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Systematic study of α decay half-lives for even–even nuclei within a deformed two-potential approach

  • Hai-Feng Gui(桂海峰) 1 ,
  • Hong-Ming Liu(刘宏铭) 1 ,
  • Xi-Jun Wu(吴喜军) 2 ,
  • Peng-Cheng Chu(初鹏程) 3 ,
  • Biao He(何彪) 4 ,
  • Xiao-Hua Li(李小华) , 1, 5, 6, 7
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  • 1School of Nuclear Science and Technology, University of South China, 421001 Hengyang, China
  • 2School of Math and Physics, University of South China, Hengyang 421001, China
  • 3The Research Center for Theoretical Physics, Science School, Qingdao University of Technology, Qingdao 266033, China
  • 4College of Physics and Electronics, Central South University, Changsha 410083, China
  • 5National Exemplary Base for International Sci & Tech. Collaboration of Nuclear Energy and Nuclear Safety, University of South China, Hengyang 421001, China
  • 6Cooperative Innovation Center for Nuclear Fuel Cycle Technology & Equipment, University of South China, 421001 Hengyang, China
  • 7Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, 410081 Changsha, China

Received date: 2021-11-27

  Revised date: 2022-04-04

  Accepted date: 2022-04-08

  Online published: 2022-05-10

Supported by

National Natural Science Foundation of China(Grants 12175100)

National Natural Science Foundation of China(Grants 11975132)

Hunan Provincial Innovation Foundation for Postgraduate(Grant CX20200909)

Copyright

© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

In this work, we systematically study the α decay half-lives of 196 even–even nuclei using a two-potential approach improved by considering nuclear deformation. The results show that the accuracy of this model has been improved after considering nuclear deformation. In addition, we extend this model to predict the α decay half-lives of Z = 118 and 120 isotopes by inputting the α decay energies extracted from the Weizsacker–Skyrme-type (WS-type) mass model, a simple nuclear mass formula, relativistic continuum Hartree–Bogoliubov theory and Duflo-Zuker-19 (DZ19) mass model. It is useful for identifying the new superheavy elements or isotopes for future experiments. Finally, the predicted α decay energies and half-lives of Z = 118 and 120 isotopes are analyzed, and the shell structure of superheavy nuclei is discussed. It shows that the shell effect is obvious at N = 184, while the shell effect at N = 178 depends on the nuclear mass model.

Cite this article

Hai-Feng Gui(桂海峰) , Hong-Ming Liu(刘宏铭) , Xi-Jun Wu(吴喜军) , Peng-Cheng Chu(初鹏程) , Biao He(何彪) , Xiao-Hua Li(李小华) . Systematic study of α decay half-lives for even–even nuclei within a deformed two-potential approach[J]. Communications in Theoretical Physics, 2022 , 74(5) : 055301 . DOI: 10.1088/1572-9494/ac6576

1. Introduction

In 1899, Rutherford firstly discovered the natural α radioactivity phenomenon from uranium and uranium compounds. Later on, Gurney and Condon [1] and Gamow [2] proposed the quantum tunneling theory to explain this process independently. α decay is the main decay mode of the unstable nuclei, the study on α decay can provide abundant nuclear structural information, such as the ground-state properties, level structure, shell closure effect and so on [37]. In addition, α decay plays an important role in identifying the new elements and synthesising superheavy nuclei [817]. Therefore, α decay has become one of the hottest topics in nuclear physics. Based on the description of Gamow et al, up to now, a lot of empirical formulas or theoretical models have been developed to study α decay, such as Coulomb and proximity potential model [18], density-dependent cluster model [19, 20], universal decay law [21], Royer formula [22], Gamow-like model [23], two-potential approach (TPA) [24] and others [25, 26].
The study of superheavy nuclei has always been a hot topic [2739]. In recent years, the effect of deformation has attracted more and more attention to the study of superheavy nuclei [4050]. Some research shows that the deformation has a certain influence on the calculation of α decay half-lives [46, 47, 51]. Recently, Hassanzad et al [52] systematically studied the favored α decay half-lives of deformed nuclei with 93 ≤ Z ≤ 118 using deformed Coulomb and proximity potential model. Their results indicated that the deformation has a certain effect on α decay half-lives. Therefore, it is necessary to modify the TPA by considering the effect of nuclear deformation, this is our first motivation of the article. In addition, the magic numbers of superheavy nuclei has also attracted the attention of researchers. Recently, by exploring the properties of α decay of Z = 118 and 120 isotopes, an obvious shell effect at N = 178 is found besides that at N = 184. It is well known that the model dependence on the shell effect at N = 184 [17, 5355] is not strong. However, the new magic number N = 178 has not been confirmed by other models. In order to test whether N = 178 is a new magic number, it is necessary to use the improved TPA to study the alpha-decay properties of Z = 118 and 120 isotopes. This is the second motivation of this paper. Driven by the two above motivations, we will improve the TPA by taking into account the nuclear deformation and further investigate the α decay properties of unstable nuclei, especially for the case of the superheavy nuclei.
The article is organized as follows. The theoretical framework of the deformed TPA is briefly presented in section 2. The detailed calculations and discussion are given in section 3. A summary is given in section 4.

