An improved semi-empirical relationship for cluster radioactivity

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Yanzhao Wang, Fengzhu Xing, Yang Xiao and Jianzhong Gu. An improved semi-empirical relationship for cluster radioactivity[J]. Chinese Physics C.
Yanzhao Wang, Fengzhu Xing, Yang Xiao and Jianzhong Gu. An improved semi-empirical relationship for cluster radioactivity[J]. Chinese Physics C. shu
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An improved semi-empirical relationship for cluster radioactivity

    Corresponding author: Yanzhao Wang, yanzhaowang09@126.com
    Corresponding author: Jianzhong Gu, jzgu1963@ciae.ac.cn
  • 1. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
  • 2. Institute of Applied Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
  • 3. China Institute of Atomic Energy, P. O. Box 275 (10), Beijing 102413, China

Abstract: An improved semi-empirical relationship for cluster radioactivity half-lives is proposed by introducing an accurate charge radius formula and an analytic expression of the preformation probability. Then, the cluster radioactivity half-lives for the daughter nuclei around 208Pb or its neighbors and the 12C radioactivity half-life of 114Ba are calculated within the improved semi-empirical relationship. It is shown that the accuracy of the new relationship is improved significantly compared to its predecessor. In addition, the cluster radioactivity half-lives that are experimentally unavailable for the trans-lead and trans-tin nuclei are predicted by the new semi-empirical formula. These predictions might be useful for searching for the new cluster emitters of the two islands in future experiments.

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    I.   INTRODUCTION
    • The cluster radioactivity of heavy nuclei was first predicted by Sandulescu, Poenaru, and Greiner in 1980 [1]. It was usually called as the heavy-ion radioactivity because the emitted clusters are heavier than $ \alpha $-particle and lighter than fission fragments. In 1984, the cluster radioactivity was first observed by Rose and Jones through the decay of $ ^{223} $Ra by emitting $ ^{14} $C [2]. From then on, many clusters heavier than $ ^{14} $C have been experimentally observed in many parent nuclei from $ ^{221} $Fr to $ ^{242} $Cm, such as $ ^{20} $O, $ ^{23} $F, $ ^{22,24-26} $Ne, $ ^{28,30} $Mg and $ ^{32,34} $Si and the daughter nuclei are the doubly magic nucleus $ ^{208} $Pb or its neighbors [37]. This implies that the shell effect plays a crucial role in cluster emitting for heavy nuclei.

      To describe the cluster radioactivity, various theoretical models are proposed [832]. Generally, these models can be divided into two categories: cluster-like model [816] and fission-like model [1732]. For the cluster-like model, the cluster is assumed to be preformed in the parent nucleus before it penetrates the barrier. As to the fission-like model, the nucleus deforms continuously as it penetrates the nuclear barrier and reaches the scission configuration after running down the Coulomb barrier. The experimental half-lives can be reproduced more or less satisfactorily by the two kinds of models [832]. In addition, many scaling laws and semi-empirical relationships have been developed for the systematic calculations of the cluster decay half-lives [3341].

      Recently, a semi-empirical relationship of the charged particles and exotic cluster radioactivity was developed by Sahu et al. based on the basic phenomenon of resonances occurring in quantum scattering process under Coulomb-nuclear potential [42]. It was usually called as the Sahu formula and its form is similar to the universal decay law (UDL) proposed by Qi et al [33, 34]. Although the coefficients are obtained naturally and the angular momentum dependence is included in the Sahu formula, there exists some deviations between the calculated half-lives of the charged particles of the parent nuclei and the experimental values. For example, the calculated $ \alpha $-decay half-life of $ ^{257} $Md is 10$ ^{3} $ times as short as its corresponding experimental value. So it is necessary to improve the accuracy of the Sahu relationship by taking into account some physical factors. Recently, we proposed an improved Sahu (ImSahu) semi-empirical relationship for $ \alpha $-decay half-lives by introducing a precise charge radius formula and an analytic expression for preformation probability [43, 44]. It is shown that the 421 experimental $ \alpha $-decay half-lives can be reproduced accurately with the ImSahu relationship. In this article, we will continue to improve the Sahu relationship for the study of the cluster radioactivity on the basis of our recent work.

      This article is organized in the following way. Sec. 2 gives the theoretical approaches. In Sec. 3 numerical results are presented and discussions are made. In the last section, some conclusions are drawn.

    II.   SAHU AND IMSAHU SEMI-EMPIRICAL RELATIONSHIPS
    • The Sahu relationship is derived by considering the metastable parent nucleus as a quantum two-body system of the emitted particle and the daughter nucleus exhibiting resonance scattering phenomena under the combined effect of nuclear, Coulomb and centrifugal forces [42].

      In a series papers of Sahu [4548], the cluster+nucleus system is considered as a Coulomb-nuclear potential scattering problem and the resonance energy of the quasibound state is taken as the $Q _{c} $-value (released energy) of the decaying system. The width or life-time of the resonance state accounts for the decay half-life. The normalized regular solution u(r) of the modified Schrödinger equation is matched at radius r = R to the outside Coulomb-Hankel outgoing spherical wave f(r) = $G _{l}(\eta ,kr) $+$iF _{l}(\eta ,kr) $ such that

      $ u(r) = N_{0}\left[ G_{l}\left( \eta ,kR\right) +iF_{l}\left( \eta ,kR\right) \right], $

      (1)

      where R is the radial position outside the range of the nuclear field.

      For a typical cluster–daughter system, the cluster particle serves as the projectile and the daughter nucleus as the target. The mean life T (or width $ {\varGamma } $) of the decay is expressed in terms of the amplitude $N _{0} $ as

      $ T = \frac{\hbar }{\Gamma } = \frac{1}{\left\vert N_{0}\right\vert ^{2}}\frac{\mu }{\hbar k}, $

      (2)

      where $ \mu $ represents the reduced mass of the system. k and $ \eta $ stand for the wave number and Coulomb parameter, respectively.

