Density-dependent Pairing Interaction Research Articles

Isovector giant dipole resonances in proton-rich Ar and Ca isotopes * *This work is partly Supported by the National Natural Science Foundation of China (11975096, 11775014) and the Fundamental Research Funds for the Central Universities (2020NTST06)

The isovector giant dipole resonances (IVGDR) in proton-rich Ar and Ca isotopes have been systematically investigated using the resonant continuum Hartree-Fock+BCS (HF+BCS) and quasiparticle random phase approximation (QRPA) methods. The Skyrme SLy5 and density-dependent contact pairing interactions are employed in the calculations. In addition to the giant dipole resonances at energy around 18 MeV, pygmy dipole resonances (PDR) are found to be located in the energy region below 12 MeV. The calculated energy-weighted moments of PDR in nuclei close to the proton drip-line exhaust about 4% of the TRK sum rule. The strengths decrease with increasing mass number in each isotopic chain. The transition densities of the PDR states show that motions of protons and neutrons are in phase in the interiors of nuclei, while the protons give the main contribution at the surface. By analyzing the QRPA amplitudes of proton and neutron 2-quasiparticle configurations for a given low-lying state, we find that only a few proton configurations give significant contributions. They contribute about 95% to the total QRPA amplitudes, which indicates that the collectivity of PDR states is not strong in proton-rich nuclei in the present study.

Proximity effect of pair correlation in the inner crust of neutron stars

Abstract We study the proximity effect of pair correlation in the inner crust of neutron stars by means of the Skyrme–Hartree–Fock–Bogoliubov theory formulated in coordinate space. We describe a system composed of a nuclear cluster immersed in neutron superfluid confined in a spherical box. Using a density-dependent effective pairing interaction that reproduces both the pair gap of neutron matter obtained in ab initio calculations and that of finite nuclei, we analyze how the pair condensate in a neutron superfluid is affected by the presence of the nuclear cluster. It is found that the proximity effect is characterized by the coherence length of the neutron superfluid measured from the edge position of the nuclear cluster. The calculation predicts that the proximity effect has a strong density dependence. In the middle layers of the inner crust with baryon density $5 \times 10^{-4}$ fm$^{-3} \mathop < \limits_ \sim \ \rho_b \mathop < \limits_ \sim \ 2\times 10^{-2}$ fm$^{-3}$, the proximity effect is strongly limited in the vicinity of the nuclear cluster, i.e., in a sufficiently smaller area than the Wigner–Seitz cell. In contrast, the proximity effect is predicted to extend to the whole volume of the Wigner–Seitz cell in shallow layers of the inner crust with $\rho_b \mathop < \limits_ \sim \ 2 \times 10^{-4}$ fm$^{-3}$, and in deep layers with $\rho_b \mathop > \limits_ \sim \ 5 \times 10^{-2}$ fm$^{-3}$.

Finite-range separable pairing interaction in Cartesian coordinates

Within a simple SO(8) algebraic model, the coexistence between isoscalar and isovector pairing modes can be successfully described using a mean-field method plus restoration of broken symmetries. In order to port this methodology to real nuclei, we need to employ realistic density functionals in the pairing channel. In this article, we present an analytical derivation of matrix elements of a separable pairing interaction in Cartesian coordinates and we correct errors of derivations available in the literature. After implementing this interaction in the code HFODD, we study evolution of pairing gaps in the chain of deformed Erbium isotopes, and we compare the results with a standard density-dependent contact pairing interaction.

Effects of deformation, pairing and tensor correlation on the evolution of bubble structure within the Skyrme-Hartree-Fock method

We study the effects of deformation, tensor and pairing correlation on the evolution of bubble structure in 46Ar, 22O and Cl isotopes using a deformed Skyrme-Hartree-Fock model with BCS and Lipkin-Nogami number projection approximation. The parameter SLy5 of the Skyrme energy functionals is adopted together with various kinds of density-dependent pairing interaction and various kinds of tensor interaction. In the spherical case, considering pairing correlation (IS + IV) or switch off pairing, there is a clear bubble structure in 46Ar with interaction SLy5 + Tw, respectively. For 46Ar, we find that deformation weakens the bubble structure strongly, and pairing correlation also weakens the bubble in 46Ar and 22O , respectively. On the other hand, the SLy5 + Tw + IS + IV interaction evolves the bubble structure of neutron-rich Cl isotopes 43-47Cl in deformed cases because the occupation probability of states $ 2s_{1/2}$ decreases from 1 to 0.5.

