Mass Staggering Research Articles

Isospin-density-dependent pairing from infinite nuclear matter to finite nuclei

The effective isospin-density dependent pairing interaction (P1) [S. S. Zhang, U. Lombardo and E. G. Zhao, Sci. Chin. Phys. Mech. Astro. {\bf 54}, 236 (2011)] extracted from neutron pairing gaps for $^1$S$_0$ in asymmetric nuclear matter calculations [S. S. Zhang, L. G. Cao, U. Lombardo, et al. Phys. Rev. C {\bf 81}, 044313 (2010)] is employed to study the bulk properties of Ca, Ni, Zr and Sn isotopes. The odd-even mass (OEM) staggering is calculated by the Skyrme Hartree-Fock plus BCS method (SHF + BCS) with the SkP interaction. For comparison, we study two other types of isovector effective pairing interactions. One is also extracted from pairing gaps of infinite nuclear matter by the Brueckner-Hartree-Fock (BHF) method but for free spectrum (P2). The other is obtained by fitting the empirical OEM (P3). An isoscalar effective pairing interaction (P4) is also adopted which is determined by fitting the empirical OEM. We find that interaction P1 can better describe the OEM staggering of Ni, Zr and Sn isotopes by $14.3 \%, 41 \%, 30.4 \%$ compared with interaction P2, in terms of root mean square deviations to the empirical OEM, respectively. On the other hand, the performance of P1 and P2 is comparable for Ca isotopes. For Ca and Ni isotopes, P1 behaves similarly as P3, but for Zr isotopes P1 is better than P3 by $\sim 34 \%$. Among the four pairing interactions studied, P4 performs the worst. Therefore, one may conclude that for neutron pairings in finite nuclei, the isovector pairings are preferred than the isoscalar one. It is quite interesting to note that the pairing interaction P1 extracted from nuclear matter calculations can describe pairing gaps of finite nuclei as well as or even better than the interaction P3 directly fitted to finite nuclei.

Consistency of two different approaches to determine the strength of a pairing residual interaction in the rare-earth region

Two fits of the pairing residual interaction in the rare-earth region are independently performed. One is made on the odd-even staggering of masses by comparing measured and explicitly calculated three-point binding-energy differences centered on odd-even nuclei. Another deals with the moments of inertia of the first 2+ states of well deformed even-even nuclei upon comparing experimental data with the results of Inglis-Belyaev moments (supplemented by a crude estimate of the so-called Thouless-Valatin corrections). The sample includes 24 even-even and 31 odd-mass nuclei selected according to two criteria: they should have good rotor properties and should not correspond to low pairing-correlation regimes in their ground states. Calculations are performed in the self-consistent Hartree-Fock plus BCS framework (implementing a self-consistent blocking in the case of odd-mass nuclei). The Skyrme SIII parametrization is used in the particle-hole channel and the fitted quantities are the strengths of |Tz|=1 proton and neutron seniority residual interactions. As a result, the two fits yield sets of strengths in excellent agreement: about 0.1% for the neutron parameters and 0.2% for protons. In contrast when one performs such a fit on odd-even staggering from quantities deduced from BCS gaps or minimal quasiparticle energies in even-even nuclei, as is traditional, one obtains results significantly different from those obtained in the same nuclei by a fit of moments of inertia. As a conclusion, beyond providing a phenomenological tool for microscopic calculations in this region, we have illustrated the proposition made in the seminal paper of Bohr, Mottelson and Pines that moments of inertia and odd-even staggering in selected nuclei were excellent measuring sticks of nuclear pairing correlations.

Extending pairing energy density functional using pairing rotational moments of inertia

The pairing energy density functionals (EDFs) that include the spatial derivative and kinetic terms of the pair densities are discussed. The coupling constants of the pairing EDF are adjusted to reproduce the experimental pairing rotational moment of inertia, and the pair-density derivative terms are shown to systematically improve the values of the pairing rotational moments of inertia in Sn and Pb isotopes. It is pointed out that the conventional average pairing gaps overestimate the experimental odd–even mass staggering in the presence of the pair-density derivative terms.

Impact of the first-forbidden β decay on the production of A∼ 195 r-process peak

We investigated the effects of first-forbidden transitions in $\beta$ decays on the production of the r-process $A \sim 195$ peak. The theoretical calculated $\beta$-decay rates with $\beta$-delayed neutron emission were examined using several astrophysical conditions. As the first-borbidden decay is dominant in $N \sim 126$ neutron-rich nuclei, their inclusion shortens $\beta$-decay lifetimes and shifts the abundance peak towards higher masses. Additionally, the inclusion of the $\beta$-delayed neutron emission results in a wider abundance peak, and smoothens the mass distribution by removing the odd-even mass staggering. The effects are commonly seen in the results of all adopted astrophysical models. Nevertheless there are quantitative differences, indicating that remaining uncertainty in the determination of half-lives for $N=126$ nuclei is still significant in order to determine the production of the r-process peak.

