quasi intro
TRANSCRIPT
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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
D. Levine and P.J. Steinhardt (1984)
Orderly arrangment . . .But QUASIPERIODICinstead of PERIODIC
Rotational Symmetry
Structure can be reduced to repeating units
QUASICRYSTALSSimilar to crystals, BUT
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D. Levine and P.J. Steinhardt (1984)
Orderly arrangment . . .But QUASIPERIODICinstead of PERIODIC
Rotational Symmetry . . .But with FORBIDDENsymmetry
Structure can be reduced to repeating units
QUASICRYSTALSSimilar to crystals, BUT
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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
D. Levine and P.J. Steinhardt (1984)
QUASICRYSTALSSimilar to crystals, BUT
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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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D. Levine and P.J. Steinhardt (1984)
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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Reasons to be skeptical:
Requires non-local interactions
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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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
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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
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Non-local Growth Rules ?
...LSLLSLSLLSLLSLSLLSLSL...
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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Non-local Growth Rules ?
...LSLLSLSLLSLLSLSLLSLSL...
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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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!
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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