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FASCINATING QUASICRYSTALS

MATERIALS SCIENCE & ENGINEERING . Part of . A Learner’s Guide. AN INTRODUCTORY E-BOOK. Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh.

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FASCINATING QUASICRYSTALS

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  1. MATERIALS SCIENCE & ENGINEERING Part of A Learner’s Guide AN INTRODUCTORY E-BOOK Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email:anandh@iitk.ac.in, URL:home.iitk.ac.in/~anandh http://home.iitk.ac.in/~anandh/E-book.htm 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 QUASICRYSTALS AMORPHOUS SIZE NANO-QUASICRYSTALS NANOCRYSTALS

  3. a a 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-Animations/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)

  9. SYMMETRY 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. QP/P QP/P QP DIMENSION OF QUASIPERIODICITY (QP) HIGHER DIMENSIONS 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 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  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…) 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. 2 3 4 1 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

  24. 1-D QC

  25. List of quasicrystals with diverse kinds of symmetries

  26. Comparison of a crystal with a quasicrystal

  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

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