Improved nuclear mass formula with an additional term from the Fermi gas model

Nuclear mass is a fundamental property of nuclear physics and a necessary input in nuclear astrophysics.Owing to the complexity of atomic nuclei and nonperturbative strong interactions,conventional physical models cannot completely describe nuclear binding energies.In this study,the mass formula was improved by considering an additional term from the Fermi gas model.All nuclear masses in the Atomic Mass Evaluation Database were reproduced with a root-mean-square deviation(RMSD)of -1.86 MeV(1.92 MeV).The new mass formula exhibits good performance in the neutron-rich nuclear region.The RMSD decreases to 0.393 MeV when the ratio of the neutron number to the proton number is≥1.6.

Comparative study of nuclear masses in the relativistic mean-field model

With experimental masses updated from AME11,the predictive power of relativistic mean-field(RMF) mass model is carefully examined and compared with HFB-17,FRDM,WS*,and DZ28 mass models.In the relativistic mean-field model,the calculation with the PC-PK1 has improved significantly in describing masses compared to the TMA,especially for the neutron-deficient nuclei.The corresponding rms deviation with respect to the known masses falls to 1.4 MeV.Furthermore,it is found that the RMF mass model better describes the nuclei with large deformations.The rms deviation for nuclei with the absolute value of quadrupole deformation parameter greater than 0.25 falls to 0.93,crossing the 1 MeV accuracy threshold for the PC-PK1,which may indicate the new model is more suitable for those largely-deformed nuclei.In addition,the necessity of new high-precision experimental data to evaluate and develop the nuclear mass models is emphasized as well.

Improved mass relations of mirror nuclei

In this study,we revisit the previous mass relations of mirror nuclei by considering 1/N-and 1/Z-dependent terms and the shell effect across a shell.The root-mean-squared deviation is 66 keV for 116 nuclei with neutron number N≥10,as com-pared with experimental data compiled in the AME2020 database.The predicted mass excesses of 173 proton-rich nuclei,including 98 unknown nuclei,are tabulated in the Supplemental Material herein with competitive accuracy.

Machine learning the nuclear mass

Background:The masses of-2500 nuclei have been measured experimentally;however,>7000 isotopes are predicted to exist in the nuclear landscape from H(Z=1)to Og(Z=118)based on various theoretical calculations.Exploring the mass of the remaining isotopes is a popular topic in nuclear physics.Machine learning has served as a powerful tool for learning complex representations of big data in many fields.Purpose:We use Light Gradient Boosting Machine(LightGBM),which is a highly efficient machine learning algorithm,to predict the masses of unknown nuclei and to explore the nuclear landscape on the neutron-rich side from learning the measured nuclear masses.Methods:Several characteristic quantities(e.g.,mass number and proton number)are fed into the LightGBM algorithm to mimic the patterns of the residual δ(Z,A)between the experimental binding energy and the theoret-ical one given by the liquid-drop model(LDM),Duflo–Zucker(DZ,also dubbed DZ28)mass model,finite-range droplet model(FRDM,also dubbed FRDM2012),as well as the Weizsacker–Skyrme(WS4)model to refine these mass models.Results:By using the experimental data of 80%of known nuclei as the training dataset,the root mean square devia-tions(RMSDs)between the predicted and the experimental binding energy of the remaining 20%are approximately 0.234±0.022,0.213±0.018,0.170±0.011,and 0.222±0.016 MeV for the LightGBM-refined LDM,DZ model,WS4 model,and FRDM,respectively.These values are approximately 90%,65%,40%,and 60%smaller than those of the corresponding origin mass models.The RMSD for 66 newly measured nuclei that appeared in AME2020 was also significantly improved.The one-neutron and two-neutron separation energies predicted by these refined models are consistent with several theoretical predictions based on various physical models.In addition,the two-neutron separation energies of several newly measured nuclei(e.g.,some isotopes of Ca,Ti,Pm,and Sm)pre-dicted with LightGBM-refined mass models are also in good agreement with the latest experimental data.Conclusions:LightGBM can be used to refine theoretical nuclear mass models and predict the binding energy of unknown nuclei.Moreover,the correlation between the input characteristic quantities and the output can be inter-preted by SHapley additive exPlanations(a popular explainable artificial intelligence tool),which may provide new insights for developing theoretical nuclear mass models.

