The Mystery of Perpendicular Fivefold Axes and the Fourth Dimension in Intermetallic Structures

Abstract

The structures of eight related known intermetallic structure types are the impetus to this paper: Li21Si5, Mg44Rh7, Zn13(Fe,Ni)2, Mg6Pd, Na6Tl, Zn91Ir11, Li13Na29Ba19, and Al69Ta39. All belong to the F43m space group, have roughly 400 atoms in their cubic unit cells, are built up at least partially from the gamma-brass structure, and exhibit pseudo-tenfold symmetric diffraction patterns. These pseudo-tenfold axes lie in the {110} directions, and thus present a paradox. The {110} set is comprised of three pairs of perpendicular directions. Yet no 3D point group contains a single pair of perpendicular fivefold axes (by Friedel's Law, a fivefold axis leads to a tenfold diffraction pattern). The current work seeks to resolve this paradox. Its resolution is based on the largest of all 4D Platonic solids, the 600-cell. We first review the 600-cell, building an intuition discussing 4D polyhedroids (4D polytopes). We then show that the positions of common atoms in the F43m structures lie close to the positions of vertices in a 3D projection of the 600-cell. For this purpose, we develop a projection method that we call intermediate projection. The introduction of the 600-cell resolves the above paradox. This 4D Platonic solid contains numerous orthogonal fivefold rotations. The six fivefold directions that are best preserved after projection prove to lie along the {110} directions of the F43m structures. Finally, this paper shows that at certain ideal projected cluster sizes related to one another by the golden mean (tau=(1+ radical 5)/2), constructive interference leading to tenfold diffraction patterns is optimized. It is these optimal values that predominate in actual F43m structures. Explicit comparison of experimental cluster sizes and theoretically derived cluster sizes shows a clear correspondence, both for isolated and crystalline pairs of projected 600-cells.