1. Introduction
Cluster radioactivity is an exotic decay mode observed in actinide nuclei, where the clusters emitted from the parent nuclei are heavier than α-particle and lighter than spontaneous fission fragments. So, the cluster radioactivity is also called heavy-ion radioactivity. It was first predicted by Sandulescu, Poenaru, and Greiner in 1980 [1]. Four years later, cluster radioactivity was first observed experimentally through 14C emitted from 223Ra. Furthermore, its decimal logarithm value of the half-life was measured as 15.20 s [2]. Since then, various emitted clusters have been observed, such as 20O, 22,24−26Ne, 28,30Mg, and 32,34Si [3-7]. These clusters are emitted from the actinide nuclei ranging from 221Fr to 242Cm, while the remaining residue is the double magic nucleus 208Pb or its neighborhoods [3-7]. It implies that the nuclear shell effect plays a crucial role in the cluster radioactivity of the heavy nuclei.
Nowadays, various theoretical models have been developed to study the cluster radioactivity [8-34]. Generally, these models are divided into the following two categories. One is the fission-like model, which means the nucleus deforms continuously as it penetrates the nuclear barrier and reaches the scission configuration after running down the Coulomb barrier [8-24]. The other one refers to the preformed cluster model. It assumes that clusters have been formed in the parent nucleus before tunneling the barrier [25-34]. In addition, some semi-empirical formulas or relationships have been proposed to calculate the half-lives of the cluster radioactivity [35-43]. The experimental half-lives of the cluster radioactivity are reproduced more or less satisfactorily by the models and the semi-empirical formulas or relationships [8-43].
Among the models, the ELDM is a successful phenomenological model for describing the cluster radioactivity [15]. In the framework of the ELDM, the cluster radioactivity is assumed as a super-asymmetric fission process [15, 44-46]. Besides the cluster radioactivity, one-proton emission, two-proton radioactivity, α-decay and cold fission can be described in a unified framework of the ELDM [15, 44-52]. In order to evaluate the half-lives of different radioactivity, the Gamow penetrability factor is obtained by considering the pre-scission phase of the two intersecting spherical fragments in the effective one-dimensional potential barrier based on four kinds of combinations of the mass transfer descriptions and the inertia coefficients. Therefore, the experimental half-lives of various radioactivity could be reproduced well [15, 44-52].
However, for each combination of any type of radioactivity, only two constants (the nuclear charge radius parameter r0 and the assaulting frequency ν0) are adjusted to obtain the agreement with the experimental half-lives [15, 44-52]. As a result, the predictive power of the ELDM is not so strong, owing to the following two reasons: (i) Since the half-life of the charged-particle radioactivity is sensitive to the nuclear charge radius [53-57] a minor change will result in a large deviation of the decay half-life. So, a more accurate charge radius expression should be introduced into the ELDM instead of a simple empirical charge radius formula (R = r0A1/3). (ii) The assaulting frequency ν0 should not be a constant, which may be correlated to the nuclear structure [57, 58]. Thus, the assaulting frequency including the nuclear structure should be introduced into the ELDM. In our previous work, the ELDM was improved by introducing an accurate nuclear charge radius formula and an analytic expression for ν0 [59]. It was shown that the accuracy of the IMELDM was improved evidently compared with its predecessor [59]. So, in this work, we will extend the IMELDM to study the cluster radioactivity of the heavy nuclei whose daughter nuclei are around 208Pb. This constitutes the motivation of this article.
This article is organized as follows. In section 2 , the framework of the IMELDM is presented. The corresponding results and discussions are made in section 3 . In the last section, some conclusions are drawn.
2. IMELDM
Figure 1. Schematic representation of the dinuclear decaying system. R1 and R2 show the radii of the emitted cluster and the daughter nucleus, respectively. ζ is the distance between their geometric centers. The variable ξ represents the distance between the plane of the intersection and the center of the daughter nucleus. |
During the nuclear decaying process, four independent coordinates (R1, R2, ζ and ξ) are used to describe the dinuclear system. Then, three constraints are introduced to reduce the spherical four-dimensional problem to an equivalent one-dimensional case.
