$\alpha$-decay width of ${}^{212}\mathrm{Po}$ from a quartetting wave function approach

A microscopic calculation of $\alpha$-cluster preformation probability and $\alpha$-decay width in the typical $\alpha$ emitter ${}^{212}\mathrm{Po}$ is presented. Results are obtained by improving a recent approach to describe $\alpha$ preformation in ${}^{212}\mathrm{Po}$ [Phys. Rev. C 90, 034304 (2014)] implementing four-nucleon correlations (quartetting). Using the actually measured density distribution of the ${}^{208}\mathrm{Pb}$ core, the calculated $\alpha$-decay width of ${}^{212}\mathrm{Po}$ agrees fairly well with the measured one.

Figure 1

Intrinsic part $W^{\mathrm{intr}}\left(r\right)$ of the effective c.m. potential $W\left(r\right)$. The empirical density distribution [9] for the ${}^{208}\mathrm{Pb}$ core has been used. The critical radius $r_{\mathrm{cluster}}=7.4383$ fm where the bound state are dissolved is indicated (dotted line). The four-nucleon Fermi energy $E_{4,\mathrm{cont}}\left(r\right)$ for $r<r_{\mathrm{cluster}}$ is taken in Thomas–Fermi approximation (free fermion gas).

Figure 2

Effective c.m. potential $W\left(r\right)$, potential A. The empirical baryon density distribution $n_B\left(r\right)$ [9] for the ${}^{208}\mathrm{Pb}$ core is also shown. The chemical potential ${\mu }_4$ coincides with the binding energy $E_{\mathrm{tunnel}}$.

Figure 3

Effective c.m. potential $W\left(r\right)$ for the $\alpha$ decay of ${}^{212}\mathrm{Po}$. Both versions A and B lead to the empirical bound state energy $E_{\mathrm{tunnel}}=-19.346$ MeV; see Table 1. The repulsive potential is given by the Pauli blocking term. The total c.m. potential $W\left(r\right)$ contains, in addition, the Coulomb part and the bound state energy $E_{\alpha }^{\left(0\right)}=-28.3$ MeV. The chemical potential ${\mu }_4$ for potential B is slightly deeper than the bound state energy $E_{\mathrm{tunnel}}=-19.346$ MeV; see Table 1.

Figure 4

The bound state wave function $\mathrm{\Phi }\left(r\right)$ and the scattering state wave function $\chi \left(r\right)$ calculated by separating potential B into two parts based on the two-potential approach. The separation point is chosen to be $r_{\mathrm{sep}}=15$ fm.

Figure 5

Model calculations for various parametrizations $c,d$ of the direct term $v\left(s\right)$. (a) Variation of the decay half-life with the energy difference $E_{\mathrm{tunnel}}-{\mu }_4$. (b) Variation of preformation factor with the energy difference $E_{\mathrm{tunnel}}-{\mu }_4$. (c) Variation of the decay half-life with the preformation factor.