Nuclear Binding Energies Research Articles

Spectral Explanation for Statistical Odd-Even Staggering in Few Fermions Systems

Odd-even statistical staggering in a Lipkin-like few fermions model has been recently encountered. Of course, staggering in nuclear binding energies is a well established fact. Similar effects are detected in other finite fermion systems as well, as for example, ultra small metallic grains and metal clusters. We work in this effort with the above-mentioned Lipkin-like, two-level fermion model and show that statistical staggering effects can be detailedly explained by recourse to a straightforward analysis of the associated energy-spectra.

Role of pair-vibrational correlations in forming the odd-even mass difference

In the random phase approximation (RPA)-amended Nilsson-Strutinskij method of calculating nuclear binding energies [1, 2], the conventional shell correction terms derived from the independent-nucleon model and the Bardeen-Cooper-Schrieffer pairing theory are supplemented by a term which accounts for the pair-vibrational correlation energy. This term is derived by means of the RPA from a pairing Hamiltonian which includes a neutron-proton pairing interaction. The method was used previously in studies of the pattern of binding energies of nuclei with approximately equal numbers N and Z of neutrons and protons and even mass number A = N + Z. Here it is applied to odd-A nuclei. Three sets of such nuclei are considered: (i) The sequence of nuclei with Z = N – 1 and 25 ⩽ A ⩽ 99. (ii) The odd-A isotopes of In, Sn, and Sb with 46 ⩽ N ⩽ 92. (iii) The odd-A isotopes of Sr, Y, Zr, Nb, and Mo with 60 ⩽ N ⩽ 64. The RPA correction is found to contribute significantly to the calculated odd-even mass differences, particularly in the light nuclei. In the upper sd shell this correction accounts for almost the entire odd-even mass difference for odd Z and about half of it for odd N. The size and sign of the RPA contribution varies, which is explained qualitatively in terms of a closed expression for a smooth RPA counter term.

Utilizing the Quark-Like Model (QLM) to Determine the $$Q_{\beta }^{ - }$$-Value in the Range 2 ≤ Z ≤ 97

In this research, the negative beta decay energy ( $$Q_{\beta }^{ - }$$ ) is expressed in terms of the nuclear binding energies instead of the mass values of a wide spectrum of nuclei that extend between 2 ≤ Z ≤ 97. The quark-like model (QLM), which can efficiently identify nuclear binding energies, has been used to obtain a new formula to express a negative beta decay energy ( $$Q_{\beta }^{ - }$$ ), and new formula was modeled through a method of plotting between the practical and theoretical values. The calibration equations are obtained, and mathematical derivations were performed to obtain a final and new formula for $$Q_{\beta }^{ - }$$ . The results showed that there are an acceptable match between practical and theoretical values, particularly for medium and heavy nuclei, and less acceptable match for light nuclei due to the presence of magic numbers and their interference in that spectrum. A standard deviation of no more than 1.73 was obtained.

Statistical aspects of nuclear mass models

We study the information content of nuclear masses from the perspective of global models of nuclear binding energies. To this end, we employ a number of statistical methods and diagnostic tools, including Bayesian calibration, Bayesian model averaging, chi-square correlation analysis, principal component analysis and empirical coverage probability. Using a Bayesian framework, we investigate the structure of the four-parameter liquid drop model by considering discrepant mass domains for calibration. We then use the chi-square correlation framework to analyze the 14-parameter Skyrme energy density functional calibrated using homogeneous and heterogeneous datasets. We show that quite a dramatic parameter reduction can be achieved in both cases. The advantage of Bayesian model averaging for improving uncertainty quantification is demonstrated. The statistical approaches used are pedagogically described; in this context this work can serve as a guide for future applications.

