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

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)

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)
1 m m
Al
6
Mn

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

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
• Orderly arrangement
• Rotational Symmetry
• Structure can be reduced to repeating units
D. Levine and P.J. Steinhardt (1984)

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

QUASICRYSTALS
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
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
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

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 !
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 !
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)

Shechtman et al. (1984) evidence for icosahedral symmetry

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 ?
...
L S LL S L S LL S LL S L S LL S L S L ...
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 ?
...
L S LL S L S LL S LL S L S LL S L S L ...
L S L S LL S L S LL S L
?
L
S L
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 ?
...
L S LL S L S LL S LL S L S LL S L S L ...
?
L
L S L S LL S L S LL S L
S L
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 ?
...
L S LL S L S LL S LL S L S LL S L S L ...
L S L S LL S L S LL S L
L
S L
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 ?
...
L S LL S L S LL S LL S L S LL S L S L ...
L
L S L S LL S L S LL S L
S L
L
L S LL S LL S L S LL S LL S L S LL S LL S
L S
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
G. Onoda, P.J. Steinhardt, D. DiVincenzo, J. Socolar (1988)
UNFORCED
FORCED

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

Al
70
Ni
15
Co
15

Al
60
Li
30
Cu
10

Zn
56.8
Mg
34.6
Ho
8.7

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 units with complex rules for how to join:
Too complicated?

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

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

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

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)

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

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)

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
Blue = Al
Red = Ni
Purple = Co

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

New Physical Properties  New Applications

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

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

Discovery of a Natural Quasicrystal
L 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 (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