Self-consistent description of the SHFB equations for 112Sn

Abstract The Hartree-Fock (HF) method is an excellent approximation of the closed shell magic nuclei. Pair correlation is essential for the description of open shell nuclei and has been derived for even-even, odd-odd and even-odd nuclei. These effects are reported by Hartree-Fock with BCS (HFBCS) or Hartree-Fock-Bogolyubov (HFB). These issues have been investigated, especially in the nuclear charts, and such studies have been compared with the observed information. We compute observations such as total binding energy, charge radius, densities, separation energies, pairing gaps and potential energy surfaces for neutrons and protons, and compare them with experimental data and the result of the spherical codes. In spherical even-even neutron-rich nuclei are considered in the Skyrme-Hartree-Fock-Bogolyubov (SHFB) method with density-dependent pairing interaction. Zero-range density-dependent interactions is used in the pairing channel. We solve SHF or SHFB equations in the spatial coordinates with spherical symmetry for tin isotopes such as 112Sn. The numerical accuracy of solving equations in the coordinate space is much greater than the fundamental extensions, which yields almost precise results.

Proton-neutron structure of first and second quadrupole excitations of 90Sr

Starting from the Skyrme interaction $f_{-}$ together with the density-dependent pairing interaction, we study the g factors for the $2_{1,2}^{+}$ excitations of 88, 90Sr and 90, 92Zr. The coupling between one- and two-phonon terms in the wave functions of excited states is taken into account within the finite-rank separable approximation. Using the same set of parameters we describe available experimental data and give the prediction for 90Sr, $g(2_{2}^{+})=+0.03$ in comparison to $ +0.31$ in the case of 92Zr.

Pairing-induced speedup of nuclear spontaneous fission

Background: Collective inertia is strongly influenced at the level crossing at which the quantum system changes its microscopic configuration diabatically. Pairing correlations tend to make the large-amplitude nuclear collective motion more adiabatic by reducing the effect of these configuration changes. Competition between pairing and level crossing is thus expected to have a profound impact on spontaneous fission lifetimes.Purpose: To elucidate the role of nucleonic pairing on spontaneous fission, we study the dynamic fission trajectories of $^{264}\mathrm{Fm}$ and $^{240}\mathrm{Pu}$ using the state-of-the-art self-consistent framework.Methods: We employ the superfluid nuclear density functional theory with the Skyrme energy density functional ${\mathrm{SkM}}^{*}$ and a density-dependent pairing interaction. Along with shape variables, proton and neutron pairing correlations are taken as collective coordinates. The collective inertia tensor is calculated within the nonperturbative cranking approximation. The fission paths are obtained by using the least action principle in a four-dimensional collective space of shape and pairing coordinates.Results: Pairing correlations are enhanced along the minimum-action fission path. For the symmetric fission of $^{264}\mathrm{Fm}$, where the effect of triaxiality on the fission barrier is large, the geometry of the fission pathway in the space of the shape degrees of freedom is weakly impacted by pairing. This is not the case for $^{240}\mathrm{Pu}$, where pairing fluctuations restore the axial symmetry of the dynamic fission trajectory.Conclusions: The minimum-action fission path is strongly impacted by nucleonic pairing. In some cases, the dynamical coupling between shape and pairing degrees of freedom can lead to a dramatic departure from the static picture. Consequently, in the dynamical description of nuclear fission, particle-particle correlations should be considered on the same footing as those associated with shape degrees of freedom.

