α particle preformation factor in heavy and superheavy nuclei

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Song Luo, Dong-Meng Zhang, Lin-Jing Qi, Xun Chen, Peng-Cheng Chu and Xiao-Hua Li. α particle preformation factor in heavy and superheavy nuclei[J]. Chinese Physics C.
Song Luo, Dong-Meng Zhang, Lin-Jing Qi, Xun Chen, Peng-Cheng Chu and Xiao-Hua Li. α particle preformation factor in heavy and superheavy nuclei[J]. Chinese Physics C. shu
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α particle preformation factor in heavy and superheavy nuclei

  • 1. School of Nuclear Science and Technology, University of South China, Hengyang 421001, China
  • 2. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266033, China
  • 3. National Exemplary Base for International Sci & Tech. Collaboration of Nuclear Energy and Nuclear Safety,
  • 4. Cooperative Innovation Center for Nuclear Fuel Cycle Technology & Equipment, University of South China, Hengyang 421001, China
  • 5. Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, Changsha 410081, China

Abstract: In the present work, the α-particle preformation in heavy and superheavy nuclei from $ ^{220}{\rm{Th}} $ to $ ^{294}{\rm{Og}} $ have been investigated. Combing the experimental α decay energies and half-lives, the α-particle preformation factors $ P_{\alpha} $ are extracted from the ratios between theoretical α decay half-lives calculated by Two-Potential Approach (TPA) and experimental data. It is found that the α-particle preformation factors show the obvious odd-even staggering behavior and unpaired nucleons will inhibit α-particle preformation. Meanwhile, it is also found that both the α decay energy and the mass number of parent nucleus show considerable regularity with the extracted experimental α-particle preformation factors. After considering the major physical factors, a local phenomenological formula with only five valid parameters for α-particle preformation factors $ P_{\alpha} $ is proposed. This analytic expression not only has clear physical meaning but also has good precision. As an application, this analytic formula is extended to estimate the α-particle preformation factors and further predict the α decay half-lives for unknown even-even nuclei with Z = 118 and 120.

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    I.   INTRODUCTION
    • α decay has long been regarded as a trusted pathway to obtain nuclear structure information such as shell effects, nuclear deformation, spin and parity, neutron-proton interaction and so on[19]. This decay mode was first observed as an unknown radioactive phenomenon and further described as a process that parent nucleus emits a $ ^{4} $$ {\rm{He}} $ particle by Rutherford[10]. Until the 1920s, the corresponding decay mechanism was explained as a quantum tunneling effect by Gamow[11] and by Condon and Gurney[12]. Since then, numerous models and/or phenomenological formulae based on tunneling theory have been proposed to calculate α decay half-lives and further reconstruct the corresponding decay process[1321].

      Generally, the numerous approaches on α decay can be divided into two categories, the cluster-like theory and the fission-like theory. The main difference between these two theories appears in the treatment that whether α particle is preformed inside the parent nucleus [22]. Consequently, the corresponding decay constant usually have different forms. In the fission-like theory, the decay constant λ can be defined as the product of two terms: the penetration probability and the effective assault frequency. Note that the effective assault frequency actually contains the certain α-particle preformation information [23, 24]. Correspondingly, the decay constant λ in the preformed cluster models can be described as the product of three physical terms: the α-particle preformation factors, the penetration probability and the collision probability. Among these three terms, the α-particle preformation factors is an indispensable physical concept, which represents the relative probability that α particle exists as an entity on the surface or inside the parent nucleus. In fact, it is largely dependent on the structure and state of parent and daughter nuclei. Therefore, the preformation factor plays an important role in the exploration of nuclear structure[25]. However, the trick issue is how to reasonably evaluate the α-particle preformation factors since it is not a valid observation quantity. Several methods have been proposed to evaluate or extract α-particle preformation factors such as R-matrix method[2628], microscopic wave function approach[29, 30], cluster-formation model[3133], density-dependent cluster model[34], generalized liquid drop model[3538], Two-Potential Approach (TPA)[3941], etc. In general, the methods mentioned above basically obtain the preformation factors through the ratios between theoretical α decay half-lives and experimental data. In these cases, most of the obtained preformation factors are model dependent since the matched integration form are different for various models.

      It is well known that the TPA is one of the excellent phenomenological models to investigate the α decay process. At present, the α-particle preformation factors $ P_{\alpha} $ are poorly studied in the framework of TPA, especially for heavy and superheavy region. On the one hand, exploring the hidden nuclear structure information can provide valuable reference for further exploration. On the other hand, it is urgent to refine this model and provide more reliable predictions for future experiments. For this purpose, we systemically study the preformation factors in heavy and superheavy region from $ ^{220}{\rm{Th}} $ to $ ^{294}{\rm{Og}} $. Referring to the relevant studies [42, 43], this work mainly focus on the ground-state α transitions. Then, we have a detailed discussion about the corresponding nuclear structure information. In the following, a local phenomenological formula with only five valid parameters for α-particle preformation factors $ P_{\alpha} $ has been proposed. This analytical expression can not only successfully evaluate the α-particle preformation factors, but also help the accurate calculation of half-lives. With the help of this analytic formula, the α decay half-lives for unknown even-even nuclei with Z = 118 and 120 are predicted.

      This article is organized as follows. In Section 2, the theoretical framework of TPA is briefly described. The detailed results and discussion are presented in Section 3. Finally, a succinct summary is given in Section 4.

    II.   THEORETICAL FRAMEWORK

      A.   The framework of TPA

    • In TPA, the α decay half-lives $ T_{1/2} $ can be defined as

      $ T_{1/2} = \frac{\rm ln2}{{\rm{\lambda}}} = \frac{\hbar\rm ln2}{{\rm{\Gamma}}}, $

      (1)

      where λ and Γ represent the decay constant and decay width, respectively. In general, the decay constant and decay width are related to three parts, which are the normalized factor F, the penetration probability P and the α-particle preformation factor $ P_{\alpha} $, respectively. Within the Two-Potential Approach framework, the decay width can be written as

      $ {\rm{\Gamma}}= \frac{\hbar^{2}FPP_{\alpha}}{4\mu}. $

      (2)

      The F is an important physical quantity related to the collision probability or assault frequency, which satisfies the condition

      $ F\int_{r_1}^{r_2}\frac{1}{2k(r)}{dr}=1, $

      (3)

      where r is the mass center distance between daughter nucleus and preformed α particle. It worth mentioning that the F appears as the normalized factor of the bound state wave function, which plays an important role in describing the frequency of particle motion in the quasi-classical period. And the condition Eq.(3) is a key constraint for the bound state wave function. The details about integral form and introduction can be found in Ref. [44]. The $ k(r) $ represents the wave number, which can be given as

      $ k(r) = \sqrt{\frac{2\mu}{{\hbar}^2} \vert Q_{\alpha}-V(r) \vert}, $

      (4)

      where $ V(r) $ and $ Q_{\alpha} $ denote total interaction potential and α decay energy, respectively. μ is the reduced mass of the α particle and daughter nucleus and can be described as μ = $ \dfrac{m_{\alpha}m_d}{m_{\alpha}+m_d} $, where $ m_{\alpha} $ and $ m_d $ represent the nucleon mass of α particle and daughter nucleus, respectively. The penetration probability P represents the probability that α particle penetrates the barrier. It can be given as

      $ P={{\rm{exp}}}\left[-2\int_{r_2}^{r_3}k(r){dr}\right], $

      (5)

      where $ r_{1} $, $ r_{2} $ and $ r_{3} $ are the classical turning points and satisfy the conditions $ V(r_1) $ = $ V(r_2) $ = $ V(r_3) $ = $ Q_{\alpha} $. Inside the parent nucleus ($ r_1 \;\; < \;\; r \;\; < \;\; r_2 $), the corresponding interactions are dominated by the nuclear potential, while in the region outside the nucleus ($ r_2 \;\; < \;\; r \;\; < \;\; r_3 $) the Coulomb potential plays an important role.

      The total interaction potential includes three parts: the nuclear potential $ V_{N}(r) $, the Coulomb potential $ V_{C}(r) $ and the centrifugal potential $ V_{l}(r) $. It can be written as

      $ \begin{array}{*{20}{l}} V(r) = V_{N}(r) + V_{C}(r) +V_{l}(r). \end{array} $

      (6)

      In this work, the nuclear potential is described by the type of cosh parametrized form, which can be expressed as

      $ V_{N}(r) = -V_{0}\frac{1+\rm cosh (R/a)}{\rm cosh(r/a)+\rm cosh(R/a)}, $

      (7)

      where $ V_{0} $ and a are parameters of the depth and diffuseness of the nuclear potential, respectively. In the previous work of our group[39], a series of parameter values have been obtained with a = 0.5958 fm and $ V_{0} $ = 192.42 +31.059$ \frac{N-Z}{A} $ MeV, where N, Z and A are the neutron, proton and mass number of daughter nucleus, respectively. $ V_{C}(r) $ represents the Coulomb potential, which is based on the assumption of the uniformly charged sphere and can be described as

      $ \begin{array}{*{20}{l}} V_{C}(r)= \left\{ \begin{array}{lr} \dfrac{Z_dZ_{\alpha}e^2}{2R}\left[3-\dfrac{r^2}{R^2}\right], & \quad r\leq R, \\ \dfrac{Z_dZ_{\alpha}e^2}{r}, & \quad r>R, \end{array} \right. \end{array} $

      (8)

      where $ Z_{d} $ and $ Z_{\alpha} $ are the charge number of daughter nucleus and α particle, respectively. The radius R is given by

      $ \begin{array}{*{20}{l}} R = 1.28A^{1/3} - 0.76 + 0.8A^{-1/3}. \end{array} $

      (9)

      For unfavored α decay ($ \ell \;\; \neq $ 0 decays, while $ \ell $ represents the angular momentum taken away by the emitted α particle), the corresponding centrifugal potential generated by the nonzero angular momentum can be considered as

      $ V_{l}(r)=\frac{\hbar^2(l+\frac{1}{2})^2}{2\mu r^2}. $

      (10)

      In fact, the Eq.(10) is related to the Langer modified form since the term $ l(l+1) \;\; \rightarrow \;\; (l+1/2)^{2} $ is important correction for one-dimensional problems [45]. The minimum angular momentum $ \ell_{min} $ taken away by the α particle are selected in accordance with the conservation laws of spin-parity, which is given by

      $ \ell_{min}=\left\{\begin{array}{*{20}{l}} \Delta_{j}, & {\rm{for~~ even}}~~ \Delta_{j} ~~{\rm{and}}~~ \pi_{p} = \pi_{d} ,\\ \Delta_{j}+1, &{\rm{for ~~even}}~~ \Delta_{j} ~~{\rm{and}}~~ \pi_{p} \neq \pi_{d} ,\\ \Delta_{j}, & {\rm{for ~~odd}}~~ \Delta_{j} ~~{\rm{and}}~~ \pi_{p} =\pi_{d},\\ \Delta_{j}+1, &{\rm{for ~~odd}}~~ \Delta_{j} ~~{\rm{and}}~~ \pi_{p} \neq \pi_{d} , \end{array}\right. $

      (11)

      where $ \Delta_{j} $ = $ \lvert {j_{p}-j_{d}} \rvert $, $ j_{p} $, $ \pi_{p} $, $ j_{d} $, $ \pi_{d} $ represents spin and parity values of parent and daughter nuclei, respectively. The details about the conservation laws of spin-parity can be found in the Ref. [46].

