Cluster radioactivity of neutron-deficient nuclei in trans-tin region

The possibility of cluster radioactivity (CR) of the neutron-deficient nuclei in the trans-tin region is explored by using the effective liquid drop model (ELDM), generalized liquid drop model (GLDM), and several sets of analytic formulas. It is found that the minimal half-lives are at Nd = 50 (Nd is the neutron number of the daughter nucleus) for the same kind cluster emission because of the Q value (released energy) shell effect at Nd = 50. Meanwhile, it is shown that the half-lives of α-like (Ae = 4n, Ze = Ne. Ze and Ne are the charge number and neutron number of the emitted cluster, respectively.) cluster emissions leading to the isotopes with Zd = 50 (Zd is the proton number of the daughter nucleus) are easier to measure than those of non-α-like (Ae = 4n + 2) cases due to the large Q values in α-like cluster emission processes. Finally, some α-like CR half-lives of the Nd = 50 nuclei and their neighbours are predicted, which are useful for searching for the new CR in future experiments.

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. 10-16 . In the unified fission model the partial half-life of a cluster emitter is simply defined as 0 where ν 0 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 ν 0 value is taken as 1.0 × 10 22 s −1 10-13 , and P is calculated by The effective one-dimensional total potential energy is given by [10][11][12][13] = + + .
The Coulomb contribution V c 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 where S e and S d are the surface areas of the two spherical fragments. σ eff is the effective surface tension, which is defined as where R 2 is the final radius of the daughter fragment. The centrifugal potential energy beyond the scission point has an usual expression In the framework of the GLDM, ν 0 is givn by the following classic method [14][15][16] where E e and M e are the kinetic energy and mass of cluster, respectively. P is calculated by using the WKB approximation, which is written by The deformation energy (relative to the sphere) is small up to the rupture point between the fragments. R in is the distance between the mass centers of the portions of the initial sphere separated by a plane perpendicular to t h e d e for m at i on a x i s to a ssu m e t h e volu m e c ons e r v at i on of t h e f utu re f r ag m e nt s .
e d e d e d 10 1/2 e d e 10 1/2 www.nature.com/scientificreports www.nature.com/scientificreports/ e d e d 10 1/2 where T 1/2 is the CR half-life, which is measured in seconds. μ = A e A d /(A e + A d ) is the reduced mass. A e and A d represent the mass numbers of the emitted particle and daughter nucleus, respectively. Z e and Z d denote the charge numbers of the two fragments. In Eq. (10), r = R t /R b , R t and R b stand for the first and second turning points of the barrier, respectively. The two turning points are defined as R t = 1.2249(A e 1/3 + A d 1/3 ) and R b = 1.43998Z e Z d /Q. The frequency of assaults ν 0 is taken as 10 22.01 s −1 . In Eq. (13), η (η z ) represents the mass (charge) asymmetry, whose form is written as The parameters in Eqs. (9)(10)(11)(12)(13)(14)(15) are determined by fitting the experimental half-lives and Q values [60][61][62][63][64][65][66] , which are listed in Table 1.

Results and discussions
It is well known that the CR half-lives are dependent on the Q values, which can be extracted by  where M, M d and M e 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 model 98 because relevant studies showed that the WS4 mass model can predict the experimental nuclear masses and decay energies accurately 98,99 . 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 SHN 100 . Firstly, we calculate the 12 C decay half-life of 114 Ba 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 3 101 . Columns 4-12 give the 12 C decay half-lives of 114 Ba extracted from all the models and formulas. The last column lists the experimental half-life of the 12 C decay from 114 Ba 94 . 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 = 87-96 65,66 . 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 12 C, 20 Ne and 28 Si emissions as examples, the CR half-lives will be predicted by all the models (formulas) except for the NRDX and VSS formulas.
The half-lives of the 12 C, 20 Ne and 28 Si emissions of some isotopes within the ELDM, GLDM, UDL, UNIV, Horoi, TM, and BKAG models (formulas) as functions of the daughter neutron number N d 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 models 10-16 . 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 www.nature.com/scientificreports www.nature.com/scientificreports/ 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 approximation [10][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 models [14][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, respectively [59][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 formula 100 . But for the half-lives given by the Horoi 62 , TM 63 , and BKAG 64 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 laws [62][63][64] , which are similar to the NRDX and VSS formulas 65,66 . 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 N d is 50 for each model. For example, the minimal half-lives of the 12 C emission occur for the parent nuclei 110 Xe, 111 Cs, 112 Ba, 113 La, and 114 Ce. Among these minimal half-lives, the half-life with the daughter nucleus 100 Sn (the parent nucleus 112 Ba) 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 N d = 50, and the shell effect at 100 Sn is strongest. To explain the shell effect of the CR half-lives shown in Figs. 1-3, the Q values of the 12 C, 20 Ne, and 28 Si emissions of these isotopic chains as functions of N d are shown in Fig. 4. As can be seen from Fig. 4, the shell effect at N d = 50 is very obvious and the shell effect at 100 Sn is most pronounced. In the half-life calculations the shell effects are included through the Q values. The Q value shell effects at N d = 50 and 100 Sn 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.
The clusters 12 C, 20 Ne and 28 Si can be seen as α-like ones 76,78 . In addition to the half-lives of the α-like CR, the half-lives of the non-α-like 78 ( 26 Mg and 30 Si) CR are calculated as well. For comparing the similarities and differences between the two sorts of cluster emissions, the half-lives of the 24,26 Mg and 28,30 Si emissions leading to Sn. 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 24,26 Mg and 28,30 Si emissions decaying to the Z d = 50 daughter nuclei versus N d . As can be seen from Fig. 6, the Q values of the 24 Mg ( 28 Si) emission are much larger than those of the 26 Mg ( 30 Si) emission in addition to the strong shell effect at 100 Sn. Small Q values of the non-α-like cluster decay lead to the long half-lives.
According to the above discussions, one can see that a CR most probably occurs in the decay process where the daughter nucleus has N d = 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 N d around 50 based on the ELDM, GLDM, UDL and UNIV models (formulas), which include the 8 Be, 12 C, 16 O, 20 Ne, 24 Mg, and 28 Si 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.

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 12 C decay half-life of 114 Ba 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 12 C, 20 Ne, and 28 Si 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   Table 3. The 8 Be, 16 C, 16 O, 20 Ne, 24 Mg, and 28 Si emission half-lives in the decay processes where the daughter nuclei with N 10 T 1/2 around 50 within the ELDM, GLDM, UDL and UNIV models (formulas) are shown in columns 5-8. The predicted half-lives of some emitted clusters within the DNSM 101 are listed in the last column.
The Q values and half-lives are measured in MeV and seconds, respectively.