Excitation-energy dependence of fission in the mercury region

Background: Recent experiments on beta-delayed fission reported an asymmetric mass yield in the neutron-deficient nucleus 180Hg. Earlier experiments in the mass region A=190-200 close to the beta-stability line, using the (p,f) and (\alpha,f) reactions, observed a more symmetric distribution of fission fragments. While the beta-delayed fission of 180Hg can be associated with relatively low excitation energy, this is not the case for light-ion reactions, which result in warm compound nuclei. Purpose: To elucidate the roles of proton and neutron numbers and excitation energy in determining symmetric and asymmetric fission yields, we compute and analyze the isentropic potential energy surfaces of 174,180,198Hg and 196,210Po. Methods: We use the finite-temperature superfluid nuclear density functional theory, for excitation energies up to E*=30MeV and zero angular momentum. For our theoretical framework, we consider the Skyrme energy density functional SkM* and a density-dependent pairing interaction. Results: For 174,180Hg, we predict fission pathways consistent with asymmetric fission at low excitation energies, with the symmetric fission pathway opening very gradually as excitation energy is increased. For 198Hg and 196Po, we expect the nearly-symmetric fission channel to dominate. 210Po shows a preference for a slightly asymmetric pathway at low energies, and a preference for a symmetric pathway at high energies. Conclusions: Our self-consistent theory suggests that excitation energy weakly affects the fission pattern of the nuclei considered. The transition from the asymmetric fission in the proton-rich nuclei to a more symmetric fission in the heavier isotopes is governed by the shell structure of pre-scission configurations.

Spontaneous fission lifetimes from the minimization of self-consistent collective action

The spontaneous fission lifetime of ${}^{264}$Fm has been studied within nuclear density functional theory by minimizing the collective action integral for fission in a two-dimensional quadrupole collective space representing elongation and triaxiality. The collective potential and inertia tensor are obtained self-consistently using the Skyrme energy density functional and density-dependent pairing interaction. The resulting spontaneous fission lifetimes are compared with the static result obtained with the minimum-energy pathway. We show that fission pathways strongly depend on assumptions underlying collective inertia. With the nonperturbative mass parameters, the dynamic fission pathway becomes strongly triaxial and it approaches the static fission valley. On the other hand, when the standard perturbative cranking inertia tensor is used, axial symmetry is restored along the path to fission; an effect that is an artifact of the approximation used.

Pairing correlations in exotic nuclei

The BCS and HFB theories which can accommodate the pairing correlations in the ground states of atomic nuclei are presented. As an application of the pairing theories, we investigate the spatial extension of weakly bound Ne and C isotopes by taking into account the pairing correlation with the Hartree-Fock-Bogoliubov (HFB) method and a 3-body model, respectively. We show that the odd-even staggering in the reaction cross sections of 30,31,32Ne and 14,15,16C are successfully reproduced, and thus the staggering can be attributed to the unique role of pairing correlations in nuclei far from the stability line. A correlation between a one-neutron separation energy and the anti-halo effect is demonstrated for s- and p-waves using the HFB wave functions. We also propose effective density-dependent pairing interactions which reproduce both the neutron-neutron (nn) scattering length at zero density and the neutron pairing gap in uniform matter. Then, we apply these interactions to study pairing gaps in semi-magic finite nuclei, such as Ca, Ni, Sn and Pb isotopic chains.

Gamow-Teller resonances and a separable approximation for Skyrme tensor interactions

A finite rank separable approximation for the quasiparticle random phase approximation (QRPA) with Skyrme interactions is applied to study properties of the Gamow-Teller (GT) resonances in the neutron-rich Cd isotopes. This approximation enables one to reduce considerably the dimension of matrix that must be diagonalized to perform QRPA calculations in a very large configuration space. Our results from the SGII Skyrme interaction with the tensor interactions and the density-dependent zero-range pairing interaction show that the GT distribution is noticeably modified when the tensor correlations are taken into account. In particular, for 130Cd the dominant peak is moved 3.6 MeV downward and 10% of the GT distribution is shifted to the high excitation energy region near E=50MeV.