      Since the wave function u(r) decreases rapidly with radius outside the daughter nucleus, it can be normalized by requiring that $ \displaystyle\int_{0}^{R}\left\vert u(r)\right\vert ^{2}dr = 1 $. Further, using the fact that for a sufficiently large value of radial distance, the value of $G _{l} $(l,kR) could be several orders of magnitude larger than that of $F _{l} $(l,kR), so Eq. (2) can be expressed as

      $ T = \frac{\mu }{\hbar k}\frac{\left\vert G_{l}(\eta ,kR)\right\vert ^{2}}{ \left\vert u(R)\right\vert ^{2}}. $

      (3)

      Eq. (3) gives the mean life T or half-life $T _{1/2} $ = 0.693T of the charged particle decay. This formulation is valid for the emission of all types of positively charged particles.

      In the derivation of the Sahu relationship, the function $G _{l} $ is consulted from a handbook of mathematical functions [49]. The derived details can be found in Ref. [42]. We will not present it here. The obtained expression of the Sahu relationship is written as

      $ \log_{10}[T_{1/2}({\texttt{s}})] = aZ_{c}Z_d\sqrt{\frac{{\cal{A}}}{Q_{c}}}+b\sqrt{{\cal{A}}Z_{c}Z_d}+c+d, $

      (4)

      where $ Z_{c} $ and $ Z_{d} $ are the proton numbers of the emitted cluster and daughter nucleus, respectively. ${\cal{A}} = A_{c}A_d/ (A_{c}+A_d)$, where $ A_{c}(A_{d}) $ denotes the mass number of the emitted cluster (daughter nucleus). The coefficients a, b, c and d in Eq. (4) are derived naturally, which are different from those of the UDL formula. These parameters are given as follows [42]

      $ a = \frac{2a_0e^2\sqrt{2m}}{\hbar ln10}, $

      (5)

      $ b = \frac{-b_f\sqrt{2me^2R}}{\hbar ln10}, $

      (6)

      $ c= \dfrac{ln\left[\left(0.231\times10^{-23}\right)\sqrt{ \dfrac{mR{\cal{A}}}{ 2e^2Z_{c}Z_{d}}} \dfrac{1}{P}\right]}{ln10}, $

      (7)

      $ d = \frac{{lnM_l-\displaystyle\sum\limits_{l = 0}^lln(\eta_l)}}{ln10}, $

      (8)

      where

      $ \begin{aligned}[b] b_{f} =& 2+a_{0}-2a_{1}+\left(\frac{a_{0}}{4}+a_{1}-2a_{2}\right)t^{1/2} \\ & +\left(\frac{a_{0}}{8}+\frac{a_{1}}{4}+a_{2}-2a_{3}-1\right)t \\ & +\left(\frac{5a_{0}}{64}+\frac{a_{1}}{8}+\frac{a_{2}}{4}+a_{3}\right)t^{3/2}\\ &+\left(\frac{5a_{1}}{64}+\frac{a_{2}}{8}+\frac{a_{3}}{4}-\frac{1}{4}\right)t^{2} \\ & +\left(\frac{5a_{2}}{64}+ \frac{a_{3}}{8}\right)t^{5/2} \\ &+\left(\frac{5a_{3}}{64}-\frac{1}{8}\right)t^{3}, \end{aligned} $

      (9)

      $ \begin{aligned}[b] \sqrt{M_{l}} = &1+\frac{4(2l+1)^{2}-1}{16(2me^{2}RZ_{c }Z_{d}{\cal{A}} /\hbar ^{2})^{1/2}} \\ &+\frac{[4(2l+1)^{2}-1][4(2l+1)^{2}-9]}{2[16^{2}(2me^{2}RZ_{c }Z_{d} {\cal{A}}/\hbar ^{2})]} \\ &+[4(2l+1)^{2}-1][4(2l+1)^{2}-9] \\ &\times \frac{\lbrack 4(2l+1)^{2}-25]}{2[16^{3}(2me^{2}RZ_{c }Z_{d} {\cal{A}}/\hbar ^{2})^{3/2}]}, \end{aligned} $

      (10)

      $ \eta _{l} = 1+\frac{l^{2}}{(e^{2}Z_{c }Z_{d}/\hbar )^{2}m{\cal{A}} /2Q_{c }}. $

      (11)

      In the expression of $ b_{f} $, $ t = Q_{c}R/(e^{2}Z_{c }Z_{d}) $. Here the nucleon mass $ m = 931.5 $ MeV, $ e^{2} = 1.4398 $ MeV fm, $ \hbar = 197.329 $ MeV fm, $ a_{0} = 1.5707288 $, $ a_{1} = -0.2121144 $, $ a_{2} = 0.074240 $ and $ a_{3} = -0.018729 $ [49]. The parameter d is l-dependent, where l is the orbital angular momentum carried by the emitted particles. In calculations, R is approximated as R = $r _{0}(A_{d}^{1/3}+A_{c}^{1/3})\approx $ 9.5 fm. P is taken as $ 10^{-3} $ [42].

      We once pointed out that the accuracy of the Sahu formula was low because R and P are treated simply as constants [43, 44]. In our recent work, R and P were assumed as the charged radius and the preformation probability, respectively. Moreover, an accurate charge radius formula on R and an analytic expression for the preformation probability on P were introduced for improving the accuracy of the Sahu formula [43, 44]. Based on our recent work [43, 44], for the cluster radioactivity, the accurate charge radius formula and analytic preformation probability for clusters inside the parent nuclei are also introduced.