Principal components of nuclear mass models

Principal component analysis(PCA)is employed to extract the principal components(PCs)present in nuclear mass models for the first time.The effects from different nuclear mass models are reintegrated and reorganized in the extracted PCs.These PCs are recombined to build new mass models,which achieve better accuracy than the original theoretical mass models.This comparison indicates that using the PCA approach,the effects contained in different mass models can be collaborated to improve nuclear mass predictions.

Nuclear mass predictions based on a deep neural network and finite-range droplet model(2012)

A neural network with two hidden layers is developed for nuclear mass prediction,based on the finiterange droplet model(FRDM12).Different hyperparameters,including the number of hidden units,choice of activation functions,initializers,and learning rates,are adjusted explicitly and systematically.The resulting mass predictions are achieved by averaging the predictions given by several different sets of hyperparameters with different regularizers and seed numbers.This can provide not only the average values of mass predictions but also reliable estimations in the mass prediction uncertainties.The overall root-mean-square deviations of nuclear mass are reduced from 0.603 MeV for the FRDM12 model to 0.200 MeV and 0.232 MeV for the training and validation sets,respectively.

Role of magnetic fields on the outer crust in a magnetar

We explore the properties of 4110 nuclides from Z=5 to Z=82 with the Sky3D code and the composition of the outer crust in magnetars under extreme magnetic fields.The effects of the variation in nuclear masses due to magnetic fields on the outer crust are comprehensively studied.The neutron-drip transition pressure,equation of state,and neutron fraction in the outer crust are also discussed.

High precision nuclear mass predictions towards a hundred kilo-electron-volt accuracy

Mass is a fundamental property and an important fingerprint of atomic nucleus.It provides an extremely useful test ground for nuclear models and is crucial to understand energy generation in stars as well as the heavy elements synthesized in stellar explosions.Nuclear physicists have been attempting at developing a precise,reliable,and predictive nuclear model that is suitable for the whole nuclear chart,while this still remains a great challenge even in recent days.Here we employ the Fourier spectral analysis to examine the deviations of nuclear mass predictions to the experimental data and to present a novel way for accurate nuclear mass predictions.In this analysis,we map the mass deviations from the space of nucleon number to its conjugate space of frequency,and are able to pin down the main contributions to the model deficiencies.By using the radial basis function approach we can further isolate and quantify the sources.Taking a pedagogical mass model as an example,we examine explicitly the correlation between nuclear effective interactions and the distributions of mass deviations in the frequency domain.The method presented in this work,therefore,opens up a new way for improving the nuclear mass predictions towards a hundred kilo-electron-volt accuracy,which is argued to be the chaos-related limit for the nuclear mass predictions.

Ground-state mass of ^(22)Al and test of state-of-the-art ab initio calculations

The ground-state mass excess of the T_(z)=−2 drip-line nucleus ^(22)Al is measured for the first time as 18103(10)keV using the newly-developed Bρ-defined isochronous mass spectrometry method at the cooler storage ring in Lanzhou.The new mass excess value allowed us to determine the excitation energies of the two low-lying 1+states in ^(22)Al with significantly reduced uncertainties of 51 keV.When compared to the analogue states in its mirror nucleus ^(22)F,the mirror energy differences of the two 1^(+)states in the ^(22)Al-^(22)F mirror pair are determined to be−625(51)keV and−330(51)keV.The excitation energies and mirror energy differences are used to test the state-of-the-art ab initio valence-space in-medium similarity renormalization group calculations with four sets of interactions derived from the chiral effective field theory.The mechanism leading to the large mirror energy differences is investigated and attributed to the occupation of theπs_(1/2) orbital.

Shape Decoupling Effects and Rotation of Deformed Halo Nuclei

With the development of radioactive-ion-beam facilities,many exotic phenomena have been discovered or predicted in the nuclei far from the stability line,including cluster structure,shell structure,deformed halo,and shape decoupling effects.The study of exotic nuclear phenomena is at the frontier of nuclear physics nowadays.The covariant density functional theory(CDFT)is one of the most successful microscopic models in describing the structure of nuclei in almost the whole nuclear chart.Within the framework of CDFT,toward a proper treatment of deformation and weak binding,the deformed relativistic Hartree-Bogoliubov theory in continuum(DRHBc)has been developed.In this contribution,we review the applications and extensions of the DRHBc theory to the study of exotic nuclei.The DRHBc theory has been used to investigate the deformed halos in B,C,Ne,Na,and Mg isotopes and the theoretical descriptions are reasonably consistent with available data.A DRHBc Mass Table Collaboration has been founded,aiming at a high precision nuclear mass table with deformation and continuum effects included,which is underway.By implementing the angular momentum projection based on the DRHBc theory,the rotational excitations of deformed halos have been investigated and it is shown that the deformed halos and shape decoupling effects also exist in the low-lying rotational excitation states of deformed halo nuclei.

