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 3 Crystal Geometry and Structure Determination

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

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

5. 5/87 Cubic Crystals? a=b=c;  =  =  =90 

6. 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 Unit cell description : 1

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

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

9. 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    a b c Unit cell description : 4

10. 7 crystal Systems Crystal SystemConventional Unit Cell 1. Cubica=b=c,  =  =  =90  2. Tetragonala=b  c,  =  =  =90  3. Orthorhombica  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,  Unit cell description : 5

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

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

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

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

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 crystal + Motif

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

22. 22/87 Crystal SystemBravais Lattices 1.CubicPIF 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP P: Simple; I: body-centred; F: Face-centred; C: End-centred 7 Crystal Systems and 14 Bravais Lattices TABLE 3.1

23. 14 Bravais lattices divided into seven crystal systems Crystal systemBravais lattices 1.CubicPIF Simple cubic Primitive cubic Cubic P Body-centred cubic Cubic I Face-centred cubic Cubic F

24. Orthorhombic C End-centred orthorhombic Base-centred orthorhombic

25. 14 Bravais lattices divided into seven crystal systems Crystal systemBravais lattices 1.CubicPIF 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP ?

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

27. 14 Bravais lattices divided into seven crystal systems Crystal systemBravais lattices 1.CubicPIFC 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP

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

29. 14 Bravais lattices divided into seven crystal systems Crystal systemBravais lattices 1.CubicPIFC 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP

30. Couldn’t find his photo on the net 1811-1863 Auguste Bravais 1850: 14 lattices 1835: 15 lattices ML Frankenheim 1801-1869 2012 Civil Engineers: 13 lattices !!! AML120 IIT-D X 1856: 14 lattices History:

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 systems and 14 Bravais lattices?

33. Primitive cell Non- primitive cell A unit cell of a lattice is NOT unique. If the lattice points are only at the corners, the unit cell is primitive otherwise non- primitive UNIT CELLS OF A LATTICE 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. If an object is brought into self- coincidence after some operation it said to possess symmetry with respect to that operation. Symmetry

37. NOW NO SWIMS ON MON

38. Rotational Symmetries Z 180  120  90  72  60  2345 6 45  8 Angles: Fold: 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: 23456789…

40. Reflection (or mirror symmetry)

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

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

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

44. 44/87 7 crystal Systems Cubic Defining Crystal system Conventional symmetry unit cell 4 1 3 1 1 1 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, 

45. 45/87 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 cubic Primitive cubic Cubic P Body-centred cubic Cubic I Face-centred cubic Cubic F

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

48. Feynman’s Lectures on Physics Vol 1 Chap 1 Fig. 1-4 “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.” Hexagonal symmetry

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

51. QUESTIONS?

52. Miller Indices of directions and planes William Hallowes Miller (1801 – 1880) University of Cambridge Miller Indices 1

53. 1. Choose a point on the direction as the origin. 2. Choose a coordinate system with axes parallel to the unit cell edges. x y3. Find the coordinates of another point on the direction in terms of a, b and c 4. Reduce the coordinates to smallest integers. 5. Put in square brackets Miller Indices of Directions [100] 1a+0b+0c z 1, 0, 0 Miller Indices 2

54. y z Miller indices of a direction represents only the orientation of the line corresponding to the direction and not its position or sense All parallel directions have the same Miller indices [100] x Miller Indices 3

55. x y z O A 1/2, 1/2, 1 [1 1 2] OA=1/2 a + 1/2 b + 1 c P Q x y z PQ = -1 a -1 b + 1 c -1, -1, 1 Miller Indices of Directions (contd.) [ 1 1 1 ] __ -ve steps are shown as bar over the number

56. Miller indices of a family of symmetry related directions [100] [001] [010] = [uvw] and all other directions related to [uvw] by the symmetry of the crystal = [100], [010], [001] = [100], [010] Cubic Tetragonal [010] [100] Miller Indices 4

