Calculations of the α-decay properties of Z = 120, 122, 124, 126 isotopes

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Zhishuai Ge, Gen Zhang, Shihui Cheng, Yu. S. Tsyganov and Feng-Shou Zhang. Calculations of the α-decay properties of Z = 120, 122, 124, 126 isotopes[J]. Chinese Physics C.
Zhishuai Ge, Gen Zhang, Shihui Cheng, Yu. S. Tsyganov and Feng-Shou Zhang. Calculations of the α-decay properties of Z = 120, 122, 124, 126 isotopes[J]. Chinese Physics C. shu
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Calculations of the α-decay properties of Z = 120, 122, 124, 126 isotopes

    Corresponding author: Feng-Shou Zhang, fszhang@bnu.edu.cn
  • 1. Key Laboratory of Beam Technology of Ministry of Education, Beijing Radiation Center, Beijing 100875, China
  • 2. Key Laboratory of Beam Technology of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China
  • 3. Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, RU- 141980 Dubna, Russian Federation
  • 4. Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China

Abstract: The $\alpha$-decay properties of even-Z nuclei with Z = 120, 122, 124, 126 have been predicted. We employ the generalized liquid-drop model (GLDM), Royer's formula, the universal decay law (UDL) to calculate the $\alpha$-decay half-lives. By comparing the theoretical with experimental data of known nuclei from Fl to Og, we confirm that all the methods we used could reproduce the $\alpha$-decay half-lives well. The preformation factor $P_{\alpha}$ and $\alpha$-decay energy $Q_{\alpha}$ show that $^{298,304,314,316,324,326,338,348}$120, $^{304,306,318,324,328,338}$122, $^{328,332,340,344}$124 might be stable. The $\alpha$-decay half-lives show a peak at Z = 120, N = 184, and the peak would vanish when Z = 122, 124, 126. With detailed analysis of the competition of $\alpha$-decay and spontaneous fission, we predict that nuclei nearby the N = 184 undergo $\alpha$-decay. The decay-modes of $^{287-339}$120, $^{294-339}$122, $^{300-339}$124, $^{306-339}$126 have also been presented.

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    1.   Introduction
    • $ \alpha $-decay is one of the main decay modes of the superheavy nuclei (SHN). It was first observed by Rutherford and was explained as quantum tunneling process independently by Gamow [1] and by Condon and Gurney[2]. $ \alpha $-decay properties reflect the information on nuclear structure and nuclear stability. In the experiments, the $ \alpha $-decay chains are commonly used to identify the new synthesized SHN. To detect the “island of stability” [3-14], many SHN have been synthesized by the hot fusion reaction [15] and the cold fusion reaction [16]. Since the existence and stability SHN are mainly due to the shell effects, it is important to evaluate the magic numbers carefully and to calculate $ \alpha $-decay properties accurately [17-21].

      Many theoretical approaches have been proposed to describe $ \alpha $-decay process, such as the shell model, fission-like model, and cluster model[22-25]. There were many semiclassical models used to reproduce $ \alpha $-decay half-lives, such as the Generalized liquid drop model (GLDM) [26-28], the Coulomb and proximity potential model (CPPM) [29], the Unified fission model (UFM) [30], the Density-dependent cluster model (DDCM) [31], etc. Based on the Geiger-Nuttall law [32], there also exist many empirical relationships, such as the Viola-Seaborg formula [33, 34], the Brown formula [35], the Royer's formula [36], the universal decay law (UDL) [37, 38] to calculate the $ \alpha $-decay half-life. These methods described the tunnelling of the $ \alpha $-particle across the Coulomb barrier for heavy and super heavy nuclei very well. To describe $ \alpha $-decay in a fully microscopic way was difficult, but there were still many works considering microscopic modifications in the $ \alpha $-decay calculations [39-46].

      In this work, we use the GLDM with shell correction, the Royer's formula and the UDL to calculate the $ \alpha $-decay half-lives of even-Z superheavy nuclei with Z = 120,122,124,126. In the framework of GLDM, we adopt two methods to calculate $ \alpha $-preformation factor. The first way is to consider $ \alpha $-preformation factor as constant, which is fitted from experimental half-lives, for each type of nuclei (even-even, odd-A, odd-odd). The second method is to use cluster formation model (CFM) [47-50]. We adopt the updated Weizsäcker-Skyrme-4 (WS4) model to calculate $ Q_{\alpha} $ [51], since the accuracy of WS4 model has been generally certificated[52]. To predict the decay-modes, we use two modified shell-induced Swiatecki's formula to calculate theoretical SF half-lives. One empirical relation is produced by Santhosh and Nithya (KPS) [53, 54], the other one is modified by Bao et al.[55].

      The paper is assembled as follows. The sec. II introduces the theoretical framework. The results and corresponding discussions are presented in Sec. III. In the last section, the conclusions are given.

