Nuclear clusters bound to doubly magic nuclei: The case of ${}^{212}\mathrm{Po}$

An effective $\alpha$-particle equation is derived for cases where an $\alpha$ particle is bound to a doubly magic nucleus. As an example, we consider ${}^{212}\mathrm{Po}$ with the $\alpha$ on top of the ${}^{208}\mathrm{Pb}$ core. We consider the core nucleus infinitely heavy, so that the $\alpha$ particle moves with respect to a fixed center; that is, recoil effects are neglected. The fully quantal solution of the problem is discussed. The approach is inspired by the Tohsaki-Horiuchi-Schuck-Röpke wave function concept that has been successfully applied to light nuclei. Shell-model calculations are improved by including four-particle ($\alpha$-like) correlations that are of relevance when the matter density becomes low. In the region where the $\alpha$-like cluster penetrates the core nucleus, the intrinsic bound-state wave function transforms at a critical density into an unbound four-nucleon shell-model state. Exploratory calculations for ${}^{212}\mathrm{Po}$ are presented. Such preformed cluster states are very difficult to describe with shell-model calculations. Reasons for the different physics behavior of an $\alpha$-like cluster with respect to a deuteron-like cluster are discussed.

Figure 1

Internal four-nucleon energy (no c.m. motion) in a medium with nucleon density $n_B=n_n+n_p$. The continuum edge of free single-particle states is given by $4E_{\mathrm{Fermi}}$ [Eq. (41)]. At zero baryon density, the four-nucleon energy is given by the binding energy of the $\alpha$ particle, $E_{\alpha }^0=-B_{\alpha }^0=-28.3$ MeV. With increasing density, the binding energy $B_{\alpha }^0$ is decreasing owing to the Pauli blocking [Eq. (42)] (stars). The four-nucleon bound state disappears at $n_B\approx 0.03 {\mathrm{fm}}^{-3}$. A fit to the calculated values, Eq. (45), is also shown.

Figure 2

Width parameter $b$ according to Eq. (37) as function of the baryon density. Below $n_B\approx 0.03 {\mathrm{fm}}^{-3}$ a bound state ($\alpha$-like) arises (full line). Above $n_B\approx 0.03 {\mathrm{fm}}^{-3}$, a resonance occurs; see Fig. 1 (dotted line).

Figure 3

Coulomb potential and isospin-dependent Woods-Saxon potentials for the ${}^{208}\mathrm{Pb}$ core.

Figure 4

Local effective potential $W\left(\mathbf{R}\right)$ (62) (red solid line) with respect to the lead ${}^{208}\mathrm{Pb}$ core for the Woods-Saxon-like distribution (59), $A=208$. The Thomas-Fermi density and the Fermi energy $4E_{\mathrm{Fermi}}$ of the four added nucleons is shown, as is the measured energy ($Q$ value) of the emitted $\alpha$ particle. The distance at which the density becomes the critical value $n_B=0.0292 {\mathrm{fm}}^{-3}$ where the $\alpha$ particle is dissolved is indicated.

Figure 5

Insertion of Fig. 4.

Figure 6

Nucleon density $n_B\left(r\right)$ (blue solid lines) according to Shlomo [40] compared with the Thomas-Fermi approach (blue dashed lines). The local effective potential $W\left(\mathbf{R}\right)$ (62) (red dashed lines) according to the Thomas-Fermi approach is compared with an effective potential (red solid lines), which describes the Pauli blocking of the long-ranged density tails at the surface of the lead core. The ground-state energy level (green dash-dotted line) is also shown.