Systematic study of decay halflives based on Gamow–like model with a screened electrostatic barrier
Abstract
In the present work we systematically study decay halflives of nuclei using the modified Gamowlike model which includes the effects of the centrifugal potential and electrostatic shielding. For the case of eveneven nuclei, this model contains two adjustable parameters: the parameter related to the screened electrostatic barrier and the radius constant , while for the case of oddodd and oddA nuclei, it is added a new parameter i.e. hindrance factor which is used to describe the effect of an oddproton and/or an oddneutron. Our calculations can well reproduce the experimental data. In addition, we use this modified Gamowlike model to predict the decay halflives of seven eveneven nuclei with and some unsynthesized nuclei on their decay chains.
keywords:
decay, Gamowlike model, electrostatic shielding, unsynthesized nuclei1 Introduction
decay, the spontaneous emission of a He by the nucleus and the formation of a new nuclides, was first defined by Rutherford in 1899. Since then, great efforts have been made in the realm of both theory and experiment, e.g., from the discovery of the atomic nucleus by scattering to the GeigerNuttall law describing a relationship between decay halflife and decay energy PhysRevC.85.044608 (); Dong2005 (); VIOLA1966741 (); PhysRevLett.103.072501 (), from the barrier tunneling theory based on the quantum mechanics to the investigation of superheavy nuclei(SHN) Gamow1928 (); 00344885783036301 (); RevModPhys.84.567 (); PhysRevLett.110.242502 (); SOBICZEWSKI2007292 (); RevModPhys.70.77 (); PhysRevLett.104.142502 (). decay, as an important tool to investigate SHN, provides abundant information about the nuclear structure and stability of SHN00344885783036301 (); Hofmann2016 (); PhysRevC.85.044608 (); Yang2015 (). Nowadays, there are many theoretical models used to study decay including the cluster model PhysRevLett.65.2975 (); PhysRevC.74.014304 (); XU2005303 (), the unified model for decay and capture PhysRevC.73.031301 (); PhysRevC.92.014602 (), the liquid drop model 03054616510005 (); PhysRevC.48.2409 (); 09543899268305 (); PhysRevC.74.017304 (); GUO2015110 (), the twopotential approach 16741137411014102 (); PhysRevC.94.024338 (); PhysRevC.93.034316 (); PhysRevC.95.014319 (); PhysRevC.95.044303 (); PhysRevC.96.024318 (); PhysRevC.97.044322 (), the empirical formulas PhysRevC.85.044608 (); 09543899391015105 (); PhysRevC.80.024310 (); 09543899425055112 () and others PhysRevLett.59.262 (); SANTHOSH201528 (); PhysRevC.87.024308 (); 09543899312005 (); PhysRevC.81.064318 (); QI2014203 ().
Recently, K. Pomorski et al. proposed a Gamowlike model which is a simple phenomenological model based on the Gamow theory for the evaluations of halflife for decay PhysRevC.87.024308 (); 140248962013T154014029 (), while the nuclear potential is chosen as the square potential well, the centrifugal potential is ignored and the Coulomb potential is taken as the potential of a uniformly charged sphere with radius defined as Eq.(7). They also extended this model to study the proton radioactivityZdeb2016 (), for the proton radioactivity shares the same mechanism as the decay. In 2016, Niu Wan et al. systematically calculated the screened decay halflives of the emitters with proton number by considering the electrons in different external environments such as neutral atoms, a metal, and so on. They found that the decay energy and the interaction potential between particle and daughter nucleus are both changed due to the electrostatic shielding effect. And the electrostatic shielding effect is found to be closely related to the decay energy and its proton numberWan2016 (). In 2017, R. Budaca and A. I. Budaca proposed a simple analytical model based on the WKB approximation for the barrier penetration probability which includes the centrifugal and overlapping effects besides the electrostatic repulsionBudaca2017 (). In their model, there is only one parameter which is used to describe the electrostatic shielding effect of Coulomb potential by using the Hulthen potentialBudaca2017 (). They systematically calculated the halflives of proton emission for nuclei. The results can well reproduce the experimental data. Combining these points, in this work we modify the Gamowlike model proposed by K. Pomoski et al., considering the shielding effect of the Coulomb potential and the influence of the centrifugal potential, to systematically study the decay halflives. All the database are taken from the latest atomic nucleus parameters from NUBASE 2016 16741137413030001 (). We also extend our model to predict the decay halflives of seven eveneven superheavy nuclei with and some unsynthesized nuclei on their decay chains.
