Quasicrystals have well-defined discrete Fourier transforms due to their quasiperiodic long-range order
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Available scientific literature indicates that quasiperiodic structures produce discrete diffraction patterns and Fourier transforms, but the retrieved evidence only partially covers the specific analytical properties of discrete Fourier transforms for quasicrystals.
To demonstrate that crystallographic methods can be applied to index and interpret diffraction patterns from well-ordered quasicrystals that display non-crystallographic 5-fold symmetry, we have characterized the properties of a series of periodic two-dimensional lattices built from pentagons, called Fibonacci pentilings, which resemble aperiodic Penrose tilings. The computed diffraction patterns from periodic pentilings with moderate size unit cells show decagonal symmetry and are virtually indistinguishable from that of the infinite aperiodic pentiling. We identify the vertices and centers of the pentagons forming the pentiling with the positions of transition metal atoms projected on the plane perpendicular to the decagonal axis of quasicrystals whose structure is related to crystalline eta phase alloys. The characteristic length scale of the pentiling lattices, evident from the Patterson (autocorrelation) function, is approximately tau 2 times the pentagon edge length, where tau is the golden ratio. Within this distance there are a finite number of local atomic motifs whose structure can be crystallographically refined against the experimentally measured diffraction data.
Formation of a quasicrystalline Pb monolayer on the ten-fold surface of the decagonal Al-Ni-Co quasicrystal
Lead has been deposited on the ten-fold surface of decagonal Al72Ni11Co17 to form an epitaxial quasicrystalline single-element monolayer. The overlayer grows through nucleation of nanometer-sized irregular islands and the coverage saturates at 1 ML. The overlayer is well-ordered quasiperiodically as evidenced by LEED and Fourier transforms of STM images. Annealing the film to 600 K improves the structural quality, but causes the evaporation of some material such that the film develops pores. Electronic structure measurements using X-ray photoemission spectroscopy indicate that the chemical interaction of the Pb atoms with the substrate is weak.
DOI: 10.1016/j.susc.2008.05.029
arXiv categories: cond-mat.other
Ledieu Department of Physics and Surface Science Research Centre, The University of Liverpool, Liverpool L69 3BX, UK Ames Laboratory, Iowa State University, Ames, IA 50011, USA LSG2M, CNRS UMR 7584, Ecole des Mines, Parc de Saurupt, 54042 Nancy Cedex, France Abstract Lead has been deposited on the ten-fold surface of decagonal Al 72 Ni 11 Co 17 to form an epitaxial quasicrystalline single-element monolayer. The overlayer grows through nucleation of nanometer-sized irregular islands and the coverage saturates at 1 ML. The overlayer is well-ordered quasiperiodically as evidenced by LEED and Fourier transforms of STM images.
The most common kinds of quasicrystal can exhibit five-fold (icosahedral) or ten-fold (decagonal) rotational symmetries as evidenced using diffraction techniques, indicating that these materials are well-ordered but aperiodic materials. A well-known aperiodic mathematical analogue to a quasicrystal is the Penrose tiling, which has previously been successfully mapped onto scanning tunnelling microscopy images from various quasicrystals [ 1 , 2 ] . These unusual materials have been the subject of intense study for the past two decades in a concerted attempt to unravel the mysteries of their structure and properties.
Such structures have been found for Cu adsorbed on the five-fold surface of icosahedral Al-Pd-Mn ( i 𝑖 i -Al-Pd-Mn) [ 10 ] and for Co adsorbed on both i 𝑖 i -Al-Pd-Mn and the ten-fold surface of decagonal Al-Ni-Co ( d 𝑑 d -Al-Ni-Co) [ 11 ] . They consist of small domains of crystalline material oriented in five directions corresponding to high symmetry orientations on the substrate. The domains themselves have a quasiperiodic structural modulation consistent with terms of the Fibonacci sequence [ 10 , 12 , 13 ] . A third class are systems where the adsorbing species adopts the structure of the quasicrystal substrate.
2 Experimental details The Al 72 Ni 11 Co 17 quasicrystal samples, produced at Ames laboratory using the melt decantation method, were polished successively with 6 μ 𝜇 \mu m, 1 μ 𝜇 \mu m and 1/4 μ 𝜇 \mu m diamond paste before introduction to vacuum and thereafter was prepared in cycles consisting of 45 minutes sputtering with 3 keV Ar + ions followed by 4 hours annealing to 1070 K, using electron-beam heating, up to a total annealing time of 20 hours. Following this preparation, low energy electron diffraction (LEED) patterns had well defined peaks and impurities were undetectable by Auger electron spectroscopy.
The radii of the two most intense rings are related by a factor τ 𝜏 \tau 1 1 1 The number τ = 1 + 5 2 = 1.618 … 𝜏 1 5 2 1.618 … \tau=\frac{1+\sqrt{5}}{2}=1.618... , known as the golden ratio, is intrinsic to the geometry of pentagons, to Penrose tilings and the Fibonacci sequence. ; this is indicative of quasiperiodic ordering. The surface has a step-terrace morphology with terrace widths of order 20 nm as observed using STM. STM images yield a dense ten-fold fast Fourier transform (FFT) (not shown), again indicative of quasiperiodic ordering. Figure 2 shows AES measurements of the film during deposition.
5 (a) shows that the shape of the Al 2 p 2 𝑝 2p core level measured is identical to that of the clean quasicrystal surface for the monolayer as deposited at room temperature or annealed to 653 K. The Pb 4 f 4 𝑓 4f core level recorded from the quasiperiodic Pb monolayer has an identical shape to that measured from one 1ML of Pb grown on Al(111) (Fig. 5 (b)). The lack of new components and chemical shifts within the core level peaks is consistent with the immiscibility of Pb and Al [ 29 , 30 ] . No changes were detected in the Ni 2 p 𝑝 p core level either.
However the observation of 4.9 Å pentagonal clusters in the 1 ML film suggests that these features are a key building block of a quasicrystalline monolayer, whether on an icosahedral or a decagonal quasicrystalline substrate. The saturation in coverage at 1 ML is similar to that found for Pb adsorption on i 𝑖 i -Al-Pd-Mn [ 23 ] . There are a number of possible explanations for this unusual behaviour. There could be a vanishingly low binding energy for Pb on top of the quasiperiodic Pb monolayer, perhaps due to a one layer quantum size effect. Quantum size effects in growth have been previously observed for adsorption on quasicrystal surfaces [ 31 ] .
In a study of Ag growth on Fe(100) for films up to N = 15 𝑁 15 N=15 monolayers, films of N = 1 , 2 𝑁 1 2 N=1,2 and 5 monolayer thicknesses were found to have exceptional structural stability [ 32 ] . This
Although annealing to 673 K improves the order of a i 𝑖 i -Al-Pd-Mn/Pb monolayer, annealing to 600 K of a d 𝑑 d -Al-Ni-Co/Pb monolayer results in some desorption of Pb, leaving a porous, though well-ordered monolayer. Both of these observations suggest that the substrate/adsorbate interaction is reduced from that for i 𝑖 i -Al-Pd-Mn/Pb [ 23 ] . 5 Conclusions Lead has been found to form a well-ordered quasiperiodic overlayer on the ten-fold surface of the decagonal Al-Ni-Co quasicrystal. This work extends the number of known systems where an adsorbate adsorbs pseudomorphically on a quasicrystal substrate.
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