2. Theoretical framework

In α decay process, half-life T1/2 is an important indicator for nuclear stability, which can be calculated by the decay width Γ and -Γ expressed as
$\begin{eqnarray}{T}_{1/2}=\displaystyle \frac{{\hslash }\mathrm{ln}2}{{\rm{\Gamma }}},\end{eqnarray}$
where ℏ is the reduced Planck constant. In the framework of TPA [24, 56], the α decay width Γ can be calculated as
$\begin{eqnarray}{\rm{\Gamma }}=\displaystyle \frac{{{\hslash }}^{2}{P}_{\alpha }{FP}}{4\mu },\end{eqnarray}$
where $\mu =\tfrac{{m}_{d}{m}_{\alpha }}{{m}_{d}+{m}_{\alpha }}$ is the reduced mass between α particle and the daughter nucleus with md and mα being the mass of daughter nucleus and α particle, respectively. F is the normalized factor, describing the α particle assault frequency, which can be approximately given by
$\begin{eqnarray}F=\displaystyle \frac{1}{{\int }_{r1}^{r2}\tfrac{1}{2k(r)}{\rm{d}}r}.\end{eqnarray}$
Here $k(r)=\sqrt{\left(\tfrac{2\mu }{{{\hslash }}^{2}}| {Q}_{\alpha }-V(r)| \right)}$ is the wave number of the α particle. V(r) is the total interaction potential between the daughter nucleus and the emitted α particle. Qα is the α decay energy. r1, r2 and the following r3 are classical turning points that satisfy the conditions V(r1) = V(r2) = V(r3) = Qα. The penetration probability P is obtained by the Wentzel–Kramers–Brillouin method and expressed as
$\begin{eqnarray}P=\exp [-2{\int }_{r2}^{r3}k(r){\rm{d}}r].\end{eqnarray}$
Considering the influence of deformation, P and F can be expressed as [57]
$\begin{eqnarray}P=\displaystyle \frac{1}{2}{\int }_{0}^{\pi }\exp [-2{\int }_{r2}^{r3}k(r(\theta )){\rm{d}}r]\sin \theta {\rm{d}}\theta ,\end{eqnarray}$
$\begin{eqnarray}F=\displaystyle \frac{1}{2}{\int }_{0}^{\pi }\displaystyle \frac{1}{{\int }_{r1}^{r2}\tfrac{1}{2k(r(\theta ))}{\rm{d}}r}\sin \theta {\rm{d}}\theta ,\end{eqnarray}$
where θ is the angle between the emission direction of α particle and the axis of rotational symmetry of deformed nucleus.
The preformation probability Pα required in the decay width Γ is calculated using the cluster formation model, which was first proposed to calculate the preformation probability of even–even heavy nuclei by Ahmed et al [58, 59] in 2013. Recently, they extended this model to calculate the odd-A and odd–odd nuclei [60]. Meanwhile, Deng et al [61] also extended this model to determine the Pα of odd-A and odd–odd nuclei. In the framework of the cluster formation model, the preformation probability Pα is calculated by
$\begin{eqnarray}{P}_{\alpha }=\displaystyle \frac{{E}_{f\alpha }}{E}.\end{eqnarray}$
Here Efα is α cluster-formation energy and E is total energy. For even–even nuclei they can be expressed as [59, 62]
$\begin{eqnarray}\begin{array}{rcl}{E}_{f\alpha } & = & 3B(A,Z)+B(A-4,Z-2)\\ & & -\,2B(A-1,Z-1)-2B(A-1,Z),\end{array}\end{eqnarray}$
$\begin{eqnarray}E=B(A,Z)-B(A-4,Z-2).\end{eqnarray}$
Here B(A, Z) is the binding energy of the nucleus with the mass number A and proton number Z.
The V(r), α-daughter total interaction potential, consists of three parts: nuclear potential VN(r), Coulomb potential VC(r) and centrifugal potential Vl(r). It can be written as
$\begin{eqnarray}V(r)={V}_{N}(r)+{V}_{C}(r)+{V}_{l}(r).\end{eqnarray}$
For nuclear potential VN(r), we choose a type of cosh parametrized form [63]. It can be expressed as
$\begin{eqnarray}{V}_{N}(r)=-{V}_{0}\displaystyle \frac{1+\cosh ({R}_{d}/a)}{\cosh (r/a)+\cosh ({R}_{d}/a)},\end{eqnarray}$
where V0 and a are parameters of the depth and diffuseness for the nuclear potential, respectively. We choose ${V}_{0}=(192.42+31.059\tfrac{{N}_{d}-{Z}_{d}}{{A}_{d}})\mathrm{MeV}$ and a = 0.5958 fm taken from [64, 65] with Nd, Zd and Ad being the neutron, proton and mass number of daughter nucleus. By considering the deformations, the Rd of daughter nucleus is given by [57]
$\begin{eqnarray}{R}_{d}(\theta )={R}_{d}^{{\prime} }(1+{\beta }_{2}{Y}_{20}(\theta )+{\beta }_{4}{Y}_{40}(\theta )+{\beta }_{6}{Y}_{60}(\theta )),\end{eqnarray}$
where β2, β4 and β6 denote the calculated quadrupole, hexadecapole and hexacontatetrapole deformation of the nuclear ground-state, respectively. Ylm is shperical harmonics function. ${R}_{d}^{{\prime} }$ is the spherical radius of daughter nucleus. It can be expressed as [66]
$\begin{eqnarray}{R}_{d}^{{\prime} }=1.28{A}_{d}^{1/3}-0.67+0.8{A}_{d}^{-1/3}.\end{eqnarray}$
Considering the effect of deformations, the Coulomb potential between α particle and daughter nucleus can be expressed as [43]
$\begin{eqnarray}{V}_{C}(\vec{r},\theta )=\int \int \displaystyle \frac{{\rho }_{d}(\vec{{r}_{1}}){\rho }_{\alpha }(\vec{{r}_{2}})}{| \vec{r}+\vec{{r}_{2}}-\vec{{r}_{1}}| }{\rm{d}}\vec{{r}_{2}}{\rm{d}}\vec{{r}_{1}},\end{eqnarray}$
where $\vec{r}$ is the vector between the centres of the α particle and daughter nucleus, $\vec{{r}_{1}}$ and $\vec{{r}_{2}}$ are the radius vectors in the charge distributions of α particle and daughter nucleus. θ represents the ensemble of intrinsic coordinates, which are implicit in ρα and ρd. Properly simplified by Fourier transform [43, 45, 67], the Coulomb potential can be approximated as
$\begin{eqnarray}{V}_{C}(\vec{r},\theta )={{V}^{(0)}}_{C}(r)+{{V}^{(1)}}_{C}(\vec{r},\theta )+{{V}^{(2)}}_{C}(\vec{r},\theta ),\end{eqnarray}$
where ${{V}^{(0)}}_{C}(r)$, ${{V}^{(1)}}_{C}(\vec{r})$ and ${{V}^{(2)}}_{C}(\vec{r})$ are the bare Coulomb interaction, linear Coulomb coupling and second-order Coulomb coupling, respectively. The detailed calculation formula of ${{V}^{(0)}}_{C}(r)$, ${{V}^{(1)}}_{C}(\vec{r})$ and ${{V}^{(2)}}_{C}(\vec{r})$ are given in the appendix.
The centrifugal potential Vl(r) is choose as
$\begin{eqnarray}{V}_{l}(r)=\displaystyle \frac{{{\hslash }}^{2}{\left(l+\tfrac{1}{2}\right)}^{2}}{2\mu {r}^{2}},\end{eqnarray}$
for $l(l+1)\to {\left(l+1/2\right)}^{2}$ being a necessary correction for one-dimensional problems [68]. Here l is the orbital angular momentum taken away by the emitted α particle, l = 0 for favored α decays, while l ≠ 0 for unfavored decays. Based on the conservation laws of party and angular momentum [69], the minimum angular momentum lmin take away by the α particle can be determined by [70]
$\begin{eqnarray}{l}_{\min }=\left\{\begin{array}{ll}{{\rm{\nabla }}}_{j}, & {\rm{for}}\,{\rm{even}}{{\rm{\nabla }}}_{j}\,{\rm{}}\,{\rm{and}}\,{\pi }_{p}={\pi }_{d};\\ {{\rm{\nabla }}}_{j}+1, & {\rm{for}}\,{\rm{even}}{{\rm{\nabla }}}_{j}\,{\rm{}}\,{\rm{and}}\,{\pi }_{p}\ne {\pi }_{d};\\ {{\rm{\nabla }}}_{j}, & {\rm{for}}\,{\rm{odd}}{{\rm{\nabla }}}_{j}\,{\rm{}}\,{\rm{and}}\,{\pi }_{p}\ne {\pi }_{d};\\ {{\rm{\nabla }}}_{j}+1, & {\rm{for}}\,{\rm{odd}}{{\rm{\nabla }}}_{j}\,{\rm{}}\,{\rm{and}}\,{\pi }_{p}={\pi }_{d};\end{array}\right.\end{eqnarray}$
where ∇j = ∣jpjd∣. jp, πp, jd, πd represent spin and parity values of the parent and daughter nuclei, respectively.

