Search for decay modes of heavy and superheavy nuclei

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A. Soylu. Search for decay modes of heavy and superheavy nuclei[J]. Chinese Physics C, 2019, 43(7): 074102. doi: 10.1088/1674-1137/43/7/074102
A. Soylu. Search for decay modes of heavy and superheavy nuclei[J]. Chinese Physics C, 2019, 43(7): 074102.  doi: 10.1088/1674-1137/43/7/074102 shu
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Search for decay modes of heavy and superheavy nuclei

    Corresponding author: A. Soylu, asimsoylu@gmail.com
  • Department of Physics, Nigde Ömer Halisdemir University, Nigde 51240, Turkey

Abstract: Spontaneous fission (SF) with a new formula based on a liquid drop model is proposed and used in the calculation of the SF half-lives of heavy and superheavy nuclei (Z = 90–120). The predicted half-lives are in agreement with the experimental SF half-lives. The half-lives of $ \alpha $ decay (AD) for the same nuclei are obtained by using the Wentzel-Kramers-Brillouin (WKB) method together with Bohr-Sommerfeld (BS) quantization condition considering the isospin-dependent effects for the cosh potential. The decay modes and branching ratios of superheavy nuclei (Z = 104-118) with experimental decay modes are obtained, and the modes are compared with the experimental ones and with the predictions found in the literature. Although some nuclei have predicted decay modes that are different from their experimental decay modes, decay modes same as the experimental ones are predicted for many nuclei. The SF and AD half-lives, branching ratios, and decay modes are obtained for superheavy nuclei (Z = 119–120) with unknown decay modes and compared with the predictions obtained in a previous study. The present results provide useful information for future experimental studies performed on both the AD and SF of superheavy nuclei.

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    1.   Introduction
    • The $ \alpha $ decay (AD) of nuclei is an important tool to study the structure of light, medium, and heavy, as well as superheavy nuclei produced at the accelerator centers around the world [18]. When superheavy nuclei are produced in the laboratory, they transition from the excited state to the ground state through the $ \alpha $ decay chains. The observation and counting of these $ \alpha $ particles provides information about the identification of the new synthesized superheavy nuclei. The spontaneous fission (SF) is another key decay energetically feasible for heavy and superheavy nuclei ($ Z\geqslant 90 $) [9, 10]. It was first proposed by Bohr and Wheeler [11], and subsequently observed by Flerov and Petrjak [12]. Since the SF of 238U was discovered, many actinide nuclei with this type of radioactive decay have been reported in the experiments [13]. More recently, the SF half-lives of many superheavy nuclei were observed in different laboratories [1416]. In fact, SF is an important limiting factor that describes the stability of synthesized superheavy nuclei. Theoretically, AD and SF comprise the same physical mechanism, i.e., the quantum mechanical tunneling effect.

      In recent years, the studies on the AD of heavy and superheavy nuclei have become interesting and popular [1726]. Many models and methods have been applied to investigate the AD of nuclei, such as the liquid drop model [27, 28], the cluster model [29], empirical formulas [3032], and others [3335]. It is difficult to model the interaction between nucleons in nuclei, as nucleus is a many-body system that contains numerous nucleons. In the cluster model, especially in terms of binary clustering, the many-body system can be reduced to a two-body (the core and surrounding $ \alpha $ particle) system, and the problem can be easily solved [36, 37]. In this sort of model, $ \alpha $ is assumed as to already exist in the nucleus before the decay, and it can be tunneled through the Coulomb barrier. This phenomena can be described as quantum tunneling. In the Gamow model, a formula between the half-life and Q-value was proposed, and followingly this relation and the formula were also produced by Geiger and Nuttall [3840]. More recently, a two-potential approach has been applied to calculate the half-lives of ADs for even-even nuclei [41], odd-A nuclei [42], and doubly odd nuclei [43]. In these studies, authors have used cosh-type nuclear potential including the isospin effects to calculate the AD half-lives of nuclei.

      The SF half-lives were obtained by using macroscopic-microscopic methods over the deformation parameters and nuclear shapes [4447]. Because the case of SF is more complex than the AD, and many difficulties in the fission arise such as the mass and charge numbers of the two fragment nuclei and the number of emitted neutrons [48], the complete microscopic explanation of such a multidimensional system is extremely hard. The most realistic calculations of the SF half-lives can be performed by investigation of the multidimensional deformation space [45, 46]. Another method applied to calculating the SF half-lives is the phenomenological technique. A systematic study of the relation between the proton number (Z) and the mass number (A), as well as the half-lives, should make it possible to achieve a deep understanding of this phenomenon. There are different models employed to compute the SF half-lives in the literature [49, 50]. A semi-empirical formula was proposed by Swiatecki [51], upon which it was applied to obtain the SF half-lives of even-even, odd-A, and odd-odd nuclei. By using this formula, the author successfully reproduced the experimental data. Recently, a generalized Swiatecki formula [52, 53] with a set of new parameters was used to reproduce the experimental SF half-lives of the heavy and superheavy nuclei. Another possible decay mode in this region is the multicluster-accompanied fission, investigated in Ref. [54].

