Empirical formula for β~--decay half-lives of r-process nuclei

Experimental data ofβ--decay half-lives of nuclei with atomic number between 20 and 190 are investigated.A systematic formula has been proposed to calculateβ~--decay half-lives of neutron-rich nuclei,with a particular consideration on shell and pair effects,the decay energy Q as well as the nucleon numbers(Z,N).Although the formula has relatively few parameters,it reproduces the experimentalβ~--decay half-lives of neutron-rich nuclei very well.The predicted half-lives for the r-process relevant nuclei obtained with the current formula serve as reliable input in the r-process model calculations.     

β~--Decay Half-Lives for Waiting Point Nuclei Around N=126

We have systematically analyzed the experimental β--decay half-lives of waiting point heavy nuclei around neutron number N = 126. A new set of parameters for the exponential formula of β~--decay half-lives is proposed. The forbidden transition effects are included in the new set of parameters self-consistently. Theoretical β~--decay half-lives of nuclei around N = 126 are compared with recent theoretical results and experimental data. It is found that the new theoretical results are in better agreement with experimental data. The unknown β~--decay half-lives of some nuclei in this region are predicted for studies on nuclear structure far from stability and the nucleosynthesis in stars.     

Half-lives of double β~+-decay with two neutrinos

Nuclear doubleβ--decays with two neutrinos were observed for many years and a systematic law describing the relation between their half-lives and decay energies was also proposed recently[Phys Rev C,2014,89:064603].However,doubleβ+-decay(β+β+)with emission of both two positrons and two neutrinos has not been observed up to date.In this article,we perform a systematic analysis on the candidates of doubleβ+-decay,based on the 2012 nuclear mass table.Eight nuclei are found to be the good candidates for doubleβ+-decay and their half-lives are predicted according to the generalization of the systematic law to doubleβ+-decay.As far as we know,there is no theoretical result on doubleβ+-decay of nucleus154Dy and our result is the first prediction on this nucleus.This is also the first complete research on eight doubleβ+-decay candidates based on the available data of nuclear masses.It is expected that the calculated half-lives of doubleβ+-decay in this article will be useful for future experimental search of doubleβ+-decay.     

Calculation of double-β decay half-lives using an improved Primakoff-Rosen formula

A systematic analysis on experimental data of the half-lives of nuclear double-β decays with two neutrinos(2vββ) is performed based on the analytical formula proposed by Primakoff and Rosen.We improve the formula by considering the shell effects and refining the energy dependence of the phase-space factor.This improved formula can closely describe all available experimental half-lives of 2vβ~-β~-decays,both for ground-state transitions and transitions from ground states of parent nuclei to the first 0~+ excited states of daughter nuclei.The calculated half-lives agree with the experimental data of ground-state transitions of all known eleven nuclei with an average factor of 2.3.Further predictions are provided for 2vββ-decay candidates with decay energies above 0.5 MeV.We compare different theoretical predictions and emphasize the importance of experimental measurements on the half-lives of double-β transitions between the ground state of ~(48)Ca,~(76)Ge,and ~(136)Xe and the first 0~+ excited states of their corresponding daughter nuclei,which will be very useful for understanding the underlying mechanisms of double-β decays and for further studying the shell effects on nuclear transition matrix elements.     

r-过程路径核β——衰变半衰期的估算

分析了20 < A < 190范围内丰中子核β衰变的实验数据,根据半衰期随质子数、中子数以及衰变能变化所呈现的壳效应和对效应等特点,提出了一种有效估算丰中子核β衰变寿命的公式。新的计算公式形式简单包含了较少的参数、计算量小。用该公式能较为准确地再现丰中子核的β衰变半衰期。用RIKEN最新测量丰中子核半衰期检验了该公式的外推能力,本工作可以为r-过程研究提供可靠的输入数据。Experimental data of the β--decay half-lives for the nuclei with atomic number between 20 and 190 are investigated. We have systematically studied the shell effects and pairing effects on β--decay half-lives versus the decay energy Q and nucleon numbers (Z, N). An empirical formula has been proposed to calculate the β--decay half-lives of neutron-rich nuclei. The empirical formula is simple and has relatively few parameters. Experimental β--decay half-lives of the neutron-rich nuclei are well reproduced by the new formula. In addition, the extrapolating capacity of this formula has been checked with the very recent experimental data from RIKEN. The predicted half-lives for r-process relevant nuclei with the current formula can be served as the reliable input of r-process model calculations.     