Mass Measurement ofSc56Reveals a SmallA=56Odd-Even Mass Staggering, Implying a Cooler Accreted Neutron Star Crust

We present the mass excesses of (52-57)Sc, obtained from recent time-of-flight nuclear mass measurements at the National Superconducting Cyclotron Laboratory at Michigan State University. The masses of 56Sc and 57Sc were determined for the first time with atomic mass excesses of -24.85(59)((-54)(+0)) MeV and -21.0(1.3) MeV, respectively, where the asymmetric uncertainty for 56Sc was included due to possible contamination from a long-lived isomer. The 56Sc mass indicates a small odd-even mass staggering in the A = 56 mass chain towards the neutron drip line, significantly deviating from trends predicted by the global FRDM mass model and favoring trends predicted by the UNEDF0 and UNEDF1 density functional calculations. Together with new shell-model calculations of the electron-capture strength function of 56Sc, our results strongly reduce uncertainties in model calculations of the heating and cooling at the 56Ti electron-capture layer in the outer crust of accreting neutron stars. We find that, in contrast to previous studies, neither strong neutrino cooling nor strong heating occurs in this layer. We conclude that Urca cooling in the outer crusts of accreting neutron stars that exhibit superbursts or high temperature steady-state burning, which are predicted to be rich in A≈56 nuclei, is considerably weaker than predicted. Urca cooling must instead be dominated by electron capture on the small amounts of adjacent odd-A nuclei contained in the superburst and high temperature steady-state burning ashes. This may explain the absence of strong crust Urca cooling inferred from the observed cooling light curve of the transiently accreting x-ray source MAXI J0556-332.

Properties of nuclei in the nobelium region studied within the covariant, Skyrme, and Gogny energy density functionals

We calculate properties of the ground and excited states of nuclei in the nobelium region for proton and neutron numbers of 92≤Z≤104 and 144≤N≤156, respectively. We use three different energy-density-functional (EDF) approaches, based on covariant, Skyrme, and Gogny functionals, each with two different parameter sets. A comparative analysis of the results obtained for quasiparticle spectra, odd–even and two-particle mass staggering, and moments of inertia allows us to identify single-particle and shell effects that are characteristic to these different models and to illustrate possible systematic uncertainties related to using the EDF modelling.

Competition between pairing correlations and deformation from the odd-even mass staggering of francium and radium isotopes

The masses of $^{222,224,226\text{--}233}\mathrm{Fr}$ and $^{233,234}\mathrm{Ra}$ have been determined with the Penning-trap mass spectrometer ISOLTRAP at the ISOLDE facility at CERN, including the previously unknown mass and half-life of $^{233}\mathrm{Fr}$. We study the evolution of the odd-even staggering of binding energies along the francium and radium isotopic chains and of its lowest-order estimator, ${\ensuremath{\Delta}}^{3}(N)$. An enhancement of the staggering of ${\ensuremath{\Delta}}^{3}(N)$ is observed towards neutron number $N=146$, which points to contributions beyond pairing correlations. These contributions are investigated in the Hartree-Fock and Hartree-Fock-Bogoliubov approaches, emphasizing the connections to the single-particle level density and nuclear deformation.

Precision Mass Measurements beyondSn132: Anomalous Behavior of Odd-Even Staggering of Binding Energies

Atomic masses of the neutron-rich isotopes (121-128)Cd, (129,131)In, (130-135)Sn, (131-136)Sb, and (132-140)Te have been measured with high precision (10 ppb) using the Penning-trap mass spectrometer JYFLTRAP. Among these, the masses of four r-process nuclei (135)Sn, (136)Sb, and (139,140)Te were measured for the first time. An empirical neutron pairing gap expressed as the odd-even staggering of isotopic masses shows a strong quenching across N = 82 for Sn, with a Z dependence that is unexplainable by the current theoretical models.

Partial-wave contributions to pairing in nuclei

We present a detailed study of partial-wave contributions of nuclear forces to pairing in nuclei. For T = 1, J = 0 pairing, partial waves beyond the standard 1S0 channel play an interesting role for the pair formation in nuclei. We explore the impact of including partial waves beyond the 1S0 channel on the odd-even mass staggering in semi-magic isotopic chains. The additional contributions are dominated by the repulsive 3P1 partial wave.

