Chapter 3 Crystal Geometry and Structure Determination

1. “There are two things to aim at in life: first, to get what you want; and, after that, to enjoy it. Only the wisest of mankind achieve the second.”Logan Pearsall Smith, Afterthought (1931), “Life and Human Nature”

2. Chapter 3Crystal Geometry and Structure Determination

3. Contents Crystal Crystal, Lattice and Motif Miller Indices Symmetry Crystal systems Bravais lattices Structure Determination

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

5. Cubic Crystals? a=b=c; ===90

6. 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

7. UNIT CELL: Unit cell description : 2 The most common shape of a unit cell is a parallelopiped.

8. Unit cell description : 3 The description of a unit cell requires: 1. Its Size and shape (lattice parameters) 2. Its atomic content (fractional coordinates)

9. a  c   b Unit cell description : 4 Size and shape of the unit cell: 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

10. 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, 

11. Lattice? A 3D translationally periodic arrangement of points in space is called a lattice.

12. Crystal Lattice A 3D translationally periodic arrangement of atoms A 3D translationally periodic arrangement of points

13. What is the relation between the two? Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated with each lattice point

14. Crystal=lattice+basis Lattice: the underlying periodicity of the crystal, Basis: atom or group of atoms associated with each lattice points Lattice: how to repeat Motif: what to repeat

15. Lattice A 3D translationally periodic arrangement of points Each lattice point in a lattice has identical neighbourhood of other lattice points.

16. Lattice + Motif = Crystal = + Love Lattice + Heart = Love Pattern

17. Air, Water and Earth by M.C. Esher

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

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

21. Classification of lattice The Seven Crystal System And The Fourteen Bravais Lattices

22. TABLE 3.1 7 Crystal Systems and 14 Bravais Lattices • 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

23. 14 Bravais lattices divided into seven crystal systems • 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. ? 14 Bravais lattices divided into seven crystal systems • 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

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

27. 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

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

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. ML Frankenheim Auguste Bravais 1801-1869 1811-1863 1835: 15 lattices 1850: 14 lattices Couldn’t find his photo on the net History: AML120IIT-D X 1856: 14 lattices 2012 Civil Engineers: 13 lattices !!!

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

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

33. UNIT CELLS OF A LATTICE If the lattice points are only at the corners, the unit cell is primitive otherwise non-primitive Non-primitive cell Primitivecell A unit cell of a lattice is NOT unique. Primitivecell Unit cell shape CANNOT be the basis for classification of Lattices

34. Lattices are classified on the basis of their symmetry

35. What is symmetry?

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

37. NOW NO SWIMS ON MON

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

39. Crsytallographic Restriction 5-fold symmetry or Pentagonal symmetry is not possible for crystals Symmetries higher than 6-fold also not possible Only possible rotational symmetries for periodic tilings and crystals: 2 3 4 5 6 7 8 9…

40. Reflection (or mirror symmetry)

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

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

43. classification of lattices Based on the rotational and reflection symmetry alone (excluding translations)  7 types of lattices  7 crystal systems Based on the complete symmetry, i.e., rotational, reflection and translational symmetry  14 types of lattices  14 Bravais lattices

44. 7 crystal Systems Defining Crystal system Conventionalsymmetryunit cell a=b=c, ===90 4 Cubic a=bc,===90 1 Tetragonal abc, ===90 3 Orthorhombic a=bc, == 90, =120 Hexagonal 1 a=b=c, ==90 1 Rhombohedral abc, ==90 1 Monoclinic abc,  none Triclinic

45. Tetragonal symmetry Cubic symmetry Cubic C = Tetragonal P Cubic F  Tetragonal I

46. The three Bravais lattices in the cubic crystal system have the same rotational symmetry but different translational symmetry. Simple cubicPrimitive cubicCubic P Body-centred cubicCubic I Face-centred cubicCubic F

47. Richard P. Feynman Nobel Prize in Physics, 1965

48. 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.”

49. Correction: Shift the box One suggested correction: But gives H:O = 1.5 : 1 http://www.youtube.com/watch?v=kUuDG6VJYgA

50. The errata has been accepted by Michael Gottlieb of Caltech and the corrections will appear in future editions Website www.feynmanlectures.info