The contribution of the first forbidden transitions to the nuclear β~--decay half-life

β-decay half-life is a key quantity for nuclear structure and nucleosynthesis studies.There exist large uncertainties in the contributions of allowed and forbidden transitions to the total β-decay life,which limits the resolution of the predicted β-decay half-life.We systematically study the contribution of the first forbidden(FF) transitions to the β~--decay half-life,and quantify it with a formula based on simple physics considerations.We also propose a new formula for calculation of the β~--decay half-life that includes the FF contribution.It is shown that the inclusion of the contribution of FF transitions significantly improves the precision of calculations of the β~--decay half-life.By fitting of the RQRPA results for neutron-rich Z=47,57 isotopes and N=80,94 isotones,the formula for the contribution of the FF transitions gives similar results as the RQRPA calculations.However,because of limited experimental data for the branching ratios of unstable nuclei,the fit parameters are not fully constrained.Therefore,the proposed formula for the β~--decay half-life is more suitable for calculations of half-lives than of the FF contributions.The formula could be used to predict the β~--decay half-life in nuclear structure studies as well as nucleosynthesis calculations in stars.     

Systematical law of β+-decay half-lives of nuclei far from β-stable line

Experimental data of nuclear β⁺-decay half-lives are systematically analyzed and investigated. We present an exponential law between the half-life of β⁺-decay with the same forbiddenness and the nucleon number (Z,N) of parent nucleus far from the β-stable line. A formula with four parameters is proposed to describe the β⁺-decay half-lives of nuclei far from stability. Experimental β⁺-decay half-lives of the first and second forbidden transitions are well reproduced by this simple formula. The physics of the exponential law is related to the statistical properties of β⁺-decay far from β-stable line.     

Competition between α-decay and β-decay for heavy and superheavy nuclei

In this work, the β-stable region for Z 90 is proposed based on a successful binding energy formula.The calculated β-stable nuclei in the β-stable region are in good agreement with the ones obtained by Mo¨ller et al. The half-lives of the nuclei close to the β-stable region are calculated and the competition between α-decay andβ-decay is systematically investigated. The calculated half-lives and the suggested decay modes are well in line with the experimental results. The decay modes are mostly β--decay above the β-stable region. Especially for Z 111,all the decay modes are β--decay. Regarding the nuclei above the β-stable region, α-decay and β--decay(α+β-)can occur simultaneously when Z 112. This is a very interesting phenomenon. The competition between α-decay and β-decay is very complex and drastic below the β-stable region. The predictions for half-lives and decay modes of the nuclei with Z =107–110 are presented in detail.     

Systematic Study on Alpha Decay Half-Lives of Superheavy Nuclei

The α-decay half-lives of a set of superheavy nuclear isotope chain from Z = 105 to 120 have been analyzed systematically within the WKB method, and some nuclear structure features are found. The decay barriers have been determined in the quasi-molecular shape path within the Generalized Liquid Drop Model (GLDM) including the proximity effects between nucleons in a neck and the mass and charge asymmetry. The results are in reasonable agreement with the published experimental data for the alpha decay half-lives of isotopes of charge 112, 114, and 116, of the element 294118 and of some decay products. A comparison of present calculations with the results by the DDM3Y effective interaction and by the Viola-Seaborg-Sobiczewski (VSS) formulae is also made. The experimental α decay half lives all stand in between the GLDM calculations and VSS formula results. This demonstrates the possibility of these models to provide reasonable estimates for the half-lives of nuclear decays by α emissions for the domain of SHN. The half-lives of these new nuclei are thus well tested from the reasonable consistence of the macroscopic, the microscopic, the empirical formulae and the experimental data. This also shows that the present data of SHN themselves are consistent.It could suggest that the present experimental claims on the existence of new elements Z = 110 ～ 118 are reliable.It is expected that greater deviations of a few SHN between the data and the model may be eliminated by further improvements on the precision of the measurements.     

