Proton Pairing Gaps Research Articles

Subshell gaps and onsets of collectivity from proton and neutron pairing gap correlations

Throughout the nuclear chart, particle-hole correlations give rise to giant resonances and, together with the proton–neutron interaction, deformation and rotational bands. In order to shed light on many-body correlations in open-shell nuclei, I explore macroscopic properties that could manifest from the collective behaviour of protons and neutrons. Intuitively, the correlation of proton and neutron Cooper pairs can be inferred from the respective pairing gaps, that can precisely be extracted from the AME 2020 atomic mass evaluation through odd-even atomic mass differences. This work shows that the combination of large and close-lying proton and neutron pairing gaps is sensitive to onsets of collectivity and subshell gaps in superfluid nuclei, away from major shell closures. Trends of reduced transition probabilities or B(E2) values — which describe the collective overlap between the wave functions of initial and final nuclear states — are revealed in overall agreement with data. Specially interesting is the peak of collectivity in the tin isotopes at 110Sn, instead of at midshell, as expected by large-scale shell-model calculations; a situation that has astounded the nuclear physics community for quite some time.

Proton and Neutron Pairing Properties within a Mixed Volume-Surface Pairing Type Using the Hartree–Fock–Bogolyubov Theory

This work aims at systematic investigations of the proton and neutron pairing properties and Fermi energies in the region from the proton drip-line to the neutron drip-line. In order to obtain a more accurate mass formula with the Skyrme (SKI3) force, the global descriptive power of the Skyrme–Hartree–Fock–Bogoliubov model for pairing properties is applied. Systematic Skyrme–HFB calculations with a mixed volume-surface pairing are carried out to study the ground-state proton pairing gap, neutron and proton pairing energies, and the neutron and proton Fermi energies for about 2095 even-even nuclei ranging from 2 ≤ Z ≤ 110 to 2 ≤ N ≤ 236 . The calculated values of proton pairing gaps are compared with experimental data, by using the difference-point formulas Δ(3), Δ(4), and Δ(5), and compared with the proton pairing gap in the Lipkin–Nogami model. It is shown that the Skyrme (SKI3) force with the mixed volume-surface pairing can be successfully used for describing the ground-state proton pairing gap, proton and neutron pairing energies, and proton and neutron Fermi properties of the investigated nuclei, in particular, the neutron-rich nuclei and the exotic nuclei near the neutron drip-line. On the other hand, the calculated proton pairing gap shows the acceptable agreement with the available experimental values of the proton pairing gap with the use of the difference-point formulas Δ(3), Δ(4), and Δ(5) and with the data of the Lipkin–Nogami model over the whole nuclear chart.

A Hartree-Fock-Bogoliubov Study on the Pairing Correlations of the Isotopes of Cobalt

Background: The phenomena of nucleon pairing could be outlined from the Bethe-Weizäcker semi-empirical formula, from which the nuclear properties, viz. the binding energy, stability, shape etc. could be clearly sketched. Though the pairing correlation seems to be a small correction to the binding energy term, it plays a determinative role in defining the structure of nuclear systems. The addition to the binding energy in turn affects the position of the isotope on the dripline and hence increases the stability.
 Purpose: To study the effects of pairing on the ground state properties of the isotopes of Cobalt.
 Methods: We use Hartree-Fock-Bogoliubov (HFB) theory for the study. The general wave functions for the HFB approach are determined from variational principle. The eigen functions for the Hamiltonian are connected with the particle operators through the Bogoliubov transformations. The Hartree-Fock energy is obtained through the minimization of the variational parameter and the HFB equation is solved by iterative diagonalization by restoring the particle number symmetry.
 Results: The HFB analysis substantiates the effect of pairing correlation on binding energies, neutron and proton pairing energies, neutron and proton pairing gaps and one- and two-neutron separation energies of the Cobalt isotopes. The binding energies and one and two-neutron separation energies match with the experimental values and for pairing energies and pairing gaps, the regions where pairing is significant and the effects of shell closure at the vicinity of magic configuration of neutrons could be recognized.
 Conclusion: The Hartree-Fock-Bogoliubov calculations of the effects of pairing could be used as an efficient tool to study the nuclear structure. It can be ascertained that pairing plays an important role in determining the ground state properties of atomic nuclei.

