THE FASCINATING WORLD OF THE FASCINATING WORLD OF QUASICRYSTALS QUASICRYSTALS Anandh Subramaniam Materials Science and Engineering INDIAN INSTITUTE OF TECHNOLOGY KANPUR INDIAN INSTITUTE OF TECHNOLOGY KANPUR Kanpur- 208016 Email: [email protected]http://home.iitk.ac.in/~anandh Oct 2011
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THE FASCINATING WORLD OF THE FASCINATING WORLD OF QUASICRYSTALSQUASICRYSTALS
Anandh SubramaniamMaterials Science and Engineering
INDIAN INSTITUTE OF TECHNOLOGY KANPURINDIAN INSTITUTE OF TECHNOLOGY KANPURKanpur-
"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)
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