The first one,
$\begin{eqnarray}{R}_{1}^{2}-{\left(\zeta -\xi \right)}^{2}={R}_{2}^{2}-{\xi }^{2},\end{eqnarray}$
keeps a circular shape for the neck connecting the nascent fragments.The second one,
$\begin{eqnarray}\begin{array}{l}2({R}_{1}^{3}+{R}_{2}^{3})+3\left[{R}_{1}^{2}\left(\zeta -\xi \right)+{R}_{2}^{2}\xi \right]\\ -\left[{\left(\zeta -\xi \right)}^{3}+{\xi }^{3}\right]\\ \,=4{R}^{3},\end{array}\end{eqnarray}$
denotes the incompressibility of the nuclear flow, where R is the radius of the parent nucleus.The last constraint is related with the mass transfer descriptions chosen to treat the process. In the case of the varying mass asymmetry shape (VMAS), the radius of the lighter fragment is fixed as
$\begin{eqnarray}{R}_{1}={\overline{R}}_{1},\end{eqnarray}$
where ${\overline{R}}_{1}$ is the final radius of the lighter fragment. In the other case of the constant mass asymmetry shape (CMAS), the volume of both fragments is kept as a constant. For the lighter fragment, volume conservation gives $\begin{eqnarray}2{R}_{1}^{3}+3{R}_{1}^{2}{(\zeta -\xi )-(\zeta -\xi )}^{3}-4{\overline{R}}_{1}^{3}=0.\end{eqnarray}$
Thus, the nuclear decay is conveniently simplified as the effective one-dimensional potential barrier penetrability problem. In order to evaluate the half-life of a given parent nucleus, the Gamow penetrability probability is calculated by
$\begin{eqnarray}P=\exp \left[-\displaystyle \frac{2}{{\hslash }}{\int }_{{\zeta }_{0}}^{{\zeta }_{c}}\sqrt{2\mu \left[V(\zeta )-{Q}_{c}\right]}{\rm{d}}\zeta \right].\end{eqnarray}$
Here, the limits ζ0 and ζc of the integral are the inner and outer turning points, respectively. μ is the inertia coefficient. V (ζ) is the one-dimensional total potential energy and given as
$\begin{eqnarray}V(\zeta )={V}_{{\rm{C}}}(\zeta )+{V}_{{\rm{S}}}(\zeta )+{V}_{l}(\zeta ),\end{eqnarray}$
where VC(ζ), VS(ζ) and Vl(ζ) represent the Coulomb energy, effective surface energy and centrifugal potential energy, respectively. The analytic expression and details of them can be seen from [15, 44, 46]. Qc denotes the decay energy and is calculated by the following relation: Qc = M(Z, N) − Md(Z − Zc, N − Nc) − M(Zc, Nc), where M(Z, N), Md(Z − Zc, N − Nc) and M(Zc, Nc) represent the mass excesses of the parent nucleus, daughter nucleus and emitted cluster, respectively. In the ELDM, two approximations are used to calculate the inertia coefficients of the dynamical evolution of the separating dinuclear system. The two types of inertia coefficients are the Werner-Wheeler's inertia coefficient (WW) and the Effective inertia coefficient (Eff). Combining two mass transfer descriptions and two inertial coefficients, the following four kinds of descriptions: (VMAS, WW), (VMAS, Eff), (CMAS, WW), and (CMAS, Eff) are contained naturally in the ELDM. For each description, the radius parameter r0 together with ν0 are determined to be constants by fitting the experimental data. The values of these parameters can be seen in table A of [46].The charge radius of the parent nucleus is obtained by
$\begin{eqnarray}R={r}_{0}{A}^{1/3},\end{eqnarray}$
where A is the mass number of the parent nucleus.For the half-life of the cluster radioactivity, it is calculated by
$\begin{eqnarray}{T}_{1/2}=\displaystyle \frac{\mathrm{ln}2}{{\nu }_{0}P}.\end{eqnarray}$
However, the parent nuclear radius by equation (7 ) is not accurate enough, which has been tested by relevant studies [54]. Meanwhile, ν0 is correlated to the structure of the nuclei and should not be a constant. Therefore, a more accurate formula for R and a more reasonable ν0 are necessary to be introduced into the ELDM.
In the next paragraphs, we will describe the details of the IMELDM. Firstly, equation (7 ) is replaced by a five-parameter formula to estimate more accurate R values [54]
$\begin{eqnarray}R={r}_{0}\left(1-a\displaystyle \frac{N-Z}{A}+b\displaystyle \frac{1}{A}+c\displaystyle \frac{K}{A}+d\displaystyle \frac{\delta }{A}\right){A}^{1/3},\end{eqnarray}$
where K and δ represent the Casten factor and the odd-even staggering factor, respectively. The parameters r0, a, b, c and d can be found from table 1 of [54].Table 1. Fitting parameters of equation ( |
| Combinations | Even-even (n = 11) | Odd-A (n = 9) | ||||||
|---|---|---|---|---|---|---|---|---|
| ${a}^{{\prime} }$ | ${b}^{{\prime} }$ | ${c}^{{\prime} }$ | ${d}^{{\prime} }$ | ${a}^{{\prime} }$ | ${b}^{{\prime} }$ | ${c}^{{\prime} }$ | ${d}^{{\prime} }$ | |
| VMAS, WW | −2425.3932 | 13.3622 | 2.3789 | −172.4131 | 3361.8992 | 7.9036 | −7.7885 | 236.1250 |
| VMAS, Eff | −2649.7264 | 3.0173 | −0.9225 | −188.8816 | 2226.0255 | 4.1084 | −20.0646 | 155.2288 |
| CMAS, WW | −2803.1143 | 9.7249 | −0.6366 | −199.3774 | 2762.8298 | 7.8227 | −16.4551 | 193.4948 |
| CMAS, Eff | −3287.3863 | −2.9681 | −1.3410 | −234.3730 | 1892.2046 | 0.9886 | −26.2232 | 131.1708 |
Combining equations (5 ) and (9 ) the penetration probability P can be estimated. Then the empirical value of ν0 for each nucleus can be extracted by the following expression
$\begin{eqnarray}{\nu }_{0}=\displaystyle \frac{\mathrm{ln}2}{{T}_{1/2}^{\mathrm{expt}.}P},\end{eqnarray}$
where ${T}_{1/2}^{\mathrm{expt}.}$ is the experimental half-life of the cluster radioactivity.Usually, the values of ν0 are calculated by the following classic method
$\begin{eqnarray}{\nu }_{0}=\displaystyle \frac{v}{2R}=\displaystyle \frac{1}{2R}\sqrt{\displaystyle \frac{2{E}_{{\rm{c}}}}{{M}_{{\rm{c}}}}},\end{eqnarray}$
where v is the velocity of the cluster inside a parent nucleus. Ec (Ec = [(A − Ac)/A]Qc), Ac and Mc represent the kinetic energy, the mass number of the emitted cluster and the atomic mass of the emitted cluster, respectively. Its decimal logarithm form is written as $\begin{eqnarray}{\mathrm{log}}_{10}{\nu }_{0}=-{\mathrm{log}}_{10}{({M}_{{\rm{c}}})}^{1/2}+{\mathrm{log}}_{10}{(2{E}_{{\rm{c}}})}^{1/2}-{\mathrm{log}}_{10}2R.\end{eqnarray}$
Because Mc = Acu, here u is the atomic mass unit, based on equation (12 ) a simple analytic expression for ν0 is expressed as 10 ). They are listed in table 1. At last, combining equations (5 ), (9 ), and (13 ), the cluster radioactivity half-life is calculated by equation (8 ).
