THE FASCINATING WORLD OF QUASICRYSTALS - iitk.ac.in ...

THE FASCINATING WORLD OF
QUASICRYSTALS
Anandh Subramaniam
Materials Science and Engineering
INDIAN INSTITUTE OF TECHNOLOGY KANPUR
Kanpur- 208016
Email: anandh@iitk.ac.in
http://home.iitk.ac.in/~anandh
Oct 2011

Daniel Shechtman

7 April 1982
8 April 1982
GLASS
(AMORPHOUS)
GLASS
(AMORPHOUS)
SOLIDS
Based on Structure
SOLIDS
Based on Structure
QUASI CRYSTALS
CRYSTALS
CRYSTALS

A leaf from a diary…
Daniel Shechtman
Born: January 24, 1941
Enter the Decagon!
7 April 1982
8 April 1982
12 Nov 1984

Painting by Dr. Alok Singh, 1993

“If you are a scientist and believe in in your results, then fight for the
truth”. “Listen to to others, but fight for what you believe in…”
-
DAN SHECHTMAN
"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
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!)
Element 117 (with 177 neutrons) has a half life of 78 ms
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: THE PRESAGES!

Gunbad-i
Kabud
tomb in Maragha, Iran, 1197 AD
Darb-I Imam shrine, Isfaha, Iran, 1453 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
5-fold because the SAD pattern
has inversion symmetry)
The tiling has regions of
local 5-fold symmetry
R. Penrose, “Pentaplexity”, Eureka, 39, 16, 1978
M. Gardner, Sci. Am. 236 (1977) 110

A brief history of aperiodic
Berger, 1966 20,000
Robinson, 1971 6 tiles
Penrose 1
Penrose 2
, 1974 4 (6) tiles
, 1978 2
tiles
tilings
tiles (then to 104 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
Kepler
concluded that the pattern would never repeat-
Penrose’s Pattern
Kepler’s Pattern
there would always be “surprises”
had anticipated the concept of aperiodic tiling by 350 years!

Wonders of Numbers:
Adventures in Mathematics,
Mind and Meaning
Clifford A Pickover
A Circle has been placed on each quasi-lattice point
of the 2D pattern to model a possible atomic structure

WHAT IS A CRYSTAL?

Crystal =
Space group (how to repeat)
+
Asymmetric unit (Motif’: what to repeat)
+
Wyckoff positions
WHAT IS A CRYSTAL?
a
=
a
+
Glide reflection
operator
Symbol g may also be used
+Wyckoff label ‘a’
Positions entities
with respect to
symmetry operators
Usually asymmetric units are regions of space- which contain the entities (e.g. atoms, molecules)

Crystals have certain symmetries
t Translation
R Rotation
R Roto-inversion
G Glide reflection
S Screw axis
Symmetry operators
t
R Inversion R Mirror
Takes object to the same form
Takes object to the enantiomorphic form
m

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
These symmetries (rotation,
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/
Plato wrote about these solids in the dialogue Timaeus c.360 B.C.
Fluorite
Octahedron
http://en.wikipedia.org/wiki/Crystal_habit http://www.galleries.com/minerals/property/crystal.htm

HOW IS A QUASICRYSTAL DIFFERENT
FROM A CRYSTAL?

Dodecahedral
single
quasicrystal
[1] I.R. Fisher et al., Phil Mag B 77 (1998) 1601
[2] Rüdiger diger Appel, Appel http://www.3quarks.com/GIF-Animations/PlatonicSolids/
FOUND!
THE MISSING PLATONIC SOLID
m
35
[1]
Mg-Zn-Ho
Octahedron and icosahedron were discovered by Theaetetus, a contemporary of Plato
[2]

QUASICRYSTALS (QC)
ORDERED PERIODIC QC ARE
ORDERED
STRUCTURES
WHICH ARE
NOT
PERIODIC
CRYSTALS
QC
AMORPHOUS

t translation
inflation
SYMMETRY
CRYSTAL QUASICRYSTAL
RC rotation 2, crystallographic
3, 4, 6
RCQ RC + 5, other 8, 10, 12
t
R C R CQ
* Quasicrystals can have allowed and disallowed crystallographic symmetries
QC are characterized by Inflationary
Symmetry and can have disallowed
crystallographic symmetries*

DIMENSION OF QUASIPERIODICITY (QP)
QC as a crystal?
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
QC can have quasiperiodicity along 1,2 or 3
dimensions
(at least one dimension should be
quasiperiodic)
QP XAL
1 4
2 5
3 6