Local relations of nuclear charge radii

In this paper we study patterns of nuclear charge radii from three perspectives. The first is based on the so-called $\ensuremath{\delta}{R}_{i\mathrm{n}\ensuremath{-}j\mathrm{p}}$ relations, the root-mean-square deviation of which was reported to be 0.0072 fm for nuclei with both neutron number $N$ and proton number $Z$ $\ensuremath{\ge}8$, and are pointed out here to have actually a much smaller value, which is $\ensuremath{\approx}0.003$ fm for most nuclei with $N\ensuremath{\ge}28$ and $Z\ensuremath{\ge}28$. The second and third perspectives are new relations proposed in this paper. One is called the $\ensuremath{\delta}{R}_{\mathrm{nn}}$ relation enlightened from the so-called ``nonpairing'' interaction $\ensuremath{\delta}{V}_{\mathrm{nn}}$ in studies of nuclear binding energies, and the other is a linear dependence of nuclear charge radii in terms of valence nucleon numbers. All these relations are very accurate in most regions, and sizable deviations from these relations could be used to shed light on structural evolutions such as phase transitions and shape coexistence. About 830 unknown nuclear charge radii are enclosed in our Supplemental Material in this paper, with theoretical uncertainty below 0.03 fm.

Toward Classical Models of Nucleons and Nuclei

Nuclear models can be complicated and far from first principles. Inspired by Mills' Grand Unified Theory of Classical Physics, we will present a model of the interior of nucleons that predicts their masses well considering its simplicity. Then we will apply Mills' nucleon model to the problem of computing nuclear binding energies. Because the theory is constrained to agree with classical physics including Maxwell's Equations and Newton's Laws, for instance lacking special forces that only operate in the nucleus, the possible compatible nuclear theories are strictly limited which should make finding them easier.

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.

Beyond the proton drip line: Bayesian analysis of proton-emitting nuclei

The limits of the nuclear landscape are determined by nuclear binding energies. Beyond the proton drip lines, where the separation energy becomes negative, there is not enough binding energy to prevent protons from escaping the nucleus. Predicting properties of unstable nuclear states in the vast territory of proton emitters poses an appreciable challenge for nuclear theory as it often involves far extrapolations. In addition, significant discrepancies between nuclear models in the proton-rich territory call for quantified predictions. With the help of Bayesian methodology, we mix a family of nuclear mass models corrected with statistical emulators trained on the experimental mass measurements, in the proton-rich region of the nuclear chart. Separation energies were computed within nuclear density functional theory using several Skyrme and Gogny energy density functionals. We also considered mass predictions based on two models used in astrophysical studies. Quantified predictions were obtained for each model using Bayesian Gaussian processes trained on separation-energy residuals and combined via Bayesian model averaging. We obtained a good agreement between averaged predictions of statistically corrected models and experiment. In particular, we quantified model results for one- and two-proton separation energies and derived probabilities of proton emission. This information enabled us to produce a quantified landscape of proton-rich nuclei. The most promising candidates for two-proton decay studies have been identified. The methodology used in this work has broad applications to model-based extrapolations of various nuclear observables. It also provides a reliable uncertainty quantification of theoretical predictions.

The origin of the elements and other implications of gravitational wave detection for nuclear physics

The neutron-star collision revealed by the event GW170817 gave us a first glimpse of a possible birthplace of most of our heavy elements. The multi-messenger nature of this historical event combined gravitational waves, a gamma-ray burst and optical astronomy of a “kilonova”, bringing the first observations of rapid neutron capture (r process) nucleosynthesis after 60 years of speculation. Modeling the r process requires a prodigious amount of nuclear-physics ingredients: practically all the quantum state and interaction properties of virtually all neutron-rich nuclides, many of which may never be produced in the laboratory! Another essential contribution of nuclear physics to neutron stars (and their eventual coalescence) is the equation of state (EoS) that defines their structure and composition. The EoS, combined with the knowledge of nuclear binding energies, determines the elemental profile of the outer crust of a neutron star and the relationship between its radius and mass. In addition, the EoS determines the form of the gravitational wave signal. This article combines a tutorial presentation and bibliography with recent results that link nuclear mass spectrometry to gravitational waves via neutron stars.

