Systematic study of α decay half-lives within the Generalized Liquid Drop Model with various versions of proximity energies

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Jun-Gang Deng and Hong-Fei Zhang. Systematic study of α decay half-lives within the Generalized Liquid Drop Model with various versions of proximity energies[J]. Chinese Physics C.
Jun-Gang Deng and Hong-Fei Zhang. Systematic study of α decay half-lives within the Generalized Liquid Drop Model with various versions of proximity energies[J]. Chinese Physics C. shu
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Systematic study of α decay half-lives within the Generalized Liquid Drop Model with various versions of proximity energies

    Corresponding author: Hong-Fei Zhang, zhanghongfei@lzu.edu.cn
  • 1. School of Nuclear Science and Technology, Lanzhou University, 730000 Lanzhou, People's Republic of China
  • 2. Joint Department for Nuclear Physics, Institute of Modern Physics, CAS and Lanzhou University, Lanzhou 730000, People's Republic of China
  • 3. Engineering Research Center for Neutron Application, Ministry of Education, Lanzhou University, 730000 Lanzhou, People's Republic of China

Abstract: It is universally acknowledged that the Generalized Liquid Drop Model (GLDM) has two advantages that over other α decay theoretical models: introducing the quasimolecular shape mechanism and proximity energy. In the past few decades, the original proximity energy has been improved by numerous works. In the present work, the different improvements of proximity energy are examined when they are applied to GLDM for enhancing the calculation accuracy and prediction ability of α decay half-lives for known and unsynthesized superheavy nuclei. The calculations of α half-lives have systematic improvements in reproducing experimental data after choosing a more suitable proximity energy applied to GLDM. Encouraged by this, the α decay half-lives of even-even superheavy nuclei with Z=112-122 are predicted by the GLDM with a more suitable proximity energy. The predictions are consistent with calculations by the improved Royer formula and the universal decay law. In addition, the features of predicted α decay half-lives imply that the next double magic nucleus after 208Pb is 298Fl.

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    I.   INTRODUCTION
    • α decay, one of the most important decay modes of heavy and superheavy nuclei, attracts constant attentions [1-4], because it can be a probe to reveal some important nuclear structure information, such as the properties of ground state, nuclear deformation, nuclear shape coexistence, energy levels and so on [5-13], and can be an important tool to identify the new synthesized superheavy nuclei [14-21].

      Constructing a reasonable nuclear potential between α-particle and daughter nucleus is the most key issue in many α decay theoretical models, because the α decay half-life is mainly decided by the barrier penetrating probability [22]. There are many α decay models choosing different nuclear potentials between α-particle and daughter nucleus, such as Coulomb and proximity potential model with proximity potential [23-26], two-potential approach with a cosh parametrized form nuclear potential [27-29], density-dependent cluster model with double-folding integral of the renormalized M3Y nucleon-nucleon potential [30-34], preformed cluster model with SLy4 Skyrme-like effective interaction [9,35,36], and so on.

      Unlike other models, the Generalized Liquid Drop Model (GLDM) has two major advantages: introducing the quasimolecular shape mechanism [37], which can describe complex deformation process from parent nucleus continuous transition to the appearance of a deep and narrow neck, and finally into two tangential fragments, and adding the proximity energy including an accurate radius and the mass asymmetry. When a neck or a gap appears in one-body shapes or between separated fragments, proximity energy plays a key role in taking into account the effects of the nuclear forces between the close surface, balancing the repulsion of the Coulomb barrier, and reasonably constructing the barrier heights and positions of the nucleus in complex deformation process [37-40]. Therefore GLDM can successfully deal with proton radioactivity [41], cluster radioactivity [42], fusion [43], fission [44] and α decay process [22,37,40,45-48].

      The proximity energy was first proposed by Blocki et al. [49] for describing the interaction energy associated with the crevice or neck in the nuclear configuration that would be expected immediately after contact of two nuclei in heavy-ion reactions. Therefore it was also introduced into GLDM by Royer [37] to take account the effects of the nucleon–nucleon force inside the neck or the gap between the nascent or separated α-particle and daughter nucleus. The proximity energies are also used to study the fusion reaction cross sections and nuclear decay (including proton radioactivity, α decay and cluster radioactivity), because these decay modes proceed in the opposite direction of fusion between particle or cluster and daughter nucleus [50,51]. The proximity energy is based on the proximity force theorem [49,52], which is described as the product of a factor depending on the mean curvature of the interaction surface and a universal function (depending on the separation distance) and is independent of the masses of colliding nuclei [53,54]. In the past few decades, numerous works are devoted to improving the original proximity energy (Prox. 77) [49], by adopting either a better form of the surface energy coefficients [55-62], or introducing an improved universal function or another nuclear radius parameterization [52,53,63-74].

      In order to obtain more precise calculations of α decay half-lives for known nuclei, and enhance the prediction ability of α decay half-lives and island of stability for superheavy nuclei, it is very important and interesting to develop the GLDM by adopting a more suitable proximity energy for constructing the reasonable potential barrier based on available α decay experimental data. This is the purpose of present work. We also notice that there have been some works using GLDM with proximity energy Prox. 81 [52] or proximity energy Denisov [73] instead of the original proximity energy formalism in GLDM to study the α decay [75-78]. The calculations can reproduce the experimental data better than ones calculated by original GLDM. However, in the present work, we find that proximity energy Prox. 81 [52] is not the most suitable substitute for original one and the proximity energy Denisov [73] has different nuclear radius formalism from GLDM, which is not self-consistent for calculation. In this work, for self-consistency, we choose 16 various versions of proximity energies that have the same radii forms as GLDM, and systematically study the these applicability that applied to GLDM. The calculations indicate that GLDM with the proximity energy Prox. 77-Set 13 gives the lowest root-mean-square (RMS) deviation in reproducing experimental α half-lives. Using GLDM with proximity energy Prox. 77-Set 13 we predict α decay half-lives of superheavy even-even nuclei with $ Z = 112–122 $. The predictions are compared with the ones calculated by the improved Royer formula [79] and universal decay law (UDL) [80].

      This article is organized as follows. In Sec. 2, theoretical frameworks of the α decay half-life and the Generalized Liquid Drop Model are briefly presented. The detailed calculations and discussion are given in Sec. 3. Sec. 4 is a brief summary.

