Symmetry analysis of the monoclinic Pd₆B superstructure: Long- and short-range orders

Abstract

A symmetry analysis of the monoclinic (space group C2/c) Pd₆B superstructure formed in the cubic (with structure B1) boron solid solution PdB _y in palladium has been performed. The formation of this superstructure occurs as a first-order phase transition via the disorder-order transition channel including nine nonequivalent superstructure vectors of four stars {k ₁₀}, {k ₄}, {k ₃}, and {k ₀}. For the monoclinic (space group C2) Pd₆B superstructure, the distribution function of boron atoms has been calculated and the interval of admissible values of long-range order parameters has been determined. It has been shown that the found transition channel is identical to the channel of the formation of the monoclinic (space group C2/c) M ₆ X superstructure; therefore, the Pd₆B superstructure can be described with the same accuracy in space group C2. The higher symmetry of the monoclinic (space group C2/c) model suggests that it more accurately describes the structure of the phase Pd₆B (Pd₆B□₅) and mutually inverse phases M ₆ X□₅ and M ₆ X ₅□ than the model with space group C2. It has been demonstrated that there are two types of the nearest environment of metal atoms with non-metal sublattice sites arranged in the first and second coordination spheres in M ₆ X□₅-type superstructures (space groups C2/c, C2, C2/m, and P3₁) and in inverse M ₆ X ₅□-type superstructures with the same space groups.