FASCINATING QUASICRYSTALS

1
FASCINATING QUASICRYSTALS
Based on atomic order quasicrystals are one of
the 3 fundamental phases of matter
2
Where are quasicrystals in the scheme of things?
UNIVERSE
STRONG WEAK ELECTROMAGNETIC GRAVITY
HYPERBOLIC EUCLIDEAN SPHERICAL
ENERGY
SPACE
nD t
FIELDS
PARTICLES
METAL SEMI-METAL SEMI-CONDUCTOR INSULATOR
NON-ATOMIC
ATOMIC
BAND STRUCTURE
STATE / VISCOSITY
LIQUID CRYSTALS
GAS
SOLID
LIQUID
STRUCTURE
CRYSTALS
RATIONAL APPROXIMANTS
AMORPHOUS
QUASICRYSTALS
SIZE
NANO-QUASICRYSTALS
NANOCRYSTALS
3
Let us first revise what is a crystal before
defining a quasicrystal
WHAT IS A CRYSTAL?
Crystal Lattice (Where to repeat) Motif
(What to repeat)



4
Crystals have certain symmetries
Symmetry operators
t ? Translation
R ? Inversion
R ? Mirror
R ? Rotation
?R ? Roto-inversion
G ? Glide reflection
S ? Screw axis
? Takes object to the same form
? Takes object to the enantiomorphic form
5
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
PyriteCube
Fluorite Octahedron
Rüdiger Appel, http//www.3quarks.com/GIF-Animatio
ns/PlatonicSolids/
6
HOW IS A QUASICRYSTAL DIFFERENT FROM A CRYSTAL?
7
FOUND! THE MISSING PLATONIC SOLID
2
Dodecahedral single crystal
Mg-Zn-Ho
1
1 I.R. Fisher et al., Phil Mag B 77 (1998)
1601 2 Rüdiger Appel, http//www.3quarks.com/GIF
-Animations/PlatonicSolids/
8
QUASICRYSTALS (QC)
ORDERED PERIODIC QC ARE ORDERED STRUCTURES WHICH ARE NOT PERIODIC
CRYSTALS ? ? QC ARE ORDERED STRUCTURES WHICH ARE NOT PERIODIC
QC ? ? QC ARE ORDERED STRUCTURES WHICH ARE NOT PERIODIC
AMORPHOUS ? ? QC ARE ORDERED STRUCTURES WHICH ARE NOT PERIODIC
9
SYMMETRY
CRYSTAL QUASICRYSTAL
t ? ?
RC ? RCQ
t ? translation
? ? inflation
RC ? rotation crystallographic
RCQ ? RC other
QC are characterized by Inflationary Symmetry and
can have disallowed crystallographic symmetries
2, 3, 4, 6
5, 8, 10, 12
Quasicrystals can have allowed and disallowed
crystallographic symmetries
10
DIMENSION OF QUASIPERIODICITY (QP)
QC can have quasiperiodicity along 1,2 or 3 dimensions(at least one dimension should be quasiperiodic)
HIGHER DIMENSIONS
QP
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
11
HOW TO CONSTRUCT A QUASICRYSTAL?

• QUASILATTICE MOTIF(Construction of a
  quasilattice followed by the decoration of the
  lattice by a motif)
• ? 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)

12
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
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
13
Penrose tiling
Construction of a 1D Quasilattice
Deflated sequence ?
A
B
a
B
A
b
B
A
B
Rational Approximants
ba
B
A
B
B
A
B
A
B
B
A
B
A
B
bab
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 ABor BAB BAB
BAB)
B
A
B
B
A
B
A
B
B
A
B
B
A
babba
Note the deflated sequence is identical to the
original sequence
1-D QC
In the limit we obtain the 1D quasilattice
2D analogue of the 1D quasilattice
Schematic diagram showing the structural analogue
of the Fibonacci sequence leading to a 1-D QC
14
PENROSE TILING
The inflated tiles can be used to create an
inflated replica of the original tiling
? Inflated 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
A 2D Quasilattice
15
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.

