A comparison of multilayered coordination numbers in FCC and HCP crystals

Abstract

A comparative study of multilayered coordination numbers between the face center cubic (FCC) and the hexagonal closest packing (HCP) was conducted. First, by the Diophantine equation models, we derived generating functions to calculate the multilayered coordination radii $r_k$ and coordination numbers $N_k$ for FCC and HCP, respectively, Calculated results for FCC and HCP were further compared, including their coordination radii, coordination numbers and mean coordination numbers. Finally, we strictly proved that the sequences $\{N_k\}$ for both FCC and HCP are unbounded, and for FCC, there is a subsequence $N_{k_n}=6$, while for HCP, there is a subsequence $N_{k_n}=2$.