STRUCTURE OF QUASICRYSTALS AND STRUCTURE OF QUASICRYSTALS AND RELATED PHASESRELATED PHASES

ANANDH SUBRAMANIAMGuest Scientist (Alexander Von Humboldt Fellow)

Electron Microscopy GroupMax-Planck-Institut für Metallforschung

STUTTGARTPh: (+49) (0711) 689 3683, Fax: (+49) (0711) 689 3522

anandh@mf.mpg.dehttp://www.geocities.com/anandh4444/

November 2004

OUTLINEOUTLINE

OVERVIEWOVERVIEW

DEFINITIONDEFINITION

PROJECTION FORMALISMPROJECTION FORMALISM

CLUSTER BASED CONSTRUCTIONCLUSTER BASED CONSTRUCTION

MgMg--ZnZn--(Y, La) SYSTEMS(Y, La) SYSTEMS

DISCUSSION

“Babuji” 1899-1983

UNIVERSE

PARTICLES

ENERGYSPACE

FIELDS

STRONG WEAKELECTROMAGNETICGRAVITY

METALSEMI-METAL

SEMI-CONDUCTORINSULATOR

nD + t

HYPERBOLICEUCLIDEANSPHERICAL

GAS

BAND STRUCTURE

AMORPHOUS

ATOMIC NON-ATOMIC

STATE / VISCOSITY

SOLID LIQUIDLIQUID

CRYSTALS

QUASICRYSTALS CRYSTALSRATIONAL APPROXIMANTS

STRUCTURE

NANO-QUASICRYSTALS NANOCRYSTALS

SIZE

VARIOUS SPACES INVOLVED

1D, 2D, 3D 4D, 5D, 6D 7D, ....., ND

PHYSICAL SPACES

QC HYPERSPACES

GENERALIZED HYPERSPACES

REAL SPACE RECIPROCAL SPACE

PARALLEL SPACE (E|| )

PERPENDICULAR SPACE (E

)

QUASICRYSTALS (QC)

ORDERED PERIODIC QC ARE ORDERED

STRUCTURES WHICH ARE

NOT PERIODIC

CRYSTALS

QC

AMORPHOUS

CRYSTALS (XAL)

MODULATED STRUCTURES

(MS)

INCOMMENSURATELY MODULATED STRUCTURES

(IMS)

QC Can be thought of as IMS which cannot be constructed with a single “unit cell”

but can be thought of as covering with a single prototile

SYMMETRY

XAL QC

t

RC RCQ

QC are characterized by inflationary

symmetry and can have disallowed crystallographic

symmetries

t translation

inflation

RC rotation crystallographic

RCQ RC + other

QC can have quasiperiodicity along 1,2 or 3 dimensions

DIMENSION OF QUASIPERIODICITY (QP)

HIGHER DIMENSIONS

QP QP/P

QP/P

QP XAL

1 4

2 5

3 6

QC can be thought of as crystals in higher dimensions

(which are projected on to lower dimensions)

2, 3, 4, 65, 8, 10, 12

THE FIBONACCI SEQUENCE

Fibonacci 1 1 2 3 5 8 13 21 34 ...

Ratio 1/1 2/1 3/2 5/3 8/5 13/8 21/13 34/21 ...

= ( 1+5)/2

Convergence of Fibonacci Ratios

1

1.2

1.4

1.6

1.8

2

2.2

1 2 3 4 5 6 7 8 9 10

n

Rat

io

WHERE

IS THE ROOT OF THE EQUATION x2 – x – 1 = 0

Schematic diagram showing the structural analogue of the Fibonacci sequence leading to a 1-D QC

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

1-D QC

a

b

ba

bab

babba

Deflated sequence

Penrose tiling

Rational Approximants

LIST OF QC.ppt

FOUND!FOUND! THE MISSING PLATONIC SOLID

[1] I.R. Fisher et al., Phil Mag B 77 (1998) 1601

[2] RRüüdigerdiger AppelAppel, http://www.3quarks.com/GIF-Animations/PlatonicSolids/Mg-Zn-Ho

[1]

[2]

DISCUSSIONDISCUSSION

STRUCTURE OF QUASICRYSTALS

QUASILATTICE APPROACH

(Construction of a quasilattice followed by the decorationof the lattice by atoms)

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)

HIGHER DIMENSIONS ARE NEATHIGHER DIMENSIONS ARE NEAT

E2

REGULAR PENTAGONS

GAPS

S2

E3

SPACE FILLING

PROJECTION METHOD

QC considered a crystal in higher dimension

Additional basis vectors needed to index the diffraction pattern

Slope = Tan ()