      For P, an analytic expression proposed by Blendowske and Walliser, P = $P _{\alpha }^{(A_{c}-1)/3} $ [50], is usually adopted. Considering the model dependence, the analytic expression of P is slightly modified as

      $ \log _{10}P = \left( \frac{A_{c}-1}{3}\right) \log _{10}P_{\alpha }+c^{\prime }, $

      (12)

      where $P _{\alpha } $ is the $ \alpha $-particle preformation factor inside the parent nucleus. The second term $c ^{\prime } $ of Eq. (12) is a constant.

      For R, it is defined as

      $ R = R_d+R_c, $

      (13)

      where $ R_d $ and $ R_c $ are the root-mean-square charge radii of the daughter nucleus and the cluster, respectively. The form of $ R_d $ or $ R_c $ is [51]

      $ R_{d(c)} = r_{0}\left(1-r_{1}\frac{N_{d(c)}-Z_{d(c)}}{A_{d(c)}}+{r}_{2}\frac{ 1}{A_{d(c)}}\right){A}_{d(c)}{}^{1/3}, $

      (14)

      where $N _{d(c)} $ is the neutron number of the daughter nucleus or the emitted cluster, and $ r_0 $ = 1.2331 fm, $ r_1 $ = 0.1461 and $ r_2 $ = 2.3310 [51].

      In addition to Eq. (14), some other charged radius formulas were proposed [5154]. To see the influence on the cluster radioactivity half-lives from a different radius formula, another charge radius formula that includes the Casten factor is also used in the article, which is written as [51]

      $ R_{d(c)} = r_{0}\left(1-r_{1}\frac{N_{d(c)}-Z_{d(c)}}{A_{d(c)}}+{r}_{2}\frac{ 1}{A_{d(c)}}+r_{3}\frac{C}{A_{d(c)}}\right){A}_{d(c)}{}^{1/3}, $

      (15)

      where C is the Casten factor, whose form is C = $N _{p} $$N _{n} $/($N _{p} $+$N _{n} $). Here $N _{p} $ and $N _{n} $ represent the valence proton number and valence neutron number, respectively. The parameters are $ r_0 $ = 1.2262 fm, $ r_1 $ = 0.1473, $ r_2 $ = 1.9876 and $ r_2 $ = 0.3993. In this article, the ImSahu formula with Eq. (14) (Eq. (15)) is named as ImSahuA (ImSahuB) relationship.

    III.   RESULTS AND DISCUSSIONS
    • Firstly, the empirical preformation factors P of clusters in nuclei are extracted within Eqs. (1-14) or Eqs. (1-13, 15) by inputting the experimental $Q _{c} $ and half-life values. The extracted P values are listed in columns 6 and 7 of Table 1, respectively. In Table 1, the first and second columns denote the parent nuclei and the emitted clusters, respectively. The l value is shown in column 3 and its effect is taken into account in calculations. Columns 4 and 5 represent the experimental $Q _{c} $ and ${\rm{log}} _{10}T_{1/2} $ values. Note that the experimental $Q _{c} $ values are extracted by the following expression

      Parent nucleiEmitted clustersl$Q _{c} $(MeV)${\rm{log}} _{10}T_{1/2} $ (s)PP${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s)
      Expt.Expt.ImSahuAImSahuBSahuImSahuAImSahuB
      Even-even nuclei
      212Po4He08.950−6.521.63 $ \times 10^{-4} $3.43 $ \times 10^{-4} $−6.25−6.21−6.23
      214Po4He07.833−3.781.90 $ \times 10^{-4} $4.15 $ \times 10^{-4} $−3.38−3.40−3.41
      238Pu4He05.5909.596.36 $ \times 10^{-5} $1.16 $ \times 10^{-4} $10.149.499.41
      222Ra14C033.0511.001.62 $ \times 10^{-8} $5.70 $ \times 10^{-8} $10.4211.3211.70
      224Ra14C030.5415.923.51 $ \times 10^{-9} $1.32 $ \times 10^{-8} $14.9215.5815.99
      226Ra14C028.2021.345.63 $ \times 10^{-10} $2.23 $ \times 10^{-9} $19.7820.2020.64
      228Th20O044.7220.724.51 $ \times 10^{-11} $7.38 $ \times 10^{-11} $20.2220.9021.05
      230U22Ne061.4019.572.40 $ \times 10^{-12} $2.52 $ \times 10^{-12} $19.1219.2819.28
      $ ^{230} $Th$ ^{24} $Ne057.5724.641.34 $ \times 10^{-13} $6.76 $ \times 10^{-13} $23.4623.9024.61
      232U24Ne062.3120.403.06 $ \times 10^{-13} $1.44 $ \times 10^{-12} $19.5020.0220.70
      234U24Ne058.8425.921.82 $ \times 10^{-14} $9.31 $ \times 10^{-14} $24.2824.3125.03
      234U26Ne059.4725.072.26 $ \times 10^{-13} $4.37 $ \times 10^{-13} $24.9325.3625.69
      234U28Mg074.1125.747.66 $ \times 10^{-16} $1.37 $ \times 10^{-15} $24.2224.3624.70
      236U28Mg071.6927.585.90 $ \times 10^{-15} $7.47 $ \times 10^{-15} $27.3527.0927.27
      236Pu28Mg079.6721.672.25 $ \times 10^{-15} $3.91 $ \times 10^{-15} $20.4620.7621.08
      238Pu28Mg075.9125.701.38 $ \times 10^{-15} $2.48 $ \times 10^{-15} $24.8824.5724.91
      238Pu32Si091.1925.282.49 $ \times 10^{-16} $1.38 $ \times 10^{-15} $25.1325.0225.92
      236U30Mg072.5127.585.17 $ \times 10^{-15} $5.11 $ \times 10^{-15} $27.6727.8327.95
      238Pu30Mg077.0025.676.80 $ \times 10^{-16} $6.73 $ \times 10^{-16} $24.8725.0425.16
      242Cm34Si096.5123.154.04 $ \times 10^{-16} $2.13 $ \times 10^{-15} $23.6923.9024.81
      Odd-A nuclei
      213Po4He08.540−5.371.04 $ \times 10^{-4} $2.21 $ \times 10^{-4} $−5.27−5.06−5.14
      215At4He08.178−4.007.59 $ \times 10^{-5} $1.60 $ \times 10^{-4} $−3.98−3.82−3.91
      221Fr14C331.3214.528.93 $ \times 10^{-10} $3.24 $ \times 10^{-9} $12.7014.4614.53
      221Ra14C332.4013.391.10 $ \times 10^{-9} $3.95$ \times 10^{-9} $11.6713.4213.48
      223Ra14C431.8315.201.74 $ \times 10^{-10} $6.35 $ \times 10^{-10} $12.7814.4314.50
      225Ac14C430.4817.341.97 $ \times 10^{-9} $6.92 $ \times 10^{-9} $16.2317.6217.68
      231Pa23F151.8426.025.72 $ \times 10^{-15} $2.77 $ \times 10^{-14} $22.9624.9925.11
      231Pa24Ne160.4223.386.12 $ \times 10^{-15} $2.94 $ \times 10^{-14} $20.7722.8422.96
      233U24Ne260.4924.821.97 $ \times 10^{-15} $9.96 $ \times 10^{-15} $21.9823.7923.91
      235U24Ne157.3627.426.01 $ \times 10^{-14} $3.19 $ \times 10^{-13} $26.5127.8828.03
      233U25Ne260.7824.832.77 $ \times 10^{-15} $7.60 $ \times 10^{-15} $22.3424.4224.28
      235U25Ne357.7627.427.58 $ \times 10^{-14} $2.19 $ \times 10^{-13} $26.8228.4528.33
      235U26Ne358.1127.458.20 $ \times 10^{-14} $1.61 $ \times 10^{-13} $27.0928.9828.68

