Quasicrystal model based on a series of supercubic crystal structures consisting of triacontahedra or Mackay icosahedra

Abstract

The Al-Mn or Al-Mn-Si icosahedral quasicrystal structure is shown to consist of Mackay icosahedra and double Mackay icosahedra packed closely together. The packing fraction f of the quasicrystal is found to be almost the same as that of a crystal structure with the same composition as the quasicrystal. An inflation rule with a scale of ϕ³ (ϕ = 1·618…) can be used to construct quasicrystal structures of any required order. The same packing method can be used to construct the (Al, Zn)₄₉Mg₃₂, quasicrystal structure if Mackay icosahedra and double Mackay icosahedra of the icosahedral phase are replaced with triacontahedra and double triacontahedra. If we arrange an nth-order structure at the vertices and body centre of a cubic unit cell, we obtain a series of supercubic crystal structures with lattice constants a = (ϕ³ⁿ a ₀ and a space group Pm3, where a _ois the lattice constant of the α-Al-Mn-Si crystal structure. The above structural model is in good agreement with previous higher resolution electron microscopy images of the Al-Mn icosahedral quasicrystal by other authors.