On the six-dimensional structure of vacancy-ordered phases

Abstract

A six-dimensional ordered orthogonal hypercell has been proposed for the first time to describe the structures of all vacancy-ordered τ-phases (which occur in Al-TM (TM = transition metal) systems such as Al-Cu-Ni) including that in the incommensurate limit. The cell is ordered and has lattice parameters a₁ = 0.286 nm along three axes and a₂ = 0124nm along the other three axes. The ordered cell contains eight nearly cubic fundamental cells with parameters a₁/2 and a₂. Line motifs representing aluminium atoms are located at for example [000000], [0½000½, [½0000½] and [½½½0000]. Composite motifs representing transitional-metal atoms and structural vacancies then occur at [½½000½], [½00000], [0½0000] and [00000½]. It is shown that a tilt of the motifs from the threefold symmetry axis in pseudo-space is essential for reproducing the known interatomic distances in the case of τ₃-type phases. All the structural features and the diffraction patterns have been explained on this basis. Extinction rules due to the ordering are formulated.