Quasiperiodic-periodic structures by projection in two stages: an illustrative example Alexander Quandt * ,I and Shelomo I. Ben-Abraham II I Institut fu ¨r Physik, Ernst-Moritz-Arndt Universita ¨t Greifswald, Domstraße 10a, 17489 Greifswald, Germany II Department of Physics, Ben-Gurion University of the Negev, POB 553, 84105 Beer-Sheba, Israel Received June 20, 2006; accepted July 18, 2006 Quasiperiodic structures / Cut-and-project method / Fibonacci chain Abstract. Recently we have modified a two-stage variant of the cut-and-project method used to generate layered 8- fold (Ben-Abraham, 2004) and 12-fold (Ben-Abraham et al., 2004) quasiperiodic structures in three dimensions, and obtained 8-fold, 10-fold and 12-fold quasiperiodic structures via cut-and-project from various four dimen- sional lattices (Ben-Abraham and Quandt, 2006). However four dimensions are certainly not the most suitable play- ground to illustrate the basics of our general two-stage projection method. To this end we cut and project a peri- odic structure in three dimensions onto two-dimensional physical space, such that a second cut and projection onto a line yields a quasiperiodic structure. This simple exam- ple contains all of the basic elements of our two-stage projection method, but now the essential steps may di- rectly be visualized. Introduction Structure models for octagonal, decagonal (including pen- tagonal) and dodecagonal quasicrystalline alloys are some- what different from structure models for icosahedral quasi- crystalline alloys (Janot, 1994), as the latter are genuinely quasiperiodic in all three dimensions of physical space, whereas for the former, the basic structure models are gen- uinely two-dimensional, and the periodic third direction is treated separately. For practical purposes this does not seem to be a major problem. But from a purely theoretical point of view, the situation is somewhat unsatisfactory, be- cause the two-dimensional layers of octagonal, decagonal and dodecagonal quasicrystals must be filled with atoms, whose chemistry takes place in three physical dimensions, and that will lead to a direct coupling of the periodic third direction to the quasiperiodic layers. Thus in two earlier papers we have developed a “two- stage cut-and-project method” that would generate octago- nal (Ben-Abraham, 2004) and dodecagonal (Ben-Abraham et al., 2004) structures, where the periodic third direction will be directly coupled to the quasiperiodic layers. Our two-stage cut-and-project method is a simple variant of the general cut-and-project procedure, and it mainly consists of a suitable projection of periodic structures in some high-dimensional space onto a three-dimensional phyical space, such that a second projection along the periodic third direction yields a two-dimensional quasiperiodic layer. Apart from the coupling of different directions in phy- sical space, another interesing feature of our method is the fact that four dimensions seem to be enough to generate octagonal, pentagonal and several versions of dodecagonal quasiperiodic structures (Ben-Abraham and Quandt, 2006). This means that the acceptance domain for a projection from a four dimensional lattice onto three-dimensional physical space is just a simple strip. Therefore our method allows for the construction of complex quasiperiodic struc- tures by a cut-and-project procedure that is as simple as the basic cut-and-project method to generate a Fibonacci- chain from a two-dimensional square lattice (Janot, 1994). But four-dimensional lattices are certainly not the most suitable objects to illustrate the basics of our method. Therefore we will now present a three-dimensional version of our method, where the essential technical steps may easily be visualized. The high-dimensional lattice will just be a simple cubic lattice in three dimensions, and the re- sult of a suitable two-stage projection will be a one-dimen- sional quasiperiodic structure consisting of Fibonacci- chains (correspoding to the two-dimensional quasiperiodic layers in 3D), which are stacked in a periodic fashion (corresponding to the periodic third direction in 3D). The rest of this paper is organized as follows: In Sec- tion 1 we briefly describe the canonical, as well as the two-stage cut-and-project scheme. Sections 2 is devoted to the application of our two-stage cut-and-project method, starting from a simple cubic lattice in three dimensions. In Section 3, we will summarize our findings. Two-stage cut and project scheme The general cut-and-project scheme has been described in some detail in (Moody, 1997), but for our purposes it will Z. Kristallogr. 221 (2006) 759–761 / DOI 10.1524/zkri.2006.221.12.759 759 # by Oldenbourg Wissenschaftsverlag, Mu ¨nchen * Correspondence author (e-mail: alexander.quandt@physik.uni-greifswald.de) Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/24/15 2:13 AM