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Sec 5.10 Quasicrystal “perfectly ordered materials that never repeat themselves”

Dept of Phys. M.C. Chang. Sec 5.10 Quasicrystal “perfectly ordered materials that never repeat themselves”. Electron diffraction pattern of Al-Mn alloy (cooling rate 10 6 k/s). 2011. An example of 1D quasicrystal. substitution rule ( → self-similarity). S. 0+1=1 1+1=2 1+2=3

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Sec 5.10 Quasicrystal “perfectly ordered materials that never repeat themselves”

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  1. Dept of Phys M.C. Chang Sec 5.10 Quasicrystal “perfectly ordered materials that never repeat themselves” Electron diffraction pattern of Al-Mn alloy (cooling rate 106 k/s). 2011

  2. An example of 1D quasicrystal substitution rule (→ self-similarity) S 0+1=1 1+1=2 1+2=3 2+3=5 3+5=8 5+8=13 L S→L L → LS L S L S L L S L L S L S L L S L S L L S L L S L S L L S L L S … Fibonacci sequence (1202) • No periodicity, but with perfect order (i.e. locations of S and L are predictable) http://home.iitk.ac.in/~anandh/E-book/Quasicrystals.ppt

  3. L S L L S L S L L S L L S … 1D quasicrystal as a projection of 2D periodic crystal : The "cut and project" construction Slope=1/τ an irrational slice (a slice that avoids any lattice plane) (L/S=τ) • Within a large segment, the ratio of numbers NL/Ns approaches τ

  4. Diffraction pattern of a Fibonacci quasicrystal • The peaks are countably infinite and dense (in the real numbers) (aka singular continuous) Buczek et al, Acta Physica Polonica B 2005

  5. 1 τ 1/τ 2π/5 2π/10 2D quasicrystal Can one find a set of shapes that can cover the plane non-periodically? → 1st example requires more than 20000 different tiles (R. Berger, 1966) → Penrose tiling(1974) • need only 2 tiles (rhombus type) 菱形 Crucial marks • substitution rule

  6. A shifted copy will never match the original exactly. • Any finite region in a tiling appears infinitely many times. M. Senechal, Quasicrystals and Geometry, p.54

  7. Are they the same? local 5-fold symmetry M. Senechal, Quasicrystals and Geometry, p.200 • They are different. • A finite patch appears infinitely many times in a tiling and, in any other tiling. Therefore, a finite patch cannot differentiate between the uncountably many Penrose tilings. http://www.alienscientist.com/penrose.html

  8. At most one point of global 5-fold symmetry Only 2 Penrose tiling have global 5-fold symmetry From Thomas Fernique’s lectures

  9. → Indication of long-range translation order Levine and Steinhardt, PRL 1985

  10. DIFFRACTION PATTERN Indication of long range rotational order 5-fold diffraction pattern from Mg23Zn68Y9 alloy (icosahedral) Computed diffraction pattern for an ideal icosahedral quasicrystal (in a plane normal to a fivefold axis),displaying only peaks above some given intensity. (Levine and Steinhardt, PRL 1985) http://home.iitk.ac.in/~anandh/E-book/Quasicrystals.ppt

  11. Ammann lines Another hidden order in Penrose tiling Decoration lines • The spacings of lines in any given direction is described by 1-dim Fibonacci sequence! • De Bruijn (1981) showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures. (wiki) (note that there are 5 bundles of parallel lines above)

  12. Real (artificial) quasicrystals • Quasicrystals are found most often in aluminium alloys Penrose tiling 蔡安邦 http://www.tagen.tohoku.ac.jp/labo/tsai/qc.html Scanning tunneling microscope image of the 2D quasicrystal Al65Cu15Co20 Natural Quasicrystals? See Bindi et al, Science 2009

  13. Where are the atoms, actually? decagonal Al70Co12Ni18reconstructed from 360 image plate scanner frames at each temperature Steurer, Journal of Non-Crystalline Solids, 2004

  14. Where are the atoms, actually? Steurer, Philos. Mag. 2007 Section perpendicular to the decagonal axis of Al-Co-Ni36.

  15. Quasi-Crystalline Tilings in Medieval Islamic Architecture Gunbad-i Kabud tomb tower in Maragha, Iran (1197 C.E.) Lu and Steinhardt, Science 2007 and comments/responses followed.

  16. old new Technology Assessment & Transfer, Inc. 2010 http://www.mdatechnology.net/update.aspx?id=a5580 APPLICATIONS OF QUASICRYSTALS • hard and brittle • low surface energy (non-stick) • high electrical resistivity • high thermal resistivity • high thermoelectric power • … fine but strong Philips and Sandvik Materials Tech → Photonic and phononic quasicrystals

  17. Another example of 2D quasicrystal Pinwheel tiling(C. Radin, 1994) substitution rule Wiki: Pinwheel tiling

  18. Pinwheel tiling The Federation Square buildings in Melbourne, Australia • Can not be obtained by the "cut and project“ construction • Diffraction pattern is fully rotation invariant

  19. Kite-Domino quasicrystal Tilings Encyclopedia: http://tilings.math.uni-bielefeld.de/

  20. QC without n-fold rotation symmetry? • QC beyond cut and projection? • QC without substitution rule? • In general, what classes of point set have diffraction spots? mathematics Some basic questions • What governs the formation of QC? (growth rule) • Where are the atoms? (structure determination) • QC without 5-fold symmetry? yes → 5, 8, 10, 12-fold have been observed so far. Why not more in real life? • QC without diffraction spots? No. Commission on Aperiodic Crystals, terms of reference (1992): By crystal we mean any solid having an essentially discrete diffraction diagrams. yes yes

  21. × Absence of diffraction spots→disordered solid ! • Bernoulli sequence: random • Rudin-Shapiro sequence: Aperiodic but deterministic Same (diffuse) diffraction pattern ! Baake and Grimm [math-ph]1105.0095

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