Quasi Intro

8/2/2019 Quasi Intro

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What are Quasicrystals?Prologue


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rystals can only exhibit certain symmetrie

crystals, atoms or atomic clusters repeat periodicallanalogous to a tesselation in 2Dconstructed from a single typeof tile.

Try tiling the plane with identical units only certain symmetries are possible


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YES


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YES


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YES


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YES


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YES


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So far so good

t what about five-fold, seven-fold or other symmetries


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?

No!


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?

No!


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


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rystals can only exhibit certain symmetrie

Crystals can only exhibit these

same rotational symmetries*

..and the symmetries determine manyof 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)


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Which leads us to


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Quasicrystals (Impossible Crystals)

were first discoveredin the laboratory by

iel Shechtman, Ilan Blech, Denis Gratias and John C

in a beautiful study of an alloy of Al and Mn


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D. Shechtman, I. Blech, D. Gratias, J.W. Cahn (1984)

Al6

Mn

1 mm


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Their surprising claim:

Al6Mn

Diffracts electrons like a crystal . . .But with a symmetry strictly forbiddenfor crystals


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By rotating the sample, they found the new alloy hasicosahedral symmetry

the symmetry of a soccer ball the most forbidden symmetry for crystals!


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five-foldsymmetry

axis

three-foldsymmetry

axis

two-foldsymmetry

axis

Their symmetry axes of an icosahedron


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QUASICRYSTALSSimilar 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


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QUASICRYSTALS


Orderly arrangment . . .But QUASIPERIODICinstead of PERIODIC

Rotational Symmetry


QUASICRYSTALSSimilar to crystals, BUT


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Orderly arrangment . . .But QUASIPERIODICinstead of PERIODIC

Rotational Symmetry . . .But with FORBIDDENsymmetry




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Orderly arrangmenet . . .But QUASIPERIODICinstead of PERIODIC

Rotational Symmetry . . .But with FORBIDDENsymmetry

Structure can be reduced to a finite numberof repeating units




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QUASICRYSTALS

Inspired by Penrose Tiles

Invented by Sir Roger Penrose in 1974

Penroses goal:

Can you find a set of shapes

that can only tile the plane non-periodically?


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With these two shapes,Peirod or non-periodic is possi


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But these rulesForce non-periodicity:

Must match edges & lines


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And these Ammann lines revethe hidden symmetry

of thenon-periodic pattern


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They are not simplynon-periodic:

They are quasiperiodic!(in this case, the lines form a

Fibonacci lattice of long and shointervals


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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 dubbedQuasicrystals = Quasiperiodic Crystals

D. Levine and PJS (1984)J. Socolar, D. Levine, and PJS (1985)


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Surprise: with quasiperiodicity,a whole new class of solids is possible!

Not just 5-fold symmetry any symmetry in any # of dimensions !

Including QuasicrystalsWith Icosahedral Symmetry in 3D:

D. Levine and PJS (1984)J. Socolar, D. Levine, and PJS (1985)


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First comparison of diffraction patterns (1984)between experiment (right) and theoretical prediction (left)


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Shechtman et al. (1984) evidence for icosahedral symmetry


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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?


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in order to grow?


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Non-local Growth Rules ?

...LSLLSLSLLSLLSLSLLSLSL...

?Suppose you are given a bunch of L and S links (top).

ASSIGNMENT: make a Fibonacci chain of L and S links (bottom) using a set of LOCAL(only allowed to check the chain a finite way back from the end to decide what to add n

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


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LSLSLLSLSLLSL

? L

SL

So, what should be added next, L or SL?

Comparing to an ideal pattern. it seems like you can choose either


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LSLSLLSLSLLSL

? L

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


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LSLSLLSLSLLSLL

SL

That never occurs in a real Fibonacci pattern, so it is ruled out

you could only discover the problem by studying the ENTIRE chain (not LOCA


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LSLSLLSLSLLSLL

SL

LSLLSLLS LSLLSLLS LSLLSLLSL

LS

The same occurs for ever-longer chains LOCAL rules are impossible in 1D


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Penrose Rules Dont Guaranteea 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


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FORCED

UNFORCED

Question:

Can we find local rulesfor adding tiles thatmake perfect QCs?