    • B.   The α-particle preformation factors $ P_{\alpha} $

    • Referring the Eq. (1) and (2), the experimental decay constant $ \lambda_{\rm exp} $ can be calculated by

      $ \lambda_{\rm exp} = \frac{\rm ln2}{T^{\rm exp}_{1/2}} = \frac{\hbar P^{\rm exp}_{\alpha}FP}{4\mu}. $

      (12)

      Correspondingly, if the α particle preformation factor is fixed as $ P_{0} = 1 $, the theoretical decay constant $ \lambda_{\rm cal} $ can be given as

      $ \lambda_{\rm cal} = \frac{\rm ln2}{T^{\rm cal}_{1/2}} = \frac{\hbar P_{0}FP}{4\mu}. $

      (13)

      Combining the Eq. (12) and (13), the α-particle preformation factors can be extracted from the rations between theoretical α decay half-lives and experimental data. It can be written as

      $ P^{\rm exp}_{\alpha} = \frac{\lambda_{\rm exp}}{\lambda_{\rm cal}} = \frac{T^{\rm cal}_{1/2}}{T^{\rm exp}_{1/2}}. $

      (14)

      It worth mentioning that the extracted α-particle preformation factors with the help of Eq. (14) are not the absolute value, the corresponding value only reflect the relative probability of α-particle forming inside or the surface of the parent nucleus.

    III.   RESULTS AND DISCUSSION
    • The TPA is widely used to study the α decay process[39, 47, 48] and proton radioactivity[49], the corresponding results are in good agreement with experimental data. In this work, the α-particle preformation factors $ P_{\alpha} $ for heavy and superheavy nuclei are extracted from the ratios between theoretical α decay half-lives calculated by TPA and experimental data. The detailed results for 65 even-even nuclei, 87 odd-A nuclei and 28 odd-odd nuclei are listed in Table 13, respectively. These three tables share the same framework, the first five columns denote α transition, corresponding decay energies ($ Q_{\alpha} $), spin and parity for parent and daughter nuclei as well as the minimum angular momentum taken away by the emitted α particle, respectively. The middle five columns represent the logarithmic value of experimental α decay half-lives, the normalized factor F, the penetration probability P, the extracted experimental α-particle preformation factors from Eq.(14) denoted as $ P_{\alpha}^{\rm exp} $ and the estimated ones by Eq.(15) denoted as $ P_{\alpha}^{\rm Eq} $, respectively. The last two columns denote the calculated α decay half-lives in logarithmic form with the corresponding preformation factors are derived from the fixed $ P_{0} = 1.0 $ and Eq.(15), denoted as $ {\rm{log}}_{10}{T_{1/2}^{{\rm{cal1}}}} $ and $ {\rm{log}}_{10}{T_{1/2}^{{\rm{cal2}}}} $, respectively.

      ${\alpha}$ transition $Q_{\alpha}$ $j_{p}^{\pi}$ $j_{d}^{\pi}$ $\ell_{min}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,exp}}}$ F P $\rm P_{\alpha}^{{\rm{\,exp}}}$ $\rm P_{\alpha}^{{\rm{\,Eq}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal1}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal2}}}$
      $ ^{220}$Th$ \to ^{216}$Ra 8.97 $0^{+}$ $0^{+}$ $0$ $ -5.01 $ $ 0.859 $ $3.455 \times 10^{-17}$ 0.5902 0.4399 −5.24 −4.88
      $ ^{222}$Th$ \to ^{218}$Ra 8.13 $0^{+}$ $0^{+}$ $0$ $ -2.69 $ $ 0.861 $ $1.595 \times 10^{-19}$ 0.6108 0.4692 −2.90 −2.58
      $ ^{224}$Th$ \to ^{220}$Ra 7.30 $0^{+}$ $0^{+}$ $0$ $ 0.12 $ $ 0.861 $ $3.007 \times 10^{-22}$ 0.5017 0.5102 −0.18 0.11
      $ ^{226}$Th$ \to ^{222}$Ra 6.45 $0^{+}$ $0^{+}$ $0$ $ 3.39 $ $ 0.876 $ $1.324 \times 10^{-25}$ 0.6021 0.5723 3.17 3.41
      $ ^{228}$Th$ \to ^{224}$Ra 5.52 $0^{+}$ $0^{+}$ $0$ $ 7.93 $ $ 0.805 $ $3.482 \times 10^{-30}$ 0.7179 0.6823 7.79 7.95
      $ ^{230}$Th$ \to ^{226}$Ra 4.77 $0^{+}$ $0^{+}$ $0$ $ 12.49$ $ 0.874 $ $7.533\times 10^{-35}$ 0.8421 0.8158 12.42 12.50
      $ ^{232}$Th$ \to ^{228}$Ra 4.08 $0^{+}$ $0^{+}$ $0$ $ 17.76 $ $ 0.825 $ $2.867 \times 10^{-40}$ 1.2594 1.0123 17.86 17.85
      $ ^{222}$U$ \to ^{218}$Th 9.48 $0^{+}$ $0^{+}$ $0$ $ -5.33 $ $ 0.848 $ $1.373 \times 10^{-16}$ 0.3146 0.4090 −5.83 −5.44
      $ ^{224}$U$ \to ^{220}$Th 8.63 $0^{+}$ $0^{+}$ $0$ $ -3.39 $ $ 0.824 $ $8.512 \times 10^{-19}$ 0.5992 0.4336 −3.61 −3.25
      $ ^{226}$U$ \to ^{222}$Th 7.70 $0^{+}$ $0^{+}$ $0$ $ -0.57 $ $ 0.869 $ $1.169 \times 10^{-21}$ 0.6269 0.4751 −0.77 −0.45
      $ ^{228}$U$ \to ^{224}$Th 6.80 $0^{+}$ $0^{+}$ $0$ $ 2.90 $ $ 0.874 $ $5.231 \times 10^{-25}$ 0.4720 0.5332 2.57 2.85
      $ ^{230}$U$ \to ^{226}$Th 5.99 $0^{+}$ $0^{+}$ $0$ $ 6.43 $ $ 0.856 $ $1.115 \times 10^{-28}$ 0.6669 0.6076 6.25 6.47
      $ ^{232}$U$ \to ^{228}$Th 5.41 $0^{+}$ $0^{+}$ $0$ $ 9.50 $ $ 0.867 $ $8.207 \times 10^{-32}$ 0.7615 0.6733 9.38 9.55
      $ ^{234}$U$ \to ^{230}$Th 4.86 $0^{+}$ $0^{+}$ $0$ $ 13.04 $ $ 0.883 $ $2.621\times 10^{-35}$ 0.6755 0.7595 12.87 12.99
      $ ^{236}$U$ \to ^{232}$Th 4.57 $0^{+}$ $0^{+}$ $0$ $ 15.00 $ $ 0.876 $ $2.174 \times 10^{-37}$ 0.9005 0.7982 14.95 15.05
      $ ^{238}$U$ \to ^{234}$Th 4.27 $0^{+}$ $0^{+}$ $0$ $ 17.25 $ $ 0.755 $ $8.980 \times 10^{-40}$ 1.4219 0.8515 17.40 17.47
      $ ^{230}$Pu$ \to ^{226}$U 7.18 $0^{+}$ $0^{+}$ $0$ $ 2.01 $ $ 0.875 $ $2.401 \times 10^{-24}$ 0.7971 0.4951 1.91 2.22
      $ ^{232}$Pu$ \to ^{228}$U 6.72 $0^{+}$ $0^{+}$ $0$ $ 4.13 $ $ 0.873 $ $3.399\times 10^{-26}$ 0.4279 0.5148 3.76 4.05
      $ ^{234}$Pu$ \to ^{230}$U 6.31 $0^{+}$ $0^{+}$ $0$ $ 5.89 $ $ 0.876 $ $5.122 \times 10^{-28}$ 0.4920 0.5344 5.58 5.85
      $ ^{236}$Pu$ \to ^{232}$U 5.87 $0^{+}$ $0^{+}$ $0$ $ 8.11 $ $ 0.873 $ $3.382 \times 10^{-30}$ 0.4506 0.5649 7.76 8.01
      $ ^{238}$Pu$ \to ^{234}$U 5.59 $0^{+}$ $0^{+}$ $0$ $ 9.59 $ $ 0.876 $ $1.044 \times 10^{-31}$ 0.4822 0.5774 9.27 9.51
      $ ^{240}$Pu$ \to ^{236}$U 5.26 $0^{+}$ $0^{+}$ $0$ $ 11.45 $ $ 0.877 $ $1.170 \times 10^{-33}$ 0.5929 0.6035 11.22 11.44
      $ ^{242}$Pu$ \to ^{238}$U 4.98 $0^{+}$ $0^{+}$ $0$ $ 13.18 $ $ 0.887 $ $1.830 \times 10^{-35}$ 0.6979 0.6262 13.02 13.23
      $ ^{244}$Pu$ \to ^{240}$U 4.67 $0^{+}$ $0^{+}$ $0$ $ 15.50 $ $ 0.875 $ $1.155 \times 10^{-37}$ 0.5366 0.6627 15.23 15.41
      $ ^{234}$Cm$ \to ^{230}$Pu 7.37 $0^{+}$ $0^{+}$ $0$ $ 2.11 $ $ 0.825 $ $2.057 \times 10^{-24}$ 0.7923 0.4519 2.00 2.35
      $ ^{236}$Cm$ \to ^{232}$Pu 7.07 $0^{+}$ $0^{+}$ $0$ $ 3.35 $ $ 0.873 $ $1.472 \times 10^{-25}$ 0.5958 0.4536 3.13 3.47
      $ ^{238}$Cm$ \to ^{234}$Pu 6.67 $0^{+}$ $0^{+}$ $0$ $ 5.51 $ $ 0.846 $ $3.143 \times 10^{-27}$ 0.1992 0.4677 4.81 5.14
      $ ^{240}$Cm$ \to ^{236}$Pu 6.40 $0^{+}$ $0^{+}$ $0$ $ 6.52 $ $ 0.826 $ $1.953\times 10^{-28}$ 0.3210 0.4711 6.03 6.35
      $ ^{242}$Cm$ \to ^{238}$Pu 6.22 $0^{+}$ $0^{+}$ $0$ $ 7.28 $ $ 0.879 $ $2.846 \times 10^{-29}$ 0.3599 0.4658 6.84 7.17
      $ ^{244}$Cm$ \to ^{240}$Pu 5.90 $0^{+}$ $0^{+}$ $0$ $ 8.87 $ $ 0.820 $ $6.784 \times 10^{-31}$ 0.4159 0.4793 8.49 8.81
      $ ^{246}$Cm$ \to ^{242}$Pu 5.48 $0^{+}$ $0^{+}$ $0$ $ 11.26 $ $ 0.874 $ $2.905\times 10^{-33}$ 0.3711 0.5119 10.83 11.12
      $ ^{248}$Cm$ \to ^{244}$Pu 5.16 $0^{+}$ $0^{+}$ $0$ $ 13.16 $ $ 0.835 $ $2.943 \times 10^{-35}$ 0.4832 0.5368 12.84 13.11
      $ ^{238}$Cf$ \to ^{234}$Cm 8.13 $0^{+}$ $0^{+}$ $0$ $ -0.08 $ $ 0.806 $ $1.889 \times 10^{-22}$ 1.3545 0.3722 0.05 0.48
      $ ^{240}$Cf$ \to ^{236}$Cm 7.71 $0^{+}$ $0^{+}$ $0$ $ 2.03 $ $ 0.866 $ $7.027 \times 10^{-24}$ 0.2629 0.3795 1.45 1.87
      $ ^{242}$Cf$ \to ^{238}$Cm 7.52 $0^{+}$ $0^{+}$ $0$ $ 2.54 $ $ 0.867 $ $1.520 \times 10^{-24}$ 0.3755 0.3720 2.11 2.54
      $ ^{244}$Cf$ \to ^{240}$Cm 7.33 $0^{+}$ $0^{+}$ $0$ $ 3.06 $ $ 0.812 $ $3.078 \times 10^{-25}$ 0.5978 0.3654 2.84 3.27
      $ ^{246}$Cf$ \to ^{242}$Cm 6.86 $0^{+}$ $0^{+}$ $0$ $ 5.11 $ $ 0.830 $ $3.741 \times 10^{-27}$ 0.4288 0.3820 4.74 5.16
      $ ^{248}$Cf$ \to ^{244}$Cm 6.36 $0^{+}$ $0^{+}$ $0$ $ 7.56 $ $ 0.827 $ $1.947 \times 10^{-27}$ 0.2936 0.4067 7.03 7.42
      $ ^{250}$Cf$ \to ^{246}$Cm 6.13 $0^{+}$ $0^{+}$ $0$ $ 8.69 $ $ 0.860 $ $1.460 \times 10^{-30}$ 0.2790 0.4085 8.14 8.52
      $ ^{252}$Cf$ \to ^{248}$Cm 6.22 $0^{+}$ $0^{+}$ $0$ $ 8.01 $ $ 0.834 $ $4.700 \times 10^{-30}$ 0.4276 0.3780 7.64 8.06
      Continued on next page

