α clustering with a hollow structure: Geometrical structure of α clusters from platonic solids to fullerene shape

We study $\ensuremath{\alpha}$-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 $\ensuremath{\alpha}$-cluster models. Here, special attention is focused upon the $\ensuremath{\alpha}$ clustering with a hollow structure; all the $\ensuremath{\alpha}$ 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, $\ensuremath{\alpha}$ clusters create certain local energy pockets. As a consequence, we insist on the possible existence of $\ensuremath{\alpha}$ 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.