Proton pairing in neutron stars from chiral effective field theory

We study the $^{1}S_{0}$ proton pairing gap in $\ensuremath{\beta}$-equilibrated neutron star matter within the framework of chiral effective field theory. We focus on the role of three-body forces, which strongly modify the effective proton-proton spin-singlet interaction in dense matter. We find that three-body forces generically reduce both the size of the pairing gap and the maximum density at which proton pairing may occur. The pairing gap is computed within Bardeen-Cooper-Schrieffer theory using a single-particle dispersion relation calculated up to second order in perturbation theory. Model uncertainties are estimated by varying the nuclear potential (its order in the chiral expansion and high-momentum cutoff) and the choice of single-particle spectrum in the gap equation. We find that a second-order perturbative treatment of the single-particle spectrum suppresses the proton $^{1}S_{0}$ pairing gap relative to the use of a free spectrum. We estimate the critical temperature for the onset of proton superconductivity to be ${T}_{c}=(3.2\text{--}5.1)\ifmmode\times\else\texttimes\fi{}{10}^{9}$ K, which is consistent with previous theoretical results in the literature and marginally within the range deduced from a recent Bayesian analysis of neutron star cooling observations.

Nuclear Pairing Gaps and Neutron Star Cooling

We study the cooling of isolated neutron stars with particular regard to the importance of nuclear pairing gaps. A microscopic nuclear equation of state derived in the Brueckner-Hartree-Fock approach is used together with compatible neutron and proton pairing gaps. We then study the effect of modifying the gaps on the final deduced neutron star mass distributions. We find that a consistent description of all current cooling data can be achieved and a reasonable neutron star mass distribution can be predicted employing the (slightly reduced by about 40%) proton 1S0 Bardeen-Cooper-Schrieffer (BCS) gaps and no neutron 3P2 pairing.

Transport coefficients of leptons in superconducting neutron star cores

I consider the thermal conductivity and shear viscosity of leptons (electrons and muons) in the nucleon NS cores where protons are in the superconducting state. I restrict the consideration to the case of not too high temperatures $T\lesssim 0.35T_{\mathrm{c}p}$, where $T_{\mathrm{c}p}$ is the critical temperature of the proton pairing. In this case, lepton collisions with protons can be neglected. Charged lepton collision frequencies are mainly determined by the transverse plasmon exchange and are mediated by the character of the transverse plasma screening. In our previous works [Shternin \& Yakovlev, Phys. Rev. D {\bf 75} 103004 (2007); {\bf 78} 063006 (2008)] the superconducting proton contribution to the transverse screening was considered in the Pippard limit $\Delta \ll \hbar q v_{\mathrm{F}p}$, where $\Delta$ is the proton pairing gap, $v_{\mathrm{F}p}$ is the proton Fermi velocity, and $\hbar q$ is the typical transferred momentum in collisions. However, for large critical temperatures (large $\Delta$) and relatively small densities (small $q$) the Pippard limit may become invalid. In the present study I show that this is indeed the case and that the older calculations severely underestimated the screening in a certain range of the parameters appropriate to the neutron star cores. As a consequence, the kinetic coefficients at $T\ll T_{\mathrm{c}p}$ are found to be smaller than in previous calculations.

Nuclear properties for astrophysical and radioactive-ion-beam applications (II)

We tabulate the ground-state odd-proton and odd-neutron spins and parities, proton and neutron pairing gaps, one- and two-neutron separation energies, quantities related to β-delayed one- and two-neutron emission probabilities, average energy and average number of emitted neutrons, β-decay energy release and half-life with respect to Gamow–Teller decay with a phenomenological treatment of first-forbidden decays, one- and two-proton separation energies, and α-decay energy release and half-life for 9318 nuclei ranging from 16O to 339136 and extending from the proton drip line to the neutron drip line. This paper is a new and improved version of Atomic Data And Nuclear Data Tables [66 131 (1997)]. The starting point of our present work is the new study (FRDM(2012)) of nuclear ground-state masses and deformations based on the finite-range droplet model and folded-Yukawa single-particle potential published in a previous issue of Atomic Data And Nuclear Data Tables [109–110, 1 (2016)]. The β-delayed neutron-emission probabilities and Gamow–Teller β-decay rates are obtained from a quasi-particle random-phase approximation with single-particle levels and wave functions at the calculated nuclear ground-state shapes as input quantities. A development since 1997 is we now use a Hauser–Feshbach approach to account for (n, γ) competition and treat first-forbidden decay in a phenomenological approach.

Shape evolution of 72,74Kr with temperature in covariant density functional theory**Supported by National Natural Science Foundation of China (11105042, 11305161, 11505157), Open Fund of Key Laboratory of Time and Frequency Primary Standards, CAS, and Support from Henan Administration of Foreign Experts Affairs

The rich phenomena of deformations in neutron-deficient krypton isotopes, such as shape evolution with neutron number and shape coexistence, have attracted the interest of nuclear physicists for decades. It is interesting to study such shape phenomena using a novel way, e.g. by thermally exciting the nucleus. In this work, we develop the finite temperature covariant density functional theory for axially deformed nuclei with the treatment of pairing correlations by the BCS approach, and apply this approach for the study of shape evolution in 72,74Kr with increasing temperature. For 72Kr, with temperature increasing, the nucleus firstly experiences a relatively quick weakening in oblate deformation at temperature T ∼0.9 MeV, and then changes from oblate to spherical at T ∼2.1 MeV. For 74Kr, its global minimum is at quadrupole deformation β2 ∼ −0.14 and abruptly changes to spherical at T∼ 1.7 MeV. The proton pairing transition occurs at critical temperature 0.6 MeV following the rule Tc=0.6Δp(0), where Δp(0) is the proton pairing gap at zero temperature. The signatures of the above pairing transition and shape changes can be found in the specific heat curve. The single-particle level evolutions with temperature are presented.

