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Basic crystallography

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Basic crystallography

  1. 1. DST-SERC School on Texture and Microstructure BASIC CRYSTALLOGRAPHY Rajesh Prasad Department of Applied Mechanics Indian Institute of Technology New Delhi 110016 rajesh@am.iitd.ac.in
  2. 2. 2 Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices Reciprocal lattice
  3. 3. 3 A 3D translationaly periodic arrangement of atoms in space is called a crystal. Crystal ?
  4. 4. 4 Lattice? A 3D translationally periodic arrangement of points in space is called a lattice.
  5. 5. 5 A 3D translationally periodic arrangement of atoms Crystal A 3D translationally periodic arrangement of points Lattice
  6. 6. 6 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
  7. 7. 7 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
  8. 8. 8 A 3D translationally periodic arrangement of points Each lattice point in a lattice has identical neighbourhood of other lattice points. Lattice
  9. 9. 9 + Love Pattern (Crystal) Love Lattice + Heart (Motif) = = Lattice + Motif = Crystal Love Pattern
  10. 10. 10 Air, Water and Earth by M.C. Esher
  11. 11. 11 Every periodic pattern (and hence a crystal) has a unique lattice associated with it
  12. 12. 12
  13. 13. 13 Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices Reciprocal lattice
  14. 14. 14 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
  15. 15. 15 The most common shape of a unit cell is a parallelopiped with lattice points at corners. UNIT CELL: Primitive Unit Cell: Lattice Points only at corners Non-Primitive Unit cell: Lattice Point at corners as well as other some points
  16. 16. 16 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 α, β, γ are called the LATTICE PARAMETERS α β γ a b c Unit cell description : 4
  17. 17. 17 7 crystal Systems Crystal System Conventional Unit Cell 1. Cubic a=b=c, α=β=γ=90° 2. Tetragonal a=b≠c, α=β=γ=90° 3. Orthorhombic a≠b≠c, α=β=γ=90° 4. Hexagonal a=b≠c, α=β= 90°, γ=120° 5. Rhombohedral a=b=c, α=β=γ≠90° OR Trigonal 6. Monoclinic a≠b≠c, α=β=90°≠γ 7. Triclinic a≠b≠c, α≠β≠γ Unit cell description : 5
  18. 18. 18 The description of a unit cell requires: 1. Its Size and shape (the six lattice parameters) 2. Its atomic content (fractional coordinates): Coordinates of atoms as fractions of the respective lattice parameters Unit cell description : 6
  19. 19. 19x y Projection/plan view of unit cells ½ ½ 0 Example 1: Cubic close-packed (CCP) crystal e.g. Cu, Ni, Au, Ag, Pt, Pb etc. 2 1 2 1 2 1 2 1 x y z Plan description : 1
  20. 20. 20 The six lattice parameters a, b, c, α, β, γ The cell of the lattice lattice crystal + Motif
  21. 21. 21 Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices Reciprocal lattice
  22. 22. 22 14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices 1. Cubic P I F 2. Tetragonal P I 3. Orthorhombic P I F C 4. Hexagonal P 5. Trigonal P 6. Monoclinic P C 7. Triclinic P P: Primitive (lattice points only at the 8 corners of the unit cell) I: Body-centred (lattice points at the corners + one lattice point at the centre of the unit cell) F: Face-centred (lattice points at the corners + lattice points at centres of all faces of the unit cell) C: End-centred or base-centred (lattice points at the corners + lattice points at the centres of a pair of opposite faces)
  23. 23. 23 The three cubic Bravais lattices Crystal system Bravais lattices 1. Cubic P I F Simple cubic Primitive cubic Cubic P Body-centred cubic Cubic I Face-centred cubic Cubic F
  24. 24. 24 Orthorhombic C End-centred orthorhombic Base-centred orthorhombic
  25. 25. 25 Monatomic Body-Centred Cubic (BCC) crystal Lattice: bcc CsCl crystal Lattice: simple cubic BCC Feynman! Corner and body-centres have the same neighbourhood Corner and body-centred atoms do not have the same neighbourhood Motif: 1 atom 000 Motif: two atoms Cl 000; Cs ½ ½ ½ Cs Cl
  26. 26. 26 ½ ½½ ½ Lattice: Simple hexagonal hcp lattice hcp crystal Example: Hexagonal close-packed (HCP) crystal x y z Corner and inside atoms do not have the same neighbourhood Motif: Two atoms: 000; 2/3 1/3 1/2
  27. 27. 27 14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices 1. Cubic P I F 2. Tetragonal P I 3. Orthorhombic P I F C 4. Hexagonal P 5. Trigonal P 6. Monoclinic P C 7. Triclinic P ?
  28. 28. 28 End-centred cubic not in the Bravais list ? End-centred cubic = Simple Tetragonal 2 a 2 a
  29. 29. 29 14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices 1. Cubic P I F C 2. Tetragonal P I 3. Orthorhombic P I F C 4. Hexagonal P 5. Trigonal P 6. Monoclinic P C 7. Triclinic P
  30. 30. 30 Face-centred cubic in the Bravais list ? Cubic F = Tetragonal I ?!!!
  31. 31. 31 14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices 1. Cubic P I F C 2. Tetragonal P I 3. Orthorhombic P I F C 4. Hexagonal P 5. Trigonal P 6. Monoclinic P C 7. Triclinic P
  32. 32. 32 Couldn’t find his photo on the net 1811-1863 Auguste Bravais 1850: 14 lattices1835: 15 lattices ML Frankenheim 1801-1869 24 March 2008: 13 lattices !!! DST-SERC School X 1856: 14 lattices History:
  33. 33. 33 Why can’t the Face- Centred Cubic lattice (Cubic F) be considered as a Body-Centred Tetragonal lattice (Tetragonal I) ?
