FASCINATING QUASICRYSTALS Based on atomic order quasicrystals are one of the 3 fundamental phases of...
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FASCINATING QUASICRYSTALS FASCINATING QUASICRYSTALS n atomic order quasicrystals are one of the 3 fundamental phases of MATERIALS SCIENCE MATERIALS SCIENCE & ENGINEERING ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: [email protected], URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide A Learner’s Guide
FASCINATING QUASICRYSTALS Based on atomic order quasicrystals are one of the 3 fundamental phases of matter MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam
FASCINATING QUASICRYSTALS Based on atomic order quasicrystals
are one of the 3 fundamental phases of matter MATERIALS SCIENCE
&ENGINEERING Anandh Subramaniam & Kantesh Balani Materials
Science and Engineering (MSE) Indian Institute of Technology,
Kanpur- 208016 Email: [email protected], URL:
home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of
http://home.iitk.ac.in/~anandh/E-book.htm A Learners Guide
Slide 2
UNIVERSE PARTICLES ENERGY SPACE FIELDS STRONG WEAK
ELECTROMAGNETIC GRAVITY METAL SEMI-METAL SEMI-CONDUCTOR INSULATOR
nD + t HYPERBOLIC EUCLIDEAN SPHERICAL GAS BAND STRUCTURE AMORPHOUS
ATOMIC NON-ATOMIC STATE / VISCOSITY SOLIDLIQUID LIQUID CRYSTALS
QUASICRYSTALS CRYSTALS RATIONAL APPROXIMANTS STRUCTURE
NANO-QUASICRYSTALSNANOCRYSTALS SIZE Where are quasicrystals in the
scheme of things?
Slide 3
Crystal = Lattice (Where to repeat) + Motif (What to repeat) =
+ aa WHAT IS A CRYSTAL? Let us first revise what is a crystal
before defining a quasicrystal
Slide 4
R Rotation G Glide reflection Symmetry operators R
Roto-inversion S Screw axis t Translation R InversionR Mirror Takes
object to the same form Takes object to the enantiomorphic form
Crystals have certain symmetries
Slide 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 Fluorite Octahedron
Pyrite Cube Rdiger 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
Slide 6
HOW IS A QUASICRYSTAL DIFFERENT FROM A CRYSTAL?
Slide 7
FOUND! FOUND! THE MISSING PLATONIC SOLID [1] I.R. Fisher et
al., Phil Mag B 77 (1998) 1601 Rdiger Appel [2] Rdiger Appel,
http://www.3quarks.com/GIF-Animations/PlatonicSolids/ Mg-Zn-Ho [1]
[2] Dodecahedral single crystal
Slide 8
QUASICRYSTALS (QC) ORDEREDPERIODIC QC ARE ORDERED STRUCTURES
WHICH ARE NOT PERIODIC CRYSTALS QC AMORPHOUS
Slide 9
SYMMETRY CRYSTALQUASICRYSTAL t RCRC R CQ QC are characterized
by Inflationary Symmetry and can have disallowed crystallographic
symmetries* t translation inflation RCRC rotation crystallographic
R CQ R C + other 2, 3, 4, 6 5, 8, 10, 12 * Quasicrystals can have
allowed and disallowed crystallographic symmetries
Slide 10
QC can have quasiperiodicity along 1,2 or 3 dimensions (at
least one dimension should be quasiperiodic) DIMENSION OF
QUASIPERIODICITY (QP) HIGHER DIMENSIONS QP QP/P QPXAL 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
Slide 11
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) HOW TO CONSTRUCT A QUASICRYSTAL?
Slide 12
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 21/1334/21... = ( 1+ 5)/2 Where is the
root of the quadratic equation: x 2 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
Slide 13
Schematic diagram showing the structural analogue of the
Fibonacci sequence leading to a 1-D QC A B BA BAB BABBA BABBABAB
BABBABABBABBA 1-D QC a b ba bab babba Deflated sequence Penrose
tiling Rational Approximants 2D analogue of the 1D quasilattice
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 ABor BAB BAB BAB)
Slide 14
PENROSE TILING Inflated tiling The inflated tiles can be used
to create an inflated replica of the original 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)
Slide 15
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.
Slide 16
HOW IS A DIFFRACTION PATTERN FROM A CRYSTAL DIFFERENT FROM THAT
OF A QUASICRYSTAL?
Slide 17
SAD patterns from a BCC phase (a = 10.7 ) in as-cast Mg 4 Zn 94
Y 2 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
Slide 18
SAD patterns from as-cast Mg 23 Zn 68 Y 9 showing the formation
of Face Centred Icosahedral QC [1 0] [1 1 1] [0 0 1] [ 1 3 + ] The
spots show inflationary symmetry Explained in the next slide Now
let us look at the SAD pattern from a quasicrystal from the same
alloy system (Mg-Zn-Y)
Slide 19
22 33 44 1 DIFFRACTION PATTERN 5-fold SAD pattern from as-cast
Mg 23 Zn 68 Y 9 alloy Successive spots are at a distance inflated
by Note the 10-fold pattern Inflationary symmetry
Slide 20
THE PROJECTION METHOD TO CREATE QUASILATTICES
Slide 21
HIGHER DIMENSIONS ARE NEAT E2 REGULAR PENTAGONS GAPS S2 E3
SPACE FILLING Regular pentagons cannot tile E2 space but can tile
S2 space (which is embedded in E3 space)
Slide 22
For this SAD pattern we 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 We require more than two basis vectors to index the
diffraction pattern in 2D
Slide 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
Slope = Tan ( ) Irrational QC Rational RA (XAL) E || EE Window e1e1
e2e2 2D 1D E || In the work presented approximations are made in E
(i.e to )
Slide 24
BABBABABBABBA 1-D QC
Slide 25
List of quasicrystals with diverse kinds of symmetries
Slide 26
CRYSTALQUASICRYSTAL Translational symmetryInflationary symmetry
Crystallographic rotational symmetriesAllowed + some disallowed
rotational symmetries Single unit cell to generate the structureTwo
prototiles are required to generate the structure 3D
periodicPeriodic in higher dimensions Sharp peaks in reciprocal
space with translational symmetry Sharp peaks in reciprocal space
with inflationary symmetry Underlying metric is a rational
numberIrrational metric Comparison of a crystal with a
quasicrystal
Slide 27
WEAR RESISTANT COATING (Al-Cu-Fe-(Cr)) 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) APPLICATIONS OF QUASICRYSTALS
Slide 28
As-cast Mg 37 Zn 38 Y 25 alloy showing a 18 R modulated phase
SAD pattern BFI High-resolution micrograph