$\begin{eqnarray}{\mathrm{log}}_{10}{\nu }_{0}={a}^{{\prime} }+{b}^{{\prime} }{\mathrm{log}}_{10}{A}_{c}^{1/2}+{c}^{{\prime} }{\mathrm{log}}_{10}{E}_{c}^{1/2}+{d}^{{\prime} }{\mathrm{log}}_{10}R,\end{eqnarray}$
where the parameters ${a}^{{\prime} }$, ${b}^{{\prime} }$, ${c}^{{\prime} }$, and ${d}^{{\prime} }$ are determined by the following steps: (i) 20 experimental pieces of data of cluster radioactivity are divided into even-even and odd-A subsets, which are taken from [4, 60, 61]. (ii) Using the multiple linear regression, the parameters ${a}^{{\prime} }$, ${b}^{{\prime} }$, ${c}^{{\prime} }$ and ${d}^{{\prime} }$ for each combination of each subset are determined by fitting the empirical ν0 values extracted from equation (3. Results and discussion
Within the IMELDM, the half-lives of ground-state to ground-state cluster radioactivity of 20 nuclei have been calculated by inputting the experimental Qc values and the minimum angular momenta l${}_{\min }$ carried by the emitted clusters. Note that the l${}_{\min }$ values are determined by the spin-parity selection rule. The calculated results are listed in table 2. From table 2, the first three columns indicate the parent nuclei, the emitted clusters and the minimum angular momenta, respectively. The fourth and fifth columns give the experimental Qc values and the decimal logarithms of cluster radioactivity half-lives [4, 60, 61]. The sixth to ninth columns show the calculated cluster radioactivity half-lives for each combination within the ELDM. In the last four columns, the cluster radioactivity half-lives of each combination within the IMELDM are listed. From table 2, it is seen that the results by the IMELDM are closer to the experimental data than those by the ELDM.
Table 2. The experimental and calculated half-lives of the cluster radioactivity of the heavy nuclei. The experimental half-lives and Qc values are taken from [4, 60, 61], whose values are measured in seconds and MeV, respectively. |
| Parent | Emitted | l${}_{\min }$ | Qc(MeV) | log10T1/2 (s) | log${}_{10}{T}_{1/2}^{\mathrm{ELDM}}$(s) | log${}_{10}{T}_{1/2}^{\mathrm{IMELDM}}$(s) | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| nuclei | clusters | Expt. | Expt. | VMAS | VMAS | CMAS | CMAS | VMAS | VMAS | CMAS | CMAS | |
| WW | Eff | WW | Eff | WW | Eff | WW | Eff | |||||
| Even-even | nuclei | |||||||||||
| 222Ra | 14C | 0 | 33.05 | 11.00 | 12.39 | 12.24 | 11.29 | 11.45 | 10.78 | 10.96 | 10.85 | 10.91 |
| 224Ra | 14C | 0 | 30.54 | 15.92 | 17.14 | 16.90 | 16.00 | 16.10 | 15.88 | 15.83 | 15.85 | 15.77 |
| 226Ra | 14C | 0 | 28.20 | 21.34 | 22.20 | 21.88 | 21.02 | 21.06 | 21.26 | 21.00 | 21.14 | 20.93 |
| 228Th | 20O | 0 | 44.72 | 20.72 | 22.40 | 22.53 | 21.22 | 21.20 | 21.21 | 21.51 | 21.37 | 21.68 |
| 230U | 22Ne | 0 | 61.40 | 19.57 | 20.95 | 21.03 | 19.52 | 19.36 | 20.34 | 20.16 | 20.28 | 20.20 |
| 230Th | 24Ne | 0 | 57.57 | 24.64 | 25.45 | 25.71 | 24.13 | 23.91 | 24.65 | 24.62 | 24.63 | 24.61 |
| 232U | 24Ne | 0 | 62.31 | 20.40 | 20.85 | 21.27 | 19.56 | 19.51 | 20.24 | 20.48 | 20.36 | 20.65 |
| 234U | 28Mg | 0 | 74.11 | 25.74 | 25.24 | 25.88 | 23.87 | 23.64 | 25.21 | 25.03 | 25.12 | 24.93 |
| 236Pu | 28Mg | 0 | 79.67 | 21.67 | 20.70 | 21.55 | 19.37 | 19.38 | 20.85 | 20.97 | 20.91 | 21.07 |
| 238Pu | 32Si | 0 | 91.19 | 25.28 | 24.81 | 25.98 | 23.46 | 23.34 | 25.64 | 25.34 | 25.49 | 25.13 |
| 242Cm | 34Si | 0 | 96.51 | 23.15 | 22.18 | 23.84 | 20.98 | 21.16 | 23.37 | 23.55 | 23.44 | 23.56 |
| | ||||||||||||
| Odd-A | nuclei | |||||||||||
| | ||||||||||||
| 221Fr | 14C | 3 | 31.29 | 14.52 | 14.89 | 14.71 | 13.79 | 13.91 | 15.00 | 14.87 | 14.93 | 14.82 |
| 221Ra | 14C | 3 | 32.40 | 13.39 | 13.73 | 13.59 | 12.64 | 12.80 | 13.69 | 13.74 | 13.72 | 13.76 |
| 223Ra | 14C | 4 | 31.83 | 15.20 | 14.78 | 14.62 | 13.68 | 13.83 | 14.49 | 14.53 | 14.51 | 14.54 |
| 225Ac | 14C | 4 | 30.48 | 17.21 | 18.39 | 18.18 | 17.26 | 17.38 | 17.65 | 17.57 | 17.61 | 17.53 |
