Systematic study of favored $\alpha$-decay half-lives of closed shell odd-$A$ and doubly-odd nuclei related to ground and isomeric states

In this work, we systematically investigate the favored $\alpha$-decay half-lives and $\alpha$ preformation probabilities of both odd-$A$ and doubly-odd nuclei related to ground and isomeric states around the doubly magic cores at $Z=82, N=82$ and at $Z=82, N=126$, respectively, within a two-potential approach from the view of the valence nucleon (or hole). The results show that the $\alpha$ preformation probability is linearly related to $N_{\text{p}}{N}_{\text{n}}$ or $N_{\text{p}}{N}_{\text{n}}I$, where $N_{\text{p}}, N_{\text{n}}$, and $I$ are the number of valence protons (or holes), the number of valence neutrons (or holes), and the isospin of the parent nucleus, respectively. Fitting the $\alpha$ preformation probabilities data extracted from the differences between experimental data and calculated half-lives without a shell correction, we give two analytic formulas of the $\alpha$ preformation probabilities and the values of corresponding parameters. Using those formulas and the parameters, we calculate the $\alpha$-decay half-lives for those nuclei. The calculated results can well reproduce the experimental data.

Figure 1

Logarithm of the $\alpha$ preformation probabilities ratio $P_{\alpha }^{*}/P_{\alpha }$ as a function of proton numbers of the parent nuclei. The $P_{\alpha }^{*}$ and $P_{\alpha }$ denote the $\alpha$ preformation probability of the isomeric and ground states, respectively.

Figure 2

The colormap of $N_{\text{p}}{N}_{\text{n}}I$ as a function of neutron numbers $N$ and proton numbers $Z$ of the parent nuclei. The rectangles denote the nuclear shell closures at $Z=82$ and $N=82$, 126, and the nuclide chart is divided into five regions.

Figure 3

The $\alpha$ preformation probabilities as a function of $\frac{N_{\text{p}}{N}_{\text{n}}}{N_0+Z_0}$, where $N_{\text{p}}, N_{\text{n}}$ represent valence proton numbers and valence neutron numbers of the parent nucleus, respectively, and $N_0\left(Z_0\right)$ express neutron (proton) magic numbers. The blue dashed and solid lines denote the fit of nuclei in Regions IV and III, respectively. The red short dashed line denotes the fit of nuclei in Region I.

Figure 4

Same as Fig. 3, but as a function of the product of valence proton numbers and valence neutron numbers and isospin asymmetry $N_{\text{p}}{N}_{\text{n}}I$.

Figure 5

Same as Fig. 3, but for doubly-odd nuclei as a function of $\frac{N_{\text{p}}{N}_{\text{n}}}{N_0+Z_0}$.

Figure 6

Same as Fig. 3, but for doubly-odd nuclei as a function of $N_{\text{p}}{N}_{\text{n}}I$.