PHILOSOPHICAL MAGAZINE A, 2002, VOL. 82, NO. 2, 255±268 Stacking sequences and symmetry properties of trigonal vacancy-ordered phases (s phases) Eric A. Lordy , S. Ranganathan and Anandh Subramaniam Department of Metallurgy, Indian Institute of Science, Bangalore 560012, India [Received 9 November 2000 and accepted 11 May 2001] 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 ‰rr qŠ. The structure has alternating ®lled and empty lamellae parallel to planes of type …rr q). 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. } 1. Introduction The 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 alu- minium atoms and those of the other are occupied by transition-metal 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 p ˆ 5; 6; 7; 8; 11; 13; 15 and 17. The X-ray di Œraction 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 well- known quasiperiodic sequence generated by the iteration rule 0 ! 1, 1 ! 10 (Elser Philosophica l Magazine A ISSN 0141±8610 print/ISSN 1460-6992 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080 /0141861011006775 2 y Email: Lord@metalrg.iisc.ernet.in