The effective single particle potential and the tadpole

The Energy Density Functional (EDF) theory is extremely successful within the effective force approach, noticeably the Skyrme or Gogny forces, in reproducing the nuclear binding energies and other nuclear properties along the whole mass table. The EDF is in this case represented formally as the Hartree-Fock (HF) mean field of an effective force, and the associated single particle states can be considered the eigenstates of the corresponding effective mean field. In general, the phenomenological single particle spectrum is not accurately reproduced. To overcome this difficulty one can improve the functional in order to incorporate in the mean field additional correlations in an effective way. In an alternative viable scheme one can introduce explicitly many-body correlations which affect the single particle motion at both dynamic and static levels. In particular, the particle-vibration coupling scheme modifies the positions of the individual single particle energies. In this case one also introduce the fragmentation of the single particle states, a feature that cannot be described at the mean field level. At the same time the so-introduced correlations modify the static single particle potential, and the single particle states that diagonalize the corresponding density matrix are not the orbitals of a mean field with occupation numbers 1 (occupied orbitals) and 0 (unoccupied orbitals) anymore. In this paper we show that both static and dynamic effects on the single particle motion can be introduced within a previously developed scheme, based on the conserving approximations, and that the static part can be identified with the so-called tadpole term introduced by Khodel in the self-consistent theory of finite Fermi systems. The treatment remains at the formal level, but we hope that several aspects of the particle-vibration coupling scheme will be clarified.

Role of pair-vibrational correlations in forming the odd-even mass difference

In the random-phase-approximation-amended (RPA-amended) Nilsson-Strutinskij method of calculating nuclear binding energies, the conventional shell correction terms derived from the independent-nucleon model and the Bardeen-Cooper-Schrieffer pairing theory are supplemented by a term which accounts for the pair-vibrational correlation energy. This term is derived by means of the RPA from a pairing Hamiltonian which includes a neutron-proton pairing interaction. The method was used previously in studies of the pattern of binding energies of nuclei with approximately equal numbers $N$ and $Z$ of neutrons and protons and even mass number $A = N + Z$. Here it is applied to odd-$A$ nuclei. Three sets of such nuclei are considered: (i) The sequence of nuclei with $Z = N - 1$ and $25 \le A \le 99$. (ii) The odd-$A$ isotopes of In, Sn, and Sb with $46 \le N \le 92$. (iii) The odd-$A$ isotopes of Sr, Y, Zr, Nb, and Mo with $60 \le N \le 64$. The RPA correction is found to contribute significantly to the calculated odd-even mass differences, particularly in the light nuclei. In the upper $sd$ shell this correction accounts for almost the entire odd-even mass difference for odd $Z$ and about half of it for odd $N$. The size and sign of the RPA contribution varies, which is explained qualitatively in terms of a closed expression for a smooth RPA counter term.

Basic characteristics of nuclear landscape by improved Weizsäcker-Skyrme-type nuclear mass model * * Supported by the National Natural Science Foundation of China (11675066 and 11705055) by the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot04) and Feitian Scholar Project of Gansu province

Atomic Mass Evaluation (AME2016) has replenished the latest nuclear binding energy data. Other physical observables, such as the separated energies, decay energies, and the pairing gaps, were evaluated based on the new mass table. An improved Weizsäcker-Skyrme-type (WS-type) nuclear mass model with only 13 parameters was presented, including the correction from two combinatorial radial basis functions (RBFs), where shell and pairing effects are simultaneously dealt with using a Strutinsky-like method. The RBFs code had 2267 updated experimental binding energies as inputs, and their correspondent root-mean square (rms) deviations dropped to 149 keV. For the training of other mass models by RBFs correction, rms deviations are clustered between 100 keV to 200 keV. Compared with other experimental quantities, the rms deviations calculated within the improved WS-type model falls between 100 keV and 250 keV. We extrapolate the binding energies to 12435 nuclei, which covers the ranges and in the framework of the WS-type model with RBFs correction. Simultaneously, the ground state deformations and all parts in the WS-type mass formula are presented in this paper. Finally, we tabulated all calculated characteristics within the improved formula and linked them to https://github.com/lukeronger/NuclearData-LZU: nuclear binding energies, one-nucleon and two-nucleon separation energies ( ), and -decay energies ( and ), and the pairing gap and .

