The Kepler tiling as the oldest complex surface structure in history: X-ray structure analysis of a two-dimensional oxide quasicrystal approximant
-
Sumalay Roy
and
Wolf Widdra
Abstract:
We have carried out a surface X-ray diffraction (SXRD) analysis of the approximant structure related to the recently discovered two-dimensional dodecagonal oxidic quasicrystal. The structure is characterized by the 32.4.3.4 Archimedean tiling, first described by Kepler in 1619. The tiling network is related to titanium atoms observed as protrusions in scanning tunneling microscopy images. All four titanium atoms within one two-dimensional unit cell (a0=13.1 Å, b0=12.9 Å, γ=90.5°) are surrounded by three oxygen atoms. The TiO3 units are separated by barium atoms. The total stoichiometry is given by Ba4 Ti4 O10.
Acknowledgment
This work is supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich SFB 762 (Funktionalität oxidischer Grenzflächen). We thank Frank Weiss and Ralf Kulla for technical support. We also thank the staff of the ESRF for their hospitality and help during the experiments.
References
[1] D. Shechtman, I. Blech, D. Gratias, J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.1984, 53, 1951.10.1103/PhysRevLett.53.1951Search in Google Scholar
[2] W. Steurer, S. Deloudi, Crystallography of Quasicrystals, Springer Verlag, Heidelberg, Dordrecht, London and New York 2009.Search in Google Scholar
[3] N. Wang, H. Chen, K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry. Phys. Rev. Lett.1987, 59, 1010.10.1103/PhysRevLett.59.1010Search in Google Scholar PubMed
[4] X. Zeng, G. Ungar, Y. Liu, V. Percec, A. E. Dulcey, J. K. Hobbs, Supramolecular dendritic liquid quasicrystals. Nature2004, 428, 157.10.1038/nature02368Search in Google Scholar PubMed
[5] J. Mikhael, J. Roth, L. Helden, C. Bechinger, Archimedean-like tiling on decagonal quasicrystalline surfaces. Nature2008, 454, 501.10.1038/nature07074Search in Google Scholar PubMed
[6] S. Fischer, A. Exner, K. Zielske, J. Perlich, S. Deloudi, W. Steurer, P. Lindner, S. Förster, Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry. Proc. Natl. Acad. Sci.2011, 108, 1810.10.1073/pnas.1008695108Search in Google Scholar PubMed PubMed Central
[7] S. Förster, K. Meinel, R. Hammer, M. Trautmann, W. Widdra, Quasicrystalline structure formation in a classical crystalline thin-film system. Nature2013, 502, 215.10.1038/nature12514Search in Google Scholar PubMed
[8] A. I. Goldman, R. F. Kelton, Quasicrystals and crystalline approximants. Rev. Mod. Phys.1993, 65, 213.10.1103/RevModPhys.65.213Search in Google Scholar
[9] J. Kepler, Harmonices Mundi, Linz (1619).10.5479/sil.135810.39088002800316Search in Google Scholar
[10] B. Grünbaum, G. C. Shepard, Tilings by regular polygons. Math. Magazine1977, 50, 227.10.1080/0025570X.1977.11976655Search in Google Scholar
[11] S. Förster, M. Trautmann, S. Roy, W. A. Adeagbo, E. M. Zollner, R. Hammer, F. O. Schumann, S. Meinel, S. K. Nayak, K. Mohseni, W. Hergert, H. L. Meyerheim, W. Widdra, Observation and strcuture determination of an oxide quasicrystal approximant. Phys. Rev. Lett.2016, 117, 095501.10.1103/PhysRevLett.117.095501Search in Google Scholar PubMed
[12] P. A. Stampfli, Dodecagonal quasi-periodic lattice in two dimensions. Helv. Phys. Acta1986, 59, 1260.Search in Google Scholar
[13] F. Gähler, Quasicrystalline Materials (Eds. C. Janot, J. M. Dubois) World Scientific, Singapore, p. 272, 1988.Search in Google Scholar
[14] E. Cockayne, M. Mihalkovic, C. L. Henley, Structure of periodic crystals and quasicrystals in ultrathin films of Ba-Ti-O. Phys. Rev. B2016, 93, 020101(R).10.1103/PhysRevB.93.020101Search in Google Scholar PubMed PubMed Central
[15] I. K. Robinson, Crystal truncation rods and surface roughness. Phys. Rev. B1986, 33, 3830.10.1103/PhysRevB.33.3830Search in Google Scholar
[16] The unweighted residual (Ru) is defined as:
[17] M. J. Buerger, Kristallographie, Walter de Gryuter Verlag, Berlin and New York 1977.10.1515/9783110842425Search in Google Scholar
[18] U. H. Zucker, E. Perenthaler, W. F. Kuhs, R. Bachmann, H. Schulz, PROMETHEUS. A program system for investigation of anharmonic thermal vibrations in crystals. J. Appl. Crystallogr.1983, 16, 358.10.1107/S0021889883010560Search in Google Scholar
[19] F. W. Kuhs, Generalized atomic displacements in crystallographic structure analysis. Acta Cryst.1992, A48, 80.10.1107/S0108767391009510Search in Google Scholar
©2016 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Graphical Synopsis
- Inorganic Crystal Structures
- On the unit cell volume expansion in homogeneous series of isotypic structures of “ionic” inorganic compounds
- Organic and Metalorganic Crystal Structures
- Syntheses and structures of two [Au(I)PPh3]-complexes with 2-aceneoxazolethiolates as ligands
- Three variants of novel Co(II) complex with NSAID mefenamic acid and N,N′-donor ligand neocuproine
- Invariom modeling of disordered structures: case studies on a dipeptide, an amino acid, and cefaclor, a cephalosporin antibiotic
- Mono urotropine adducts of some binary zinc xanthates and dithiocarbamates: solid-state molecular structures and supramolecular self-assembly
- Letter
- The Kepler tiling as the oldest complex surface structure in history: X-ray structure analysis of a two-dimensional oxide quasicrystal approximant
Articles in the same Issue
- Frontmatter
- Graphical Synopsis
- Inorganic Crystal Structures
- On the unit cell volume expansion in homogeneous series of isotypic structures of “ionic” inorganic compounds
- Organic and Metalorganic Crystal Structures
- Syntheses and structures of two [Au(I)PPh3]-complexes with 2-aceneoxazolethiolates as ligands
- Three variants of novel Co(II) complex with NSAID mefenamic acid and N,N′-donor ligand neocuproine
- Invariom modeling of disordered structures: case studies on a dipeptide, an amino acid, and cefaclor, a cephalosporin antibiotic
- Mono urotropine adducts of some binary zinc xanthates and dithiocarbamates: solid-state molecular structures and supramolecular self-assembly
- Letter
- The Kepler tiling as the oldest complex surface structure in history: X-ray structure analysis of a two-dimensional oxide quasicrystal approximant