Influence of binding energies of electrons on nuclear mass predictions

Nuclear mass contains a wealth of nuclear structure information, and has been widely employed to extract the nuclear effective interactions. The known nuclear mass is usually extracted from the experimental atomic mass by subtracting the masses of electrons and adding the binding energy of electrons in the atom. However, the binding energies of electrons are sometimes neglected in extracting the known nuclear masses. The influence of binding energies of electrons on nuclear mass predictions are carefully investigated in this work. If the binding energies of electrons are directly subtracted from the theoretical mass predictions, the rms deviations of nuclear mass predictions with respect to the known data are increased by about 200 keV for nuclei with Z, N ~〉 8. Furthermore, by using the Coulomb energies between protons to absorb the binding energies of electrons, their influence on the rms deviations is significantly reduced to only about 10 keV for nuclei with Z, N ≥ 8. However, the binding energies of electrons are still important for the heavy nuclei, about 150 keV for nuclei around Z = 100 and up to about 500 keV for nuclei around Z = 120. Therefore, it is necessary to consider the binding energies of electrons to reliably predict the masses of heavy nuclei at an accuracy of hundreds of keV.

Exploring the uncertainties in theoretical predictions of nuclear β-decay half-lives

Nuclear β-decay half-lives are predicted based on an empirical formula and the mass predictions from various nuclear models.It is found that the empirical formula can reproduce the nuclearβ-decay half-lives well,especially for short-lived nuclei with T_(1/2)<1s.The theoretical half-life uncertainties fromβ-decay energies and the parameters of the empirical formula are further investigated.It is found that the uncertainties of the half-lives are relatively large for heavy nuclei and nuclei near the neutron-drip line.For nuclei on the r-process path,the uncertainties for those with N=126 are about one order of magnitude,which are much larger than the uncertainties for those with N=50 and 82.However,theoretical uncertainties from the parameters of the empirical formula are relatively small for the nuclei on the r-process path,which indicates that the empirical formula is very suitable for predicting theβ-decay half-lives in r-process simulations.

Extending the nuclear chart by continuum:From oxygen to titanium

Nuclear masses ranging from O to Ti isotopes are systematically investigated with relativistic continuum Hartree-Bogoliubov(RCHB)theory,which can provide a proper treatment of pairing correlations in the presence of the continuum.From O to Ti isotopes,there are 402 nuclei predicted to be bound by the density functional PC-PK1.For the 234 nuclei with mass measured,the root mean square(rms)deviation is 2.23 MeV.It is found that the proton drip-lines predicted with various mass models are roughly the same and basically agree with the observation.The neutron drip-lines predicted,however,are quite diferent.Due to the continuum couplings,the neutron drip-line nuclei predicted are extended further neutron-rich than other mass models.By comparison with finite-range droplet model(FRDM),the neutron drip-line nucleus predicted by RCHB theory has respectively2(O),10(Ne),10(Na),6(Mg),8(Al),6(Si),8(P),6(S),14(K),10(Ca),10(Sc),and 12(Ti)more neutrons.

Ability of the radial basis function approach to extrapolate nuclear mass

The ability of the radial basis function(RBF)approach to extrapolate the masses of nuclei in neutron-rich and superheavy regions is investigated in combination with the Duflo-Zuker(DZ31),Hartree–Fock-Bogoliubov(HFB27),finite-range droplet model(FRDM12)and Weizsäcker-Skyrme(WS4)mass models.It is found that when the RBF approach is employed with a simple linear basis function,different mass models have different performances in extrapolating nuclear masses in the same region,and a single mass model may have different performances when it is used to extrapolate nuclear masses in different regions.The WS4 and FRDM12 models(two macroscopic–microscopic mass models),combined with the RBF approach,may perform better when extrapolating the nuclear mass in the neutron-rich and superheavy regions.

