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 inter-nucleon 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. As a consequence, we insist on the possible existence of stable $\alpha$-clustering with a hollow structure for all the configurations. Especially, two configurations, that is, dual polyhedra of dodecahedron-icosahedron and fullerene, have a prominent hollow structure compared with other six configurations.