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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
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