What are Quasicrystals? Prologue. Crystals can only exhibit certain symmetries In crystals, atoms or atomic clusters repeat periodically, analogous to.

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

YES

YES

YES

YES

YES

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 asquare, rectangle, parallelogram triangle or hexagon,but not others, such as pentagons.

Crystals can only exhibit certain symmetries

Crystals can only exhibit thesesame rotational symmetries*

..and the symmetries determine many of their physical properties and applications

*in 3D, there can be different rotational symmetriesAlong different axes, but they are restricted to the same set (2-, 3, 4-, and 6- fold)

Which leads us to…

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

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

Al6Mn

1 m

Their surprising claim:

Al6Mn

“Diffracts electrons like a crystal . . .But with a symmetry strictly forbidden for crystals”

By rotating the sample, they found the new alloy has icosahedral symmetry

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

five-foldsymmetry

axis

three-foldsymmetry

axis

two-foldsymmetry

axis

Their symmetry axes of an icosahedron

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…

QUASICRYSTALS

D. Levine and P.J. Steinhardt (1984)

• Orderly arrangment . . . But QUASIPERIODIC instead of PERIODIC

• Rotational Symmetry

• Structure can be reduced to repeating units

QUASICRYSTALSSimilar to crystals, BUT…

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

QUASICRYSTALSSimilar 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)

QUASICRYSTALSSimilar to crystals, BUT…

QUASICRYSTALS

Inspired by Penrose TilesInvented by Sir Roger Penrose in 1974

Penrose’s goal:

Can you find a set of shapesthat can only tile the plane non-periodically?

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

But these rulesForce non-periodicity:

Must match edges & lines

And these “Ammann lines” revealthe 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 shortintervals

L

L

L

S

S

LS

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 dubbedQuasicrystals = 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)

D. Levine and P.J. Steinhardt (1984)

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

Shechtman et al. (1984) evidence for icosahedral symmetry

Reasons to be skeptical:

Requires non-local interactions in order to grow?

Two or more repeating unitswith 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 rulesFor example, you learn from this that S is always followed by L

Non-local Growth Rules ?

...LSLLSLSLLSLLSLSLLSLSL ...

LSLSLLSLSLLSL

? L

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 ...

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

Non-local Growth Rules ?

...LSLLSLSLLSLLSLSLLSLSL ...

LSLSLLSLSLLSLL

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 ...

LSLSLLSLSLLSLL

SL

LSLLSLLS LSLLSLLS LSLLSLLSL

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 5Dbecause there are 5 Fibonacci sequences of Ammann lines to be constructed

FORCED

UNFORCED

Question:

Can we find local rulesfor adding tiles thatmake perfect QCs?

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

But not Penrose’s rule;instead

Only add at forced sites

Penrose tiling has 8 typesof vertices

Forced = only one wayto add consistent w/8 types

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 sizeCcn be constructed

with just local (short-range) interactions

Since then, highly perfect quasicrystalswith many different symmetries havebeen discovered in the laboratory …

Al70 Ni15 Co15

Al60Li30Cu10

Zn56.8 Mg34.6 Ho8.7

AlMnPd

Faceting was predicted: Example of prediction of facets

Reasons to be skeptical:

Requires non-local interactions in order to grow?

Two or more repeating unitswith complex rules for how to join:

Too complicated?

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

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!

Gummelt Tile

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

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

Another Surprise:Overlap Rules Maximizing Cluster Density

Clusters energetically favored Quasicrystal has minimum energy

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

AlAl7272NiNi2020CoCo88

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

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

Example of decagon

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

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

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

Blue = AlRed = NiPurple = 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

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

New Physical Properties New Applications

• Diffraction• Faceting

• Elastic Properties

• Electronic Properties

A commercial application: Cookware with Quasicrystal Coating

(nearly as slippery as Teflon)

Epilogue 1:

A new application -- synthetic quasicrystals

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

Weining Man, M. Megans, P. Chaikin, & PJS, Nature (2005)

Photonic Quasicrystal for Microwaves

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

Epilogue 2:

The first “natural quasicrystal”

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

Science 324, 1306 (2009)

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 (CuAl2) and cupalite (CuAl) but also includes granules, like the one in (B), with composition Al63Cu24Fe13. 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