Laves Phases, γ‐Brass, and 2×2×2 Superstructures: A New Class of Quasicrystal Approximants and the Suggestion of a New Quasicrystal

Abstract

Of the most common cubic intermetallic structure types, several (MgCu(2), Cu(5)Zn(8), Ti(2)Ni, and alpha-Mn) have superstructures with unusual symmetry properties. These superstructures (Be(5)Au, Li(21)Si(5), Sm(11)Cd(45), and Mg(44)Ir(7)) have the unusual property of pairs of perpendicular pseudo fivefold axes, most apparent in their X-ray diffraction patterns. The current work shows that an 8D to 3D projection method cleanly describes most (and in one case, all) of the atomic positions in the four superstructures mentioned above. This type of projection, which maps the E(8) lattice (a mathematically simple 8D crystal) into 3D space, combines the desired higher dimensional point group's perpendicular fivefold rotations with 3D translational symmetry-exactly what we see in the experimental crystal structures. The projection method successfully accounts for all heavy atom positions in the four superstructures, and at least 60-70 % of the light atom positions. The results suggest that all of these structures, previously known to be connected only by qualitative similarities in their atomic "clusters", are approximants of a single, as-yet unknown, class of quasicrystal.

Pseudo‐Fivefold Diffraction Symmetries in Tetrahedral Packing

“…As we and others have shown, these larger objects are often 4D quasicrystals, [43,47] but these quasicrystals have the same 4D point group symmetry as the 600-cell itself. In such cases, the fivefold rotational symmetry corresponding to the 720 edges of the 600-cell, retain their character as fivefold pseudosymmetry axes in their large crystal unit cell 3D projections.…”

“…[34][35][36][37][38] In the other two cases, the clusters will prove to be of T d and D 3h symmetry and will result in, respectively, cubic and hexagonal crystalline structures. The literature has found common ground for these structures as projections of six- [39][40][41][42] or eight-dimensional [43][44][45][46][47] Bravais lattices, curved topology, [21,48,49] and networks of disclinations. [50][51][52] The current work pares these mathematical approaches to their bare minimum, that is, to concepts which describe concrete 3D crystals: the 3D pseudosymmetries, the pseudo-equivalences of their 3D diffraction reflections, and the tetrahedral organization of both their 3D real and reciprocal space clusters.…”

The Intermetalloid Cluster [(Cp*AlCu)6H4], Embedding a Cu6 Core Inside an Octahedral Al6 Shell: Molecular Models of Hume–Rothery Nanophases

“…The 26-atom nested polyhedral clusters of g-brass (Figure 1) consist of successive layers of an inner tetrahedron (IT, orange), an outer tetrahedron (OT, green), an octahedron (OH, pale blue), and a distorted cuboctahedron (CO, gray). [16] The polyhedron formed by combination of the tetrahedra IT and OT is also known as "stella-quadrangula". In Cu 5 Zn 8 , OT and OH are occupied by Cu atoms, while IT and CO represent Zn positions.…”

A Higher Dimensional Description of the Structure of β-Mn

“…The principle is simple: In six dimensional space, it is possible to generate three mutually perpendicular 8 fold axes, and on projection to a five dimensional subspace, one of these may be preserved, while a projection to three-dimensional space may preserve only 4 fold axis. The procedure we are using owns a lot to that used by Lee et al in a paper dealing with the unexpected occurrence of mutually perpendicular 5 fold axis in large cubic intermetallic structures [7].…”