Stacking sequences and symmetry properties of trigonal vacancy-ordered phases (τ phases)

Abstract

The vacancy-ordered phases known as τ phases are described and the literature dealing with the observed stacking sequences is reviewed. It is shown that the stacking sequences along the threefold axis can be derived from a projection method involving projection on to an axis of type [rrq]. The structure has alternating filled and empty lamellae parallel to planes of type (rrq). The particular cases in which r and q are consecutive numbers of the Fibonacci sequence can be regarded as rational approximants to a one-dimensional quasiperiodic structure. Some mathematical properties of the sequences, and their relationship with the three-dimensional structures, are presented.