$\alpha$ clustering with a hollow structure: Geometrical structure of $\alpha$ clusters from platonic solids to fullerene shape

We study $\alpha$-cluster structure based on the geometric configurations with a microscopic framework, which takes full account of the Pauli principle, and which also employs an effective internucleon force including finite-range three-body terms suitable for microscopic $\alpha$-cluster models. Here, special attention is focused upon the $\alpha$ clustering with a hollow structure; all the $\alpha$ clusters are put on the surface of a sphere. All the platonic solids (five regular polyhedra) and the fullerene-shaped polyhedron coming from icosahedral structure are considered. Furthermore, two configurations with dual polyhedra, hexahedron-octahedron and dodecahedron-icosahedron, are also scrutinized. When approaching each other from large distances with these symmetries, $\alpha$ clusters create certain local energy pockets. As a consequence, we insist on the possible existence of $\alpha$ clustering with a geometric shape and hollow structure, which is favored from Coulomb energy point of view. Especially, two configurations, that is, dual polyhedra of dodecahedron-icosahedron and fullerene, have a prominent hollow structure compared with the other six configurations.

Figure 1

Schematic figures for the prepared configurations, where vertices on the polyhedra show the positions of the $\alpha$ clusters: (a) tetrahedron, (b) hexahedron (cube), (c) octahedron, (d) dodecahedron, (e) icosahedron, (f) hexahedron-octahedron, (g) dodecahedron-icosahedron, and (h) fullerene-shape polyhedron configurations.

Figure 2

Energy for one $\alpha$ as a function of the radius ($\rho$ in the text): (a) tetrahedron, (b) hexahedron (cube), (c) octahedron, (d) dodecahedron, and (e) icosahedron.

Figure 3

Energy for one $\alpha$ as a function of the radius ($\rho$ in the text), (f) hexahedron-octahedron, (g) dodecahedron-icosahedron, and (h) fullerene.

Figure 4

Nucleon number density of the fullerene configuration with $\rho =7.7$ fm as a function of the radius. The integration over the radius is normalized to the number of nucleons.