This article is organized as follows. In Sec. II the theoretical framework for decay halflife is described in detail including Gamowlike model and other models such as Coulomb potential and Proximity potential model (CPPM) with Bass73 formalism, the ViolaâSeaborgâSobiczewski (VSS) empirical formula, the Universal curve (UNIV), Royer formula, the Universal decay law (UDL) and the NiRenDongXu empirical formula (NRDX). In Sec. III, the detailed calculations, discussion and predictions are provided. A brief summary is given in Sec. IV.
2 Theoretical Framework
2.1 Gamowlike model
decay halflife , an important indicator of nuclear stability, can be calculated by the decay constant as
(1) 
where is the socalled hindrance factor of decay due to the effect of an oddproton and/or an oddneutron. For the eveneven nuclei, = 0, while for nuclei with an odd number of nucleons i.e. even, odd nuclei or odd, even , odd, odd nuclei . The decay constant is given by PhysRevC.83.014601 ()
(2) 
where represents the preformation probability of particles in decay. According to Ref. PhysRevC.87.024308 (), it can be known that the value of the preformation probability can be changed by adjusting the radius constant appropriately. The results show that the best fitting result can be obtained with =1, meanwhile fm, which also confirms the conclusion of Refs. PhysRevC.83.014601 (); 0954389917S045 (). Then we choose =1 in this work.
given in Eq. (2) represents the penetration probability of the particle crossing the barrier, calculated by the classical WKB approximation. Its concrete representation in the Gamowlike model is expressed as
(3) 
here is the kinetic energy of particle emitted during decay. and are decay energy and the mass number of the parent nucleus, respectively. is the classical turning point. It satisfies the condition . is the reduced mass of the particle and the daughter nucleus in the centerofmass coordinate with and being masses of the daughter nucleus and particle. is the total –daughter nucleus interaction potential.
In general, the –daughter nucleus electrostatic potential is by default of the Coulomb type as
(4) 
where and are the proton numbers of particle and daughter nucleus. Whereas, in the process of decay, for the superposition of the involved charges, movement of the emitted particle which generates a magnetic field and the inhomogeneous charge distribution of the nucleus, the emitted daughter nucleus electrostatic potential behaves as a Coulomb potential at short distance and drop exponentially at large distance i.e. the screened electrostatic effectBudaca2017 (). This behavior of electrostatic potential can be described as the Hulthen type potential which is widely used in nuclear, atomic, molecular and solid state physicsdoi:10.1063/1.4995175 (); PhysRevC.91.034614 () and defined as
(5) 
where is the screening parameter. In this framework, the total –daughter nucleus interaction potential is given by
(6) 
where is the depth of the square well. and are the Hulthen type of screened electrostatic Coulomb potential and centrifugal potential, respectively. The spherical square well radius is equal to the sum of the radii of both daughter nucleus and particle, it is expressed as
(7) 
where and are the mass number of the daughter nucleus and particle, respectively. , the radius constant, is the adjustable parameter in our model.
Because is a necessary correction for onedimensional problem Gur31 (), the centrifugal potential is chose as the Langer modified form in this work. It can be expressed as
(8) 
where is the orbital angular momentum taken away by the particle. for the favored decays, while for the unfavored decays. Based on the conservation laws of party and angular momentum PhysRevC.82.059901 (), the minimum angular momentum taken away by the particle can be determined by
(9) 
where . , , , represent spin and parity values of the parent and daughter nuclei, respectively.
The represents the collision frequency of particle in the potential barrier. It can be calculated with the oscillation frequency and expressed as PhysRevC.81.064309 ()
(10) 
where is the nucleus rootmeansquare (rms) radius and is the radius of the parent nucleus. is the main quantum number with and being the radial quantum number and the angular quantity quantum number, respectively. In the work of Ref. PhysRevC.69.024614 (), for decay, can be obtained by
(11) 
2.2 Other models
2.2.1 Coulomb potential and Proximity potential model with proximity potential Bass73 formalism (CPPMBass73)
In CPPM, the decay halflife is related to the decay constant as
(12) 
where the decay constant can be obtained by
(13) 
The assault frequency can be calculated with the oscillation frequency and expressed as
(14) 
where is the Planck constant. The zeropoint vibration energy can be calculated with and expressed as Poenaru1986 ()
(15) 
denote the semiclassical WKB barrier penetration probability, which is expressed as
(16) 
where and are the classical turning points which satisfy the conditions . The total interaction potential , between the emitted proton and daughter nucleus, including nuclear, Coulomb and centrifugal potential barriers. It can be expressed as
(17) 
are same as Eq.(8), can be expressed as
(18) 
We select proximity potential Bass73 to calculate the nuclear potential BASS1973139 (); BASS197445 (), which is given by
(19) 
where the fm is the range parameter, and the surface term in the liquid drop model mass formula MeV. The is the specific surface energy of the liquid drop model. represents the the sum of the halfmaximum density radii with fm.
