Improved empirical formula for α particle preformation factor

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Yan He, Xuan Yu and Hongfei Zhang. Improved empirical formula for α particle preformation factor[J]. Chinese Physics C.
Yan He, Xuan Yu and Hongfei Zhang. Improved empirical formula for α particle preformation factor[J]. Chinese Physics C. shu
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Improved empirical formula for α particle preformation factor

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

Abstract: In this contribution, the $\alpha$ preformation factors of 606 nuclei are extracted within the framework of generalized liquid drop model (GLDM). Through the systematically analysis of the $\alpha$ preformation factors of even-even Po-U isotopes, we found there is a significant weakening of influence of $N=126$ shell closure in uraninum, which is consistent with the result of a recent experiment [J. Khuyagbaatar et al., Phys. Rev. Lett. 115.242502 (2015)], implying that $N=126$ may be not the magic number for U isotopes. Furthermore, we propose an improved formula with only 7 parameters to calculate $\alpha$ preformation factors suitable for all types of $\alpha$-decay, which has fewer parameters than the original formula proposed by Zhang et al. [H. F. Zhang et al., Phys. Rev. C 80.057301 (2009)] with high precision. The standard deviation of the $\alpha$ preformation factors calculated by our formula with extracted values for all 606 nuclei is 0.365 with a factor of 2.3, indicating that our improved formula can accurately reproduce the $\alpha$ preformation factors. Encouraged by this, the $\alpha$-decay half-lives of actinide elements are predicted, which could be useful in future experiments. Noticeably, the predicted $\alpha$-decay half-lives of two new isotopes $^{220}$Np [Z.Y. Zhang, et al., Phys. Rev. Lett. 122. 192503 (2019)] and $^{219}$Np [H. B. Yang et al., Phys. Lett. B 777, 212 (2018)] are in good agreement with the experimental $\alpha$-decay half-lives.

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    I.   INTRODUCTION
    • Generally, the perception of $ \alpha $-decay is that $ \alpha $ particle exists as an entity inside the parent nucleus before emission from it explained by Gamow as a quantum-tunneling effect, $ \alpha $ preformation factors $ P_\alpha $ which is used to describe this formed amplitude may be determined by the ratios between experimental and calculated $ \alpha $-decay half-lives depending on various theoretical models. Behaviors of $ P_\alpha $ incorporate nuclear structure information such as the shell closure effect of nucleons, blocking effect of unpaired nucleons in the open shells of parent nucleus, interaction of proton-neutron and deformation of the daughter nucleus[1], etc. $ \alpha $ preformation factor is expected to be a superposition of the different nuclear structure effects.

      One of recent evidences shows $ \alpha $ preformation factor has a clearly dependence on the product of valence protons (holes) and valence neutrons (holes) $ N_pN_n $ as well as isospin asymmetry[2-4]. In the work of Sun et al. [5], the result considering the isospin asymmetry effect for nuclear potential has an obvious improvement to reproduce $ \alpha $-decay half-lives of 164 even-even nuclei in the two-potential model(TPM). Furthermore, it shows that the $ \alpha $-decay width in the neutron skin-type (NST) case considering the effect of isospin asymmetry for proton and neutron density is about 30% greater than that in the Hartree-Fock formalism (HF) case[6]. Therefore, the effect of isospin asymmetry may play a key role in determination of $ \alpha $ preformation factors. Accumulating evidences show that the interaction of valence nucleons and the effect of isospin asymmetries are incredibly important while determing $ \alpha $ preformation factors.

      There are many firsthand approaches to obtain $ \alpha $ preformation factors. The cluster formation model (CFM) [7-11] is based on quantum-mechanical dynamic states of the $ \alpha $ cluster and daughter nucleus to calculate $ P_\alpha $ and to study shell effect and clusterization mechanisms around nucleus surface. The preformed cluster model (PCM) [12-15] uses the solution of the stationary Schrödinger equation to determine $ P_\alpha $ in $ \eta $ coordinates. The $ \alpha $ preformation factors are also studied in the double-folding model of the density dependence (DMDD)[16], the cluster model (CM), Bardeen-Cooper-Schrieffer (BCS) approach, and potential model like the density-dependent Michigan 3-range Yukawa (DDM3Y)[17, 18] from different perspectives. It is also suggested that the trendency of $ \alpha $ preformation factors is not dependent on the model[19].