    2.   Theoretical Framework

      2.1.   $ \alpha $-Decay

      2.1.1.   GLDM
    • In the framework of the GLDM, the decay width is defined as $ \lambda $ = $ P_{\alpha} $$ \nu_{0} $P. The Wenzel-Kramers-Brillouin (WKB) approximation is used to calculate the barrier penetrability P,

      $ P = \exp \left[ { - \frac{2}{h}\int\limits_{R_{in} }^{R_{out} } {\sqrt {2B\left( r \right)\left( {E\left( r \right) - E\left( {sphere} \right)} \right)} dr} } \right], $

      (1)

      where $ E_{sphere} $ is the ground state energy of the parent nucleus. $ E(R_{in}) $ = $ E(R_{out}) $ = $ Q_{\alpha}^{exp} $, $ B(r) $ = $ \mu $, the parameter $ \mu $ is the reduced mass of the daughter nucleus and $ \alpha $-particle.

      The $ P_{\alpha} $ is $ \alpha $-preformation factor. The $ \nu_{0} $ is assault frequency which is calculated by [56]

      $ \nu _0 = \frac{1}{{2R}}\sqrt {\frac{{2E_\alpha }}{M_\alpha}}, $

      (2)

      where $ M_{\alpha} $ is mass, $ E_{\alpha} $ is the kinetic energy of the $ \alpha $ particle which has been corrected for recoil, and R is the radius of the parent nucleus.

      The model has considered shell corrections which is shape-dependent as follows, [57]

      $ E_{shell} = E_{shell}^{sphere} \left( {1 - 2.6\alpha ^2 } \right)e^{ - \alpha ^2 }, $

      (3)

      where $ \alpha ^2 = \left( {\delta R} \right)^2 /a^2 $ is the root mean square of the deviation, which includes all types of deformation, for the particle surface from the sphere. With the distortion of the nucleus increases, the whole shell correction energy comes to zero, owing to the attenuating factor $ e^{ - \alpha ^2 } $.

      The $ E^{sphere}_{shell} $ is defined as

      $ E_{shell}^{sphere} = cE_{sh}, $

      (4)

      which is the shell correction for a spherical nucleus. $ E_{sh} $ is the shell correction energy of which is calculated by the Strutinsky process [58]. The Strutinsky calculation use the smoothing parameter $ \gamma = 1.2\hbar \omega _0 $ and the order p = 6 of the Gauss-Hermite polynomials, $ \hbar \omega _0 = 41A^{ - 1/3} $ is the mean distance between the gross shells. The parameter c is scaled to adapt the separation of the binding energy between the macroscopic and the microscopic correction[59].

    • 2.1.2.   The $ \alpha $-preformation factor
    • The $ \alpha $-preformation factor $ P_{\alpha} $ are adopted from two methods. The first one is to take the same preformation factor for a certain kind of nuclei [60, 61]. We extract experimental $ P_{\alpha} $ values from nuclei with N $ \geqslant $ 152, Z $ \geqslant $ 82, make least squares fit to the experimental $ \alpha $-decay half-lives, then obtain the $ P_{\alpha} $ values: $ P_{\alpha} $ = 0.33 (even-even), $ P_{\alpha} $ = 0.05 (odd-A) $ P_{\alpha} $ = 0.01 (odd-odd). The results are consistent with results also extracted from GLDM by Ref. [62].

      Another method to obtain $ \alpha $-preformation factor is to use Cluster Formation Model (CFM)[47-50],

      $ P_\alpha = \frac{{E_{f\alpha } }}{E}, $

      (5)

      where $ E_{f\alpha} $ is the formation energy of the $ \alpha $ particle, E is the total energy combining the intrinsic energy for the $ \alpha $ particle and the interaction energy between the $ \alpha $ particle and the daughter nucleus.

      The $ E_{f\alpha} $ is calculated from separation energies[48, 49],

      $ E_{f\alpha } = \left\{ \begin{array}{l} 2S_p + 2S_n - S_c \left( {{\rm{even - even}}} \right), \\ 2S_p + S_{2n} - S_c \left( {{\rm{even - odd}}} \right), \\ S_{2p} + 2S_n - S_c \left( {{\rm{odd - even}}} \right), \\ S_{2p} + S_{2n} - S_c \left( {{\rm{odd - odd}}} \right), \\ \end{array} \right. $

      (6)

      $ E = S_c \left( {A,Z} \right), $

      (7)

      where $ S_{2n} $ is the two-neutron separation energy, $ S_{2p} $ is the two-proton separation energy, $ S_{c} $ is the $ \alpha $-particle separation energy,

      $ S_{2n} \left( {A,Z} \right) = B\left( {A,Z} \right) - B\left( {A - 2,Z} \right), $

      (8)

      $ S_{2p} \left( {A,Z} \right) = B\left( {A,Z} \right) - B\left( {A - 2,Z - 2} \right), $

      (9)

      $ S_{c} \left( {A,Z} \right) = B\left( {A,Z} \right) - B\left( {A - 4,Z - 2} \right), $

      (10)

      where B is the binding energy. The binding energy can be calculated from nucleus excess mass $ \Delta M $. Hence $ S_{2p} $, $ S_{2n} $ and $ S_{c} $ can be written as,

      $ S_{2p} \left( {A,Z} \right) = \Delta M\left( {A - 2,Z - 2} \right) - \Delta M\left( {A,Z} \right) + 2\Delta M_p , $

      (11)