Search of islands of stability for hypothetical superheavy nuclei using covariant density functional theory

A systematic search for the location of islands of stability has been performed for the proton number $100\le Z\le 220$ and the neutron number $Z+30\le N\le 2Z+30$ using covariant density functional theory (CDFT), a relativistic Hartree-Bogoliubov (RHB) formalism with separable pairing, for two different force parameters, DD-ME2 and NL3*. Locations of the islands of stability are identified by analysis of the two-neutron and two-proton separation energy, two-nucleon shell gaps, vanishing neutron and proton pairing gap, energy surface, and the single particle states. The results show that beyond $^{292}120$ only Z = 154 and N $>$ 220 can be a center of a new island of stability.

Upper critical field and (non)-superconductivity of magnetars

We construct equilibrium models of compact stars using a realistic equation of state and obtain the density range occupied by the proton superconductor in strong B-fields. We do so by combining the density profiles of our models with microscopic calculations of proton pairing gaps and the critical unpairing field H c2 above which the proton type-II superconductivity is destroyed. We find that magnetars with interior homogeneous field within the range 0.1 ≤ B 16 ≤ 2, where B 16 = B/1016 G, are partially superconducting, whereas those with B 16 > 2 are void of superconductivity. We briefly discuss the neutrino emissivity and superfluid dynamics of magnetars in the light of their (non)-superconductivity.

A pilgrimage through superheavy valley

We searched for the shell closure proton and neutron numbers in the superheavy region beyond Z = 82 and N = 126 within the framework of non-relativistic Skryme–Hartree–Fock (SHF) with FITZ, SIII, SkMP and SLy4 interactions. We have calculated the average proton pairing gap Δp, average neutron pairing gap Δn, two-nucleon separation energy S2q and shell correction energy Eshell for the isotopic chain of Z = 112–126. Based on these observables, Z = 120 with N = 182 is suggested to be the magic numbers in the present approach.

A strong correlation between nuclear collectivity and atomic masses

This work explores how nuclear collectivity arises in singly-closed shell nuclei where particle–particle correlations may be the dominant degree of freedom. The trends of values in the Ni and Sn superfluids with N⪆Z are predicted from proton and neutron pairing gaps extracted from atomic masses. Nuclear collectivity is enhanced for the combination of large and close-lying proton and neutron pairing gaps. The agreement with the Ni isotopes is excellent, and recent experimental results support the enhanced trend predicted for the neutron-deficient Sn isotopes. This work shows that many-body microscopic properties such as nuclear collectivity could be inferred directly from macroscopic average properties such as atomic masses.

Study on 222–226Th Isotopes in the Cranking Framework

The yrast spectra, quadrupole moments, quadrupole deformation parameters (β 2), non-axiality parameters (γ), root mean-square radii for protons and neutrons, occupation probabilities, moment of inertia (I), and B(E2) transition probabilities are calculated for 222–226Th in the cranked Hartree–Bogoliubov framework. The calculations employ a quadrupole-quadrupole plus pairing model of residual interaction operating in a reasonably large valence space outside the 164Pb core. Our calculations reproduce qualitatively the observed yrast spectra in 222–226Th up to spin 20+. The calculated results indicate that the non-axiality parameter decreases as one moves along the yrast states. The observed increase in deformation from 222Th to 226Th is due to the increase in the occupation of low-k components of $(2g_{9/2})_{\pi }$ and $(1j_{{15/2}})_{\nu }$ orbits. The model parameters reproduce not only the moment of inertia, deformation, and transition probabilities but also the proton and neutron pairing gaps and are the most appropriate for cranking studies in this region.

Isoscalar and isovector density dependence of the pairing functional determined from global fitting

We optimize the density dependence of a local energy density functional for pairing correlations (pair-DF), taking into account both the isoscalar and the isovector densities. All the experimental pairing gaps in even-even nuclei with $N,Zg8$ are analyzed by performing the Hartree-Fock-Bogoliubov calculation. The new pairing energy density functionals give the best fit for all the pairing gaps with the rms deviation of ${\ensuremath{\sigma}}_{\text{tot}}=(260$--360) keV, depending on adopted Skyrme interactions in the particle-hole channel. It is shown that the isoscalar-density dependence in the pair-DF can be determined almost uniquely for the several Skyrme forces by adjusting the neutron and proton pairing gaps. An empirical relation among the parameters in the isoscalar part of pair-DF is extracted. The uncertainty of the isovector part is also examined by calculating the pairing gaps of neutron-rich and proton-rich nuclei in a wider region of the nuclear chart. We point out also that the linear isovector-density dependence in the pair-DF can mimic well the Coulomb effect for pairing correlations in neutron-rich nuclei. We discuss finally the predictive power of the pair-DF for pairing properties around drip-line regions.

