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

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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 = (ϕ3na 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.