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additional and advanced information about crystallography.

FASCINATING QUASICRYSTALSBased on atomic order quasicrystals are one of the 3 fundamental phases of matter

UNIVERSEPARTICLESENERGYSPACEFIELDSSTRONG WEAKELECTROMAGNETICGRAVITYMETALSEMI-METALSEMI-CONDUCTORINSULATORnD + tHYPERBOLICEUCLIDEANSPHERICALGASBAND STRUCTUREAMORPHOUSATOMICNON-ATOMICSTATE / VISCOSITYSOLIDLIQUIDLIQUID CRYSTALSQUASICRYSTALSCRYSTALSRATIONAL APPROXIMANTSSTRUCTURENANO-QUASICRYSTALSNANOCRYSTALSSIZEWhere are quasicrystals in the scheme of things?

Crystal = Lattice (Where to repeat) + Motif (What to repeat)=+WHAT IS A CRYSTAL?Let us first revise what is a crystal before defining a quasicrystal

R RotationG Glide reflectionSymmetry operatorsR Roto-inversionS Screw axist TranslationR InversionR Mirror Takes object to the same form Takes object to the enantiomorphic formCrystals have certain symmetries

3 out of the 5 Platonic solids have the symmetries seen in the crystalline world i.e. the symmetries of the Icosahedron and its dual the Dodecahedron are not found in crystalsFluoriteOctahedronPyrite CubeRdiger Appel, http://www.3quarks.com/GIF-Animations/PlatonicSolids/These symmetries (rotation, mirror, inversion) are also expressed w.r.t. the external shape of the crystal

HOW IS A QUASICRYSTAL DIFFERENT FROM A CRYSTAL?

FOUND! THE MISSING PLATONIC SOLID[1] I.R. Fisher et al., Phil Mag B 77 (1998) 1601[2] Rdiger Appel, http://www.3quarks.com/GIF-Animations/PlatonicSolids/Mg-Zn-Ho[1][2]Dodecahedral single crystal

QUASICRYSTALS (QC)

ORDEREDPERIODICQC ARE ORDERED STRUCTURES WHICH ARE NOT PERIODICCRYSTALSQCAMORPHOUS

SYMMETRYQC are characterized by Inflationary Symmetry and can have disallowed crystallographic symmetries*2, 3, 4, 65, 8, 10, 12* Quasicrystals can have allowed and disallowed crystallographic symmetries

CRYSTALQUASICRYSTALtRCRCQ

ttranslationinflationRCrotation crystallographicRCQRC + other

DIMENSION OF QUASIPERIODICITY (QP)HIGHER DIMENSIONSQC can be thought of as crystals in higher dimensions(which are projected on to lower dimensions lose their periodicity*)* At least in one dimension

QC can have quasiperiodicity along 1,2 or 3 dimensions (at least one dimension should be quasiperiodic)

QPXAL142536

QUASILATTICE + MOTIF (Construction of a quasilattice followed by the decoration of the lattice by a motif) PROJECTION FORMALISM TILINGS AND COVERINGS CLUSTER BASED CONSTRUCTION (local symmetry and stagewise construction are given importance) TRIACONTAHEDRON (45 Atoms) MACKAY ICOSAHEDRON (55 Atoms) BERGMAN CLUSTER (105 Atoms)

HOW TO CONSTRUCT A QUASICRYSTAL?

THE FIBONACCI SEQUENCEWhere is the root of the quadratic equation: x2 x 1 = 0The Fibonacci sequence has a curious connection with quasicrystals* via the GOLDEN MEAN ()The ratio of successive terms of the Fibonacci sequence converges to the Golden Mean* There are many phases of quasicrystals and some are associated with other sequences and other irrational numbers

Fibonacci 1 1 2 3 5 8 13 21 34... Ratio 1/1 2/1 3/2 5/3 8/5 13/8 21/1334/21... = ( 1+5)/2

Chart3

1

2

1.5

1.6666666667

1.6

1.625

1.6153846154

1.619047619

1.6176470588

1.6181818182

n

Ratio

Convergence of Fibonacci Ratios

Sheet1

1

11

22

31.5

51.6666666667

81.6

131.625

211.6153846154

341.619047619

551.6176470588

891.6181818182

1441.6179775281

2331.6180555556

3771.6180257511

6101.6180371353

9871.6180327869

15971.6180344478

25841.6180338134

41811.6180340557

67651.6180339632

109461.6180339985

Sheet1

n

Ratio

Convergence of Fibonacci Ratios

Sheet2

Sheet3

Sheet4

Sheet5

Schematic diagram showing the structural analogue of the Fibonacci sequence leading to a 1-D QC1-D QCDeflated sequence Penrose tilingRational Approximants2D analogue of the 1D quasilatticeNote: the deflated sequence is identical to the original sequenceIn the limit we obtain the 1D quasilatticeEach one of these units (before we obtain the 1D quasilattice in the limit) can be used to get a crystal (by repetition: e.g. AB AB ABor BAB BAB BAB)

