Recalculated Viola-Seaborg Coefficients for Partial Alpha Half-lives Based on AME2016

Received: February 1, 2021 Accepted: April 12, 2021 Published Online: August 31, 2021 In this paper, the systematics for obtaining the Viola-Seaborg formula (VSF) for logarithmic partial alpha half-lives T1 2 / a ( ) have been undertaken based on the NUBASE2016 evaluation. The constants Az and Bz in Geiger-Nuttal law for determination of T1 2 / a , are obtained using gs-gs transitions data, of even-even nuclei for two sets of nuclei with Z = 84 102 and Z = 86 98 with N > 126. The ViolaSeaborg co-efficients are determined for both the sets. The obtained parameters for both sets are tested on even-even nuclei for Z ranging from 104 118 and it is observed that first set parameters fare better. This formula for estimating α-decay half-lives of heavy nuclei can be extrapolated to predict those of super-heavy nuclei. The logarithmic half-lives T1 2 / a obtained for isotopes of Z = 121 and 122 using current modified VSF (AME2016) are compared with those obtained from theoretical considerations using Coulomb and proximity potential model (CPPM) and observed to be much larger. They are also much larger than those obtained from the previous coefficients based on AME2003 data.


Introduction
Theoretical models to predict various properties of Super Heavy Elements (SHE) are becoming more and more important as they provide inputs for designing experiments. SHE predominantly undergo alpha decay followed by spontaneous fission (SF). So, properties of alpha decay are studied using various macro-micro methods based on cluster model, generalized liquid drop model (GLDM) and Coulomb and proximity potential model (CPPM) to name a few [1]. It is customary to match the predictions of these models with those of empirical ones such as Viola-Seaborg formula (VSF) [3] and UNIV2 [4]. The main objective of this paper is to recalculate the VSF coefficients using the latest atomic mass evaluation (AME2016) data [5]. The VSF parameters were last calculated by Dong and Ren [6] in 2005 using AME2003, wherein they have not considered the fine-structure of alpha-decay. In this paper, we include the following: 1. Effect of intensity of alpha transitions occuring to various energy levels in the daughter, on determination of experimental partial alpha half-lives. 2. Electron screening correction for determination of alpha-decay energies [7].

Geiger-Nuttel Law
The empirical relationship between partial alpha half-life of isotopic nuclei T 1 2 / a and alpha decay energies Q a has been established by Geiger-Nuttel(GN) in 1911 as where T 1 2 / a is determined from experimental data as Here, T 1/2 (sec) is total half-life due to all possible decay channels. B.R. is branching ratio for alpha-decay. Q a is obtained by taking difference between rest mass of parent and rest masses of daughter and He nuclei. It is expressed in MeV and is given by All the half-lives for a given isotopic sequence for different parent nuclei, identified by atomic number Z, are observed to fall on straight lines with slope A Z and intercept B Z .
where h log is a set of constants to compensate for the hindrance factor HF . For all odd-A and odd-odd nuclei, the logarithmic partial half-lives calculated using a, b, c, d coefficients of VSF are subtracted from experimental T 1 2 / a values to obtain the absolute error. The mean of these absolute errors give the corresponding h log constants for oddZ-oddN, oddZ-evenN and evenZ-oddN nuclei.

VSF with Fine-Structure of a -Transitions
In case of gs-gs transitions of even-even nuclei, which are considered to be due to barrier penetration without change in angular momentum between parent and the daughter nuclei, the value of HF = 1. In case of odd-A and odd-odd nuclei, the favored transition is to one of the excited states of the daughter for which there is no angular momentum change and HF is defined as ratio of experimental partial alpha half-life to that obtained using a semi-empirical formula (SEF) such as VSF without h log .

Contribution from Electron Screening
The VSF is further fine tuned by considering the electron screening correction term which is responsible for the decrease in the alpha-particle energy by about 30 -40keV in heavy elements. This electron screening energy could be a result of a) alpha particle having to do work against Coulomb attractive force due to electrons. b) loss in total electron binding energy due to reduction of charge of parent nucleus from Z to Z-2.
The electron screening energy as discussed in Rasmussen[7], is given by where Z D = Z -2, is the atomic number of daughter nucleus. Q a is replaced by effective alpha energy E a * which is where E a is the alpha transition energy to a particular level and A is mass number of parent.