      Table 1.  Calculations of α-particle preformation factors and α decay half-lives for 65 even-even nuclei. The experimental α decay half-lives, spin and parity are taken from Ref. [46, 63, 66]. The α decay energy are taken from the latest evaluated atomic mass table AME2020 [64, 65]. The decay energy and half-lives are in the unit of MeV and s, respectively.

      Table 1-continued from previous page
      ${\alpha}$ transition $Q_{\alpha}$ $j_{p}^{\pi}$ $j_{d}^{\pi}$ $\ell_{min}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,exp}}}$ F P $\rm P_{\alpha}^{{\rm{\,exp}}}$ $\rm P_{\alpha}^{{\rm{\,Eq}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal1}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal2}}}$
      $ ^{254}$Cf$ \to ^{250}$Cm 5.93 $0^{+}$ $0^{+}$ $0$ $ 9.31 $ $ 0.866 $ $1.505 \times 10^{-31}$ 0.6448 0.3869 9.12 9.53
      $ ^{244}$Fm$ \to ^{240}$Cf 8.55 $0^{+}$ $0^{+}$ $0$ $ -0.51 $ $ 0.850 $ $9.260 \times 10^{-22}$ 0.7051 0.3105 −0.66 −0.15
      $ ^{246}$Fm$ \to ^{242}$Cf 8.38 $0^{+}$ $0^{+}$ $0$ $ 0.17 $ $ 0.842 $ $2.892 \times 10^{-22}$ 0.4765 0.3023 −0.15 0.37
      $ ^{248}$Fm$ \to ^{244}$Cf 7.99 $0^{+}$ $0^{+}$ $0$ $ 1.66 $ $ 0.832 $ $1.504 \times 10^{-23}$ 0.3002 0.3063 1.14 1.65
      $ ^{250}$Fm$ \to ^{246}$Cf 7.56 $0^{+}$ $0^{+}$ $0$ $ 3.38 $ $ 0.792 $ $4.302\times 10^{-25}$ 0.2100 0.3144 2.70 3.20
      $ ^{252}$Fm$ \to ^{248}$Cf 7.15 $0^{+}$ $0^{+}$ $0$ $ 5.04 $ $ 0.812 $ $1.063 \times 10^{-26}$ 0.1814 0.3238 4.30 4.79
      $ ^{254}$Fm$ \to ^{250}$Cf 7.31 $0^{+}$ $0^{+}$ $0$ $ 4.14 $ $ 0.790 $ $5.371 \times 10^{-26}$ 0.2929 0.2967 3.61 4.13
      $ ^{256}$Fm$ \to ^{252}$Cf 7.03 $0^{+}$ $0^{+}$ $0$ $ 5.14 $ $ 0.838 $ $3.985 \times 10^{-27}$ 0.3724 0.2984 4.71 5.24
      $ ^{252}$No$ \to ^{248}$Fm 8.55 $0^{+}$ $0^{+}$ $0$ $ 0.74 $ $ 0.845 $ $2.179 \times 10^{-22}$ 0.1696 0.2638 −0.03 0.55
      $ ^{254}$No$ \to ^{250}$Fm 8.23 $0^{+}$ $0^{+}$ $0$ $ 1.82 $ $ 0.848 $ $2.047 \times 10^{-23}$ 0.1496 0.2640 0.99 1.57
      $ ^{256}$No$ \to ^{252}$Fm 8.58 $0^{+}$ $0^{+}$ $0$ $ 0.53 $ $ 0.828 $ $3.309\times 10^{-22}$ 0.1848 0.2357 −0.20 0.42
      $ ^{256}$Rf$ \to ^{252}$No 8.93 $0^{+}$ $0^{+}$ $0$ $ 0.32 $ $ 0.869 $ $6.876 \times 10^{-22}$ 0.1375 0.2353 −0.54 0.09
      $ ^{258}$Rf$ \to ^{254}$No 9.20 $0^{+}$ $0^{+}$ $0$ $ -0.61 $ $ 0.853 $ $4.952 \times 10^{-21}$ 0.1656 0.2139 −1.39 −0.72
      $ ^{260}$Sg$ \to ^{256}$Rf 9.90 $0^{+}$ $0^{+}$ $0$ $ -2.04 $ $ 0.860 $ $9.918 \times 10^{-20}$ 0.2379 0.1932 −2.66 −1.95
      $ ^{266}$Hs$ \to ^{262}$Sg 10.35 $0^{+}$ $0^{+}$ $0$ $ -2.41 $ $0.852 $ $3.574 \times 10^{-19}$ 0.1450 0.1634 −3.25 −2.46
      $ ^{268}$Hs$ \to ^{264}$Sg 9.77 $0^{+}$ $0^{+}$ $0$ $ -0.39 $ $ 0.836 $ $1.116 \times 10^{-20}$ 0.0452 0.1680 −1.73 -0.96
      $ ^{270}$Hs$ \to ^{266}$Sg 9.07 $0^{+}$ $0^{+}$ $0$ $ 1.18 $ $ 0.826 $ $1.035 \times 10^{-22}$ 0.1327 0.1775 0.30 1.05
      $ ^{270}$Ds$ \to ^{266}$Hs 11.12 $0^{+}$ $0^{+}$ $0$ $ -2.70 $ $0.841 $ $6.663 \times 10^{-18}$ 0.0154 0.1404 −4.51 −3.66
      $ ^{282}$Ds$ \to ^{278}$Hs 9.15 $0^{+}$ $0^{+}$ $0$ $ 1.82 $ $ 0.837 $ $5.506\times 10^{-23}$ 0.0565 0.1363 0.57 1.44
      $ ^{286}$Cn$ \to ^{282}$Ds 9.24 $0^{+}$ $0^{+}$ $0$ $ 1.48 $ $ 0.820 $ $2.180\times 10^{-23}$ 0.3208 0.1282 0.98 1.88
      $ ^{286}$FI$ \to ^{282}$Cn 10.36 $0^{+}$ $0^{+}$ $0$ $ -0.66 $ $0.845 $ $7.558 \times 10^{-21}$ 0.1231 0.1141 −1.57 −0.63
      $ ^{288}$FI$ \to ^{284}$Cn 10.08 $0^{+}$ $0^{+}$ $0$ $ -0.19 $ $0.758 $ $1.401 \times 10^{-21}$ 0.2507 0.1130 −0.79 0.16
      $ ^{290}$Lv$ \to ^{286}$FI 11.00 $0^{+}$ $0^{+}$ $0$ $ -2.05 $ $0.838 $ $8.488 \times 10^{-20}$ 0.2712 0.0997 −2.62 −1.62
      $ ^{292}$Lv$ \to ^{288}$FI 10.79 $0^{+}$ $0^{+}$ $0$ $ -1.80 $ $0.839 $ $2.728 \times 10^{-20}$ 0.4743 0.0976 −2.12 −1.11
      $ ^{294}$Og$ \to ^{290}$Lv 11.87 $0^{+}$ $0^{+}$ $0$ $ -3.15 $ $0.769 $ $2.592 \times 10^{-18}$ 0.1219 0.0855 −4.06 −3.00
      ${\alpha}$ transition $Q_{\alpha}$ $j_{p}^{\pi}$ $j_{d}^{\pi}$ $\ell_{min}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,exp}}}$ F P $\rm P_{\alpha}^{{\rm{\,exp}}}$ $\rm P_{\alpha}^{{\rm{\,Eq}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal1}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal2}}}$
      $ ^{221}$Th$ \to ^{217}$Ra 8.63 $7/2^{+}{\#}$ $(9/2^{+})$ $2$ $ -2.76 $ $ 0.873 $ $ 2.532\times 10^{-18}$ $ 0.0446 $ 0.1123 −4.11 −3.16
      $ ^{223}$Th$ \to ^{219}$Ra 7.57 $(5/2)^{+}$ $(7/2)^{+}$ $2$ $ -0.22 $ $ 0.876 $ $1.465 \times 10^{-21}$ $ 0.2216 $ 0.1254 −0.87 0.03
      $ ^{225}$Th$ \to ^{221}$Ra 6.92 $3/2^{+}$ $5/2^{+}*$ $2$ $ 2.77 $ $ 0.866 $ $6.527 \times 10^{-24}$ $ 0.0515 $ 0.1336 1.48 2.36
      $ ^{227}$Th$ \to ^{223}$Ra 6.15 $(1/2^{+})$ $3/2^{+}*$ $2$ $ 6.21 $ $ 0.875 $ $3.325 \times 10^{-27}$ $ 0.0363 $ 0.1494 4.77 5.60
      $ ^{229}$Th$ \to ^{225}$Ra 5.17 $5/2^{+}*$ $1/2^{+}$ $2$ $ 11.39$ $ 0.863 $ $1.752 \times 10^{-32}$ $ 0.0462 $ 0.1843 10.05 10.79
