Simple Formula of β^+-Decay Half-Lives of Nuclei Far From β-Stable Line

We generalize the formula of β^+-decay half-lives [Phys.Rev.C 78 (2006) 014305] to include all availabledata of nuclei far from the β-stable line. Good agreement with experimental data is obtained by taking into accountthe even-odd (e-o) effect.It is shown that the e-o effect on β^+-deeay half-life is apparent for the allowed β^+-transition,whereas it is not apparent for the first and second forbidden transitions.This demonstrates that the allowed β^+-transitionis more sensitive to the nuclear even-odd effect than the forbidden β^+-transition.The underlying physics is discussed.This formula can be used to predict the β^+-decay half-lives of the nuclei far from β-stability.It is useful to experimentalphysicists for analyzing the data of β^+-decay.

Benford's Law and β-Decay Half-Lives

The experimental values of 2059 β-decay half-lives are systematically analyzed and investigated. We have found that they are in satisfactory agreement with Benford＇s law, which states that the frequency of occurrence of each figure, 1-9, as the first significant digit in a surprisingly large number of different data sets follows a logarithmic distribution favoring the smaller ones. Benford＇s logarithmic distribution of β-deeay half-lives can be explained in terms of Neweomb＇s justification of Benford＇s law and empirical exponential law of β-decay half-lives. Moreover, we test the calculated values of 6721 β-decay half-lives with the aid of Benford＇s law. This indicates that Benford＇s law is useful for theoretical physicists to test their methods for calculating β-decay half-lives.

Systematic Calculations of α-Decay Half-Lives Using Microscopic Deformed Potentials

A new version of the generalized density-dependent cluster model （GDDCM） is developed to describe an α particle tunneling through a deformed potential barrier. The microscopic deformed potential is numerically constructed in the double-folding model using the multipole ex- pansion method. The decay width of an α-cluster state is evaluated using the integral of the quasi-bound state wave function, the scattering state wave function, and the difference of poten- tials. We perform a systematic calculation of α-decay half-lives for favored transitions in even-even nuclei ranging from Z=52 to Z=104. The calculated half-lives are in good agreement with the experimental values. The relation between nuclear deformations and α-decay half-lives is also discussed in details.

Half-Lives for Alpha Radioactivity of Elements with 104 ≤Z≤120 within Unified Fission Model

α decay half-lives are calculated using the Qα values obtained by Semi-empirical Shell Model in the framework of the Unified Fission Model （UFM） with the Coulomb repulsion, nuclear attraction due to proximity potential, and rotational energy due to angular momentum transfer of a particle. In addition, the calculated and experimental half-lives of 425 nuclei are compared to check the validity of the model applied on α decay. The calculated half-lives of decay are in good agreement with the experimental data. Finally, some useful predications on the α decay half-lives are provided for future experiments.

Theoretical calculations of proton emission half-lives based on a deformed Gamow-like model

In this study,proton emission half-lives were investigated for deformed proton emitters with 53≤Z≤83based on the presented deformed Gamow-like model,where the deformation effect was included in the Coulomb potential.The experimental half-lives of proton emitters can be reproduced within a factor of 3.45.For comparison,the results from the universal decay law and the new Geiger-Nuttall law are also presented.Furthermore,the relevance of the half-lives to the angular momentum l for ^(117)La,^(121)Pr,^(135)Tb,and ^(141)Ho were analyzed,and the corresponding possible values of l were proposed:l=3,3,4,4.

An improved simple model for α decay half-lives

In this paper,using theαparticle preformation probabilities Pαfrom Xu et al.[Xu and Ren,Nucl.Phys.A 760,303(2005)],which were extracted by fitting experimental half-lives ofαdecay,based on a phenomenological harmonic oscillator potential model(HOPM)[Bayrak,J Phys G 47,025102(2020)],refitting 178αdecay half-lives of even-even nuclei obtained from the latest nuclear property table NUBASE2020,we obtain the only one adjustable parameter V0-162.6 MeV in the HOPM,i.e.,the depth of nuclear potential.The corresponding root-mean-square(rms)deviation isσ-0.322.Furthermore,to consider the contribution of centrifugal potential to unfavoredαdecay half-lives,adding a new term■(d and l are the adjustable parameter and orbital angular momentum carried away by emittedαparticle)to the logarithmic form of favoredαdecay half-lives under the HOPM framework,we propose an improved simple model(ISM)for calculating favored and unfavoredαdecay half-lives.Fitting the experimental half-lives of 205 unfavoredαdecay,we obtain d-0.381.The ISM is used to calculate the unfavoredαdecay half-lives of 128 odd-A and 77 odd-odd nuclei.The results improve by 54.2%and 53.6%,respectively,compared with HOPM.In addition,we extend the ISM to predict theαdecay half-lives of 144 nuclei with Z=117,118,119,and 120.For comparison,the improved model with eight parameters(DUR)proposed by Deng et al.[Deng,Phys.Rev.C 101,034307(2020)]and the modified universal decay law(MUDL)proposed by Soylu et al.[Soylu,Nucl.Phys.A 1013,122221(2021)]are also used.The predictions of these models and/or formulas are generally consistent with each other.