Number-conserving theory of nuclear pairing gaps: A global assessment

We study odd-even mass staggering of nuclei, also called pairing gaps, using a Skyrme self-consistent mean-field theory and a numerically exact treatment of the pairing Hamiltonian. We find that the configuration-space Monte Carlo method proposed by Cerf and Martin [Phys. Rev. C 47, 2610 (1993)] offers a practical computational procedure to carry out the numerical solutions in large-dimensional model spaces. Refitting the global strength of the pairing interaction for 443 neutron pairing gaps in our number-conserving treatment, we find the correction to the pairing correlation energies and pairing gaps to have rms values of 0.6 and 0.12 MeV, respectively. The exact treatment provides a significant improvement in the fit to experimental gaps, although it is partially masked by a larger rms error due to deficiencies in other aspects of the theory, such as the mean-field energy functional.

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.

Isospin Dependence in the Odd-Even Staggering of Nuclear Binding Energies

The FRS-ESR facility at GSI provides unique conditions for precision measurements of large areas on the nuclear mass surface in a single experiment. Values for masses of 604 neutron-deficient nuclides (30 < or = Z < or = 92) were obtained with a typical uncertainty of 30 microu. The masses of 114 nuclides were determined for the first time. The odd-even staggering (OES) of nuclear masses was systematically investigated for isotopic chains between the proton shell closures at Z = 50 and Z = 82. The results were compared with predictions of modern nuclear models. The comparison revealed that the measured trend of OES is not reproduced by the theories fitted to masses only. The spectral pairing gaps extracted from models adjusted to both masses, and density related observables of nuclei agree better with the experimental data.

On the linear term in the nuclear symmetry energy

The nuclear symmetry energy calculated in the RPA from the pairing plus symmetry force Hamiltonian with equidistant single-nucleon levels is for mass number A=48 approximately proportional to T(T+1.025), where T is the isospin quantum number. An isovector character of the pair field assumed to produce the observed odd–even mass staggering is essential for this result. The RPA contribution to the symmetry energy cannot be simulated by adding to the Hartree–Fock–Bogolyubov energy a term proportional to the isospin variance in the Bogolyubov quasiparticle vacuum, and there are significant corrections to the approximation which consist in adding half the isocranking angular velocity. The present calculation employs a smaller single-nucleon level spacing than used in a previous investigation of the model.

Pairing correlations. II. Microscopic analysis of odd-even mass staggering in nuclei

The odd-even mass staggering in nuclei is analyzed in the context of self-consistent mean-field calculations, for spherical as well as for deformed nuclei. For these nuclei, the respective merits of the energy differences $\Delta^{(3)}$ and $\Delta^{(5)}$ to extract both the pairing gap and the time-reversal symmetry breaking effect at the same time are extensively discussed. The usual mass formula $\Delta^{(3)}$, is shown to contain additional mean-field contributions when realistic pairing is used in the calculation. A simple tool is proposed in order to remove time-reversal symmetry breaking effects from $\Delta^{(5)}$. Extended comparisons with the odd-even mass staggering obtained in the zero pairing limit (schematic model and self-consistent calculations) show the non-perturbative contribution of pairing correlations on this observable. PACS: 21.10Dr; 21.10.Hw; 21.30.-x. Keywords: Mean-field theories; Pairing correlations; Odd-even mass staggering;

Odd-Even Staggering of Nuclear Masses: Pairing or Shape Effect?

The odd-even staggering of nuclear masses was recognized in the early days of nuclear physics. Recently, a similar effect was discovered in other finite fermion systems, such as ultrasmall metallic grains and metal clusters. It is believed that the staggering in nuclei and grains is primarily due to pairing correlations (superconductivity), while in clusters it is caused by the Jahn-Teller effect. We find that, for light- and medium-mass nuclei, the staggering has two components. The first originates from pairing while the second, comparable in magnitude, has its roots in the deformed mean field.

I = 2 Staggering in Ground Rotational Bands and A = 2 Mass Staggering in the Deformed Nuclei

We find a ΔI = 2 staggering or ΔI = 4 bifurcation phenomenon in many even-even mass ground rotational bands in rare-earth and actinide nuclei by performing staggering parameter analysis. We take the Interacting Boson Model and reproduce the ΔI = 4 bifurcation by slightly modifying the SU(3) limit of the IBM wave function. The IBM then predicts ΔA = 2 mass staggering in the deformed nuclei, which is also found experimentally.