Challenging nuclear structure of the heaviest – opportunities at S3

After more than half a century of research addressing the synthesis and nuclear structure of superheavy nuclei (SHN) a boost for its progress is expected from the advent of new instrumentation. An order of magnitude in beam intensity increase is envisaged to be provided by new powerful accelerators like the new DC280 cyclotron at the SHE factory of FLNR/JINR or the superconducting linac at SPIRAL2 of GANIL. In addition new ion-optical installations like the separator-spectrometer set-up S3 with two complementary detection systems SIRIUS and LEB will provide a substantial sensitivity increase for classically pursued routes like decay spectroscopy after separation (DSAS), and alternative and complementary methods like high precision mass measurements and laser spectroscopy. Decay spectroscopy has proven in the past to be a powerful tool to study the low lying nuclear structure of heavy and superheavy nuclei. Single particle levels and other structure features likeK isomerism, being important in the fermium-nobelium region as well as for tracing deformation towards spherical shell stabilised SHN, have been investigated almost up to the limit posed by the sensitivity of present-day instrumentation. Precision mass measurements and laser spectroscopy will offer the possibility to study alternative features like atomic and nuclear binding energies, nuclear charge radii and quadrupolemoments.

Advanced Statistical Methods to Fit Nuclear Models

We discuss advanced statistical methods to improve parameter estimation of nuclear models. In particular, using the Liquid Drop Model for nuclear binding energies, we show that the area around the global $\chi^2$ minimum can be efficiently identified using Gaussian Process Emulation. We also demonstrate how Markov-chain Monte-Carlo sampling is a valuable tool for visualising and analysing the associated multidimensional likelihood surface.

Skyrmions and Clustering in Light Nuclei.

One of the outstanding problems in modern nuclear physics is to determine the properties of nuclei from the fundamental theory of the strong force, quantum chromodynamics (QCD). Skyrmions offer a novel approach to this problem by considering nuclei as solitons of a low energy effective field theory obtained from QCD. Unfortunately, the standard theory of Skyrmions has been plagued by two significant problems: (1)It yields nuclear binding energies that are an order of magnitude larger than experimental nuclear data, and (2)it predicts intrinsic shapes for nuclei that fail to match the clustering structure of light nuclei. Here we show that extending the standard theory of Skyrmions, by including the next lightest subatomic meson particles traditionally neglected, dramatically improves both of these aspects. We find Skyrmion clustering that now agrees with the expected structure of light nuclei, with binding energies that are much closer to nuclear data.

Quantitative analysis of tensor effects in the relativistic Hartree-Fock theory

Tensor force is identified in each meson-nucleon coupling in the relativistic Hartree-Fock theory. It is found that all the meson-nucleon couplings, except the $\sigma$-scalar one, give rise to the tensor force. The effects of tensor force on various nuclear properties can now be investigated quantitatively, which allows fair and direct comparisons with the corresponding results in the non-relativistic framework. The tensor effects on nuclear binding energies and the evolutions of the $Z,\,N = 8,\,20$, and $28$ magic gaps are studied. The tensor contributions to the binding energies are shown to be tiny in general. The $Z,\,N = 8$ and $20$ gaps are sensitive to the tensor force, but the $Z,\,N = 28$ gaps are not.

Many-body correlations in shell model effective interactions derived by ab initio methods and the Vlow-k approach**Supported by National Natural Science Foundation of China (11505056, U1732138, 11605054, 11790325, 11305108, 11575290, 11675148, 11747312, 11775316) and the Outstanding Young Talent Research Fund of Zhengzhou University (1521317002)

We investigate many-body correlations caused by two- and three-body (2-, 3bd) forces. Shell-model effective interactions derived from ab initio methods (coupled-cluster method, no-core shell model) are adopted. Vlow-k potentials, based on many-body perturbation theory, are also tested, especially for their cut-off dependence. We compare the central, tensor and spin-orbit interactions from microscopic theory to the fitted interactions. After the inclusion of the three-body force, the matrix elements become fairly close to those fitted directly to experimental data. Calculations of neutron-rich oxygen isotopes are performed, to clarify the effects of 3bd forces, tensor, and spin-orbit interactions on the nuclear binding and excitation energies. We find that the 3bd force can influence the binding energies greatly, which also determines the drip line position, while its effect on excitation energies is not very pronounced. The spin-orbit force, which is part of the 2bd force, can affect the shell structure explicitly, at least for neutron-rich systems.