5-fold 1 ? 0
Note the occurrence of irrational Miller indices
3-fold 2?1 ? 0
A 3D Quasilattice seen in perspective
2-fold ?1 ? 1
16
HOW IS A DIFFRACTION PATTERN FROM A CRYSTAL
DIFFERENT FROM THAT OF A QUASICRYSTAL?
17
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
111
011
Superlattice spots
SAD patterns from a BCC phase (a 10.7 Å) in
as-cast Mg4Zn94Y2 alloy showing important zones
18
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
? 1 ?3 ?
0 0 1
SAD patterns from as-cast Mg23Zn68Y9 showing the
formation of Face Centred Icosahedral QC
19
DIFFRACTION PATTERN
5-fold SAD pattern from as-cast Mg23Zn68Y9 alloy
Note the 10-fold pattern
Successive spots are at a distance inflated by ?
Inflationary symmetry
20
THE PROJECTION METHOD TO CREATE QUASILATTICES
21
HIGHER DIMENSIONS ARE NEAT
E2
GAPS
S2 ? E3
REGULAR PENTAGONS
Regular pentagons cannot tile E2 space but can
tile S2 space (which is embedded in E3 space)
SPACE FILLING
22
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
23
PROJECTION METHOD
QC considered a crystal in higher dimension ?
projection to lower dimension can give a crystal
or a quasicrystal
Additional basis vectors needed to index the
diffraction pattern
2D ? 1D










Window
E?
E
E
?
?
e2
In the work presentedapproximations are madein
E? (i.e to ?)
e1
?
Slope Tan (?) Irrational ? QC
Slope Tan (?)
Slope Tan (?) Rational ? RA (XAL)
24
B
A
B
B
A
B
A
B
B
A
B
B
A
1-D QC
25
List of quasicrystals with diverse kinds of
symmetries
26
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
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
27
APPLICATIONS OF QUASICRYSTALS

• WEAR RESISTANT COATING (Al-Cu-Fe-(Cr))
• NON-STICK COATING (Al-Cu-Fe)
• THERMAL BARRIER COATING (Al-Co-Fe-Cr)
• HIGH THERMOPOWER (Al-Pd-Mn)
• IN POLYMER MATRIX COMPOSITES (Al-Cu-Fe)
• SELECTIVE SOLAR ABSORBERS (Al-Cu-Fe-(Cr))
• HYDROGEN STORAGE (Ti-Zr-Ni)

28
High-resolution micrograph
SAD pattern
BFI
As-cast Mg37Zn38Y25 alloy showing a 18 R
modulated phase
29
Rational Approximants

• 3.14 (314/100) is a rational approximant to ?.
  Similarly, 1.414 is a rational approximant to ?2
  (to 3 decimal places).
• Rational approximants are structural analogues of
  such number approximants.
• Rational approximants locally resemble the
  quasicrystal (to which it is an approximant), but
  it is a crystal based on long range order. The
  higher the order of the approximant, the longer
  the patch in the structure which looks like a
  quasicrystal.

As mentioned before if we two length segments (A
and B) such that Length of A ?2? (length of
B) and make the structure as below (In the ?
limit we get the quasicrystal_.
B
A
B
B
A
B
A
B
B
A
B
B
A
1-D QC
However if a we take a part of the same and
repeat it to make a crystal, it is a rational
approximant
B
A
B
Repeat
If we take a bigger patch, it will resemble a QC
more (locally)
30
Rational Approximants

• A nice example of a rational approximant amongst
  crystal structures is the Al12W Frank-Kasper
  phase.
• This phase locally has icosahedra (as in figure
  below), but is overall a crystal having a BCC
  lattice. This crystal structure can be considered
  as a rational approximant to the icosahedral
  quasicrystal.

Motif 12Al W (consistent with stoichiometry)
Lattice Body Centred Cubic
Al12W (Frank-Kasper)
Lattice parameter(s) a 7.58 Å
Space Group Im?3 (204)
Strukturbericht notation
Pearson symbol cI26
Other examples with this structure Al12Mn, Al12Mo
Wyckoff position SiteSymmetry x y z Occupancy
Al 24g m 0 0.184 0.309 1
W 2a m-3 0 0 0 1