Introduction

In recent years, the CR of unstable heavy nuclei has received attention by many researchers1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. The CR was first predicted in 1980 by Sandulescu, Poenaru and Greiner27, and then it was confirmed by Rose and Jones in 1984 for the 14C radioactivity from 223Ra28. From then on, the emissions of 14C, 20O, 23F, 22,24−26Ne, 28,30Mg and 32,34Si, have been experimentally observed in the mass region where the parent nuclei with their charge numbers Z = 87–9629,30,31,32,33. In this region all cluster emissions have closed shell daughters, i.e. the daughter nuclei are 208Pb or its neighbors. It is well known that α-decay is an important decay mode for unstable heavy nuclei34,35,36,37, which can be described by the quantum tunneling effect through a potential barrier38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56. Usually the CR is seen as a cold asymmetric fission process, whose case is similar to α-decay. On the basis of the fission knowledge57,58 and the quantum tunneling effect, many phenomenological and microscopic models were developed to construct the potential barrier of CR, and furthermore to estimate the half-life1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. In addition to these models, many analytic formulas were proposed by fitting the experimental half-lives and Q values of CR processes, such as the UDL59,60, UNIV61, Horoi62, TM63, BKAG64, NRDX65, and VSS66 formulas.

Besides the CR of the parent nuclei with Z = 87–96, two new islands of cluster emitters have been predicted by many models67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90. One is in the superheavy nuclei (SHN) region67,68,69,70,71,72,73,74,75, the other is the in the trans-tin region decaying into the daughter nuclei close to 100Sn76,77,78,79,80,81,82,83,84,85,86,87,88,89,90. For the CR of the SHN, Poenaru et al. changed the concept of the CR to allow emitted particles with Ze > 28 from the parents with Z > 110 (daughter around 208Pb). They found that the CR is one of the most important decay modes and its branching ratio is larger than that of the α-decay for Z ≥ 121 nuclei by the analytic superasymmetric fission (ASAF) model69,70,71. Additionally, it is shown that the shell effects at 208Pb and N = 184 strongly influence the CR half-lives69,70,71. Later, the calculations within several models gave similar predictions to that of the ASAF model72,73. For the CR in the trans-tin region, the half-life of the 12C emission of 114Ba has been measured by Oganessian et al. at Dubna (Dubna94)91 and by Guglielmetti et al. at GSI (GSI95)92,93. The obtained experimental half-lives of Dubna94 and GSI95 were ≥103 s91 and ≥1.1 × 103 s (1.7 × 104 s)92,93, respectively. However, the 12C decay of 114Ba was not observed in the later measurement of Guglielmetti et al.(GSI97)94, which suggested the branching ratio for the 12C decay is lower than the limit obtained in the GSI95 experiment. By consulting the NUBASE2016 Table the experimental lower limit of the half-life of the 12C emission from 114Ba is found to be >4.13 s (in logarithmic scale)95. So the half-life of the 12C radioactivity from 114Ba has not yet been determined accurately.

As a matter of fact, the CR of the trans-tin region has been predicted since 198996. In recent decades, the CR half-lives of the emitters from 8Be to 32S have been estimated within many models by inputting different kinds of Q values76,77,78,79,80,81,82,83,84,85,86,87,88,89,90. Very recently, the CR of the SHN was studied systematically by several models. It is shown that the CR half-lives are strongly dependent on the models used97. This drives us to wonder that in the trans-tin region whether the CR island exists if other models are employed. Furthermore, whether the CR half-lives extracted from different models are similar to each other if we input the same Q values. This constitutes the motivation of this article. In this article, we will explore the CR of neutron-deficient nuclei in the trans-tin region and examine the model dependence of half-lives using the ELDM, GLDM, and several sets of analytic formulas (UDL, UNIV, Horoi, TM, BKAG, NRDX and VSS formulas). The paper is organized as follows. In section 2, the theoretical approaches are introduced. The numerical results and discussions are presented in section 3. Some conclusions are drawn in the last section.