      $ ^{221}$Pa$ \to ^{217}$Ac 9.24 $9/2^{-}$ $9/2^{-}$ $0$ $ -5.23 $ $ 0.838 $ $7.637 \times 10^{-17}$ $ 0.4534 $ 0.1438 −5.57 −4.73
      $ ^{223}$Pa$ \to ^{219}$Ac 8.35 $9/2^{-}$ $9/2^{-}$ $0$ $ -2.15 $ $ 0.860 $ $3.065 \times 10^{-19}$ $ 0.0918 $ 0.1539 −3.19 −2.37
      $ ^{225}$Pa$ \to ^{221}$Ac 7.41 $5/2^{-}{\#}$ $9/2^{-}{\#}$ $2$ $ 0.23 $ $ 0.872 $ $1.647 \times 10^{-22}$ $ 0.7018$ 0.1254 0.08 0.98
      $ ^{227}$Pa$ \to ^{223}$Ac 6.58 $(5/2^{-})$ $(5/2^{-})$ $0$ $ 3.43 $ $ 0.872 $ $1.712 \times 10^{-25}$ $ 0.4260 $ 0.1895 3.06 3.78
      $ ^{229}$Pa$ \to ^{225}$Ac 5.84 $5/2^{+}$ $3/2^{-}$ $1$ $ 7.40 $ $ 0.882 $ $4.832 \times 10^{-29}$ $ 0.1601 $ 0.1792 6.60 7.35
      $ ^{231}$Pa$ \to ^{227}$Ac 5.15 $3/2^{-}*$ $3/2^{-}*$ $0$ $ 12.00 $ $ 0.869 $ $7.156 \times 10^{-33}$ $ 0.0276$ 0.2462 10.44 11.05
      $ ^{223}$U$ \to ^{219}$Th 9.17 $7/2^{+}{\#}$ $9/2^{+}{\#}$ $2$ $ -4.74 $ $ 0.791 $ $1.392 \times 10^{-17}$ $ 0.8552$ 0.1037 −4.81 −3.82
      $ ^{225}$U$ \to ^{221}$Th 8.01 $5/2^{+}$ $7/2^{+}{\#}$ $2$ $ -1.16 $ $ 0.869 $ $6.834 \times 10^{-21}$ $ 0.4170 $ 0.1163 −1.54 −0.61
      $ ^{227}$U$ \to ^{223}$Th 7.24 $(3/2^{+})$ $(5/2^{+})$ $2$ $ 1.82$ $ 0.868 $ $1.569 \times 10^{-23}$ $ 0.1904$ 0.1259 1.10 2.00
      $ ^{229}$U$ \to ^{225}$Th 6.48 $3/2^{+}$ $3/2^{+}$ $0$ $ 4.24 $ $ 0.823 $ $2.288 \times 10^{-26}$ $ 0.5238 $ 0.1888 3.96 4.68
      $ ^{231}$U$ \to ^{227}$Th 5.58 $5/2^{+}{\#}$ $(1/2^{+})$ $2$ $ 9.82 $ $ 0.828 $ $4.231 \times 10^{-31}$ $ 0.0740 $ 0.1652 8.69 9.47
      $ ^{233}$U$ \to ^{229}$Th 4.91 $5/2^{+}*$ $5/2^{+}*$ $0$ $ 12.69$ $ 0.891 $ $5.536 \times 10^{-35}$ $ 0.7096 $ 0.2610 12.54 13.12
      $ ^{223}$Np$ \to ^{219}$Pa 9.66 $(9/2^{-})$ $9/2^{-}$ $0$ $ -5.66 $ $ 0.858 $ $1.728 \times 10^{-16}$ $ 0.5281 $ 0.1355 −5.94 −5.05
      $ ^{225}$Np$ \to ^{221}$Pa 8.83 $9/2^{-}{\#}$ $9/2^{-}$ $0$ $ -2.44 $ $ 0.851 $ $1.347 \times 10^{-18}$ $ 0.0412$ 0.1429 −3.83 −2.98
      $ ^{227}$Np$ \to ^{223}$Pa 7.82 $5/2^{+}{\#}$ $9/2^{-}{\#}$ $3$ $ -0.29 $ $ 0.873 $ $3.964 \times 10^{-22}$ $0.9648$ 0.1028 −0.31 0.68
      $ ^{229}$Np$ \to ^{225}$Pa 7.02 $5/2^{+}$ $5/2^{-}{\#}$ $1$ $ 2.55 $ $ 0.702 $ $1.250 \times 10^{-24}$ $ 0.5501 $ 0.1455 2.29 3.13
      $ ^{231}$Np$ \to ^{227}$Pa 6.37 $5/2^{+}{\#}$ $(5/2^{-})$ $1$ $ 5.16 $ $ 0.839 $ $2.161 \times 10^{-27}$ $ 0.6539 $ 0.1584 4.98 5.78
      $ ^{233}$Np$ \to ^{229}$Pa 5.63 $5/2^{+}{\#}$ $5/2^{+}$ $0$ $ 8.49 $ $ 0.847 $ $4.674 \times 10^{-31}$ $ 1.4010 $ 0.2160 8.64 9.30
      $ ^{235}$Np$ \to ^{231}$Pa 5.19 $5/2^{+}$ $3/2^{-}*$ $1$ $ 12.12 $ $ 0.861 $ $9.755 \times 10^{-34}$ $ 0.1548 $ 0.1954 11.31 12.02
      $ ^{237}$Np$ \to ^{233}$Pa 4.96 $5/2^{+}*$ $3/2^{-}$ $1$ $ 13.83 $ $ 0.854 $ $3.241 \times 10^{-35}$ $ 0.0917 $ 0.1993 12.79 13.49
      $ ^{229}$Pu$ \to ^{225}$U 7.59 $3/2^{+}{\#}$ $5/2^{+}$ $2$ $ 1.95 $ $ 0.858 $ $4.507 \times 10^{-23}$ $ 0.0497 $ 0.1181 0.65 1.57
      $ ^{231}$Pu$ \to ^{227}$U 6.84 $(3/2^{+})$ $(3/2^{+})$ $0$ $ 3.71 $ $ 0.860 $ $1.053 \times 10^{-25}$ $ 0.3692 $ 0.1754 3.28 4.03
      $ ^{233}$Pu$ \to ^{229}$U 6.41 $5/2^{+}{\#}$ $3/2^{+}$ $2$ $ 6.00 $ $ 0.853 $ $8.244\times 10^{-28}$ $ 0.2436 $ 0.1346 5.39 6.26
      $ ^{235}$Pu$ \to ^{231}$U 5.95 $(5/2^{+})$ $5/2^{+}{\#}$ $0$ $ 7.73 $ $ 0.852 $ $8.520 \times 10^{-30}$ $ 0.4395 $ 0.1934 7.37 8.09
      $ ^{241}$Pu$ \to ^{237}$U 5.14 $5/2^{+}*$ $1/2^{+}$ $2$ $ 13.26 $ $ 0.847 $ $1.172 \times 10^{-34}$ $ 0.0949 $ 0.1529 12.24 13.05
      $ ^{229}$Am$ \to ^{225}$Np 8.14 $5/2^{-}{\#}$ $9/2^{-}{\#}$ $2$ $ -0.01 $ $ 0.867 $ $1.294 \times 10^{-21}$ $ 0.1563$ 0.1104 −0.82 0.14
      $ ^{233}$Am$ \to ^{229}$Np 7.07 $5/2^{-}{\#}$ $5/2^{+}$ $1$ $ 3.63 $ $ 0.884 $ $3.000 \times 10^{-25}$ $ 0.1515 $ 0.1366 2.81 3.67
      $ ^{235}$Am$ \to ^{231}$Np 6.58 $5/2^{-}{\#}$ $5/2^{+}{\#}$ $1$ $ 5.19 $ $ 0.862 $ $2.642 \times 10^{-27}$ $ 0.4863$ 0.1435 4.88 5.72
      $ ^{237}$Am$ \to ^{233}$Np 6.20 $5/2^{-}$ $5/2^{+}{\#}$ $1$ $ 7.24 $ $ 0.853 $ $4.601\times 10^{-29}$ $ 0.2512 $ 0.1486 6.64 7.47
      $ ^{239}$Am$ \to ^{235}$Np 5.92 $5/2^{-}$ $5/2^{+}$ $1$ $ 8.63 $ $ 0.840 $ $1.840 \times 10^{-30}$ $ 0.2600 $ 0.1510 8.04 8.87
      $ ^{241}$Am$ \to ^{237}$Np 5.64 $5/2^{-}*$ $5/2^{+}*$ $1$ $ 10.13 $ $ 0.861 $ $5.715 \times 10^{-32}$ $ 0.2583 $ 0.1544 9.54 10.35
      $ ^{243}$Am$ \to ^{239}$Np 5.44 $5/2^{-}*$ $5/2^{+}$ $1$ $ 11.37 $ $ 0.870 $ $4.146 \times 10^{-33}$ $ 0.2027 $ 0.1550 10.68 11.49
      $ ^{233}$Cm$ \to ^{229}$Pu 7.48 $3/2^{+}{\#}$ $3/2^{+}{\#}$ $0$ $ 2.06 $ $ 0.807 $ $5.111 \times 10^{-24}$ $ 0.3619 $ 0.1547 1.62 2.43
      $ ^{235}$Cm$ \to ^{231}$Pu 7.28 $5/2^{+}{\#}$ $(3/2^{+})$ $2$ $ 4.47 $ $ 0.872 $ $5.613 \times 10^{-25}$ $ 0.0119$ 0.1120 2.54 3.49
      $ ^{241}$Cm$ \to ^{237}$Pu 6.19 $1/2^{+}$ $7/2^{-}$ $3$ $ 8.45 $ $ 0.867 $ $6.323 \times 10^{-30}$ $ 0.1110 $ 0.1066 7.50 8.47
      Continued on next page

      Table 2.  Same as Table 1 but for 87 odd-A nuclei. '*', '#' and '()' means values estimated from trends in neighboring nuclei and uncertain spin and/or parity, respectively.