      $ S_{2n} \left( {A,Z} \right) = \Delta M\left( {A - 2,Z} \right) - \Delta M\left( {A,Z} \right) + 2\Delta M_n , $

      (12)

      $ S_c \left( {A,Z} \right) = \Delta M\left( {A - 4,Z - 2} \right) - \Delta M\left( {A,Z} \right) + 2\Delta M_p + 2\Delta M_n . $

      (13)
    • 2.1.3.   Empirical formulas
    • The Royer's formula fit different kinds of nuclei to calculate $ \alpha $-decay half-lives [36]. For the even-even nuclei, the formula fit 131 even-even nuclei, the root mean square (RMS) deviation is 0.285,

      $ \log _{10} [T_{1/2} (s)] = - 25.31 - 1.1629A^{1/6} Z^{1/2} + 1.5864Z/\sqrt {Q_\alpha }. $

      (14)

      For the subset of 106 even-odd nuclei, the following equation was obtained (RMS deviation = 0.39)

      $ \log _{10} [T_{1/2} (s)] = - 26.65 - 1.0859A^{1/6} Z^{1/2} + 1.5848Z/\sqrt {Q_\alpha }. $

      (15)

      For the odd-even nuclei, 86 nuclei were adopted with RMS deviation of 0.36

      $ \log _{10} [T_{1/2} (s)] = - 25.68 - 1.1423A^{1/6} Z^{1/2} + 1.592Z/\sqrt {Q_\alpha }. $

      (16)

      For the odd-odd nuclei, 50 nuclei were used (RMS deviation = 0.35)

      $ \log _{10} [T_{1/2} (s)] = - 29.48 - 1.113A^{1/6} Z^{1/2} + 1.6971Z/\sqrt {Q_\alpha }. $

      (17)

      The UDL are also adopted to calculate $ \alpha $-decay half-lives [37, 38],

      $ \log _{10} [T_{1/2} (s)] = aZ_\alpha Z_d \sqrt {\frac{A}{{Q_\alpha }}} + b\sqrt {AZ_\alpha Z_d \left( {A_d^{1/3} + A_\alpha ^{1/3} } \right)} + c, $

      (18)

      where $ A = \frac{{A_d A_\alpha }}{{A_d + A_\alpha }} $, a = 0.4314, b = -0.4087, and c = -25.7725 which are determined from the experimental data.

    • 2.2.   Spontaneous fission

    • The spontaneous fission half-lives are calculated by semi-empirical formulas which are based on Swiatecki formula[63]. One formula is modified by Santhosh and Nithya[54] (KPS), the other one is from Bao et al. [55]. The two empirical relations both considered the isospin effect ($ \frac{{N - Z}}{{N + Z}} $), the fissionability parameter ($ \frac{{Z^2 }}{A} $) and the shell effect[53-55, 64].

      The KPS formula is defined as follows[53, 54],

      $ \begin{split} {\log _{10}}\left( {{T_{1/2}}/yr} \right) =& a\frac{{{Z^2}}}{A} + b{\left( {\frac{{{Z^2}}}{A}} \right)^2} + c\left( {\frac{{N - Z}}{{N + Z}}} \right)\\ &+ d{\left( {\frac{{N - Z}}{{N + Z}}} \right)^2} + e{E_{shell}} + f, \end{split}$

      (19)

      where a = -43.25203, b = 0.49192, c = 3674.3927, d = -9360.6, e = 0.8930 and f = 578.56058. $ E_{shell} $ is the shell correction energy from FRDM [65].

      The modified empirical formula from Bao et al. is determined as follows[55],

      $ \begin{split} \log _{10} [T_{1/2} (yr)] =& c_1 + c_2 \left( {\frac{{Z^2 }}{{\left( {1 - kI^2 } \right)A}}} \right)\\& + c_3 \left( {\frac{{Z^2 }}{{\left( {1 - kI^2 } \right)A}}} \right)^2 + c_4 E_{sh} + h_i, \end{split} $

      (20)

      where $ Z^{2}/(1 - kI^{2})A $ is the fissionability parameter considering the isospin effect. The constant value k = 2.6[36]. The coefficients $ c_1 $ = 1174.353441, $ c_2 $ = -47.666855, $ c_3 $ = 0.471307 and $ c_4 $ = 3.378848, which were fitted from 45 even-even nuclei. The blocking effect has also been considered by parameter $ h_{i} $, where $ h_{eo} $ = 2.609374 (even-odd), $ h_{oe} $ = 2.619768 (odd-even), $ h_{oo} $ = $ h_{eo} $ + $ h_{oe} $ (odd-odd), $ h_{ee} $ = 0 (even-even). The shell correction energy $ E_{sh} $ also takes from Ref. [65].

