1. What are Quasicrystals? Prologue

2. Crystals can only exhibit certain symmetries In crystals, atoms or atomic clusters repeat periodically, analogous to a tesselation in 2D constructed from a single type of tile. Try tiling the plane with identical units … only certain symmetries are possible

3. YES

8. So far so good … but what about five-fold, seven-fold or other symmetries??

9. ? No!

10. ?

11. According to the well-known theorems of crystallography, only certain symmetries are allowed: the symmetry of a square, rectangle, parallelogram triangle or hexagon, but not others, such as pentagons.

12. Crystals can only exhibit certain symmetries Crystals can only exhibit these same rotational symmetries*..and the symmetries determine many of their physical properties and applications *in 3D, there can be different rotational symmetries Along different axes, but they are restricted to the same set (2-, 3, 4-, and 6- fold)

15. Which leads us to…

17. Quasicrystals (Impossible Crystals) were first discoveredin the laboratory by Daniel Shechtman, Ilan Blech, Denis Gratias and John Cahn in a beautiful study of an alloy of Al and Mn

18. D. Shechtman, I. Blech, D. Gratias, J.W. Cahn (1984) Al 6 Mn 1  m

19. Their surprising claim: Al 6 Mn “Diffracts electrons like a crystal... But with a symmetry strictly forbidden for crystals”

20. By rotating the sample, they found the new alloy has icosahedral symmetry the symmetry of a soccer ball – the most forbidden symmetry for crystals!

21. five-fold symmetry axis three-fold symmetry axis two-fold symmetry axis Their symmetry axes of an icosahedron

22. QUASICRYSTALS Similar to crystals D. Levine and P.J. Steinhardt (1984) Orderly arrangement Rotational Symmetry Structure can be reduced to repeating units As it turned out, a theoretical explanation was waiting in the wings…

23. QUASICRYSTALS D. Levine and P.J. Steinhardt (1984) Orderly arrangment... But QUASIPERIODIC instead of PERIODIC Rotational Symmetry Structure can be reduced to repeating units QUASICRYSTALS Similar to crystals, BUT…

24. D. Levine and P.J. Steinhardt (1984) Orderly arrangment... But QUASIPERIODIC instead of PERIODIC Rotational Symmetry... But with FORBIDDEN symmetry Structure can be reduced to repeating units QUASICRYSTALS Similar to crystals, BUT…

25. Orderly arrangmenet... But QUASIPERIODIC instead of PERIODIC Rotational Symmetry... But with FORBIDDEN symmetry Structure can be reduced to a finite number of repeating units D. Levine and P.J. Steinhardt (1984) QUASICRYSTALS Similar to crystals, BUT…

26. QUASICRYSTALS Inspired by Penrose Tiles Invented by Sir Roger Penrose in 1974 Penrose’s goal: Can you find a set of shapes that can only tile the plane non-periodically?

27. With these two shapes, Peirod or non-periodic is possible

28. But these rules Force non-periodicity: Must match edges & lines

29. And these “Ammann lines” reveal the hidden symmetry of the “non-periodic” pattern

30. They are not simply “non-periodic”: They are quasiperiodic! (in this case, the lines form a Fibonacci lattice of long and short intervals

31. L L L S S L S L

32. Fibonacci = example of quasiperiodic pattern Surprise: with quasiperiodicity, a whole new class of solids is possible! Not just 5-fold symmetry – any symmetry in any # of dimensions ! New family of solids dubbed Quasicrystals = Quasiperiodic Crystals D. Levine and PJS (1984) J. Socolar, D. Levine, and PJS (1985)

37. Surprise: with quasiperiodicity, a whole new class of solids is possible! Not just 5-fold symmetry – any symmetry in any # of dimensions ! Including Quasicrystals With Icosahedral Symmetry in 3D: D. Levine and PJS (1984) J. Socolar, D. Levine, and PJS (1985)

39. D. Levine and P.J. Steinhardt (1984) First comparison of diffraction patterns (1984) between experiment (right) and theoretical prediction (left)

40. Shechtman et al. (1984) evidence for icosahedral symmetry

41. Reasons to be skeptical: Requires non-local interactions in order to grow? Two or more repeating units with complex rules for how to join: Too complicated?

42. Reasons to be skeptical: Requires non-local interactions in order to grow?

43. Non-local Growth Rules ?...LSLLSLSLLSLLSLSLLSLSL... ? Suppose you are given a bunch of L and S links (top). YOUR ASSIGNMENT: make a Fibonacci chain of L and S links (bottom) using a set of LOCAL rules (only allowed to check the chain a finite way back from the end to decide what to add next) N.B. You can consult a perfect pattern (middle) to develop your rules For example, you learn from this that S is always followed by L

44. Non-local Growth Rules ?...LSLLSLSLLSLLSLSLLSLSL... LSLSLLSLSLLSL ? L SLSL So, what should be added next, L or SL? Comparing to an ideal pattern. it seems like you can choose either…

45. Non-local Growth Rules ?...LSLLSLSLLSLLSLSLLSLSL... LSLSLLSLSLLSL ? L SLSL Unless you go all the way back to the front of the chain – Then you notice that choosing S+L produces LSLSL repeating 3 times in a row

46. Non-local Growth Rules ?...LSLLSLSLLSLLSLSLLSLSL... LSLSLLSLSLLSL L SLSL That never occurs in a real Fibonacci pattern, so it is ruled out… But you could only discover the problem by studying the ENTIRE chain (not LOCAL) !

