Atomic order and metallic nanoclusters in Na$_{4}$Ir$_{3}$O$_{8}$

A theory for forming Ir-atomic nanoclusters in a dielectric matrix of Na$_{4}$Ir$_{3}$O$_{8}$ structure and spinel-like structures is suggested. The atomic order in the Na$_{4}$Ir$_{3}$O$_{8}$ structure is investigated by group-theoretical methods of phase transition theory. The critical irreducible representation $\tau$, generating appearance of enantiomorphic P 4$_{1}$32 (P 4$_{3}$32)-phases from high symmetry spinel-like phase with space group Fd3m, is six dimensional irreducible representation k$_{10}$($\tau_{1}$) (in Kovalev designation). Ir and Na atoms form an intriguing atom ordering, giving rise to a network of corner shared Ir triangles, called a hyperkagome lattice. It is shown that inside the hyperkagome lattice, there are closed metal contours of chemical bonds formed by Ir-clusters - decagons. Unusual physical properties of solid solutions on the basis of Na$_{4}$Ir$_{3}$O$_{8}$ are expected. The existence of hyperkagome lattices in six types in ordered spinel structures is theoretically predicted.