3. Results and discussion

In this work, we systematically study the α decay half-lives of 196 even–even nuclei using TPA improved by considering nuclear deformation. The experimental α decay half-lives ${T}_{1/2}^{\exp }(s)$, α decay energies Qα and binding energy B(A, Z) are taken from the NUBASE2020 [71] and AME2020 [72]. At the first, we calculate the α decay half-lives of 196 even–even nuclei without considering the deformation. The results denoted as ${{lgT}}_{1/2}^{{cal}(1)}$ are given in the eighth column of table 1. In this table, the first six columns are the α decay type, deformation factors β2, β4 and β6, Qα and preformation factor Pα, respectively. The last three columns are the experimental α decay half-livesand the calculated ones obtained by TPA without considering nuclear deformation in logarithmic form denoted as ${{lgT}}_{1/2}^{\exp }$ and ${{lgT}}_{1/2}^{{cal}(1)}$, respectively. Rencently, there are many evidences show that the deformation has certain influence on the calculation of α decay half-lives [46, 47, 51]. In order to study the deformation effect of daughter nucleus on the α decay half-lives. We then systematically calculate the α decay half-lives of these 196 even–even nuclei, and the effect of deformations is considered. These calculated results are denoted as ${{lgT}}_{1/2}^{{cal}(2)}$ and listed in the last column of table 1. To intuitively compare the calculated results with experimental data, we introduce the standard deviation σ. In this work, it can be expressed as
$\begin{eqnarray}{\sigma }^{(i)}=\sqrt{\sum {\left[{\mathrm{log}}_{10}({T}_{1/2}^{\mathrm{cal}(i)}/{T}_{1/2}^{\exp })\right]}^{2}/n}.\end{eqnarray}$
Table 1. Comparison of the experimental data for α decay nuclei with calculated results. The deformation parameters β2, β4 and β6 are take from FRDM2012 [78]. The Qα are taken from the AME2020 [72]. The experimental α decay half-lives $\mathrm{lg}{T}_{1/2}^{\exp }(s)$ are taken from the NUBASE2020 [71]. $\mathrm{lg}{T}_{1/2}^{\mathrm{cal}(1)}$ is the logarithm of half-life calculated by TPA without considering deformation, and $\mathrm{lg}{T}_{1/2}^{\mathrm{cal}(2)}$ is the logarithm of half-life calculated by TPA considering deformation.
α transition β2 β4 β6 Qα (MeV) Pα $\mathrm{lg}{T}_{1/2}^{\exp }(s)$ $\mathrm{lg}{T}_{1/2}^{\mathrm{cal}(1)}(s)$ $\mathrm{lg}{T}_{1/2}^{\mathrm{cal}(2)}(s)$
${}^{104}{\rm{Te}}{\to }^{100}{\rm{Sn}}$ 0.000 0.000 0.000 5.100 0.341 −8.398 −7.006 −7.006
${}^{106}{\rm{Te}}{\to }^{102}{\rm{Sn}}$ 0.119 0.066 0.018 4.290 0.247 −4.108 −3.688 −3.809
${}^{108}{\rm{Te}}{\to }^{104}{\rm{Sn}}$ 0.139 0.056 −0.003 3.420 0.255 0.628 0.987 0.859
108Xe → 104Te 0.162 0.073 −0.008 4.570 0.390 −4.143 −4.038 −4.212
110Xe → 106Te 0.173 0.063 −0.018 3.872 0.258 −0.840 −0.480 −0.665
112Xe → 108Te 0.195 0.054 −0.019 3.330 0.277 2.324 2.837 2.616
114Ba → 110Xe 0.218 0.059 −0.025 3.592 0.271 1.694 2.386 2.108
144Nd → 140Ce 0.000 0.000 0.000 1.901 0.196 22.859 23.613 23.613
146Sm → 142Nd 0.000 0.000 0.000 2.529 0.198 15.332 15.887 15.887
148Sm → 144Nd 0.172 0.060 0.009 1.987 0.195 23.298 23.938 23.605
148Gd → 144Sm 0.000 0.000 0.000 3.271 0.199 9.352 9.691 9.691
150Gd → 146Sm 0.172 0.047 0.017 2.807 0.202 13.752 14.161 13.862
150Dy → 146Gd 0.000 0.000 0.000 4.351 0.236 3.107 3.223 3.223
152Gd → 148Sm 0.205 0.053 −0.001 2.204 0.221 21.532 22.033 21.642
152Dy → 148Gd 0.172 0.035 0.005 3.727 0.239 6.930 7.233 6.997
152Er → 148Dy −0.084 0.014 −0.001 4.934 0.239 1.057 1.201 1.159
154Dy → 150Gd 0.205 0.041 −0.004 2.945 0.238 13.976 14.079 13.732
154Er → 150Dy 0.150 0.045 0.006 4.280 0.257 4.677 4.715 4.507
154Yb → 150Er −0.104 0.028 0.007 5.474 0.235 −0.355 −0.268 −0.335
156Er → 152Dy 0.205 0.028 −0.006 3.481 0.252 9.989 10.422 10.111
156Yb → 152Er 0.139 0.044 0.006 4.810 0.257 2.408 2.822 2.637
156Hf → 152Yb −0.084 0.014 −0.001 6.026 0.270 −1.638 −1.623 −1.664
158Hf → 154Yb 0.128 0.030 0.003 5.405 0.246 0.808 0.963 0.830
158W → 154Hf 0.085 0.003 0.000 6.613 0.241 −2.845 −2.810 −2.850
160Hf → 156Yb 0.161 0.022 −0.007 4.902 0.255 3.276 3.367 3.180
160W → 156Hf 0.128 0.018 0.002 6.066 0.273 −0.989 −0.914 −1.029
162Hf → 158Yb 0.183 0.013 −0.009 4.416 0.241 5.687 6.134 5.902
162W → 158Hf 0.150 0.020 0.002 5.678 0.244 0.420 0.684 0.521
162Os → 158W 0.107 −0.008 0.009 6.768 0.237 −2.678 −2.524 −2.590
164W → 160Hf 0.173 −0.001 −0.001 5.278 0.251 2.218 2.459 2.273
164Os → 160W 0.129 0.006 0.000 6.479 0.271 −1.662 −1.618 −1.722
166W → 162Hf 0.184 0.001 −0.011 4.856 0.231 4.738 4.638 4.420
166Os → 162W 0.151 −0.004 −0.001 6.143 0.247 −0.593 −0.356 −0.493
166Pt → 162Os −0.105 0.004 0.000 7.292 0.238 −3.532 −3.446 −3.503
168W → 164Hf 0.206 −0.009 −0.003 4.501 0.236 6.200 6.663 6.398
168Os → 164W 0.173 −0.013 −0.003 5.816 0.247 0.685 1.041 0.767
168Pt → 164Os 0.118 0.005 0.000 6.990 0.272 −2.695 −2.484 −2.673
170Os → 166W 0.184 −0.012 −0.003 5.537 0.237 1.889 2.260 1.956
170Pt → 166Os 0.129 −0.006 −0.001 6.707 0.242 −1.856 −1.511 −1.712
170Hg → 166Pt −0.094 −0.020 0.002 7.770 0.283 −3.509 −4.110 −4.254
172Os → 168W 0.195 −0.010 −0.003 5.224 0.240 3.207 3.728 3.394
172Pt → 168Os 0.140 −0.005 −0.001 6.463 0.254 −0.994 −0.693 −0.914
172Hg → 168Pt −0.105 −0.019 0.002 7.524 0.277 −3.636 −3.421 −3.575