2.2.2 The ViolaâSeaborgâSobiczewski (VSS) semiempirical relationship
The ViolaâSeaborgâSobiczewski semiempirical relationship, one of the commonly used formulas for calculating the halflife of decay, is proposed by Viola and Seaborg and the value given by Sobiczewski instead of the original value given by Viola and Seaborg VIOLA1966741 (). It can be expressed as
(20) 
where is the atomic number of the parent nucleus and is hindrance factor. The values of parameters are and
(21) 
2.2.3 The Universal curve (NUIV)
Poenaru et al. proposed the Universal (UNIV) curve for calculating the decay halflives by extending a fission theory to larger asymmetry, which can be expressed as PhysRevC.83.014601 (); PhysRevC.85.034615 ()
(22) 
The penetrability of an external Coulomb barrier may be obtained analytically asSANTHOSH201833 ()
(23) 
where fm with fm and fm being the two classic turning points. The logarithmic form of the preformation factor is given by
(24) 
is the additive constant PhysRevC.83.014601 (); PhysRevC.85.034615 ().
2.2.4 Royer formula
Royer proposed the analytical formula for determining decay halflives by fitting emitters experimental data 09543899268305 (). It can be written as
(25) 
The parameters a, b and c are given by
(26) 
2.2.5 The Universal decay law (UDL)
Qi et al. given a new universal decay law (UDL) for describing decay and cluster decay modes starting from like matrix theory and the microscopic mechanism of the chargedparticle emissionPhysRevLett.103.072501 (); PhysRevC.80.044326 (). It can be expressed as
(27) 
where and . Here the parameters and are determined by fitting to experiments of and cluster decays.PhysRevLett.103.072501 (); PhysRevC.80.044326 ()
2.2.6 The NiRenDongXu empirical formula (NRDX)
Ni et al. proposed a new general formula with three parameters for determining halflives and decay energies of decay and cluster radioactivity PhysRevC.78.044310 ().This new formula is directly deduced from the WKB barrier penetration probability with some approximations. Their calculations by using this formula show excellent agreement between the experimental data and the calculated values. It can be given by,
(28) 
The parameters a, b and c are given by
(29) 
This formula successfully combines the phenomenological laws of decay and cluster radioactivity.
3 Results and Discussion
In this work, we use the least squares principle to fit the adjustable parameters, while the database are taken from the latest evaluated nuclear properties table NUBASE2016 16741137413030001 (). At first, for the parameter being used to describe the effect of an oddproton and/or an oddneutron, we choose the experimental data of decay halflives of 169 eveneven nuclei as the database to obtain the parameters and , while . Then choosing the experimental data of decay halflives of 132 odd, even nuclei, 94 even, odd nuclei and 66 doublyodd nuclei as the database to determine the parameter , while fixed the parameters and , using the relationship . The values of 3 adjustable parameters are given as
(30) 
The value of is small but it observably impacts on the classical turning point , whereas the decay halflife is sensitive to . For intuitively display the effects, in Fig. 1 we show the different kinetic energy values correspond to difference in values for the pure Coulomb and Hulthen potential, i.e., no centrifugal potential contribution, where and represent the value calculated using Coulomb and using the Hulthen potential, respectively. From this figure, we can find that the smaller decay energy and larger proton number of the daughter nucleus are, the greater difference in the value between the pure Coulomb and the Hulthen potential be.