      Table 1.  The preformation factors P of the cluster radioactivity extracted in the present calculation and the calculated cluster radioactivity half-lives within the Sahu, ImSahuA and ImSahuB relationships. The P values and half-lives of $ \alpha $-decay are calculated by the method used in our recent work on Sahu formula [43, 44]. $Q _{c} $ and ${\rm{log}} _{10}T_{1/2} $ are measured in MeV and second, respectively.

      $ Q_{c} = M-(M_{d}+M_{c}), $

      (16)

      where M, $M _{d} $ and $M _{c} $ represent the mass excesses of the parent nucleus, daughter nucleus and emitted particle, respectively. The experimental nuclear mass excesses are taken from Ref. [55]. The experimental half-lives are taken from NUBASE2016 Table [56] and NNDC [57].

      In Table 1, it is seen that the P values extracted from the ImSahuA relationship and those extracted from the ImSahuB relationship are different because of the differences of the inputting the charge radius formulas. It indicates that the P values are dependent on the charge radius to a certain extent, which is consistent with the calculations of Qian et al [58]. Furthermore, here it is necessary to point out that the P values are dependent strongly on the theoretical models. Relevant study suggests that the P value differences from different models amount to several orders of magnitude [14, 15, 5861].

      As can be observed from Eq. (12), ${\rm{log}} _{10} P$ will be linear with ($A _{c} $-1)/3 if ${\rm{log}} _{10} $$P _{\alpha } $ is assumed as a constant. $P _{\alpha } $ may be the average preformation probability of $ \alpha $-particle for different nuclei. In Fig. 1, we plot -${\rm{log}} _{10} P$ versus ($A _{c} $-1)/3 for even-even (e-e) parent nuclei. According to Eq. (12), a best fitting line with $P _{\alpha } $ = 0.0624 for Fig. 1 (a) is obtained. For Fig. 1 (b), the obtained $P _{\alpha } $ value is 0.0552. In relevant studies, the $P _{\alpha } $ values were determined [14, 15, 60, 61]. The $P _{\alpha } $ values within a generalized liquid drop model (GLDM) and a unified fission model (UFM) were 0.0290 [14] and 0.0338 [15], respectively. By a systematic study on the correlation between the $ \alpha $-decay and cluster radioactivity, the $P _{\alpha } $ value given by Poenaru et al. was 0.0161 [60]. Bhattacharya and Gangopadhyay determined the $P _{\alpha } $ within the DDM3Y1 model combining the relativistic mean filed model, whose value was 0.0193 [61]. Therefore, the obtained $P _{\alpha } $ values of our work are comparable to those of the other models.

      Figure 1.  (Color online) Negative of logarithm of preformation factor (-${\rm{log}} _{10} P$) for e-e parent nuclei as a function of ($A _{c} $-1)/3. The left and right panels refer to the cases of the ImSahuA and ImSahuB relationships, respectively.

      Using the same method, our study can be extended to the case of the odd-A parent nuclei. The empirical -${\rm{log}} _{10} P$ values as functions of ($A _{c} $-1)/3 for the odd-A parent nuclei and the best fitting lines are shown in Fig. 2. The obtained $P _{\alpha } $ values by the ImSahuA and ImSahuB relationships are 0.0392 and 0.0414, respectively. The two values are close to the $P _{\alpha } $ values of 0.0214 within the GLDM [14], 0.0262 within the UFM [15] and 0.0135 from Ref. [61].

      Figure 2.  (Color online) Same as Fig. 1, but for the odd-A parent nuclei.

      Within Eqs. (1-14) and Eqs. (1-13, 15) by inputting the fitting $P _{\alpha } $ values, the cluster radioactivity half-lives for the nuclei in the trans-lead region are calculated. The calculated half-lives are listed in the last two columns of Table 1. For comparison with the values by the Sahu formula, the half-lives extracted by the Sahu formula are also given in column 8. Compared with the half-lives within the Sahu formula, it is seen that the calculated half-lives by the ImSahuA and ImSahuB relationships are closer to the experimental ones. This indicates that the accuracy of the Sahu relationship is improved by using the accurate charge radius formulas and the fitting expressions of the cluster preformation probability.