Simple model for cluster radioactivity half-lives in trans-lead nuclei

In this study,considering the modified preformation probability P_(c)to be log_(10)P_(c)=(A_(c)−1)/3log_(10)P_(α)+c′,where P_(α)and c′are theα-particle preformation probability and an adjustable parameter proposed by Wang et al.[Chin.Phys.C 45,044111(2021)],respectively,we extend a new simple model put forward by Bayrak[J.Phys.G 47,025102(2020)]to systematically study the cluster radioactivity half-lives of 28 trans-lead nuclei ranging from^(222)Fr to^(242)Cm,which is based on the Wentzel-Kramers-Brillouin approximation and Bohr–Sommerfeld quantization condition.For comparison,a universal decay law proposed by Qi et al.[Phys.Rev.C 80,044326(2009)],a three-parameter model-independent formula put forward by Balasubramaniam et al.[Phys.Rev.C 70,017301(2004)],and the semi-empirical model proposed by Tavares et al.[Eur.Phys.J.A 49,1(2013)]are used.Our calculated results reproduce the experimental data well,with a standard deviation of 0.818.Furthermore,we use this model to predict the cluster radioactivity half-lives of 51 possible cluster radioactive candidates whose cluster radioactivities are energetically allowed or observed but not yet quantified in NUBASE2020.

New Geiger-Nuttall law for cluster radioactivity half-lives

In this study,derived from Balasubramaniam’s formula[Phys.Rev.C 70,017301(2004)]and further considering the effect of the parent nucleus mass,blocking effect,and effect of reduced mass on cluster radioactivity half-lives,we propose a new Geiger-Nuttall law that is model-independent to systematically evaluate the halflives of this process for 16 even-even nuclei and 10 odd-A nuclei.For comparison,a single universal curve for cluster radioactivity andαdecay proposed by Poenaru[Phys.Rev.C 83,014601(2011)],a scaling law proposed by Horoi[J.Phys.G:Nucl.Part.Phys.30,945(2004)],an extension of the Viola-Seaborg formula fromαdecay to cluster radioactivity proposed by Ren et al.[Phys.Rev.C 70,034304(2004)],a new semi-empirical formula for exotic cluster decay proposed by Balasubramaniam et al.[Phys.Rev.C 70,017301(2004)],and a unified formula for the half-lives ofαdecay and cluster radioactivity proposed by Ni et al.[Phys.Rev.C 78,044310(2008)]are also used.The calculated results of our new Geiger-Nuttall law are in good agreement with the experimental half-lives,with the least rms being 0.606,and are better than the compared values.Moreover,we extend this formula to predict the cluster radioactivity half-lives of 51 nuclei whose decay energies are energetically allowed or observed but not yet quantified in NUBASE2020.

Systematic calculations of cluster radioactivity half-lives in trans-lead nuclei

In the present work,based on the Wentzel-Kramers-Brillouin(WKB)theory,considering the cluster preformation probability(Pc),we systematically investigate the cluster radioactivity half-lives of 22 trans-lead nuclei ranging from221Fr to242Cm.When the mass number of the emitted cluster Ac<28,Pcis obtained by the exponential relationship of Pcto theαdecay preformation probability(Pα)proposed by R.Blendowskeis et al.[Phys.Rev.Lett.61,1930(1988)],while Pαis calculated through the cluster-formation model(CFM).When Ac≥28,Pcis calculated through the charge-number dependence of Pcon the decay products proposed by Ren et al.[Phys.Rev.C70,034304(2004)].The half-lives of cluster radioactivity have been calculated by the density-dependent cluster model[Phys.Rev.C 70,034304(2004)]and by the unified formula of half-lives for alpha decay and cluster radioactivity[Phys.Rev.C 78,044310(2008)].For comparison,a universal decay law(UDL)proposed by Qi et al.[Phys.Rev.C 80,044326(2009)],a semi-empirical model for bothαdecay and cluster radioactivity proposed by Santhosh[J.Phys.G:Nucl.Part.Phys.35,085102(2008)],and a unified formula of half-lives for alpha decay and cluster radioactivity[Phys.Rev.C 78,044310(2008)]are also used.The calculated results of our work,Ni’s formula,and the UDL can well reproduce the experimental data and are better than those of Santhosh’s model.In addition,we extend this model to predict the half-lives for 51 nuclei,whose cluster radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020.

