1. Quasicrystals: What are they, and why do they exist? Outline What is a crystal? –Symmetries, periodicity, (quasiperiodicity) How can you tell for sure? –Diffraction patterns, indexing Higher-dimensional representation –Cut and project –phason fluctuations + diffuse scattering Thermodynamic stability

2. What is a crystal? Hillman Hall of Minerals and Gems Carnegie Museum of Natural History, Pittsburgh SulfurTopaz Quartz

3. Symmetries Rotations: 2-fold: 3-fold: 4-fold: 6-fold:

4. Symmetries Translations:Reflections:

5. Symmetries Combinations of reflections and rotations:

6. Symmetries Combinations of translations, reflections and rotations: The symmetry “space group” 6-fold symmetry Translations Group is closed under combinations

7. Symmetries What happened to 5-fold symmetry? Rotation Translation (shortest) Combination: New translation (shorter) Translationally periodic structure cannot have 5x axis  = (1+√5)/2 = 1.61803… is the Golden Mean, the “most irrational number”.

8. Flux-grown Quasicrystals (Ian Fisher, et al.) i-AlGaPdMn i-ZnMgHo d-AlCoNi

9. Penrose Tiling (1974) I L I S I L I L I S I

10. Quasiperiodicity Two or more incommensurate periods present simultaneously Periodic LS pattern: LSLSLSLSLSLSLSLS Periods: 2 (LS), 4 (LSLS), 6 (LSLSLS), … Quasiperiodic Fibonacci pattern: SF 0 =1 LF 1 =1 LSF 2 =2 LSLF 3 =3 LSLLSF 4 =5 LSLLSLSLF 5 =8 LSLLSLSLLSLLSF n =F n-1 +F n-2 …… Ratios of Fibonacci numbers: Lim F n+1 /F n → 

11. Penrose Matching Rules Shared tile edge types must match to achieve perfect quasiperiodicity Levine and Steinhardt proposed as mechanism of stability

12. Bachelor Hall, Miami University of Ohio, 1979

13. Storey Hall, Royal Melbourne Institute of Technology (1998)

14. Penrose Quilt, Newbold (2005) Penrosette doily, Jason (1999) Penrose arts & crafts

15. (C.S. Kaplan) “The Pentalateral Commission” “Busby Berkeley Chickens” Penrose “Escher” designs

16. Toys: ZomeTool ® and SuperMag ®

17. “True technological advances are welcome in any field. Cybernox stick- resistant cookware is such an advance. The cooking surface of Cybernox pans is Quasi-Crystal, a patented metal alloy that is super hard (10 times harder than stainless steel), extremely durable, distributes heat rapidly and evenly, and has low adhesion properties. The French government owns the patent, and ……” Copyright©1998-2002 A Cook's Wares®

18. Crystal Diffraction Pattern kiki koko 0 r a a

19. Outgoing wave =  r Incoming wave scattered by atom at r Relative phase of incoming wave reaching r ~ exp(+ik i ·r) Relative phase of outgoing wave from r ~ exp(-ik o ·r) Net phase of wave scattered from r ~ exp(-i  k·r),  k=k o -k i Total outgoing wave ~ {  r exp(-i  k·r)} exp[i(k o ·r-  t)] Diffraction pattern is Fourier Transform! Incoming wave ~ exp[i(k i ·r-  t)] Vanishes unless exp((-i  k·r)=1 for all atom positions r. Bragg peaks at  k=G, where G=(2  /a){hx+ky+lz}. (h,k,l) are Miller indices.

20. Crystal Diffraction Patterns Ta 97 Te 60 (tetragonal, 2x and 4x rotations) Diffraction pattern == Reciprocal lattice closed under rotations and translations (600) (060) (660)

21. Quasicrystal Diffraction Patterns Decagonal Al-Co-Ni (10000 0) (01000 0) (00001 0) (01001 0)

22. R || Cut and project method Atomic surfaces RR

23. Fourier Transform

24. Reciprocal Space Q || QQ

25. Fibonacci diffraction grating (Ferralis, Szmodis and Diehl (2004))

26. R || “Phason” degrees of freedom Atomic surfaces RR

27. 4 3 2 1 0 Tiling of plane by 60° rhombi Phason freedom: Add/remove block

28. 4 3 2 1 0 Entropy calculation via quantum mechanical world lines Space Time Lines never start nor stop: particles conserved Lines never cross: particles are “fermions”

29. 4D hypercube (tesseract) Octagonal Tiling Projected from 4D Squares and 45° rhombi

30. Octagonal Tiling Projection from 4D

31. Penrose Tiling Projected from 5D

32. Phason Diffuse Scattering Decagonal AlCoNi (Estermann & Steurer)

33. Simulated Atomic Surfaces 5D body-centered hypercubic lattice Aluminum Cobalt Nickel Combined

34. Tiling model of 10x Al-Co-Ni Aluminum Cobalt Nickel

35. Phason Diffuse Scattering Elastic neutron scattering i-AlMnPd (Schweika) Predicted phason diffuse scattering

36. Phason Diffuse Scattering X-ray diffuse scattering (046046) peak (Colella) Phason prediction

37. Summary and conclusions Quasicrystals are quasiperiodic structures of high rotational symmetry They possess sharp Bragg diffraction peaks with additional diffuse backgrounds Structural models exist, but they do not minimize the total energy Intrinsic “phason” fluctuations contribute entropy that may lend thermodynamic stability at high temperatures

38. Thanks! Marek Mihalkovic (CMU/Slovakia) Siddartha Naidu (CMU → Google Bangalore) Veit Elser (Cornell) Chris Henley (Cornell) John Moriarty (Livermore National Lab) Yang Wang (Pittsburgh Supercomputer Center) Ibrahim Al-Lehyani (CMU → Saudi Arabia) Remy Mosseri (Paris) Nicolas Destainville (Toulouse) and many more.....