      To observe the global deviation between the experimental and the calculated half-lives intuitively, the ${\rm{log}} _{10} $HF (${\rm{log}} _{10} $HF = ${\rm{log}} _{10}\frac{T_{Cal.}}{T_{Expt.}} = \log _{10}T_{cal.}-\log_{10}T_{Expt.} $) values as functions of the neutron number N for 28 parent nuclei, 17 e-e nuclei and 11 odd-A nuclei are plotted in Fig. 3. Generally, ones believe that if the ${\rm{log}} _{10} $HF value is within a factor of 1.0, the calculated half-lives will be in agreement with the experimental data [6264]. From Fig. 3, it is seen that the accuracies of the ImSahuA and ImSahuB relationships are improved evidently. To show the global deviation quantitatively, the average deviation $ \bar{\delta} $ and standard deviation $ \sqrt{\bar{\delta^2}} $ are calculated. The $ \bar{\delta} $ and $ \sqrt{\bar{\delta^2}} $ values for 28 (n = 28. n means the numbers of nuclei.) heavy nuclei as well as e-e and odd-A subsets of the full data set using the three empirical formulas are listed in Table 2. As can be seen from Table 2, the $ \bar{\delta} $ and $ \sqrt{\bar{\delta^2}} $ values of the total, e-e and odd-A nuclei within the ImSahuA and ImSahuB formulas are much smaller than those within the Sahu formula. It suggests that the accuracies of the ImSahuA and ImSahuB formulas become higher, which is consistent with the conclusion of Fig. 3. Therefore, the reasonableness of Eq. (12) is tested by the high accuracies of the two new relationships and the extracted $P _{\alpha } $ values. In addition, by comparing the ${\rm{log}} _{10} $HF distributions ($ \bar{\delta} $ values or $ \sqrt{\bar{\delta^2}} $ values) between the ImSahuA and the ImSahuB relationships of Fig. 3 (Table 2), it is easy to know that the accuracy of the ImSahuB formula is higher than that of the ImSahuA formula since Eq. (15) is more precise than Eq. (14). This indicates that an accurate charge radius formula is important for estimating the cluster decay half-lives.

      Formulas $ \bar{\delta} $ $ \sqrt{\bar{\delta^2}} $
      Total (n = 28)e-e (n = 17)odd-A (n = 11) Total (n = 28)e-e (n = 17)odd-A (n = 11)
      Sahu1.0480.7261.5431.3510.8701.863
      ImSahuA0.5920.5940.5890.7290.7280.733
      ImSahuB0.5360.5340.5390.6500.6490.652

      Table 2.  The $ \bar{\delta} $ and $ \sqrt{\bar{\delta^2}} $ values between the experimental and calculated cluster radioactivity half-lives for 28 heavy nuclei as well as for e-e and odd-A subsets of the full data set using the Sahu, ImSahuA and ImSahuB relationships.

      Figure 3.  (Color online) The ${\rm{log}} _{10} $HF values as functions of N of (a) 28 parent nuclei, (b) the e-e nuclei and (c) the odd-A nuclei.

      For the cluster radioactivity of some heavy nuclei, only the lower limit of the half-lives are measured. Thus, the experimental cluster radioactivity half-lives with the lower limit have been a ground to test the ImSahuA and ImSahuB relationships. The experimental half-lives and the calculated ones within the Sahu, ImSahuA and ImSahuB formulas are listed in Table 3, which can be seen from columns 5-8 of Table 3. In Table 3, the parent nuclei, emitted clusters, l values and experimental $Q _{c} $ values are shown on the top four columns. The experimental $Q _{c} $ values and half-lives are taken from the same references as Table 1. To see the agreement between the experimental half-lives and calculated ones clearly, the half-lives within the Sahu, ImSahuA and ImSahuB relationships and the corresponding experimental half-lives with the lower limit are shown in Fig. 4. From Fig. 4, it is seen that the half-lives of most parent nuclei are enhanced after the modification except for the half-lives of the $ ^{24} $Ne emission from $ ^{236} $U and $ ^{28} $Mg emission from $ ^{232} $U. For the $ ^{24} $Ne radioactivity of $ ^{236} $U, its half-life within the ImSahuA relationship decreases comparing to the half-life within the Sahu relationship. However, the half-life within the ImSahuB relationship is lower than that within the Sahu one as to the $ ^{28} $Mg radioactivity of $ ^{232} $U. Nevertheless, the calculated half-lives are larger than the corresponding experimental lower limit. In addition, as can be seen form Fig. 4 and Table 3, for the decays of $ ^{232} $Th$ \longrightarrow ^{208} $Hg+$ ^{24} $Ne, $ ^{233} $U$ \longrightarrow ^{205} $Hg+$ ^{28} $Mg and $ ^{235} $U$ \longrightarrow ^{207} $Hg+$ ^{28} $Mg, although the half-lives within the Sahu relationship are not in agreement with the experimental ones, the half-lives estimated by the ImSahuA and ImSahuB relationships are larger than or closer to the lower limit of the experimental half-lives. Thus, according to the above analysis one can see that the experimental half-lives in Table 3 can be reproduced better by the ImSahuA and ImSahuB relationships. The validity of the two new relationships is then tested again. As to the accuracies of the ImSahuA and ImSahuB relationships, it is difficult to determine which is higher by the current experimental data.