174Hf → 170Yb 0.288 −0.043 −0.016 2.494 0.144 22.800 24.525 23.927
174Os → 170W 0.217 −0.007 −0.004 4.871 0.226 5.251 5.598 5.203
174Pt → 170Os 0.162 −0.003 −0.001 6.183 0.225 0.061 0.396 0.133
174Hg → 170Pt −0.105 −0.019 0.002 7.233 0.249 −2.699 −2.510 −2.664
176Pt → 172Os 0.239 0.021 0.002 5.885 0.261 1.197 1.504 1.022
176Hg → 172Pt −0.115 −0.030 0.004 6.897 0.267 −1.651 −1.460 −1.627
178Pt → 174Os 0.250 0.011 −0.001 5.573 0.227 2.428 2.930 2.434
178Hg → 174Pt −0.125 −0.017 0.003 6.577 0.248 −0.526 −0.317 −0.496
178Pb → 174Hg 0.011 0.000 0.000 7.789 0.227 −3.602 −3.432 −3.530
180W → 176Hf 0.243 −0.065 −0.007 2.515 0.162 25.701 25.679 25.292
180Pt → 176Os 0.251 −0.002 −0.004 5.276 0.205 4.028 4.299 3.914
180Hg → 176Pt −0.135 −0.005 −0.008 6.259 0.265 0.730 0.751 0.654
180Pb → 176Hg 0.000 0.000 0.001 7.419 0.248 −2.387 −2.476 −2.475
182Pt → 178Os 0.219 −0.031 −0.008 4.951 0.187 5.623 6.068 5.779
182Hg → 178Pt −0.146 −0.004 0.001 5.996 0.254 1.892 1.825 1.711
182Pb → 178Hg 0.011 0.000 0.000 7.066 0.234 −1.260 −1.324 −1.326
184Pt → 180Os 0.219 −0.044 −0.001 4.599 0.186 7.768 8.146 7.857
184Hg → 180Pt −0.146 −0.004 0.001 5.660 0.242 3.442 3.328 3.210
184Pb → 180Hg 0.000 0.012 0.000 6.774 0.236 −0.213 −0.333 −0.334
186Os → 182W 0.209 −0.083 0.003 2.821 0.157 22.800 22.936 22.620
186Pt → 182Os 0.208 −0.058 −0.002 4.320 0.173 9.728 10.004 9.727
186Hg → 182Pt −0.146 −0.004 0.001 5.204 0.247 5.701 5.588 5.466
186Pb → 182Hg 0.000 0.000 0.000 6.471 0.230 1.072 0.790 0.790
186Po → 182Pb 0.330 0.016 −0.012 8.501 0.314 −4.469 −5.172 −5.668
188Pt → 184Os 0.186 −0.061 −0.001 4.007 0.216 12.528 12.199 11.958
188Hg → 184Pt −0.146 −0.015 0.003 4.709 0.239 8.722 8.453 8.332
188Pb → 184Hg 0.000 0.000 0.000 6.109 0.221 2.468 2.265 2.265
188Po → 184Pb 0.308 −0.002 −0.006 8.082 0.275 −3.569 −3.991 −4.433
190Pt → 186Os 0.163 −0.051 0.002 3.269 0.203 19.183 18.984 18.784
190Pb → 186Hg 0.000 0.000 0.000 5.698 0.216 4.245 4.111 4.111
190Po → 186Pb −0.217 0.017 −0.001 7.693 0.262 −2.611 −2.799 −3.028
192Pb → 188Hg 0.000 0.000 0.000 5.222 0.204 6.546 6.551 6.551
192Po → 188Pb −0.217 0.017 −0.001 7.320 0.257 −1.492 −1.635 −1.870
194Pb → 190Hg 0.000 0.000 0.000 4.738 0.195 9.944 9.412 9.412
194Po → 190Pb −0.207 0.015 −0.001 6.987 0.235 −0.407 −0.478 −0.702
194Rn → 190Po −0.237 0.020 −0.002 7.862 0.261 −3.108 −2.590 −2.862
196Pb → 192Hg 0.000 0.000 0.000 4.238 0.177 9.869 12.911 12.911
196Po → 192Pb 0.085 0.003 0.000 6.658 0.216 0.775 0.780 0.725
196Rn → 192Po −0.227 0.019 −0.002 7.617 0.256 −2.328 −1.851 −2.110
198Po → 194Pb 0.075 0.002 0.000 6.310 0.203 2.266 2.176 2.133
198Rn → 194Po −0.227 0.019 −0.002 7.349 0.239 −1.163 −0.975 −1.239
200Po → 196Pb −0.063 0.013 −0.001 5.982 0.184 3.793 3.622 3.593
200Rn → 196Po −0.207 0.004 0.001 7.043 0.221 0.070 0.104 −0.116
202Po → 198Pb −0.063 0.001 0.000 5.701 0.177 5.143 4.936 4.908
202Rn → 198Po −0.115 0.017 0.008 6.774 0.209 1.090 1.217 1.035
202Ra → 198Rn −0.227 0.007 0.001 7.880 0.248 −2.387 −1.886 −2.222
204Po → 200Pb −0.042 0.001 0.000 5.485 0.158 6.275 6.140 6.033
204Rn → 200Po −0.115 0.017 0.008 6.547 0.191 2.012 2.117 1.934
204Ra → 200Rn −0.207 −0.007 0.003 7.637 0.231 −1.222 −1.110 −1.403
206Po → 202Pb 0.000 0.000 0.000 5.327 0.145 7.144 6.983 6.983
206Rn → 202Po −0.094 0.015 0.009 6.384 0.180 2.737 2.785 2.627
206Ra → 202Rn −0.125 0.018 0.008 7.415 0.219 −0.620 −0.353 −0.547
208Po → 204Pb 0.000 0.000 0.000 5.216 0.135 7.961 7.590 7.590
208Rn → 204Po −0.063 −0.010 0.001 6.261 0.163 3.367 3.320 3.203
208Ra → 204Rn −0.125 0.018 0.008 7.273 0.195 0.104 0.149 −0.046
208Th → 204Ra −0.197 −0.020 0.005 8.200 0.270 −2.620 −2.218 −2.483
210Pb → 206Hg 0.000 0.000 0.000 3.792 0.107 16.567 16.610 16.610
210Po → 206Pb 0.000 0.000 0.000 5.408 0.105 7.078 6.591 6.591
210Rn → 206Po 0.000 0.000 0.000 6.159 0.152 3.954 3.763 3.763
210Ra → 206Rn −0.084 −0.009 0.001 7.151 0.187 0.602 0.574 0.438
210Th → 206Ra −0.135 0.007 0.009 8.069 0.234 −1.796 −1.775 −1.973
212Po → 208Pb 0.000 0.000 0.000 8.954 0.221 −6.531 −6.541 −6.541
212Rn → 208Po 0.000 0.000 0.000 6.385 0.121 3.157 2.825 2.825
212Th → 208Ra −0.094 −0.008 0.001 7.958 0.196 −1.499 −1.385 −1.527
214Po → 210Pb 0.000 0.000 0.000 7.834 0.213 −3.786 −3.542 −3.542
214Rn → 210Po 0.000 0.000 0.000 9.208 0.228 −6.587 −6.475 −6.475
214Ra → 210Rn 0.000 0.000 0.000 7.273 0.138 0.387 0.182 0.182
214Th → 210Ra −0.063 −0.022 −0.008 7.827 0.195 −1.060 −1.001 −1.118
216Po → 212Pb 0.000 0.000 0.000 6.906 0.205 −0.842 −0.485 −0.485
216Rn → 212Po 0.000 0.000 0.000 8.198 0.237 −4.538 −3.893 −3.893
216Ra → 212Rn 0.000 0.000 0.000 9.526 0.239 −6.764 −6.567 −6.567
216Th → 212Ra 0.000 0.000 0.000 8.072 0.159 −1.580 −1.730 −1.730
216U → 212Th −0.073 −0.021 −0.008 8.531 0.206 −2.161 −2.426 −2.548
218Po → 214Pb 0.056 0.028 0.007 6.115 0.196 2.269 2.682 2.551
218Rn → 214Po 0.079 0.054 0.012 7.263 0.234 −1.472 −0.983 −1.199
218Ra → 214Rn 0.078 0.054 0.010 8.540 0.243 −4.587 −4.133 −4.334
218Th → 214Ra 0.000 0.000 0.000 9.849 0.251 −6.914 −6.664 −6.664
218U → 214Th 0.000 0.000 0.000 8.775 0.188 −3.451 −3.117 −3.117
220Rn → 216Po 0.110 0.068 0.014 6.405 0.221 1.745 2.288 1.936
220Ra → 216Rn 0.111 0.081 0.026 7.594 0.239 −1.742 −1.310 −1.740
