Frank's 'cubic' hexagonal phase: an intermetallic cluster compound as an example

Frank introduced in 1965 the novel idea of projection from a four-dimensional cube to recover the points of a hexagonal lattice with a special c/a ratio of $(^3 - 2^1^/^2$. This was called a cubic hexagonal crystal, as there was a similarity to the conventional cubic crystals in that directions were perpendicular to planes with the same Miller-Bravais indices. While a number of crystals in the $NiAs-Ni_2In$ family have been reported with this special axial ratio, the number of atoms in the unit cell is small. As the first example of an intermetallic cluster compound, we identify $\mu-Al_4Mn, \mu-Al_4Cr, Zn-Mg-Sm$ and a host of related intermetallics featuring a special aggregate of icosahedra as Frank's 'cubic' hexagonal phase or its variant. The metric is generated by the Friauf polyhedra and the icosahedral linkages and leads to a multimetric crystal and several interesting connections with hexagonal quasicrystals, hexagonal phases and derivative orthorhombic and lower-symmetry structures