Systematic Study on Alpha Decay Half-Lives of Superheavy Nuclei

The α-decay half-lives of a set of superheavy nuclear isotope chain from Z = 105 to 120 have been analyzed systematically within the WKB method, and some nuclear structure features are found. The decay barriers have been determined in the quasi-molecular shape path within the Generalized Liquid Drop Model （GLDM） including the proximity effects between nucleons in a neck and the mass and charge asymmetry. The results are in reasonable agreement with the published experimental data for the alpha decay half-llves of isotopes of charge 112, 114, and 116, of the element 294118 and of some decay products. A comparison of present calculations with the results by the DDM3Y effective interaction and by the Viola-Seaborg-Sobiczewski （VSS） formulae is also made. The experimental a decay half lives all stand in between the GLDM calculations and VSS formula results. This demonstrates the possibility of these models to provide reasonable estimates for the half-lives of nuclear decays by a emissions for the domain of SHN. The half-lives of these new nuclei are thus well tested from the reasonable consistence of the macroscopic, the microscopic, the empirical formulae and the experimental data. This also shows that the present data of SHN themselves are consistent. It could suggest that the present experimental claims on the existence of new elements Z =110 - 118 are reliable. It is expected that greater deviations of a few SHN between the data and the model may be eliminated by further improvements on the precision of the measurements.     

Systematic study of two-proton radioactivity within a Gamow-like model

In this study,based on the Gamow-like model,we systematically analyze two-proton(2p) radioactivity half-lives of nuclei near or beyond the proton drip line.It is found that the calculated results can reproduce experimental data well.Furthermore,using this model,we predict the half-lives of possible 2p radioactivity candidates whose 2p radioactivity is energetically allowed or observed but not yet quantified in the latest table of evaluated nuclear properties,i.e.,NUBASE2016.The predicted results are in good agreement with those from other theoretical models and empirical formulas,namely the effective liquid drop model(ELDM),generalized liquid drop model(GLDM),Sreej a formula,and Liu formula.     

Influence of nuclear physics inputs and astrophysical conditions on r-process

The rapid neutron-capture process(r-process) is one of the main mechanisms to explain the origin of heavy elements in the universe. Although the past decades have seen great progress in understanding this process, the related nuclear physics inputs to r-process models include significant uncertainty. In this study, ten nuclear mass models, including macroscopic, macroscopicmicroscopic, and microscopic models, are used to calculate the β-decay rates and neutron-capture rates of the neutron-rich isotopes for the r-process simulations occurring in three classes of astrophysical conditions. The final r-process abundances include uncertainties introduced by the nuclear mass model mainly through the variation of neutron-capture rates, whereas the uncertainties of β-decay rates make a relatively small contribution. The uncertainties in different astrophysical scenarios are also investigated,and are found to be connected to the diverse groups of nuclei produced during nucleosynthesis.     

Theoretical approaches to alpha decay half-lives of super-heavy nuclei

We consider the systematics of α-decay half-lives of super-heavy nuclei versus the decay energy and the total α-kinetic energy. We calculate the half-lives using the experimental Q_α values. The computed half-lives are compared with the experimental data and with existing empirical estimates and are found to be in good agreement.Also, we obtain α-preformation factors from the ratio between theoretical and experimental results for some superheavy nuclei and evaluate the standard deviation. The results indicate the acceptability of the approach.     

α-decay half-lives of superheavy nuclei and general predictions

The generalized liquid drop model （GLDM） and the cluster model have been employed to calculate the α-decay half-lives of superheavy nuclei （SHN） using the experimental α-decay Q values. The results of the cluster model are slightly poorer than those from the GLDM if experimental Q values are used. The prediction powers of these two models with theoretical Q values from Audi et al. （QAudi） and Muntian et al. （QM） have been tested to find that the cluster model with QAudi and QM could provide reliable results for Z 〉 112 but the GLDM with QAudi for Z 112. The half-lives of some still unknown nuclei are predicted by these two models and these results may be useful for future experimental assignment and identification.     

Investigation of proton radioactivity with the effective liquid drop model

Proton radioactivity has been investigated using the effective liquid drop model with varying mass asymmetry shapes and effective inertial coefficients. An effective nuclear radius constant formula replaces the old empirical one in the calculations. The theoretical half-lives are in good agreement with the available experimental data. All the deviations between the calculated logarithmic half-lives and the experimental values are less than 0.8.The root-mean-square(rms) deviation is 0.523. Predictions for the half-lives of proton radioactivity are made for elements across the periodic table. From the theoretical results, there are 11 candidate nuclei for proton radioactivity in the region Z <51. In the region Z >83, no nuclei are suggested as probable candidate nuclei for proton radioactivity within the selected range of half-lives studied.     