    II.   THEORETICAL FRAMEWORK

      A.   The Generalized Liquid Drop Model

    • The α decay half-life can be calculated with decay constant $ \lambda $ as

      $ T_{1/2} = \frac{\ln{2}}{\lambda} . $

      (1)

      In the framework of the Generalized Liquid Drop Model (GLDM) [22,37,40,45-48], the α decay constant $ \lambda $ can be obtained by the product of α-particle preformation factor $ P_{\alpha} $, the assault frequency $ \nu $ and the barrier penetrating probability $ P $:

      $ \lambda = P_{\alpha}{\nu}P . $

      (2)

      The α-particle preformation factor $ P_{\alpha} $ can be estimated by the analytic formula put forwarded in our previous work [81]. It is expressed as

      $ \log_{10}P_{\alpha} = a+bA^{1/6}\sqrt{Z}+c\frac{Z}{\sqrt{Q_{\alpha}}}-d\chi^{\prime}-e\rho^{\prime}+f\sqrt{l(l+1)} , $

      (3)

      where $ \chi^{\prime} \!=\! Z_1Z_2\sqrt{\frac{A_1A_2}{(A_1+A_2)Q_{\alpha}}} $ and $ \rho^{\prime} \!\!=\!\! \sqrt{\frac{A_1A_2}{A_1\!+\!A_2}Z_1Z_2(A_1^{1/3}\!+\!A_2^{1/3})} $. $ A $, $ Z $ represent the mass number and proton number of α decay parent nucleus. $ l $ is the angular momentum taken away by α-particle. The values of adjustable parameters $ a $, $ b $, $ c $, $ d $, $ e $, and $ f $ are fitted by extracted α-particle preformation factor from the ratios of calculated α decay half-lives with $ P_{\alpha} = 1 $ to experimental data, and listed in Table 3.

      Nuclei Region $a$ $b$ $c$ $d$ $e$ $f$
      Even-even Nuclei $N\le126$ −9.4985 −8.9005 4.0450 1.0432 −2.9731
      $N>126$ −2.1047 0.1230 4.2051 1.0681 0.0533
      Odd-A Nuclei $N\le126$ −24.5445 −13.2233 9.9493 2.5690 −4.5754 −0.0350
      $N>126$ 1.3626 −6.2523 −0.0252 −0.0155 −1.9616 −0.0937
      Doubly odd Nuclei $N\le126$ −2.7484 −4.2572 2.2748 0.5947 −1.3917 −0.0901
      $N>126$ −37.5320 −20.0571 23.5391 6.0638 −6.9409 −0.2030

      Table 3.  The parameters of Eq. (3) for estimating the α-particle preformation factor.

      The assault frequency $ \nu $ can be obtained by

      $ \nu = \frac{1}{2R_0}\sqrt{\frac{2E_{\alpha}}{M}} , $

      (4)

      with $ E_{\alpha} $ and $ M_{\alpha} $ being the kinetic energy and mass of α-particle. $ R_0 $ is the radius of α decay parent nucleus obtained by

      $ R_{i} = 1.28A_{i}^{1/3}-0.76+0.8A_{i}^{-1/3}(i = 0,1,2) . $

      (5)

      The barrier penetrating probability $ P $ can be obtained by Wentzel-Kramers-Brillouin (WKB) approximation as

      $ P = \exp[-\frac{2}{\hbar}{\int_{r_{{\rm{in}}}}^{r_{{\rm{out}}}} \sqrt{2B(r)(E_{r}-E({\rm{sphere}}))} dr}] , $

      (6)

      where $ r $ represents the center of mass distance between preformed α-particle and daughter nucleus. The classical turning points $ r_{{\rm{in}}} $ and $ r_{{\rm{out}}} $ can be obtained by: $ r_{{\rm{in}}} = R_1+R_2 $ and $ E(r_{{\rm{out}}}) = Q_{\alpha} $. $ B(r) = \mu $ represents the reduced mass between preformed α-particle and daughter nucleus.

      The total interaction potential $ E $ in GLDM is composed of five parts [37]: volume energy $ E_V $, surface energy $ E_S $, Coulomb energy $ E_C $, proximity energy $ E_{{\rm{Prox}}} $, and centrifugal potential $ E_l $.

      $ E = E_V+E_S+E_C+E_{{\rm{Prox}}}+E_l . $

      (7)

      For one-body shapes, the volume, surface and Coulomb energies are expressed as

      $ E_V = -15.494(1-1.8I^2)A , $

      (8)

      $ E_S = 17.9439(1-2.6I^2)A^{2/3}(S/4{\pi}R_0^2) , $

      (9)

      $ E_C = 0.6e^2(Z^2/R_0)\times0.5\int(V(\theta)/V_0)(R(\theta)/R_0)^3\sin\theta{d\theta} , $

      (10)

      where $ S $ denotes the surface of one-body deformed nucleus. $ I $ is the relative neutron excess. $ V(\theta) $ represents the electrostatic potential at the surface and $ V_0 $ the surface potential of the sphere.

      For two separated fragments: the volume, surface and Coulomb energies are defined as

      $ E_V = -15.494[(1-1.8I_1^2)A_1+(1-1.8I_2^2)A_2] , $

      (11)

      $ E_S = 17.9439[(1-2.6I_1^2)A_1^{2/3}+(1-2.6I_2^2)A_2^{2/3}] , $

      (12)

      $ E_C = 0.6e^2Z_1^2/R_1+0.6e^2Z_2^2/R_2+e^2Z_1Z_2/r , $

      (13)

      with $ A_i $, $ Z_i $, $ R_i $ and $ I_i $ being the mass numbers, proton numbers, radii, and the relative neutron excesses of α-particle and daughter nucleus, respectively.

      The centrifugal barrier $ E_l(r) $ can be calculated by

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

      (14)

      On the basis of the conservation laws of angular momentum and parity [82], the minimum angular momentum $ l_{\min} $ carried by α-particle can be obtained by

      $ l_{\min} = \left\{\begin{array}{llll} {\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\ne{\pi}_d,\\ {\Delta}_j,&{\rm{for\;odd}}\;{\Delta}_j\;{\rm{and}}\;{\pi}_p\ne{\pi}_d,\\ {\Delta}_j+1,&{\rm{for\;odd}}\;{\Delta}_j\;{\rm{and}}\;{\pi}_p = {\pi}_d, \end{array}\right. $

      (15)

      where $ {\Delta}_j = |j_p-j_d| $, $ j_p $, $ \pi_p $, $ j_d $, $ \pi_d $ are the spin and parity values of parent and daughter nuclei, respectively.

    • B.   The proximity energy

    • The surface energy comes from the effects of the surface tension forces in a half-space. When a neck or a gap appears in one-body shapes or between separated fragments an additional term called proximity energy must be added to take into account the effects of the nuclear forces between the close surface [37,40]. Proximity energy is described as the product of a factor depending on the mean curvature of interaction surface and an universal function depending on the separation distance [53,54].

      In the present work, for self-consistency, we choose 16 various versions of proximity energies that have the same radii forms as GLDM, including: proximity energy formalisms Prox. 77 [49] and its 12 modified forms on the basis of improving the surface energy coefficients [55-62], Bass 80 [69], Prox. 81 [52], and Guo 2013 [74]. The proximity energy Prox. 77 [49] and its 12 modified forms are expressed as,

      $ E_{\rm{Prox}}(r) = 4\pi{\gamma}b\bar{R}\phi(\xi) , $

      (16)

      where the mean curvature radius $ \bar{R} $ can be obtained by

      $ \bar{R} = \frac{C_1C_2}{C_1+C_2} , $

      (17)

      here $ C_1 $ and $ C_2 $ denote the matter radii of α-particle and daughter nucleus, respectively. They can be given by

      $ C_i = R_i[1-(\frac{b}{R_i})^2] (i = 1,2) , $

      (18)

      with the effective sharp radius $ R_i $ obtaining by Eq. (5).