      Actually, it is challenging to calculate $ \alpha $ preformation factors using the microscopic method, which is necessarily a complex quantum multibody problem. At present, the determination of $ \alpha $ preformation factors is mostly empirical. For instance, Ref. [20] proposes that $ P_\alpha $ shows an exponential relationship with a polynomial of $ \alpha $-decay energy, $ P_\alpha $ has a linear correlation with the product of valence nucleons put forward in Ref. [21] and the original formula of Zhang et al. is related to the shell closure effect[22]. These formulas incorporate some nuclear structure information, but there are some limitations such as too many parameters or limited nuclear range of application, which make it difficult to promote further. In our present work, we aim to obtain a new formula with fewer parameters, a broader range of application, and higher precision. We extract $ \alpha $ preformation factors of 606 nuclei in the framework of GLDM. Based on shell closure effect and features of $ \alpha $ preformation factors, we propose a new, improved formula for calculation of $ P_\alpha $. In the next section, the theoretical framework of extracting $ P_\alpha $ is presented. The results of our numerical calculation, discussion, and prediction are in Sec. III. A brief summary is given in Sec. IV.

    II.   METHODS
    • Classically, assuming that $ \alpha $ particle is moving back and forth inside the parent nucleus with the velocity $ v = \sqrt{\frac{2E_\alpha}{M}} $, it penetrates barrier with a classical frequence

      $ \nu_0 = \frac{1}{2R}\sqrt{\frac{2E_\alpha}{M}}, $

      (1)

      R is the radius of nucleus given by

      $ R = \left( 1.28A^\frac{1}{3}-0.76+0.8A^{-\frac{1}{3}}\right)fm , $

      (2)

      The barrier penetration probability is calculated by the Wentzel-Kramers-Brillouin(WKB) approximation:

      $ P = exp\left[ -\frac{2}{\hbar}\int_{R_{in}}^{R_{out}}\sqrt{2B(r)(E(r)-E(parent)) }\,dr\right], $

      (3)

      where $ R_{in} $ and $ R_{out} $ are the two turning points of the WKB action integral and the mass inertia $ B(r) $ is approximated to the reduced mass $ \mu $. The barriers are constructed by the GLDM including the volume, surface, Coulomb, proximity, and centrifugal energy[26]:

      $ E(r) = E_V+E_S+E_C+E_{prox}+E_{cen}, $

      (4)

      where $ E(r) $ is the function of the distance between the centroids of two fragments during evolution. $ E_V $, $ E_S $, $ E_C $, $ E_{prox} $ and $ E_{cen} $ are the volume, surface, Coulomb, proximity, and centrifugal energies, respectively. For one-body shapes[26-28], the volume, surface, and Coulomb energies are given by

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

      (5)

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

      (6)

      $ E_C = 0.6e^2(Z^2/R_0)\times 0.5\int(V(\theta)/V_0)(R(\theta)/R_0)^3sin{\theta} d\theta. $

      (7)

      where $ I = (N-Z)/A $ is the relative neutron excess. S is the surface of the deformed nucleus. $ V_0 $ is the surface potential of the sphere and $ V(\theta) $ is the electrostatic potential at the surface.

      When the nuclei are separated,

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

      (8)

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

      (9)

      $ E_C = 0.6e^2(Z_1^2/R_1)+0.6e^2(Z_2^2/R_2)+e^2Z_1Z_2/r. $

      (10)

      $ A_i, Z_i, R_i \; and \; I_i $ are the mass numbers, charge numbers, radii, and relative neutron excesses. $ R_i $ is given by Eq.(2). The proximity energy $ E_{prox} $ is given by:

      $ E_{prox}(r) = 2\gamma\int_{h_{min}}^{h_{max}}\Phi \left[D(r,h)/b \right] 2\pi hdh, $

      (11)

      where $ h = R(\theta)sin\theta $ is the distance varying from the neck radius or zero to the height of the neck border. D is the distance between the opposite infinitesimal surfaces and $ b = 0.99 $ fm the surface width. $ \Phi $ is the proximity function. The surface parameter $ \gamma $ is the geometric mean between the surface parameters of the two fragments. The centrifugal energy $ E_{cen} $ is given by

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

      (12)

      l is the minimum angular momentum taken away by the $ \alpha $ particle determined by the angular momentum and parity conservation laws:

      $ l = \left\{ {\begin{array}{*{20}{l}} {{\triangle_j}\;\;\;\;\;\;\;{\rm{for}}\;{\rm{even}}\;{\triangle_j}\;\;\;\;\;\;{\rm{and}}\;\;\;{\pi _p} = {\pi _d},}\\ {{\triangle_j} + 1\;\;\;{\rm{for}}\;{\rm{even}}\;{\triangle_j}\;\;\;\;\;{\rm{and}}\;\;\;{\pi _p}\not = {\pi _d},}\\ {{\triangle_j}\;\;\;\;\;\;\;\;\;\;\;{\rm{for}}\;{\rm{odd}}\;{\triangle_j}\;\;\;{\rm{and}}\;\;\;{\pi _p}\not = {\pi _d},}\\ {{\triangle_j} + 1\;\;\;\;\;\;{\rm{for}}\;{\rm{odd}}\;{\triangle_j}\;\;\;{\rm{and}}\;\;\;{\pi _p} = {\pi _d},} \end{array}} \right. $