Effect of Intensity on
The partial alpha half-life needs to be determined accordingly by taking into account intensity of a particular transition. Since intensities of all alpha transitions to various energy levels add up to 1, the partial alpha half-life is determined as

Results and Discussion
In this paper, 54 even-even nuclei between Z = 84 to 102 with N > 126 have been considered to fit parameters a, b, c and d of Viola-Seaborg formula. To determine h log constants that improve the estimates for (odd-Z)-(odd-N) and odd-A nuclei, 30(even-odd), 28(odd-even) and 21(odd-odd) nuclei have been selected in the same range Z = 84-102. First the constants A z and B z have been determined (See Table 1  , . , .
These sets of coefficients are utilised to obtain the logarithmic half-lives of all available even-even nuclei in the test region with Z = 103-118 and the data is presented in Table 2. Then, using these sets of co-efficients in VSF, the theoretical half-lives for the selected odd-A and odd-odd nuclei have been determined. These were subtracted from the experimental half-lives to determine the error in each case. The mean of these absolute errors have given rise to the constants h log as follows: case i Set of parameters considering even-Z, even-N nuclei from Z = 84-102 Now, these sets of VSFs need to be tested for their effectiveness by determining the logarithmic partial alpha half-lives of nuclei in the region Z = 103-118. Further, these formulae are utilised to estimate the half-lives for SHE isotopes of Z = 121 and 122. In order to verify the agreement of theoretical half-lives with experimental data, mean deviation and standard deviation are defined as follows: The mean absolute deviation is The standard deviation is  Table 3.   Table 3, one can observe that data for 54 eveneven nuclei have mean deviation of 0.1025(0.0655) and standard deviation of 0.1352(0.1111), respectively. This means that the VSF with obtained parameters reproduce the experimental data of even-even nuclei on an average within a factor of 1.4 for Z = 84-102 set (and 1.3 for Z = 86-98 set). For 30 even-odd, 28 odd-even and 21 odd-odd nuclei, on average the half-life from formula agrees with experimental data within a factor of 6.1(8.6), 5.9(7.4) and 8.2(9.0) respectively. The mean deviations and the standard ones of odd-A nuclei are relatively smaller than those of odd-odd nuclei.
In the third column of Table 3, we have presented results for test data with Z = 103-118. For even-even nuclei, mean deviations and standard deviations are 0.1913(0.1920), 0.2228(0.2270), respectively. So it reproduces the experimental data on an average within a factor of 1.67(1.69). In case of even-odd, odd-even and odd-odd, the half-lives with VSF parameters reproduce experimental data on an average within a factor of 4.2(5.9), 4.2(6.6) and 5.4(5.6) respectively. The effectiveness of data set with Z=84-102 as compared to that of Z = 86-98 becomes more obvious when we look at the number of halflives that fall under various %-error values in Table 4. respectively. This clearly shows that parameters from all available even-even nuclei upto Z = 102 perform better. Finally, we have presented the estimated half-lives in seconds units in comparision to those obtained from Coulomb and proximity potential model. It has been observed that half-life values, for isotopic sequences of Z = 121 and 122, using VSF with recalculated coefficients from AME2016 data are on a higher side as compared to those obtained from AME2003 data. This takes these values even farther away from the theoretical values using CPPM. One has to check how the other empirical formulae such as UNIV2 and GLDM would change with this new evaluation.

Conclusion
Using least-square fit for logarthimic half-lives w.r.t of even-even nuclei between Z = 84-102 (86-98), we have obtained new parameters for the Viola-Seaborg formula. With these parameters, obtained half-lives match very well with experimental ones for eveneven nuclei between Z = 84-102. Then, these parameters are used for test data between Z = 103 -118 and are found to be in agreement with experimental half-lives. Finally, a comparision of estimated half-lives of SHE (Z = 121,122) with theoretical results obtained by Coulomb and proximity potential model (CPPM) results have been found to be comparable. We hope that these new coefficients would help for making comparisions while proposing new theoretical formulations.