      Parent nucleiEmitted clustersl$Q _{c} $(MeV)${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s)
      Expt.Expt.SahuImSahuAImSahuB
      $ ^{226} $Th$ ^{14} $C030.67>15.3016.5416.9617.32
      $ ^{226} $Th$ ^{18} $O045.73>16.8016.7617.2517.45
      $ ^{232} $Th$ ^{24} $Ne054.50>29.2028.2228.2328.82
      $ ^{230} $U$ ^{24} $Ne061.55>18.2020.4720.9821.67
      $ ^{236} $U$ ^{24} $Ne055.95>25.9028.7128.3329.08
      $ ^{232} $Th$ ^{26} $Ne055.97>29.2030.3830.4730.82
      $ ^{236} $U$ ^{26} $Ne056.75>25.9029.1629.1729.51
      $ ^{232} $U$ ^{28} $Mg074.32>22.2623.9224.1923.09
      $ ^{233} $U$ ^{28} $Mg374.23>27.5924.1426.1524.36
      $ ^{235} $U$ ^{28} $Mg172.20>28.1026.6828.3427.92
      $ ^{237} $Np$ ^{30} $Mg274.82>27.6026.1928.3127.69
      $ ^{240} $Pu$ ^{34} $Si091.03>25.5226.5126.6127.55
      $ ^{241} $Am$ ^{34} $Si393.93>24.4124.9627.3527.44

      Table 3.  The calculated cluster radioactivity half-lives within the Sahu, ImSahuA and ImSahuB relationships, but for the cases that the experimental half-lives with the lower limit are measured.

      Figure 4.  (Color online) The comparison between the cluster radioactivity half-lives within the Sahu, ImSahuA and ImSahuB relationships and the corresponding experimental lower limit.

      Encouraged by the good agreement between the experimental half-lives and the calculations within the ImSahuA and ImSahuB formulas, we will attempt to predict the cluster radioactivity half-lives that have not yet been measured for the nuclei in the trans-lead region. A relevant study suggests that the l effect on the cluster decay half-life is very small because the centrifugal potential is much smaller than that of the Coulomb potential [15]. To show the influence on the cluster emission half-life visibly, by taking the decays of $ ^{221} $Fr$ \longrightarrow ^{207} $Tl+$ ^{14} $C and $ ^{233} $U$ \longrightarrow ^{209} $Pb+$ ^{24} $Ne as examples, the logarithms of half-lives within the ImSahuA and ImSahuB relationships as functions of l are plotted in Fig. 5. As can be seen from Fig. 5, it is easy to see the l effect is not so important for the cluster decay half-life. So the l contribution will not be taken into account in the subsequent predictions. By the ImSahuA and ImSahuB formulas, the predicted half-lives of the $ ^{8} $Be, $ ^{12,14} $C, $ ^{15} $N, $ ^{16-20} $O, $ ^{20-26} $Ne, $ ^{24-28} $Mg and $ ^{30-34} $Si radioactivity are shown in Table 4, which are helpful for searching for the new cluster emitters in future experiments.

      Parent nucleiEmitted clusters$Q _{c} $(MeV)${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s) Parent nucleiEmitted clusters$Q _{c} $(MeV)${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s)
      Expt.ImSahuAImSahuB Expt.ImSahuAImSahuB
      $ ^{213} $At$ ^{8} $Be12.3024.1423.11 $ ^{221} $Ac$ ^{16} $O43.0818.9217.91
      $ ^{214} $At$ ^{8} $Be13.9317.5416.70 $ ^{222} $Ac$ ^{16} $O43.6117.1016.47
      $ ^{215} $At$ ^{8} $Be14.8415.0914.80 $ ^{223} $Ac$ ^{16} $O43.6018.1018.23
      $ ^{216} $At$ ^{8} $Be14.0717.0516.90 $ ^{224} $Ac$ ^{16} $O41.7219.9220.38
      $ ^{217} $At$ ^{8} $Be13.1020.9220.56 $ ^{225} $Ac$ ^{16} $O40.0223.7223.75
      $ ^{214} $Rn$ ^{8} $Be14.5216.3416.22 $ ^{222} $Th$ ^{16} $O45.7315.0415.53
      $ ^{215} $Rn$ ^{8} $Be16.3411.5411.25 $ ^{223} $Th$ ^{16} $O46.5714.9115.01
      $ ^{216} $Rn$ ^{8} $Be17.069.259.13 $ ^{224} $Th$ ^{16} $O46.4814.0014.48
      $ ^{217} $Rn$ ^{8} $Be16.3311.5411.24 $ ^{225} $Th$ ^{16} $O44.6617.4417.56
      $ ^{218} $Rn$ ^{8} $Be15.0014.7814.66 $ ^{226} $Th$ ^{16} $O42.6619.3619.87
      $ ^{215} $Fr$ ^{8} $Be15.4314.7414.41 $ ^{223} $Pa$ ^{16} $O47.1115.0715.11
      $ ^{216} $Fr$ ^{8} $Be16.9010.2810.12 $ ^{224} $Pa$ ^{16} $O47.4713.5814.00
      $ ^{217} $Fr$ ^{8} $Be17.638.978.68 $ ^{225} $Pa$ ^{16} $O47.3414.7314.83
      $ ^{218} $Fr$ ^{8} $Be16.9110.2210.08 $ ^{226} $Pa$ ^{16} $O45.5616.0516.51
      $ ^{219} $Fr$ ^{8} $Be15.5414.3714.05 $ ^{227} $Pa$ ^{16} $O43.4320.1220.22
      $ ^{217} $Fr$ ^{12} $C28.1417.2416.40 $ ^{223} $Ac$ ^{18} $O42.4322.5121.10
      $ ^{218} $Fr$ ^{12} $C29.3114.2413.67 $ ^{224} $Ac$ ^{18} $O43.2720.0019.03
      $ ^{219} $Fr$ ^{12} $C29.6514.3414.43 $ ^{225} $Ac$ ^{18} $O43.4520.8320.61
      $ ^{220} $Fr$ ^{12} $C28.2316.2416.57 $ ^{226} $Ac$ ^{18} $O41.8422.2522.40
      $ ^{221} $Fr$ ^{12} $C26.9219.6519.67 $ ^{227} $Ac$ ^{18} $O40.2826.0725.74
      $ ^{218} $Ra$ ^{12} $C30.4413.0213.39 $ ^{224} $Th$ ^{18} $O44.5619.0019.21
      $ ^{219} $Ra$ ^{12} $C31.8511.4011.47 $ ^{225} $Th$ ^{18} $O45.5418.6818.46
      $ ^{220} $Ra$ ^{12} $C32.0210.3610.71 $ ^{226} $Th$ ^{18} $O45.7317.2517.45
      $ ^{221} $Ra$ ^{12} $C30.5813.4513.54 $ ^{227} $Th$ ^{18} $O44.2020.6120.39
      $ ^{222} $Ra$ ^{12} $C29.0515.4515.83 $ ^{228} $Th$ ^{18} $O42.2822.4822.70
      $ ^{219} $Ac$ ^{12} $C31.6212.5012.53 $ ^{225} $Pa$ ^{18} $O45.1820.1519.87
      $ ^{220} $Ac$ ^{12} $C32.6110.1610.46 $ ^{226} $Pa$ ^{18} $O45.6918.2418.38
      $ ^{221} $Ac$ ^{12} $C32.7810.6210.69 $ ^{227} $Pa$ ^{18} $O45.8719.0918.87
      $ ^{222} $Ac$ ^{12} $C31.4112.0412.38 $ ^{228} $Pa$ ^{18} $O44.5019.9520.13
      $ ^{223} $Ac$ ^{12} $C29.6915.7915.86 $ ^{229} $Pa$ ^{18} $O42.5424.1923.94
      $ ^{219} $Fr$ ^{14} $C29.6514.3414.43 $ ^{225} $Ac$ ^{20} $O41.6626.2824.70
      $ ^{220} $Fr$ ^{14} $C28.2316.2416.57 $ ^{226} $Ac$ ^{20} $O42.7723.0821.99
      $ ^{222} $Fr$ ^{14} $C25.6321.8022.05 $ ^{227} $Ac$ ^{20} $O43.0923.8023.47
      $ ^{223} $Fr$ ^{14} $C24.4625.3625.31 $ ^{228} $Ac$ ^{20} $O41.8524.5924.68
      $ ^{220} $Ra$ ^{14} $C32.0210.3610.71 $ ^{229} $Ac$ ^{20} $O40.5428.2027.75
      $ ^{225} $Ra$ ^{14} $C25.2024.4424.58 $ ^{226} $Th$ ^{20} $O43.1923.4223.58
      $ ^{221} $Ac$ ^{14} $C32.7810.6210.69 $ ^{227} $Th$ ^{20} $O44.4622.6122.28
      $ ^{222} $Ac$ ^{14} $C31.4112.0412.38 $ ^{228} $Th$ ^{20} $O44.7220.8921.05
      $ ^{223} $Ac$ ^{14} $C29.6915.7915.86 $ ^{229} $Th$ ^{20} $O43.4024.2523.93
      Continued on next page