| 231Pa | 23F | 1 | 51.84 | 26.02 | 24.88 | 25.22 | 23.69 | 23.61 | 24.71 | 24.86 | 24.79 | 24.94 |
| 231Pa | 24Ne | 1 | 60.42 | 22.89 | 22.36 | 22.71 | 21.05 | 20.94 | 22.73 | 22.90 | 22.81 | 23.00 |
| 233U | 24Ne | 2 | 60.49 | 24.84 | 23.53 | 23.87 | 22.20 | 22.09 | 23.63 | 23.79 | 23.71 | 23.89 |
| 235U | 25Ne | 1 | 57.68 | 27.44 | 28.60 | 28.97 | 27.30 | 27.10 | 28.51 | 28.44 | 28.48 | 28.40 |
| 235U | 28Mg | 1 | 72.43 | 27.44 | 27.58 | 28.13 | 26.17 | 25.87 | 28.55 | 28.25 | 28.40 | 28.07 |
To further test the agreement between the calculated results and the experimental data, the average deviation $\overline{\sigma }$ and the standard deviation $\sqrt{\overline{{\sigma }^{2}}}$ are calculated by the following expressions
$\begin{eqnarray}\overline{\sigma }=\displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}\left|{\mathrm{log}}_{10}{T}_{1/2}^{\mathrm{expt}.i}-{\mathrm{log}}_{10}{T}_{1/2}^{\mathrm{cal}.i}\right|,\end{eqnarray}$
$\begin{eqnarray}\sqrt{\overline{{\sigma }^{2}}}={\left[\displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}{({\mathrm{log}}_{10}{T}_{1/2}^{\mathrm{expt}.i}-{\mathrm{log}}_{10}{T}_{1/2}^{\mathrm{cal}.i})}^{2}\right]}^{1/2},\end{eqnarray}$
where ${\mathrm{log}}_{10}{T}_{1/2}^{\mathrm{expt}.}$ and ${\mathrm{log}}_{10}{T}_{1/2}^{\mathrm{cal}.}$ are the decimal logarithms of the experimental and calculated half-lives of cluster radioactivity, respectively. n is the number of cluster radioactivity events. Then the calculated $\overline{\sigma }$ and $\sqrt{\overline{{\sigma }^{2}}}$ values of each combination of the ELDM and IMELDM are listed in table 3.Table 3. The average deviation $\overline{\sigma }$ and the standard deviation $\sqrt{\overline{{\sigma }^{2}}}$ between the experimental and calculated half-lives of the cluster radioactivity for each combination of the ELDM and IMELDM. |
| Combinations | ELDM | IMELDM | ||
|---|---|---|---|---|
| $\overline{\sigma }$ | $\sqrt{\overline{{\sigma }^{2}}}$ | $\overline{\sigma }$ | $\sqrt{\overline{{\sigma }^{2}}}$ | |
| VMAS, WW | 0.865 | 0.965 | 0.522 | 0.655 |
| VMAS, Eff | 0.787 | 0.919 | 0.473 | 0.593 |
| CMAS, WW | 1.101 | 1.397 | 0.495 | 0.620 |
| CMAS, Eff | 1.165 | 1.444 | 0.480 | 0.579 |
From table 3, it is seen that the $\overline{\sigma }$ ($\sqrt{\overline{{\sigma }^{2}}}$) values within the IMELDM are much smaller than those within the ELDM. For each combination, the $\overline{\sigma }$ ($\sqrt{\overline{{\sigma }^{2}}}$) values within the IMELDM decrease to about 0.50 (0.60) from large $\overline{\sigma }$ ($\sqrt{\overline{{\sigma }^{2}}}$) values. Especially, for the (CMAS, Eff) combination, the $\overline{\sigma }$ ($\sqrt{\overline{{\sigma }^{2}}}$) value within the IMELDM decreases to 0.480 (0.579) from 1.165 (1.444), which implies the accuracy of the ELDM increases by 59% (60%). Therefore, the accuracy of the IMELDM becomes much higher than that of the ELDM by introducing more reasonable analytic expressions for R and ν0.
Besides $\overline{\sigma }$ and $\sqrt{\overline{{\sigma }^{2}}}$, the hindrance factor (HF) is usually applied to analyze the deviation between the experimental half-lives and theoretical ones. Its decimal logarithm form is expressed as
$\begin{eqnarray}{\mathrm{log}}_{10}{HF}={\mathrm{log}}_{10}({T}_{1/2}^{\mathrm{expt}.}/{T}_{1/2}^{\mathrm{cal}.}).\end{eqnarray}$
Usually, it is believed that if the ${\mathrm{log}}_{10}{HF}$ value is within a factor of 1.0, the calculated half-lives will be in agreement with the experimental data [47, 59, 62]. Within equation (16 ), the ${\mathrm{log}}_{10}{HF}$ values for the (VMAS, WW), (VMAS, Eff), (CMAS, WW) and (CMAS, Eff) combinations of the ELDM and IMELDM are calculated. The ${\mathrm{log}}_{10}{HF}$ values versus the neutron number N of the parent nuclei are plotted in figure 2. From figure 2, one can see that more ${\mathrm{log}}_{10}{HF}$ values within each combination of the IMELDM fall between -1.0 and 1.0. So the conclusion based on figure 2 is the same as that from table 3.