      Table 2-continued from previous page
      ${\alpha}$ transition $Q_{\alpha}$ $j_{p}^{\pi}$ $j_{d}^{\pi}$ $\ell_{min}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,exp}}}$ F P $\rm P_{\alpha}^{{\rm{\,exp}}}$ $\rm P_{\alpha}^{{\rm{\,Eq}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal1}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal2}}}$
      $ ^{243}$Cm$ \to ^{239}$Pu 6.17 $5/2^{+}*$ $1/2^{+}*$ $2$ $ 8.96 $ $ 0.822 $ $9.640 \times 10^{-30}$ $ 0.0237 $ 0.1150 7.34 8.27
      $ ^{245}$Cm$ \to ^{241}$Pu 5.62 $7/2^{+}*$ $5/2^{+}*$ $2$ $ 11.42$ $ 0.867 $ $1.081\times 10^{-32}$ $ 0.0696 $ 0.1265 10.26 11.16
      $ ^{247}$Cm$ \to ^{243}$Pu 5.35 $9/2^{-}*$ $7/2^{+}$ $1$ $ 14.69 $ $ 0.861 $ $3.967 \times 10^{-34}$ $ 0.0010 $ 0.1478 11.70 12.53
      $ ^{245}$Bk$ \to ^{241}$Am 6.45 $3/2^{-}$ $5/2^{-}*$ $2$ $ 8.55 $ $ 0.843 $ $8.184 \times 10^{-29}$ $ 0.0070 $ 0.1049 6.40 7.37
      $ ^{247}$Bk$ \to ^{243}$Am 5.89 $3/2^{-}$ $5/2^{-}*$ $2$ $ 10.64 $ $ 0.858 $ $1.214 \times 10^{-31}$ $ 0.0377 $ 0.1147 9.22 10.16
      $ ^{249}$Bk$ \to ^{245}$Am 5.52 $7/2^{+}*$ $5/2^{+}$ $2$ $ 12.32 $ $ 0.827 $ $9.715 \times 10^{-34}$ $ 0.1022 $ 0.1209 11.33 12.25
      $ ^{237}$Cf$ \to ^{233}$Cm 8.23 $5/2^{+}{\#}$ $3/2^{+}{\#}$ $2$ $ 0.06 $ $ 0.864 $ $2.272 \times 10^{-22}$ $ 0.7605 $ 0.0943 −0.06 0.97
      $ ^{239}$Cf$ \to ^{235}$Cm 7.77 $5/2^{+}{\#}$ $5/2^{+}{\#}$ $0$ $ 1.63 $ $ 0.794 $ $1.104 \times 10^{-23}$ $ 0.4588 $ 0.1312 1.27 2.17
      $ ^{243}$Cf$ \to ^{239}$Cm 7.42 $(1/2^{+})$ $7/2^{-}{\#}$ $3$ $ 3.66 $ $ 0.801 $ $2.258 \times 10^{-25}$ $ 0.2075 $ 0.0815 2.98 4.07
      $ ^{245}$Cf$ \to ^{241}$Cm 7.26 $1/2^{+}$ $1/2^{+}$ $0$ $ 3.89$ $ 0.866 $ $1.701 \times 10^{-25}$ $ 0.1500 $ 0.1225 3.07 3.98
      $ ^{247}$Cf$ \to ^{243}$Cm 6.50 $(7/2^{+})$ $5/2^{+}*$ $2$ $ 7.50 $ $ 0.904 $ $5.137 \times 10^{-29}$ $ 0.1168 $ 0.1012 6.57 7.56
      $ ^{249}$Cf$ \to ^{245}$Cm 6.29 $9/2^{-}$ $7/2^{+}*$ $1$ $ 10.04 $ $ 0.876 $ $7.643 \times 10^{-30}$ $ 0.0023 $ 0.1148 7.41 8.35
      $ ^{251}$Cf$ \to ^{247}$Cm 6.18 $1/2^{+}$ $9/2^{-}*$ $5$ $ 10.45 $ $ 0.866 $ $1.853 \times 10^{-31}$ $ 0.0379 $ 0.0675 9.03 10.20
      $ ^{243}$Es$ \to ^{239}$Bk 8.08 $(7/2^{+})$ $(7/2^{+})$ $0$ $ 1.55 $ $ 0.783 $ $6.317 \times 10^{-23}$ $ 0.0977 $ 0.1142 0.54 1.48
      $ ^{249}$Es$ \to ^{245}$Bk 6.94 $7/2^{+}$ $3/2^{-}$ $3$ $ 6.03 $ $ 0.840 $ $1.139 \times 10^{-27}$ $ 0.1673 $ 0.0788 5.25 6.36
      $ ^{251}$Es$ \to ^{247}$Bk 6.60 $3/2^{-}$ $3/2^{-}$ $0$ $ 7.37 $ $ 0.832 $ $1.059 \times 10^{-28}$ $ 0.0830 $ 0.1241 6.29 7.20
      $ ^{253}$Es$ \to ^{249}$Bk 6.74 $7/2^{+}*$ $7/2^{+}*$ $0$ $ 6.25 $ $ 0.853 $ $5.207 \times 10^{-28}$ $ 0.2173 $ 0.1137 5.59 6.53
      $ ^{243}$Fm$ \to ^{239}$Cf 8.70 $7/2^{-}{\#}$ $5/2^{+}{\#}$ $1$ $ -0.60 $ $ 0.838 $ $2.185 \times 10^{-21}$ $ 0.3729 $ 0.0888 −1.03 0.02
      $ ^{245}$Fm$ \to ^{241}$Cf 8.44 $1/2^{+}{\#}$ $7/2^{-}{\#}$ $3$ $ 0.62 $ $ 0.864 $ $1.522 \times 10^{-22}$ $ 0.3128 $ 0.0680 0.12 1.28
      $ ^{247}$Fm$ \to ^{243}$Cf 8.26 $(7/2^{+})$ $(1/2^{+})$ $4$ $ 1.68 $ $ 0.869 $ $2.141 \times 10^{-23}$ $ 0.1927 $ 0.0585 0.96 2.20
      $ ^{251}$Fm$ \to ^{247}$Cf 7.43 $9/2^{-}$ $(7/2^{+})$ $1$ $ 6.02 $ $ 0.867 $ $1.177 \times 10^{-25}$ $ 0.0016 $ 0.0895 3.23 4.27
      $ ^{257}$Fm$ \to ^{253}$Cf 6.86 $9/2^{+}$ $7/2^{+}$ $2$ $ 6.94 $ $ 0.816 $ $4.393 \times 10^{-28}$ $ 0.0550 $ 0.0756 5.68 6.80
      $ ^{245}$Md$ \to ^{241}$Es 9.02 $(7/2^{-})$ $3/2^{-}{\#}$ $2$ $ -0.48 $ $ 0.857 $ $6.472 \times 10^{-21}$ $0.0935$ 0.0723 −1.51 −0.37
      $ ^{247}$Md$ \to ^{243}$Es 8.77 $7/2^{-}{\#}$ $(7/2^{+})$ $1$ $ 0.08 $ $ 0.853 $ $1.774 \times 10^{-21}$ $ 0.0944 $ 0.0809 −0.95 0.15
      $ ^{249}$Md$ \to ^{245}$Es 8.44 $(7/2^{-})$ $(3/2^{-})$ $2$ $ 1.54 $ $ 0.855 $ $1.237 \times 10^{-22}$ $ 0.0468$ 0.0711 0.21 1.36
      $ ^{251}$Md$ \to ^{247}$Es 7.96 $(7/2^{-})$ $(7/2^{+})$ $1$ $ 3.40 $ $ 0.869 $ $4.349 \times 10^{-24}$ $ 0.0181 $ 0.0833 1.66 2.74
      $ ^{253}$Md$ \to ^{249}$Es 7.57 $(7/2^{-})$ $7/2^{+}$ $1$ $ 4.71 $ $ 0.850 $ $1.673 \times 10^{-25}$ $ 0.0235 $ 0.0849 3.08 4.15
      $ ^{255}$Md$ \to ^{251}$Es 7.91 $(7/2^{-})$ $3/2^{-}$ $2$ $ 4.36 $ $ 0.827 $ $2.468 \times 10^{-24}$ $ 0.0037 $ 0.0663 1.92 3.10
      $ ^{257}$Md$ \to ^{253}$Es 7.56 $(7/2^{-})$ $7/2^{+}*$ $1$ $ 5.12 $ $ 0.866 $ $1.861 \times 10^{-25}$ $ 0.0081 $ 0.0765 3.03 4.14
      $ ^{251}$No$ \to ^{247}$Fm 8.76 $(7/2^{+})$ $(7/2^{+})$ $0$ $ -0.02 $ $ 0.836 $ $9.666 \times 10^{-22}$ $ 0.2225$ 0.0890 −0.67 0.38
      $ ^{253}$No$ \to ^{249}$Fm 8.42 $9/2^{-}*$ $7/2^{+}$ $1$ $ 2.23 $ $ 0.781 $ $7.212 \times 10^{-23}$ $ 0.0179$ 0.0748 0.48 1.61
      $ ^{257}$No$ \to ^{253}$Fm 8.48 $(3/2^{+})$ $1/2^{+}$ $2$ $ 1.46 $ $ 0.870 $ $9.831 \times 10^{-23}$ $ 0.0696 $ 0.0585 0.30 1.54
      $ ^{259}$No$ \to ^{255}$Fm 7.85 $9/2^{+}$ $7/2^{+}$ $2$ $ 3.66 $ $ 0.863 $ $6.814 \times 10^{-25}$ $ 0.0638 $ 0.0620 2.47 3.67
      $ ^{253}$Lr$ \to ^{249}$Md 8.92 $(7/2^{-})$ $(7/2^{-})$ $0$ $ 0.12 $ $ 0.802 $ $1.355 \times 10^{-21}$ $ 0.1198 $ 0.0845 −0.80 0.27
      $ ^{255}$Rf$ \to ^{251}$No 9.06 $(9/2^{-})$ $(7/2^{+})$ $1$ $ 0.62 $ $ 0.866 $ $1.382 \times 10^{-21}$ $ 0.0344 $ 0.0675 −0.84 0.33
      $ ^{259}$Rf$ \to ^{255}$No 9.13 $3/2^{+}{\#}$ $(1/2^{+})$ $2$ $0.49 $ $ 0.800 $ $1.949 \times 10^{-21}$ $ 0.0357$ 0.0528 −0.96 0.32
      $ ^{261}$Rf$ \to ^{257}$No 8.65 $3/2^{+}{\#}$ $(3/2^{+})$ $0$ $ 1.06 $ $ 0.861$ $1.134 \times 10^{-22}$ $ 0.1531 $ 0.0733 0.25 1.38
      $ ^{257}$Db$ \to ^{253}$Lr 9.21 $9/2^{+}{\#}$ $(7/2^{-})$ $1$ $ 0.26 $ $ 0.861 $ $1.780 \times 10^{-21}$ $ 0.0616 $ 0.0642 −0.95 0.24
      $ ^{263}$Db$ \to ^{259}$Lr 8.84 $9/2^{+}{\#}$ $1/2^{-}{\#}$ $5$ $ 1.80 $ $ 0.832 $ $1.690 \times 10^{-23}$ $0.1935$ 0.0351 1.09 2.54
