THE FASCINATING WORLD OF THE FASCINATING WORLD OF QUASICRYSTALSQUASICRYSTALS

Anandh SubramaniamMaterials Science and Engineering

INDIAN INSTITUTE OF TECHNOLOGY KANPURINDIAN INSTITUTE OF TECHNOLOGY KANPURKanpur-

208016Email: anandh@iitk.ac.in

http://home.iitk.ac.in/~anandh

Oct 2011

Daniel Shechtman

GLASS (AMORPHOUS)

SOLIDS

CRYSTALS

Based on Structure

7 April 1982

GLASS (AMORPHOUS)

SOLIDS

CRYSTALS

Based on Structure

8 April 1982

QUASI CRYSTALS

A leaf from a diary…

7 April 1982

8 April 1982

12 Nov 1984

Enter the Decagon!

Daniel ShechtmanBorn: January 24, 1941

Painting by Dr. Alok

Singh, 1993

"I must have shared with you my first ever meeting with him in July

this year. I was invited to Ames Lab by Mat Kramer and I was sitting in his

office and

told him "I have been waiting to meet Prof. Shechtman

from my PhD days". That was the time one person entered his office and was asking Mat, "Mat, I have been searching for the glue for ion milling my sample and could not find it in the lab. Can you please let me know". Mat tuned towards me and told me "the man you are looking forward to meet is here". He was about to celebrate his 70th birthday

in a few days from then. That speaks volumes about the

commitment to research from this great scientist."–

B.S. MURTHY

“If you are a scientist and believe in your results, then fight for the truth”. “Listen to others, but fight for what you believe in…”

-

DAN SHECHTMAN

“If you are a scientist and believe in your results, then fight for the truth”. “Listen to others, but fight for what you believe in…”

-

DAN SHECHTMAN

Are QC only made of rare-

“hard to find”

elements?

Why did it take so long?

No! Most of them contain common elements like Al, Mn, Mg, Cu, Fe…

Do we require ‘difficult conditions for synthesis’-

High temperature, High pressure,…?

No! Many of them can be produced by simple casting (e.g. AlCuFe, MgZnY…)

Having produced them-

are they ‘unstable’

with small lifetimes?

No! Some of them are so stable (at RT) that they would survive for millennia (but for corrosion!)

Do we need extremely sensitive experimentation (like neutron diffraction…) to detect their presence/identify them?

No! All you need is a Transmission Electron Microscope (TEM) (that too without EELS, EDXS… however, HREM

would help!)

Element 117 (with 177 neutrons) has a half life of 78 ms

They even occur naturally

QUASICRYSTALS: THE QUASICRYSTALS: THE PRESAGES!PRESAGES!

1453 ADGunbad-i

Kabud

tomb in Maragha, Iran, 1197 AD Darb-I Imam

shrine, Isfaha, Iran, 1453 AD

PENROSE TILING

The tiling has regions of local 5-fold symmetry

The 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)

M. Gardner, Sci. Am. 236 (1977) 110R. Penrose, “Pentaplexity”, Eureka, 39, 16, 1978

Berger, 1966 20,000

tiles (then to 104 tiles)

Robinson, 1971 6 tiles

Penrose1

, 1974 4 (6) tiles

Penrose2

, 1978 2

tiles

A brief history of aperiodic

tilings

R. Berger, Mem. Am. Math. Soc., No.66, 1966.R.W. Robinson, Invent. Math., 12, 177, 1971.[1] R. Penrose, Bull. Inst. Math. Appl., 10, 266, 1974.[2] R. Penrose, “Pentaplexity”, Eureka, 39, 16, 1978.

Penrose versus Kepler

(Harmonice

Mundi, 1619)

Penrose’s Pattern Kepler’s Pattern

Kepler

concluded that the pattern would never repeat-

there would always be “surprises” Kepler

had anticipated the concept of aperiodic

tiling by 350 years!

A Circle has been placed on each quasi-lattice point of the 2D pattern to model a possible atomic structure

Wonders of Numbers: Adventures in Mathematics, Mind and MeaningClifford A Pickover

WHAT IS A CRYSTAL?WHAT IS A CRYSTAL?