      The study in Ref. [55] has shown that among the formulae used to calculate AD half-lives, the SemFIS2 formula performs the best in this prediction. In addition, the UNIV2 formula with the fewest parameters, as well as the VSS, SP and NRDX formulas with fewer parameters work well in the prediction of the AD half-lives of superheavy nuclei [5664]. With regard to the cluster decay, there are many different studies on the calculations of cluster decay half-lives of nuclei considering various approaches in the literature [6569].

      Xu et al. [70] systematically investigated the AD and SF half-lives for heavy and superheavy nuclei with a proton number $ Z\geqslant 90 $. The AD half-lives were obtained by the deformed version of the density-dependent cluster model (DDCM). The SF half-lives of nuclei from 232Th to 286114 were calculated with the parabolic potential approximation by considering the nuclear structure effects. The competition between the AD and SF was analyzed in detail, and the branching ratios of these two decay modes were predicted for the unknown cases.

      Bao et al. [71] obtained the AD half-lives of superheavy nuclei within the framework of the unified fission model (UFM) and the analytical formula. A modified formula based on Swiatecki's formula was proposed for explaining of the SF half-lives, which included the shell correction and isospin effect terms inside. The stability of superheavy nuclei against AD and SF, as well as the competition between them, were discussed. For nuclei with Z = 119–120, they interpreted the existing experimental decay modes and predicted decay modes of yet unknown nuclei.

      Santhosh et al. [72] attempted to reproduce the experimental AD half-lives and modes of the decay of superheavy nuclei with the Coulomb and proximity potential model for deformed nuclei (CPPMDN), which is a deformed version of the Coulomb and proximity potential model (CPPM). A modified formula was proposed to obtain the SF half-lives by including the microscopic shell correction in the formula. A complete theoretical analysis on the half-lives was conducted, and the decay modes of experimentally synthesized superheavy nuclei were obtained for the first time. More recently, Santhosh et al. have predicted the decay modes and half lives of all even Z isotopes of the superheavy elements within the range $ 104\leqslant Z\leqslant136 $, and they have compared the results of the AD half-lives with the SF half-lives [73].

      The aim of the present study is to perform a comprehensive investigation of both the AD and SF half-lives and to predict decay modes for superheavy nuclei with the known and yet unknown experimental decay modes. The half-lives are obtained for superheavy nuclei ( Z = 104–118), for the SF with the new formula and for the AD using the WKB method together with the BS quantization condition for cosh potential, including the isospin effects. The new formula is used to obtain the SF values of nuclei with Z = 108–120 and the logarithmic values of the SF. These are then compared with the results of other models. Branching ratios for the SFs and ADs are calculated, and the modes of decays are predicted for Z = 104–118 nuclei, which have the known experimental decay modes. The predictions are in good agreement with experiment. Branching ratios for the SF and ADs are likewise obtained, and the decay modes are predicted for Z = 119–120 nuclei with experimental decay modes that are still unknown. Different decay modes from the predicted ones in the literature are obtained for some nuclei.

      In Section 2, the theoretical background and equations required for the SF and AD half-lives in the WKB method together with BS are presented. The obtained numerical results and discussion can be found in Section 3. Section 4 is devoted to discussion.

    2.   Theoretical models

      2.1.   Spontaneous fission

    • Although the SF process is described as the quantum tunneling effect in physics, it is difficult to solve such a multidimensional penetration problem. This problem can be simplified to a one-dimensional WKB approach. Similarly to the AD, the only unknown term is the potential, and the so-called Hill-Wheeler formula can be obtained in a parabolic potential [70]. By modeling the potential, the following expression of spontaneous fission was given by Xu et al. [70]

      $ T_{\rm SF} = \frac{{\rm ln}2}{n.P_{\rm SF}} = e^{{2\pi[c_{0}+c_{1}A+c_{2}Z^{2}+c_{3}Z^{4}+c_{4}(N-Z)^{2}-Q_{\rm SF}]}}, $

      (1)

      where $ Q_{\rm SF} = 0.13323\displaystyle\frac{Z^{2}}{A^{1/3}}-11.64 $. Eq. (1) has five parameters that were obtained from fitting to the experimental SF half-lives of 45 even-even nuclei from 232Th to 286114. These values are given in Ref. [70] as $ c_{0} = $ $ -195.09227 $, $ c_{1} = 3.10156 $, $ c_{2} = -0.04386 $, $ c_{3} = 1.40301\times $ $10^{-6} $, $ c_{4} = -0.03199 $.

      In this study, moving from the idea of Xu et al. [70], a new function is proposed and used by establishing similarity with the nuclear liquid drop model. It is given by

      $ T_{\rm SF} = e^{2\pi[aA+bA^{2/3}+cZ(Z-1)/A^{1/3}+d(N-Z)^{2}/A+eZ^{4}+f]}, $

      (2)

      where Z, N and A are the proton, neutron and mass numbers of the parent nuclei, and a, b, c, d, e, f are the adjustable parameters that can be obtained by fitting to experimental SF half-lives. This equation is given in terms of years.