      (13)

      The decay constant is:

      $ \lambda = P_{\alpha}\nu_0P = \frac{ln2}{T_{1/2}^{exp}}. $

      (14)

      $ {\rm{T}}_{1/2}^{\rm{exp}} $ is experimental half-life of $ \alpha $ decay. $ P_{\alpha} $ can be extracted by:

      $ P_\alpha^{ext} = \frac{\lambda}{\nu_0P}. $

      (15)
    III.   RESULTS AND DISCUSSION
    • The closed shell structure is one of the fundamental pillars of nucleus which affects the behaviors of $ \alpha $ preformation factors significantly. As one consequence, the closer proton or neutron number is to the magic number, the smaller $ \alpha $ preformation factor is, and $ \alpha $ preformation factor decreases dramatically near the magic number [29, 30]. Fig. 1 is the $ \alpha $ preformation factors of five isotopic chains of even-even Po to Th nuclei versus neutron number N. The shell closure effect of $ N = 126 $ from Po ($ Z = 84 $) to Th ($ Z = 90 $) isotopes is prominent. The $ \alpha $ preformation factors decrease rapidly when N below 126, and increase dramatically above 126 with increasing neutron number. $ N = 126 $ is a traditional magic number where the $ \alpha $ preformation factor should be minimal like even-even Po-Th isotopic chains in Fig. 1, but the $ \alpha $ preformation factor of $ ^{216} $U ($ N = 124 $) is apparently smaller than that of $ ^{216} $U ($ N = 126 $). In addition, the $ \alpha $ preformation factors of U isotopes increase more gently than Po-Th isotopes from N = 126 to 130. In Fig. 2, from Po ($ Z = 84 $) to Th ($ Z = 90 $) for $ N = 126 $ isotones, the $ \alpha $ preformation factors rise smoothly with increasing Z, but a sudden increase occurs at $ ^{218} $U ($ N = 126 $). The above phenomenons indicate a obviously weakening of the $ N = 126 $ shell effect for U. This is consistent with the result of a recent experimental work by J. Khuyagbaatar[31], pointing out that there is a significant weakening of the influence of $ N = 126 $ shell closure in U by analyzing the reduced widths and neutron-shell gaps of Po-U isotopes.

      Figure 1.  The $ \alpha$ preformation factors of Po, Rn, Ra, Th, and U istopes extracted by GLDM.

      Figure 2.  The extracted $ \alpha$ preformation factors of $ N = 126$ istones for Po, Rn, Ra, Th, and U.

      Besides the shell closure effect, sufficient evidences suggest that the correlation of valence nucleons plays an important role in the $ \alpha $ preformation factors[21, 23-25]. It has previously been shown that there is a linear relationship between $ P_\alpha $ and the product of valence protons (holes) and valence neutrons (holes) $ N_pN_n $ [2-4]. At the same time, a recent calculation using the scheme of microscopic $ N_pN_n $ can accurately reproduce the extracted $ \alpha $ performation factors[32]. It is also refered that the interaction of valence nucleons reflects the effect of deformation[33, 34]. In addition, it is reported that the $ \alpha $ cluster formation energy and surface energy dependenting on differences of binding energies are critical for determining the values of $ \alpha $ preformation probability [7, 11].

      It is a consensus that the increase of valence nucleons is considered to be beneficial to the formation of $ \alpha $ particle in parent nucleus. The interaction of valence protons with valence neutrons can not be neglected in $ \alpha $ preformation, it reflects the role of proton-neutron correlation in $ \alpha $-decay. The shell closure effect for the preformation factors is intuitionistic and obvious, which is indispensable in the construction of $ \alpha $ preformation factor formula reflected well in concise formula of Zhang et al. It has been reported that the proton-neutron pair number, neutron-neutron pair number and gap energy have significant correlation with isospin [35, 36]. A more direct evidence is that $ P_\alpha $ is positively correlated with isospin asymmtry of parent nuclei having the same $ N_pN_n $ values [2]. Isospin asymmtry is an important factor affecting the density distribution of protons and neutrons which expected to play a more important role in the preformation of the $ \alpha $ nucleus[18, 37-41]. This leads us to consider that isospin asymmetry may be a key for determination of $ \alpha $ preformation factors. At the same time, protons and neutrons are in different charge states of the same nucleon, and the nuclear force is charge independent, the new formula should be able to reflect this feature. According to the discussion above, we put forward a new improved formula based on the original formula of Zhang et al [22].

