Dilute multi-$\alpha$ cluster states in nuclei

Dilute multi-$\alpha$ cluster condensed states with spherical and axially deformed shapes are studied with the Gross-Pitaevskii equation and Hill-Wheeler equation, where the $\alpha$ cluster is treated as a structureless boson. Applications to self-conjugate $4N$ nuclei show that the dilute $N\alpha$ states of ${}^{12}\mathrm{C}$ to ${}^{40}\mathrm{Ca}$ with $J^{\pi }=0^{+}$ appear in the energy region from threshold up to about $20\text{MeV}$, and the critical number of $\alpha$ bosons that the dilute $N\alpha$ system can sustain as a self-bound nucleus is estimated roughly to be $N_{cr}\sim 10$. We discuss the characteristics of the dilute $N\alpha$ states with emphasis on the $N$ dependence of their energies and rms radii.

Figure 1

Radial part of the relative wave function between the $2\alpha$ clusters in the $J^{\pi }=0^{+}$ resonant state at $E_{2\alpha }=92\text{keV}$ with use of the soft-core $\alpha \text{−}\alpha$ potential in Eq. (31). The scale of $r\varphi \left(r\right)$ is arbitrary.

Figure 2

(a) Total energies for the dilute $N\alpha$ states measured from each $N\alpha$ threshold, and (b) their nuclear rms radii, which are obtained by solving the Gross-Pitaevskii equation with the density-dependent potential.

Figure 3

Single-$\alpha$-particle potentials $U_{\alpha }\left(R\right)$ (solid line) which are obtained by solving the Gross-Pitaevskii equation with the density-dependent potential; (a) $3\alpha$, (b) $4\alpha$, (c) $5\alpha$, (d) $8\alpha$, and (e) $10\alpha$ systems. The dashed, dot-dashed, and dotted lines demonstrate, respectively, the contribution from the two-range-Gaussian term, density-dependent term, and Coulomb potential.

Figure 4

(a) Total energies for the dilute $N\alpha$ states measured from each $N\alpha$ threshold, and (b) their nuclear rms radii, which are obtained by solving the Gross-Pitaevskii equation with the phenomenological $2\alpha$ and $3\alpha$ potentials.

Figure 5

Single-$\alpha$-particle potentials $U_{\alpha }\left(R\right)$ (solid line) which are obtained by solving the Gross-Pitaevskii equation with the phenomenological $2\alpha$ and $3\alpha$ potentials; (a) $3\alpha$, (b) $4\alpha$, (c) $5\alpha$, (d) $8\alpha$, and (e) $10\alpha$ systems. The dashed, dot-dashed, and dotted lines demonstrate, respectively, the contribution from the $2\alpha$ potential, $3\alpha$ potential, and Coulomb potential.

Figure 6

(a) Total energies for the dilute $N\alpha$ states (spherical case) measured from each $N\alpha$ threshold (solid line), and (b) their nuclear rms radii (solid), which are obtained by solving the Hill-Wheeler equation with the phenomenological $2\alpha$ and $3\alpha$ potentials. For comparison, we also give the corresponding values obtained by solving the Gross-Pitaevskii equation with use of the same potentials (dotted lines).

Figure 7

(a) Total energies for the dilute $N\alpha$ states with the axial deformation measured from each $N\alpha$ threshold (solid line), and (b) their nuclear rms radii (solid) and rms distance between $2\alpha$ particles (dotted), which are obtained by solving the Hill-Wheeler equation with the phenomenological $2\alpha$ and $3\alpha$ potentials.

Figure 8

Phenomenological $2\alpha$ potential (solid line), where the $2\alpha$ nuclear potential [Eq. (31)] and its Coulomb potential [Eq. (30)] are drawn by the dashed and dotted lines, respectively.