220Th → 216Ra 0.090 0.055 0.012 8.973 0.243 −4.991 −4.578 −4.809
222Rn → 218Po 0.110 0.068 0.014 5.590 0.222 5.519 6.083 5.708
222Ra → 218Rn 0.122 0.082 0.018 6.678 0.199 1.526 2.097 1.629
222Th → 218Ra 0.111 0.069 0.015 8.133 0.232 −2.650 −2.231 −2.574
222U → 218Th 0.100 0.056 0.002 9.480 0.245 −5.328 −5.176 −5.410
224Ra → 220Rn 0.143 0.084 0.009 5.789 0.184 5.497 6.127 5.585
224Th → 220Ra 0.144 0.084 0.010 7.299 0.198 0.017 0.563 0.055
224U → 220Th 0.132 0.070 0.006 8.628 0.247 −3.402 −2.979 −3.354
226Ra → 222Rn 0.164 0.098 0.010 4.871 0.182 10.703 11.417 10.681
226Th → 222Ra 0.154 0.085 0.010 6.453 0.182 3.265 3.936 3.358
226U → 222Th 0.143 0.084 0.009 7.701 0.208 −0.570 −0.055 −0.558
228Th → 224Ra 0.174 0.100 0.011 5.520 0.183 7.781 8.520 7.736
228U → 224Th 0.186 0.126 0.035 6.800 0.190 2.748 3.320 2.246
228Pu → 224U 0.154 0.073 0.008 7.940 0.228 0.322 −0.108 −0.602
230Th → 226Ra 0.195 0.114 0.022 4.770 0.183 12.376 13.176 12.116
230U → 226Th 0.185 0.126 0.024 5.993 0.187 6.243 6.984 5.929
230Pu → 226U 0.195 0.114 0.022 7.178 0.201 2.021 2.635 1.681
232Th → 228Ra 0.205 0.103 0.010 4.082 0.160 17.645 18.633 17.629
232U → 228Th 0.206 0.116 0.013 5.414 0.169 9.337 10.143 9.100
232Pu → 228U 0.206 0.116 0.013 6.716 0.165 4.005 4.575 3.592
234U → 230Th 0.215 0.106 0.001 4.858 0.150 12.889 13.725 12.737
234Pu → 230U 0.216 0.119 0.004 6.310 0.152 5.723 6.410 5.396
234Cm → 230Pu 0.205 0.103 0.010 7.365 0.194 2.285 2.723 1.849
236U → 232Th 0.226 0.108 −0.009 4.573 0.153 14.869 15.756 14.740
236Pu → 232U 0.215 0.106 0.001 5.867 0.140 7.955 8.643 7.692
236Cm → 232Pu 0.215 0.106 0.001 7.067 0.157 3.351 3.949 3.048
238U → 234Th 0.236 0.098 −0.021 4.270 0.137 17.149 18.207 17.238
238Pu → 234U 0.226 0.095 −0.012 5.593 0.145 9.442 10.100 9.210
238Cm → 234Pu 0.226 0.095 −0.012 6.670 0.137 5.314 5.661 4.815
238Cf → 234Cm 0.226 0.082 −0.004 8.130 0.185 −0.076 0.748 −0.003
240Pu → 236U 0.237 0.086 −0.024 5.256 0.129 11.316 12.137 11.276
240Cm → 236Pu 0.237 0.085 −0.014 6.398 0.137 6.419 6.870 6.027
240Cf → 236Cm 0.237 0.085 −0.014 7.711 0.150 1.612 2.258 1.471
242Pu → 238U 0.237 0.073 −0.027 4.984 0.133 13.073 13.878 13.066
242Cm → 238Pu 0.237 0.086 −0.024 6.216 0.122 7.148 7.768 6.941
242Cf → 238Cm 0.237 0.073 −0.027 7.517 0.145 2.534 2.953 2.239
244Pu → 240U 0.237 0.061 −0.030 4.666 0.127 15.410 16.155 15.386
244Cm → 240Pu 0.249 0.063 −0.029 5.902 0.144 8.757 9.281 8.504
244Cf → 240Cm 0.249 0.063 −0.029 7.329 0.153 3.190 3.609 2.890
244Fm → 240Cf 0.249 0.063 −0.029 8.550 0.166 −0.506 0.086 −0.587
246Cm → 242Pu 0.249 0.051 −0.032 5.475 0.142 11.172 11.696 10.944
246Cf → 242Cm 0.249 0.051 −0.032 6.862 0.155 5.109 5.505 4.808
246Fm → 242Cf 0.248 0.064 −0.039 8.379 0.154 0.218 0.629 −0.044
248Cm → 244Pu 0.250 0.039 −0.035 5.162 0.146 13.079 13.634 12.905
248Cf → 244Cm 0.250 0.039 −0.035 6.361 0.138 7.460 7.843 7.154
248Fm → 244Cf 0.249 0.051 −0.032 7.995 0.164 1.538 1.865 1.205
250Cf → 246Cm 0.250 0.027 −0.037 6.129 0.136 8.616 8.985 8.321
250Fm → 246Cf 0.250 0.039 −0.035 7.557 0.149 3.270 3.479 2.830
252Cf → 248Cm 0.251 0.014 −0.030 6.217 0.146 7.935 8.455 7.821
252Fm → 248Cf 0.250 0.027 −0.037 7.154 0.154 4.961 5.041 4.408
252No → 248Fm 0.250 0.027 −0.037 8.549 0.158 0.562 0.736 0.146
254Cf → 250Cm 0.240 0.012 −0.031 5.927 0.154 9.224 9.934 9.325
254Fm → 250Cf 0.251 0.015 −0.040 7.307 0.135 4.067 4.423 3.814
254No → 250Fm 0.251 0.015 −0.040 8.226 0.140 1.755 1.825 1.239
254Rf → 250No 0.252 0.002 −0.033 9.210 0.165 −2.816 −0.593 −1.128
256Fm → 252Cf 0.240 0.000 −0.033 7.025 0.155 5.064 5.509 4.952
256No → 252Fm 0.252 0.002 −0.033 8.582 0.153 0.466 0.555 0.011
256Rf → 252No 0.252 0.002 −0.033 8.926 0.156 0.327 0.246 −0.299
258Rf → 254No 0.252 −0.010 −0.036 9.196 0.174 −0.595 −0.662 −1.183
260Sg → 256Rf 0.242 −0.024 −0.038 9.901 0.172 −1.772 −1.936 −2.417
264Hs → 260Sg 0.243 −0.050 −0.023 10.591 0.176 −0.002 −3.099 −3.541
266Hs → 262Sg 0.232 −0.052 −0.023 10.346 0.157 −2.409 −2.469 −2.896
268Hs → 264Sg 0.232 −0.065 −0.015 9.760 0.171 0.146 −0.983 −1.419
270Hs → 266Sg 0.222 −0.079 −0.017 9.070 0.155 0.954 1.072 0.617
270Ds → 266Hs 0.232 −0.066 −0.006 11.117 0.179 −3.688 −3.804 −4.196
282Ds → 278Hs 0.130 −0.043 0.005 9.150 0.196 2.401 1.303 1.074
${}^{286}{\rm{Cn}}{\to }^{282}{\rm{Ds}}$ 0.075 −0.034 0.008 9.240 0.184 1.477 1.754 1.611
${}^{286}{\rm{Fl}}{\to }^{282}{\rm{Cn}}$ 0.064 −0.010 −0.001 10.360 0.208 −0.658 −0.836 −0.953
${}^{288}{\rm{Fl}}{\to }^{284}{\rm{Cn}}$ −0.021 0.012 0.000 10.076 0.198 −0.185 −0.063 −0.147
${}^{290}{\rm{Fl}}{\to }^{286}{\rm{Cn}}$ −0.011 0.000 0.000 9.860 0.184 1.903 0.549 0.469
290Lv → 286Fl 0.064 −0.022 −0.001 11.000 0.207 −2.046 −1.883 −2.002
292Lv → 288Fl −0.073 0.002 0.000 10.791 0.202 −1.796 −1.384 −1.504
294Og → 290Lv 0.064 −0.034 0.008 11.870 0.211 −3.155 −3.375 −3.500