      The surface energy coefficient $ \gamma $ can be obtained by

      $ \gamma = \gamma_0(1-k_sA_s^2), $

      (19)

      where, $ A_s = \dfrac{N-Z}{N+Z} $ represents neutron-proton excess. The surface energy constant $ \gamma_0 $ and the surface asymmetry constant $ k_s $ are set in various improvements as

      Set 1: $ \gamma_0 = 0.9517\ ({\rm{Mev/fm}}^2) $, $ k_s = 1.7826 $ [49]

      Set 2: $ \gamma_0 = 1.01734\ ({\rm{Mev/fm}}^2) $, $ k_s = 1.79 $ [55]

      Set 3: $ \gamma_0 = 1.460734\ ({\rm{Mev/fm}}^2) $, $ k_s = 4.0 $ [56]

      Set 4: $ \gamma_0 = 1.2402\ ({\rm{Mev/fm}}^2) $, $ k_s = 3.0 $ [57]

      Set 5: $ \gamma_0 = 1.1756\ ({\rm{Mev/fm}}^2) $, $ k_s = 2.2 $ [58]

      Set 6: $ \gamma_0 = 1.27326\ ({\rm{Mev/fm}}^2) $, $ k_s = 2.5 $ [58]

      Set 7: $ \gamma_0 = 1.2502\ ({\rm{Mev/fm}}^2) $, $ k_s = 2.4 $ [58]

      Set 8: $ \gamma_0 = 0.9517\ ({\rm{Mev/fm}}^2) $, $ k_s = 2.6 $ [59]

      Set 9: $ \gamma_0 = 1.2496\ ({\rm{Mev/fm}}^2) $, $ k_s = 2.3 $ [60]

      Set 10: $ \gamma_0 = 1.25284\ ({\rm{Mev/fm}}^2) $, $ k_s = 2.345 $ [61]

      Set 11: $ \gamma_0 = 1.08948\ ({\rm{Mev/fm}}^2) $, $ k_s = 1.9830 $ [62]

      Set 12: $ \gamma_0 = 0.9180\ ({\rm{Mev/fm}}^2) $, $ k_s = 0.7546 $ [62]

      Set 13: $ \gamma_0 = 0.911445\ ({\rm{Mev/fm}}^2) $, $ k_s = 2.2938 $ [62]

      The universal function $ \phi(\xi) $ is expressed as

      $ \phi(\xi) = \left\{\begin{array}{ll} -\dfrac{1}{2}(\xi-\xi_0)^2-0.0852(\xi-\xi_0)^3,&0<\xi\leqslant{1.2511},\\ -3.347\exp\left(\dfrac{-\xi}{0.75}\right),&\xi\geqslant{1.2511}, \end{array}\right. $

      (20)

      here $ \xi_0 = 2.54 $. $ \xi = \dfrac{r-C_1-C_2}{b} $ is the distance between the near surface of the α-particle and daughter nucleus with the width parameter $ b $ taken as unity.

    III.   RESULTS AND DISCUSSION
    • The aim of the present work is to develop the Generalized Liquid Drop Model (GLDM) for enhancing calculation accuracy and prediction ability of α decay half-lives for known and unsynthesized superheavy nuclei by chosing a more suitable proximity energy in constructing a reasonable potential barrier.

      If the improved versions of original proximity energy can be applied to the GLDM, three conditions need to be met: first, both the radii formulas of the proximity energy and GLDM are the same; second, the total GLDM energy, including the proximity energy, between α-particle and daughter nucleus is reasonable; finally, the calculated α decay half-lives by GLDM with the best selected proximity energy should give the lowest RMS deviation in reproducing experimental α half-lives. Therefore, we comparatively study the abilities of 16 various versions of proximity energies when they are applied to GLDM for describing the α decay half-lives, in which proximity energies have the same radii forms as GLDM. These proximity energies include Prox. 77 [49] and its 12 modified forms on the basis of improving the surface energy coefficients [55-62], Bass 80 [69], Prox. 81 [52], and Guo 2013 [74].

      In these various versions of proximity energies, we find that the proximity energy Guo 2013 [74] is not suitable for applying to the GLDM because for some α decay nuclei such as 148Gd, the total nuclear potential distribution shows short-term decline and even less than zero after the two tangent fragments separating, which is resulted from this proximity energy determining too strong attractive interaction potential.

      For choosing the most suitable one in the remainder of proximity energies that can be applied to the GLDM, we calculate the RMS deviation between calculated α decay half-lives by GLDM with various proximity energies where α-particle preformation factor is assumed as a constant $ P_{\alpha} = 1 $, and experimental data for all 535 nuclei, including 159 even-even nuclei, 295 odd-$ A $ nuclei and 81 doubly odd nuclei using

      $ \sigma = \sqrt{\frac{1}{n}\sum ({\log_{10}T_{1/2}^{{\rm{cal}}}-\log_{10}T_{1/2}^{{\rm{exp}}}})^2} . $

      (21)

      The results are listed in Table 1. In this table, we can find that the values of all $ \sigma $ are greater than 1, indicating that there are average deviations of more than one order of magnitude between calculations and experimental data because α-particle preformation factors are assumed as $ P_{\alpha} = 1 $, which are overestimated. In addition, we can find that the values of $ \sigma $ are different caused by GLDM with various proximity energies. The minimum $ \sigma = 1.459 $ and maximal $ \sigma = 1.644 $ are caused by GLDM with proximity energies Prox. 77-Set 13 and with Bass 80, respectively.

      GLDM with proximity energy $\sigma$ GLDM with proximity energy $\sigma$
      Prox. 77-Set 1 1.472 Prox. 77-Set 2 1.488
      Prox. 77-Set 3 1.579 Prox. 77-Set 4 1.534
      Prox. 77-Set 5 1.525 Prox. 77-Set 6 1.546
      Prox. 77-Set 7 1.541 Prox. 77-Set 8 1.467
      Prox. 77-Set 9 1.542 Prox. 77-Set 10 1.543
      Prox. 77-Set 11 1.505 Prox. 77-Set 12 1.471
      Prox. 77-Set 13 1.459 Bass 80 1.644
      Prox. 81 1.500 Original one 1.605

      Table 1.  The RMS deviations between calculated α decay half-lives by GLDM with different versions of proximity energies and experimental data.