Figure 2. The ${\mathrm{log}}_{10}{HF}$ values as functions of the neutron number N within four different combinations of the ELDM and IMELDM: (a) (VMAS, WW) combination; (b) (VMAS, Eff) combination; (c) (CMAS, WW) combination; (d) (CMAS, Eff) combination. The solid black star and solid red circle represent the ${\mathrm{log}}_{10}{HF}$ values of the ELDM and IMELDM, respectively. |
From the perspective of ν0, we know that the accuracy of the IMELDM depends on the agreement between the empirical ν0 values and the fitting ones. To illustrate why the accuracy of the IMELDM becomes higher, the empirical and fitting ν0 values of the (CMAS, Eff) combination are shown in the last two columns of table 4. From table 4, good agreement between the empirical ν0 values and the fitting ones can be found. As a result, the half-lives estimated from the IMELDM are closer to the experimental data. Moreover, as can be seen from table 4, the log10ν0 values decrease with the increase of the mass of the emitted cluster. In fact, the cluster preformation probability inside a parent nucleus is included in ν0 in the framework of the IMELDM. The heavier the emitted cluster, the smaller the cluster preformation probability inside a parent nucleus. So the heavier cluster has a smaller log10ν0 value naturally. In addition, we know that the ν0 values that do not contain the preformation probabilities can be estimated by equation (11 ). Thus the cluster preformation probabilities can be estimated by the ratio between the empirical ν0 (fitting ν0) values and the calculated ν0 values within equation (11 ). By taking 221Ra ⟶ 207Pb + 14C and 231Pa ⟶ 207Tl + 24Ne as examples, we obtain the 14C and 24Ne preformation probabilities, whose values are 7.88 × 10−2 and 3.69 × 10−4, respectively.
Table 4. The empirical values of ν0 extracted by equation ( |
| Parent | Emitted | l${}_{\min }$ | Q${}_{c}^{\mathrm{Expt}.}$(MeV) | log${}_{10}{T}_{1/2}^{\mathrm{Expt}.}$(s) | log${}_{10}{{\nu }_{0}}^{\mathrm{equation}}$ ( | log${}_{10}{{\nu }_{0}}^{\mathrm{equation}}$ ( |
|---|---|---|---|---|---|---|
| nuclei | clusters | |||||
| 222Ra | 14C | 0 | 33.05 | 11.00 | 21.07 | 21.19 |
| 224Ra | 14C | 0 | 30.54 | 15.92 | 20.79 | 20.93 |
| 226Ra | 14C | 0 | 28.20 | 21.34 | 20.33 | 20.68 |
| 228Th | 20O | 0 | 44.72 | 20.72 | 20.49 | 19.54 |
| 230U | 22Ne | 0 | 61.40 | 19.57 | 19.37 | 18.78 |
| 230Th | 24Ne | 0 | 57.57 | 24.64 | 18.97 | 18.97 |
| 232U | 24Ne | 0 | 62.31 | 20.40 | 18.64 | 18.39 |
| 234U | 28Mg | 0 | 74.11 | 25.74 | 17.11 | 17.88 |
| 236Pu | 28Mg | 0 | 79.67 | 21.67 | 16.72 | 17.31 |
| 238Pu | 32Si | 0 | 91.19 | 25.28 | 16.74 | 16.84 |
| 242Cm | 34Si | 0 | 96.51 | 23.15 | 16.43 | 16.07 |
| 221Fr | 14C | 3 | 31.29 | 14.52 | 20.09 | 19.78 |
| 221Ra | 14C | 3 | 32.40 | 13.39 | 20.03 | 19.66 |
| 223Ra | 14C | 4 | 31.83 | 15.20 | 19.26 | 19.92 |
| 225Ac | 14C | 4 | 30.48 | 17.21 | 20.72 | 20.39 |
| 231Pa | 23F | 1 | 51.84 | 26.02 | 17.35 | 18.43 |
| 231Pa | 24Ne | 1 | 60.42 | 22.89 | 17.71 | 17.60 |
| 233U | 24Ne | 2 | 60.49 | 24.84 | 16.79 | 17.74 |
| 235U | 25Ne | 1 | 57.68 | 27.44 | 19.16 | 18.20 |
| 235U | 28Mg | 1 | 72.43 | 27.44 | 17.67 | 17.04 |
By consulting [4, 60], it is found that for the cluster radioactivity of several heavy nuclei, only the lower limits of the half-lives were measured. Thus, those experimental data constitutes a ground to test the IMELDM. The above discussion suggests the accuracy of the (CMAS, Eff) combination is improved most evidently. So, the (CMAS, Eff) combination is used to calculate the cluster radioactivity half-lives in this case. The experimental half-lives with a lower limit and the calculated half-lives within the ELDM and IMELDM are shown in table 5. From table 5, it can be seen that all the calculated half-lives within the IMELDM are larger than those within the ELDM. For 226Th, 232U and 240Pu, the half-lives within the ELDM and IMELDM are larger than the corresponding experimental lower limits. For 233U, 237Np and 241Am, as can be seen from table 5 the experimental half-lives are not reproduced well within the ELDM. However, the half-lives within the IMELDM are closer to or in agreement with the experimental data compared to those within the ELDM.