Based on the experimental data, using TPA, we obtain σ(1) = 0.573 without considering nuclear deformation, and σ(2) = 0.506 after considering nuclear deformation. These results show that there is indeed some improvement after considering deformation. In other words, the calculation results of TPA are in better agreement with the experimental data. For a more intuitive comparison of this standard deviations, the difference in logarithmic form of α decay half-lives between experimental data and calculated one is shown in figure 1. From this figure, we can see that the deviations in the two regions i.e. 158 ≤ A ≤ 180 and 222 ≤ A ≤ 254 are obviously larger. This clearly shows the influence of nuclear deformation on the calculation results. The overall calculation result is better, although there is a slight deviation at some points, such as A = 254, 256 and 258. We speculate that this may be caused by the difference between the theoretical value and actual value of the deformation parameters (β2, β4 and β6). In other words, if the experimental data and the model formula used are reliable, we may be able to take the experimental value as the input and use this model to reverse the deformation parameter. We will consider making an attempt in this regard and expect experimenters to measure these nuclei.
Figure 1. The difference in logarithmic form of α decay half-lives between experimental data and calculated ones. The abscissa is the mass number A and the ordinate is the value of ${\mathrm{log}}_{10}({T}_{1/2}^{\exp }/{T}_{1/2}^{\mathrm{cal}})$. The black square and red dot represent the theoretical value calculated by using the TPA without considering deformation and after considering deformation, respectively.
In order to provide a reference for the synthesis of new elements, we extend this model to predict the α decay half-lives of 14 even–even nuclei with Z = 118 and Z = 120 in the following. As we all know, in the α decay, the half-life T1/2 is extremely sensitive to Qα. Therefore, to obtain the precise predictions of α decay half-lives for the heavy and superheavy nuclei, the method of selecting a more precise Qα is at the heart of the matter. In this work, we use four mass tables i.e. WS-type [73], simple nuclear mass formula (SNMF) [74], relativistic continuum Hartree–Bogoliubov (RCHB) [75] and DZ19 [76], to obtain Qα. The calculated results are listed in table 2. In this table, the first column is α decay parent nucleus, and the second to fifth columns are the different Qα from the four mass tables, respectively. The last four columns are the logarithmic form of α decay half-lives corresponding to the four mass tables. To see more intuitively, the value of Qα and predicted α decay half-lives for even–even nuclei with Z = 118 and 120 are drawn in figure 2. In this figure, we can clearly see that N = 184 has an obvious shell effect. However, the shell effect of N = 178 is not observed. The Qα in this figure are from the mass tables: WS-type [73], SNMF [74], RCHB [75] and DZ19 [76]. Combined with the mass table quoted by Cui et al: WS4 [77], FRDM [78], KTUY [79] and GHFB, it is not difficult to draw a conclusion: N = 184 is a magic number, but the shell effect of N = 178 is dependent on models. This information may help to provide a reference for future work.
Figure 2. The value of Qα and predicted α decay half-lives for even–even nuclei with Z = 118 and 120 isotopes. The black square and red dot indicate Z = 118 and Z = 120, respectively. The abscissa is neutron number N, the ordinate in the left column is Qα in Mev, and the ordinate in the right column is logarithm ${\mathrm{log}}_{10}{T}_{1/2}$ of calculated half-life in s. The mass tables used from top to bottom in this figure are WS-type [73], SNMF [74], RCHB [75] and DZ19 [76].
Table 2. The Qα and prediction half-lives of α decay in even–even nuclei with Z = 118 and Z = 120. The deformation parameters β2, β4 and β6 are take from WS4 [77], and the binding energy or mass excess required to calculate the Pα and Qα are respectively from each mass table. ${Q}_{\alpha }^{{WS}-{type}}$, ${Q}_{\alpha }^{{SNMF}}$, ${Q}_{\alpha }^{{RCHB}}$ and ${Q}_{\alpha }^{{DZ}19}$ indicate that the data are from mass tables WS-type [73], SNMF [74], RCHB [75] and DZ19 [76], and the unit is MeV. $\mathrm{lg}{T}_{1/2}^{\mathrm{WS}-\mathrm{type}}$, $\mathrm{lg}{T}_{1/2}^{\mathrm{SNMF}}$, $\mathrm{lg}{T}_{1/2}^{\mathrm{RCHB}}$ and $\mathrm{lg}{T}_{1/2}^{\mathrm{DZ}19}$ are the logarithms of the corresponding calculated half-life, in s.
Nucleus ${Q}_{\alpha }^{\mathrm{WS}-\mathrm{type}}$ ${Q}_{\alpha }^{\mathrm{SNMF}}$ ${Q}_{\alpha }^{\mathrm{RCHB}}$ ${Q}_{\alpha }^{\mathrm{DZ}19}$ $\mathrm{lg}{T}_{1/2}^{\mathrm{WS}-\mathrm{type}}$ $\mathrm{lg}{T}_{1/2}^{\mathrm{SNMF}}$ $\mathrm{lg}{T}_{1/2}^{\mathrm{RCHB}}$ $\mathrm{lg}{T}_{1/2}^{\mathrm{DZ}19}$
292118 12.07 11.93 10.97 11.95 −3.65 −3.51 −0.98 −3.56
294118 11.89 11.72 10.92 11.56 −3.27 −3.08 −0.89 −2.69
296118 11.89 11.52 10.78 11.18 −3.29 −2.64 −0.56 −1.81
298118 11.86 11.34 10.62 10.84 −3.24 −2.21 −0.12 −0.92
300118 11.75 11.17 10.48 10.52 −3.08 −1.82 0.26 −0.10
302118 11.65 11.02 10.62 10.23 −2.83 −1.47 −0.01 0.69
304118 12.36 11.05 12.66 11.92 −4.60 −1.58 −5.15 −3.61
296120 13.16 12.65 11.88 12.21 −5.41 −4.55 −2.59 −3.59
298120 13.09 12.45 11.77 11.81 −5.28 −4.15 −2.37 −2.70
300120 13.05 12.26 11.63 11.43 −5.25 −3.74 −2.04 −1.80
302120 12.81 12.08 11.52 11.08 −4.80 −3.36 −1.78 −0.93
304120 12.66 11.91 11.73 10.75 −4.55 −3.01 −2.16 −0.08
306120 13.29 12.01 13.59 13.12 −5.95 −3.28 −6.41 −5.56
308120 13.07 11.82 13.08 12.68 −5.78 −3.11 −5.69 −4.93