      The experimental data and calculations of α decay half-lives for 148Gd as am example, are listed in Table 2. From this table, one can find that GLDM with various proximity energies calculate different α decay half-lives. And all calculations are an order of magnitude smaller than experimental data indicating that α-particle preformation factor is in the order of $ 10^{-1} $. In addition, one can see that most of calculations by GLDM with improved proximity energies are better than the ones by original GLDM in aspect of reproducing experimental data. However, GLDM with proximity energy Bass 80 give a worse calculation than one calculated by original GLDM showing that it is not appropriate for applying to GLDM. The calculations by GLDM with proximity energy Prox. 77-Set 13 is the closest one to reproduce experimental data. Why are calculations by GLDM with various proximity energies different from each other? Based on our comparative analysis we can explore what particular feature of a given potential impacts these differences between various theoretical calculations, as well as differences between theory and experiment. From Section 2.1, we can find that α decay half-life can be obtained by the α-particle preformation factor, which is assumed as a constant $ P_{\alpha} = 1 $ in comparing proximity energies, assault frequency $ \nu $, which is dependent on α decay energy $ Q_{\alpha} $, and barrier penetration probability $ P $, which is related to the total GLDM energy. However, for a α decay nucleus, $ Q_{\alpha} $ and the assault frequency $ \nu $ are fixed. Therefore, these are different total GLDM energy including various versions of the proximity energies cause the differences between calculated α decay half-lives. In order to verity this conclusion, taking 148Gd for instance, we plot its 16 versions of total GLDM energies distributions, including original proximity energy and its 15 improved versions, in Fig. 1. In this figure, we can see that the proximity energies only work in a short region of 7.8 fm to 12 fm. After α-particle and daughter nuclei are separated, the proximity energies are equal to zero. Therefore, as can be seen from Section 2.1 that the classic turning points $ r_{{\rm{in}}} = R_1+R_2 $ and $ E(r_{{\rm{out}}}) = Q_{\alpha} $ in GLDM with various proximity energies are the same. In addition, we can see that the proximity energies can lower the height of the potential barriers, and their attractive effect balance the Coulomb repulsion between the two fragments. Further, the peak of potential barriers are shifted toward more external position. So, the different proximity energies cause changes in the shape and height of the total GLDM energies distributions.

      Figure 1.  (Color online) The distributions of total GLDM energies including various versions of proximity energies for 148Gd.

      Method α decay half-lives for 148Gd (s)
      Experimental data $2.24\times10^9$
      GLDM with original proximity energy $4.83\times10^8$
      GLDM with Prox. 77- Set 1 $6.93\times10^8$
      GLDM with Prox. 77- Set 2 $6.49\times10^8$
      GLDM with Prox. 77- Set 3 $4.31\times10^8$
      GLDM with Prox. 77- Set 4 $5.30\times10^8$
      GLDM with Prox. 77- Set 5 $5.56\times10^8$
      GLDM with Prox. 77- Set 6 $5.05\times10^8$
      GLDM with Prox. 77- Set 7 $5.16\times10^8$
      GLDM with Prox. 77- Set 8 $7.03\times10^8$
      GLDM with Prox. 77- Set 9 $5.15\times10^8$
      GLDM with Prox. 77- Set 10 $5.14\times10^8$
      GLDM with Prox. 77- Set 11 $6.05\times10^8$
      GLDM with Prox. 77- Set 12 $7.04\times10^8$
      GLDM with Prox. 77- Set 13 $7.28\times10^8$
      GLDM with Bass 80 $3.90\times10^8$
      GLDM with Prox. 81 $6.33\times10^8$

      Table 2.  The α decay half-lives of calculations by GLDM with different versions of proximity energies and experimental data for 148Gd.

      The same values of $ r_{{\rm{in}}} $ and $ r_{{\rm{out}}} $ as well as the highest height of $ E(r) $ in the GLDM with proximity energy Prox. 77-Set 13 causes the minimum barrier penetration probability $ P $. Thus the calculated α decay half-life is the maximum one in Table 2. Similarly, the lowest height of $ E(r) $ in GLDM with proximity energy Bass 80 caused the minimum calculation. It is shown that the proximity energy is very important in GLDM, because it affects the shape and height of the total potential barrier, which determines the possibility of barrier penetration and in turn leads to the theoretical calculation of α decay half-life. Therefore, it is interesting and important to find the most suitable proximity energy for developing GLDM to obtain precise calculations and enhance the prediction ability of α decay half-lives. From Tables 1 and 2, it is shown that the proximity energy Prox. 77-Set 13 is the most suitable one in applying to GLDM for describing the α decay half-lives. The $ \sigma $ values indicate that compared with the original GLDM, calculated α decay half-lives using the GLDM with proximity energy Prox. 77-Set 13 are improved by $ \frac{1.605-1.459}{1.605}$ = 9.1%. Although the relative value is not large, this is a significant improvement on the Generalized Liquid Drop Model because proximity energy can affect the total interaction potential in a short region.

      In our previous work [81], we proposed an analytic formula for estimating the α-particle preformation factor, i.e. Eq. (3). In this work, because we choose a more suitable proximity energy in GLDM, the parameters of Eq. (3) should be redetermined. First, we extract the α-particle preformation factor by the ratios of calculated α decay half-life within the GLDM with proximity energy Prox. 77-Set 13 and $ P_{\alpha} = 1 $ to experimental data. Then we use the extracted α-particle preformation factor and Eq. (3) to obtain the parameters, which are listed in Table 3.

      The calculated α decay half-lives and experimental data are listed in Tables 46 for even-even nuclei, odd-$ A $ nuclei and doubly odd nuclei, respectively. In each part of these 3 tables, the first four columns represent α decay parent nucleus, daughter nucleus, experimental α decay energy, and the minimum angular momentum taken away by α-particle while the spin and parity values for α decay parent and daughter nuclei are taken from the latest evaluated nuclear properties table NUBASE2016 [83], respectively. The fifth one is experimental α decay half-life. The sixth one represents the calculated α decay half-life within original GLDM with $ P_{\alpha} = 1 $. The seventh one is calculated α decay half-life by GLDM with proximity energy Prox. 77-Set 13 and $ P_{\alpha} = 1 $. The eighth one represents obtained α-particle preformation factor by Eq. (3). The last one is calculated α decay half-life within GLDM with proximity energy Prox. 77-Set 13 and estimated α-particle preformation factor by Eq. (3). From these 3 tables, it can be found that compared with $ \lg T^{{\rm{cal1}}}_{1/2} $, $ \lg T^{{\rm{cal2}}}_{1/2} $ has a significant improvement in conformity with experimental data. However, both $ \lg T^{{\rm{cal1}}}_{1/2} $ and $ \lg T^{{\rm{cal2}}}_{1/2} $ are smaller than experimental data by more than an order of magnitude on the whole. It is due to that α preformation factor is assumed as $ P_{\alpha} = 1 $, which is overestimated. Therefore, α preformation factor $ P_{\alpha} $ should be introduced in theoretical model. After considering α-particle preformation factor calculated by Eq. (3), $ \lg T^{{\rm{cal3}}}_{1/2} $ can well reproduce experimental data.

