quasicrystals: what are they, and why do they exist? outline what is a crystal? –symmetries,...
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Quasicrystals: What are they, and why do they exist?
Outline• What is a crystal?
– Symmetries, periodicity, (quasiperiodicity)
• How can you tell for sure?– Diffraction patterns, indexing
• Higher-dimensional representation– Cut and project– phason fluctuations + diffuse scattering
• Thermodynamic stability
What is a crystal?
Hillman Hall of Minerals and GemsCarnegie Museum of Natural History, Pittsburgh
Sulfur Topaz
Quartz
Symmetries
Rotations:
2-fold:
3-fold:
4-fold:
6-fold:
SymmetriesTranslations: Reflections:
SymmetriesCombinations of reflections and rotations:
SymmetriesCombinations of translations, reflections and rotations:The symmetry “space group”
6-fold symmetryTranslationsGroup is closed under combinations
Symmetries
What happened to 5-fold symmetry?
Rotation Translation (shortest)Combination: New translation (shorter)
Translationally periodic structure cannot have 5x axis
= (1+√5)/2 = 1.61803…is the Golden Mean, the “most irrational number”.
Flux-grown Quasicrystals (Ian Fisher, et al.)
i-AlGaPdMni-ZnMgHo
d-AlCoNi
Penrose Tiling (1974)
I L I S I L I L I S I
QuasiperiodicityTwo or more incommensurate periods present simultaneously
Periodic LS pattern: LSLSLSLSLSLSLSLSPeriods: 2 (LS), 4 (LSLS), 6 (LSLSLS), …
Quasiperiodic Fibonacci pattern: S F0=1L F1=1LS F2=2LSL F3=3LSLLS F4=5LSLLSLSL F5=8LSLLSLSLLSLLS Fn=Fn-1+Fn-2
……
Ratios of Fibonacci numbers:Lim Fn+1/Fn →
Penrose Matching Rules
Shared tile edge types must match to achieve perfect quasiperiodicity
Levine and Steinhardt proposed as mechanism of stability
Bachelor Hall, Miami University of Ohio, 1979
Storey Hall, Royal Melbourne Institute of Technology (1998)
Penrose Quilt, Newbold (2005) Penrosette doily, Jason (1999)
Penrose arts & crafts
(C.S. Kaplan)
“The Pentalateral Commission”“Busby Berkeley Chickens”
Penrose “Escher” designs
Toys: ZomeTool® and SuperMag®
“True technological advances are welcome in any field. Cybernox stick-resistant cookware is such an advance. The cooking surface of Cybernox pans is Quasi-Crystal, a patented metal alloy that is super hard (10 times harder than stainless steel), extremely durable, distributes heat rapidly and evenly, and has low adhesion properties. The French government owns the patent, and ……”
Copyright©1998-2002 A Cook's Wares®
Crystal Diffraction Pattern
kiko
0
r
a
a
Crystal Diffraction Pattern
Outgoing wave = r Incoming wave scattered by atom at r
Relative phase of incoming wave reaching r ~ exp(+iki·r)
Relative phase of outgoing wave from r ~ exp(-iko·r)
Net phase of wave scattered from r ~ exp(-ik·r), k=ko-ki
Total outgoing wave ~ {r exp(-ik·r)} exp[i(ko·r-t)]
Diffraction pattern is Fourier Transform!
Incoming wave ~ exp[i(ki·r-t)]
Vanishes unless exp((-ik·r)=1 for all atom positions r.Bragg peaks at k=G, where G=(2/a){hx+ky+lz}.(h,k,l) are Miller indices.
Crystal Diffraction Patterns
Ta97Te60 (tetragonal, 2x and 4x rotations)
Diffraction pattern == Reciprocal latticeclosed under rotations and translations
(600)
(060) (660)
Quasicrystal Diffraction Patterns
Decagonal Al-Co-Ni
(10000 0)
(01000 0)
(00001 0)
(01001 0)
R||
Cut and project method
Atomic surfaces
R
2/
2/sin)(
QL
QLQS
Fourier Transform
Reciprocal Space
Q||
Q
)2/(
2/sin)( ||
LQ
LQQS
Fibonacci diffraction grating
(Ferralis, Szmodis and Diehl (2004))
R||
“Phason” degrees of freedom
Atomic surfaces
R
4
3
2
1
0
Tiling of plane by 60° rhombi
Phason freedom:Add/remove block
4
3
2
1
0
Entropy calculation via quantum mechanical world lines
Spa
ce
Time
Lines never start nor stop: particles conservedLines never cross: particles are “fermions”
4D hypercube (tesseract)
Octagonal Tiling Projected from 4D
Squares and 45° rhombi
Octagonal TilingProjection from 4D
Penrose Tiling Projected from 5D
Phason Diffuse Scattering
Decagonal AlCoNi (Estermann & Steurer)
Simulated Atomic Surfaces5D body-centered hypercubic lattice
Aluminum Cobalt
Nickel Combined
Tiling model of 10x Al-Co-Ni
Aluminum Cobalt Nickel
Phason Diffuse Scattering
Elastic neutron scatteringi-AlMnPd (Schweika)
Predicted phason diffuse scattering
Phason Diffuse Scattering
X-ray diffuse scattering(046046) peak (Colella)
Phason prediction
Summary and conclusions
Quasicrystals are quasiperiodic structures of high rotational symmetry
They possess sharp Bragg diffraction peaks with additional diffuse backgrounds
Structural models exist, but they do not minimize the total energy
Intrinsic “phason” fluctuations contribute entropy that may lend thermodynamic stability at high temperatures
Thanks!
Marek Mihalkovic (CMU/Slovakia)Siddartha Naidu (CMU → Google Bangalore)Veit Elser (Cornell)Chris Henley (Cornell)John Moriarty (Livermore National Lab)Yang Wang (Pittsburgh Supercomputer Center)Ibrahim Al-Lehyani (CMU → Saudi Arabia)Remy Mosseri (Paris)Nicolas Destainville (Toulouse)
and many more .....