4. Summary

In this work, the TPA is extended to the study of deformed nuclei. Through a systematic study of α decay half-lives for 196 even–even nuclei, we find that TPA after taking deformation into account are in better agreement with the experimental data. In addition, the α decay half-lives of 14 even–even nuclei with Z = 118 and Z = 120 isotopes are predicted by using the four mass tables of WS-type, SNMF, RCHB and DZ19. Combining the mass tables (WS4, FRDM, KTUY, GHFB) used in Cui et al's work, we can conclude that N = 184 is a magic number and N = 178 is dependent on models. This work will help to provide reference to the future research.

Supported by the National Natural Science Foundation of China (Grants No. 12 175 100 and No. 11 975 132), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 18A237), the Innovation Group of Nuclear and Particle Physics in USC, the Shandong Province Natural Science Foundation, China (Grant No. ZR2019YQ01), Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX20210942 and No. CX20200909), the Opening Project of Cooperative Innovation Center for Nuclear Fuel Cycle Technology and Equipment, University of South China (Grant No. 2019KFZ10).

Considering the deformation, the Coulomb potential ${V}_{C}(\vec{r},\theta )$ can be approximated as [43]

$\begin{eqnarray}\begin{array}{l}{V}_{C}(\vec{r},\theta )={{V}^{(0)}}_{C}(r)+{{V}^{(1)}}_{C}(\vec{r},\theta )+{{V}^{(2)}}_{C}(\vec{r},\theta ),\end{array}\end{eqnarray}$
where ${{V}^{(0)}}_{C}(r)$, ${{V}^{(1)}}_{C}(\vec{r})$ and ${{V}^{(2)}}_{C}(\vec{r})$ are the bare Coulomb interaction, linear Coulomb coupling and second-order Coulomb coupling, respectively.

The bare Coulomb interaction ${V}_{C}^{(0)}(r)$ can be written as [43]

$\begin{eqnarray}\begin{array}{l}{V}_{C}^{(0)}(r)={Z}_{d}{Z}_{\alpha }{e}^{2}\left\{\begin{array}{l}1/r,(r\gt {R}_{c});\\ 1/r-[{\left(r-{R}_{\alpha }-{R}_{d}\right)}^{4}({r}^{2}+4r({R}_{\alpha }+{R}_{d})-5({R}_{\alpha }^{2}-4{R}_{d}{R}_{\alpha }+{R}_{d}^{2}))\\ /(160{{rR}}_{\alpha }^{3}{R}_{d}^{3})],({R}_{c}\gt r\gt {R}_{{cc}});\\ \left[-5{r}^{2}+3(-{R}_{\alpha }^{2}+5{R}_{d}^{2})\right]/(10{R}_{d}^{3}),({R}_{{cc}}\gt r\gt 0);\end{array}\right.\end{array}\end{eqnarray}$
where Rc = Rd + Rα and Rcc = RdRα are introduced for simplicity.

The linear Coulomb coupling can be written as [43]

$\begin{eqnarray}\begin{array}{l}{V}_{C}^{(1)}(\vec{r},\theta )={Z}_{d}{Z}_{\alpha }{e}^{2}\,\displaystyle \sum _{n=2,4,6}{F}_{n}^{(1)}(r){\beta }_{n}{Y}_{n0}(\theta ,0),\end{array}\end{eqnarray}$
where ${F}_{n=2}^{(1)}(r)$, ${F}_{n=4}^{(1)}(r)$ and ${F}_{n=6}^{(1)}(r)$ denote the quadrupole, hexadecapole and hexacontatetrapole Coulomb coupling form factor of linear order, respectively. They can be written as [43]
$\begin{eqnarray}\begin{array}{l}{F}_{n=2}^{(1)}(r)\,=\,\left\{\begin{array}{l}3{R}_{d}^{2}/(5{r}^{3}),(r\gt {R}_{c}),\\ 3{r}^{2}/(10{R}_{d}^{3})+[3{r}^{5}-12{r}^{3}(3{R}_{\alpha }^{2}+{R}_{d}^{2})+18r(-3{R}_{\alpha }^{4}+2{R}_{\alpha }^{2}{R}_{d}^{2}+{R}_{d}^{4})]/(256{R}_{\alpha }^{3}{R}_{d}^{3})\\ +3({R}_{\alpha }^{6}-3{R}_{\alpha }^{4}{R}_{d}^{2}+3{R}_{\alpha }^{2}{R}_{d}^{4}-{R}_{d}^{6})/(64{{rR}}_{\alpha }^{3}{R}_{d}^{3})+3(-3{R}_{\alpha }^{8}+20{R}_{\alpha }^{6}{R}_{d}^{2}-90{R}_{\alpha }^{4}{R}_{d}^{4}\\ +128{R}_{\alpha }^{3}{R}_{d}^{5}-60{R}_{\alpha }^{2}{R}_{d}^{6}+5{R}_{d}^{8})/(1280{r}^{3}{R}_{\alpha }^{3}{R}_{d}^{3}),({R}_{c}\gt r\gt {R}_{{cc}}),\\ 3{r}^{2}/(5{R}_{d}^{3}),({R}_{{cc}}\gt r\gt 0).\end{array}\right.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{F}_{n=4}^{(1)}(r)\,=\,\left\{\begin{array}{l}{R}_{d}^{4}/(3{r}^{5}),(r\gt {R}_{c}),\\ {r}^{4}/(6{R}_{d}^{5})+[7{r}^{7}-18{R}^{5}(7{R}_{\alpha }^{2}+{R}_{d}^{2})+9{r}^{3}(-35{R}_{\alpha }^{4}+10{R}_{\alpha }^{2}{R}_{d}^{2}+{R}_{d}^{4})+4r\\ (35{R}_{\alpha }^{6}-45{R}_{\alpha }^{4}{R}_{d}^{2}+9{R}_{\alpha }^{2}{R}_{d}^{4}+{R}_{d}^{6})]/(2048{R}_{\alpha }^{3}{R}_{d}^{5})+9(-7{R}_{\alpha }^{8}+20{R}_{\alpha }^{6}{R}_{d}^{2}\\ -18{R}_{\alpha }^{4}{R}_{d}^{4}+4{R}_{\alpha }^{2}{R}_{d}^{6}+{R}_{d}^{8})/(2048{{rR}}_{\alpha }^{3}{R}_{d}^{5})+9({R}_{\alpha }^{10}-5{R}_{\alpha }^{8}{R}_{d}^{2}+10{R}_{\alpha }^{6}{R}_{d}^{4}-10{R}_{\alpha }^{4}{R}_{d}^{6}\\ +5{R}_{\alpha }^{2}{R}_{d}^{8}-{R}_{d}^{10})/(1024{r}^{3}{R}_{\alpha }^{3}{R}_{d}^{5})+(-7{R}_{\alpha }^{12}+54{R}_{\alpha }^{10}{R}_{d}^{2}-189{R}_{\alpha }^{8}{R}_{d}^{4}+420{R}_{\alpha }^{6}{R}_{d}^{6}\\ -945{R}_{\alpha }^{4}{R}_{d}^{8}+1024{R}_{\alpha }^{3}{R}_{d}^{9}-378{R}_{\alpha }^{2}{R}_{d}^{10}+21{R}_{d}^{12})/(6144{r}^{5}{R}_{\alpha }^{3}{R}_{d}^{5}),({R}_{c}\gt r\gt {R}_{{cc}}),\\ {r}^{4}/(3{R}_{d}^{5}),({R}_{{cc}}\gt r\gt 0).\end{array}\right.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{F}_{n=6}^{(1)}(r)\,=\,\left\{\begin{array}{l}3{R}_{d}^{6}/(13{r}^{7}),(r\gt {R}_{c}),\\ 3{r}^{6}/(26{R}_{d}^{7})+[99{r}^{9}-216{r}^{7}(11{R}_{\alpha }^{2}+{R}_{d}^{2})+84{r}^{5}(-99{R}_{\alpha }^{4}+18{R}_{\alpha }^{2}{R}_{d}^{2}+{R}_{d}^{4})\\ +24{r}^{3}(231{R}_{\alpha }^{6}-189{R}_{\alpha }^{4}{R}_{d}^{2}+21{R}_{\alpha }^{2}{R}_{d}^{4}+{R}_{d}^{6})+18r(-231{R}_{\alpha }^{8}+420{R}_{\alpha }^{6}{R}_{d}^{2}\\ -210{R}_{\alpha }^{4}{R}_{d}^{4}+20{R}_{\alpha }^{2}{R}_{d}^{6}+{R}_{d}^{8})]/(65536{R}_{\alpha }^{3}{R}_{d}^{7})+3(99{R}_{\alpha }^{10}-315{R}_{\alpha }^{8}{R}_{d}^{2}\\ +350{R}_{\alpha }^{6}{R}_{d}^{4}-150{R}_{\alpha }^{4}{R}_{d}^{6}+15{R}_{\alpha }^{2}{R}_{d}^{8}+{R}_{d}^{10})/(8192{{rR}}_{\alpha }^{3}{R}_{d}^{7})+21(-11{R}_{\alpha }^{12}+54{R}_{\alpha }^{10}{R}_{d}^{2}\\ -105{R}_{\alpha }^{8}{R}_{d}^{4}+100{R}_{\alpha }^{6}{R}_{d}^{6}-45{R}_{\alpha }^{4}{R}_{d}^{8}+6{R}_{\alpha }^{2}{R}_{d}^{10}+{R}_{d}^{12})/(16384{r}^{3}{R}_{\alpha }^{3}{R}_{d}^{7})\\ +27({R}_{\alpha }^{14}-7{R}_{\alpha }^{12}{R}_{d}^{2}+21{R}_{\alpha }^{10}{R}_{d}^{4}-35{R}_{\alpha }^{8}{R}_{d}^{6}+35{R}_{\alpha }^{6}{R}_{d}^{8}\\ -21{R}_{\alpha }^{4}{R}_{d}^{10}+7{R}_{\alpha }^{2}{R}_{d}^{12}-{R}_{d}^{14})/(8192{r}^{5}{R}_{\alpha }^{3}{R}_{d}^{7})+3(-99{R}_{\alpha }^{16}+936{R}_{\alpha }^{14}{R}_{d}^{2}\\ -400{R}_{\alpha }^{12}{R}_{d}^{4}+10296{R}_{\alpha }^{10}{R}_{d}^{6}-18018{R}_{\alpha }^{8}{R}_{d}^{8}+24024{R}_{\alpha }^{6}{R}_{d}^{10}-36036{R}_{\alpha }^{4}{R}_{d}^{12}\\ +32768{R}_{\alpha }^{3}{R}_{d}^{13}-10296{R}_{\alpha }^{2}{R}_{d}^{14}+429{R}_{d}^{16})/(851968{r}^{7}{R}_{\alpha }^{3}{R}_{d}^{7}),({R}_{c}\gt r\gt {R}_{{cc}}),\\ 3{r}^{6}/(13{R}_{d}^{7}),({R}_{{cc}}\gt r\gt 0).\end{array}\right.\end{array}\end{eqnarray}$

