affects the proton radius in nuclei. It determines the difference between neutron and proton radii in nuclei with N approx Z. It also provides an estimate for the difference between the radii of the Z core neutrons and the protons in nuclei with a large neutron excess. The results obtained from the derived formula are compared with radii calculated in a Skyrme Hartree-Fock calculation.

Symmetry energy I: Semi-infinite matter

Energy for a nucleus is considered in the macroscopic limit, in terms of nucleon numbers. Further considered for a nuclear system is the Hohenberg–Kohn energy functional, in terms of proton and neutron densities. Finally, Skyrme–Hartree–Fock calculations are carried out for a half-infinite particle-stable nuclear-matter. In each case, the attention is focused on the role of neutron–proton asymmetry and on the nuclear symmetry energy. We extend the considerations on the symmetry term from an energy formula to the respective term within the Hohenberg–Kohn functional. We show, in particular, that in the limit of an analytic functional, and subject to possible Coulomb corrections, it is possible to construct isoscalar and isovector densities out of the proton and neutron densities, that retain a universal relation to each other, approximately independent of asymmetry. In the so-called local approximation, the isovector density is inversely proportional to the symmetry energy in uniform matter, at the local isoscalar density. Generalized symmetry coefficient of a nuclear system is related, in the analytic limit of the functional, to an integral of the isovector density. We test the relations, inferred from the Hohenberg–Kohn functional, in the Skyrme–Hartree–Fock calculations of half-infinite matter. Within the calculations, we obtain surface symmetry coefficients and parameters characterizing the densities, for the majority of Skyrme parameterizations proposed in the literature. The volume-to-surface symmetry-coefficient ratio, and the displacement of nuclear isovector relative to isoscalar surfaces, both strongly increase as the slope of symmetry energy, in the vicinity of normal density, increases.