      $ ^{259}$Sg$ \to ^{255}$Rf 9.77 $(11/2^{-})$ $(9/2^{-})$ $2$ $ -0.48 $ $ 0.847 $ $2.378 \times 10^{-20}$ $ 0.0257 $ 0.0506 −2.07 −0.77
      $ ^{261}$Sg$ \to ^{257}$Rf 9.71 $(3/2^{+})$ $(1/2^{+})$ $2$ $ -0.72 $ $ 0.874 $ $1.791 \times 10^{-20}$ $ 0.0576 $ 0.0484 −1.96 −0.64
      Continued on next page
      Table 2-continued from previous page
      ${\alpha}$ transition $Q_{\alpha}$ $j_{p}^{\pi}$ $j_{d}^{\pi}$ $\ell_{min}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,exp}}}$ F P $\rm P_{\alpha}^{{\rm{\,exp}}}$ $\rm P_{\alpha}^{{\rm{\,Eq}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal1}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal2}}}$
      $ ^{263}$Sg$ \to ^{259}$Rf 9.41 $3/2^{+}{\#}$ $3/2^{+}{\#}$ $0$ $ 0.03$ $0.802$ $4.560 \times 10^{-21}$ $0.0438$ 0.0651 −1.33 −0.14
      $ ^{265}$Sg$ \to ^{261}$Rf 9.05 $11/2^{-}{\#}$ $3/2^{+}{\#}$ $5$ $ 1.26 $ $ 0.863 $ $3.526 \times 10^{-23}$ $0.3101$ 0.0330 0.75 2.23
      $ ^{261}$Bh$ \to ^{257}$Db 10.51 $(5/2^{-})$ $9/2^{+}{\#}$ $3$ $ -1.89 $ $ 0.836 $ $5.991 \times 10^{-19}$ $0.0266 $ 0.0393 −3.46 −2.06
      $ ^{265}$Hs$ \to ^{261}$Sg 10.47 $3/2^{+}{\#}$ $(3/2^{+})$ $0$ $ -2.72 $ $ 0.817 $ $6.846 \times 10^{-19}$ $0.1611 $ 0.0561 −3.51 −2.26
      $ ^{269}$Hs$ \to ^{265}$Sg 9.28 $9/2^{+}{\#}$ $11/2^{-}{\#}$ $1$ $ 1.18 $ $ 0.841 $ $3.727 \times 10^{-22}$ $0.0362$ 0.0502 −0.26 1.04
      $ ^{267}$Ds$ \to ^{263}$Hs 11.78 $3/2^{+}{\#}$ $3/2^{+}{\#}$ $0$ $ -5.00 $ $ 0.810 $ $1.664 \times 10^{-16}$ $0.1274$ 0.0478 −5.89 −4.57
      ${\alpha}$ transition $Q_{\alpha}$ $j_{p}^{\pi}$ $j_{d}^{\pi}$ $\ell_{min}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,exp}}}$ F P $\rm P_{\alpha}^{{\rm{\,exp}}}$ $\rm P_{\alpha}^{{\rm{\,Eq}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal1}}}$ ${\rm{log}}_{10}{{T}}_{1/2}^{{\rm{\,cal2}}}$
      $ ^{224}$Pa$ \to ^{220}$Ac 7.69 $(5^{-})$ $(3^{-})$ $2$ $ -0.07 $ $ 0.868 $ $1.488 \times 10^{-21}$ $ 0.1559 $ 0.0417 -0.88 0.50
      $ ^{226}$Pa$ \to ^{222}$Ac 6.99 $1^{-}{\#}$ $1^{-}$ $0$ $ 2.16 $ $ 0.873 $ $7.997 \times 10^{-24}$ $ 0.1698 $ 0.0608 1.39 2.61
      $ ^{228}$Pa$ \to ^{224}$Ac 6.27 $3^{+}$ $(0^{-})$ $3$ $ 6.63 $ $ 0.764 $ $2.338 \times 10^{-27}$ $ 0.0225 $ 0.0437 4.98 6.34
      $ ^{230}$Pa$ \to ^{226}$Ac 5.44 $2^{-}$ $(1^{-})$ $2$ $ 10.67 $ $ 0.791 $ $2.180 \times 10^{-31}$ $ 0.0212 $ 0.0581 9.00 10.23
      $ ^{224}$Np$ \to ^{220}$Pa 9.33 $2^{-}{\#}$ $1^{-}{\#}$ $2$ $ -4.42$ $ 0.842 $ $1.617 \times 10^{-17}$ $0.3308$ 0.0344 -4.90 -3.44
      $ ^{226}$Np$ \to ^{222}$Pa 8.34 $()$ $1^{-}{\#}$ $0$ $ -1.46 $ $ 0.884 $ $5.278 \times 10^{-20}$ $ 0.1059 $ 0.0508 -2.43 -1.14
      $ ^{228}$Np$ \to ^{224}$Pa 7.54 $4^{+}{\#}$ $(5^{-})$ $1$ $ 2.18 $ $ 0.841 $ $1.105 \times 10^{-22}$ $ 0.0122$ 0.0460 0.27 1.60
      $ ^{230}$Np$ \to ^{226}$Pa 6.78 $4^{+}{\#}$ $1^{-}{\#}$ $3$ $ 3.96 $ $ 0.861 $ $5.339 \times 10^{-26}$ $ 0.4087$ 0.0392 3.57 4.98
      $ ^{236}$Np$ \to ^{232}$Pa 5.01 $(6^{-})$ $2^{-}$ $4$ $ 15.48 $ $ 0.890 $ $1.206 \times 10^{-35}$ $ 0.0053 $ 0.0468 13.20 14.53
      $ ^{234}$Am$ \to ^{230}$Np 6.80 $0^{-}{\#}$ $4^{+}{\#}$ $5$ $ 5.55 $ $ 0.874 $ $1.808 \times 10^{-27}$ $0.3058$ 0.0288 5.04 6.58
      $ ^{236}$Am$ \to ^{232}$Np 6.26 $5^{-}$ $5^{-}$ $0$ $ 6.73 $ $ 0.856 $ $1.041 \times 10^{-28}$ $ 0.3582 $ 0.0610 6.28 7.50
      $ ^{234}$Bk$ \to ^{230}$Am 8.11 $3^{-}{\#}$ $1^{-}$ $2$ $ 2.24$ $ 0.871 $ $2.003 \times 10^{-22}$ $ 0.0057 $ 0.0346 -0.01 1.45
      $ ^{240}$Es$ \to ^{236}$Bk 8.27 $4^{-}{\#}$ $4^{+}$ $1$ $ 0.93 $ $ 0.857 $ $1.993 \times 10^{-22}$ $ 0.1179 $ 0.0343 1.64 1.47
      $ ^{242}$Es$ \to ^{238}$Bk 8.16 $2^{+}{\#}$ $1{\#}$ $1$ $ 1.49 $ $ 0.849 $ $9.438 \times 10^{-23}$ $ 0.0692 $ 0.0330 0.33 1.81
      $ ^{244}$Es$ \to ^{240}$Bk 7.95 $6^{+}{\#}$ $7^{-}$ $1$ $ 2.96$ $ 0.840 $ $1.964 \times 10^{-23}$ $ 0.0114 $ 0.0324 1.02 2.51
      $ ^{246}$Es$ \to ^{242}$BK 7.65 $4^{-}{\#}$ $3^{+}$ $1$ $ 3.66 $ $ 0.815 $ $1.766 \times 10^{-24}$ $ 0.0261 $ 0.0324 2.08 3.57
      $ ^{248}$Es$ \to ^{244}$Bk 7.16 $2^{-}{\#}$ $4^{-}{\#}$ $2$ $ 5.76 $ $ 0.869 $ $1.599 \times 10^{-26}$ $0.0215$ 0.0298 4.09 5.62
      $ ^{252}$Es$ \to ^{248}$Bk 6.74 $(4^{+})$ $6^{+}$ $2$ $ 7.72 $ $ 0.842 $ $2.896 \times 10^{-28}$ $ 0.0134 $ 0.0293 5.85 7.38
      $ ^{244}$Md$ \to ^{240}$Es 8.95 $3^{+}{\#}$ $4^{-}{\#}$ $1$ $ -0.42 $ $ 0.863 $ $5.377 \times 10^{-21}$ $ 0.0972$ 0.0290 -1.43 0.11
      $ ^{246}$Md$ \to ^{242}$Es 8.90 $1^{-}{\#}$ $2^{+}{\#}$ $1$ $ -0.04 $ $ 0.849 $ $4.204 \times 10^{-21}$ $0.0527$ 0.0277 -1.32 0.24
      $ ^{250}$Md$ \to ^{246}$Es 8.16 $2^{-}{\#}$ $4^{-}{\#}$ $2$ $ 2.87 $ $ 0.821 $ $1.482 \times 10^{-23}$ $ 0.0190$ 0.0247 1.15 2.76
      $ ^{256}$Md$ \to ^{252}$Es 7.75 $(1^{-})$ $(4^{+})$ $3$ $ 4.70 $ $ 0.870 $ $3.956 \times 10^{-25}$ $ 0.0099 $ 0.0199 2.70 4.40
      $ ^{258}$Md$ \to ^{254}$Es 7.27 $8^{-}{\#}$ $7^{+}$ $1$ $ 6.65 $ $ 0.858 $ $1.338 \times 10^{-26}$ $ 0.0033 $ 0.0268 4.17 5.75
      $ ^{252}$Lr$ \to ^{248}$Md 9.17 $7^{-}{\#}$ $()$ $0$ $ -0.08 $ $ 0.775 $ $7.252 \times 10^{-21}$ $ 0.0367 $ 0.0284 -1.52 0.03
      $ ^{254}$Lr$ \to ^{250}$Md 8.83 $4^{+}{\#}$ $2^{-}{\#}$ $3$ $ 1.40 $ $ 0.816 $ $2.711 \times 10^{-22}$ $0.0309$ 0.0184 -0.11 1.63
      $ ^{256}$Lr$ \to ^{252}$Md 8.86 $(0^{-},3^{-}){\#}$ $1^{+}$ $1$ $ 1.52 $ $0.843$ $8.642\times 10^{-22}$ $0.0071$ 0.0224 -0.63 1.02
      $ ^{256}$Db$ \to ^{252}$Lr 9.34 $9^{-}{\#}$ $7^{-}{\#}$ $2$ $ 0.36 $ $ 0.862 $ $2.931 \times 10^{-21}$ $0.0297$ 0.0193 -1.17 0.55
      $ ^{258}$Db$ \to ^{254}$Lr 9.44 $0^{-}{\#}$ $4^{+}{\#}$ $1$ $ 0.75 $ $ 0.845 $ $8.770 \times 10^{-21}$ $0.0041$ 0.0205 -1.64 0.05