Irrational QC

Rational RA (XAL)

E||E Window

e1

e2

2D 1D

Strip Projection Plane

Irrational Rational

Irrational Tiles Irrational dimensionsArrangement Quasiperiodic Quasicrystal (QC)

Tiles Irrational dimensions Arrangement Periodic QC Approximant

Rational Tiles Rational dimensions Arrangement Quasiperiodic Quasiperiodic Superlattice (QPSL)

Tiles Rational dimensionsArrangement periodic QPSL Approximant

KINDS OF STRUCTURES OBTAINED BY PROJECTION FORMALISM

Real Space Reciprocal Space

1. Rational Lengths L and S arranged periodically

PPeerriiooddiicc

2. Rational lengths L and S arranged in a Fibonacci chain

PPeerriiooddiicc with satellites

3. Irrational length L and S arranged periodically

Peak positions ppeerriiooddiicc Intensities aappeerriiooddiicc

4. Irrational length L and S arranged in a Fibonacci chain

AAppeerriiooddiicc

Diffraction properties of various distributions of scatterers.

Progressive lowering of dimension starting with an Progressive lowering of dimension starting with an NN--foldfold symmetry in ND spacesymmetry in ND space

N-fold symmetry Hypercubic Lattice viewed

along [111....1]N 1s

N-D

AApppprrooxxiimmaannttss

Quasiperiodic tiling 2D RRAA

Sequence of numbers Sequence of ‘a’s and ‘b’s Polynomial Equation

1D

RReeppeeaattiinngg SSeeqquueennccee

Convergence of sequence Length of ‘a’/length of ‘b’ Root of Polynomial Eq.

0D

RRaattiioonnaall NNuummbbeerr

ND-0D.ppt

GENERALIZED PROJECTION METHOD

[A] a1, a2, a3, ..., aN : a set of vectors in E||

[B] b1, b2, b3, ..., bN : a set of vectors in E

W : the acceptance region or window in E

{n1, n2, n3, ..., nN} are a set of integers in N dimensional space such that

n1a1 + n2a2 + n3a3 + ... + nNaN is accepted as a point in E|| if and only if:

n1b1 + n2b2 + n3b3 + ... + nNbN W

LLiinneeaarr ddeeffoorrmmaattiioonnss ooff EE ddoo nnoott aaffffeecctt tthhee ppaatttteerrnn pprroodduucceedd iinn EE||||,,

i.e. if E is m dimensional and T is a non singular m m matrix, then:

n1(Tb1) + n2(Tb2) + ... + nN(TbN) TW,

if and only if n1b1 + n2b2 + ... + nNbN W

The pattern in E|| will have a period n1a1 + n2a2 + n3a3 + ... + nNaN

for any {n1, n2, n3, ..., nN} such that n1b1 + n2b2 + n3b3 + ... + nNbN = 0

2D AND 3D 2D AND 3D QUASILATTICS QUASILATTICS

AND THEIR AND THEIR APPROXIMANTS APPROXIMANTS

(QC & RA)(QC & RA)

{1/1 1/1} RA to the Penrose tiling

Fourier transform of the lattice

a set of 10-fold spots are marked with circles.

RATIONAL APPROXIMANTS TO THE PENROSE TILING WITH ORTHOGONAL BASIS VECTORS

Lattice with rectangular unit cell ABCD

{1/1 2/1} {3/2 1/1}

{

2/1}

RA to the Penrose tiling

RATIONAL APPROXIMANTS WITH APPROXIMATIONS ALONG BASIS VECTORS 72

APART

Fourier transform of the lattice with remnant of the 10-fold symmetry marked by circles.

{1/1 1/1}e RA to the Penrose tiling

Lattice with rectangular unit cell ABCD and parallelogram cell EFGH

ICOSAHEDRAL QUASILATTICE

5-fold [1

0]

3-fold [2+1

0]

2-fold [+1

1]

E||

3

3

2

2 5

5

6

6 4

4

1

V1

V2

V3

B’[p/q, , ]

200110

11

pqqqqq

{1/1 }P PENTAGONAL QUASILATTICE

1

V1

24

5

5

4

6

6 3

V2

V3

E||

B' =

2 2 2

2 1 2 12 2 1 1

{1/1 }T TRIGONAL QUASILATTICE

Three dimensional covering with triacontahedra

Lord, E. A., Ranganathan, S., and Kulkarni, U. D., Current Science, 78 (2000) 64

(a) (b)

(a) Bergman, G., Waugh, J. L. T., and Pauling, L., Acta Cryst., 10 (1957) 2454(b) Ranganathan, S., and Chattopadhyay, K., Annu. Rev. Mater. Sci., 21 (1991) 437

BBeerrggmmaann cclluusstteerr MMaacckkaayy ddoouubbllee iiccoossaahheeddrroonn

= 1

Important clusters underlying the structure of quasicrystals and their approximants.