      Eq. (2) is a new semi-empirical formula proposed for spontaneous fission half-lives. It can be considered as the modified form of the formula of Xu et al. [70]. Hence, this formula was inspired by the binding energy formula of the liquid drop model. Each term in the liquid drop model is assumed to correspond to a change of the SF half-lives with Z, N, and A. The aA term, i.e., the volume effect, is used to model an increase in SF half-lives with A. The $ bA^{2/3} $ term, which depicts the surface effect, shows an increase of SF half-lives proportional to $ A^{2/3} $. The $cZ(Z-1)/ $$ A^{1/3} $ term, depicting the Coulombic effect, is used to model an increase in SF with $ Z(Z-1)/A^{1/3} $, and the $ d(N-Z)^{2}/A $ represents the isospin effects. Finally, the $ eZ^{4} $ term is added to formula to consider a higher-order correction of the Coulomb term, which describes the transition from asymmetric to symmetric charge distributions for various fission nuclei [70]. Furthermore, the f parameter is added to the expression to take into account other contributions to SF half-lives. The obtained fitting parameters are given by a = −10.0987592959, b = 119.319858732, c = −0.516609881059, d = −9.52538327068, e = 1.92155604207×10-6, and f = −1496.05967574. In the fitting used in the calculations, the curve fit function was used in Scipy in Python 2.7 program language based on Spyder 2 with Anaconda [74]. As the SF is considered to be dependent on the binding energy of the nucleus and the Q-value, this can be modeled in terms of the liquid drop model, as in Eq. (2). It is not necessary to consider $ {Q}_{\rm SF} $ separately in Eq. (2), as the equation already includes this term.

      When comparing this new formula for the SF with the formula of Xu et al. [70], there is one extra term, which depicts the surface term, in the new formula that resultantly comprises six parameters. However, this does not include the $ {Q}_{\rm SF} $ term, which is within the formula of Xu et al. [70]. Since all parameters include the atomic number A in this new form, the values of fitting parameters are changed as well. Moreover, even if most of the parameters of this formula were inspired by the terms of the binding energy formula of the liquid drop model, this model produces the experimental SF values of similar rms to the formula of Xu et al., which was obtained from the basically effective potential, including the nuclear, Coulomb, and isospin potential.

    • 2.2.   $ \alpha $ decay

    • The AD half-life can be obtained using the following formula

      $ T_{1/2} = \hbar \frac{{\ln 2}}{\Gamma }, $

      (3)

      where $ \Gamma $ denotes the decay width for the decay. According to the semi-classical WKB method, the $ \alpha $-decay width $ \Gamma $ is given by,

      $ \Gamma = P_\alpha F\frac{{\hbar ^2 }}{{4\mu }}\exp \left[ - 2\int\limits_{r_2 }^{r_3 } {k(r} ){\rm d}r\right], $

      (4)

      where $ P_\alpha $ is the preformation probability of the $ \alpha $ particle in a parent nuclei [75, 76]. In the half-life calculations, similar studies [77] and the experimental study, the preformation probabilities are specified as $ P_\alpha $ = 1.0 for even-even nuclei, $ P_\alpha $ = 0.6 for odd-A nuclei, and $ P_\alpha $ = 0.35 for odd-odd nuclei. In Eq. (4), the normalization factor is

      $ F = 1\bigg/\int\limits_{r_1 }^{r_2 } {\frac{1}{{k(r)}}} {\rm d}r\cos ^2 \left(\int\limits_{r_1 }^r {k(r'){\rm d}r'} - \frac{\pi }{4}\right), $

      (5)

      where the squared cosine term might be replaced by 1/2 without significant loss of accuracy [75, 77]. In Eqs. (4) and (5), the wave number k(r) is given by,

      $ k(r) = \sqrt {\frac{{2\mu }}{{\hbar ^2 }}\left| {Q - V_{\rm eff}(r)} \right|}, $

      (6)

      where Q is Q-value for the AD, and $ {V}_{\rm eff}(r) $ is the effective potential between the $ \alpha $ and core nuclei that stems from the binary clustering model that assumes the parent nuclei as the $ \alpha $ particle surrounding the daughter (core) nuclei. Hence, the only unknown term in these equations is the effective potential between $ \alpha $ and the core, and it is given by,

      $ V_{\rm eff}(r) = V_{N}(r)+V_{C}(r)+V_{L}(r), $

      (7)

      where r is the separation radius between the center of mass of the $ \alpha $ particle and the daughter nucleus.