Using our modified Gamow–like model, we systematically calculate the decay halflives of eveneven, oddodd, odd nuclei. The detailed results are shown in the Fig. 2 – 5. In Fig. 2, we show the 169 decay experimental data of eveneven nuclei and the theoretical values of decay halflive calculated by different methods. The Xaxis represents the mass number in the corresponding decay, the Yaxis represents the logarithmic of the decay halflife. The three coordinate points represent logarithmic form of the experimental decay halflives, logarithmic forms of the calculated decay halflives in this work denoted as and by the theoretical model and parameters in Ref. PhysRevC.87.024308 () denoted as , respectively. The cases of even, odd nuclei, odd, even nuclei and odd, odd nuclei are shown in Fig. 3, Fig. 4 and Fig. 5, respectively. The meanings of each coordinate in Fig. 3 – 5 is same as Fig. 2.
As can be seen from the Fig. 2 – 5, the can better reproduce with experimental data than . In order to intuitively compare with , we calculate the standard deviation between decay halflives of calculations and experimental data. The results , represent standard deviations between , and , which are given in the Table 1. From this table, we can clearly see that for the cases of eveneven, oddproton, oddneutron and doublyodd nuclei, our calculations improve , , and compared to , respectively. It is shown that can better reproduce with experimental data than by considering the shielding effect of the Coulomb potential and the centrifugal potential in this work. And for eveneven nuclei, are calculated by Gamowlike model proposed by K. Pomorski PhysRevC.87.024308 () which contain only one parameter , and are calculated by our improved Gamowlike model with two parameters and . So the addition of the parameter makes the more consistent with the experimental data than . In many of the Fig. 2 – 5 we can see dips and peaks in halflives, it is because , and is the magic core, and the nucleons in the core play an essential role on the preformation probabilityPhysRevC.94.024338 ().
n  

ee  169  –  0.348  0.487 
eo  132  0.3455  0.681  0.967 
oe  94  0.3455  0.598  0.789 
oo  66  0.691  0.748  1.235 
The synthesis and research of SHN have became a hot topic in nuclear physics PhysRevC.97.064609 (); 16741137417074106 (); PhysRevC.98.014618 (). Now we extend our model to predict the decay halflives of nuclei i.e. , ,,,, as well as and some unsynthesized nuclei on their decay chains. From the conclusion of decay properties for SHN in Ref. SANTHOSH201833 (), we can obtain the decay chains of these nuclei, which are , , , , , , and . In our previous studies of the superheavy nucleus16741137424044102 (); 167411374112124109 (), the decay energy is one key input for calculating the decay halflife. Meanwhile Sobiczewski PhysRevC.94.051302 () discovered that the calculation taking decay energy from WS3+ PhysRevC.84.051303 () can best reproduce experimental decay halflife. In the present work, we use decay energy from WS3+ to calculate the halflife of eveneven nuclide with proton number and nuclei on their decay chains except the five known nuclei i.e. Og, Lv, Fl, Lv and Fl are taken from NUBASE2016 16741137413030001 ().
For comparatively, we also systematically calculate the decay halflives of eveneven nuclei of proton numbers and nuclei on their decay chain using Coulomb potential and Proximity potential model with proximity potential Bass73 formalism (CPPMBass73) BASS197445 (), the ViolaSeaborgSobiczewski (VSS) empirical formula VIOLA1966741 (), the Universal (UNIV) curve PhysRevC.83.014601 (); PhysRevC.85.034615 (), Royer formula 09543899268305 (), the Universal decay law (UDL) PhysRevLett.103.072501 (); PhysRevC.80.044326 () and the NiRenDongXu (NRDX) empirical formula PhysRevC.78.044310 (), respectively. The logarithmic forms of calculated decay halflives are listed in Table 2. In this tables, the first two columns represent the parent nucleus of the decay and the decay energy, the next seven columns represent the theoretical decay halflives calculated by CPPMBass73, VSS, UNIV, Royer, UDL, NRDX and our improved Gamowlike model denoted as (s), (s), (s), (s), (s), (s) and (s), respectively. The last column represents logarithmic form of the experimental decay halflives taken from NUBASE2016 16741137413030001 (). It can be seen from Table 2 that for the decay of the same parent nuclear, the logarithmic form of theoretical decay halflife of all models are not much different, and the decay theoretical halflives of the CPPMBass73 model is smaller than other models. For the parent nuclei with known experimental halflife, the maximum difference between the logarithmic forms of experimental halflife value and the logarithmic forms of theoretical halflife obtained from the model of this work is less than 0.65. To make a more intuitive comparison of these theoretical predictions, the theoretical halflife of decay calculated using this seven theoretical models are plotted in Fig. 6. In this figure, decay chains begin with an nucleus with a proton number , the nucleus at the end of each decay chain is spontaneous fission, and the decay of the remaining nucleus is decay. The Xaxis represents the mass number in the corresponding decay chain, the Yaxis represents the logarithmic of the decay halflife.
