Pseudo‐Fivefold Diffraction Symmetries in Tetrahedral Packing

We review the way in which atomic tetrahedra composed of metallic elements pack naturally into fused icosahedra. Orthorhombic, hexagonal, and cubic intermetallic crystals based on this packing are all shown to be united in having pseudo‐fivefold rotational diffraction symmetry. A unified geometric model involving the 600‐cell is presented: the model accounts for the observed pseudo‐fivefold symmetries among the different Bravais lattice types. The model accounts for vertex‐, edge‐, polygon‐, and cell‐centered fused‐icosahedral clusters. Vertex‐centered and edge‐centered types correspond to the well‐known pseudo‐fivefold symmetries in I_h and D_5h quasicrystalline approximants. The concept of a tetrahedrally‐packed reciprocal space cluster is introduced, the vectors between sites in this cluster corresponding to the principal diffraction peaks of fused‐icosahedrally‐packed crystals. This reciprocal‐space cluster is a direct result of the pseudosymmetry and, just as the real‐space clusters, can be rationalized by the 600‐cell. The reciprocal space cluster provides insights for the Jones model of metal stability. For tetrahedrally‐packed crystals, Jones zone faces prove to be pseudosymmetric with one another. Lower and upper electron per atom bounds calculated for this pseudosymmetry‐based Jones model are shown to accord with the observed electron counts for a variety of Group 10–12 tetrahedrally‐packed structures, among which are the four known Cu/Cd intermetallic compounds: CdCu₂, Cd₃Cu₄, Cu₅Cd₈, and Cu₃Cd₁₀. The rationale behind the Jones lower and upper bounds is reviewed. The crystal structure of Zn₁₁Au₁₅Cd₂₃, an example of a 1:1 MacKay cubic quasicrystalline approximant based solely on Groups 10–12 elements is presented. This compound crystallizes in Im$bar 3$ (space group no. 204) with a=13.842(2) Å. The structure was solved with R₁=3.53 %, I>2σ;=5.33 %, all data with 1282/0/38 data/restraints/parameters.