57. 5. Enclose in parenthesis Miller Indices for planes 3. Take reciprocal 2. Find intercepts along axes in terms of respective lattice parameters 1. Select a crystallographic coordinate system with origin not on the plane 4. Convert to smallest integers in the same ratio 1 1 1 (111) x y z O

58. Miller Indices for planes (contd.) origin intercepts reciprocals Miller Indices A B C D O ABCD O 1 ∞ ∞ 1 0 0 (1 0 0) OCBE O* 1 -1 ∞ 1 -1 0 (1 1 0) _ Plane x z y O* x z E Zero represents that the plane is parallel to the corresponding axis Bar represents a negative intercept

59. Miller indices of a plane specifies only its orientation in space not its position All parallel planes have the same Miller Indices A B C D O x z y E (100) (h k l )  (h k l ) _ _ _ (100)  (100) _

60. Miller indices of a family of symmetry related planes = (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal {hkl } All the faces of the cube are equivalent to each other by symmetry Front & back faces: (100) Left and right faces: (010) Top and bottom faces: (001) {100} = (100), (010), (001)

61. {100} cubic = (100), (010), (001) {100} tetragonal = (100), (010) (001) Cubic Tetragonal Miller indices of a family of symmetry related planes x z y z x y

62. Some IMPORTANT Results Condition for a direction [uvw] to be parallel to a plane or lie in the plane (hkl): h u + k v + l w = 0 Weiss zone law True for ALL crystal systems Not in the textbook

63. CUBIC CRYSTALS [hkl]  (hkl) Angle between two directions [h 1 k 1 l 1 ] and [h 2 k 2 l 2 ]: C [111] (111)

64. d hkl Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell O x (100) B O x z E

65. [uvw]Miller indices of a direction (i.e. a set of parallel directions) (hkl)Miller Indices of a plane (i.e. a set of parallel planes) Miller indices of a family of symmetry related directions {hkl}Miller indices of a family of symmetry related planes Summary of Notation convention for Indices

66. In the fell clutch of circumstance I have not winced nor cried aloud. Under the bludgeonings of chance My head is bloody, but unbowed. From "Invictus" by William Ernest Henley (1849–1903).William Ernest Henley

67. Some crystal structures CrystalLatticeMotifLattice parameter CuFCCCu 000a=3.61 Å ZnSimple HexZn 000, Zn 1/3, 2/3, 1/2 a=2.66 c=4.95

68. Q1: How do we determine the crystal structure?

69. Incident BeamTransmitted Beam Diffracted Beam Sample Diffracted Beam X-Ray Diffraction

70. Incident Beam X-Ray Diffraction Transmitted Beam Diffracted Beam Sample Braggs Law (Part 1): For every diffracted beam there exists a set of crystal lattice planes such that the diffracted beam appears to be specularly reflected from this set of planes. ≡ Bragg Reflection

71. X-Ray Diffraction Braggs’ recipe for Nobel prize? Call the diffraction a reflection!!!

72. Braggs Law (Part 1): the diffracted beam appears to be specularly reflected from a set of crystal lattice planes. Specular reflection: Angle of incidence =Angle of reflection (both measured from the plane and not from the normal) The incident beam, the reflected beam and the plane normal lie in one plane X-Ray Diffraction i   plane r

73. X-Ray Diffraction i   r d hkl Bragg’s law (Part 2):

74. i  r Path Difference =PQ+QR P Q R   d hkl

75. Path Difference =PQ+QR i r P Q R   Constructive inteference Bragg’s law

76. Two equivalent ways of stating Bragg’s Law 1 st Form 2 nd Form

77. n th order reflection from (hkl) plane 1 st order reflection from (nh nk nl) plane  e.g. a 2 nd order reflection from (111) plane can be described as 1 st order reflection from (222) plane Two equivalent ways of stating Bragg’s Law

78. X-rays Characteristic Radiation, K  Target Mo Cu Co Fe Cr Wavelength, Å 0.71 1.54 1.79 1.94 2.29

79. Powder Method is fixed (K  radiation)  is variable – specimen consists of millions of powder particles – each being a crystallite and these are randomly oriented in space – amounting to the rotation of a crystal about all possible axes