      Table 3.  Same as Table 1 and 2 but for 28 odd-odd nuclei.

      In Table 1, one can clearly see that the normalized factor F always remains a small range, the experimental half-lives increases with the neutron number N in each isotope chain. Perhaps the increased neutrons play an important role in maintaining the nuclear stability. Wang $ et\ al. $ suggests the rising of the symmetry energy leads to this behavior[50]. On the contrary, the decay energy $ Q_{\alpha} $ and the penetration probability P are decreasing gradually in each isotope chain. There are very large discrepancies in the penetration probability for different nuclei with a given isotope chain, which means the penetration probability largely determines the α decay half-lives. More importantly, as shown in the ninth column, the penetration probability always stay a narrow range between $ 10^{-22} $ and $ 10^{-18} $ for $ Rf-Og $ (Z = 104 - 118) isotope chain. The relatively large penetration probability means that α-particle can easily escape from the parent nuclei and the superheavy nuclei should be weakly bound[51].

      As for preformation factors, combined these three tables, one can find that the sequence of nuclei in the order of decreasing $ P_{\alpha}^{\rm exp} $ are even-even nuclei, odd-A nuclei and odd-odd nuclei. The results satisfy the variation trend of α-particle preformation factors extracted by various models[5254]. For a more intuitive description, the extracted experimental α-particle preformation factors for $ Th $ and $ Pa $ isotopes are plotted as black square and red ball in Fig. 1. From this figure, one can clearly see that the preformation factors of even-even nuclei are significantly larger than corresponding ones of the adjacent odd-A nuclei from $ Th $ isotopes. Also, the corresponding preformation factors of odd-A nuclei are larger than corresponding ones of the adjacent odd-odd nuclei can be found in the $ Pa $ isotopes. This result suggests that the unpaired nucleons will inhibit the α-particle preformation. It worth mentioning that the above regularities are analyzed through the overall performance. Strictly, the preformation factors are strongly correlated with the nuclear structure. Zhang $ et\ al. $ have proved that the nuclei near the closed shell are relatively stable due to the strong bound states[51]. Thus, the preforming factors are relatively small when the nucleons occupy the closed shell or magic number. This pattern implies that the shell effect plays a key role in the α-particle preformation[55]. In addition, the obvious odd-even staggering effect in Fig. 1 can be seen as a valid object to further investigate the strong interaction. Similar odd-even effect also occurs in the one-neutron separation energy and one-proton separation energy[56]. Kaneko $ et\ al. $ have found that α-particle condensation is enhanced with the strong proton-neutron interaction[57]. Therefore, the α-particle preformation are restricted by multiple factors.

      Figure 1.  (color online) The extracted experimental α-particle preformation factor $P_{\alpha}^{\rm exp}$ from Eq.(14). The black square and red ball denote the corresponding results of Th and Pa isotopes, respectively.

      Combined with the above discussion, for those nuclei with exact experimental data, the corresponding preformation factors can be extracted by Eq.(14). However, there is no effective method to evaluate the α-particle preformation factors of unknown nuclei under the framework of TPA. Whether the preformation factor can be reasonably evaluated has aroused our attention. In this contribution, we are committed to construct a simple analytical expression to estimate preformation factors and thus the corresponding physical factors associated with the α-particle preformation factors spark our interests.

      An important feature that has to be considered is the relationship between the preforamtion factors and decay energy. As we all know, the $ Q_{\alpha} $ value determines the penetration probability and further affects the decay process. Deng $ et\ al. $ have proved that there is a nice agreement between the logarithmic values of extracted experimental α-particle preformation factors $ {{\rm {log}}_{10}}P_{\alpha}^{\rm exp} $ and the reciprocal of the square root of α decay energy $ Q_{\alpha}^{-1/2} $[58]. To have a more intuitive presentation, the corresponding results for even-even nuclei are plotted as a function of mass numbers A in Fig. 2. From this figure, one can see that the variation trend of $ {{\rm {log}}_{10}}P_{\alpha}^{\rm exp} $ and $ Q_{\alpha}^{-1/2} $ is consistent. This results indicate that the α decay energy can be seen as a well-described quantity to estimate α-particle preformation factors. Another key feature is that we need to take nuclear structure information into account. Referring to the study of proton radioactivity[59, 60], Delion $ et\ al. $ have proved that there is an obvious linear decline relationship between proton preformation factors and $ A^{1/3} $ (A denotes the mass number of the parent nuclei). They suggested that this performance is derived from the asymptotic exponential behavior of proton wave function on the nuclear surface. Whether the similar linear relationship exists in α decay process is a worthwhile exploration since these two decay modes share the same decay mechanism. For a further investigation, we plot the logarithmic values of extracted experimental α-particle preformation factors $ {{\rm {log}}_{10}}P_{\alpha}^{\rm exp} $ versus $ A^{1/3} $ in Fig. 3. From this figure, one can see that the preformation factors decrease gradually with the increase of mass number and the fitting line also satisfies the corresponding variation trend. This result means that $ A^{1/3} $ also can be seen as a valid physical quantity to describe α-particle preformation factors from another perspective.

      Figure 2.  (color online) The variations of logarithmic values of extracted experimental α-particle formation factors ${\rm {log}}_{10}P_{ \alpha}^{Exp}$ (the red ones) and the reciprocal of the square root of α decay energy $Q_{\alpha}^{-1/2}$ (the blue ones) against mass numbers for even-even nuclei.

      Figure 3.  (color online) The logarithm of the experimental α-particle formation factors versus $A^{1/3}$ for even-even nuclei.

      To sum up, we mainly describe that two physical quantities $ Q_{\alpha}^{-1/2} $ and $ A^{1/3} $ show considerable regularity in the estimation of α-particle preformation factors. And the previous discussion suggests that the unpaired nucleons will inhibit the α-particle preformation. Additionally, some studies have shown that for unfavored α decay, the corresponding centrifugal potential generated by the nonzero angular momentum also inhibit the formation of α particle on the surface of the parent nuclei[58]. Taking these analyses into account and catching the main physical factors, we further propose a simple analytical expression to estimate α-particle preformation factors, which can be expressed as

      $ \begin{array}{*{20}{l}} {\rm {log}}_{10}P_{\alpha}^{Eq} =aZQ_{\alpha}^{-1/2} +bA^{1/3} +c +d\sqrt{l(l+1)} +h, \end{array} $

      (15)

      where Z and A are the proton and mass number of parent nucleus. The first term is dependent on the α decay energy. The second and third terms jointly describe the relationship between preformation factors and $ A^{1/3} $. The fourth term represents the contribution of centrifugal potential to α-particle preformation. The last term denotes the blocking effect derived from the unpaired nucleons. The h value of even-even nuclei is equal to 0 since there are no unpaired nucleons in parent nuclei. While for odd-A nuclei and odd-odd nuclei, the value of blocking effect h of odd-odd nuclei is twice as much as the corresponding value of odd-A nuclei due to presence of unpaired nucleons. By fitting the extract experimental α-particle preformation factors, the values of adjustable parameters are obtained and listed in Table 4. In this table, one can see that the values of b, d and h are all negative. The negative value of b indicates that the preformation factors should decline as the increase of mass number A. This result is consistent with the linear decline relationship as shown in Fig. 3. The negative value of d and h also suggest that both centrifugal potential and the unpaired nucleons will inhibit the α-particle preformation. Using this formula, we systematically calculate the α-particle preformation factors denoted as $ P_{\alpha}^{\rm Eq} $ and list in Table 13 for even-even nuclei, odd-A nuclei and odd-odd nuclei, respectively.

      Number Nuclei a b c d h σ
      65 Even-Even nuclei 0.244
      87 Odd-A nuclei 0.035 −1.406 7.070 −0.054 −0.4687 0.601
      28 Odd-Odd nuclei −0.9374 0.575
      180 All nuclei 0.468

      Table 4.  The parameters of Eq.(15), and standard deviations between estimated α-particle preformation factors by Eq.(15) and extracted experimental α-particle preformation factors from Eq.(14).

