$\alpha$ clustering from the formation of a pocket structure in the $\alpha$-nucleus potential

We investigate the formation of a pocket structure in the $\alpha$-nucleus potential based on the dynamic double-folding potential model. We improve the model with the density-dependent Migdal nucleon-nucleon (NN) interaction and combine it with the nuclear medium effect. The improved potential naturally generates a pocket structure at the nuclear surface, which physically agrees with the characteristics suggested by the microscopic many-body calculations. The result reveals that the formation of the pocket structure is due to the strong Pauli repulsion caused by the variation of the NN interaction when the $\alpha$ cluster and the daughter nucleus have large density overlaps. The existence of the nuclear medium effect is essential to the physical self-consistency of the pocket position. These findings highlight the importance of the medium effect on the NN interaction and provide solid theoretical support for the understanding that $\alpha$ clustering occurs at the surface of heavy nuclei.

Figure 1

Schematic of the coordinates used in the spherical double-folding potential model. The vector between the interacting nucleons, defined as $\mathit{s}=\mathit{R}+{\mathit{r}}_{\mathit{2}}-{\mathit{r}}_{\mathit{1}}$, is zero in the schematic to elucidate the zero-range Migdal interaction.

Figure 2

The $\alpha$-nucleus potential $V\left(R\right)$ and the $\alpha$-cluster wave function $\varphi \left(R\right)$ for ${}^{212}\mathrm{Po}$: (a) without medium effect, (b) with medium effect. The $\alpha$ decay energy $Q_{\alpha }$ and the pocket positions $r_p$ are shown. The critical radius $r_c$, where the medium density is $1/5$ of the saturation density, is marked. Note that after incorporating the medium effect, the pocket and the domain of $\varphi \left(R\right)$ shift to the surface region specified by $r_c$. The first classical turning point appears very close to $r_c$, which nicely matches the physical picture indicated by the microscopic calculation [10].

Figure 3

The density overlap (grey patches) between the daughter nucleus ${}^{208}\mathrm{Pb}$ and the $\alpha$ cluster at $R=7\text{fm}$: (a) without medium effect, (b) with medium effect. The center of mass of the $\alpha$ cluster is denoted by the dashed line. The strength of the $\alpha$-nucleus potential $V$, determined by the overlaps, is given to illustrate that the repulsion felt by the $\alpha$ cluster is strengthened after incorporating the medium effect.

Figure 4

The $\alpha$-nucleus potentials $V$ for ${}^{105}\mathrm{Te}$ in $L=0\alpha$ transition. The DDFP with Migdal interaction is denoted by the blue solid line. The inset at the left upper corner shows the behavior of the DDFP at $R\to 0$, while the inset at the right upper corner, taken from Ref. [30] for comparison, shows the potentials constructed with the Skyrme energy-density formalism. The DDFP is similar to that directly derived by the SEDF (black bold line) in terms of the location of the pocket and the height of the Coulomb potential barrier.

Figure 5

(a) The determined surface diffuseness of the daughter nucleus as a function of its neutron number. (b) The experimental $\alpha$ decay energy of the parent nucleus as a function of its daughter's neutron number. The variation of diffuseness appears as a reverse pattern of the $Q_{\alpha }$, showing an evident shell effect at $N_d=126$.

Figure 6

The extracted $\alpha$-preformation factor $P_{\alpha }$ for even-even Pb, Po, and Rn isotopes. The DDFP generated with Migdal (solid lines) and CDM3Y6 (dotted lines) NN interactions yield a similar trend for the $P_{\alpha }$ variation, in which the well-known neutron $N_p=126$ shell effect and the proton $Z_p=82$ shell effect are both reproduced.

Figure 7

(a) The input $\alpha$-preformation factor $P_{\alpha }^{\text{inp.}}$ (grey bold line) in the half-life calculation and the extracted $P_{\alpha }$ (colored symbols) are shown for comparison. (b) The logarithmic deviation between the theoretical and experimental $\alpha$ decay half-lives. The range of the deviations indicates a factor of $2–3$ between the theoretical and experimental values.