Models

The ELDM and GLDM are successful models for describing the processes of proton emission, α-decay, and CR in a unified framework. The details of them can be found in refs. 10,11,12,13,14,15,16.

In the unified fission model the partial half-life of a cluster emitter is simply defined as

$$T=\frac{\mathrm{ln}\,2}{{\nu }_{0}P},$$
(1)

where ν0 is the frequency of assaults on the barrier. P is the barrier penetration probability.

For the ELDM, in the combination of the Varying Mass Asymmetry Shape and Werner-Wheeler’s inertia, the ν0 value is taken as 1.0 × 1022 s−1 10,11,12,13, and P is calculated by

$$P=\exp \left[-\frac{2}{\hslash }{\int }_{{\zeta }_{0}}^{{\zeta }_{c}}\sqrt{2\mu [V(\zeta )-Q]}d\zeta \right],$$
(2)

where μ is the Werner-Wheeler’s inertia inertial coefficient. ζ0 and ζc are the inner and outer classical turning points, respectively. The two classical turning points are expressed as \({\zeta }_{0}=R-{\bar{R}}_{1}\) and ζc = \(\frac{{Z}_{e}{Z}_{d}{e}^{2}}{2Q}+\sqrt{{\left(\frac{{Z}_{e}{Z}_{d}{e}^{2}}{2Q}\right)}^{2}+\frac{l(l+1){\hslash }^{2}}{2\mu Q}}\), respectively. Here R is the radius of the parent nucleus. \({\bar{R}}_{1}\) represents the the final radius of the emitted cluster.

The effective one-dimensional total potential energy is given by10,11,12,13

$$V={V}_{c}+{V}_{s}+{V}_{l}\mathrm{}.$$
(3)

The Coulomb contribution Vc is determined by using an analytical solution of the Poisson’s equation for a uniform charge distribution system. The effective surface potential can be calculated by

$${V}_{s}={\sigma }_{eff}({S}_{e}+{S}_{d}),$$
(4)

where Se and Sd are the surface areas of the two spherical fragments. σeff is the effective surface tension, which is defined as

$${\sigma }_{eff}=\frac{1}{\mathrm{4(}{R}^{2}-{\bar{R}}_{1}^{2}-{\bar{R}}_{2}^{2})}\left[Q-\frac{3}{20\pi {\varepsilon }_{0}}{e}^{2}\left(\frac{{Z}^{2}}{R}-\frac{{Z}_{e}^{2}}{{\bar{R}}_{1}}-\frac{{Z}_{d}^{2}}{{\bar{R}}_{2}}\right)\right],$$
(5)

where \({\bar{R}}_{2}\) is the final radius of the daughter fragment.

The centrifugal potential energy beyond the scission point has an usual expression

$${V}_{l}=\frac{{\hslash }^{2}}{2\bar{\mu }}\frac{l(l+\mathrm{1)}}{{\zeta }^{2}},$$
(6)

where l is the angular momentum of the emitted particle, \(\bar{\mu }={M}_{e}{M}_{d}/({M}_{e}+{M}_{d})\) is the reduced mass of the two separated fragments. Me and Md represent their atomic masses.

In the framework of the GLDM, ν0 is givn by the following classic method14,15,16

$${\nu }_{0}=\frac{1}{2R}\sqrt{\frac{2{E}_{e}}{{M}_{e}}},$$
(7)

where Ee and Me are the kinetic energy and mass of cluster, respectively.

P is calculated by using the WKB approximation, which is written by

$$P=\exp \left[-\frac{2}{\hslash }{\int }_{{R}_{{\rm{in}}}}^{{R}_{{\rm{out}}}}\sqrt{2B(r)(E(r)-{E}_{sph})}dr\right]\mathrm{}.$$
(8)