47. Non-local Growth Rules ?...LSLLSLSLLSLLSLSLLSLSL... LSLSLLSLSLLSL L SLSL LSLLSLLS LSLLSLLS LSLLSLLS L LSLS The same occurs for ever-longer chains – LOCAL rules are impossible in 1D

48. Penrose Rules Don’t Guarantee a Perfect Tiling In fact, it appears at first that the problem is 5x worse in 5D because there are 5 Fibonacci sequences of Ammann lines to be constructed

49. FORCED UNFORCED Question: Can we find local rules for adding tiles that make perfect QCs? Onoda et al (1988): Surprising answer: Yes! But not Penrose’s rule; instead Only add at forced sites Penrose tiling has 8 types of vertices Forced = only one way to add consistent w/8 types G. Onoda, P.J. Steinhardt, D. DiVincenzo, J. Socolar (1988)

50. In 1988, Onoda et al. provided the first mathematical proof that a perfect quasicrystal of arbitrarily large size Ccn be constructed with just local (short-range) interactions Since then, highly perfect quasicrystals with many different symmetries have been discovered in the laboratory …

51. Al 70 Ni 15 Co 15

52. Al 60 Li 30 Cu 10

53. Zn 56.8 Mg 34.6 Ho 8.7

54. AlMnPd

55. Faceting was predicted: Example of prediction of facets

56. Reasons to be skeptical: Requires non-local interactions in order to grow? Two or more repeating units with complex rules for how to join: Too complicated?

57. Gummelt Tile (discovered by Petra Gummelt) P.J. Steinhardt, H.-C. Jeong (1996) Not so! A single repeating unit suffices! The Quasi-unit Cell Picture

58. For simple proof, see P.J. Steinhardt, H.-C. Jeong (1996) Gummelt Tile Quasi-unit Cell Picture: A single repeating unit with overlap rules (A and B) produces a structure isomorphic to a Penrose tiling!

59. Gummelt Tile Quasi-unit Cell Picture Can interpret overlap rules as atomic clusters sharing atoms

60. The Tiling (or Covering) obtained using a single Quasi-unit Cell + overlap rules

61. Another Surprise: Overlap Rules  Maximizing Cluster Density Clusters energetically favored  Quasicrystal has minimum energy P.J. Steinhardt, H.-C. Jeong (1998)

62. Al 72 Ni 20 Co 8 P.J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A.P. Tsai Nature 396, 55-57 (1998) High Angle Annular Dark Field Imaging shows a real decagonal quasicrystal = overlapping decagons

63. Example of decagon

64. Fully overlapping decagons (try toggling back and forth with previous image)

65. Focus on single decagonal cluster – note that center is not 10-fold symmetric (similar to Quasi-unit Cell)

67. Blue = Al Red = Ni Purple = Co Quasi-unit cell picture constrains possible atomic decorations – leads to simpler solution of atomic structure (below) that matches well with all measurements (next slide) and total energy calculations

68. Prediction agrees with Later Higher Resolution Imaging Yan & Pennycook (2001) Mihalkovic et al (2002)

69. New Physical Properties  New Applications Diffraction Faceting Elastic Properties Electronic Properties

70. A commercial application: Cookware with Quasicrystal Coating (nearly as slippery as Teflon)

71. Epilogue 1: A new application -- synthetic quasicrystals Experimental measurement of the photonic properties of icosahedral quasicrystals Experimental measurement of the photonic properties of icosahedral quasicrystals W. Man, M. Megans, P.M. Chaikin, and P. Steinhardt, Nature (2003)

72. Weining Man, M. Megans, P. Chaikin, & PJS, Nature (2005) Photonic Quasicrystal for Microwaves

73. Y. Roichman, et al. (2005): photonic quasicrystal synthesized from colloids

74. Epilogue 2: The first “natural quasicrystal” Discovery of a Natural Quasicrystal Discovery of a Natural Quasicrystal L Bindi, P. Steinhardt, N. Yao and P. Lu Science 324, 1306 (2009)

75. LEFT: Fig. 1 (A) The original khatyrkite-bearing sample used in the study. The lighter-colored material on the exterior contains a mixture of spinel, augite, and olivine. The dark material consists predominantly of khatyrkite (CuAl 2 ) and cupalite (CuAl) but also includes granules, like the one in (B), with composition Al 63 Cu 24 Fe 13. The diffraction patterns in Fig. 4 were obtained from the thin region of this granule indicated by the red dashed circle, an area 0.1 µm across. (C) The inverted Fourier transform of the HRTEM image taken from a subregion about 15 nm across displays a homogeneous, quasiperiodically ordered, fivefold symmetric, real space pattern characteristic of quasicrystals. RIGHT: Diffraction patterns obtained from natural quasicrystal grain