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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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**What are Quasicrystals?**Prologue**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**So far so good …**but what about five-fold, seven-fold or other symmetries??**?**No!**?**No!**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.**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)**Impossible**Crystals**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**1 mm**D. Shechtman, I. Blech, D. Gratias, J.W. Cahn (1984) Al6Mn**Their surprising claim:**“Diffracts electrons like a crystal . . . But with a symmetry strictly forbidden for crystals” Al6Mn**By rotating the sample, they found the new alloy has**icosahedral symmetry the symmetry of a soccer ball – the most forbidden symmetry for crystals!**Their symmetry axes of an icosahedron**three-fold symmetry axis five-fold symmetry axis two-fold symmetry axis**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)**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)**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)**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)**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?**With these two shapes,**Peirod or non-periodic is possible**But these rules**Force non-periodicity: Must match edges & lines**And these “Ammann lines” reveal**the hidden symmetry of the “non-periodic” pattern**They are not simply**“non-periodic”: They are quasiperiodic! (in this case, the lines form a Fibonacci lattice of long and short intervals**L**S L S L L S L**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)**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)**First comparison of diffraction patterns (1984)**between experiment (right) and theoretical prediction (left) D. Levine and P.J. Steinhardt (1984)**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?**Reasons to be skeptical:**Requires non-local interactions in order to grow?**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**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…**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**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) !**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**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**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)**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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