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
The ratio of
successive terms of
the Fibonacci
sequence converges to
the Golden Mean
In 1202 Fibonacci
discussed the
number sequence in
connection with the
proliferation of
rabbits
Where
Ratio
2.2
2
1.8
1.6
1.4
1.2
is the root of the quadratic equation: x 2
1
Convergence of Fibonacci Ratios
1 2 3 4 5 6 7 8 9 10
n
–x –1 = 0
* There are many phases of quasicrystals and some are associated with other sequences and other irrational numbers
1.618… x 1
x
1 x
2
x x
10

Rational
Approximants
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 limit we obtain the
1D Quasilattice
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
Deflated
sequence
Schematic diagram showing the structural analogue of the Fibonacci sequence
leading to a 1-D QC
a
b
ba
bab
babba
Note: the deflated sequence is
identical to the original sequence

Where is the Golden Mean?
In the ratio of lengths
In the ratio of numbers
n
n
B
A
L
L
A
B
1
1 1
1
1
1
1
1 1

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?

Let us look at the Selected Area Diffraction Pattern (SAD) from a crystal → the spots/peaks are arranged periodically
Superlattice spots
The spots are
periodically
arranged
[112]
[111] [011]
SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg 4 Zn 94 Y 2 alloy showing important zones

Now let us look at the SAD pattern from a quasicrystal from the same alloy system (Mg-Zn-Y)
[1
0]
[0 0 1]
The spots
show
inflationary
symmetry
Explained in
the next slide
SAD patterns from as-cast Mg 23 Zn 68 Y 9 showing the formation of Face Centred Icosahedral QC
[1 1 1]
[
1 3 + ]

DIFFRACTION PATTERN
5-fold SAD pattern
from as-cast
alloy
Mg23Zn68Y9 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)

E2
HIGHER DIMENSIONS ARE NEAT
REGULAR
PENTAGONS
Regular pentagons cannot tile E2 space but can tile
S2 space (which is embedded in E3 space)
GAPS
S2
E3
SPACE FILLING

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 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
E Window
Slope = Tan ()
e 2
Irrational QC
e 1
Rational RA (XAL)
E ||
E ||
To get RA
approximations are made
in E (i.e to )
x' Cos Sin x R
y' Sin Cosy

1D
2D
Penrose Tiling
2 1 1
0 3 3 3 - 3
R2
1 1
0 3- 3 3 3
2 2 2 2 2
1-D QC
B A B B A B A B B A B B A
Octogonal
Tiling
1 0 1 1
2 2
0 1 1
1
2 2
R
1 1 1 0
2 2
1 1
0 1
2 2

3D
5-fold
[1
0]
Note the occurrence of
irrational Miller indices
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.
3-fold
[2+1
0]
1 0 1
0
1 0 0 1
0 1 0 1
R
1 0 1
0
1 0 1 0
0 1 0 1
2-fold [+1 1]

Cluster Based Construction
Rhombic Triacontahedron
Bergman cluster
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
(b) Ranganathan, S., and Chattopadhyay, K., Annu. Rev. Mater. Sci., 21 (1991) 437
(a) (b)
Mackay double icosahedron
Hiraga, K et al, S., Phil. Mag. B67 (1993) 193

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

SYSTEMS FORMING QUASICRYSTALS
&
TYPES OF QUASICRYSTALS

List of quasicrystals with diverse kinds of symmetries
Type QP + Rank Metric Symmetry System Reference
Icosahedral 3 D 6 (5)
Cubic 3D 6 3
Tetrahedral 3D 6 3
m3 _
5 _
_
43m
m3 _
AlMn Shechtman et al., 1984
VNiSi
AlLiCu
Feng et al., 1989
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,
Pentagonal 2D 5 (5)
5m
_
CrNiSi
AlCuFe
Wang et al., 1987
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

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
S. Ranganathan
C. Suryanarayana
http://www.iitk.ac.in/infocell/announce/metallo/collection.htm
G.V.S SASTRY

Allowed crystallographic symmetrytiled aperiodically
Discovery of the decagonal phase
Basis for synthesis of QC

1-D quasiperiodicity

Approximant to 7fold
quasilattice
= 2
= 1
Icosahedral Quasicrystal = 3
Decagonal
Hexagonal
Quasicrystal
= 1
Quasicrystal
Digonal Pentagonal Cubic R.A.S. Trigonal Hexagonal
Quasicrystal Quasicrystal Mackay Bergman Quasicrystal R.A.S.
Orthorhombic
Orthorhombic
R.A.S.
R.A.S
Taylor
Robinson
Little
R.A.S. Monoclinic
Monoclinic
R.A.S.
R.A.S.
= 90 o
= 108 o
Trigonal and Pentagonal
quasilattices
3 2
x x 2x10 Trigonal
R.A.S.
Orthorhombic
R.A.S.
Monoclinic
R.A.S.
120 o
First observation of a
relation between five-fold
and hexagonal symmetry
Unified view of quasicrystals,
rational approximants and
related structures
Fundamental
work on
Vacancy
Ordered Phases
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