Nuclear chart in covariant density functional theory with dynamic correlations: From oxygen to tin

Nuclear masses of even-even nuclei with the proton number 8≤Z≤50(O to Sn isotopes)from the proton drip line to neutron drip line are investigated using the triaxial relativistic Hartree-Bogoliubov theory with the relativistic density functional PC-PK1.Further,the dynamical correlation energies(DCEs)associated with the rotational motion and quadrupole-shaped vibrational motion are taken into account by the five-dimensional collective Hamiltonian(5DCH)method.The root-mean-square deviation with respect to the experimental masses reduces from 2.50 to 1.59 MeV after the consideration of DCEs.The inclusion of DCEs has little influence on the position of drip lines,and the predicted numbers of bound even-even nuclei between proton and neutron drip lines from O to Sn isotopes are 569 and 564 with and without DCEs,respectively.

Examination of machine learning for assessing physical effects:Learning the relativistic continuum mass table with kernel ridge regression

The kernel ridge regression(KRR)method and its extension with odd-even effects(KRRoe)are used to learn the nuclear mass table obtained by the relativistic continuum Hartree-Bogoliubov theory.With respect to the binding energies of 9035 nuclei,the KRR method achieves a root-mean-square deviation of 0.96 MeV,and the KRRoe method remarkably reduces the deviation to 0.17 MeV.By investigating the shell effects,one-nucleon and twonucleon separation energies,odd-even mass differences,and empirical proton-neutron interactions extracted from the learned binding energies,the ability of the machine learning tool to grasp the known physics is discussed.It is found that the shell effects,evolutions of nucleon separation energies,and empirical proton-neutron interactions are well reproduced by both the KRR and KRRoe methods,although the odd-even mass differences can only be reproduced by the KRRoe method.

Global analysis of Skyrme forces with higher-order density dependencies

The density-dependent term in Skyrme forces is essential to simulate three-body and many-body correlations beyond the low-momentum two-body interaction. We speculate that a single density term may be insumcient and a higher-order density dependent term is added. The present work investigates the influence of higher-order density dependencies based on extended UNEDF0 and SkM＊ forces. Global descriptions of nuclear masses and charge radii are presented. The extended UNEDF0 force gives a global rms error on binding energies of 1.29 MeV. The influence on fission barriers and equation of state are also investigated. Perspectives to improve Skyrme forces are discussed, including global center-of-mass corrections and Lipkin-Nogami pairing corrections.

Toward precision mass measurements of neutron-rich nuclei relevant to r-process nucleosynthesis

The open question of where, when, and how the heavy elements beyond iron enrich our Universe has triggered a new era in nuclear physics studies. Of all the relevant nuclear physics inputs, the mass of very neutron-rich nuclides is a key quantity for revealing the origin of heavy elements beyond iron. Although the precise determination of this property is a great challenge, enormous progress has been made in recent decades, and it has contributed significantly to both nuclear structure and astrophysical nucleosynthesis studies. In this review, we first survey our present knowledge of the nuclear mass surface, emphasizing the importance of nuclear mass precision in r-process calculations. We then discuss recent progress in various methods of nuclear mass measurement with a few selected examples. For each method, we focus on recent breakthroughs and discuss possible ways of improving the weighing of r-process nuclides.

Possible existence of bound nuclei beyond neutron drip lines driven by deformation

Based on the relativistic calculations of the nuclear masses in the transfermium region from No(Z=102)to Ds(Z=110)using the deformed relativistic Hartree-Bogoliubov theory in continuum(DRHBc),the possible existence of bound nuclei beyond the neutron drip lines is studied.The two-neutron and multi-neutron emission bound nuclei beyond the primary neutron drip line of N=258 are predicted in Z=106,108,and 110 isotopes.A detailed microscopic mechanism investigation reveals that nuclear deformation plays a vital role in the existence of bound nuclei beyond the drip line.Furthermore,not only the quadrupole deformation β_(2) but also the higher orders of deformation are indispensable in the reliable description of the phenomenon of reentrant binding.

Precision mass measurements of short-lived nuclides at HIRFL-CSR in Lanzhou

In recent years, extensive short-lived nuclear mass measurements have been carried out at the Heavy- Ion Research Facility （HIRFL） in Lanzhou using Isochronous Mass Spectrometry （IMS）. The obtained mass values have been successfully applied to nuclear structure and astrophysics studies. In this contribution, we give a brief introduction to the nuclear mass measurements at HIRFL-CSR facility. Main technical developments are described and recent results are summarized. Furthermore, we envision the future perspective for the next-generation storage ring facility HIAF in Huizhou.