    3.   Results and discussion
    • Table 1 presents the $ \alpha $-decay half-lives of known nuclei from Fl to Og calculated by GLDM, UDL and Royer's formula. Because these nuclei are regarded as “upper super heavy region”[66] and have been produced by the hot-fusion reactions. The $ P_{\alpha} $ adopted in GLDM is through a least squares fitting to the experimental half-lives for known SHN from N $ \geqslant $ 152 and Z $ \geqslant $ 82. The experimental $ Q_{\alpha} $ values are from Ref.[67]. We use standard deviation to compare the calculation results and the experimental values,

      Ele.A $Q_{\alpha}^{exp.}$(MeV)$T_{1/2}^{exp.}$(s)$T_{1/2}$(s)$T_{1/2}$(s)$T_{1/2}$(s)$T_{1/2}$(s)
      RoyerUDLGLDMGLDM$_{shell}$
      Fl28510.56 $\pm$ 0.051.00$\times$10$^{-1 }$1.60$\times$10$^{-1 }$4.27$\times$10$^{-2 }$6.61$\times$10$^{-2 }$6.57$\times$10$^{-2 }$
      28610.35 $\pm$ 0.041.20$\times$10$^{-1 }$1.08$\times$10$^{-1 }$1.62$\times$10$^{-1 }$3.16$\times$10$^{-2 }$3.37$\times$10$^{-2 }$
      28710.17 $\pm$ 0.024.80$\times$10$^{-1 }$1.68$\times$10$^{0 }$5.25$\times$10$^{-1 }$5.67$\times$10$^{-1 }$6.67$\times$10$^{-1 }$
      28810.07 $\pm$ 0.036.60$\times$10$^{-1 }$5.93$\times$10$^{-1 }$1.01$\times$10$^{0 }$1.47$\times$10$^{-1 }$1.99$\times$10$^{-1 }$
      2899.98 $\pm$ 0.021.90$\times$10$^{0 }$5.34$\times$10$^{0 }$1.82$\times$10$^{0 }$1.60$\times$10$^{0 }$2.55$\times$10$^{0 }$
      Mc28710.76 $\pm$ 0.053.70$\times$10$^{-2 }$4.70$\times$10$^{-2 }$2.60$\times$10$^{-2 }$4.06$\times$10$^{-2 }$3.85$\times$10$^{-2 }$
      28810.65 $\pm$ 0.011.74$\times$10$^{-1 }$4.49$\times$10$^{-1 }$5.05$\times$10$^{-2 }$3.57$\times$10$^{-1 }$3.47$\times$10$^{-1 }$
      28910.49 $\pm$ 0.053.30$\times$10$^{-1 }$2.23$\times$10$^{-1 }$1.37$\times$10$^{-1 }$1.67$\times$10$^{-1 }$1.82$\times$10$^{-1 }$
      29010.41 $\pm$ 0.046.50$\times$10$^{-1 }$2.00$\times$10$^{0 }$2.25$\times$10$^{-1 }$1.26$\times$10$^{0 }$1.51$\times$10$^{0 }$
      Lv29011 $\pm$ 0.078.30$\times$10$^{-3 }$8.94$\times$10$^{-3 }$1.21$\times$10$^{-2 }$3.00$\times$10$^{-3 }$2.88$\times$10$^{-3 }$
      29110.89 $\pm$ 0.071.90$\times$10$^{-2 }$8.94$\times$10$^{-2 }$2.31$\times$10$^{-2 }$3.41$\times$10$^{-2 }$3.51$\times$10$^{-2 }$
      29210.78 $\pm$ 0.021.30$\times$10$^{-2 }$3.01$\times$10$^{-2 }$4.46$\times$10$^{-2 }$8.84$\times$10$^{-3 }$1.04$\times$10$^{-2 }$
      29310.71 $\pm$ 0.025.70$\times$10$^{-2 }$2.41$\times$10$^{-1 }$6.72$\times$10$^{-2 }$8.01$\times$10$^{-2 }$1.04$\times$10$^{-1 }$
      Ts29311.32 $\pm$ 0.052.20$\times$10$^{-2 }$6.89$\times$10$^{-3 }$3.57$\times$10$^{-3 }$6.59$\times$10$^{-3 }$6.65$\times$10$^{-3 }$
      29411.18 $\pm$ 0.045.10$\times$10$^{-2 }$7.25$\times$10$^{-2 }$7.98$\times$10$^{-3 }$6.45$\times$10$^{-2 }$7.23$\times$10$^{-2 }$
      Og29411.82 $\pm$ 0.065.80$\times$10$^{-4 }$3.67$\times$10$^{-4 }$4.26$\times$10$^{-4 }$1.64$\times$10$^{-4 }$1.60$\times$10$^{-4 }$
      0.380.390.350.35

      Table 1.  The experimental and theoretical $\alpha$-decay half-lives of known SHN from Fl to Og. The theoretical results are calculated by Royer's formula, the UDL, the GLDM with and without shell corrections by inputting the experimental $Q_{\alpha}$ [67]. The $P_{\alpha}$ adopted in GLDM is constant which is fitted from experimental data ($P_{\alpha}$ = 0.33 for even-even nuclei, $P_{\alpha}$ = 0.05 for odd-A nuclei, $P_{\alpha}$ = 0.01 for odd-odd nuclei). The $\sigma$ is the standard deviation between the experiments and the theoretical calculations by eq.(21).

