Unified structure theory of icosahedral quasicrystals: Evidence from neutron powder diffraction patterns that AlCrFeMnSi, AlCuLiMg, and TiNiFeSi icosahedral quasicrystals are twins of cubic crystals containing about 820 or 1012 atoms in a primitive unit cube

A unified structure theory of icosahedral quasicrystals, combining the twinned-cubic-crystal theory and the Penrose-tiling-six-dimensional-projection theory, is described. Values of the primitive-cubic lattice constant for several quasicrystals are evaluated from x-ray and neutron diffraction data. The fact that the low-angle diffraction maxima can be indexed with cubic unit cells provides additional support for the twinned-cubic-crystal theory of icosahedral quasicrystals.     

Icosahedral quasicrystals of intermetallic compounds are icosahedral twins of cubic crystals of three kinds, consisting of large (about 5000 atoms) icosahedral complexes in either a cubic body-centered or a cubic face-centered arrangement or smaller (about 1350 atoms) icosahedral complexes in the beta-tungsten arrangement

The twofold-axis electron-diffraction photographs of icosahedral quasicrystals are of three kinds, reflecting three different structures of the cubic crystals that by icosahedral twinning form the quasicrystals. The first kind, represented by Al₁₃Cu₄Fe₃, contains two very large icosahedral complexes, each of about 4680 atoms, in the body-centered arrangement, with six smaller icosahedral complexes (104 atoms each) in the principal interstices. The second kind, represented by Al₅Mn, contains four of the very large complexes in the face-centered arrangement (cubic close packing), with four of the smaller clusters in the interstices. The third kind, represented by Al₆CuLi₃, contains eight icosahedral complexes, each of about 1350 atoms, in the β-W arrangement. The supporting evidence for these cubic structures is discussed as well as other evidence showing that the simple quasicrystal theory, which states that quasicrystals do not involve any translational identity operations, has to be modified.     

Analysis of pulsed-neutron powder diffraction patterns of the icosahedral quasicrystals Pd3Siu and AlCuLiMg (three alloys) as twinned cubic crystals with large units

The low-Q peaks on three pulsed-neutron powder patterns (total, U differential, and Pd differential) of the icosahedral quasicrystal Pd3SiU have been indexed on the basis of an assumed cubic structure of the crystals that by icosahedral twinning form the quasicrystal. The primitive unit cube is found to have edge length 56.20 A and to contain approximately 12,100 atoms. Similar analyses of pulsed-neutron patterns of Al55Cu10Li35, Al55Cu10Li30Mg5, and Al510Cu125Li235Mg130 give values of the cube edge length 58.3, 58.5, and 58.4 A, respectively, with approximately 11,650 atoms in the unit cube. It is suggested that the unit contains eight complexes in the beta-W positions, plus some small interstitial groups of atoms, with each complex consisting of a centered icosahedron of 13 clusters, each of 116 atoms with the icosahedral structure found in the body-centered cubic crystal Mg32(Al,Zn)49.     

Sigma-phase packing of icosahedral clusters in 780-atom tetragonal crystals of Cr(5)Ni(3)Si(2) and V(15)Ni(10)Si that by twinning achieve 8-fold rotational point-group symmetry

A 780-atom primitive tetragonal unit with edges 27.3, 27.3, and 12.6 Å is assigned to rapidly solidified Cu₅Ni₃Si₂ and V₁₅Ni₁₀Si by analysis of electron diffraction photographs with the assumption that the crystals contain icosahedral clusters. There are thirty 26-atom clusters at the sigma-phase positions. Apparent 8-fold symmetry results from 45° twinning on the basal plane.