Table 5. The calculated half-lives of cluster radioactivity within the (CMAS, Eff) combination of the ELDM and IMELDM, for those cases in which the lower limits of the experimental half-lives were measured. |
| Parent | Emitted | l${}_{\min }$ | Q${}_{c}^{\mathrm{Expt}.}$(MeV) | log${}_{10}{T}_{1/2}^{\mathrm{ELDM}}$(s) | log${}_{10}{T}_{1/2}^{\mathrm{IMELDM}}$(s) | log${}_{10}{T}_{1/2}^{\mathrm{Expt}.}$(s) |
|---|---|---|---|---|---|---|
| nuclei | clusters | |||||
| 226Th | 18O | 0 | 45.73 | 17.67 | 17.82 | >16.76 |
| 232U | 28Mg | 0 | 74.32 | 23.55 | 24.56 | >22.64 |
| 240Pu | 34Si | 0 | 91.06 | 24.44 | 26.46 | >24.20 |
| 233U | 28Mg | 3 | 74.23 | 23.63 | 26.10 | >27.59 |
| 237Np | 30Mg | 2 | 74.79 | 25.42 | 27.32 | >27.23 |
| 241Am | 34Si | 2 | 93.84 | 22.72 | 24.82 | >24.13 |
In addition to the experimental data of table 5, from [4, 60] it is found that there exist two kinds of cluster radioactivity for 234,236U, 238Pu and 232Th. For example, 24Ne and 26Ne can be emitted simultaneously from 234U. Although the experimental half-life of each kind of cluster radioactivity has not been determined, the total half-lives or the lower limits of the total half-lives for the two types of cluster radioactivity were given [4, 60]. Thus, these experimental half-lives provide alternative grounds for testing the IMELDM. Generally, the angular momenta l carried by the clusters are selected as 0, being only a weak impact on the cluster radioactivity half-lives [33, 43]. To show the relationship between l and the half-life, the decimal logarithms of half-lives for 221Ra → 207Pb+14C and 231Pa → 207Tl + 24Ne as functions of l are plotted in figure 3. From figure 3, it is seen that the decimal logarithms of half-lives of the two emissions grow by 0.28 and 0.15 when l rises from 0 to 5. So, the cluster radioactivity half-lives are indeed slightly affected by l. For the centrifugal potential, it is a small quantity compared to the Coulomb barrier, the Gamow penetration probability is almost unchanged when the centrifugal potential is taken into account. Thus, l values are selected as 0 in the subsequent calculations. The calculated half-lives within the (CMAS, Eff) combination of the ELDM and IMELDM are listed in table 6. In table 6, the parent nuclei and the emitted two types of clusters are listed in the first two columns. The experimental Qc values of each kind of cluster radioactivity are listed in column 3. By inputting the experimental Qc values, the partial half-lives and the total half-lives of the two types of cluster radioactivity are calculated within the ELDM and IMELDM, which are listed in columns 4-7. The experimental total half-lives are listed in the last column. From the last three columns of table 6, it is seen that the experimental total half-lives are reproduced better within the IMELDM than those within the ELDM. Therefore, by the discussion of tables 3-6, we can conclude that the IMELDM is a successful model for estimating the cluster radioactivity half-lives of the heavy nuclei.
Figure 3. For 221Ra → 207Pb + 14C and 231Pa → 207Tl + 24Ne, the decimal logarithm values of the half-lives versus the angular momenta l carried by the emitted clusters within the (CMAS, Eff) combination of the IMELDM. |
Table 6. The calculated partial half-lives and the total half-lives of two types of clusters from the same parent nucleus within the (CMAS, Eff) combination of the ELDM and IMELDM. The experimental half-lives and the Qc values are taken from [4, 60, 61]. |
| Parent | Emitted | Q${}_{c}^{\mathrm{Expt}.}$(MeV) | log${}_{10}{T}_{1/2}^{\mathrm{ELDM}}$(s) | log${}_{10}{T}_{1/2}^{\mathrm{IMELDM}}$(s) | log${}_{10}{T}_{1/2}^{\mathrm{ELDM}}$(s) | log${}_{10}{T}_{1/2}^{\mathrm{IMELDM}}$(s) | log${}_{10}{T}_{1/2}^{\mathrm{Expt}.}$(s) |
|---|---|---|---|---|---|---|---|
| nuclei | clusters | ||||||
| 234U | 24Ne | 58.83 | 24.49 | 25.86 | 24.42 | 25.79 | 25.93 |
| 26Ne | 59.41 | 25.22 | 26.62 | ||||
| 236U | 28Mg | 71.69 | 26.81 | 28.27 | 26.62 | 28.09 | 27.58 |
| 30Mg | 72.51 | 27.07 | 28.55 | ||||
| 238Pu | 28Mg | 75.91 | 23.97 | 25.90 | 23.75 | 25.69 | 25.66 |
| 30Mg | 76.79 | 24.15 | 26.10 | ||||
| 232Th | 24Ne | 54.67 | 28.49 | 29.36 | 27.98 | 28.87 | >29.20 |
| 26Ne | 55.91 | 28.13 | 29.04 | ||||
| 236U | 24Ne | 55.96 | 28.98 | 30.47 | 28.85 | 30.35 | >25.90 |
| 26Ne | 56.75 | 29.43 | 30.96 |
Encouraged by the agreement mentioned above, within the (CMAS, Eff) combination of the IMELDM we attempt to predict the cluster emission half-lives that have not yet been available. The cluster radioactivity half-lives of 8Be, 12,14C, 15N, 16−20O, 20−26Ne, 24−28Mg, and 30−34Si emitted from the heavy nuclei in the trans-lead region are predicted, which are listed in table 7. We hope these predictions are helpful for searching for new cluster emitters in future experiments.