The deformation energy (relative to the sphere) is small up to the rupture point between the fragments. Rin is the distance between the mass centers of the portions of the initial sphere separated by a plane perpendicular to the deformation axis to assume the volume conservation of the future fragments. Rout = \(\frac{{Z}_{e}{Z}_{d}{e}^{2}}{2Q}+\sqrt{{\left(\frac{{Z}_{e}{Z}_{d}{e}^{2}}{2Q}\right)}^{2}+\frac{l(l+1){\hslash }^{2}}{2\mu Q}}\). The inertia B(r) = μ(1 + 1.3 f(r)), which can simulate a rapid variation of the friction force effects only at the moment of the neck rupture between the nascent fragments. If r ≤ Rcont, \(f(r)=\sqrt{\frac{{R}_{cont}-r}{{R}_{cont}-{R}_{in}}}\). Otherwise, f(r) = 014,15,16. Here Rcont = Re + Rd, Re and Rd are the radii of the cluster and daughter nucleus, respectively.

The analytic formulas (UDL59,60, UNIV61, Horoi62, TM63, BKAG64, NRDX65, and VSS66 formulas) used in this article are expressed as

$${\log }_{10}{T}_{\mathrm{1/2}}({\rm{UDL}})=a\sqrt{\mu }{Z}_{e}{Z}_{d}{Q}^{-\mathrm{1/2}}+b{[\mu {Z}_{e}{Z}_{d}({A}_{e}^{\mathrm{1/3}}+{A}_{d}^{\mathrm{1/3}})]}^{\mathrm{1/2}}+c,$$
(9)
$$\begin{array}{rcl}{\log }_{10}{T}_{\mathrm{1/2}}({\rm{UNIV}}) & = & a{(\mu {Z}_{e}{Z}_{d}{R}_{b})}^{\mathrm{1/2}}\times [\arccos \sqrt{r}-\sqrt{r\mathrm{(1}-r)}]\\ & & +b({A}_{e}-\mathrm{1)}+[{\log }_{10}(\mathrm{ln}\,\mathrm{2)}-{\log }_{10}{\nu }_{0}],\end{array}$$
(10)
$${\log }_{10}{T}_{\mathrm{1/2}}({\rm{Horoi}})=(a{\mu }^{0.416}+b[({Z}_{e}{Z}_{d}{)}^{0.613}{Q}^{-\mathrm{1/2}}-\mathrm{7]}+(c{\mu }^{x}+d),$$
(11)
$$\begin{array}{l}{\log }_{10}{T}_{\mathrm{1/2}}({\rm{TM}})=(a{Z}_{e}+b)({Z}_{d}/Q{)}^{\mathrm{1/2}}+c{Z}_{e}+d,\end{array}$$
(12)
$$\begin{array}{l}{log}_{10}{T}_{\mathrm{1/2}}({\rm{BKAG}})=(a{A}_{e}\eta +b{Z}_{e}{\eta }_{z}){Q}^{-\mathrm{1/2}}+c,\end{array}$$
(13)
$$\begin{array}{l}{\log }_{10}{T}_{\mathrm{1/2}}({\rm{NRDX}})=a\sqrt{\mu }{Z}_{e}{Z}_{d}{Q}^{-\mathrm{1/2}}+b\sqrt{\mu }{({Z}_{e}{Z}_{d})}^{\mathrm{1/2}}+c,\end{array}$$
(14)
$$\begin{array}{l}{\log }_{10}{T}_{\mathrm{1/2}}({\rm{VSS}})=a{Z}_{e}{Z}_{d}{Q}^{-\mathrm{1/2}}+b{Z}_{e}{Z}_{d}+c+d,\end{array}$$
(15)

where T1/2 is the CR half-life, which is measured in seconds. μ = AeAd/(Ae + Ad) is the reduced mass. Ae and Ad represent the mass numbers of the emitted particle and daughter nucleus, respectively. Ze and Zd denote the charge numbers of the two fragments. In Eq. (10), r = Rt/Rb, Rt and Rb stand for the first and second turning points of the barrier, respectively. The two turning points are defined as Rt = 1.2249(\({A}_{e}^{1/3}\) + \({A}_{d}^{1/3}\)) and Rb = 1.43998ZeZd/Q. The frequency of assaults ν0 is taken as 1022.01 s−1. In Eq. (13), η (ηz) represents the mass (charge) asymmetry, whose form is written as \(\eta =\frac{{A}_{d}-{A}_{e}}{A}\) \(({\eta }_{z}=\frac{{Z}_{d}-{Z}_{e}}{Z})\). The parameters in Eqs. (9–15) are determined by fitting the experimental half-lives and Q values60,61,62,63,64,65,66, which are listed in Table 1.