Onoda et al (1988):Surprising answer: Yes!

But not Penroses rule;instead

Only add at forced sites

Penrose tiling has 8 types

of verticesForced = only one way

to add consistent w/8 types

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


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In 1988, Onoda et al. provided

the first mathematical proofthat a perfect quasicrystal of arbitrarily large sizeCcn be constructed

with just local (short-range) interactions

Since then, highly perfect quasicrystals

with many different symmetries havebeen discovered in the laboratory


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Al70 Ni15 Co15


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Al60Li30Cu10


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Zn56.8 Mg34.6 Ho8.7


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AlMnPd


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Faceting was predicted: Example of prediction of facets


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in order to grow?

Two or more repeating units

with complex rules for how to join:Too complicated?

Not so! A single repeating unit suffices!


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Gummelt Tile(discovered by Petra Gummelt)

P.J. Steinhardt, H.-C. Jeong (1996)

g p gThe Quasi-unit Cell Picture

Quasi-unit Cell Picture:


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For simple proof, see P.J. Steinhardt, H.-C. Jeong (1996)

Gummelt Tile

A single repeating unit with overlap rules (A and B) producesa structure isomorphic to a Penrose tiling!

Quasi-unit Cell Picture


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Gummelt Tile

Quasi-unit Cell PictureCan interpret overlap rules asatomic clusters sharing atoms

The Tiling (or Covering) obtained


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The Tiling (or Covering) obtainedusing a single Quasi-unit Cell + overlap rules


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Another Surprise:Overlap Rules Maximizing Cluster Density

Clusters energetically favoredQuasicrystal has minimum energy

P.J. Steinhardt, H.-C. Jeong (1998)

High Angle Annular Dark Field Imaging shows a real decagonal quasicrystal = overlapping decagons


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Al72Ni20Co8

P.J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A.P. TsaiNature 396 55-57 1998

Example of decagon


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


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


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


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Quasi-unit cell picture constrains possible atomic decorations


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Blue = AlRed = NiPurple = Co

p p leads to simpler solution of atomic structure (below) that matches well w

all measurements (next slide) and total energy calculations


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Prediction agrees with Later Higher Resolution ImagingYan & Pennycook (2001)Mihalkovic et al 2002

New Physical Properties New Applications


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New Physical Properties New Applications Diffraction

Faceting

Elastic Properties

Electronic Properties


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A commercial application:Cookware with Quasicrystal Coating

(nearly as slippery as Teflon)


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Epilogue 1:

A new application -- synthetic quasicrystals

Experimental measurement of the photonic properties of icosahedral quasicrystalW. Man, M. Megans, P.M. Chaikin, and P. Steinhardt, Nature (2003)

Weining Man, M. Megans, P. Chaikin, & PJS, Nature (2005)
http://www.physics.princeton.edu/~steinh/quasiphoton/NATURE.pdfhttp://www.physics.princeton.edu/~steinh/quasiphoton/NATURE.pdfhttp://www.physics.princeton.edu/~steinh/quasiphoton/NATURE.pdfhttp://www.physics.princeton.edu/~steinh/quasiphoton/NATURE.pdfhttp://www.physics.princeton.edu/~steinh/quasiphoton/NATURE.pdf


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Photonic Quasicrystal for Microwaves


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


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Epilogue 2:

The first natural quasicrystal

Discovery of a Natural QuasicrystalL Bindi, P. Steinhardt, N. Yao and P. Lu

Science 324, 1306 (2009)
http://www.sciencemag.org/cgi/content/full/324/5932/1306?ijkey=M.9o1c.ng5mcg&keytype=ref&siteid=scihttp://www.sciencemag.org/cgi/content/full/324/5932/1306?ijkey=M.9o1c.ng5mcg&keytype=ref&siteid=sci


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LEFT: Fig. 1 (A) The original khatyrkite-bearing sample used in the study. The lighter-colored material on the exteriorcontains 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 Al63Cu24Fe13. The diffraction patterns in Fig. 4 wereobtained from the thin region of this granule indicated by the red dashed circle an area 0 1 m across (C) The inverted

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