= 2

The structure of the Al3Mn decagonal phase

Hiraga, K., Kaneko, M., Matsuo, Y., and Hashimoto, S., Phil. Mag. B67 (1993) 193

= 3

(a) (b)

Arrangement of sub-units in complex hexagonal phases

Cluster of three dodecahedra Cluster of three dodecahedra Four vertexFour vertex--connected icosahedraconnected icosahedra

(a) Singh, A., Abe, E., and Tsai, A. P., Phil. Mag. Lett., 77 (1998) 95(b) Kreiner, G., and Franzen, H. F., J. Alloys and Compounds, 221 (1995) 15

IQC (( == 11)) DQC (( == 22))

Mackay Approximant Taylor Approximant

Little Approximant Robinson Approximant

IQC (( == 11) HQC (( == 33))

Key: shows a twinning operation

R e la tio n b e tw e e n IQ C a n d its a p p ro x im a n ts w ith D Q C , its a p p ro x im a n ts a n d H Q C v ia th e tw in n in g

o p e ra tio n

A quadrant of the stereogram of the decagonal phase with indices derived by the twinned icosahedron

model

Stereogram of the Taylor phase obtained by twinning of the Mackay approximant to the icosahedral phase

Quadrant of the stereogram corresponding to I3 cluster

= 1 = 2 Icosahedral Quasicrystal = 3

Decagonal Quasicrystal

Hexagonal Quasicrystal

= 1 Digonal

Quasicrystal Pentagonal Quasicrystal

Cubic R.A.S. Mackay Bergman

Trigonal Quasicrystal

Hexagonal R.A.S.

Orthorhombic

R.A.S. Orthorhombic

R.A.S Trigonal R.A.S.

Orthorhombic R.A.S.

Taylor Little Robinson

R.A.S.

Monoclinic Monoclinic

R.A.S. Monoclinic

R.A.S. R.A.S.

= 90o 120o

= 108o

Unification scheme based on the twinning of the icosahedral cluster

EXPERIMENTALEXPERIMENTAL

MgMg--ZnZn--(Y, La) (Y, La)

SYSTEMSSYSTEMS

METASTABLE PHASES IN Mg-BASED ALLOYS

QUASICRYSTALS

RATIONAL APPROXIMANTS & RELATED STRUCTURES

METALLIC GLASSES

NANOCRYSTALS & NANOQUASICRYSTALS

Mg-Zn-Al First Mg-Based QC(Icosahedral)

P. Ramachandrarao, G.V.S. Sastry1985

Mg-Zn-Al-Cu Quaternary System N.K. Mukhopadhyay, G.N. Subbanna, S. Ranganathan, K. Chattopadhyay

1986

Mg-Zn-Ga Stable QC W. Ohashi, F. Spaepen1987

Mg-Zn-RE Icosahedral QC Z. Luo, S. Zhang, Y. Tang, D. Zhao1993

Mg-Al Cubic QC P. Donnadieu, A. Redjaimia1995

Mg-Zn-RE Decagonal QC T.J. Sato, E. Abe, A.P. Tsai1997

Mg-Zn-RE QC without underlying atomic clusters

E. Abe, T.J. Sato, A.P. Tsai1999

MILESTONES IN Mg-BASED QUASICRYSTAL RESEARCH

IMPORTANT PHASES IN THE Mg-Zn-RE SYSTEMS

Composition e/a Phase, Symmetry Comments

Mg3Zn6RE 2.1 Icosahedral, Fm53 aR = 0.519

RE = Y, Gd, Tb, Dy, Ho, Er dia (0.352, 0.360)

Mg40Zn58RE2 2.02 Decagonal, 10/mmm RE = Y, Dy, Ho, Er, Tm, Lu dia < 0.355

Mg24Zn65RE10 (S) 2.1 Hexagonal superlattice, P63/mmc a = 1.46 nm, c = 0.86 nm

RE = Y, Sm, Gd Related to IQC

Mg24Zn65RE10 (M) 2.1 Hexagonal superlattice, P63/mmc a = 2.35 nm, c = 0.86 nm

RE = Sm, Gd Related to IQC

Mg24Zn65Y10 (L) 2.1 Hexagonal superlattice, P63/mmc a = 3.29 nm, c = 0.86 nm

RE = Sm Related to IQC

aS : aM : aL = 3 : 5 : 7

Mg12ZnY 2.07 ?