      α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{\exp}_{1/2}$ $\lg T^{{\rm{cal1}}}_{1/2}$ $\lg T^{{\rm{cal2}}}_{1/2}$ ${P_{\alpha}}$ $\lg T^{{\rm{cal3}}}_{1/2}$ α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{\exp}_{1/2}$ $\lg T^{{\rm{cal1}}}_{1/2}$ $\lg T^{{\rm{cal2}}}_{1/2}$ ${P_{\alpha}}$ $\lg T^{{\rm{cal3}}}_{1/2}$
      148Gd 144Sm 3.27 0 9.35 8.68 8.86 0.2841 9.41 150Gd 146Sm 2.81 0 13.75 13.17 13.17 0.3578 13.62
      150Dy 146Gd 4.35 0 3.07 2.17 2.21 0.1775 2.96 152Dy 148Gd 3.73 0 6.93 6.26 6.32 0.2117 6.99
      154Dy 150Gd 2.95 0 13.98 13.17 13.39 0.3075 13.9 152Er 148Dy 4.94 0 1.06 0.12 0.26 0.1635 1.05
      154Er 150Dy 4.28 0 4.68 3.72 3.95 0.1845 4.69 156Er 152Dy 3.48 0 10.24 9.49 9.61 0.2389 10.23
      154Yb 150Er 5.47 0 −0.35 −1.39 −1.15 0.1601 −0.36 156Yb 152Er 4.81 0 2.41 1.79 1.95 0.1725 2.71
      158Yb 154Er 4.17 0 6.63 5.52 5.55 0.1917 6.27 156Hf 152Yb 6.03 0 −1.63 −2.73 −2.55 0.1609 −1.75
      158Hf 154Yb 5.41 0 0.35 −0.18 −0.15 0.165 0.64 160Hf 156Yb 4.9 0 3.28 2.28 2.38 0.1672 3.15
      162Hf 158Yb 4.42 0 5.69 5.07 5.2 0.1719 5.96 158W 154Hf 6.62 0 −2.9 −4.04 −3.9 0.1645 −3.12
      160W 156Hf 6.07 0 −0.99 −2.05 −1.91 0.1616 −1.12 162W 158Hf 5.68 0 0.42 −0.46 −0.31 0.1549 0.5
      164W 160Hf 5.28 0 2.22 1.31 1.52 0.1501 2.34 166W 162Hf 4.86 0 4.74 3.49 3.58 0.1475 4.41
      168W 164Hf 4.5 0 6.2 5.57 5.71 0.1439 6.55 180W 176Hf 2.52 0 25.75 24.54 24.72 0.1572 25.52
      162Os 158W 6.77 0 −2.68 −3.76 −3.58 0.1623 −2.79 166Os 162W 6.14 0 −0.53 −1.55 −1.43 0.1412 −0.58
      168Os 164W 5.82 0 0.68 −0.23 −0.08 0.1327 0.79 170Os 166W 5.54 0 1.89 1.01 1.21 0.124 2.12
      172Os 168W 5.22 0 3.23 2.46 2.54 0.1171 3.48 174Os 170W 4.87 0 5.25 4.35 4.53 0.1122 5.48
      186Os 182W 2.82 0 22.8 21.72 21.89 0.1048 22.87 166Pt 162Os 7.29 0 −3.52 −4.73 −4.58 0.1518 −3.76
      168Pt 164Os 6.99 0 −2.69 −3.78 −3.65 0.14 −2.8 172Pt 168Os 6.46 0 −1 −2.02 −1.9 0.1185 −0.98
      174Pt 170Os 6.18 0 0.06 −0.95 −0.76 0.1096 0.2 176Pt 172Os 5.89 0 1.2 0.28 0.48 0.1019 1.47
      178Pt 174Os 5.57 0 2.43 1.63 1.71 0.0953 2.73 180Pt 176Os 5.24 0 4.27 3.28 3.47 0.0898 4.51
      182Pt 178Os 4.95 0 5.62 4.83 5.02 0.0842 6.09 184Pt 180Os 4.6 0 7.77 6.94 7.06 0.0803 8.15
      190Pt 186Os 3.27 0 19.31 17.86 17.98 0.0797 19.08 172Hg 168Pt 7.53 0 −3.64 −4.79 −4.65 0.1315 −3.77
      174Hg 170Pt 7.23 0 −2.7 −3.9 −3.7 0.1206 −2.78 176Hg 172Pt 6.9 0 −1.65 −2.79 −2.59 0.1114 −1.63
      178Hg 174Pt 6.58 0 −0.53 −1.69 −1.59 0.1029 −0.6 180Hg 176Pt 6.26 0 0.73 −0.47 −0.28 0.0952 0.74
      182Hg 178Pt 6 0 1.89 0.62 0.81 0.0876 1.87 184Hg 180Pt 5.66 0 3.44 2.12 2.23 0.0816 3.32
      186Hg 182Pt 5.2 0 5.7 4.37 4.55 0.0778 5.66 178Pb 174Hg 7.79 0 −3.64 −4.94 −4.8 0.114 −3.86
      180Pb 176Hg 7.42 0 −2.39 −3.81 −3.62 0.1049 −2.64 184Pb 180Hg 6.77 0 −0.21 −1.64 −1.53 0.0885 −0.48
      186Pb 182Hg 6.47 0 1.07 −0.54 −0.36 0.0813 0.73 188Pb 184Hg 6.11 0 2.43 0.94 1.13 0.0752 2.25
      190Pb 186Hg 5.7 0 4.24 2.81 2.93 0.0703 4.08 192Pb 188Hg 5.22 0 6.55 5.26 5.44 0.0664 6.62
      186Po 182Pb 8.5 0 −4.47 −6.36 −6.16 0.086 −5.09 190Po 186Pb 7.69 0 −2.61 −4.08 −3.98 0.0724 −2.84
      194Po 190Pb 6.99 0 −0.41 −1.79 −1.6 0.0608 −0.38 196Po 192Pb 6.66 0 0.75 −0.58 −0.47 0.0557 0.78
      198Po 194Pb 6.31 0 2.27 0.82 1 0.0512 2.29 200Po 196Pb 5.98 0 3.79 2.26 2.44 0.047 3.76
      202Po 198Pb 5.7 0 5.14 3.53 3.64 0.0431 5.01 204Po 200Pb 5.49 0 6.27 4.61 4.77 0.0393 6.18
      206Po 202Pb 5.33 0 7.14 5.44 5.61 0.0357 7.06 208Po 204Pb 5.22 0 7.96 6.06 6.17 0.0323 7.66
      212Po 208Pb 8.95 0 −6.53 −8 −7.81 0.1047 −6.83 214Po 210Pb 7.83 0 −3.79 −4.94 −4.75 0.1419 −3.91
      216Po 212Pb 6.91 0 −0.84 −1.86 −1.69 0.1921 −0.97 218Po 214Pb 6.12 0 2.27 1.28 1.45 0.2616 2.03
      194Rn 190Po 7.86 0 −3.11 −3.9 −3.72 0.0708 −2.57 196Rn 192Po 7.62 0 −2.33 −3.16 −3.08 0.0641 −1.89
      200Rn 196Po 7.04 0 0.07 −1.22 −1.04 0.053 0.23 202Rn 198Po 6.77 0 1.09 −0.27 −0.16 0.0482 1.15
      204Rn 200Po 6.55 0 2.01 0.61 0.78 0.0438 2.14 206Rn 202Po 6.38 0 2.74 1.27 1.45 0.0396 2.85
      208Rn 204Po 6.26 0 3.37 1.79 1.91 0.0358 3.35 210Rn 206Po 6.16 0 3.95 2.16 2.32 0.0323 3.81
      212Rn 208Po 6.38 0 3.16 1.15 1.31 0.0287 2.86 214Rn 210Po 9.21 0 −6.57 −7.97 −7.77 0.0891 −6.72
      Continued on next page

      Table 4.  Calculations of α decay half-lives for even-even nuclei. Experimental experimental α decay half-lives are taken from the latest evaluated nuclear properties table NUBASE2016 [83]. The α decay energies are taken from the latest evaluated atomic mass table AME2016 [85,86]. The α decay energies and half-lives are in the unit of 'MeV' and 's', respectively.