Table 7. The predicted half-lives of cluster radioactivity within the (CMAS, Eff) combination of the IMELDM. All the Qc values are extracted by the relation: Qc = M(Z, N) − Md(Z − Zc, N − Nc) − M(Zc, Nc) and the mass excesses are taken from [61]. “#”means only the empirical mass excesses for the parent and/or daughter nuclei in [61]. |
| Parent | Emitted | Qc(MeV) | log10T1/2(s) | Parent | Emitted | Qc(MeV) | log10T1/2(s) | Parent | Emitted | Qc(MeV) | log10T1/2(s) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| nuclei | clusters | nuclei | clusters | nuclei | clusters | ||||||
| Even-even | nuclei | 238Cm | 32Si | 97.31 | 21.51 | 215At | 8Be | 14.84 | 13.74 | ||
| 214Rn | 8Be | 14.52 | 14.99 | 238Cm | 32Si | 97.68 | 21.11 | 217At | 8Be | 13.1 | 18.91 |
| 216Rn | 8Be | 17.06 | 7.89 | 240Cm | 32Si | 95.39 | 23.67 | 215Fr | 8Be | 15.43 | 13.55 |
| 218Rn | 8Be | 15 | 13.90 | 240Cm | 34Si | 95.47 | 24.67 | 217Fr | 8Be | 17.63 | 8.01 |
| 218Ra | 12C | 30.44 | 12.70 | 244Cm | 34Si | 93.14 | 27.50 | 219Fr | 8Be | 15.54 | 12.75 |
| 220Ra | 12C | 32.02 | 10.06 | Odd-A | nuclei | 217Fr | 12C | 28.14 | 18.49 | ||
| 222Ra | 12C | 29.05 | 15.79 | 215Rn | 8Be | 16.34 | 10.54 | 219Fr | 12C | 29.65 | 15.46 |
| 220Ra | 14C | 32.02 | 12.59 | 217Rn | 8Be | 16.33 | 10.33 | 221Fr | 12C | 26.92 | 20.44 |
| 220Ra | 15N | 33.99 | 21.65 | 219Ra | 12C | 31.85 | 12.59 | 219Ac | 12C | 31.62 | 13.75 |
| 222Ra | 15N | 35.25 | 19.31 | 221Ra | 12C | 30.58 | 14.40 | 221Ac | 12C | 32.78 | 11.65 |
| 224Ra | 15N | 32.37 | 25.41 | 225Ra | 14C | 25.2 | 27.98 | 223Ac | 12C | 29.69 | 16.50 |
| 222Th | 15N | 37.16 | 18.09 | 221Ra | 15N | 35.12 | 21.45 | 219Fr | 14C | 29.65 | 17.94 |
| 224Th | 15N | 38.15 | 16.50 | 223Ra | 15N | 33.88 | 23.49 | 223Fr | 14C | 24.46 | 29.13 |
| 226Th | 15N | 34.96 | 22.60 | 223Th | 15N | 38.15 | 17.95 | 221Ac | 14C | 32.78 | 13.83 |
| 222Th | 16O | 45.73 | 15.45 | 225Th | 15N | 37 | 19.56 | 223Ac | 14C | 29.69 | 19.10 |
| 224Th | 16O | 46.48 | 14.50 | 223Th | 16O | 46.57 | 16.76 | 221Ac | 15N | 38.21 | 17.23 |
| 226Th | 16O | 42.66 | 20.68 | 225Th | 16O | 44.66 | 19.16 | 223Ac | 15N | 39.47 | 15.06 |
| 224Th | 18O | 44.56 | 19.62 | 225Th | 18O | 45.54 | 19.77 | 225Ac | 15N | 36.26 | 19.91 |
| 226Th | 18O | 45.73 | 17.86 | 227Th | 18O | 44.2 | 21.55 | 221Ac | 16O | 43.08 | 21.08 |
| 228Th | 18O | 42.28 | 23.87 | 227Th | 20O | 44.46 | 22.99 | 223Ac | 16O | 43.6 | 20.02 |
| 226Th | 20O | 43.19 | 24.29 | 229Th | 20O | 43.4 | 24.52 | 225Ac | 16O | 40.02 | 25.53 |
| 228Th | 20O | 44.72 | 21.68 | 227U | 20Ne | 58.54 | 24.15 | 223Pa | 16O | 47.11 | 16.98 |
| 230Th | 20O | 41.79 | 27.19 | 229U | 20Ne | 55.87 | 27.58 | 225Pa | 16O | 47.34 | 16.37 |
| 226U | 20Ne | 58.16 | 22.69 | 229U | 22Ne | 61.69 | 21.34 | 227Pa | 16O | 43.43 | 21.71 |
| 228U | 20Ne | 58.01 | 23.04 | 231U | 22Ne | 59.45 | 24.00 | 223Ac | 18O | 42.43 | 23.89 |
| 230U | 20Ne | 53.39 | 30.37 | 231U | 24Ne | 62.21 | 21.79 | 225Ac | 18O | 43.45 | 21.98 |
| 228U | 22Ne | 61.03 | 20.59 | 233U | 26Ne | 58.89 | 27.56 | 227Ac | 18O | 40.28 | 27.14 |
| 232U | 22Ne | 57.36 | 26.17 | 235U | 26Ne | 58.05 | 28.58 | 225Pa | 18O | 45.18 | 21.31 |
| 232U | 26Ne | 57.91 | 28.89 | 231Pu | 24Mg | 74.69 | 25.70 | 227Pa | 18O | 45.87 | 20.00 |
| 234U | 26Ne | 59.41 | 26.58 | 233Pu | 24Mg | 71.6 | 29.22 | 229Pa | 18O | 42.54 | 25.05 |
| 236U | 26Ne | 56.69 | 31.10 | 233Pu | 26Mg | 78.72 | 22.04 | 225Ac | 20O | 41.66 | 26.97 |
| 230Pu | 24Mg | 74.65 | 24.11 | 235Pu | 26Mg | 76.01 | 24.88 | 227Ac | 20O | 43.09 | 24.22 |
| 232Pu | 24Mg | 74.05 | 24.97 | 235Pu | 28Mg | 79.65 | 21.84 | 229Ac | 20O | 40.54 | 28.57 |