      Generally, the standard deviation can reflects the agreement between the extracted experimental α-particle preformation factors from Eq.(14) and the estimated ones by Eq.(15). In this work, it is defined as follows

      $ \begin{array}{*{20}{l}} \sigma = \sqrt{\sum{({\rm{log}}_{10}{P_{\alpha}^{{\rm{Eq}}}}-{\rm{log}}_{10}{P_{\alpha}^{{\rm{exp}}}})^2}/n}, \end{array} $

      (16)

      where $ {\rm{log}}_{10}{P_{\alpha}^{{\rm{Eq}}}} $ and $ {\rm{log}}_{10}{P_{\alpha}^{{\rm{exp}}}} $ are logarithmic form of estimated α-particle preformation factors and the experimental ones, n is the number of nuclei involved for each case. With the help of Eq.(16), the corresponding standard deviations for 65 even-even nuclei, 87 odd-A nuclei, 28 odd-odd nuclei and all nuclei are calculated and listed in Table 4. From this table, one can find that Eq.(15) with only five valid parameters can reproduce extracted experimental α-particle preformation factors well. For a more intuitive comparison, the experimental α-particle preformation factors derived from Eq.(14) and estimated ones by Eq.(15) for even-even nuclei, odd-A nuclei and odd-odd nuclei are plotted in Fig. 46, respectively. From these three figures, one can see that the estimated α-particle preformation factors are relatively close to the experimental ones for even-even nuclei. However, the results for odd-A and odd-odd nuclei are not very satisfactory. To have a clear description, we separated two regions in Fig. 5 and 6, and interestingly found that the estimated α-particle factors were in good agreement with experimental ones in the light yellow region with neutron number $ N \;\; > $ 141. The large discrepancy from another region as shown in Fig. 5 and 6 prompts us to give a reasonable explanation. Tracing back to each stages, after detailed analysis, we believe that the corresponding difference is largely due to the uneven nuclear data affecting the balance of parameter values i.e. h of the blocking effect. Considering the overall performance, perhaps the corresponding deviations are understandable and acceptable.

      Figure 4.  (color online) The extracted experimental α-particle preformation $P_{\alpha}^{\rm exp}$ from Eq.(14) and the estimated ones $P_{\alpha}^{\rm Eq}$ by Eq.(15) for even-even nuclei. The black circle and red ball denote $P_{\alpha}^{\rm Exp}$ and $P_{\alpha}^{\rm Eq}$, respectively.

      Figure 5.  (color online) Same as Fig. 4, but depicting the extracted experimental α-particle preformation $P_{\alpha}^{\rm exp}$ from Eq. (14) and the estimated ones $P_{\alpha}^{\rm Eq}$ by Eq. (15) for odd-A nuclei.

      Figure 6.  (color online) Same as Fig. 4 and 5, but depicting the extracted experimental α-particle preformation $P_{\alpha}^{\rm exp}$ from Eq.(14) and the estimated ones $P_{\alpha}^{\rm Eq}$ by Eq.(15) for odd-odd nuclei.

      In the following, we list the calculated α decay half-lives with $ P_{0} = 1 $ denoted as $ {{\rm {log}}}_{10}{T_{1/2}^{{\rm{cal1}}}} $ in Table 13 for even-even nuclei, odd-A nuclei and odd-odd nuclei, respectively. Meanwhile, the estimated α-particle preformation factors by Eq.(15) are also used to calculate α decay half-lives, which are denoted as $ {{\rm {log}}}_{10}{T_{1/2}^{{\rm{cal2}}}} $ and listed in the same tables. For a clear comparison, the deviations between experimental α decay half-lives and the two calculated ones in the logarithmic form for even-even nuclei, odd-A nuclei and odd-odd nuclei are plotted in Fig. 79, respectively. From these three figures, one can see that there are large deviations between the $ {{\rm {log}}}_{10}{T_{1/2}^{{\rm{cal1}}}} $ and experimental α decay half-lives. After considering estimated α-particle preformation factors, the $ {{\rm {log}}}_{10}{T_{1/2}^{{\rm{cal2}}}} $ can reproduce the experimental data well as shown by green shaded regions in Fig. 79. The excellent performances suggest that this analytical expression can not only effectively evaluate the α-particle preformation factors, but also facilitate the accurate calculation of half-lives.

      Figure 7.  (color online)The differences in logarithmic form between experimental α decay half-lives and two calculated ones for even-even nuclei. The green star and red ball denote the differences caused by ${\rm{log}}_{10}{T_{1/2}^{{\rm{cal1}}}}$ and ${\rm{log}}_{10}{T_{1/2}^{{\rm{cal2}}}}$, respectively.

      Figure 9.  (color online) Same as Fig. 7 and 8, but for odd-odd nuclei.

      As an application, this analytical expression is used to estimate the α-particle preformation factors and further predict corresponding half-lives for unknown even-even nuclei with Z = 118 and 120. It is well known that the $ Q_{\alpha} $ value is extremely important to predict α decay half-lives. And the estimation of corresponding preformation factors is also related to decay energy as shown in Eq.(15). To have a more scientific predictions, two credible mass models FRDM[61] and WS4+[62] are used to predict α decay energy. The detailed results are listed in Table 5. In this table, the first three columns denote α transition, experimental α decay energy and half-lives, respectively. The fourth and fifth columns represent the predicted α decay energy derived from the FRDM and WS4+ mass models, respectively. The sixth and seventh columns represent the predicted preformation factors with the help of Eq.(15) based on corresponding decay energy. The last two columns represent the predicted α decay half-lives in terms of $ Q_{\alpha} $ from the FRDM and WS4+ mass models denoted as $ {\rm{log}}_{10}{{\rm{T}}}_{1/2}^{FRDM} $ and $ {\rm{log}}_{10}{{\rm{T}}}_{1/2}^{WS4+} $, respectively. From this table, it can be seen that decay energy plays an important role in the prediction of the related half-lives. When the differences of the decay energy are kept within a small range, the predicted results of the related half-lives are almost consistent. The uncertainty of 1 MeV in the decay energy can lead to the uncertainty of the related half-lives with a factor of 10 or $ 10^{2} $. These corresponding predictions can provide valuable references for the synthesis of superheavy nuclei and/or new elements.

      $ {\alpha} $ transition $ Q_{\alpha}^{\rm exp} $ $ {\rm{log}}_{10}{{\rm{T}}}_{1/2}^{{\rm{\,exp}}} $ $ Q_{\alpha}^{\rm FRDM} $ $ Q_{\alpha}^{\rm WS4+} $ $ \rm P_{\alpha}^{FRDM} $ $ \rm P_{\alpha}^{WS4+} $ $ {\rm{log}}_{10}{{\rm{T}}}_{1/2}^{FRDM} $ $ {\rm{log}}_{10}{{\rm{T}}}_{1/2}^{\rm WS4+} $
      Nuclei with Z=118
      $ ^{288} $Og$ \to ^{284} $Lv 12.88 12.59 0.0883 0.0991 $ -5.05 $ $ -4.48 $
      $ ^{290} $Og$ \to ^{286} $Lv 12.81 12.57 0.0847 0.0868 $ -4.93 $ $ -4.46 $
      $ ^{292} $Og$ \to ^{288} $Lv 12.37 12.21 0.0845 0.0860 $ -4.07 $ $ -3.74 $
      $ ^{294} $Og$ \to ^{290} $Lv 11.87 −3.15 12.28 12.17 0.0813 0.0823 $ -3.89 $ $ -3.67 $
      $ ^{296} $Og$ \to ^{292} $Lv 12.29 11.73 0.0773 0.0825 $ -3.89 $ $ -2.71 $
      $ ^{298} $Og$ \to ^{294} $Lv 12.51 12.16 0.0719 0.0748 $ -4.39 $ $ -3.63 $
      $ ^{300} $Og$ \to ^{296} $Lv 12.72 11.93 0.0670 0.0731 $ -4.83 $ $ -3.21 $
      Nuclei with Z=120
      $ ^{296} $120$ \to ^{292} $Og 13.69 13.32 0.0700 0.0726 $ -6.05 $ $ -5.34 $
      $ ^{298} $120$ \to ^{294} $Og 13.36 12.98 0.0689 0.0716 $ -5.46 $ $ -4.70 $
      $ ^{300} $120$ \to ^{296} $Og 13.40 13.29 0.0654 0.0661 $ -5.56 $ $ -5.36 $
      $ ^{302} $120$ \to ^{298} $Og 13.72 12.87 0.0604 0.0658 $ -6.15 $ $ -4.57 $
      $ ^{304} $120$ \to ^{300} $Og 13.83 12.74 0.0569 0.0636 $ -6.33 $ $ -4.33 $
      $ ^{306} $120$ \to ^{302} $Og 14.27 13.76 0.0521 0.0546 $ -7.11 $ $ -6.26 $
      $ ^{308} $120$ \to ^{304} $Og 14.31 12.94 0.0495 0.0566 $ -7.20 $ $ -4.76 $

      Table 5.  The predicted α-particle preformation factors by Eq.(15) and half-lives for unknown even-even nuclei with Z = 118 and 120. The predicted α decay energies are derived from the FRDM[61] and WS4+[62] mass models. The experimental α decay energy and half-lives are taken from the latest evaluated atomic mass table AME2020[6365].

    IV.   SUMMARY
    • Based on the Two-Potential Approach, we systematically analyze the α-particle preformation factors of heavy and superheavy nuclei from $ ^{220}{\rm{Th}} $ to $ ^{294}{\rm{Og}} $. It is found that the α-particle preformation factors show considerable regularity i.e. odd-even staggering effect and unpaired nucleons will inhibit the α-particle preformation. In addition, the preformation factor is strongly correlated with multiple physical factors. Both decay energy and mass number of parent nucleus all exhibit the certain connection with the extracted experimental α-particle preformation factors. On this basis, a simple analytical expression has been proposed to estimate α-particle preformation factors. The excellent performances suggest that this analytical expression can not only evaluate the α-particle preformation factors well, but also help the accurate calculation of half-lives. With the help of this formula, we estimate the α-particle preformation factors and further predict the corresponding α decay half-lives for unknown even-even nuclei with Z = 118 and 120. It is desired to provide valuable references to further experiments in synthesizing superheavy nuclei.

      Figure 8.  (color online) Same as Fig. 7, but for odd-A nuclei.

    Acknowledgements
    • We would like to thank Yang-Yang Xu, Xiao-Yan Zhu and Prof. Yan-Zhao Wang for useful discussion.

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