      α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{\exp}_{1/2}$ $\lg T^{{\rm{cal1}}}_{1/2}$ $\lg T^{{\rm{cal2}}}_{1/2}$ ${P_{\alpha}}$ $\lg T^{{\rm{cal3}}}_{1/2}$ α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{\exp}_{1/2}$ $\lg T^{{\rm{cal1}}}_{1/2}$ $\lg T^{{\rm{cal2}}}_{1/2}$ ${P_{\alpha}}$ $\lg T^{{\rm{cal3}}}_{1/2}$
      149Tb 145Eu 4.08 2 4.95 3.68 3.8 0.0841 4.88 151Tb 147Eu 3.5 2 8.82 7.76 7.79 0.1413 8.64
      151Dy 147Gd 4.18 0 4.28 3.19 3.25 0.0986 4.25 153Dy 149Gd 3.56 0 8.39 7.54 7.67 0.1652 8.45
      151Ho 147Tbm 4.64 0 2.2 1.09 1.18 0.0838 2.25 151Hom 147Tb 4.74 0 1.79 0.6 0.7 0.0774 1.81
      153Hom 149Tb 4.12 0 5.47 4.14 4.27 0.1119 5.23 153Er 149Dy 4.8 0 1.84 0.78 0.91 0.0796 2
      155Er 151Dy 4.12 0 6.15 4.74 4.86 0.119 5.79 153Tm 149Ho 5.25 0 0.21 −0.87 −0.75 0.076 0.37
      153Tmm 149Hom 5.24 0 0.43 −0.85 −0.72 0.0763 0.39 155Tm 151Ho 4.57 0 3.38 2.55 2.69 0.1032 3.67
      155Yb 151Er 5.34 0 0.3 −0.8 −0.65 0.0782 0.46 157Yb 153Er 4.62 0 3.89 2.76 2.81 0.1057 3.78
      155Lu 151Tm 5.8 0 −1.12 −2.3 −2.14 0.0789 −1.04 155Lum 151Tmm 5.73 0 −0.74 −2 −1.84 0.0821 −0.76
      155Lun 151Tm 7.58 8 −2.57 −4.48 −4.29 0.0181 −2.55 157Lum 153Tm 5.13 0 1.89 0.64 0.72 0.0967 1.73
      157Hf 153Yb 5.89 0 −0.91 −2.22 −2.08 0.0827 −0.99 157Tan 153Lu 7.95 8 −2.77 −4.75 −4.46 0.0246 −2.85
      159Ta 155Lum 5.66 0 0.48 −0.83 −0.6 0.1006 0.39 159Tam 155Lu 5.75 0 0.01 −1.19 −0.97 0.0964 0.05
      159W 155Hf 6.45 0 −2 −3.46 −3.23 0.0926 −2.2 161W 157Hf 5.92 0 −0.25 −1.46 −1.31 0.0963 −0.29
      163W 159Hf 5.52 0 1.27 0.24 0.4 0.0961 1.42 159Rem 155Ta 6.97 0 −3.54 −4.8 −4.56 0.1005 −3.57
      161Rem 157Tam 6.43 0 −1.8 −2.97 −2.82 0.1018 −1.82 163Re 159Ta 6.01 0 0.08 −1.41 −1.24 0.0998 −0.24
      163Rem 159Tam 6.07 0 −0.49 −1.63 −1.46 0.0975 −0.45 165Re 161Ta 5.69 0 1.25 −0.13 0.08 0.0949 1.1
      165Rem 161Tam 5.66 0 1.12 0.02 0.22 0.0963 1.24 167Rem 163Ta 5.41 0 2.77 1.17 1.29 0.0898 2.34
      169Re 165Ta 5.01 3 5.18 3.79 3.99 0.0681 5.15 169Rem 165Tam 5.16 3 3.88 3 3.19 0.0632 4.39
      161Os 157W 7.07 0 −3.19 −4.74 −4.57 0.1068 −3.6 163Os 159W 6.69 0 −2.26 −3.5 −3.31 0.101 −2.32
      165Os 161W 6.34 0 −1.1 −2.28 −2.08 0.0951 −1.05 167Os 163W 5.99 0 0.21 −0.93 −0.79 0.0903 0.25
      169Os 165W 5.71 0 1.4 0.21 0.42 0.0837 1.49 165Irm 161Rem 6.89 0 −2.57 −3.82 −3.6 0.1031 −2.62
      167Ir 163Re 6.51 0 −1.17 −2.51 −2.36 0.0971 −1.35 167Irm 163Rem 6.56 0 −1.55 −2.71 −2.56 0.0954 −1.54
      169Ir 165Re 6.14 0 −0.18 −1.13 −0.92 0.0916 0.11 169Irm 165Rem 6.27 0 −0.45 −1.63 −1.42 0.0877 −0.37
      171Ir 167Rem 5.87 0 1.31 −0.05 0.06 0.0842 1.14 171Irm 167Re 6.16 2 0.43 −0.94 −0.84 0.0625 0.37
      173Irm 169Re 5.94 2 1.26 −0.09 0.04 0.0562 1.29 175Ir 171Re 5.43 2 3.02 2.23 2.42 0.0567 3.67
      177Ir 173Re 5.08 0 4.69 3.67 3.79 0.0664 4.97 167Pt 163Os 7.16 0 −3.1 −4.33 −4.17 0.104 −3.18
      171Pt 167Os 6.61 0 −1.3 −2.54 −2.43 0.0851 −1.36 173Pt 169Os 6.36 0 −0.35 −1.64 −1.5 0.0767 −0.38
      175Pt 171Os 6.16 2 0.58 −0.57 −0.38 0.056 0.87 177Pt 173Os 5.64 0 2.27 1.36 1.48 0.0676 2.65
      179Pt 175Os 5.41 2 3.94 2.72 2.81 0.0503 4.11 181Pt 177Os 5.15 0 4.85 3.74 3.93 0.0564 5.17
      183Pt 179Os 4.82 0 6.61 5.57 5.69 0.0535 6.96 171Aum 167Irm 7.16 0 −2.76 −4.04 −3.95 0.0947 −2.93
      173Au 169Ir 6.84 0 −1.53 −2.96 −2.82 0.0863 −1.75 173Aum 169Irm 6.9 0 −1.86 −3.17 −3.02 0.085 −1.95
      175Au 171Ir 6.59 0 −0.64 −2.08 −1.88 0.0773 −0.77 175Aum 171Irm 6.59 0 −0.75 −2.08 −1.88 0.0773 −0.77
      177Au 173Ir 6.3 0 0.56 −1 −0.89 0.0702 0.26 177Aum 173Irm 6.26 0 0.25 −0.85 −0.74 0.0709 0.41
      179Au 175Ir 5.98 1 1.51 0.35 0.44 0.0575 1.68 181Au 177Ir 5.75 2 2.7 1.56 1.75 0.0475 3.07
      183Au 179Ir 5.47 0 3.89 2.59 2.7 0.053 3.98 185Au 181Ir 5.18 0 4.98 4.05 4.23 0.0488 5.54
      171Hg 167Pt 7.67 2 −4.15 −4.9 −4.8 0.0898 −3.75 173Hg 169Pt 7.38 2 −3.1 −4.04 −3.88 0.0804 −2.79
      177Hg 173Pt 6.74 2 −0.82 −1.92 −1.82 0.0657 −0.64 179Hg 175Pt 6.36 0 0.14 −0.86 −0.76 0.0737 0.37
      181Hg 177Pt 6.28 2 1.12 −0.27 −0.09 0.0519 1.2 183Hg 179Pt 6.04 0 1.9 0.42 0.53 0.0566 1.78
      185Hg 181Pt 5.77 0 2.91 1.58 1.77 0.0511 3.06 177Tl 173Au 7.07 0 −1.61 −2.97 −2.88 0.0951 −1.86
      177Tlm 173Aum 7.66 0 −3.44 −4.91 −4.78 0.0849 −3.71 179Tl 175Au 6.71 0 −0.36 −1.75 −1.64 0.0863 −0.57
      179Tlm 175Aum 7.38 0 −2.85 −4.07 −3.93 0.0756 −2.81 181Tlm 177Aum 6.97 2 −0.46 −2.42 −2.23 0.0567 −0.99
      Continued on next page