      Table 4.  The predicted cluster radioactivity half-lives within the ImSahuA and ImSahuB relationships. All the $Q _{c} $ values are extracted by Eq. (16). In the extraction, the mass excesses are taken from Ref. [55]. “#” means only the empirical mass excesses for the parent and/or daughter nuclei are given in Ref. [55].

      Figure 5.  (Color online) The logarithms of the half-lives for $ ^{221} $Fr$ \longrightarrow ^{207} $Tl+$ ^{14} $C and $ ^{233} $U$ \longrightarrow ^{209} $Pb+$ ^{24} $Ne versus l within the ImSahuA and ImSahuB relationships.

      Besides the cluster radioactivity of the trans-lead region, a new island of cluster emitters around the doubly magic nucleus $ ^{100} $Sn has been paid attention by many researchers [6579]. For the cluster radioactivity in the trans-tin region, only the half-life of $ ^{12} $C emission from $ ^{114} $Ba was measured, whose value was determined as $ \geq 10^{3} $ s at Dubna (Dubna94) [80] and $ \geq $1.1$ \times 10^{3} $ s (1.7$ \times 10^{4} $ s) at GSI (GSI95) [81, 82], respectively. However, the $ ^{12} $C decay of $ ^{114} $Ba was not observed in the subsequent experiment [83], which suggested the branching ratio for the $ ^{12} $C decay is lower than the limit obtained in the GSI95 measurement. By consulting the NUBASE2016 Table the experimental lower limit of the half-life of the $ ^{12} $C emission from $ ^{114} $Ba is found to be $ > $10$ ^{4.13} $ s [56]. So the half-life of the $ ^{12} $C radioactivity from $ ^{114} $Ba has not yet been determined accurately. Nevertheless, the experimental half-life with the lower limit of the $ ^{12} $C emission from $ ^{114} $Ba can be used to test various cluster decay models.

      Then, we extend our method to study the $ ^{12} $C radioactivity of $ ^{114} $Ba. The calculated decimal logarithm half-lives within the Sahu, ImSahuA and ImSahuB formulas are 3.994 s, 9.079 s and 9.355 s, respectively. By a comparison between the calculated half-lives and the experimental lower limit of the half-life, it is easy to see that the experimental half-life of the $ ^{12} $C radioactivity of $ ^{114} $Ba can be reproduced better within the ImSahuA and ImSahuB formulas. Therefore, we have enough confidence to predict the half-lives of the cluster radioactivity for the nuclei in the trans-tin region. Very recently, the cluster radioactivity half-lives of the neutron-deficient nuclei in the trans-tin region are investigated by the effective liquid drop model, the GLDM and several sets of analytic formulas [84]. It was found that the $ \alpha $-like cluster radioactivity, such as $ ^{8} $Be, $ ^{12} $C, $ ^{16} $O, $ ^{20} $Ne, $ ^{24} $Mg and $ ^{28} $Si emissions, decaying to the $ N_{d} $ = 50 daughter nuclei, was the most probable. Thus, the half-lives of these most probable cluster radioactivity are predicted by the ImSahuA and ImSahuB formulas, which are listed in Table 5. In Table 5, the $Q _{c} $ values are still extracted by Eq. (16). For the unknown nuclear masses, whose values are from the empirical masses of the NUBASE2016 table [56] or the WS4 mass table [85] because relevant studies showed that the WS4 mass model can predict the experimental nuclear masses and decay energies accurately [8588]. We hope these predicted half-lives are useful for identifying the new cluster emissions of the trans-tin region in future measurements.