      In this study, the modified form proposed by Brink and Takigawa in Ref. [78] was used instead of the Coulomb potential to solve the discontinuity in the Coulomb potential in WKB semi-classical calculations as follows

      $ {\widetilde{V}}_{ \rm{C}}(r) = \frac{Z_{d} Z_{\alpha}e^{2}}{r}(1-e^{-\varphi r-\frac{1}{2} (\varphi r)^2-0.35(\varphi r)^3}),\\ \;\;\;\; \varphi R = \frac{3}{2}, $

      (8)

      where $ Z_\alpha $ and $ Z_d $ are the charge numbers of the $ \alpha $ and daughter nuclei, and R is the Coulomb radius. In Eq. (7), the last term is Langer modified centrifugal barrier potential [79] that is given by

      $ V_{L}(r) = \frac{\hbar^{2}(L+1/2)^{2}}{2\mu r^{2}}, $

      (9)

      with the WKB being valid for one-dimensional problems, the above modification from $ L(L+1)\rightarrow(L+\frac{1}{2})^{2} $ is essential to ensure the correct behavior of the WKB wave function near the origin as well as the validity of the connection formulas used in Ref. [80]. In this study, L = 0 is used in the calculations.

      Although the forms of the Coulomb and centrifugal potentials are known very well, the shape of the nuclear potential in Eq. (7) is the only unknown term. As the analytical formula for the nuclear interaction between the $ \alpha $ and core nuclei cannot be written, various potential models, phenomenological or microscopic, should be used to determine the nuclear interaction. In this study, the nuclear potential is considered as phenomenological cosh potential similar to Ref. [41]. The cosh potential was proposed by Buck and Pilt [81], and it is a symmetrized form of the Woods-Saxon form,

      $ V_{N}(r) = -V_{0} \frac{1+{\rm cosh}\left(\dfrac{\lambda}{a}\right)}{{\rm cosh}\left(\dfrac{r}{a}\right)+{\rm cosh}\left(\dfrac{\lambda}{a}\right)}, $

      (10)

      where $ V_{0} $ and a are the depth of the nuclear potentials and diffuseness parameters, respectively. The studies using this form of potential were conducted to obtain both the AD and exotic decay half-lives of heavy nuclei [77, 82, 83]. Furthermore, $ \lambda $, the renormalization factor, is obtained by the Bohr-Sommerfeld quantization. As the isospin effect plays a important role in nuclear physics, one should take into account isospin effect in AD calculations as well. If protons and neutrons in the nucleus have a different nucleon density, the asymmetry of the isospin might affect the motion of alpha particles on the surface, and the nuclear interaction potential between the $ \alpha $ and core nuclei would be isospin-dependent. In Ref. [41], the authors have added a parameter related to the isospin in the depth of the nuclear potential to include it in their considerations. They have used this potential form to investigate the isospin effects on the $ \alpha $-decay half-lives for the even-even nuclei from Z = 62 to Z = 118 using the two-potential approach. Considering this effect improved the results by 6.8% in Ref. [41]. In this study, to be able to consider the isospin effects on the AD as well as the SF, the isospin-dependent potential parameter $ V_{0} = 192.42+ $$31.059(N-Z)/A $ MeV similar as Ref. [41] and a = 0.75 fm were used. $ V_{0} $ and a were obtained phenomenologically to obtain the best AD half-life values that are close to the experiment.

      Moreover, $ \lambda $ is determined separately for each decay by applying the Bohr-Sommerfeld quantization condition. The $ \lambda $ in the Eq. (10) can be calculated for every single decay by using the Bohr-Sommerfeld quantization rule,

      $ \int\limits_{r_1 }^{r_2 } {\sqrt {\frac{{2\mu }}{{\hbar ^2 }}(Q - V_{\rm eff}(r))} {\rm d}r = (G - L + 1)\frac{\pi }{2}}, $

      (11)

      where G are global quantum numbers coming from the Wildermuth condition [84], and they are used as follows [75, 76]

      $ \begin{array}{l} G = 22 \;\;\;\;(N > 126), \\ G = 20 \;\;\;\;(82 < N \leqslant 126), \\ G = 18 \;\;\;\;(N \leqslant 82). \\ \end{array} $

      (12)

      In the semiclassical WKB approximation, there are three classical turning points, which are $ r_1 $, $ r_2 $, and $ r_3 $. They are obtained by numerical solutions of the equation of $ V_{\rm eff}(r) = Q $, where Q is the $ \alpha $-decay energy for special decays [77, 85].

    3.   Results and discussion
    • The SF half-lives of even-even nuclei with Z = 90–114 were calculated using the proposed formula in Eq. (2) to find how the present formula obtains experimental SF half-lives. The obtained results and results of other models (by Xu et al. [70], by Bao et al. [71], and by Santhosh et al. [72]) are listed in Table 1. In Table 1, the first column depicts the nuclei, the second column depicts the proton number Z and neutron number N for parent nuclei, respectively. The Exp. column shows the experimental $ \log_{10}T_{\rm SF} $ values (in years) of spontaneous fission (SF) of nuclei with Z = 90–114, which are taken from Refs. [13, 86]. The results obtained by Xu et al. [70] (Xu), Bao et al. [71] (Bao), and Santhosh et al. [72] (KPS) are also presented in Table 1.

