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What are Quasicrystals? Prologue

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

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What are Quasicrystals? Prologue

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  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

  4. YES

  5. YES

  6. YES

  7. YES

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

  9. ? No!

  10. ? No!

  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)

  13. Which leads us to…

  14. Impossible Crystals

  15. Quasicrystals (Impossible Crystals) were first discoveredinthe laboratory by Daniel Shechtman, IlanBlech, Denis Gratias and John Cahn in a beautiful study of an alloy of Al and Mn

  16. 1 mm D. Shechtman, I. Blech, D. Gratias, J.W. Cahn (1984) Al6Mn

  17. Their surprising claim: “Diffracts electrons like a crystal . . . But with a symmetry strictly forbidden for crystals” Al6Mn

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

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

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

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

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

  23. QUASICRYSTALS Similar to crystals, BUT… • 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)

  24. 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?

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

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

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

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

  29. L S L S L L S L

  30. 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)

  31. 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)

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

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

  34. 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?

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

  36. 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

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

  38. Non-local Growth Rules ? ...LSLLSLSLLSLLSLSLLSLSL... ? L LSLSLLSLSLLSL SL 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

  39. Non-local Growth Rules ? ...LSLLSLSLLSLLSLSLLSLSL... L LSLSLLSLSLLSL SL 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) !

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

  41. 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

  42. 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 UNFORCED FORCED G. Onoda, P.J. Steinhardt, D. DiVincenzo, J. Socolar (1988)

  43. 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 …

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