Table 1 The parameter sets of UDL, UNIV, Horoi, TM, BKAG, NRDX, and VSS formulas.
Full size table

Results and discussions

It is well known that the CR half-lives are dependent on the Q values, which can be extracted by

$$Q=M-({M}_{d}+{M}_{e}),$$
(16)

where M, Md and Me represent the masses of the parent nucleus, daughter nucleus and emitted particle, respectively. The experimental nuclear masses are taken from ref. 95. For the unknown nuclear masses, in the CR half-life calculations whose values can be replaced by the theoretical nuclear masses extracted from the WS4 mass model98 because relevant studies showed that the WS4 mass model can predict the experimental nuclear masses and decay energies accurately98,99. Especially for our recent work on SHN, it suggested that the WS4 mass model is the most accurate one to reproduce the experimental α-decay energies of the SHN100.

Firstly, we calculate the 12C decay half-life of 114Ba using the ELDM, GLDM and some analytic formulas (UDL, UNIV, Horoi, TM, BKAG, NRDX and VSS formulas) and further test the predicted accuracies of these models by comparing to the experimental half-life. The calculated and experimental half-lives are presented in Table 2. The first and second columns are the parent nucleus and daughter nucleus, respectively. The released energy Q is listed in column 3101. Columns 4–12 give the 12C decay half-lives of 114Ba extracted from all the models and formulas. The last column lists the experimental half-life of the 12C decay from 114Ba94. According to Table 2, one can see that only the calculated half-lives by the NRDX and VSS formulas are below the experimental lower limit. The two formulas are simple scaling laws and the coefficients are determined by fitting the experimental data with the parent charge number Z = 87–9665,66. When they are extended to calculate the CR half-lives in trans-tin region, the predicted half-lives deviate from the experimental data. This indicates that the two scaling laws are not so universal and not suitable for estimating the CR half-lives in the trans-tin region. So, the two formulas will not be used to predict the CR half-lives in later calculations. In the following paragraphs by taking 12C, 20Ne and 28Si emissions as examples, the CR half-lives will be predicted by all the models (formulas) except for the NRDX and VSS formulas.