      nuclei Z N Exp. Xu[70] Bao[71] KPS[72] present nuclei Z N Exp. Xu [70] Bao[71] KPS[72] present
      232Th 90 142 21.08 21.88 22.22 21.87 21.13 250Fm 100 150 −0.10 −1.57 −0.67 −0.35 −1.37
      234U 92 142 16.18 16.03 16.04 16.44 15.87 252Fm 100 152 2.10 −0.92 0.89 0.36 −0.77
      236U 92 144 16.40 16.56 16.26 16.36 16.42 254Fm 100 154 −0.20 −0.98 −1.04 −0.26 −0.92
      238U 92 146 15.91 16.38 16.04 15.35 16.17 256Fm 100 156 −3.48 −1.76 −3.71 −1.61 −1.83
      236Pu 94 142 9.18 9.71 9.65 10.24 9.81 252No 102 150 −6.54 −6.04 −5.38 −4.70 −6.00
      238Pu 94 144 10.68 10.99 10.24 11.18 11.18 254No 102 152 −3.04 −4.65 −3.28 −3.12 −4.61
      240Pu 94 146 11.06 11.55 10.84 11.40 11.74 256No 102 154 −4.77 −3.97 −4.72 −2.90 −3.99
      242Pu 94 148 10.83 11.40 10.92 10.81 11.51 254Rf 104 150 −12.14 −10.62 −9.35 −9.14 −10.74
      244Pu 94 150 10.82 10.54 11.08 9.57 10.52 256Rf 104 152 −9.71 −8.48 −6.98 −6.73 −8.57
      240Cm 96 144 6.28 5.02 4.52 5.40 5.28 258Rf 104 154 −9.35 −7.06 −7.74 −5.63 −7.17
      242Cm 96 146 6.85 6.33 5.34 6.62 6.65 260Rf 104 156 −9.2 −6.36 −8.87 −5.24 −6.54
      244Cm 96 148 7.12 6.92 6.69 7.00 7.23 262Rf 104 158 −7.18 −6.36 −8.32 −5.20 −6.65
      246Cm 96 150 7.26 6.80 7.35 6.74 7.03 258Sg 106 152 −10.04 −12.34 −9.63 −10.19 −12.48
      248Cm 96 152 6.62 5.96 7.41 5.67 6.06 260Sg 106 154 −9.65 −10.17 −9.80 −8.31 −10.31
      250Cm 96 154 4.05 4.41 4.61 3.37 4.35 262Sg 106 156 −9.32 −8.72 −10.41 −7.13 −8.91
      242Cf 98 144 −1.33 −1.27 −1.17 −0.71 −1.13 264Sg 106 158 −8.93 −7.98 −9.42 −6.30 −8.26
      246Cf 98 148 3.26 2.12 2.09 2.75 2.43 266Sg 106 160 −7.86 −7.96 −7.48 −5.80 −8.35
      248Cf 98 150 4.51 2.74 3.27 3.42 3.02 264Hs 108 156 −10.2 −11.02 −12.10 −9.14 −11.10
      250Cf 98 152 4.23 2.65 4.31 3.25 2.84 270Ds 110 160 −8.6 −9.46 −10.22 −7.23 −9.39
      252Cf 98 154 1.93 1.84 2.11 1.76 1.90 282112 112 170 −10.58 −9.39 −11.28 −7.21 −9.40
      254Cf 98 156 −0.78 0.32 −0.82 −0.33 0.23 284112 112 172 −8.5 −11.43 −9.65 −8.14 −11.52
      246Fm 100 146 −6.60 −5.01 −4.15 −4.14 −4.94 286114 114 172 −8.08 −7.12 −5.95 −4.45 −6.44
      248Fm 100 148 −2.94 −2.93 −2.43 −1.92 −2.76

      Table 1.  Calculated $ \log_{10}T_{\rm SF} $ (in years) and results of other models for SF half-lives of nuclei with Z = 90–114.

      To compare the results, the rms deviations of the decimal logarithmic values are calculated using the following equation,

      $ \sigma = [{\frac{1}{{n- 1}}\sum\limits_{k = 1}^n {[\log _{10} (T_{\rm SF}^{\rm cal} ) - } \log _{10} (T_{\rm SF}^{\rm exp } )]^2}]^{1/2}, $

      (13)

      where n denotes the number of the related nuclei [32]. The rms deviation ($ \sigma $) was computed for the present model calculations. The obtained value is presented in Table 2. In Table 2, $ \sigma $ values were also presented for Xu [70], Bao [71], and KPS [72]. As depicted in Table 2, $ \sigma = 1.22 $ was obtained for this present model.

      $ \sigma $ (rms values)
      Xu et al.[70] Bao et al.[71] KPS [72] present
      1.27 1.12 1.69 1.22

      Table 2.  Rms values for all models.

      The parameters obtained by fitting and Eq. (2) have been used to calculate the SF half-lives and compare them with the results of three different models for the even-even superheavy nuclei with Z = 108, 110, 112, 114, 116, 118, and 120 as listed in Table 3, 4, and 5, respectively. In these Tables, the Z, N, A depict the proton, neutron and mass number of nuclei, respectively. The Xu column lists the results of Xu et al. [70], the Bao lists the results of Bao et al. [71], the KPS shows the results of Santhosh et al. [72], and the "present" column depicts the obtained $ \log_{10}T_{\rm SF} $ values in terms of second in this study.