      $ \sigma {\rm{ = }}\left[ {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {\left( {\log _{10} T_{1/2}^{theo.} - \log _{10} T_{1/2}^{\exp .} } \right)^2 } } \right]^{1/2} . $

      (21)

      The $ \sigma $ values of the Royer's formula, the UDL, the GLDM and GLDM with shell correction, are 0.38, 0.39, 0.35, 0.35 respectively. The effect of shell correction is more obvious for nuclei near the predicted shell-closure[68]. Take $ ^{289} $Fl for example, the $ T_{\alpha}^{1/2} $ would be increased from 0.32s to 0.51s.

      The results by GLDM are systematically lower than the experimental data. After shell correction, the calculated $ \alpha $-decay half-lives would be slightly increased. The $ \sigma $ values indicate that the with experimental fit constant $ P_{\alpha} $, models with and without shell correction would all present accuracy calculation of $ \alpha $-decay half-lives.

    • 3.1.   $ \alpha $-preformation factor

    • The $ \alpha $-preformation factors have been calculated by cluster formation model (CFM)[48, 49]. The $ Q_{\alpha} $ and $ P_{\alpha} $ are both extracted from WS4 model [51]. The $ Q_{\alpha} $ and $ P_{\alpha} $ values of even-even nuclei from Z = 120 to 126 have been plotted in figure 1. The $ P_{\alpha} $ values of even-even nuclei are aroud 0.1 - 0.3, which satisfy the experimental general features [49, 69]. The figure shows that $ Q_{\alpha} $ values decrease with larger neutron numbers, which means the stability against $ \alpha $-decay of the nucleus is increasing. The $ Q_{\alpha} $ and $ P_{\alpha} $ have much similar trends.

      Figure 1.  The preformation factors of Z = 120,122,124,126 even-even nuclei.

      The discontinuity of $ Q_{\alpha} $ represents the position of magic numbers. Moreover, in the region where $ P_{\alpha} $ value is relatively small, the nuclei are regarded to be stable[70]. However the positions of $ P_{\alpha} $ discontinuity and $ Q_{\alpha} $ discontinuity are not particularly the same, as shown from Z = 120 even-even isotopes in fig 1(a). This is because the $ P_{\alpha} $ value of one nucleus is calculated by five nuclei around it. The $ P_{\alpha} $ values may contain complex structure information of several nearby nuclei.

      We use $ Q_{\alpha} $ as well as $ P_{\alpha} $ values to predict the stable nuclei for Z = 122 - 126 elements. Figure 1(a) shows that for Z = 120, nuclei around N = 178,184,194,196,204,206,218,228 might be stable. For Z = 122, nuclei with N = 182,184,196,202,206,216 show higher stability. For Z = 124 nuclei, the nuclei with N = 204,208,216,220 maybe stable against $ \alpha $-decay. Figure 1(d) indicates that the Z = 126 even-even nuclei have no obvious shell structures. Because the $ Q_{\alpha} $ of Z = 126 isotopes are smoothly continuous, and the $ P_{\alpha} $ distribution has no dips. We find that when the atomic number increases, the neutron numbers of stable nuclei would also increase. It seems that with larger proton numbers, the nucleus need more neutrons to keep stable.

    • 3.2.   $ \alpha $-decay properties of Z = 120,122,124,126 isotopes

    • Figure 2 present the $ \alpha $-decay half-lives of Z = 120,122,124,126 even-even isotopes. This figure shows that at N $ < $ 186, the $ \alpha $-decay half-lives are getting large with increasing nuclear mass. This phenomenon indicates that this might be a shell closure at N $ < $ 186. For Z = 120 nuclei, there is one obvious peak at N = 184. However this peak would be gradually vanished when Z values getting large. The $ \alpha $-decay half-lives indicate that the neutron magic number at N = 184 could not be observed at Z = 122,124,126. This phenomenon is consistent with the results shown by $ P_{\alpha} $ and $ Q_{\alpha} $ in Fig. 1. For Z = 122, nuclei with N = 182 and 184 both have relatively longer half-lives, see Fig. 2(b). The corresponding $ Q_{\alpha} $ and $ P_{\alpha} $ values in Fig. 1(b) are relatively small. Hence for elements Z = 120,122,124, and 126 isotopes, $ ^{304} $120 would be probably stable and might be a shell closure.

      Figure 2.  The $ \alpha $-decay half-lives of even-even isotopes of Z = 120,122,124,126.

      The $ \alpha $-decay half-lives and SF half-lives of $ ^{287-339} $120, $ ^{294-339} $122, $ ^{300-339} $124, $ ^{306-339} $126 are presented in Table 2. To identify the decay-modes of unknown nuclei, the competition of $ \alpha $-decay and spontaneous fission is studied [71-77]. The predicted decay-modes of nuclei in the last column of Table 2. The two SF equations both consider the shell correction. However the SF half-lives calculated by eq.(20) would be more sensitive to the nuclear structures[78]. The results show that most nuclei at around N = 184 would undergo $ \alpha $-decay. With larger Z, the competition of $ \alpha $-decay and SF would be more obvious. By comparing $ \alpha $-decay and SF half-lives, we predict that $ ^{287-307} $120 would have $ \alpha $-decay, $ ^{308-309} $120 have both $ \alpha $-decay and SF, $ ^{310-339} $120 go to SF. The $ ^{294-309} $122 isotopes would have $ \alpha $-decay, $ ^{310-314} $122 have two decay modes, $ ^{315-339} $122 have SF. For Z = 124 nuclei, $ ^{300-315} $124 are $ \alpha $-decay, $ ^{316-320,326,327,331} $124 would have both $ \alpha $-decay and SF, $ ^{321-325,328-330,332-339} $124 undergo SF. The competition of two decay modes for $ ^{328-339} $126 isotopes is very obvious, hence $ ^{328-335,337,339} $126 have both $ \alpha $-decay and SF, $ ^{336,338} $126 would have SF, $ ^{306-327} $126 undergo $ \alpha $-decay.

