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

2. … An example of 1D quasicrystal Fibonacci sequence (1202)
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)

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

8. Only 2 Penrose tiling have global 5-fold symmetry
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 5-fold diffraction pattern from Mg23Zn68Y9 alloy
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)

11. There was some confusion at first due to the ``local 5-fold symmetry'‘ of the Penrose tilings; the Penrose tilings have arbitrarily large regions with centers of 5-fold rotational symmetry. Any such symmetry in the real-space pattern of the scatterers will produce the same rotational symmetry in the diffraction pattern in certain directions. But it was eventually realized that the key feature of the Penrose tilings is their ``statistical 10-fold symmetry'‘ the fact that every finite pattern of tiles in the tiling appears in 10 different rotational orientations with the same frequencies, the equality of the frequencies giving rise to the 10-fold rotational symmetry of the diffraction patterns.
(C. Radin, J. Stat. Phys. 1999)

12. Another hidden order in Penrose tiling
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)

13. Real (artificial) quasicrystals
Quasicrystals are found most often in aluminium alloys
Penrose tiling
蔡安邦
Scanning tunneling microscope image of the 2D quasicrystal Al65Cu15Co20
Natural Quasicrystals?
See Bindi et al, Science 2009

14. Where are the atoms, actually?
decagonal Al70Co12Ni18 reconstructed from 360 image plate scanner frames at each temperature
Steurer, Journal of Non-Crystalline Solids, 2004

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

16. 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.

17. APPLICATIONS OF QUASICRYSTALS
hard and brittle
low surface energy (non-stick)
high electrical resistivity
high thermal resistivity
high thermoelectric power …
fine but strong
Technology Assessment & Transfer, Inc. 2010
old new
Philips and Sandvik Materials Tech
→ Photonic and phononic quasicrystals

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

19. Can not be obtained by the "cut and project“ construction
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

20. Kite-Domino quasicrystal
Tilings Encyclopedia:

21. What governs the formation of QC? (growth rule)
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.
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
yes
yes

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

23. Penrose tiling What else?
local rules (self-similarity)
More circles?
Cut and projection
Diffraction spots
Penrose tiling What else?
some substitution tilings can be obtained by projecting certain points from higher dimensional point lattices.
A model set is a discrete point set arising from a cut and project scheme. A theorem of Hof, generalized by Schlottmann, states that each model set is pure point diffractive..