A

B

B

A

B

A

B

B

A

B

B

A

B

A

B

B

A

B

A

B

B

A

B

B

A

B

A

B

B

A

B

B

A

a

b

ba

bab

babba

PENROSE TILING Inflated tiling The inflated tiles can be used to create an inflated replica of the original tilingThe tiling has regions of local 5-fold symmetryThe tiling has only one point of global 5-fold symmetry (the centre of the pattern)However if we obtain a diffraction pattern (FFT) of any broad region in the tiling, we will get a 10-fold pattern! (we get a 10-fold instead of a 5-fold because the SAD pattern has inversion symmetry)

ICOSAHEDRAL QUASILATTICE5-fold [1 0]3-fold [2+1 0]2-fold [+1 1]Note the occurrence of irrational Miller indicesThe icosahedral quasilattice is the 3D analogue of the Penrose tiling.It is quasiperiodic in all three dimensions.The quasilattice can be generated by projection from 6D.It has got a characteristic 5-fold symmetry.

HOW IS A DIFFRACTION PATTERN FROM A CRYSTAL DIFFERENT FROM THAT OF A QUASICRYSTAL?

SAD patterns from a BCC phase (a = 10.7 ) in as-cast Mg4Zn94Y2 alloy showing important zones[111][011][112]The spots are periodically arrangedLet us look at the Selected Area Diffraction Pattern (SAD) from a crystal the spots/peaks are arranged periodicallySuperlattice spots

SAD patterns from as-cast Mg23Zn68Y9 showing the formation of Face Centred Icosahedral QC[1 0][1 1 1][0 0 1][ 1 3+ ]The spots show inflationary symmetryExplained in the next slideNow let us look at the SAD pattern from a quasicrystal from the same alloy system (Mg-Zn-Y)

DIFFRACTION PATTERN 5-fold SAD pattern from as-cast Mg23Zn68Y9 alloySuccessive spots are at a distance inflated by Note the 10-fold patternInflationary symmetry

THE PROJECTION METHODTO CREATE QUASILATTICES

HIGHER DIMENSIONS ARE NEATE2REGULAR PENTAGONSGAPSS2 E3SPACE FILLINGRegular pentagons cannot tile E2 space but can tile S2 space (which is embedded in E3 space)

For this SAD patternwe require 5 basis vectors (4 independent)to index the diffraction pattern in 2DFor crystals We require two basis vectors to index the diffraction pattern in 2DFor quasicrystals We require more than two basis vectors to index the diffraction pattern in 2D

PROJECTION METHODQC considered a crystal in higher dimension projection to lower dimension can give a crystal or a quasicrystalAdditional basis vectors needed to index the diffraction patternE||EWindowe1e22D 1DE||In the work presented approximations are made in E (i.e to )

Slope = Tan ()Irrational QC

Rational RA (XAL)

1-D QC

B

A

B

B

A

B

A

B

B

A

B

B

A

List of quasicrystals with diverse kinds of symmetries

Type of quasicrystal

QP+

Rank

Metric

Symmetry

System

First

Report

Icosahedral

3 D

6

(

((5)

AlMn

Shechtman et al.

1984

Cubic

3D

6

(3

VNiSi

Feng et al

1989

Tetrahedral

3D

6

(3

EMBED Equation.2

AlLiCu

Donnadieu

1994

Decagonal

2D

5

(

((5)

10/mmm

AlMn

Chattopadhyay et al., 1985a and Bendersky, 1985

Dodecagonal

2D

5

(3

12/mmm

NiCr

Ishimasa et al.

1985

Octagonal

2D

5

(2

8/mmm

VNiSi,

CrNiSi

Wang et al.

1987

Pentagonal

2D

5

(

((5)

AlCuFe

Bancel

1993

Hexagonal

2D

5

(3

6/mmm

AlCr

Selke et al.

1994

Trigonal

1D

4

(3

AlCuNi

Chattopadhyay et al.,

1987

Digonal

1D

4

(2

222

AlCuCo

He et al.

1988

_923185781.unknown

_923185783.unknown

_923185785.unknown

_923185786.unknown

_923185784.unknown

_923185782.unknown

_923185780.unknown

Comparison of a crystal with a quasicrystal

CRYSTALQUASICRYSTALTranslational symmetryInflationary symmetryCrystallographic rotational symmetriesAllowed + some disallowed rotational symmetriesSingle unit cell to generate the structureTwo prototiles are required to generate the structure3D periodicPeriodic in higher dimensionsSharp peaks in reciprocal space with translational symmetrySharp peaks in reciprocal space with inflationary symmetryUnderlying metric is a rational numberIrrational metric

WEAR RESISTANT COATING (Al-Cu-Fe-(Cr)) NON-STICK COATING (Al-Cu-Fe) THERMAL BARRIER COATING (Al-Co-Fe-Cr) HIGH THERMOPOWER (Al-Pd-Mn) IN POLYMER MATRIX COMPOSITES (Al-Cu-Fe) SELECTIVE SOLAR ABSORBERS (Al-Cu-Fe-(Cr)) HYDROGEN STORAGE (Ti-Zr-Ni)APPLICATIONS OF QUASICRYSTALS

As-cast Mg37Zn38Y25 alloy showing a 18 R modulated phaseSAD patternBFIHigh-resolution micrograph

*Found the Missing Platonic Solid