Semi-microscopic model for the effective pairing interaction in atomic nuclei

A model has been proposed in which the ab initio effective pairing interaction in atomic nuclei is supplemented by a small phenomenological term containing one parameter universal for all intermediate-mass and heavy nuclei. The neutron and proton pairing gaps have been calculated for several chains of semimagic nuclei; these calculations demonstrate the applicability of this model.

Odd-even mass difference and isospin dependent pairing interaction

The neutron and proton odd-even mass differences are studied with Hartree-Fock + BCS (HF + BCS) calculations with Skyrme interactions and an isospin dependent contact pairing interaction, which is recently derived from a microscopic nucleon-nucleon interaction. To this end, we perform HF + BCS calculations for even and odd semi-magic tin and lead isotopes together with even and odd Z isotones with N = 50 and 82. The filling approximation is applied to the last unoccupied particle in odd nuclei. Comparisons with the experimental data show a clear manifestation of the isospin dependent pairing correlations in both proton and neutron pairing gaps.

Non-empirical pairing energy density functional

We perform systematic calculations of pairing gaps in semi-magic nuclei across the nuclear chart using the Energy Density Functional method and a {\it non-empirical} pairing functional derived, without further approximation, at lowest order in the two-nucleon vacuum interaction, including the Coulomb force. The correlated single-particle motion is accounted for by the SLy4 semi-empirical functional. Rather unexpectedly, both neutron and proton pairing gaps thus generated are systematically close to experimental data. Such a result further suggests that missing effects, i.e. higher partial-waves of the NN interaction, the NNN interaction and the coupling to collective fluctuations, provide an overall contribution that is sub-leading as for generating pairing gaps in nuclei. We find that including the Coulomb interaction is essential as it reduces proton pairing gaps by up to 40%.

Neutron–proton pairing competition in [formula omitted] nuclei: Metastable state decays in the proton dripline nuclei 8241Nb and 8643Tc

The low-lying structures of the self-conjugate (N=Z) nuclei 8241Nb41 and 8643Tc43 have been investigated using isomeric-decay spectroscopy following the projectile fragmentation of a 107Ag beam. These represent the heaviest odd–odd N=Z nuclei in which internal decays have been identified to date. The resulting level schemes shed light on the shape evolution along the N=Z line between the doubly-magic systems 5628Ni and 10050Sn and support a preference for T=1 states in Tz=0 odd–odd nuclei at low excitation energies associated with a T=1 neutron–proton pairing gap. Comparison with Projected Shell Model calculations suggests that the decay in 82Nb may be interpreted as an isospin-changing K isomer.

Proton and neutron 1S0 superfluidity in asymmetric nuclear matter

The proton and neutron 1S0 pairing gaps and their isospin dependence in isospin asymmetric nuclear matter have been studied by the isospin dependent Brueckner-Hartree-Fock approach and the BCS theory. We have focused on investigating and discussing the effect of three-body force. The calculated results indicate that as the isospin asymmetry increases，the density range of the 1S0 neutron superfluidity is narrowed slightly and the maximum value of the neutron pairing gap increases，while the density domain for the proton superfluidity enlarges rapidly and the peak value of the proton gap decreases remarkably. The three-body force turns out to affect only weakly the neutron 1S0 superfluidity and its isospin dependence，i.e.，it leads to a small reduction of the neutron 1S0 paring gap. However, the three-body force not only reduces largely the strength of the proton 1S0 gaps at high densities in highly asymmetric nuclear matter but also suppresses strongly the density domain for the proton 1S0 superfluidity phase.

S01proton superfluidity in neutron star matter: Impact of bulk properties

We study the ^{1}S_{0} proton pairing gap in neutron star matter putting emphasis on influence of the Dirac effective mass and the proton fraction on the gap within the relativistic Hartree-Bogoliubov model. The gap equation is solved using the Bonn-B potential as a particle-particle channel interaction. It is found that the maximal pairing gap \Delta_{max} is 1-2 MeV, which has a strong correlation with the Dirac effective mass. Hence we suggest that it serves as a guide to narrow down parameter sets of the relativistic effective field theory. Furthermore, the more slowly protons increase with density in the core region of neutron stars, the wider the superfluid range and the slightly lower the peak of the gap become.