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 URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of A Learner’s Guide

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?

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

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

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

HOW IS A QUASICRYSTAL DIFFERENT FROM A CRYSTAL?

FOUND! FOUND! THE MISSING PLATONIC SOLID [1] I.R. Fisher et al., Phil Mag B 77 (1998) 1601 Rüdiger Appel [2] Rüdiger Appel, Mg-Zn-Ho [1] [2] Dodecahedral single crystal

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

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

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

 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?

THE FIBONACCI SEQUENCE Fibonacci   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

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 AB…or BAB BAB BAB…)

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)

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.

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 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

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)

 22 33 44 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

THE PROJECTION METHOD TO CREATE QUASILATTICES

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)

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

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 || EE Window e1e1 e2e2 2D  1D    E || In the work presented approximations are made in E  (i.e to  )

BABBABABBABBA 1-D QC

List of quasicrystals with diverse kinds of symmetries

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

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

As-cast Mg 37 Zn 38 Y 25 alloy showing a 18 R modulated phase SAD pattern BFI High-resolution micrograph