Abstract: The existence of “magic number” is the fundamental characteristic of clusters. For clusters bound by van der Waals force, they can be described by the icosahedral model, which can well reproduce the magic numbers. Although the coordination number of the central atom of icosahedral clusters is 12, it is not close-packed. To support the cluster, the “atomic spheres” are tangent along the vertex direction. Currently, the detailed calculation about the packing fraction of a regular icosahedron cluster can be hardly found. In this paper, we calculate it by spherical geometry. It decreases as the number of atomic layer increases. When infinite layers are concerned, the packing fraction is 68.82%, which is between the body-centered cubic and face-centered cubic (hexagonal close-packed) structures. The result can be referenced by colleagues engaged in teaching and investigation on clusters. 

Key words: clusters, icosahedron, packing fraction