Systematic calculations of cluster radioactivity half-lives with a screened electrostatic barrier

In this study,based on Wentzel-Kramers-Brillouin theory,we systematically investigate the cluster radioactivity half-lives of 22 nuclei ranging from^(221)Fr to^(242)Cm using a phenomenological model that considers the screened electrostatic effect of the Coulomb potential.In this model,there are two adjustable parameters,t and g,which are related to the screened electrostatic barrier and the strength of the spectroscopic factor,respectively.The calculated results indicate that this model can effectively reproduce the experimental data,with a corresponding rootmean-square deviation of 0.660.In addition,we extend this model to predict the half-lives of possible cluster radioactive candidates whose cluster radioactivities are energetically allowed or observed but not yet quantified in the evaluated nuclear properties table NUBASE2020.The predicted results are consistent with those obtained using other theoretical models and/or empirical formulas,including the universal decay law proposed by Qi et al.[Phys.Rev.C 80,044326(2009)],a semi-empirical model for bothαdecay and cluster radioactivity proposed by Santhosh et al.[J.Phys.G 35,085102(2008)],and a unified formula for the half-lives ofαdecay and cluster radioactivity proposed by Ni et al.[Phys.Rev.C 78,044310(2008)].

Half-lives and fine structure for the α decay of deformed even-even nuclei

The α-decay properties of well-deformed even-even nuclei are systematically calculated within the multichannel cluster model (MCCM). Instead of working in the WKB framework, the quasibound solution to the coupled Schro¨dinger equation is presented with outgoing wave boundary conditions, and the coupling potential is taken into full account in terms of the general quantum theories. The calculated α-decay half-lives are found to agree well with the experimental data with a mean factor of less than 2. The fine structure observed in α decay is also well reproduced by the four-channel microscopic calculation. Very strikingly, the MCCM can give relatively precise descriptions of the branching ratio to excited 4+ states, which is often overestimated in the usual WKB calculations. We expect it to be a significant development of theoretical models toward quantitative descriptions of α transitions to high-spin daughter states.

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.

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.

Calculations of β-decay half-lives of proton-rich nuclei

We investigate the half-lives of β + /EC(electron capture) decay using the proton-neutron quasiparticle random-phase approximation(pnQRPA) with a δ-form Gamow-Teller residual interaction.Both particle-hole and particle-particle residual interactions are consistently introduced in dealing with the pnQRPA matrix equation.The sensitivity of the calculated half-lives to some physical quantities used in the calculations is examined.Calculations are performed for even-even neutron-deficient isotopes ranging from Z = 10 to Z = 76.Good agreement between experiment and theory is achieved especially for the nuclei far from stability,and the results of our calculations are discussed with comparison with other theoretical results.Predictions on the β-decay half-lives of some very neutron-deficient nuclei are also given for reference in future experiments.

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 double β^--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, double β^＋ -decay （β^＋β^＋） with emission of both two positrons and two neutrinos has not been observed up to date. In this article, we pcrform a systematic analysis on the candidates of double β^＋-decay, based on the 2012 nuclear mass table. Eight nuclei are found to be the good candidates for double β^＋-decay and their half-lives are predicted according to the generalization of the systematic law to double β^＋-decay. As far as we know, there is no theoretical result on double β^＋-decay of nucleus ^154Dy and our result is the first prediction on this nucleus. This is also the first complete research on eight double β^＋-decay candidates based on the available data of nuclear masses. It is expected that the calculated half-lives of double β^＋-decay in this article will be useful for future experimental search of double β^＋-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 aver-age 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 48Ca,76 Ge,and 136Xe 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.

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.

αDecay in extreme laser fields within a deformed Gamow-like model

In this study, the effect of extreme laser fields on the α decay process of ground-state even–even nuclei was investigated.Using the deformed Gamow-like model, we found that state-of-the-art lasers can cause a slight change in the α decay penetration probability of most nuclei. In addition, we studied the correlation between the rate of change of the α decay penetration probability and angle between the directions of the laser electric field and α particle emission for different nuclei. Based on this correlation, the average effect of extreme laser fields on the half-life of many nuclei with arbitrary α particle emission angles was calculated. The calculations show that the laser suppression and promotion effects on the α decay penetration probability of the nuclei population with completely random α particle-emission directions are not completely canceled.The remainder led to a change in the average penetration probability of the nuclei. Furthermore, the possibility of achieving a higher average rate of change by altering the spatial shape of the laser is explored. We conclude that circularly polarized lasers may be helpful in future experiments to achieve a more significant average rate of change of the α decay half-life of the nuclei population.

Systematic study on the proton radioactivity of spherical proton emitters

In this study, based on a two-potential approach, we systematically investigated the proton radioactivity half-lives of spherical proton emitters with 69≤Z≤81 from the ground and/or isomeric state, choosing the nuclear potential to be a modified Woods–Saxon potential that contains the isospin effect of the daughter nucleus. It was found that the calculated half-lives could reproduce the experimental data well. Furthermore, we extended this model to predict the half-lives of 17 protonemitting candidates whose radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020. For comparison, the unified fission model, Coulomb potential and proximity potential model, universal decay law for proton emission, and new Geiger–Nuttall law were also used. All the predicted results are consistent with each other.