Neutron-diffraction study of icosahedral Al-Cu-Fe single quasicrystals

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Abstract
This paper reports neutron-diffraction results on a single icosahedral quasicrystal of Al-Cu-Fe. The basic properties of the structure have been extracted using six-dimensional (6D) Patterson analysis from 219 independent orbits of reflections. Described in 6D space, the structure has hyperspace group F⊗m35 and is defined by the three atomic surfaces located at special points with full icosahedral symmetry of the F lattice. These points are the two inequivalent nodes of the underlying primitive lattice plus one of the two inequivalent body centers, the remaining one being empty. The atomic surfaces are embedded in perpendicular space and are well approximated by polyhedra bounded by two-fold planes. These are a large triacontahedron located at the origin, a triacontahedron of the same size truncated along the five-fold directions at the other node, and a small polyhedron bounded by twofold planes at the occupied body center. Although no speculation has been made for distributing the atomic species within these atomic surfaces, the raw reliability factor between experimental and calculated diffraction intensities is already 0.20 with no fitting parameters and the density is found only 2.9% lower than the experimental one. The model presented here can be considered as a zero-order structure to be used for subsequent modeling. The atomic surfaces generate no unacceptably short distances between atoms. Both interatomic distances and coordination numbers of the three first shells are in good agreement with the most recent extended x-ray-absorption fine-structure results. The atomic surfaces are connected together by 3D pieces embedded in the parallel space. They define a partition of the 6D space in hyperprisms, which can be decomposed in direct products of 3D facets located in perpendicular and parallel spaces similar to the oblique cell decomposition of the 3D Penrose tiling. Phasons can propagate along the five-fold and two-fold directions by atomic jumps of 0.1705 and 0.179 nm, respectively.