Probing the pairing interaction through two-neutron transfer reactions

The treatment of the pairing interaction in mean-field-based models is addressed. In particular, the possibility to use pair transfers as a tool to better constrain this interaction is discussed. First, pairing inter- actions with various density dependencies (surface/volume mixing) are used in the microscopic Hartree-Fock- Bogoliubov + quasiparticle random-phase approximation model to generate the form factors to be used in reac- tion calculations. Cross sections for (p,t) two-neutron transfer reactions are calculated in the one-step zero-range distorted-wave Born approximation for some Tin isotopes and for incident proton energies from 15 to 35 MeV. Three different surface/volume mixings of a zero-range density-dependent pairing interaction are employed in the microscopic calculations and the sensitivity of the cross sections to the different mixings is analyzed. Differences among the three different theoretical predictions are found espacially for the nucleus 136 Sn and they are more important at the incident proton energy of 15 MeV. We thus indicate (p,t) two-neutron transfer reactions with very neutron-rich Sn isotopes and at proton energies around 15 MeV as good experimental cases where the sur- face/volume mixing of the pairing interaction may be probed. In the second part of the manuscript, ground-state to ground-state transitions are investigated. Approximations made to estimate two-nucleon transfer probabilities in ground-state to ground-state transitions and the physical interpretation of these probabilities are discussed. Probabilities are often calculated by approximating both ground states of the initial nucleus A and of the final nucleus A ±2 by the same quasiparticle vacuum. We analyze two improvements of this approach. First, the effect of using two different ground states with average numbers of particles A and A ±2 is quantified. Second, by using projection techniques, the role of particle number restoration is analyzed. Our analysis shows that the improved treatment plays a role close to magicity, leading to an enhancement of the pair-transfer probability. In midshell regions, part of the error made by approximating the initial and final ground states by a single vacuum is com- pensated by projecting onto a good particle number. Surface effects are analyzed by using pairing interactions with a different volume/surface mixing. Finally, a simple expression of the pair-transfer probability is given in terms of occupation probabilities in the canonical basis. We show that, in the canonical basis formulation, surface effects that are visible in the transfer probability are related to the fragmentation of single-particle occupancies close to the Fermi energy. This provides a complementary interpretation with respect to the standard quasiparti- cle representation where surface effects are generated by the integrated radial profiles of the contributing wave functions.

Probing the pairing interaction through two-neutron transfer reactions

Cross sections for ($p,t$) two-neutron transfer reactions are calculated in the one-step zero-range distorted-wave Born approximation for the tin isotopes $^{124}$Sn and $^{136}$Sn and for incident proton energies from 15 to 35 MeV. Microscopic quasiparticle random-phase approximation form factors are provided for the reaction calculation and phenomenological optical potentials are used in both the entrance and the exit channels. Three different surface/volume mixings of a zero-range density-dependent pairing interaction are employed in the microscopic calculations and the sensitivity of the cross sections to the different mixings is analyzed. Since absolute cross sections cannot be obtained within our model, we compare the positions of the diffraction minima and the shapes of the angular distributions. No differences are found in the position of the diffraction minima for the reaction $^{124}$Sn($p,t$)$^{122}$Sn. On the other side, the angular distributions obtained for the reaction $^{136}$Sn($p,t$)$^{134}$Sn with surface and mixed interactions differ at large angles for some values of the incident proton energy. For this reaction, we compare the ratios of the cross sections associated to the ground state and the first excited state transitions. Differences among the three different theoretical predictions are found and they are more important at the incident proton energy of 15 MeV. As a conclusion, we indicate ($p,t$) two-neutron transfer reactions with very neutron-rich Sn isotopes and at proton energies around 15 MeV as good experimental cases where the surface/volume mixing of the pairing interaction may be probed.

MICROSCOPIC DESCRIPTION OF TEMPERATURE AND PAIRING EFFECTS IN NUCLEI

Finite temperature Hartree-Fock-Bogoliubov calculations are performed in finite nuclei using Skyrme and zero-range, density-dependent pairing interactions. The value of the critical temperature follows approximatively the empirical rule Tc ≃ 0.5 ΔT=0. The specific heat is calculated within the FT-HFB framework. Preliminary results obtained with the FT-QRPA are discussed.

PAIRING CORRELATIONS IN UNSTABLE NUCLEI AND BCS-BEC CROSSOVER

We propose new types of density dependent contact pairing interaction which reproduce the pairing gaps in symmetric and neutron matters obtained by a microscopic treatment of the realistic nucleon-nucleon interaction. The BCS-BEC crossover of neutrons pairs in symmetric and asymmetric nuclear matters is studied by using these contact interactions. It is shown that the bare and screened pairing interactions lead to different features of the BCS-BEC crossover in symmetric nuclear matter. We perform Hartree-Fock-Bogoliubov (HFB) calculations for semi-magic Calcium, Nickel, Tin and Lead isotopes using these density-dependent pairing interactions. It is pointed out that the isospin dependent pairing interaction reproduces well the neutron number dependence of experimental data for the binding energy, two neutrons separation energy, and odd-even mass staggering of these isotopes.