      Z N A Xu[70] Bao [71] KPS [72] present
      108 150 258 −12.26 −7.71 −10.18 −12.46
      108 152 260 −8.63 −5.32 −6.31 −8.72
      108 154 262 −5.72 −4.77 −3.61 −5.77
      108 156 264 −3.53 −4.47 −1.64 −3.60
      108 158 266 −2.05 −3.18 −0.04 −2.19
      108 160 268 −1.29 −0.89 1.23 −1.53
      108 162 270 −1.24 0.69 1.74 −1.61
      108 164 272 −1.90 −0.04 1.09 −2.40
      108 166 274 −3.28 −3.43 −0.79 −3.90
      108 168 276 −5.38 −6.43 −3.07 −6.10
      108 170 278 −8.18 −6.58 −5.08 −8.98
      108 172 280 −11.70 −5.40 −7.19 −12.52
      110 154 264 −8.63 −7.5 −6.68 −8.41
      110 156 266 −5.69 −5.25 −3.94 −5.47
      110 158 268 −3.47 −4.76 −1.63 −3.30
      110 160 270 −1.96 −3.1 0.27 −1.89
      110 162 272 −1.17 −1.04 1.61 −1.22
      110 164 274 −1.1 −1.03 1.62 −1.27
      110 166 276 −1.73 −2.41 0.38 −2.04
      110 168 278 −3.09 −4.79 −1.26 −3.51
      110 170 280 −5.15 −4.53 −2.4 −5.67
      110 172 282 −7.93 −10.27 −3.99 −8.50
      110 174 284 −11.42 −7.65 −5.97 −12.00

      Table 3.  Comparison of calculated $ \log_{10}T_{\rm SF} $ (s) with other models for Z = 108 and Z = 110.

      Z N A Xu[70] Bao [71] KPS [72] present
      112 158 270 −4.67 −8.33 −3.34 −3.95
      112 160 272 −2.42 −5.9 −0.87 −1.79
      112 162 274 −0.88 −3.51 0.98 −0.37
      112 164 276 −0.06 −3.53 1.64 0.31
      112 166 278 0.04 −5.7 1.17 0.28
      112 168 280 −0.57 −6.02 0.67 −0.46
      112 170 282 −1.89 −4.02 0.29 −1.90
      112 172 284 −3.93 −2.29 −0.65 −4.02
      112 174 286 −6.68 −0.84 −2.11 −6.81
      114 160 274 −2.56 −8.4 −1.8 −1.08
      114 162 276 −0.29 −6.24 0.53 1.08
      114 164 278 1.28 −2.49 2.7 2.51
      114 166 280 2.12 −1.03 3.71 3.20
      114 168 282 2.25 −0.4 3.97 3.19
      114 170 284 1.67 −0.14 3.62 2.47
      114 172 286 0.38 1.13 3.05 1.06
      114 174 288 −1.64 2.95 2.16 −1.02
      116 168 284 5.47 0.21 6.04 7.58
      116 170 286 5.63 1.77 6.52 7.59
      116 172 288 5.07 2.73 6.33 6.89
      116 174 290 3.81 3.58 5.64 5.52
      116 176 292 1.82 5.34 4.74 3.48
      116 178 294 −0.87 5.84 3.06 0.78
      116 180 296 −4.28 5.39 0.72 −2.57
      116 182 298 −8.41 4.55 −2.13 −6.56
      116 184 300 −13.24 2.64 −5.65 −11.16
      116 186 302 −18.79 −2.84 −10.48 −16.39
      116 188 304 −25.06 −9.05 −15.86 −22.21

      Table 4.  Comparison of calculated $ \log_{10}T_{\rm SF} $ (s) with other models for Z = 112, Z = 114 and Z = 116.

      Z N A Xu[70] Bao [71] KPS [72] present
      118 170 288 10.07 1.97 9.12 13.60
      118 172 290 10.25 3.24 9.47 13.62
      118 174 292 9.73 4.3 9.27 12.95
      118 176 294 8.48 4.29 8.33 11.61
      118 178 296 6.53 5.49 7.27 9.60
      118 180 298 3.86 4.38 5.16 6.94
      118 182 300 0.48 2.98 2.57 3.64
      118 184 302 −3.62 0.92 −0.59 −0.29
      118 186 304 −8.43 −4.5 −5.01 −4.84
      120 172 292 16 3.25 12.69 21.38
      120 174 294 16.21 4.63 13 21.42
      120 176 296 15.71 4.45 12.42 20.78
      120 178 298 14.5 4.78 11.53 19.46
      120 180 300 12.57 3.58 9.8 17.49
      120 182 302 9.93 2.07 7.57 14.87
      120 184 304 6.57 −0.23 4.73 11.62
      120 186 306 2.5 −5.65 0.69 7.74

      Table 5.  Comparison of calculated $ \log_{10}T_{\rm SF} $ (s) with other models for Z = 118 and Z = 120.