Table 2 Comparison between the experimental half-life of the 12C radioactivity of 114Ba and the estimated ones by the ELDM, GLDM and 7 formulas (The UDL, UNIV, Horoi, TM, BKAG, NRDX and VSS formulas).
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The half-lives of the 12C, 20Ne and 28Si emissions of some isotopes within the ELDM, GLDM, UDL, UNIV, Horoi, TM, and BKAG models (formulas) as functions of the daughter neutron number Nd are plotted in Figs. 1–3. Note that in the calculations by the ELDM and GLDM, the angular momenta carried by emitted particles are selected as 0. From Figs. 1–3, we can see that for each isotopic chain the CR half-lives calculated by the ELDM, GLDM, UDL and UNIV are almost the same. In the ELDM and GLDM, the cluster decay process is assumed as a super-asymmetric fission. The shape evolution process from one spherical nucleus to two separated fragments can be described well by the two models10,11,12,13,14,15,16. The shape evolution described by the two models contains more important nuclear structure information. In the ELDM the contributions of the Coulomb and surface energies to the potential barrier are considered more reasonably. The Coulomb energy is obtained by the exact solution of the Poisson’s equation for the system with a uniform charge distribution. For the surface potential energy, an effective surface tension is introduced. In addition, the inertial coefficient in the prescission phase is calculated with the Werner-Wheeler’s approximation10,11,12,13. In the GLDM, with the quasimolecular shape sequence and nuclear proximity energy, a reasonable configuration of the potential barrier can be obtained. Besides these factors, the accurate nuclear radius, decay asymmetry and assumed decay path are used as well. Thus, the charged particle emissions and nuclear fission can be described successfully by the two models14,15,16. Due to these advantages of the ELDM and GLDM, the predicted half-lives by them for yet unmeasured cluster emissions are more reliable than those by other phenomenological models. So to some extent the ELDM and GLDM can be seen as the standard models for estimating the half-lives of cluster emissions. As to the UDL and UNIV formulas, they are derived from from the α-like R-matrix theory and the fission-like theory, respectively59,60,61. Reasonable physical bases are behind them so that the CR half-lives extracted from the ELDM and GLDM are reproduced with a comparable accuracy by both of the formulas. Here it is worth mentioning that the experimental α-decay half-lives of SHN can be reproduced well by the UNIV formula100. But for the half-lives given by the Horoi62, TM63, and BKAG64 formulas, it is seen from Fig. 1 that they deviate from those by the ELDM and GLDM. Because the three formulas are the simple scaling laws62,63,64, which are similar to the NRDX and VSS formulas65,66. Although a little nuclear structure information is taken into account, their prediction power is not so strong. Moreover, from Fig. 1 the shortest half-lives appear when Nd is 50 for each model. For example, the minimal half-lives of the 12C emission occur for the parent nuclei 110Xe, 111Cs, 112Ba, 113La, and 114Ce. Among these minimal half-lives, the half-life with the daughter nucleus 100Sn (the parent nucleus 112Ba) is shorter than any other minimal half-life. Similar phenomena can also be observed in Figs. 2 and 3. These facts reveal that the CR half-lives are related to the shell effect at Nd = 50, and the shell effect at 100Sn is strongest. To explain the shell effect of the CR half-lives shown in Figs. 1–3, the Q values of the 12C, 20Ne, and 28Si emissions of these isotopic chains as functions of Nd are shown in Fig. 4. As can be seen from Fig. 4, the shell effect at Nd = 50 is very obvious and the shell effect at 100Sn is most pronounced. In the half-life calculations the shell effects are included through the Q values. The Q value shell effects at Nd = 50 and 100Sn lead to the above phenomena. In addition, from Figs. 1–3, it is found that the half-lives by the TM and BKAG formulas become closer and closer to the ones by the ELDM and GLDM with the increase of the emitted cluster mass. This suggests that the TM and BKAG are just suitable for studying heavier cluster emissions.

Figure 1
figure 1

The 12C decay half-lives of the Xe, Cs, Ba, La, and Ce isotopes within the ELDM, GLDM, UDL, UNIV, Horoi, TM, and BKAG models (formulas) versus the neutron numbers of the daughter nuclei Nd.

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Figure 2
figure 2

Same as Fig. 1, but for the 20Ne decay half-lives in the Ce, Pr, Nd, Pm, and Sm isotopes.

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Figure 3
figure 3

Same as Figs. 1 and 2, but for the 28Si decay half-lives in the Sm, Eu, Gd, Tb, and Dy isotopes.

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Figure 4
figure 4

The Q values of the 12C, 20Ne, and 28Si emissions in some isotopic chains versus Nd.

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The clusters 12C, 20Ne and 28Si can be seen as α-like ones76,78. In addition to the half-lives of the α-like CR, the half-lives of the non-α-like78 (26Mg and 30Si) CR are calculated as well. For comparing the similarities and differences between the two sorts of cluster emissions, the half-lives of the 24,26Mg and 28,30Si emissions leading to the daughter nuclei with Zd = 50 are shown in Fig. 5 as functions of Nd, which are calculated with all the models (formulas) except for the NRDX and VSS formulas. From Fig. 5, we can see that for each model the half-lives of the 26Mg and 30Si emissions are much longer than those of the 24Mg and 28Si emissions besides the shell effect at 100Sn. This implies that the non-α-like cluster emissions are more difficult to observe than the α-like ones, which is consistent with the conclusion of refs. 76,78. In Fig. 6, we plot the Q values of the 24,26Mg and 28,30Si emissions decaying to the Zd = 50 daughter nuclei versus Nd. As can be seen from Fig. 6, the Q values of the 24Mg (28Si) emission are much larger than those of the 26Mg (30Si) emission in addition to the strong shell effect at 100Sn. Small Q values of the non-α-like cluster decay lead to the long half-lives.