      Table 5.  Same as Table 4, but for α decay of odd-A nuclei. Elements with upper suffixes 'm', 'n', 'p' or 'x' indicate assignments to excited isomeric states (defined as higher states with half-lives greater than 100 ns). Suffixes 'p' also indicate non-isomeric levels, but used in the AME2016 [85,86].

      α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{\exp}_{1/2}$ $\lg T^{{\rm{cal1}}}_{1/2}$ $\lg T^{{\rm{cal2}}}_{1/2}$ ${P_{\alpha}}$ $\lg T^{{\rm{cal3}}}_{1/2}$ α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{\exp}_{1/2}$ $\lg T^{{\rm{cal1}}}_{1/2}$ $\lg T^{{\rm{cal2}}}_{1/2}$ ${P_{\alpha}}$ $\lg T^{{\rm{cal3}}}_{1/2}$
      148Eu 144Pm 2.69 0 14.98 13.77 13.95 0.0795 15.04 152Ho 148Tb 4.51 0 3.12 1.83 1.95 0.0634 3.14
      152Hom 148Tbm 4.58 0 2.66 1.44 1.56 0.0635 2.76 154Ho 150Tb 4.04 0 6.56 4.65 4.88 0.0582 6.11
      154Tm 150Ho 5.09 0 1.17 −0.16 0.08 0.059 1.3 154Tmm 150Hom 5.18 0 0.75 −0.54 −0.31 0.0592 0.92
      156Tm 152Ho 4.35 0 5.12 3.88 4.03 0.0533 5.3 156Lum 152Tmm 5.72 0 −0.68 −1.96 −1.78 0.0558 −0.53
      158Ta 154Lu 6.13 0 −1.29 −2.71 −2.62 0.0526 −1.34 158Tam 154Lum 6.21 0 −1.42 −3.01 −2.91 0.0528 −1.63
      160Re 156Ta 6.7 2 −2.26 −3.6 −3.46 0.0303 −1.94 162Re 158Ta 6.25 0 −0.95 −2.32 −2.15 0.0456 −0.81
      162Rem 158Tam 6.28 0 −1.07 −2.43 −2.26 0.0457 −0.92 164Rem 160Tam 5.77 0 1.46 −0.43 −0.22 0.0411 1.16
      164Irm 160Rem 7.06 0 −2.78 −4.36 −4.15 0.0445 −2.8 166Ir 162Re 6.73 0 −1.95 −3.29 −3.15 0.0405 −1.76
      166Irm 162Rem 6.73 0 −1.81 −3.29 −3.15 0.0405 −1.76 168Ir 164Re 6.38 0 −0.64 −2.03 −1.9 0.0368 −0.46
      168Irm 164Rem 6.48 0 −0.68 −2.41 −2.27 0.0371 −0.84 170Ir 166Rep 5.96 0 1.24 −0.37 −0.17 0.0332 1.31
      170Irm 166Re 6.27 2 0.35 −1.32 −1.12 0.0205 0.57 172Ir 168Re 5.99 3 2.34 0.06 0.16 0.0151 1.98
      172Irm 168Re 6.13 0 1.36 −1.16 −1.04 0.0314 0.46 174Ir 170Re 5.63 2 3.17 1.31 1.5 0.0168 3.27
      174Irm 170Re 5.82 2 2.29 0.44 0.63 0.0172 2.4 170Au 166Ir 7.18 0 −2.58 −4.03 −3.82 0.0362 −2.38
      Continued on next page

      Table 6.  Same as Tables 4 and 5, but for α decay of doubly odd nuclei.

      The differences between logarithmic values of three calculated α decay half-lives and experimental data are denoted as black open square, red solid square as well as blue solid circle, and plotted in Figs. 24 for even-even nuclei, odd-$ A $ nuclei and doubly odd nuclei, respectively. From these figures, we can find that $ \lg T^{{\rm{cal1}}}_{1/2} $ is significantly less than experimental value. After adopting the GLDM with proximity energy Prox. 77-Set 13, compared with $ \lg T^{{\rm{cal1}}}_{1/2} $, $ \lg T^{{\rm{cal2}}}_{1/2} $ are significantly improved in reproducing experimental data. And when neutron numbers are close to the $ N = 126 $ closed shell and superheavy nuclei region, the deviations caused by $ \lg T^{{\rm{cal1}}}_{1/2} $ and $ \lg T^{{\rm{cal2}}}_{1/2} $ have maximum values, indicating that there are important physics, i.e. shell effect needed to be considered. It is also indicated that changing proximity energy will not affect the revealing of microscopic shell effect, which is important for predicting the island of stability for superheavy nuclei. After considering α-particle preformation factors obtained by Eq. (3), the deviations caused by $ \lg T^{{\rm{cal3}}}_{1/2} $ are around zero indicating the accuracy of calculations has been significantly improved. For all 535 nuclei, the RMS deviation between $ \lg T^{{\rm{cal3}}}_{1/2} $ and $ \lg T^{\exp}_{1/2} $ is $ \sigma = 0.258 $, indicating that calculated α decay half-lives using GLDM with proximity energy Prox. 77-Set 13 and α-particle preformation factor estimated by Eq. (3) can reproduce experimental data within a factor of $ 10^{0.258} = 1.81 $. In addition, as can be seen from Figs. 24, $ \lg T^{{\rm{cal2}}}_{1/2} $ are approximately 0.2 larger than $ \lg T^{{\rm{cal1}}}_{1/2} $ on the whole, indicating that the introduction of proximity energy Prox. 77-Set 13 systematically improve the calculated accuracy of the GLDM to describe the α decay half-lives. Fig. 5 shows the deviations between the calculations by GLDM with the proximity energy Prox. 77-Set 13 and with original one for even-even heavy and superheavy nuclei. In this figure, one can find that the deviations are around 0.2 and 0.14 in heavy and superheavy nuclei regions, respectively. It is indicated that compared with the heavy nuclei, in the superheavy nuclei region, α decay half-life is less sensitive to the proximity energy, which will help us to predict α decay half-lives of unsynthesized superheavy nuclei.