Systematic description of nuclear electric quadrupole moments

The nuclear electric quadrupole moment(NQM) is one of the fundamental bulk properties of the nucleus with which nuclear deformations can be investigated. The number of measured NQMs is significantly less than that of known masses, and there is still no global NQM formula for all bound nuclei. In this paper, we propose an analytical formula, which includes the shell corrections and which is the function of the charge number, mass number, spin,charge radius, and nuclear deformation, for calculating the NQMs of all bound nuclei. Our calculated NQMs of 524 nuclei in their ground states are reasonable compared to the experimental data based on the nuclear deformation parameters derived from the Weizs¨acker-Skyrme(WS) nuclear mass models. Smaller rms deviations between the calculated NQMs and experimental data indicate that the deformation parameters predicted from the WS mass models are reasonable. In addition, 161 unmeasured NQMs with known spins are also predicted with the proposed formula.     

Global α-decay study based on the mass table of the relativistic continuum Hartree-Bogoliubov theory

The α-decay energies(Qα) are systematically investigated with the nuclear masses for 10 Z 120 isotopes obtained by the relativistic continuum Hartree-Bogoliubov(RCHB) theory with the covariant density functional PC-PK1, and compared with available experimental values. It is found that the α-decay energies deduced from the RCHB results present a similar pattern to those from available experiments. Owing to the large predicted Qαvalues( 4 Me V), many undiscovered heavy nuclei in the proton-rich side and super-heavy nuclei may have large possibilities for α-decay. The influence of nuclear shell structure on α-decay energies is also analysed.     

Competition between α-decay and β-decay for heavy andsuperheavy nuclei

In this work, the β-stable region for Z≥90 is proposed based on a successful binding energy formula. The calculated β-stable nuclei in the β-stable region are in good agreement with the ones obtained by Möller et al. The half-lives of the nuclei close to the β-stable region are calculated and the competition between α-decay and β-decay is systematically investigated. The calculated half-lives and the suggested decay modes are well in line with the experimental results. The decay modes are mostly β^--decay above the β-stable region. Especially for Z≤111, all the decay modes are β^--decay. Regarding the nuclei above the β-stable region, α-decay and β^--decay (α+β^-) can occur simultaneously when Z≥112. This is a very interesting phenomenon. The competition between α-decay and β-decay is very complex and drastic below the β-stable region. The predictions for half-lives and decay modes of the nuclei with Z=107-110 are presented in detail.     

Calculations of the β-decay half-lives of neutron-deficient nuclei

In this work,β~+/EC decays of some medium-mass nuclei are investigated within the extended quasiparticle random-phase approximation(QRPA),where neutron-neutron,proton-proton and neutron-proton(np) pairing correlations are taken into consideration in the specialized Hartree-Fock-Bogoliubov(HFB) transformation.In addition to the pairing interaction,the Br¨uckner G-matrix obtained with the charge-dependent Bonn nucleon-nucleon force is used for the residual particle-particle and particle-hole interactions.Calculations are performed for even-even proton-rich isotopes ranging from Z =24 to Z =34.It is found that the np pairing interaction plays a significant role inβ-decay for some nuclei far from stability.Compared with other theoretical calculations,our calculations show good agreement with the available experimental data.Predictions of β-decay half-lives for some very neutron-deficient nuclei are made for reference.     

Fully self-consistent calculation ofβ-decay half-lives within Skyrme energy density functional

β-decay half-lives of some magic and semi-magic nuclei have been studied in a fully self-consistent Skyrme Hartree-Fock(HF) plus charge-exchange random phase approximation(RPA).The self-consistency is addressed,in that the same Skyrme energy density functional is adopted in the calculation of ground states and Gamow-Teller excited states.First,the impact of J2 terms on the β-decay half-lives is investigated by using the SGII interaction,revealing a large influence.Subsequently,numerical calculations are performed for the selected nuclei with Skyrme energy density functionals SGII,LNS,SKX,and SAMi.Finally,comparisons to available experimental data and predictions of different theoretical models are discussed.