Figure 5
figure 5

The half-lives of the 24,26Mg and 28,30Si emissions leading to the daughter nuclei with Zd = 50 within the ELDM, GLDM, UDL, UNIV, Horoi, TM, and BKAG models (formulas) versus Nd.

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Figure 6
figure 6

The Q values of the 24,26Mg and 28,30Si emissions leading to the daughter nuclei with Zd = 50 versus Nd.

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According to the above discussions, one can see that a CR most probably occurs in the decay process where the daughter nucleus has Nd = 50 and its half-life is shortest. Moreover, an α-like cluster decay is more probable than a non-α-like cluster decay. Therefore, the predicted half-lives of some α-like cluster emissions decaying to the daughter nuclei with Nd around 50 based on the ELDM, GLDM, UDL and UNIV models (formulas), which include the 8Be, 12C, 16O, 20Ne, 24Mg, and 28Si emissions, are listed in Table 3. We hope our predictions are useful for searching for new CR in trans-tin region in future experiments. At last, to compare these predictions with those of other models, the half-lives of some clusters within a dinuclear system model (DNSM)102 are listed in the last column. Meanwhile, the Q values used in the DNSM calculations are given in the penultimate column. By observing Table 3, it is found that the difference is large between our predicted half-lives and those within the DNSM, which is caused by the differences of the Q values and models. In other words, the predicted CR half-lives are dependent strongly on the Q values and the models. Therefore, it is important to improve the predicted abilities of the nuclear mass models and the approaches of CR by including more reasonable factors of nuclear structure.

Table 3 The 8Be, 16C, 16O, 20Ne, 24Mg, and 28Si emission half-lives in the decay processes where the daughter nuclei with N10T1/2 around 50 within the ELDM, GLDM, UDL and UNIV models (formulas) are shown in columns 5-8.
Full size table

Conclusions

In this article, the CR of the neutron-deficient nuclei in the trans-tin region has been explored within the ELDM, GLDM and several analytic formulas (UDL, UNIV, Horoi, TM, BKAG, NRDX and VSS formulas). Firstly, the 12C decay half-life of 114Ba has been calculated by all the models. By the comparison between the calculated half-lives and the experimental half-life, it is found that the NRDX and VSS formulas are not so suitable for predicting the CR half-lives in the trans-tin region because the calculated half-lives by the two formulas are less than the experimental lower limit. Next by taking the 12C, 20Ne, and 28Si emissions as examples, their half-lives are predicted by the ELDM, GLDM, and the UDL, UNIV, Horoi, TM, and BKAG formulas. Because the UDL formula originates from the α-like R-matrix theory and the UNIV formula roots in the fission-like theory, their predicted accuracies are close to the ones by the ELDM and GLDM. However, the half-lives by the ELDM and GLDM are not reproduced with a comparable accuracy by the simple scaling laws (Horoi, TM, and BKAG formulas). With the increase of the emitted cluster mass, only the half-lives by the TM and BKAG formulas become closer and closer to the ones by the ELDM and GLDM. Meanwhile, it is found that the Q value shell effects at Nd = 50 and 100Sn crucially influence the half-lives, and the daughter nuclei with Nd = 50 have therefore the minimal half-lives. Furthermore, the half-life at 100Sn is lower than any other minimal half-life for the same kind cluster emission. It is observed that the half-lives of the non-α-like CR decaying to the Zd = 50 daughter nuclei are much longer than those of the α-like CR due to the low Q values in the non-α-like CR process. At last, the half-lives of some α-like cluster emissions, such as the 8Be, 12C, 16O, 20Ne, 24Mg, and 28Si emission half-lives, are predicted by the ELDM, GLDM, UDL, and UNIV models (formulas). We hope these predictions are helpful for future experiments.