      ZA$Q_{\alpha}^{WS4}$(MeV)$T_{1/2}^{\alpha}\;({\rm s})$$T_{1/2}^{\alpha}\;({\rm s})$$T_{1/2}^{\alpha}\;({\rm s})$$T_{1/2}^{\alpha}\;({\rm s})$$T_{1/2}^{SF}\;({\rm s})$$T_{1/2}^{SF}\;({\rm s})$Decay-mode
      RoyerUDLGLDMGLDM$_{P_{\alpha}}$eq.(20)[55]KPS[54]
      12028713.857.90E-078.96E-081.12E-064.46E-073.39E+031.03E+10$\alpha$
      28813.732.18E-071.53E-072.62E-073.39E-071.68E+015.83E+10$\alpha$
      28913.711.31E-061.55E-071.79E-067.17E-071.70E+054.73E+11$\alpha$
      29013.702.23E-071.59E-072.82E-073.72E-073.45E+021.16E+12$\alpha$
      29113.512.96E-063.73E-073.75E-061.50E-065.63E+053.19E+12$\alpha$
      29213.475.65E-074.36E-076.66E-078.62E-071.76E+034.86E+12$\alpha$
      29313.404.41E-065.77E-075.28E-062.26E-061.65E+071.20E+13$\alpha$
      29413.241.43E-061.19E-061.52E-062.06E-064.25E+049.94E+12$\alpha$
      29513.277.26E-069.91E-077.85E-063.29E-061.50E+081.10E+13$\alpha$
      29613.348.30E-076.79E-078.70E-071.20E-062.84E+042.66E+12$\alpha$
      29713.141.21E-051.72E-061.15E-055.32E-062.37E+071.18E+12$\alpha$
      29813.013.56E-063.24E-062.90E-064.24E-066.02E+043.40E+11$\alpha$
      29913.266.56E-069.08E-076.53E-062.72E-062.58E+077.66E+10$\alpha$
      30013.327.82E-076.59E-077.40E-071.01E-063.80E+036.33E+09$\alpha$
      30113.061.48E-052.18E-061.23E-055.16E-061.67E+068.84E+08$\alpha$
      30212.895.21E-065.02E-063.73E-065.17E-061.17E+023.73E+07$\alpha$
      30312.814.53E-057.25E-063.15E-051.25E-053.32E+042.91E+06$\alpha$
      30412.768.79E-068.89E-065.13E-067.12E-065.87E-015.42E+04$\alpha$
      30513.284.74E-066.64E-073.56E-061.45E-063.40E-015.27E+02$\alpha$
      30613.797.76E-085.94E-087.03E-089.47E-082.24E-064.88E+00$\alpha$
      30713.521.48E-061.94E-071.15E-064.72E-076.53E-058.70E-02$\alpha$
      30812.972.84E-062.76E-061.44E-061.78E-063.20E-081.65E-03$\alpha$/SF
      30912.169.06E-041.81E-042.97E-041.09E-049.87E-063.64E-05$\alpha$/SF
      31011.504.88E-037.72E-031.10E-031.60E-031.32E-093.28E-07SF
      31111.201.68E-014.73E-023.47E-021.71E-022.43E-074.25E-09SF
      31211.222.27E-024.02E-024.20E-036.97E-031.79E-112.22E-11SF
      31311.024.26E-011.29E-017.60E-023.96E-025.39E-092.24E-13SF
      31410.763.29E-016.99E-014.84E-021.75E-015.15E-138.66E-16SF
      3159.431.73E+041.04E+047.27E+033.99E+031.86E-106.38E-18SF
      3169.191.71E+047.33E+048.70E+032.09E+043.41E-142.04E-20SF
      3179.934.26E+022.05E+021.28E+027.12E+017.92E-129.44E-23SF
      3189.936.57E+012.01E+021.88E+013.53E+011.74E-152.26E-25SF
      3199.847.35E+023.70E+022.20E+021.28E+024.86E-137.81E-28SF
      3209.683.68E+021.28E+031.18E+022.16E+021.75E-161.53E-30SF
      3219.536.77E+033.97E+032.34E+031.36E+033.02E-136.21E-33SF
      3229.373.44E+031.40E+041.28E+032.42E+031.12E-168.92E-36SF
      3239.121.48E+051.07E+056.75E+043.73E+046.68E-142.01E-38SF
      Continued on next page

      Table 2.  The theoretical $\alpha$-decay half-lives and SF half-lives of $^{287-339}$120, $^{294-339}$122, $^{300-339}$124, $^{306-339}$126 isotopes. The $Q_{\alpha}^{th.}$ values are extracted from WS4 model [51]. The columns (4-7) are $\alpha$-decay half-lives calculated by the Royer's formula, the UDL, the GLDM with shell correction, the GLDM with shell correction and CFM $P_{\alpha}$. The column (8-9) present SF half-lives calculated by eq.(20)[55] and by KPS equation [54]. The last column lists predicted decay-modes.