      $ \begin{aligned}[b] lgP_{\alpha} = &\{ a+b\left[(Z-Z_1)(Z_2-Z)^3+(N-N_1)(N_2-N)^3\right] \\ &+c/\left[(Z-Z_1)(N-N_1)+1\right]+a_p+a_ll\} I+dA. \end{aligned} $

      (16)

      where Z, N and A are the charge, neutron, and mass number of the parent nucleus. $ Z_1 $ and $ Z_2 $ are the proton magic numbers around $ Z\;(Z_1<Z\leqslant Z_2) $, $ N_1 $ and $ N_2 $ are the neutron magic numbers around $ N\;(N_1<N\leqslant N_2) $, respectively. Proton magic numbers can be 50, 82,120 and neutron magic numbers can be 82,126,162,184. I is the isospin asymmtry, $ I = (N-Z)/(N+Z) $. The second term represents shell closure effect of protons and neutrons, third term represent interaction of valence protons with valence neutrons. $ a_p $ is a constant representing the blocking effect of unpaired nucleons for odd-A and odd-odd nuclei. l is the minimum angular momentum transferred to $ \alpha $ particle in $ \alpha $-decay.

      We found that the values of $ a_p $ for even-odd and odd-even nuclei are approximately equal because the fact that proton and neutron are in different charge states of the same nucleon, the pairing force is same for proton-neutron and neutron-neutron interaction, and an unpaired proton or neutron has the same effect on the calculation of $ \alpha $ preformation factors. Therefore, the parameter $ a_p $ for even-odd and odd-even nuclei are fixed to the same value. In the second term of Eq. (16), the representations of the proton and neutron shell closure effect are the same form with one parameter b, also reflecting that nuclear force is charge independent. Though the effect of centrifugal potential has involved in GLDM method, we put the influence of angular momentum into consideration in our formula. On the one hand, it is difficult to produce $ \alpha $ particle with large angular momentum on the surface of nucleus according to the principle of least action. On the other hand, the centrifugal potential is calculated in the condition of sphere daughter nucleus. Therefore, it is necessary to consider the role of angular momentum in the formula.

      The coefficients of Eq. (16) for fitting 606 nuclei's extracted preformation factors are listed in Table. 1, and the parameter $ a_p $ for odd-odd nuclei is smaller than that for odd-A and even-even nuclei, revealing the blocking effect is stronger for odd-odd nuclei, and it is difficult for formation of $ \alpha $ particle in odd-odd nuclei including two unpaired nucleons. Fig. 3 shows the variation of $ \alpha $ preformation factors for different types of nuclei versus Z and N. The $ \alpha $ preformation factors calculated by the formula Eq. (16) can reproduce the trend of the extracted ones.

      ParameterValueParameterValue
      a2.65991$a_l$−0.69111
      b1.07$\times 10^{-5}$$a_p$(e-e)0
      c−2.66753$a_p$(odd A)−1.82248
      d−0.00964$a_p$(o-o)−4.68543

      Table 1.  Coefficient sets of Eq. (16) determined by fitting to 606 nuclei's extracted preformation factors.

      Figure 3.  (a)(e), (b)(f), (c)(g) and (d)(h) are extracted and calculated [from Eq. (16)] $\alpha$ preformation factors of 172 even Z-even N, 167 even Z-odd N, 163 even Z-odd N and 105 odd Z-odd N nuclei, respectively.

      To measure the agreement of the theoretical values with the experimental values, the standard deviation between the extracted and calculated $ \alpha $ preformation factors is defined as:

      $ \sigma = \sqrt{\sum (lgP_{\alpha}^{ext}-lgP_{\alpha}^{cal})^2/n}. $

      (17)

      The standard deviation for even-even, even-odd, odd-even and odd-odd nuclei are 0.280, 0.417, 0.359 and 0.397, respectively. As for all 606 nuclei the value of $ \sigma $ is 0.365 with a factor of $ 10^{0.365} = 2.32 $, which is better than the result of Seif et al. [45] $ \sigma $ = 0.529 for 445 nuclei utilizing 16 parameters. Compared to the previous formula of Zhang et al. [22] with 48 parameters, the importance of angular momentum is firstly taken into account in our present work and the effect for unfavored $ \alpha $-decay is prominent especially for odd-A and odd-odd nuclei while neutron number $ N>126 $. The detailed comparison of our improved formula [Eq. (16)] with the formulas of Zhang et al. and Seif et al. is listed in Table 2. To show the precision of Eq. (16) explicitly, Fig. 4 display the extracted and calculated $ \alpha $ preformation factors of Po, Rn and Pa isotopes. The preformation factors calculated from Eq. (16) are very close to the extracted data, and can reflecte the nuclear microscopic properties, such as the neutron magic number $ N = 126 $ and the odd-even effect.