| 234Pu | 24Mg | 69.01 | 31.96 | 237Pu | 28Mg | 77.73 | 23.78 | 227Pa | 20O | 43.09 | 26.39 |
| 232Pu | 26Mg | 78.36 | 21.14 | 237Cm | 30Si | 96.13 | 22.43 | 229Pa | 20O | 44.36 | 23.96 |
| 234Pu | 26Mg | 78.31 | 21.29 | 239Cm | 30Si | 93.19 | 25.19 | 231Pa | 20O | 41.49 | 28.79 |
| 236Pu | 26Mg | 73.84 | 27.03 | 239Cm | 32Si | 97.55 | 21.63 | 225Pa | 20Ne | 55.2 | 27.94 |
| 234Pu | 28Mg | 79.15 | 21.61 | 241Cm | 32Si | 93.61 | 25.49 | 227Pa | 20Ne | 54.91 | 28.08 |
| 236Cm | 30Si | 96.07 | 21.66 | 241Cm | 34Si | 96.11 | 23.73 | 229Pa | 20Ne | 50.58 | 34.69 |
| 238Cm | 30Si | 95.63 | 22.20 | 243Cm | 34Si | 94.75 | 24.93 | 227Np | 20Ne | 59.66 | 23.82 |
| 240Cm | 30Si | 90.89 | 27.63 | 213At | 8Be | 12.3 | 22.23 | 229Np | 20Ne | 59.08 | 24.28 |
| 231Np | 20Ne | 54.53 | 30.52 | 229Np | 24Mg | 71.53 | 28.60 | 235Am | 30Si | 92.88 | 24.76 |
| 227Pa | 22Ne | 58.68 | 24.45 | 231Np | 24Mg | 70.59 | 29.50 | 237Am# | 30Si | 92.04 | 25.35 |
| 229Pa | 22Ne | 58.96 | 23.79 | 233Np | 24Mg | 65.53 | 36.25 | 239Am | 30Si | 87.47 | 30.26 |
| 231Pa | 22Ne | 55.09 | 29.12 | 231Am# | 24Mg | 76.4 | 24.97 | 237Bk# | 30Si | 97.68 | 22.21 |
| 229Np | 22Ne | 61.86 | 22.25 | 233Am# | 24Mg | 75.46 | 25.74 | 239Bk# | 30Si | 96.94 | 22.63 |
| 231Np | 22Ne | 61.91 | 21.89 | 235Am | 24Mg | 70.42 | 31.82 | 241Bk# | 30Si | 92.32 | 27.23 |
| 233Np | 22Ne | 57.83 | 27.18 | 231Np | 26Mg | 75.66 | 24.60 | 237Am# | 32Si | 94.47 | 23.76 |
| 229Pa | 24Ne | 59.67 | 24.31 | 233Np | 26Mg | 75.2 | 24.86 | 239Am | 32Si | 94.5 | 23.44 |
| 233Pa | 24Ne | 57.09 | 27.52 | 235Np | 26Mg | 70.9 | 30.04 | 241Am | 32Si | 90.66 | 27.39 |
| 231Np | 24Ne | 61.63 | 23.71 | 233Am# | 26Mg | 79.53 | 22.37 | 241Bk# | 32Si | 98.37 | 21.83 |
| 233Np | 24Ne | 62.16 | 22.72 | 235Am# | 26Mg | 79.1 | 22.54 | 243Bk | 32Si | 94.63 | 25.47 |
| 235Np | 24Ne | 58.85 | 27.07 | 237Am | 26Mg | 74.64 | 27.56 | 239Am | 34Si | 93.17 | 25.83 |
| 231Pa | 26Ne | 56.76 | 29.83 | 233Np | 28Mg | 76.79 | 24.18 | 241Am | 34Si | 93.93 | 24.71 |
| 233Pa | 26Ne | 58.04 | 27.60 | 235Np | 28Mg | 77.1 | 23.54 | 243Am | 34Si | 90.78 | 28.01 |
| 235Pa | 26Ne | 55.45 | 31.44 | 237Np | 28Mg | 73.54 | 27.67 | 241Bk# | 34Si | 96.04 | 25.20 |
| 233Np | 26Ne | 57.52 | 30.95 | 235Am | 28Mg | 79.7 | 23.02 | 243Bk | 34Si | 96.91 | 23.97 |
| 235Np | 26Ne | 58.82 | 28.66 | 237Am# | 28Mg | 79.85 | 22.56 | 245Bk | 34Si | 93.63 | 27.32 |
| 237Np | 26Ne | 56.25 | 32.48 | 239Am | 28Mg | 76.27 | 26.56 |
4. Conclusions
In this article, the ELDM has been improved by introducing an accurate formula for R and an analytic expression for ν0, which is called the IMELDM in the article. Within the IMELDM, the experimental cluster radioactivity half-lives of 20 nuclei in the trans-lead region are calculated. It is shown that the accuracy of the IMELDM becomes much higher than that of the ELDM. Next, the IMELDM has been tested by the experimental cluster radioactivity half-lives with lower limits and the experimental total half-lives with two types of cluster radioactivity from the same parent nuclei. It indicates that the IMELDM is a successful model for studying the cluster radioactivity of the heavy nuclei. Then, the (CMAS, Eff) combination of the IMELDM is adopted to predict the cluster radioactivity half-lives of the heavy nuclei in the trans-lead region. These predictions may be helpful for searching for new candidates for cluster radioactivity in future experiments. Finally, it is necessary to point out that the proton radioactivity is an important decay mode for the extremely proton-rich nuclei [51, 52, 62-66]. So it is interesting to extend our approach to study the proton radioactivity, which is underway.