Effective pairing interactions with isospin density dependence

We perform Hartree-Fock-Bogoliubov (HFB) calculations for semi-magic calcium, nickel, tin, and lead isotopes and $N=20,28,50$, and 82 isotones using density-dependent pairing interactions recently derived from a microscopic nucleon-nucleon interaction. These interactions have an isovector component so that the pairing gaps in symmetric and neutron matter are reproduced. Our calculations well account for the experimental data for the neutron number dependence of binding energy, two-neutron separation energy, and odd-even mass staggering of these isotopes. This result suggests that by introducing the isovector term in the pairing interaction, one can construct a global effective pairing interaction that is applicable to nuclei in a wide range of the nuclear chart. It is also shown with the local density approximation that the pairing field deduced from the pairing gaps in infinite matter reproduces qualitatively well the pairing field for finite nuclei obtained with the HFB method.

BCS-BEC crossover of neutron pairs in symmetric and asymmetric nuclear matter

We propose new types of density-dependent contact pairing interactions which reproduce the pairing gaps in symmetric and neutron matters obtained by a microscopic treatment based on realistic nucleon nucleon interaction. These interactions are able to simulate the pairing gaps of either the bare interaction or the interaction screened by the medium polarization effects. It is shown that the medium polarization effects cannot be cast into the usual density power law function of the contact interaction and require the introduction of another isoscalar term. The BCS-BEC crossover of neutron pairs in symmetric and asymmetric nuclear matters is studied by using these contact interactions. This work shows that the bare and screened pairing interactions lead to different features of the BCS-BEC crossover in symmetric nuclear matter. For the screened pairing interaction, a two-neutron BEC state is formed in symmetric matter at ${k}_{\mathit{Fn}}~0.2$ fm${}^{\ensuremath{-}1}$ (neutron density ${\ensuremath{\rho}}_{n}/{\ensuremath{\rho}}_{0}~{10}^{\ensuremath{-}3}$). In contrast, the bare interaction does not form the BEC state at any neutron density.

Large-amplitudeQn-Qpcollectivity in the neutron-rich oxygen isotopeO20

By means of HFB calculations with independent constraints on axial neutron and proton quadrupole moments ${\stackrel{^}{Q}}_{n}$ and ${\stackrel{^}{Q}}_{p}$, we investigate the large amplitude isoscalar and isovector deformation properties of the neutron-rich isotope $^{20}\mathrm{O}$. Using the particle-number and angular-momentum projected generator coordinate method, we analyze the collective dynamics in the ${\ensuremath{\langle}{\stackrel{^}{Q}}_{n}\ensuremath{\rangle},\ensuremath{\langle}{\stackrel{^}{Q}}_{p}\ensuremath{\rangle}}$ plane. The parametrization SLy4 of the Skyrme interaction is used for all calculations in connection with a density-dependent zero-range pairing interaction. Our results show that already for this moderately neutron-rich nucleus the transition moments are modified when independent neutron and proton collective dynamics are allowed.

Pairing interactions and vanishing pairing correlations in hot nuclei

Finite temperature Hartree–Fock–Bogoliubov calculations are performed in Sn isotopes using Skyrme and zero-range, density-dependent pairing interactions. For both stable and very neutron-rich nuclei the critical temperature at which pairing correlations vanish is independent of the volume/surface nature of the pairing interaction. The value of the critical temperature follows approximatively the empirical rule T c ≃ 0.5 Δ T = 0 for all the calculated isotopes, showing that the critical temperature could be deduced from the pairing gap at zero temperature. On the other hand, the pairing gap at temperatures just below T c is strongly sensitive to the volume/surface nature of the pairing interaction.

Spectroscopy and single-particle structure of the odd- Z heavy elements 255Lr, 251Md and 247Es

The odd-Z isotope 255Lr, its daughter 251Md and grand-daughter 247Es were studied in two experiments performed at GANIL and the University of Jyvaskyla. The 255Lr nuclei were produced using the cold fusion-evaporation reaction 209Bi(48Ca, 2n)255Lr at a bombarding energy of 217MeV. The single-particle structure and decay properties were investigated using α, γ and electron spectroscopy. The ground-state spin and parity could be assigned for 255Lr (1/2-) and 251Md (7/2-). States corresponding to the 7/2+[633], 7/2-[514] and 1/2-[521] Nilsson orbitals were observed. Results are compared to experimental data obtained in neighbouring isotopes and to Hartree-Fock-Bogoliubov calculations using the Skyrme interaction SLy4 and a density-dependent pairing interaction. The position of the 1/2-[521] orbital from the spherical 2f 5/2 shell is discussed.