      Nucleifactor [Eq. (16)]factor [22]factor [45]
      even-even1.901.40-1.65
      odd-A ($N<126$)1.981.95-2.34
      odd-A ($N>126$)3.024.96-4.99
      odd-odd ($N<126$)2.342.00-2.25
      odd-odd ($N>126$)2.824.09-6.53
      All nuclei2.323.38
      Number of nuclei606445445
      Number of parameters74816

      Table 2.  The comparison of our improved formula [Eq. (16)] with the formula of Zhang et al. [22] and Seif et al. [45].

      Figure 4.  Extracted and calculated $\alpha$ preformation factors [from Eq. (16) ${\rm{lg P}}_\alpha^{\rm{cal}}$] for Po (Z = 84), At (Z = 85) and Rn (Z = 86) isotopes.

      We also compare our calculations with the result of Ren et al. [23-25, 46], as shown in Fig. 5. Both theoretical $ P_{\alpha} $ of N = 125,126 and 127 isotones are consistent with the trend of extracted values, and the values calculated by our improved formula show significant agreement with the extracted ones. Noticeablly, with the increase of proton number when $ Z>92 $, the $ \alpha $ preformation factors calculated by Ren et al. still increase, but the values calculated by our formula gradually decrease. This is because that the formula of Ren et al. mainly considers the role of valence nucleons rather than the shell closure effect. Our formula is closely related to the shell closure effect and can give the trend more accuratly that the preformation factors increase when Z away from the magic number and then decrease while closing to the next one. In the all, our formula can accurately describe the features of $ \alpha $ preformation factors and reproduce the extracted values with only 7 parameters.

      Figure 5.  Extracted and calculated $\alpha$ preformation factors [from Ren et al. [46] ${\rm{lg P}}_\alpha^{\rm{Ren}}$ and Eq. (16) ${\rm{lg P}}_\alpha^{\rm{cal}}$] for $N = 125,126$ and $127$ isotones

      Considering $ \alpha $ preformation factor, the calculated and experimental $ \alpha $-decay half-lives of actinide elements are listed in the Table. 3. In this table, the first four columns denote $ \alpha $-decay channel, experimental decay energy, spin and parity transition as well as the minimum angular momentum taken away by $ \alpha $ particle, respectively. The next one is the half-lives from GLDM with $ P_\alpha = 1 $ and The sixth column is the calculated $ \alpha $ preformation factors by Eq. (16), parameters are listed in Table. 1. By taking $ P_\alpha^{cal} $ into account, it can be seen that the calculated $ \alpha $-decay half-lives $ T_{1/2}^{cal} $ can accurately reproduce most of $ \alpha $-decay experimental half-lives. But for some nuclei such as $ ^{212} $Pa $ ^{250} $Cm, $ ^{249} $Md, $ ^{236} $Am, $ ^{256} $Cf, $ ^{237} $Cf, $ ^{233} $Np, $ ^{243} $Es, $ ^{253} $Cf, $ ^{247} $Fm, $ ^{225} $Pa, $ ^{255} $Es, $ ^{220} $Pa, $ ^{254} $Cf, $ ^{238} $U, $ ^{231} $Np, $ ^{227} $Np and $ ^{218} $Ac, the differences significantly increase. Firstly, this is maybe because we didn't consider the effect of different decay channels while extracting $ \alpha $ preformation factors, the branching ratio transmitting to the first excited state can be as large as 30%[47]. Secondly, these experimental data like half-lives, spin and parity used to these special nuclei are not accurate enough and should be examined again.

