THETHE FASCINATINGFASCINATING WORLDWORLD OFOF QUASICRYSTALSQUASICRYSTALS

Anandh Subramaniam Materials Science and Engineering INDIAN INSTITUTE OF TECHNOLOGY KANPUR Kanpur- 208016 Email: [email protected]

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

Oct 2011 Daniel Shechtman 7 April 1982 SOLIDS Based on Structure

GLASS CRYSTALS (AMORPHOUS)

8 April 1982 SOLIDS Based on Structure

GLASS CRYSTALS (AMORPHOUS) QUASI CRYSTALS

ry 24, 1941  Born: Janua 8 April 1982 12 Nov 1984 7 April 1982 Daniel Shechtman Enter the Decagon!

A leaf from a diary… Painting by Dr. Alok Singh, 1993

“If“If youyou areare aa scientistscientist andand believebelieve inin youryour results,results, thenthen fightfight forfor thethe truth”.truth”. “Listen“Listen toto others,others, butbut fightfight forfor whatwhat youyou believebelieve in…”in…” --DANDAN SHECHTMANSHECHTMAN

"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

Why did it take so long?

 Are QC only made of rare- “hard to find” elements? No! Most of them contain common elements like Al, Mn, Mg, Cu, Fe…

 Do we require ‘difficult conditions for synthesis’- High temperature, High pressure,…? They even No! Many of them can be produced by simple casting (e.g. AlCuFe, MgZnY…)occur naturally

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

QUASICRYSTALS:QUASICRYSTALS: THETHE PRESAGES!PRESAGES! Darb-I Imam shrine, Isfaha, Iran, 1453 AD

Gunbad-i Kabud tomb in Maragha, Iran, 1197 AD 1453 AD

PENROSE TILING

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 The tiling has regions of 5-fold because the SAD pattern local 5-fold symmetry has inversion symmetry)

R. Penrose, “Pentaplexity”, Eureka, 39, 16, 1978 M. Gardner, Sci. Am. 236 (1977) 110 A brief history of aperiodic tilings

 Berger, 1966  20,000 tiles (then to 104 tiles)  Robinson, 1971  6 tiles 1  Penrose , 1974  4 (6) tiles 2  Penrose , 1978  2 tiles

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)

Kepler concluded that the pattern would never repeat- there would always be “surprises”  Kepler had anticipated the concept of aperiodic tiling by 350 years!

Penrose’s Pattern Kepler’s Pattern 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 Meaning Clifford A Pickover WHATWHAT ISIS AA CRYSTAL?CRYSTAL? Crystal = Space group (how to repeat) + Asymmetric unit (Motif’: what to repeat) + Wyckoff positions

a = Glide reflection L? TA a operator YS CR Symbol g may also be used S A T I Positions entities HA + with respect to W symmetry operators

+Wyckoff label ‘a’

Usually asymmetric units are regions of space- which contain the entities (e.g. atoms, molecules)

Crystals have certain symmetries Symmetry operators

t t  Translation     

  R  Rotation R Inversion R Mirror m R  Roto-inversion

G  Glide reflection

S  Screw axis

 Takes object to the same form  Takes object to the enantiomorphic form

Plato wrote about these solids in the dialogue Timaeus c.360 B.C.

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 crystals

Fluorite These symmetries (rotation, Octahedron mirror, inversion) are also expressed w.r.t. the external shape of the crystal

Pyrite Cube

Rüdiger Appel, http://www.3quarks.com/GIF-Animations/PlatonicSolids/ http://en.wikipedia.org/wiki/Crystal_habit http://www.galleries.com/minerals/property/crystal.htm

HOWHOW ISIS AA QUASICRYSTALQUASICRYSTAL DIFFERENTDIFFERENT FROMFROM AA CRYSTAL?CRYSTAL? FOUND! THE MISSING PLATONIC SOLID

[2]

Dodecahedral single quasicrystal m 35 [1] Mg-Zn-Ho

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

[1] I.R. Fisher et al., Phil Mag B 77 (1998) 1601 [2] Rüdiger Appel,Appel http://www.3quarks.com/GIF-Animations/PlatonicSolids/

QUASICRYSTALS (QC)

ORDERED PERIODIC QC ARE ORDERED CRYSTALS   STRUCTURES WHICH ARE QC   NOT AMORPHOUS   PERIODIC SYMMETRY

CRYSTAL QUASICRYSTAL t  

RC  RCQ

t  translation   inflation QC are characterized by Inflationary Symmetry and can have disallowed RC  rotation2, crystallographic 3, 4, 6 crystallographic symmetries*

RCQ  RC + 5,other 8, 10, 12

* Quasicrystals can have allowed and disallowed crystallographic symmetries

DIMENSION OF QUASIPERIODICITY (QP)

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

QC as a crystal?

QC can be thought of as crystals in higher QP XAL dimensions 1  4 (which are projected on to lower dimensions → lose their periodicity*) 2  5 3  6

* At least in one dimension QUASIPERIODICITY & INFLATIONARY SYMMETRY The Fibonacci sequence has a curious connection with quasicrystals* via the GOLDEN MEAN ()

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

Where  is the root of the quadratic equation: x2 –x –1 = 0

Convergence of Fibonacci Ratios The ratio of successive terms of 2.2 the Fibonacci sequence converges to 2 the Golden Mean 1.8 1.618… xx1 1.6 

Ratio 1 x In 1202 Fibonacci 1.4 discussed the 1.2 xx2  10 number sequence in connection with the 1 proliferation of 12345678910 rabbits n

* There are many phases of quasicrystals and some are associated with other sequences and other irrational numbers