Crystal = Space group

(how to repeat)

+ Asymmetric unit

(Motif’: what to repeat)

+Wyckoff positions

=

+

a

aGlide reflection operator

Usually asymmetric units are regions of space-

which contain the entities (e.g. atoms, molecules)

Symbol g may also be used

+Wyckoff label ‘a’

Positions entities with respect to

symmetry operatorsWHAT IS A CRYSTAL?

R Rotation

G Glide reflection

Symmetry operators

R Roto-inversion

S Screw axis

t Translation

R Inversion R Mirror

Takes object to the same form

Takes object to the enantiomorphic

form

Crystals have certain symmetries

m

t

3 out of the 5 Platonic solids have the symmetries seen in the crystalline worldi.e. the symmetries of the Icosahedron

and its dual the Dodecahedron are

not found in crystals

FluoriteOctahedron

Pyrite Cube

Rüdiger

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

http://en.wikipedia.org/wiki/Crystal_habit http://www.galleries.com/minerals/property/crystal.htm

Plato wrote about these solids in the dialogue Timaeus

c.360 B.C.

HOW IS A QUASICRYSTAL DIFFERENT HOW IS A QUASICRYSTAL DIFFERENT FROM A CRYSTAL?FROM A CRYSTAL?

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]

Dodecahedral single

quasicrystal 3 5m

Octahedron and icosahedron were discovered by Theaetetus, a contemporary of Plato

QUASICRYSTALS (QC)

ORDERED PERIODIC QC ARE ORDERED

STRUCTURES WHICH ARE

NOT PERIODIC

CRYSTALS

QC

AMORPHOUS

SYMMETRY

CRYSTAL QUASICRYSTAL

t

RC RCQ

QC are characterized by Inflationary Symmetry

and can have disallowed crystallographic symmetries*

t translation

inflation

RC rotation crystallographic

RCQ RC

+ other2, 3, 4, 6

5, 8, 10, 12

* Quasicrystals can have allowed and disallowed crystallographic

symmetries

QC can have quasiperiodicity along 1,2 or 3 dimensions

(at least one dimension should be

quasiperiodic)

DIMENSION OF QUASIPERIODICITY (QP)

QC as a crystal?

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 → lose their periodicity*)

* At least in one dimension

QUASIPERIODICITY & INFLATIONARY SYMMETRY

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 quadratic equation: x2

– x – 1 = 0

The 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

1.618… 11x x

x

2 1 0x x In 1202 Fibonacci discussed the number sequence in connection with the proliferation of rabbits

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

Rational Approximants

Note: the deflated sequence is identical to the original sequence

In the limit we obtain the 1D Quasilattice

Each 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

AB…or BAB

BAB

BAB…)

In the ratio of lengths

In the ratio of numbers

Where is the Golden Mean?

B

A

nn

A

B

LL

11 11 11 111

Inflationary symmetry in the Penrose tiling

Inflated tiling

The inflated tiles can be used to create an inflated replica of the original tiling

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

SAD patterns from a BCC phase

(a = 10.7 Å) in as-cast Mg4

Zn94

Y2

alloy showing important zones

[111] [011][112]

The spots are periodically

arranged

Let us look at the Selected Area Diffraction Pattern (SAD) from a crystal → the spots/peaks are arranged periodically

Superlattice spots

SAD patterns from as-cast Mg23

Zn68

Y9

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 slide

Now let us look at the SAD pattern from a quasicrystal from the same alloy system (Mg-Zn-Y)

2 3 41

DIFFRACTION PATTERN DIFFRACTION PATTERN 5-fold SAD pattern

from as-cast Mg23

Zn68

Y9

alloy

Successive spots are at a distance inflated by

Note the 10-fold pattern

Inflationary symmetry

STRUCTURE OF QUASICRYSTALS

QUASILATTICE APPROACH

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

PROJECTION FORMALISM

TILINGS AND COVERINGS

CLUSTER BASED CONSTRUCTION

(local symmetry and stage-wise 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 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 2D