      Figure 2.  (Color online) The logarithmic differences between three calculated $\alpha$ decay half-lives and experimental data of even-even nuclei. The black open square, red solid square, and blue solid circle denote the differences caused by $\lg T^{{\rm{cal1}}}_{1/2}$, $\lg T^{{\rm{cal2}}}_{1/2}$ and $\lg T^{{\rm{cal3}}}_{1/2}$, respectively.

      Figure 3.  (Color online) Same as Fig. 2, but it depicts the logarithmic differences between three calculated α decay half-lives and experimental data of odd-A nuclei.

      Figure 4.  (Color online) Same as Fig. 2, but it depicts the logarithmic differences between three calculated α decay half-lives and experimental data of doubly odd nuclei.

      Figure 5.  (Color online) The differences between calculated α decay half-lives $\lg T^{{\rm{cal2}}}_{1/2}$ and $\lg T^{{\rm{cal1}}}_{1/2}$ for even-even nuclei with Z = 84, 94,104, and 114.

      Encouraging by the good precise of calculated α decay half-lives for known nuclei, α decay half-lives of even-even superheavy nuclei with $ Z = 112–122 $ are predicted using the GLDM with the proximity energy Prox. 77-Set 13 and α-particle preformation factors obtained by Eq. (3), the improved Royer formula [79] and the universal decay law (UDL) [80]. The α decay energies are taken from WS4+ mass model [84], which is the most accurate nuclear mass model at present. The predictions are listed in Table 7. In each part of this table, the first, second and fourth columns are the same as Tables 46. The third one is the α decay energy obtained by WS4+ mass model [84]. The last three columns are the predicted α decay half-lives by the improved Royer formula, UDL formula, and GLDM with proximity energy Prox. 77-Set 13 and α-particle preformation factor obtained by Eq. (3). From this table, one can see that three calculations can be consistent with each other, and the change trends of half-lives are consistent. For intuitively, the logarithms of half-lives by three methods are plotted in Fig 6. In this figure, one can see that GLDM and UDL formula give the longest and shortest predictions of α decay half-lives, respectively. The predictions by GLDM are very close to the ones predicted by improved Royer formula. Noticeably, for 286Fl, 288Fl,290Lv, 292Lv and 294Og, the predictions can reproduce experimental data well, indicating that predictions are reliable. In particular, one can find that when neutron numbers $ N $ cross $ N = 184 $, predicted α decay half-lives decrease sharply, and at $ N = 186 $, α decay half-lives reduce by more than 2 orders of magnitude. It is indicated that strong shell effects is reflected, implying that the next neutron magic number after $ N = 126 $ is $ N = 184 $. The Fig. 7 plots the logarithms of half-lives by three methods for $ N = 184 $ isotones. In this figure, one can see that when proton number $ Z>114 $, predicted α decay half-lives drop dramatically by 8 orders of magnitude at $ Z = 116 $, indicating that there is a major proton shell and the next proton magic number after $ Z = 82 $ is $ Z = 114 $. In addition, the half-life of 296Og shows a peak, which nucleus is the closest one to the heaviest nucleus 295Og at present and may be as a next candidate for synthesizing superheavy nucleus experiment.

      Figure 6.  (Color online) The predicted α decay half-lives of even-even nuclei with $ Z = 112–122 $ using the GLDM with the proximity energy Prox. 77-Set 13 and α-particle preformation factors obtained by Eq. (3), the improved Royer formula [79] and the universal decay law (UDL) [80] taking the $ Q_{\alpha} $ obtained by WS4+ mass model [84]. The purple square, blue star and green triangle denote the experimental α decay half-lives taken from Refs. [14,21,83].

      α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{{\rm{Royer}}}_{1/2}$ $\lg T^{{\rm{UDL}}}_{1/2}$ $\lg T^{{\rm{calc3}}}_{1/2}$ α transition $Q_{\alpha}$ $l_{\min}$ $\lg T^{{\rm{Royer}}}_{1/2}$ $\lg T^{{\rm{UDL}}}_{1/2}$ $\lg T^{{\rm{calc3}}}_{1/2}$
      Nuclei with Z = 112
      272Cn 268Ds 12.05 0 −5.32 −5.62 −5.04 274Cn 270Ds 11.52 0 −4.18 −4.5 −4.04
      276Cn 272Ds 11.9 0 −5.08 −5.38 −4.85 278Cn 274Ds 11.74 0 −4.75 −5.06 −4.52
      280Cn 276Ds 10.83 0 −2.63 −2.96 −2.53 282Cn 278Ds 10.11 0 −0.76 −1.11 −0.69
      284Cn 280Ds 9.52 0 0.94 0.57 0.97 286Cn 282Ds 9.01 0 2.51 2.13 2.53
      288Cn 284Ds 9.09 0 2.24 1.86 2.27 290Cn 286Ds 8.85 0 2.98 2.59 3.01
      292Cn 288Ds 8.27 0 5.06 4.65 5.08 294Cn 290Ds 8.06 0 5.83 5.41 5.81
      296Cn 292Ds 7.7 0 7.24 6.81 7.21 298Cn 294Ds 8.77 0 3.12 2.73 3.18
      300Cn 296Ds 8.42 0 4.36 3.96 4.42 302Cn 298Ds 7.49 0 8.05 7.61 8.08
      Continued on next page

      Table 7.  Predicted α decay half-lives of even-even nuclei with $Z = 112–122$ using the GLDM with proximity energy Prox. 77-Set 13, the improved Royer formula [79] and the universal decay law (UDL) [80]. The α decay energies are calculated by WS4+ mass model [84]. The α decay energies and half-lives are in the unit of 'MeV' and 's', respectively.

      Figure 7.  (Color online) Same as Fig. 6, but it depicts predicted α decay half-lives of even-even nuclei with N = 184 isotones.

    IV.   SUMMARY
    • In summary, we systematically study the abilities of various versions of proximity energies when they are applied to Generalized Liquid Drop Model for enhancing the calculation accuracy and prediction ability of α decay half-lives for known and unsynthesized superheavy nuclei. By choosing a more suitable proximity energy applied to GLDM, calculations of α half-lives have systematic improvements in reproducing experimental data. In addition, calculations indicate that changing proximity energy will not affect the revealing of microscopic shell effect, which is important for predicting the island of stability for superheavy nuclei. Encouraged by this, the α decay half-lives of even-even superheavy nuclei with $ Z = 112–122 $ are predicted by the GLDM with a more suitable proximity energy. The predictions are conform to ones calculated by the improved Royer formula and the universal decay law. In addition, the features of predicted α decay half-lives imply that the next double magic nucleus after 208Pb is 298Fl.

Reference (86)

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