A Deflated sequence B 

B A a

B A B b Rational Approximants B A B B A ba B A B B A B A B Each one of these units bab (before we obtain the 1D quasilattice in the limit) B A B B A B A B B A B B A can be used to get a babba crystal (by repetition: e.g. AB AB AB…or BAB BAB BAB…) Note: the deflated sequence is identical to the original sequence 1-D QC In the limit we obtain the 1D Quasilattice

Schematic diagram showing the structural analogue of the Fibonacci sequence leading to a 1-D QC Where is the Golden Mean? 1  1  In the ratio of lengths 1 1  In the ratio of numbers 1 1 1 1 1 n L B  A  nA LB Inflationary symmetry in the Penrose tiling The inflated tiles can be used to create an inflated replica of the original tiling

 Inflated tiling HOW IS A DIFFRACTION PATTERN FROM A CRYSTAL DIFFERENT FROM THAT OF A QUASICRYSTAL? Let us look at the Selected Area Diffraction Pattern (SAD) from a crystal → the spots/peaks are arranged periodically

The spots are periodically arranged

[112] Superlattice spots [111] [011]

SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg4Zn94Y2 alloy showing important zones

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

The spots show inflationary symmetry Explained in the next slide [1 1 1] [1  0]

[0 0 1] [ 1 3+ ]

SAD patterns from as-cast Mg23Zn68Y9 showing the formation of Face Centred Icosahedral QC

DIFFRACTION PATTERN

5-fold SAD pattern from as-cast

Mg23Zn68Y9 alloy

Note the 10-fold pattern

1  2 3 4

Successive spots are at a distance inflated by 

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 NEAT

E2

GAPS

S2  E3

REGULAR PENTAGONS

Regular pentagons cannot tile E2 space but can tile SPACE FILLING S2 space (which is embedded in E3 space) For crystals  We require two basis vectors to index the diffraction pattern in 2D

For quasicrystals  We require more than two basis vectors to index the diffraction pattern in 2D

For this SAD pattern we require 5 basis vectors (4 independent) to index the diffraction pattern in 2D PROJECTION METHOD QC considered a crystal in higher dimension → projection to lower dimension can give a crystal or a quasicrystal

2D  1D

ow E ind E  W E || ||  

To get RA  e2 approximations are made  e1 in E (i.e to )

Irrational  QC

Slope = Tan () x ' Cos Sin  x R   Rational  RA (XAL) ySinCosy'    1D 1-D QC  B A B B A B A B B A B B A  

2D

Penrose Tiling Octogonal Tiling 2 11      1011 03  3 3-3   22    0111 R2 11    22  R       11 10 03 -3 33   22     11 01 22 2 2 2  22  3D ICOSAHEDRAL QUASILATTICE

. 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.

 10 1 0    10 0 1  0101    R    10  10 10 10   5-fold [1  0]    0101

Note the occurrence of irrational Miller indices 3-fold [2+1  0]

2-fold [+1  1]

Cluster Based Construction

(a) (b)

Rhombic Triacontahedron Bergman cluster Mackay double icosahedron

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

(a) Bergman, G., Waugh, J. L. T., and Pauling, L., Acta Cryst., 10 (1957) 2454 Hiraga, K et al, S., Phil. Mag. B67 (1993) 193 (b) Ranganathan, S., and Chattopadhyay, K., Annu. Rev. Mater. Sci., 21 (1991) 437

Comparison of a crystal with a quasicrystal

CRYSTAL QUASICRYSTAL Translational 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 Sharp peaks in reciprocal space with translational symmetry inflationary symmetry Underlying metric is a rational number Irrational metric

Usually made of ‘small’ clusters Large clusters

SYSTEMSSYSTEMS FORMINGFORMING QUASICRYSTALSQUASICRYSTALS && TYPESTYPES OFOF QUASICRYSTALSQUASICRYSTALS List of quasicrystals with diverse kinds of symmetries

Type QP+ Rank Metric Symmetry System Reference

Icosahedral 3 D 6 _ _ AlMn Shechtman et al., 1984  (5) m35

Cubic 3D 6 3 _ VNiSi Feng et al., 1989 43m

Tetrahedral 3D 6 3 _ AlLiCu Donnadieu, 1994 m3

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, Wang et al., 1987

CrNiSi

Pentagonal 2D 5  (5) _ AlCuFe Bancel, 1993 5m

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

Trigonal 1D 4 3 _ AlCuNi Chattopadhyay et al., 1987 3m

Digonal 1D 4 2 222 AlCuCo He et al., 1988 Naturally Occurring QC

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

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

Conference in Honour of Prof. T.R. Anantharaman

IITK

S. Lele

C. Suryanarayana G.V.S SASTRY

S. Ranganathan

http://www.iitk.ac.in/infocell/announce/metallo/collection.htm Allowed crystallographic symmetry- tiled aperiodically

Discovery of the decagonal phase

Basis for synthesis of QC 1-D quasiperiodicity  = 1  = 2 Icosahedral Quasicrystal  = 3 Decagonal Hexagonal Quasicrystal Quasicrystal  = 1 Digonal Pentagonal Cubic R.A.S. Trigonal Hexagonal Quasicrystal Quasicrystal Mackay Bergman Quasicrystal R.A.S.

Orthorhombic Orthorhombic Trigonal Orthorhombic R.A.S. R.A.S R.A.S. R.A.S. Taylor Little Robinson Monoclinic R.A.S. Monoclinic R.A.S. Monoclinic R.A.S.   120o Unified view of quasicrystals, R.A.S.  = 90o rational approximants and  = 108o related structures

Trigonal and Pentagonal quasilattices Fundamental work on Vacancy Ordered Phases xx32210 x

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

Approximant to 7- Uniform deformation along the arrow of the [0 0 1] 2-fold pattern fold quasilattice from IQC giving rise to a pattern similar to the [ 1 3+ ] pattern