The Nobel Prize in Chemistry 2011 Dan Shechtman Technion – Israel Institute of Technology, Haifa, Israel Prize motivation: "for the discovery of quasicrystals"

Matter Solid LiquidGas Plasma Liquid Crystal Amorphous Crystalline QUASICRYSTALLINE 1984

L+  T Stable liquid Under Cooled liquid log t TmTm TTT Diagram for liquid-to-solid transformation Coarse grained crystals Fine grained crystals glass

Crystal Amorphous SiO 2

Crystal 3D Periodic arrangement of atoms Amorphous Random arrangement of atoms Long-range translational order Short-range order No Long-range order Short-range order Sharp diffraction pattern Diffuse diffraction pattern

Sharp Crystalline Amorphous Diffuse Diffraction Patterns

Electron Diffraction and symmetry Beam :

9/87 7 crystal Systems Cubic Defining Crystal system Conventional symmunit cell none Tetragonal Orthorhombic Hexagonal Rhombohedral Triclinic Monoclinic a=b=c,  =  =  =90  a=b  c,  =  =  =90  a  b  c,  =  =  =90  a=b  c,  =  = 90 ,  =120  a=b=c,  =  =  90  a  b  c,  =  =90  a  b  c, 

Rotational Symmetries Z 180  120  90  72  60   8 Angles: Fold: Graphic symbols

Crsytallographic Restriction 5-fold symmetry or Pentagonal symmetry is not possible for Periodic Tilings Symmetries higher than 6-fold also not possible Only possible rotational symmetries for periodic tilings …

A crystal with 10-Fold symmetry???

Five-fold Two-fold Five-fold Two-fold Three-fold Icosahedral Symmetry Icosahedron

Regular Polygons: All sides equal all angles equal How many regular polygons are possible? There are infinitely many regular polygons Triangle square pentagon hexagon…

3D: Regular Polyhedra or Platonic Solids Cube How many regular solids? Tetrahedron All faces regular congruent polygons, all corners identical.

There are 5 and only 5 Platonic or regular solids ! Icosahedron Octahedron Tetrahedron Cube Dodecahedron

What is the structure of Quasicrystal?

One Dimensional Quasicrystal Fibbonacci Chain

Two-dimensional Quasicrystal Penrose Pattern

Hexagons always tile periodically Square can tile periodically or aperiodically. Is there a tile or a set of tile that will tile only aperiodically?

Thank you