Microscopically based energy density functionals for nuclei using the density matrix expansion. II. Full optimization and validation

We seek to obtain a usable form of the nuclear energy density functional that is rooted in the modern theory of nuclear forces. We thus consider a functional obtained from the density matrix expansion of local nuclear potentials from chiral effective field theory. We propose a parametrization of this functional carefully calibrated and validated on selected ground-state properties that is suitable for large-scale calculations of nuclear properties. The first component of this functional is a non-local functional of the density and corresponds to the direct part (Hartree term) of the expectation value of local chiral potentials on a Slater determinant. A second component is a local functional of the density and is obtained by applying the density matrix expansion to the exchange part (Fock term) of the expectation value of the local chiral potential. We apply the UNEDF2 optimization protocol to determine the coupling constants of this energy functional. We obtain a set of microscopically-constrained functionals for local chiral potentials from leading-order up to next-to-next-to-leading order with and without three-body forces and contributions from $\Delta$ excitations. These functionals are validated on the calculation of nuclear and neutron matter, nuclear mass tables, single-particle shell structure in closed-shell nuclei and the fission barrier of $^{240}$Pu. Quantitatively, they perform noticeable better than the more phenomenological Skyrme functionals. The inclusion of higher-order terms in the chiral perturbation expansion seems to produce a systematic improvement in predicting nuclear binding energies. This result is especially promising since all the fits have been performed at the single reference level of the energy density functional approach, where important collective correlations such as center-of-mass correction have not been taken into account yet.

Estimating Parameter Uncertainty in Binding-Energy Models by the Frequency-Domain Bootstrap.

We propose using the frequency-domain bootstrap (FDB) to estimate errors of modeling parameters when the modeling error is itself a major source of uncertainty. Unlike the usual bootstrap or the simple χ^{2} analysis, the FDB can take into account correlations between errors. It is also very fast compared to the Gaussian process Bayesian estimate as often implemented for computer model calibration. The method is illustrated with a simple example, the liquid drop model of nuclear binding energies. We find that the FDB gives a more conservative estimate of the uncertainty in liquid drop parameters than the χ^{2} method, and is in fair accord with more empirical estimates. For the nuclear physics application, there are no apparent obstacles to apply the method to the more accurate and detailed models based on density-functional theory.

Weizsäcker-Skyrme-type nuclear mass formula incorporating two combinatorial radial basis function prescriptions and their application

Within the improved Weizs\acker-Skyrme (WS)-type nuclear mass formulas, we systematically calculated one-nucleon and two-nucleon separated energy, \ensuremath{\alpha}-decay and \ensuremath{\beta}-decay energies, and the odd-even staggering (OES) of nuclear binding energies. As a result, the root-mean-square (rms) deviations of 2267 nuclei within the new improved WS-type mass formula are dropped from 493 to 167 keV, where 2267 nuclei are extracted from the atomic mass evaluation of 2012. Simultaneously, all the rms deviations of one-nucleon and two-nucleon separation energies and decay energies ${Q}_{\ensuremath{\alpha}},{Q}_{{\ensuremath{\beta}}^{\ensuremath{-}}},{Q}_{{\ensuremath{\beta}}^{+}}$, and ${Q}_{\mathrm{EC}}$ for more than 3000 nuclei are cut down by about 100--400 keV. Further, some basic physical observations of 988 boundary nuclei are predicted for providing reference to experiments. Finally, the overall neutron OESs and proton OESs have been systemically investigated and the residual error satisfies a normal distribution. The pairing gaps ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{n}}$ and ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{p}}$ of the isotopes of O, Ca, Ni, Zr, Sn, Gd, Qs, Pb, Pa, Ds and the isotonic magic chains of $N=28,50,82,126$ and even-even nuclei are also studied with dramatic improvements obtained. Especially, the rms of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{n}}$ and ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{p}}$ in these nuclei have been reduced by about 200 keV. The above physical quantities show important information for nuclear charts and the features of nuclear structure.