80. 2121 Incident beam Transmitted beam Diffracted beam 1 Diffracted beam 2 X-ray detector Zero intensity Strong intensity sample Powder diffractometer geometry i  plane r   t 2121 2222 22 Intensity

81. X-ray tube detector Crystal monochromator X-ray powder diffractometer

82. The diffraction pattern of austenite Austenite = fcc Fe

83. x y z d 100 = a 100 reflection= rays reflected from adjacent (100) planes spaced at d 100 have a path difference /2 No 100 reflection for bcc Bcc crystal No bcc reflection for h+k+l=odd

84. Extinction Rules: Table 3.3 Bravais LatticeAllowed Reflections SCAll BCC(h + k + l) even FCCh, k and l unmixed DC h, k and l are all odd Or if all are even then (h + k + l) divisible by 4

85. Diffraction analysis of cubic crystals   2 sin  222 )lkh( constant Bragg’s Law: Cubic crystals (1) (2) (2) in (1) =>

86. h 2 + k 2 + l 2 SCFCCBCCDC 1100 2110 3111 4200 5210 6211 7 8220 9300, 221 10310 11311 12222 13320 14321 15 16400 17410, 322 18411, 330 19331

87. Crystal StructureAllowed ratios of Sin 2 (theta) SC1: 2: 3: 4: 5: 6: 8: 9… BCC1: 2: 3: 4: 5: 6: 7: 8… FCC3: 4: 8: 11: 12… DC3: 8: 11:16…

88.  19.0 22.5 33.0 39.0 41.5 49.5 56.5 59.0 69.5 84.0 sin 2  0.11 0.15 0.30 0.40 0.45 0.58 0.70 0.73 0.88 0.99 2 4 6 8 10 12 14 16 18 20 bcc h 2 +k 2 +l 2 1 2 3 4 5 6 8 9 10 11 sc h 2 +k 2 +l 2 3 4 8 11 12 16 19 20 24 27 fcc This is an fcc crystal Ananlysis of a cubic diffraction pattern p sin 2  1.0 1.4 2.8 3.8 4.1 5.4 6.6 6.9 8.3 9.3 p=9.43 p sin 2  2.8 4.0 8.1 10.8 12.0 15.8 19.0 20.1 23.9 27.0 p=27.3 p sin 2  2 2.8 5.6 7.4 8.3 10.9 13.1 13.6 16.6 18.7 p=18.87

89. a 4.05 4.02 4.02 4.04 4.02 4.04 4.03 4.04 4.01 4.03 hkl 111 200 220 311 222 400 331 420 422 511  19.0 22.5 33.0 39.0 41.5 49.5 56.5 59.0 69.5 84.0 h 2 +k 2 +l 2 3 4 8 11 12 16 19 20 24 27 Indexing of diffraction patterns The diffraction pattern is from an fcc crystal of lattice parameter 4.03 Å Ananlysis of a cubic diffraction pattern contd.   2 2 2 222 sin a4 )lkh(

90. Education is an admirable thing, but it is well to remember from time to time that nothing that is worth knowing can be taught. -Oscar Wilde

91. William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971) Nobel Prize (1915) A father-son team that shared a Nobel Prize

92. One of the greatest scientific discoveries of twentieth century

93. Max von Laue, 1879-1960 Nobel 1914

94. Two Questions Q1: X-rays waves or particles? Father Bragg: Particles Son Bragg: Waves “Even after they shared a Nobel Prize in 1915, … this tension persisted…” – Ioan James in Remarkable Physicists Q2:Crystals: Perodic arrangement of atoms? X-RAY DIFFRACTION: X-rays are waves and crystals are periodic arrangement of atoms

95. If it is permissible to evaluate a human discovery according to the fruits which it bears then there are not many discoveries ranking on par with that made by von Laue. -from Nobel Presentation Talk