$\alpha$ decay to a doubly magic core in the quartetting wave function approach

We present a microscopic calculation of $\alpha$-cluster formation in heavy nuclei ${}^{104}\mathrm{Te}$ $(\alpha +{}^{100}\mathrm{Sn})$, ${}^{212}\mathrm{Po}$ $(\alpha +{}^{208}\mathrm{Pb})$, and their neighbors ${}^{102}\mathrm{Sn},{}^{102}\mathrm{Te},{}^{210}\mathrm{Pb}$, and ${}^{210}\mathrm{Po}$ by using the quartetting wave function approach. To improve the local density approximation, the shell structure of the core nucleus is considered, and the center-of-mass (c.m.) effective potential for the quartet is obtained self-consistently from the shell model wave functions. The $\alpha$-cluster formation and decay probabilities are obtained by solving the bound-state of the c.m. motion of the quartet and the scattering state of the formed $\alpha$ cluster in the Gurvitz approach. Striking shell effects on the $\alpha$-cluster formation probabilities are analyzed for magic numbers 50, 82, and 126. The computed $\alpha$-decay half-lives of these special nuclei are compared with the newest experimental data.

Figure 1

The contributing single-particle wave functions of protons and neutrons in the quartets of ${}^{102}\mathrm{Sn},{}^{102}\mathrm{Te},{}^{104}\mathrm{Te},{}^{210}\mathrm{Pb},{}^{210}\mathrm{Po}$, and ${}^{212}\mathrm{Po}$.

Figure 2

The c.m. effective potentials of the $\alpha$ cluster in (a) ${}^{102}\mathrm{Sn},{}^{102}\mathrm{Te}$, and ${}^{104}\mathrm{Te}$ and (b) ${}^{210}\mathrm{Pb},{}^{210}\mathrm{Po}$, and ${}^{212}\mathrm{Po}$. The sketch of small box with filled circles denotes the core nucleus considered as a mean field. The position of the energy eigenvalues $E$ for the c.m. motion is marked.

Figure 3

Comparison of the normalized bound state wave functions for (a) ${}^{102}\mathrm{Sn},{}^{102}\mathrm{Te}$, and ${}^{104}\mathrm{Te}$ and (b) ${}^{210}\mathrm{Pb},{}^{210}\mathrm{Po}$, and ${}^{212}\mathrm{Po}$. The critical radius $R_c$ of each nucleus is marked. The two peaks or the shift of the maximum are caused by the formation of a pocket in Fig. 2.

Figure 4

The scattering wave functions $\chi \left(R\right)$ for $\alpha$-emitters ${}^{104}\mathrm{Te}$ and ${}^{212}\mathrm{Po}$ in the two-potential approach. The separating point is chosen to be $R_{\mathrm{sep}}=15$ fm.