The second-order Coulomb coupling can be expressed as [43]

$\begin{eqnarray}\begin{array}{rcl}{V}_{C}^{(2)}(\vec{r},\theta ) & = & {Z}_{d}{Z}_{\alpha }{e}^{2}\left[{F}_{n=2}^{(2)}(r)\displaystyle \frac{\sqrt{5}}{7\sqrt{\pi }}{Y}_{20}(\theta ,0)\right.\left.+{F}_{n=4}^{(2)}(r)\displaystyle \frac{3}{7\sqrt{\pi }}{Y}_{40}(\theta ,0)\right]{\beta }_{2}^{2},\end{array}\end{eqnarray}$
where the ${F}_{n=2}^{(2)}(r)$ and ${F}_{n=4}^{(2)}(r)$ denote the quadrupole and hexadecapole Coulomb coupling form factor of second order, respectively. They can be written as [43]
$\begin{eqnarray}{F}_{n=2}^{(2)}(r)=\left\{\begin{array}{l}6{R}_{d}^{2}/(5{r}^{3}),(r\gt {R}_{c}),\\ -3{r}^{2}/(20{R}_{d}^{3})-[3{r}^{5}-12{r}^{3}(3{R}_{\alpha }^{2}-{R}_{d}^{2})-18r(3{R}_{\alpha }^{4}+2{R}_{\alpha }^{2}{R}_{d}^{2}+3{R}_{d}^{4})]/(512{R}_{\alpha }^{3}{R}_{d}^{2})\\ +3(-{R}_{\alpha }^{6}-3{R}_{\alpha }^{4}{R}_{d}^{2}+9{R}_{\alpha }^{2}{R}_{d}^{4}-5{R}_{d}^{6})/(128{{rR}}_{\alpha }^{3}{R}_{d}^{3})+3(3{R}_{\alpha }^{8}+20{R}_{\alpha }^{6}{R}_{d}^{2}\\ -270{R}_{\alpha }^{4}{R}_{d}^{4}+512{R}_{\alpha }^{3}{R}_{d}^{5}-300{R}_{\alpha }^{2}{R}_{d}^{6}+35{R}_{d}^{8})/(2560{r}^{3}{R}_{\alpha }^{3}{R}_{d}^{3}),({R}_{c}\gt r\gt {R}_{{cc}}),\\ -3{r}^{2}/(10{R}_{d}^{3}),({R}_{{cc}}\gt r\gt 0),\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}{F}_{n=4}^{(2)}(r)=\left\{\begin{array}{l}{R}_{d}^{4}/({r}^{5}),(r\gt {R}_{c}),\\ -{r}^{4}/(4{R}_{d}^{5})-[21{r}^{7}-18{r}^{5}(21{R}_{\alpha }^{2}+{R}_{d}^{2})-9{r}^{3}(105{R}_{\alpha }^{4}-10{R}_{\alpha }^{2}{R}_{d}^{2}+{R}_{d}^{4})-12r(-35{R}_{\alpha }^{6}\\ +15{R}_{\alpha }^{4}{R}_{d}^{2}+3{R}_{\alpha }^{2}{R}_{d}^{4}+{R}_{d}^{6})]/(4096{R}_{\alpha }^{3}{R}_{d}^{5})+9(21{R}_{\alpha }^{8}-20{R}_{\alpha }^{6}{R}_{d}^{2}-18{R}_{\alpha }^{4}{R}_{d}^{4}+12{R}_{\alpha }^{2}{R}_{d}^{6}\\ +5{R}_{d}^{8})/(4096{{rR}}_{\alpha }^{3}{R}_{d}^{5})+9(-3{R}_{\alpha }^{10}+5{R}_{\alpha }^{8}{R}_{d}^{2}+10{R}_{\alpha }^{6}{R}_{d}^{4}-30{R}_{\alpha }^{4}{R}_{d}^{6}+25{R}_{\alpha }^{2}{R}_{d}^{8}\\ -7{R}_{d}^{10})/(2048{r}^{3}{R}_{\alpha }^{3}{R}_{d}^{5})+(7{R}_{\alpha }^{12}-18{R}_{\alpha }^{10}{R}_{d}^{2}-63{R}_{\alpha }^{8}{R}_{d}^{4}+420{R}_{\alpha }^{6}{R}_{d}^{6}-1575{R}_{\alpha }^{4}{R}_{d}^{8}\\ +2048{R}_{\alpha }^{3}{R}_{d}^{9}-882{R}_{\alpha }^{2}{R}_{d}^{10}+63{R}_{d}^{12})/(4096{r}^{5}{R}_{\alpha }^{3}{R}_{d}^{5}),({R}_{c}\gt r\gt {R}_{{cc}}),\\ -{r}^{4}/(2{R}_{d}^{5}),({R}_{{cc}}\gt r\gt 0).\end{array}\right.\end{eqnarray}$

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