Theory of the formation of P4132(P4332)-phase spinels

A group-theoretical, thermodynamic, and structural study of the formation of P4132(P4332) spinel modification has been performed. In particular, the occurrence of unique hyper-kagome atomic order is analyzed. The critical order parameter inducing a phase transition is established. It is shown that the calculated structure of the low-symmetry P4132(P4332) phase is formed as a result of displacements of atoms of all types and due to the cation and anion ordering. The problem of the occurrence of unique hyper-kagome atomic order in the structures of P4132(P4332) spinel modifications is considered theoretically. It is proven within the Landau theory of phase transitions that the P4132(P4332) phase can be formed from the high-symmetry Fd3m phase with an ideal spinel structure only as a result of first-order phase transition. Therefore, the formation of hyper-kagome sublattice in the P4132(P4332) phase is accompanied by a significant transformation of the spinel structure.