      We also use FRDM $ Q_{\alpha} $ values to calculate $ \alpha $-decay half-lives and put the results in Table 3. For Z = 120 isotopes, $ ^{296-307} $120 undergo $ \alpha $-decay, $ ^{308} $120 may has both $ \alpha $-decay and SF, $ ^{309-327} $ would have SF. For Z = 122 nuclei, $ ^{300-309,311} $122 would probably undergo $ \alpha $-decay, $ ^{310,312-315} $122 may have both two decay-modes, $ ^{316-331} $122 have SF. The $ ^{304-315,317} $124 isotopes are probably $ \alpha $-decay, $ ^{316,318-320,327} $124 have both $ \alpha $-decay and SF, $ ^{321-335} $124 would undergo SF. For Z = 126, $ ^{308-322,325} $ may have $ \alpha $-decay, $ ^{323,326-335,337,339} $126 would probably have two decay-modes, and $ ^{324,336,338} $126 would have SF decay mode. Since the adopted $ Q_{\alpha} $ values are different for Table 2 and Table 3, the theoretical $ \alpha $-decay half-lives are slightly different. However, the predicted decay-modes from two sets of results are mostly similar. Both FRDM and WS4 model are capable to provide accurate $ Q_{\alpha} $ values for $ \alpha $-decay calculations.

      ZA$Q_{\alpha}^{FRDM}$(MeV)$T_{1/2}^{\alpha}\;({\rm s}$$T_{1/2}^{\alpha}\;({\rm s}$$T_{1/2}^{SF}\;({\rm s}$$T_{1/2}^{SF}\;({\rm s}$Decay-mode
      GLDMGLDM$_{P_{\alpha}}$eq.(20)[55]KPS [54]
      12029613.593.80E-077.40E-072.84E+042.66E+12$\alpha$
      29713.651.99E-061.30E-062.37E+071.18E+12$\alpha$
      29813.241.42E-062.43E-066.02E+043.40E+11$\alpha$
      29913.741.31E-065.87E-072.58E+077.66E+10$\alpha$
      30013.692.20E-073.75E-073.80E+036.33E+09$\alpha$
      30113.621.83E-061.01E-061.67E+068.84E+08$\alpha$
      30213.563.18E-075.64E-071.17E+023.73E+07$\alpha$
      30313.522.23E-061.24E-063.32E+042.91E+06$\alpha$
      30413.552.38E-074.41E-075.87E-015.42E+04$\alpha$
      30514.261.16E-075.92E-083.40E-015.27E+02$\alpha$
      30614.271.49E-082.06E-082.24E-064.88E+00$\alpha$
      30713.628.93E-072.30E-076.53E-058.70E-02$\alpha$
      30812.971.58E-061.58E-063.20E-081.65E-03$\alpha$/SF
      30911.762.43E-038.53E-049.87E-063.64E-05SF
      31011.284.18E-037.11E-031.32E-093.28E-07SF
      31110.765.08E-013.32E-012.43E-074.25E-09SF
      31210.719.12E-021.87E-011.79E-112.22E-11SF
      31310.502.09E+001.33E+005.39E-092.24E-13SF
      Continued on next page

      Table 3.  The theoretical $\alpha$-decay half-lives and SF half-lives of $^{296-327}$120, $^{300-331}$122, $^{304-335}$124, $^{308-339}$126 isotopes. The $Q_{\alpha}^{th.}$ values are extracted from FRDM [65]. The columns (4,5) are $\alpha$-decay half-lives calculated by the GLDM with shell correction, the GLDM with shell correction and CFM $P_{\alpha}$. The columns (6,7) present SF half-lives calculated by eq.(20)[55] and by KPS equation [54]. The last column lists predicted decay-modes.

    • 3.3.   Comparison with other works

    • We compare our results with results calculated by phenomenological models [78, 79]. For the $ \alpha $-decay half-lives using FRDM $ Q_{\alpha} $ values, we compare our results with results from Ref.[78]. The $ \alpha $-decay and SF half-lives are shown in Figure 3. The results show that the SF half-lives calculated by modified equation by Bao et al.[55, 78] have even-odd effect. This is because in eq.(20), the blocking effect of unpaired nucleon has been considered. The SF half-lives show a trend that with increasing A, the $ \log_{10} $$ T_{1/2}^{SF} $ values are decreasing. It seems that the SF equation modified by Refs. [55, 78] is more sensitive to nuclear strucure[78]. The $ \alpha $-decay half-lives and SF half-lives by this work and by Ref.[78] are slightly different. This is because we use FRDM2016 [65] to calculate the $ Q_{\alpha} $ and shell correction, the results from Ref.[78] uses FRDM1995 [80]. However, the predicted decay-modes for most nuclei are the same.