      As shown in the tables, even if the logarithmic values of Xu, Bao, KPS, and the present study exhibit similar behaviors of change according to the mass number of the parent nuclei, their size is different. However, the results of Bao et al. show slightly different behavior in comparison to the others.

      A successful model should produce both experimental SF half-lives and predict the decay modes of nuclei. Superheavy nuclei decay through the AD, followed by the SF. If the half-lives of AD are shorter than the SF, then nuclei survive the fission and therefore decay through the AD. The $ \alpha $ decay half-lives for even-even nuclei from Z = 104 to Z = 118 were calculated within the framework of the WKB method and BS quantization rule by considering the isospin-dependent effects and the SF half-lives using the proposed formula (Eq. (2)). The obtained results are shown in Table 6. To make predictions about which decay is dominant for each nuclei, the branching ratios for SF (%) (($ T_{\alpha}/(T_{\rm SF}+T_{\alpha} $))$ \times $100) and $ \alpha $ decay (%) (($ T_{\rm SF}/(T_{\rm SF}+T_{\alpha} $))$ \times $100) were calculated, and subsequently the modes of decays were predicted and compared with the decay modes in Ref. [72] as well as the experimental ones, as seen in Table 6. In Table 6, the nuclei column shows the related superheavy nuclei, $ {Q}_{\alpha}^{\rm exp.} $ shows the experimental Q-value taken from Ref. [87]. $ T_{\rm SF} $ and $ T_{\alpha} $ are the calculated values for SF and AD, respectively. $ {\rm{BR}}_{\rm SF}$(%) and $ {\rm{BR}}_{\alpha} $(%) show the calculated branching ratio values for SFs and ADs, respectively. The present column shows the dominant decay modes in present calculations; Ref. [72] shows the predicted decay modes for nuclei in Ref. [72], and the Exp. column shows the dominant decay modes in the experiment taken from Ref. [87]. In the "present" column, the parenthesis is used to depict the dominant decay mode. As can be seen in Table 6, the decay modes predicted in present calculations are in very good agreement with the predicted decay modes in Ref. [72] and the experimental ones, with the exception of some nuclei. When the present results are compared to Ref. [72], the predictions are different for some nuclei even if all other predictions obtained in this study are agreement with the ones in Ref. [72]. The SF values of half-lives are observed to increase with the Z number of parent nuclei, whereas the AD half-lives are tend to decrease.

      nuclei${Q} _{\alpha}^{\rm exp.} $ [87]$ {T}_{\rm SF}{\rm (s)} $$ {T}_{\alpha}{\rm (s)} $${\rm{BR}} _{\rm SF} $(%)$ {\rm{BR}}_{\alpha} $(%)presentRef. [72]Exp. [87]
      29411811.824.044e+111.976e-030.000 100.000$ \alpha $$ \alpha $$ \alpha $
      29411711.182.118e+068.768e-020.000 100.000$ \alpha $$ \alpha $$ \alpha $
      29311711.322.638e+072.340e-020.000 100.000$ \alpha $$ \alpha $$ \alpha $
      29311610.711.613e+023.704e-010.229 99.771$ \alpha $$ \alpha $$ \alpha $
      29211610.782.998e+031.458e-010.005 99.995$ \alpha $$ \alpha $$ \alpha $
      29111610.893.816e+041.271e-010.000 100.000$ \alpha $$ \alpha $$ \alpha $
      29011611.003.314e+054.029e-020.000 100.000$ \alpha $$ \alpha $$ \alpha $
      29011510.411.084e+011.814e+0014.33685.664$ \alpha/{\rm SF} (\alpha) $$ \alpha $$ \alpha $
      28911510.491.410e+026.436e-010.454 99.546$ \alpha $$ \alpha $$ \alpha $
      28811510.631.250e+034.719e-010.038 99.962$ \alpha $$ \alpha $$ \alpha $
      28711510.767.526e+031.270e-010.002 99.998$ \alpha $$ \alpha $$ \alpha $
      2891149.984.935e-037.492e+0099.9340.066SF$ \alpha $$ \alpha $
      28811410.079.620e-022.490e+0096.2803.720SF$ \alpha $$ \alpha $
      28711410.171.279e+002.175e+0062.95637.044$ \alpha/{\rm SF} ({\rm SF}) $$ \alpha $$ \alpha $
      28611410.351.158e+014.215e-013.511 96.489$ \alpha $$ \alpha $$ \alpha = 0.6,{\rm SF} = 0.4 $
      2851149.4927.104e+011.977e+0273.56126.439$ \alpha/{\rm SF} ({\rm SF}) $$ \alpha $$ \alpha $
      2861139.792.013e-032.018e+0199.9900.010$ \alpha/{\rm SF} ({\rm SF}) $$ \alpha $$ \alpha $
      28511310.012.739e-022.771e+0099.0210.979SF$ \alpha $$ \alpha $
      28411310.122.531e-012.338e+0090.2309.770SF$ \alpha $$ \alpha $
      28311310.381.582e+002.711e-0114.62685.374$ \alpha $$ \alpha $$ \alpha $

      Table 6.  Calculated $ {T}_{\rm SF}{\rm (s)} $, $ {T}_{\alpha}{\rm (s)} $, $ {\rm{BR}}_{\rm SF} $(%), ${\rm{BR}} _{\alpha} $(%), and the predicted decay modes of superheavy nuclei and their experimental modes for nuclei with Z = 104 to Z = 118.

