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®

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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 .....