      Figure 3.  The $ \alpha $-decay half-lives and SF half-lives of $ ^{296-308} $120, $ ^{300-310} $122, $ ^{304-312} $124. The $ \log_{10} $$ T_{1/2}^{\alpha} $ values calculated by the UDL and the GLDM are from Ref.[78].

      We compare $ \alpha $-decay half-lives calculated by WS4 $ Q_{\alpha} $ values with results from Ref.[79]. The $ \alpha $-decay half-lives from this work and Ref.[79] are presented in Figure 4. The $ \log_{10} $$ T_{1/2}^{\alpha} $ values by using the Coulomb and proximity potential model for deformed nuclei (CPPMDN) and the Coulomb and proximity potential model (CPPM) are from Ref.[79]. The SF half-lives calculated by KPS equation [54] are exactly the same, and decrease smoothly with increasing A for Z = 120,122 isotopes. For Z = 124 and 126 nuclei, the SF half-lives also show the similar trend, which is consistent with results presented in Figure 3. For $ ^{319-322} $124, $ ^{326-329} $126 isotopes, the competition between $ \alpha $-decay and SF is obvious, indicating that these nuclei may have two decay-modes. The results show that with similar $ Q_{\alpha} $ values, different phenomenological models show good consistency.

      Figure 4.  The $ \alpha $-decay half-lives and SF half-lives of $ ^{295-309} $120, $ ^{301-314} $122, $ ^{307-323} $124, $ ^{313-331} $126. The $ \log_{10} $$ T_{1/2}^{\alpha} $ values calculated by the Coulomb and proximity potential model(CPPM), and the Coulomb and proximity potential model for deformed nuclei (CPPMDN) are from Ref.[79].

      Since we use fully phenomenological approach, we compare our results with results by calculations considering microscopic modifications [45]. As generally known the $ Q_{\alpha} $ values deduced would influence the calculated $ \alpha $-decay half-lives obviously. With 1 MeV change of $ Q_{\alpha} $ value may lead to a change of $ \log_{10} $$ T_{1/2}^{\alpha} $ value of around 3 orders of magnitude or more. In Ref.[45], they use different mass tables to calculate $ Q_{\alpha} $, including WS4 mass table. Hence, we compare our $ \log_{10} $$ T_{1/2}^{\alpha} $ with $ \log_{10} $$ T_{1/2}^{\alpha} $ calculated from WS4 mass model by Ref.[45]. In Fig. 3 from Ref.[45], the $ \log_{10} $$ T_{1/2}^{\alpha} $ values of Z = 120,122,124 nuclei would have dips at $ N_{d} $ = 184, where $ N_{d} $ represents the neutron number of daughter nuclear. The Fig. 2 in this work produce the same trend for the $ \alpha $-decay half-lives. The discussion above indicate that with similar $ Q_{\alpha} $ values, the results by phenomenological approach is much consistent with the results by calculations considering microscopic modifications[81].

    4.   Summary
    • We use shell correction induced GLDM to calculate the $ \alpha $-decay half-lives of Z = 120,122,124,126 isotopes. The preformation factor $ P_{\alpha} $ in the model have two types, one is constant for each type of nuclei which is adopted from least-quares fitting to the known experimental half-lives (N $ \geqslant $ 152, Z $ \geqslant $ 82). The other type is calculated by CFM. We compare our calculations with experimental data for the known nuclei from Fl to Og, find that all the methods we used would reproduce $ \alpha $-decay half-lives well. We then expend our method to predict the $ \alpha $-decay properties of the even-Z SHN from Z = 120 to 126.

      The theoretical $ P_{\alpha} $ values calculated by CFM are much sensitive to nuclear structure. The $ P_{\alpha} $ values and $ Q_{\alpha} $ values show similar trends. They both reflect the position of shell structures. However the $ P_{\alpha} $ contains more complex shell structure information since it is adopted from several nearby nuclei. From $ Q_{\alpha} $ and $ P_{\alpha} $ values, we present some nuclei which might be stable, i.e., Z = 120, N = 178,184,194,196,206,218,228; Z = 122, N = 182,184,196,202,206,216; and Z = 124, N = 204,208,216,220. With larger proton numbers, the more neutrons is needed for a nucleus to keep stable.

      With the information of $ \alpha $-decay half-lives, we find that at N = 184, there is no obvious shell structure for Z = 122,124,126 isotopes. $ ^{304} $120 nucleus is predicted to be stable comparing with the nearby nuclei. The competition between $ \alpha $-decay and SF is more and more evident from Z = 120 to 126. However, nuclei at around N = 184 would mostly have $ \alpha $-decay. The predicted decay modes for $ ^{287-339} $120, $ ^{294-339} $122, $ ^{300-339} $124, $ ^{306-339} $126 are presented in Table 2.

      We compare our results with other works, including the results by microscopic calculations. The comparisons show that the phenomenological method and the microscopic method would give much similar $ \alpha $-decay half-lives, when similar $ Q_{\alpha} $ values are adopted. It is suggested to select suitable $ Q_{\alpha} $ values, since the $ Q_{\alpha} $ values would influence the calculations obviously.

      The authors acknowledge support by Key Laboratory of Beam Technology of Ministry of Education, Beijing Normal University.

Reference (81)

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