      To be able to make the predictions for the unknown decay modes of superheavy nuclei, the AD half-lives in the WKB method considering the isospin-dependent potential and BS quantization condition, as well as the SF half-lives using the new formula proposed in this study, have been calculated for possible AD chains from isotopes superheavy nuclei with Z = 119–120. The calculations of the SF and AD for Z = 119,120 are obtained and presented in Table 7. In Table 7, the nuclei column depicts the superheavy nuclei, ${{Q}} _{\alpha}^{\rm exp.} $ shows the experimental Q-value taken from Ref. [87], and $ T_{\rm SF} $ and $ T_{\alpha} $ are calculated values for SF and AD, respectively. With regard to $ BR_{\rm SF} $ and $ BR_{\alpha} $, they show the calculated branching ratio values for SF and AD, respectively. The "present" column depicts the dominant decay modes in calculations, and the Bao [71] column shows the obtained decay modes by Bao et al. [71]. In the "present" column, the parenthesis is used to depict the dominant decay mode. As seen in the table, similar behaviors have been shown for nuclei with Z numbers ranging from 104 to 118.

      nuclei$ {Q}_{\alpha}^{\rm exp.} $$ {T}_{\rm SF}{\rm (s)} $$ {T}_{\alpha}{\rm (s)} $$ {\rm{BR}}_{\rm SF} $(%)$ {\rm{BR}}_{\alpha} $(%)presentBao [71]
      30012013.313.096e+176.485e-060.000100.000$ \alpha $$ \alpha $
      29611811.753.973e+092.872e-030.000100.000$ \alpha $$ \alpha $
      29912013.253.613e+181.384e-050.000100.000$ \alpha $$ \alpha $
      29511811.904.841e+102.195e-030.000100.000$ \alpha $$ \alpha $

      Table 7.  Calculated $ {T}_{\rm SF}{\rm (s)} $, $ {T}_{\alpha}{\rm (s)} $, $ {\rm{BR}}_{\rm SF} $(%), $ {\rm{BR}}_{\alpha} $(%), and the predicted decay modes of superheavy nuclei without known experimental decay modes and predictions of Bao et al. [71].

      It should be also underlined that the reason why the predictions for some nuclei are different from each other in the presented SF values in Tables 6 and 7 would come from the fact that the present model does not consider shell effects, magic numbers, and also whether the mass, proton and neutron numbers of the related nuclei are odd or even.

    4.   Summary
    • The half-lives of the spontaneous fission (SF) were obtained for heavy and superheavy nuclei with the new formula and $ \alpha $ decay (AD) by using the Wentzel-Kramers-Brillouin (WKB) method together with Bohr-Sommerfeld (BS) quantization condition for cosh type potential including the isospin effects. By comparing the SF results with experimental values, rms values were calculated. When the new SF function is used in the calculations to obtain the experimental SF half-lives, the rms values become significantly better. The new formula is applied to obtain the SF half-lives of Z = 108–120 nuclei, and the logarithmic values of the SF are obtained and subsequently compared with the results of other models. Even if the logarithmic half-lives obtained by the formulas of Xu et al. [70], Bao et al. [71], KPS, Santhosh et al. [72], and the present study exhibit a similar behavior of change according the mass number of the parent nuclei, their size is different. However, the results of Bao et al. [71] show a slightly different behavior in comparison to others. The proposed formula, comprising six parameters and excluding the $ {{Q}}_{\rm SF} $ value, produces the experimental value of SF of nuclei in reasonable rms values.

      The branching ratios for SFs and ADs were obtained, and the modes of decays were predicted for Z = 104–118 nuclei with known experimental decay modes. The decay modes extracted in calculations are in very good agreement with the experimental ones. Although theoretical predictions of the decay modes of many nuclei are the same as experimental ones, the predictions of decay modes for some nuclei are nevertheless different from experimental results. Furthermore, the branching ratios for SFs and ADs were obtained, and the modes of decay for Z = 119 and Z = 120 nuclei, which have the unknown experimental decay modes, were predicted. The decay modes are predicted for 45 nuclei that do not have experimental decay modes, and they compared with the predictions of Bao et al. [71]. Even if the predictions of two different models are same for many nuclei, the predictions of decay modes obtained in present study for some nuclei are different from the results of Bao et al. [71]. These are the decay modes of 280112, 278112, 292115, 288113, 291115, and 287113 nuclei.

      In this study, different decay modes are obtained for some nuclei in comparison to the predictions provided in the literature for some superheavy nuclei. The present results provide useful information and knowledge for improving theoretical models and possible future experimental studies on superheavy nuclei in terms of both half-lives of the $ \alpha $ decays and spontaneous fission.

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