Group-theoretical and thermodynamic methods of the Landau theory of phase transitions are used to investigate the hyper-kagome atomic order in structures of ordered spinels and a spinel-like Na4Ir3O8 crystal. The formation of an atom hyper-kagome sublattice in Na4Ir3O8 is described theoretically on the basis of the archetype (hypothetical parent structure/phase) concept. The archetype structure of Na4Ir3O8 has a spinel-like structure (space group Fd\bar 3m) and composition [Na1/2Ir3/2](16d)[Na3/2](16c)O(32e)4. The critical order parameter which induces hypothetical phase transition has been stated. It is shown that the derived structure of Na4Ir3O8 is formed as a result of the displacements of Na, Ir and O atoms, and ordering of Na, Ir and O atoms, ordering dxy, dxz, dyz orbitals as well. Ordering of all atoms takes place according to the type 1:3. Ir and Na atoms form an intriguing atom order: a network of corner-shared Ir triangles called a hyper-kagome lattice. The Ir atoms form nanoclusters which are named decagons. The existence of hyper-kagome lattices in six types of ordered spinel structures is predicted theoretically. The structure mechanisms of the formation of the predicted hyper-kagome atom order in some ordered spinel phases are established. For a number of cases typical diagrams of possible crystal phase states are built in the framework of the Landau theory of phase transitions. Thermodynamical conditions of hyper-kagome order formation are discussed by means of these diagrams. The proposed theory is in accordance with experimental data.
Keywords: atom and orbital orders; decagons; enantiomorphic modifications; hyper-kagome order; ordered spinels.