Mg3Zn3Y2 2.25 cF16, Fm3m

Mg7Zn3 2 oI142, Immm 1/1 RA to IQC

Mg4Zn7 2 mC110, B2/m Related to DQC

MgZn2 2 hp12, P63/mmc Related to S, L & M phases

SEM micrograph of as-cast Mg51 Zn41 Y8 alloy showing (a) Eutectic Microstructure (b) Four-fold dendrite

(a) (b)

55--FOLD TO 6FOLD TO 6--FOLDFOLD

5-FOLD

DEVELOPING INTO 6-FOLD

SEM micrograph of as-cast Mg51 Zn41 Y8 alloy showing distorted 5-fold dendrite growing into hexagonal shape

Initial stages of growth

As-cast Mg37 Zn38 Y25 alloy showing the formation of a cubic phase (a = 7.07 Å):

BFI [111]

[110] [113]

SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg4 Zn94 Y2 alloy showing important zones

[111] [011][112]

As-cast Mg37 Zn38 Y25 alloy showing a 18 R modulated phase

SAD pattern BFI

High-resolution micrograph

SAD patterns from as-cast Mg23 Zn68 Y9 showing the formation of FCI QC

[1

0] [1 1 1]

[0 0 1] [

1 3+ ]

Uniform deformation along the arrow of the [0 0 1] 2-fold pattern from IQC giving rise to a pattern similar to the [

1 3+ ] pattern

TEM micrograph of as-cast Mg4 Zn94 Y2 alloy showing the formation of nanocrystalline Mg3 Zn6 Y phase

Mg4 Zn94 Y2 as-cast alloy heat treated at 350oC for 20 hrs (corresponding to the MgZn5.51 phase)

BFI SAD

BFI SAD

BFI from as-cast Mg46 Zn46 La8 alloy showing patterns from APBs

Melt-spun Mg50 Zn45 Y5 alloy showing the formation of a cubic phase (a = 6.63 Å)

BFI [001]

[113] [111]

Comparison of the [001] two-fold of the FCI QC (a) with the two-fold from other phase in the MgZnY (b), (c) and MgZnLa (d) systems

ACKNOWLEDGEMENTS ACKNOWLEDGEMENTS

Dr. Eric A Lord

Prof. S. Ranganathan

Dr. K. Ramakrishnan

Dr. Sandip Bysakh

Dr. Steffen Weber

CONCLUSIONSCONCLUSIONS

A variety of Quasiperiodic and Rational Approximant structures can be realized using the Strip Projection Method, which serves to unify these structures using higher dimensions

Structures with diverse kinds of symmetries can be generated using the Twinned Icosahedron Model, which further can be used to construct a unified framework based on the orientations of the icosahedron and the lowering of symmetry

The Mg-Zn-RE systems serves a new ‘model system’ for the study of quasicrystals and related phases

Study of quasicrystals is fun

WEAR RESISTANT COATING (AlWEAR RESISTANT COATING (Al--CuCu--FeFe--(Cr))(Cr))

NONNON--STICK COATING (AlSTICK COATING (Al--CuCu--Fe)Fe)

THERMAL BARRIER COATING (AlTHERMAL BARRIER COATING (Al--CoCo--FeFe--Cr)Cr)

HIGH THERMOPOWER (AlHIGH THERMOPOWER (Al--PdPd--MnMn))

IN POLYMER MATRIX COMPOSITES (AlIN POLYMER MATRIX COMPOSITES (Al--CuCu--Fe)Fe)

SELECTIVE SOLAR ABSORBERS (AlSELECTIVE SOLAR ABSORBERS (Al--CuCu--FeFe--(Cr))(Cr))

HYDROGEN STORAGE (TiHYDROGEN STORAGE (Ti--ZrZr--Ni)Ni)

APPLICATIONS OF QUASICRYSTALSAPPLICATIONS OF QUASICRYSTALS

PENROSE TILING PENROSE TILING

Inflated tiling

2 3 41

DIFFRACTION PATTERN DIFFRACTION PATTERN

5-fold SAD pattern from as-cast

Mg23 Zn68 Y9 alloy