In view of the fact that $\ensuremath{\alpha}$ decay and cluster radioactivity are physically analogical processes, we propose a general formula of half-lives and decay energies for $\ensuremath{\alpha}$ decay and cluster radioactivity. This new formula is directly deduced from the WKB barrier penetration probability with some approximations. It is not only simple in form and easy to see the physical meanings but also shows excellent agreement with the experimental values. Moreover, the difference between two sets of parameters to separately describe $\ensuremath{\alpha}$ decay and cluster radioactivity is small. Therefore, we use only one set of adjustable parameters to simultaneously describe the $\ensuremath{\alpha}$ decay and cluster radioactivity data for even-even nuclei. The results are also satisfactory. This indicates that this formula successfully combines the phenomenological laws of $\ensuremath{\alpha}$ decay and cluster radioactivity. We expect it to be a significant step toward a unified phenomenological law of $\ensuremath{\alpha}$ decay and cluster radioactivity.

Unified formula of half-lives for α decay and cluster radioactivity

One single line of universal (UNIV) curve for $\ensuremath{\alpha}$ decay and cluster radioactivities is obtained by plotting the sum of the decimal logarithm of the half-life and cluster preformation probability versus the decimal logarithm of the penetrability of external barrier. This fission-like theory is compared to the universal decay law (UDL) derived using $\ensuremath{\alpha}$-like $R$-matrix theory. The experimental data on heavy cluster decay in three groups of even-even, even-odd, and odd-even parent nuclei are reproduced with comparable accuracy by both types of universal curves, UNIV and UDL.

Single universal curve for cluster radioactivities and α decay

A linear relation for charged-particle emissions is presented starting from the microscopic mechanism of the radioactive decay. It relates the logarithms of the decay half-lives with two variables, called ${\ensuremath{\chi}}^{'}$ and ${\ensuremath{\rho}}^{'}$, which depend upon the $Q$ values of the outgoing clusters as well as the masses and charges of the nuclei involved in the decay. This relation explains well all known cluster decays. It is found to be a generalization of the Geiger-Nuttall law in $\ensuremath{\alpha}$ radioactivity, and therefore we call it the universal decay law. Predictions of the most likely emissions of various clusters are presented by applying the law over the whole nuclear chart. It is seen that the decays of heavier clusters with nonequal proton and neutron numbers are mostly located in the trans-lead region. The emissions of clusters with equal protons and neutrons, like $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$, are possible in some neutron-deficient nuclei with $Z\ensuremath{\geqslant}54$.

Microscopic mechanism of charged-particle radioactivity and generalization of the Geiger-Nuttall law

Cluster radioactivity is the emission of a fragment heavier than an $\ensuremath{\alpha}$ particle and lighter than mass 50. The range of clusters observed in experiments goes from ${}^{14}$C to ${}^{32}$Si while the heavy mass residue is always a nucleus in the neighborhood of the doubly-magic ${}^{208}$Pb nucleus. Cluster radioactivity is described in this paper as very asymmetric nuclear fission. A new fission valley leading to a decay with large fragment mass asymmetry matching the cluster radioactivity products is found. The mass octupole moment is found to be more convenient than the standard quadrupole moment as the parameter driving the system to fission. The mean-field Hartree-Fock-Bogoliubov theory with the phenomenological Gogny interaction has been used to compute the cluster emission properties of a wide range of even-even actinide nuclei from ${}^{222}$Ra to ${}^{242}$Cm, where emission of the clusters has been experimentally observed. Computed half-lives for cluster emission are compared with experimental results. The noticeable agreement obtained between the predicted properties of cluster emission (namely, cluster masses and emission half-lives) and the measured data confirms the validity of the proposed methodology in the analysis of the phenomenon of cluster radioactivity. A continuous fission path through the scission point has been described using the neck parameter constraint.

/pdf/microscopic-description-of-cluster-radioactivity-in-actinide-2pzspui9ta.pdf

Microscopic description of cluster radioactivity in actinide nuclei

The generalized density-dependent cluster model (GDDCM) is extended to study cluster radioactivity in even-even and odd-$A$ nuclei decaying to the doubly magic nucleus $^{208}\mathrm{Pb}$ or its neighboring nuclei. The microscopic cluster-daughter potential is numerically constructed in the double-folding model with M3Y nucleon-nucleon interactions plus proton-proton Coulomb interactions. Instead of the WKB barrier penetration probability, the exact solution of the Schr\"odinger equation with outgoing Coulomb wave boundary conditions is presented. The cluster preformation factor is well taken into account based on some available experimental cases. The calculated half-lives are found to be in good agreement with the experimental data. This indicates that a unified description of $\ensuremath{\alpha}$ decay and cluster radioactivity has been achieved by the GDDCM. Predictions of cluster emission half-lives are made for promising emitters, which may guide future experiments.

Half-lives and cluster preformation factors for various cluster emissions in trans-lead nuclei

A new semi-empirical model is proposed for determining the half lives of radioactive nuclei exhibiting cluster radioactivity. The parameters of the formula are obtained by making a least-square fit to the available experimental cluster decay data. The calculated half-life time for cluster decay is compared with the corresponding experimental values and with the values proposed by earlier scaling laws and with those predicted by the Coulomb and proximity potential model. The semi-empirical formula is applied to alpha decay of parents with Z = 85–102 and is compared with experimental data and with other semi-empirical formula predictions. The predicted alpha and cluster decay half-life time values are found to be in good agreement with the experimental data. The calculated alpha decay half-life time is also compared with the values predicted by Viola–Seaborg–Sobiczewski and Horoi systematics.

https://iopscience.iop.org/article/10.1088/0954-3899/35/8/085102/pdf

A semi-empirical model for alpha and cluster radioactivity

Alpha radioactivity in heavy and super heavy elements

The cluster decay half lives for 294?326122 isotopes are computed for various clusters ranging from an alpha particle to 70Ni within the Coulomb and proximity potential model. The computed alpha decay half lives agree with the values calculated using the Viola?Seaborg?Sobiczewski (VSS) systematic formulas. The half lives for spontaneous fission of these isotopes are carried out using the formula of Ren et al. It is found that the alpha emission is the dominant mode of decay for isotopes with mass number A 306 spontaneous fission is found to be dominant. The demarcation between alpha decay and spontaneous fission is at 306122, which shows the presence of a spherical neutron shell closure at N = 184.

https://iopscience.iop.org/article/10.1088/0954-3899/36/1/015107/pdf

Alpha decay, cluster decay and spontaneous fission in 294?326122 isotopes

Cluster formation probability in the trans-tin and trans-lead nuclei

The competition between the alpha decay and spontaneous fission half-lives of even?even elements with Z = 100?122 is studied using the Coulomb and proximity potential model and the phenomenological formula for spontaneous fission. The isotopes situated within the spontaneous fission half-life parabola will have alpha decay as the dominant mode of decay. We have studied the Seaborg plots connecting the spontaneous fission half-lives with the fissionability parameter (Z2/A) in the superheavy region for isotopes with Z = 100?138. The role of relative neutron excess in spontaneous fission half-lives is also studied. The isotopes in the first quadrant of the plot for the ratio of spontaneous fission to alpha decay half-lives against relative neutron excess will survive fission and can be synthesized and detected through alpha decay. Our computed alpha decay half-lives for all isotopes agree with the values calculated by the Viola?Seaborg empirical formula and with the experimental data.

https://iopscience.iop.org/article/10.1088/0954-3899/36/11/115101/pdf

The systematic study of spontaneous fission versus alpha decay of superheavy nuclei

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