      parent $\rightarrow$daughter$Q\alpha$$j_p^\pi\rightarrow j_d^\pi$l$T^{cal}_{1/2,P_\alpha=1}$$P_\alpha^{cal}$$T_{1/2}^{exp}$$T_{1/2}^{cal}$
      $^{216}$Ac$\rightarrow^{212}$Fr9.24(1$^-$)$\rightarrow$5$^+$$^*$51.3$\times10^{-6}$2.3$\times10^{-3}$4.4$\times10^{-4}$5.8$\times10^{-4}$
      $^{215}$Ac$\rightarrow^{211}$Fr7.759/2$^-$$\rightarrow$9/2$^-$$^*$01.9$\times10^{-3}$9.9$\times10^{-3}$1.7$\times10^{-1}$1.9$\times10^{-1}$
      $^{207}$Ac$\rightarrow^{203}$Fr7.859/2$^-$$\#$$\rightarrow$9/2$^-$$^*$01.2$\times10^{-3}$2.9$\times10^{-2}$3.1$\times10^{-2}$4.3$\times10^{-2}$
      $^{216}$Ac$^m$$\rightarrow^{212}$Fr9.28(9$^-$)$\rightarrow$5$^+$$^*$51.0$\times10^{-6}$2.3$\times10^{-3}$4.4$\times10^{-4}$4.5$\times10^{-4}$
      $^{211}$Ac$\rightarrow^{207}$Fr7.629/2$^-$$\rightarrow$9/2$^-$$^*$05.7$\times10^{-3}$2.8$\times10^{-2}$2.1$\times10^{-1}$2.0$\times10^{-1}$
      $^{227}$Ac$\rightarrow^{223}$Fr5.043/2$^-$$\rightarrow$3/2$^-$$^*$02.5$\times10^{9}$7.1$\times10^{-2}$5.0$\times10^{10}$3.5$\times10^{10}$
      $^{206}$Ac$^m$$\rightarrow^{202}$Fr$^m$7.91(10$^-$)$\rightarrow$10$^-$$^*$08.4$\times10^{-4}$1.2$\times10^{-2}$4.1$\times10^{-2}$7.1$\times10^{-2}$
      $^{217}$Ac$\rightarrow^{213}$Fr9.839/2$^-$$\rightarrow$9/2$^-$$^*$03.6$\times10^{-9}$3.8$\times10^{-2}$6.9$\times10^{-8}$9.6$\times10^{-8}$
      $^{206}$Ac$\rightarrow^{202}$Fr7.96(3$^+$)$\rightarrow$3$^+$$^*$05.7$\times10^{-4}$1.2$\times10^{-2}$2.5$\times10^{-2}$4.8$\times10^{-2}$
      $^{225}$Ac$\rightarrow^{221}$Fr5.943/2$^-$$\#$$\rightarrow$5/2$^-$$^*$24.8$\times10^{4}$3.7$\times10^{-2}$8.6$\times10^{5}$1.3$\times10^{6}$
      $^{217}$Ac$^m$$\rightarrow^{213}$Fr11.84(29/2)$^+$$\rightarrow$9/2$^-$$^*$111.4$\times10^{-8}$1.6$\times10^{-3}$1.7$\times10^{-5}$8.4$\times10^{-6}$
      $^{205}$Ac$\rightarrow^{201}$Fr$^n$7.909/2$^-$$\#$$\rightarrow$(13/2$^+$)32.8$\times10^{-3}$1.6$\times10^{-2}$8.0$\times10^{-2}$1.8$\times10^{-1}$
      $^{219}$Ac$\rightarrow^{215}$Fr8.839/2$^-$$\rightarrow$9/2$^-$09.4$\times10^{-7}$5.1$\times10^{-2}$1.2$\times10^{-5}$1.8$\times10^{-5}$
      $^{222}$Ac$\rightarrow^{218}$Fr7.141$^-$$\rightarrow$1$^-$02.0$\times10^{-1}$1.8$\times10^{-2}$5.0$\times10^{0}$1.1$\times10^{1}$
      $^{223}$Ac$\rightarrow^{219}$Fr6.78(5/2$^-$)$\rightarrow$9/2$^-$27.8$\times10^{0}$3.6$\times10^{-2}$1.3$\times10^{2}$2.2$\times10^{2}$
      $^{221}$Ac$\rightarrow^{217}$Fr7.789/2$^-$$\#$$\rightarrow$9/2$^-$01.1$\times10^{-3}$6.2$\times10^{-2}$5.2$\times10^{-2}$1.9$\times10^{-2}$
      $^{218}$Ac$\rightarrow^{214}$Fr9.371$^-$$\#$$\rightarrow$(1$^-$)03.8$\times10^{-8}$1.3$\times10^{-2}$1.0$\times10^{-6}$2.9$\times10^{-6}$
      $^{226}$Ac$\rightarrow^{222}$Fr5.51(1)($^-$$\#$)$\rightarrow$2$^-$$^*$28.8$\times10^{6}$9.0$\times10^{-3}$1.7$\times10^{9}$9.8$\times10^{8}$
      $^{220}$Ac$\rightarrow^{216}$Fr8.35(3$^-$)$\rightarrow$(1$^-$)23.5$\times10^{-5}$8.8$\times10^{-3}$2.6$\times10^{-2}$4.0$\times10^{-3}$
      $^{214}$Th$\rightarrow^{210}$Ra7.830$^+$$\rightarrow$0$^+$02.7$\times10^{-3}$5.3$\times10^{-2}$8.7$\times10^{-2}$5.0$\times10^{-2}$
      $^{219}$Th$\rightarrow^{215}$Ra9.519/2$^+$$\#$$\rightarrow$9/2$^+$$\#$03.7$\times10^{-8}$4.3$\times10^{-2}$1.0$\times10^{-6}$8.6$\times10^{-7}$
      $^{222}$Th$\rightarrow^{218}$Ra8.130$^+$$\rightarrow$0$^+$02.1$\times10^{-4}$1.3$\times10^{-1}$2.2$\times10^{-3}$1.6$\times10^{-3}$
      $^{218}$Th$\rightarrow^{214}$Ra9.850$^+$$\rightarrow$0$^+$06.3$\times10^{-9}$7.6$\times10^{-2}$1.2$\times10^{-7}$8.3$\times10^{-8}$
      $^{212}$Th$\rightarrow^{208}$Ra7.960$^+$$\rightarrow$0$^+$01.1$\times10^{-3}$5.1$\times10^{-2}$3.2$\times10^{-2}$2.1$\times10^{-2}$
      $^{224}$Th$\rightarrow^{220}$Ra7.300$^+$$\rightarrow$0$^+$01.2$\times10^{-1}$1.5$\times10^{-1}$1.0$\times10^{0}$8.1$\times10^{-1}$
      Continued on next page