      Parent nucleiEmitted clusters$Q _{c} $${\rm{log}} _{10}T_{1/2} $ (s)${\rm{log}} _{10}T_{1/2} $ (s)
      (MeV)ImSahuAImSahuB
      $ ^{108} $Xe$ ^{\ast } $$ ^{8} $Be10.406.065.92
      $ ^{109} $Cs$ ^{\ast } $$ ^{8} $Be10.008.698.37
      $ ^{110} $Ba$ ^{\ast } $$ ^{8} $Be10.188.468.32
      $ ^{110} $Xe$ ^{12} $C15.7215.1415.44
      $ ^{111} $Cs$ ^{\#} $$ ^{12} $C18.569.819.81
      $ ^{112} $Ba$ ^{\ast } $$ ^{12} $C21.733.894.13
      $ ^{113} $La$ ^{\ast } $$ ^{12} $C21.575.765.74
      $ ^{114} $Ce$ ^{\ast } $$ ^{12} $C22.264.694.95
      $ ^{114} $Ba$ ^{16} $O26.4712.7713.17
      $ ^{115} $La$ ^{\ast } $$ ^{16} $O29.809.359.35
      $ ^{116} $Ce$ ^{\ast } $$ ^{16} $O33.224.665.00
      $ ^{117} $Pr$ ^{\ast } $$ ^{16} $O33.056.826.80
      $ ^{118} $Nd$ ^{\ast } $$ ^{16} $O33.256.506.86
      $ ^{118} $Ce$ ^{\ast } $$ ^{20} $Ne35.0316.3016.42
      $ ^{119} $Pr$ ^{\ast } $$ ^{20} $Ne37.5215.1014.74
      $ ^{120} $Nd$ ^{\ast } $$ ^{20} $Ne40.6810.7810.89
      $ ^{121} $Pm$ ^{\ast } $$ ^{20} $Ne40.5313.4213.05
      $ ^{122} $Sm$ ^{\ast } $$ ^{20} $Ne41.3212.2812.40
      $ ^{122} $Nd$ ^{\ast } $$ ^{24} $Mg46.6515.6615.59
      $ ^{123} $Pm$ ^{\ast } $$ ^{24} $Mg49.7514.8414.20
      $ ^{124} $Sm$ ^{\ast } $$ ^{24} $Mg53.8610.2010.14
      $ ^{125} $Eu$ ^{\ast } $$ ^{24} $Mg53.9812.8012.15
      $ ^{126} $Gd$ ^{\ast } $$ ^{24} $Mg54.9011.5011.44
      $ ^{126} $Sm$ ^{\ast } $$ ^{28} $Si60.0914.1014.76
      $ ^{127} $Eu$ ^{\ast } $$ ^{28} $Si63.5314.0313.98
      $ ^{128} $Gd$ ^{\ast } $$ ^{28} $Si67.9910.0010.61
      $ ^{129} $Tb$ ^{\ast } $$ ^{28} $Si68.6312.2012.13

      Table 5.  The $ ^{8} $Be, $ ^{12} $C, $ ^{16} $O, $ ^{20} $Ne, $ ^{24} $Mg and $ ^{28} $Si emission half-lives in the decay processes where the daughter nuclei with $N _{d} $ around 50 within the ImSahuA and ImSahuB formulas are shown in columns 4 and 5. “#” and “*” denote the $Q _{c} $ values are extracted from the empirical mass excesses of the NUBASE2016 table [55] and the theoretical ones of the WS4 mass model [85], respectively.

    IV.   CONCLUSIONS
    • In this article, the Sahu relationship has been improved by introducing two accurate root-mean-square charge radius formulas and an analytic expression of the cluster preformation probability. The improved Sahu relationships with the two charge radius formulas are called as the ImSahuA relationship and the ImSahuB relationship, respectively. Within the Sahu, ImSahuA and ImSahuB relationships, the cluster radioactivity half-lives of the trans-lead nuclei and the half-life of the $ ^{12} $C radioactivity of $ ^{114} $Ba are calculated. The calculated results allow us to draw the following conclusions:

      (i) The experimental half-lives of the trans-lead nuclei and the experimental half-life lower limit of the $ ^{12} $C decay from $ ^{114} $Ba within the ImSahuA and ImSahuB relationships are reproduced better than those within the Sahu relationship;

      (ii) By the linear correlation between the cluster preformation probability and the mass number of the emitted cluster, the extracted $P _{\alpha } $ value is close to those given by other models;

      (iii) The high accuracies of the ImSahuA and ImSahuB relationships and the extracted $P _{\alpha } $ values demonstrate the validity of the analytic expression of the cluster preformation probability;

      (iv) The accuracy of the ImSahuB formula is higher than that of the ImSahuA formula since the charge radius formula with the Casten factor is more accurate. This indicates that an accurate charge radius formula is important for calculating the half-life of the cluster radioactivity.

      Finally, the cluster radioactivity half-lives of the trans-lead and trans-tin nuclei that are not experimentally available are predicted by the ImSahuA and ImSahuB formulas, which might be helpful for searching for the new cluster emitters in the two regions in future experiments.

    ACKNOWLEDGMENTS
    • We thank professor Shangui Zhou, professor Ning Wang and professor Fengshou Zhang for helpful discussions.

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