Most Important Element in life

1. Most Important Element in life Most important Element in Engineering C Most important element in Nanotechnology

2. Allotropes of C Graphite Diamond Buckminster Fullerene1985 Graphene2004 Carbon Nanotubes1991

3. Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Classification of Lattices:7 crystal systems 14 Bravais lattices Miller Indices & Miller-Bravais IndicesDirections and Planes Reciprocal lattice

4. Crystal, Lattice and Motif Crystal ? A 3D translationaly periodic arrangement of atoms in space is called a crystal.

5. Crystal, Lattice and Motif Lattice? A 3D translationally periodic arrangement of points in space is called a lattice.

6. Crystal, Lattice and Motif Crystal Lattice A 3D translationally periodic arrangement of atoms A 3D translationally periodic arrangement of points

7. Crystal, Lattice and Motif Motif? Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated with each lattice point

8. “Nothing that is worth knowing can be taught.” Oscar Wilde

9. Lattice + Motif = Crystal = + Love Lattice + Heart (Motif) = Love Pattern(Crystal) Love Pattern

10. Maurits Cornelis Escher 1898-1972 Dutch Graphic Artist Air, Water and Earth

11. Every periodic pattern (and hence a crystal) has a unique lattice associated with it

13. Crystal, Lattice and Motif Crystal Cu Crystal NaCl Crystal FCC FCC Lattice Motif 1 Cu+ ion 1 Na+ ion + 1 Cl- ion

14. Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Classification of Lattices:7 crystal systems 14 Bravais lattices Miller Indices & Miller-Bravais IndicesDirections and Planes

15. Unit cell description : 1 Translational Periodicity One can select a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps) Unit Cell

16. UNIT CELL: The most common shape of a unit cell is a parallelopiped with lattice points at corners. Primitive Unit Cell: Lattice Points only at corners Non-Primitive Unit cell: Lattice Point at corners as well as other some points

17. a  c   b Unit cell description : 4 Lattice Parameters: 1. A corner as origin 2. Three edge vectors {a, b, c} from the origin define a CRSYTALLOGRAPHIC COORDINATE SYSTEM 3. The three lengths a, b, c and the three interaxial angles , ,  are called the LATTICE PARAMETERS

18. Wigner-Seitz Unit Cells FCC Rhombic Dodcahedron BCC Tetrakaidecahedron

19. The six lattice parameters a, b, c, , ,  The cell of the lattice lattice + Motif crystal

20. Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Classification of Lattices:7 crystal systems 14 Bravais lattices Miller Indices & Miller-Bravais IndicesDirections and Planes

21. Unit cell description : 5 7 crystal Systems CrystalSystem Conventional Unit Cell 1. Cubic a=b=c, ===90 2. Tetragonal a=bc,===90 3. Orthorhombic abc, ===90 4. Hexagonal a=bc, == 90, =120 5. Rhombohedral a=b=c, ==90 OR Trigonal 6. Monoclinic abc, ==90 7. Triclinic abc, 

22. Crystal System Bravais Lattices • Cubic P I F • Tetragonal P I • Orthorhombic P I F C • Hexagonal P • Trigonal P • Monoclinic P C • Triclinic P ? P: Simple; I: body-centred;F: Face-centred; C: End-centred Why so many empty boxes? E.g. Why cubic C is absent?

23. The three cubic Bravais lattices • Crystal system Bravais lattices • Cubic P I F Simple cubicPrimitive cubicCubic P Body-centred cubicCubic I Face-centred cubicCubic F

24. Orthorhombic CEnd-centred orthorhombicBase-centred orthorhombic

25. Monatomic Body-Centred Cubic (BCC) crystal CsCl crystal Cl Cs Corner and body-centres have the same neighbourhood Corner and body-centred atoms do not have the same neighbourhood Lattice: simple cubic Lattice: bcc Feynman! BCC Motif: 1 atom 000 Motif: two atoms Cl 000; Cs ½ ½ ½

26. Example: Hexagonal close-packed (HCP) crystal z ½ ½ y ½ ½ Corner and inside atoms do not have the same neighbourhood x Lattice: Simple hexagonal Motif: Two atoms: 000; 2/3 1/3 1/2 hcp lattice hcp crystal

27. Crystal System Bravais Lattices • Cubic P I F • Tetragonal P I • Orthorhombic P I F C • Hexagonal P • Trigonal P • Monoclinic P C • Triclinic P ? P: Simple; I: body-centred;F: Face-centred; C: End-centred Why so many empty boxes? E.g. Why cubic C is absent?

28. End-centred cubic not in the Bravais list ? End-centred cubic = Simple Tetragonal

29. 14 Bravais lattices divided into seven crystal systems • Crystal system Bravais lattices • Cubic P I F C • Tetragonal P I • Orthorhombic P I F C • Hexagonal P • Trigonal P • Monoclinic P C • Triclinic P

30. Face-centred cubic in the Bravais list ? Cubic F = Tetragonal I ?!!!

31. 14 Bravais lattices divided into seven crystal systems • Crystal system Bravais lattices • Cubic P I F C • Tetragonal P I • Orthorhombic P I F C • Hexagonal P • Trigonal P • Monoclinic P C • Triclinic P

32. ML Frankenheim Auguste Bravais 1801-1869 1811-1863 1835: 15 lattices 1850: 14 lattices Couldn’t find his photo on the net History: AML750 IIT-D 06 Aug 2009: 13 lattices !!! X 1856: 14 lattices

33. Why can’t the Face-Centred Cubic lattice (Cubic F) be considered as a Body-Centred Tetragonal lattice (Tetragonal I) ?

34. UNIT CELLS OF A LATTICE Non-primitive cell A unit cell of a lattice is NOT unique. Primitivecell Unit cell shape CANNOT be the basis for classification of Lattices Primitivecell

35. What is the basis for classification of lattices into 7 crystal systemsand 14 Bravais lattices?

36. Lattices are classified on the basis of their symmetry

37. What is symmetry?

38. Symmetry If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.

39. NOW NO SWIMS ON MON

40. Translational symmetry Lattices also have translational symmetry In fact this is the defining symmetry of a lattice

41. Rotation Axis If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where =180 2-fold rotation axis n=2 n=4 4-fold rotation axis =90

42. Rotational Symmetries Z Angles: 180 120 90 72 60 45 Fold: 6 2 3 4 5 8 Graphic symbols

43. 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 2 3 4 5 6 7 8 9…

44. Proof of The Crystallographic Restriction A rotation can be represented by a matrix If T is a rotational symmetry of a lattice then all its elements must be integers (wrt primitive basis vectors)

45. Feynman’s Lectures on Physics Vol 1 Chap 1 Fig. 1-4 Hexagonal symmetry “Fig. 1-4 is an invented arrangement for ice, and although it contains many of the correct features of the ice, it is not the true arrangement. One of the correct features is that there is a part of the symmetry that is hexagonal. You can see that if we turn the picture around an axis by 120°, the picture returns to itself.”

46. Correction: Shift the box Michael Gottlieb’s correction: But gives H:O = 1.5 : 1

47. QUASICRYSTALS (1984) Icosahedral symmetry (5-fold symmetry) Lack strict translational periodicity -> Quasiperiodic Icosahedron Penrose Tiling Diffraction Pattern External Morphology

48. Reflection (or mirror symmetry)

49. Symmetry of lattices Lattices have Translational symmetry Rotational symmetry Reflection symmetry

50. Point Group and Space Group The group of all symmetry elements of a crystal except translations (e.g. rotation, reflection etc.) is called its POINT GROUP. The complete group of all symmetry elements including translations of a crystal is called its SPACE GROUP