      Table 3.  Calculations of $\alpha$-decay half-lives of actinide nuclei. The experimental $\alpha$-decay half-lives, spin and parity are taken from the latest evaluated nuclear properties table NUBASE2016[42]. The $\alpha$-decay energies are taken from the latest evaluated atomic mass table AME2016[43]. Elements with superscripts "m," "n," "p," or "x" indicate assignments to excited isomeric states. "$()$" means uncertain spin or parity, "$\#$" means spin and parity values estimated from trends in neighboring nuclides with the same Z and N parities. The $\alpha$-decay energy and half-lives are in the unit of MeV and s, respectively. Experimental data is consistent with Ref. [44]. $T_{1/2}^{cal}$ is the calculations in the framework of GLDM with $P_\alpha^{cal}$.

      The predictions of $ \alpha $-decay half-lives of actinide elements within GLDM adding $ \alpha $ preformation factor calculated by our improved formula are shown in Table. 4, which may be useful in future experiments. Specially, the experiment $ \alpha $-decay half-lives of two new isotopes $ ^{219} $Np, $ ^{220} $Np are 1.50$ \times10^{-4} $s [50] and 2.50$ \times10^{-5} $s [51], the calculated $ \alpha $-decay half-lives are 2.52$ \times10^{-4} $s and 3.4$ \times10^{-5} $s, respectively. It shows amazing accuracy, indicating that predictions with our formula are more reliable.

