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

Abstract

The vacancy-ordered phases known as t 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 qŠ. The structure has alternating ®lled and empty lamellae parallel to planes of type …rrq q). The particular cases in which r and q are consecutive numbers of the Fibonacci sequence can be regarded as rational approximants to a onedimensional quasiperiodic structure. Some mathematical properties of the sequences, and their relationship with the three-dimensional structures, are presented. } 1. IntroductionThe vacancy-ordere d phases known as t-phases are B2 structures in which the vertices of one of the two constituent primitive cubic lattices are occupied by aluminium atoms and those of the other are occupied by transition-meta l atoms or are vacant sites. The (111) planes are either completely ®lled or completely empty, with characteristic periodic stacking sequences along the [111] direction.The stacking sequences for a large number of t phases are now known. A t 5 phase Al 5 Cu 2 Ni was reported by Bingham and Haughton (1923) and the structure of the t 3 phase Al 3 Ni 2 was elucidated by Bradley and Taylor (1937). The most extensive investigation is the work of Lu and Chang (1957) in which the Al±Cu±Ni system was explored and stacking sequences determined, for t p phases with pˆ5; 6; 7; 8; 11; 13; 15 and 17. The X-ray diOE raction analyses of van Sande et al. (1978) con®rmed the evidence for t 2 ; t 3 ; t 5 ; t 8 and t 13 …1 † but did not encounter the other members on list given by Lu and Chang.A very striking feature of the list (1) is as follows: the lengths of the repeat units of the stacking sequences are terms in the Fibonacci sequence. Motivated by this observation, Chattopadhyay et al. (1987) were led to the discovery that the actual stacking sequences of these phases are in fact rational approximants to the wellknown quasiperiodic sequence generated by the iteration rule 0 ! 1, 1 ! 10 (Elser

Instability of Al‐Cu‐Co single quasicrystals at low temperature

“…The [110] direction of the τ-phases is parallel to [00001] direction of decagonal phase of Al-Cu-Co alloys [10,22,23]. It may be easily noticed that in the decagonal phase the two-fold planes of (01100) Let us assume that the decagonal phase domain boundaries are parallel to the plane of the (01100) type.…”

“…In vacancy-ordered τ-phases the planes of type (111) are either completely filled for atoms or completely empty, with characteristic periodic stacking sequences along the [111] -type direction [10,22]. The [110] direction of the τ-phases is parallel to [00001] direction of decagonal phase of Al-Cu-Co alloys [10,22,23].…”

Intergrowth Compounds in the Zn-Rich Zn−Pd System: Toward 1D Quasicrystal Approximants

“…Consequently, there are two different ways to achieve the atomic structure. Perez Mato et al and Lord et al investigated this idea by trying to correlate the ordering of either vacancies or successions of octahedra/trigonal prisms with the Fibonacci sequence in their respective systems. In the Zn−Pd system, which follows Hume-Rothery's principles of electron compounds, the cubic γ-brass phase is preferred for a valence electron concentration (vec = valence s, p electrons / atom ratio) of 1.61 valence s and p electrons/atom but shows a wide range in composition that varies from its published phase diagram…”