For crystals We require two basis vectors to index the diffraction pattern in 2D

For quasicrystals For quasicrystals

We require more than two basis vectors to index the diffWe require more than two basis vectors to index the diffraction pattern in 2Draction pattern in 2D

PROJECTION METHODPROJECTION METHOD

QC considered a crystal in higher dimension → projection to lower dimension can give a crystal or a quasicrystal

Slope = Tan ()

Irrational QC

Rational RA (XAL)

E||E Window

e1

e2

2D 1D

''

x Cos Sin xR

y Sin Cos y

E||

To get RA

approximations are made

in E

(i.e

to )

B A B B A B A B B A B B A

1-D QC

1D

2D

Octogonal

Tiling1 12

0 3 3 3 - 31 12

0 3 - 3 3 3

2 2 2 2 2

R

1 11 02 2

1 10 12 2

1 1 1 02 2

1 1 0 12 2

R

Penrose Tiling

ICOSAHEDRAL QUASILATTICE

5-fold

[1

0]

3-fold

[2+1

0]

2-fold

[+1

1]

Note the occurrence of irrational Miller indices

The 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.

3D

1 0 1 01 0 0 1

0 1 0 11 0 1 0

1 0 1 00 1 0 1

R

(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

Cluster Based Construction

Rhombic Triacontahedron

Hiraga, K et al, S., Phil. Mag. B67 (1993)

193

Kreiner, G., and Franzen, H. F., J. Alloys and Compounds, 221 (1995) 15

CRYSTAL QUASICRYSTALTranslational symmetry Inflationary

symmetry

Crystallographic rotational symmetries Allowed + some disallowed

rotational symmetries

Single unit cell to generate the structure Two

prototiles

are required to generate the structure (covering possible with one tile!)

3D periodic Periodic in higher dimensions

Sharp peaks in reciprocal space with translational symmetry

Sharp peaks in reciprocal space with inflationary symmetry

Underlying metric is a rational number Irrational

metric

Usually made of ‘small’

clusters Large clusters

Comparison of a crystal with a quasicrystal

SYSTEMS FORMING QUASICRYSTALSSYSTEMS FORMING QUASICRYSTALS

&&

TYPES OF QUASICRYSTALSTYPES OF QUASICRYSTALS

List of quasicrystals with diverse kinds of symmetries

Type QP+ Rank Metric Symmetry System Reference

Icosahedral 3 D 6 (5) m3_5_ AlMn Shechtman et al., 1984

Cubic 3D 6 3 43m_

VNiSi Feng et al., 1989

Tetrahedral 3D 6 3 m3_

AlLiCu Donnadieu, 1994

Decagonal 2D 5 (5) 10/mmm AlMn Chattopadhyay et al., 1985

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) 5m_

AlCuFe Bancel, 1993

Hexagonal 2D 5 3 6/mmm AlCr Selke et al., 1994

Trigonal 1D 4 3 3m_

AlCuNi Chattopadhyay et al., 1987

Digonal 1D 4 2 222 AlCuCo He et al., 1988

First naturally occurring QC was reported associated with the mineral Khatyrkite.

Naturally Occurring QC

“However, India missed some opportunities in this area. Early work of T.R. Anantharaman

on Mn-Ga

alloys and G.V.S. Sastry

and C. Suryanarayana

(BHU) on Al-Pd

alloys came tantalizingly close

to the discovery of quasicrystals”.

http://www.iucr.org/news/newsletter/volume-15/number-4/crystallography-in-india

Indian Contributions

http://www.iitk.ac.in/infocell/announce/metallo/collection.htm

IITKIITK

G.V.S

SASTRYC. Suryanarayana

S. Ranganathan

S. Lele

Conference in Honour

of Prof. T.R. Anantharaman

Allowed crystallographic symmetry-

tiled aperiodically

Discovery of the decagonal phase

Basis for synthesis of QC

1-D quasiperiodicity

= 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

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

First observation of a relation between five-fold and hexagonal symmetry

Unified view of quasicrystals, rational approximants and

related structures

Approximant to 7-

fold quasilattice

Fundamental work on Vacancy

Ordered Phases

Trigonal

and Pentagonal quasilattices

3 2 2 1 0x x x