      parent$\rightarrow$daughter$j_p\rightarrow j_d$Q$_\alpha$l$T_{1/2}^{cal}$parent$\rightarrow$daughter$j_p\rightarrow j_d$Q$_\alpha$l$T_{1/2}^{cal}$
      $^{195}$Ac$\rightarrow$$^{191}$Fr5/2$^+$$\rightarrow$5/2$^+$9.3202.7$\times10^{-6}$$^{251}$Cm$\rightarrow$$^{247}$Pu1/2$^+$$\rightarrow$7/2$^+$5.2146.9$\times10^{14}$
      $^{197}$Ac$\rightarrow$$^{193}$Fr11/2$^-$$\rightarrow$5/2$^+$9.2332.9$\times10^{-5}$$^{252}$Cm$\rightarrow$$^{248}$Pu0$^+$$\rightarrow$0$^+$5.0906.4$\times10^{13}$
      $^{199}$Ac$\rightarrow$$^{195}$Fr5/2$^-$$\rightarrow$5/2$^+$9.0312.7$\times10^{-5}$$^{253}$Cm$\rightarrow$$^{249}$Pu11/2$^-$$\rightarrow$1/2$^+$4.7753.2$\times10^{18}$
      $^{201}$Ac$\rightarrow$$^{197}$Fr5/2$^-$$\rightarrow$11/2$^+$8.932.0$\times10^{-4}$$^{254}$Cm$\rightarrow$$^{250}$Pu0$^+$$\rightarrow$0$^+$4.4901.7$\times10^{18}$
      $^{203}$Ac$\rightarrow$$^{199}$Fr7/2$^-$$\rightarrow$11/2$^+$8.5831.5$\times10^{-3}$$^{255}$Cm$\rightarrow$$^{251}$Pu1/2$^+$$\rightarrow$11/2$^-$4.1852.3$\times10^{23}$
      $^{209}$Ac$\rightarrow$$^{205}$Fr7/2$^-$$\rightarrow$13/2$^+$7.5133.4$\times10^{0}$$^{231}$Bk$\rightarrow$$^{227}$Am3/2$^-$$\rightarrow$5/2$^-$8.8124.2$\times10^{-2}$
      $^{213}$Ac$\rightarrow$$^{209}$Fr3/2$^-$$\rightarrow$5/2$^-$7.4422.3$\times10^{0}$$^{233}$Bk$\rightarrow$$^{229}$Am3/2$^-$$\rightarrow$5/2$^-$8.1724.1$\times10^{0}$
      $^{229}$Ac$\rightarrow$$^{225}$Fr3/2$^+$$\rightarrow$3/2$^+$4.4406.4$\times10^{14}$$^{235}$Bk$\rightarrow$$^{231}$Am3/2$^-$$\rightarrow$5/2$^-$7.8723.8$\times10^{1}$
      $^{231}$Ac$\rightarrow$$^{227}$Fr1/2$^-$$\rightarrow$3/2$^+$3.5313.1$\times10^{23}$$^{237}$Bk$\rightarrow$$^{233}$Am3/2$^-$$\rightarrow$5/2$^-$7.5227.3$\times10^{2}$
      $^{233}$Ac$\rightarrow$$^{229}$Fr3/2$^+$$\rightarrow$1/2$^+$3.1821.1$\times10^{28}$$^{239}$Bk$\rightarrow$$^{235}$Am3/2$^-$$\rightarrow$5/2$^-$7.1422.4$\times10^{4}$
      $^{203}$Th$\rightarrow$$^{199}$Ra1/2$^+$$\rightarrow$3/2$^+$8.8323.4$\times10^{-4}$$^{241}$Bk$\rightarrow$$^{237}$Am3/2$^-$$\rightarrow$5/2$^-$6.827.2$\times10^{5}$
      $^{204}$Th$\rightarrow$$^{200}$Ra0$^+$$\rightarrow$0$^+$8.7701.2$\times10^{-4}$$^{243}$Bk$\rightarrow$$^{239}$Am3/2$^-$$\rightarrow$5/2$^-$6.5121.6$\times10^{7}$
      Continued on next page

      Table 4.  The $\alpha$-decay energy are calculated by FRDM model, the spin and parity of nuclei are taken from the calculations of odd-nucleon spin and parity at the nuclear ground state by P. Möller which are from Ref. [48, 49]. The $\alpha$-decay energy and half-lives are in the units of MeV and s, respectively.

    IV.   SUMMARY AND OUTLOOK
    • In summary, we extracted $ \alpha $ preformation factors for 606 nuclei in the framework of GLDM and detaily analyzed the characteristics of $ \alpha $ preformation factors for even-even nuclei from Po to U isotopes. The $ N = 126 $ is a well known magic number, which shows significant shell closure effect for Po to Th isotopes. However, as for U isotopes the signal of $ N = 126 $ shell closure effect is weak implying that $ N = 126 $ maybe not the magic number for U isotopes. Based on previous formula of Zhang et al., we propose a unified formula with 7 parameters to calculate the $ \alpha $ preformation factor, which is adapted to different types of $ \alpha $-decay. Coefficients are obtained by fiting the extracted $ \alpha $ preformation factors of 606 $ \alpha $-decay nuclei. The standard deviation of total 606 nuclei is 0.37 with a factor of 2.3, shows a significant precision. Comparing with the work of Zhang et al. and Seif et al., our improved formula Eq. (16) has the fewest parameters with high precision, also has a wide application scope. Finally, $ \alpha $-decay half-lives of actinides are calculated in the framework of GLDM taken $ \alpha $ preformation factors into account with our improved formula, theoretical half-lives are in good agreement with experimental data, indicating our improved formula are reliable. The $ \alpha $-decay half-lives of even-even and odd-A nuclei of actinide are predicted. Specially, the predicted $ \alpha $-decay half-lives of the new isotopes $ ^{219,220}Np $ are also in good agreement with the experimental values. The predictions of $ \alpha $-decay half-lives can provide references for synthesis of new elements and microscopic studies of nuclear structure.

Reference (51)

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