Nucleus  (MeV)  (s)  (s)  (s)  (s)  (s)  (s)  (s)  (s) 

120  13.187  6.189  5.613  5.884  5.774  5.842  5.398  5.662  – 
Og  12.015  4.264  3.662  4.044  3.842  3.809  3.516  3.832  – 
Lv  11.105  2.698  2.082  2.527  2.275  2.165  1.994  2.326  – 
Fl  10.666  2.202  1.568  2.018  1.767  1.647  1.515  1.831  – 
Cn  10.911  3.471  2.797  3.183  2.999  2.975  2.744  3.006  – 
Ds  10.976  4.259  3.555  3.891  3.76  3.802  3.508  3.723  – 
Hs  9.54  1.077  0.406  0.823  0.603  0.477  0.425  0.678  – 
120  12.9  5.643  5.032  5.371  5.231  5.258  4.826  5.148  – 
Og  11.835  3.889  3.254  3.688  3.471  3.408  3.115  3.474  2.939 
Lv  11.005  2.482  1.832  2.319  2.062  1.932  1.748  2.12  2.097 
Fl  10.365  1.431  0.771  1.28  1.008  0.833  0.731  1.097  0.456 
Cn  10.106  1.375  0.695  1.186  0.934  0.776  0.677  1.014  – 
Ds  10.31  2.601  1.882  2.315  2.122  2.057  1.862  2.15  – 
120  13.287  6.461  5.811  6.13  6.045  6.116  5.591  5.907  – 
Og  11.561  3.279  2.612  3.109  2.867  2.759  2.484  2.895  – 
Lv  10.775  1.922  1.243  1.784  1.51  1.338  1.168  1.587  1.602 
Fl  10.065  0.624  0.06  0.506  0.214  0.017  0.087  0.325  0.125 
120  12.878  5.671  4.986  5.391  5.259  5.273  4.781  5.166  – 
Og  12.118  4.607  3.893  4.358  4.182  4.148  3.741  4.144  – 
Lv  10.451  1.083  0.379  0.981  0.683  0.453  0.319  0.785  – 
120  12.745  5.43  4.71  5.162  5.019  5.011  4.509  4.937  – 
Og  11.905  4.162  3.414  3.935  3.741  3.672  3.27  3.719  – 
Lv  10.777  2.002  1.248  1.853  1.588  1.406  1.172  1.654  – 
120  13.823  7.59  6.836  7.169  7.175  7.296  6.595  6.949  – 
Og  11.995  4.404  3.618  4.16  3.98  3.92  3.47  3.944  – 
120  13.036  6.102  5.309  5.784  5.689  5.709  5.096  5.559  – 
Og  13.104  6.789  5.96  6.389  6.354  6.434  5.769  6.178  – 
From Fig. 6, we can clearly see that theoretical calculations of decay halflives by different models of the same nucleus are different due to the model dependent. But all theoretically calculated decay halflife curves have the same trend. In order to intuitively compare different theories, we calculate the standard deviation between decay halflives of calculations and experimental data of different theories in Table 3.
A of nucleus  

298  0.817  0.299  0.656  0.443  0.360  0.276  0.482 
300  0.419  0.286  0.299  0.091  0.212  0.342  0.142 
We can clearly see that NRDX model reproduces experimental half lives well in superheavy region in case , Royer formula reproduces experimental half lives well in superheavy region in case . In particular, our calculations are sandwiched in other decay chains, and closed to the known experimental data for the halflives, which shows that the model and calculated parameters of present work are believable.
4 Summary
In summary, we modify the Gamowlike model by considering the effects of screened electrostatic for Coulomb potential and the centrifugal potential and use this model systematically to study decay halflives for nuclei. In addition, we extend this model to the superheavy nuclei, and predict the halflives of seven eveneven nuclei with a proton number and some unsynthesized nuclei on their decay chains. This work is useful for the future research of superheavy nuclei.
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China (Grant No. 11205083), the construct program of the key discipline in Hunan province, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 15A159 and 18A237), the Natural Science Foundation of Hunan Province, China (Grant No. 2015JJ3103 and No. 2015JJ2123), the Innovation Group of Nuclear and Particle Physics in USC, the double first class construct program of USC.
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