  34. 34. 34 Primitive cell Primitive cell Non- primitive cell A unit cell of a lattice is NOT unique. UNIT CELLS OF A LATTICE Unit cell shape CANNOT be the basis for classification of Lattices
  35. 35. 35 What is the basis for classification of lattices into 7 crystal systems and 14 Bravais lattices?
  36. 36. 36 Lattices are classified on the basis of their symmetry
  37. 37. 37 What is symmetry?
  38. 38. 38 If an object is brought into self- coincidence after some operation it said to possess symmetry with respect to that operation. Symmetry
  39. 39. 39 NOW NO SWIMS ON MON
  40. 40. 40 Rotational symmetry http://fab.cba.mit.edu/
  41. 41. 41 If an object come into self-coincidence through smallest non-zero rotation angle of θ then it is said to have an n- fold rotation axis where θ 0 360 =n θ=180° θ=90° Rotation Axis n=2 2-fold rotation axis n=4 4-fold rotation axis
  42. 42. 42 Reflection (or mirror symmetry)
  43. 43. 43 Lattices also have translational symmetry Translational symmetry In fact this is the defining symmetry of a lattice
  44. 44. 44 Symmetry of lattices Lattices have Rotational symmetry Reflection symmetry Translational symmetry
  45. 45. 45 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 of a crystal including translations is called its SPACE GROUP Point Group and Space Group
  46. 46. 46 Classification of lattices Based on the space group symmetry, i.e., rotational, reflection and translational symmetry ⇒ 14 types of lattices ⇒ 14 Bravais lattices Based on the point group symmetry alone ⇒ 7 types of lattices ⇒ 7 crystal systems
  47. 47. 47 7 crystal Systems System Required symmetry • Cubic Three 4-fold axis • Tetragonal one 4-fold axis • Orthorhombic three 2-fold axis • Hexagonal one 6-fold axis • Rhombohedral one 3-fold axis • Monoclinic one 2-fold axis • Triclinic none
  48. 48. 48 Tetragonal symmetry Cubic symmetry Cubic C = Tetragonal P Cubic F ≠≠≠≠ Tetragonal I
  49. 49. 49 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
  50. 50. 50 QUESTIONS?
  51. 51. 51 Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices Reciprocal lattice
  52. 52. 52 Miller Indices of directions and planes William Hallowes Miller (1801 – 1880) University of Cambridge Miller Indices 1
  53. 53. 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 y 3. 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 1, 0, 0 Miller Indices 2 a b c
  54. 54. 54 y z Miller indices of a direction: only the orientation not its position or sense All parallel directions have the same Miller indices [100]x Miller Indices 3
  55. 55. 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 Direction OA Direction PQ
  56. 56. 56 Miller indices of a family of symmetry related directions [100] [001] [010] uvw = [uvw] and all other directions related to [uvw] by the symmetry of the crystal cubic 100 = [100], [010], [001] tetragonal 100 = [100], [010] Cubic Tetragonal [010] [100] Miller Indices 4
  57. 57. 57 Slip System A common microscopic mechanism of plastic deformation involves slip on some crystallographic planes known as slip planes in certain crystallographic directions called slip directions. A combination of slip direction lying in a slip plane is called a slip system
  58. 58. 58 Miller indices of slip directions in CCP y z [110] [110] [101] [ 101] [011] [101] Slip directions = close-packed directions = face diagonals x Six slip directions: [110] [101] [011] [110] [101] [101] All six slip directions in ccp: 110 Miller Indices 5
  59. 59. 59 5. Enclose in parenthesis Miller Indices for planes 3. Take reciprocal 2. Find intercepts along axes 1. Select a crystallographic coordinate system with origin not on the plane 4. Convert to smallest integers in the same ratio 1 1 1 1 1 1 1 1 1 (111) x y z O
  60. 60. 60 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
  61. 61. 61 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) _
  62. 62. 62 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)
  63. 63. 63 Slip planes in a ccp crystal Slip planes in ccp are the close-packed planes AEG (111) A D FE D G x y z B C y ACE (111) A E D x z C ACG (111) A G x y z C A D FE D G x y z B C All four slip planes of a ccp crystal: {111} D E G x y z C CEG (111)
  64. 64. 64 {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
  65. 65. 65 [100] [010] Symmetry related directions in the hexagonal crystal system cubic 100 = [100], [010], [001] hexagonal 100 Not permutations Permutations [110] x y z = [100], [010], [110]
  66. 66. 66 (010) (100) (110) x {100}hexagonal = (100), (010), {100}cubic = (100), (010), (001) Not permutations Permutations Symmetry related planes in the hexagonal crystal system (110) y z
  67. 67. 67 Problem: In hexagonal system symmetry related planes and directions do NOT have Miller indices which are permutations Solution: Use the four-index Miller- Bravais Indices instead
  68. 68. 68 x1 x2 x3 (1010) (0110) (1100) (hkl)=>(hkil) with i=-(h+k) Introduce a fourth axis in the basal plane x1 x3 Miller-Bravais Indices of Planes x2 Prismatic planes: {1100} = (1100) (1010) (0110) z
  69. 69. 69 Miller-Bravais Indices of Directions in hexagonal crystals x1 x2 x3 Basal plane =slip plane =(0001) [uvw]=>[UVTW] wWvuTuvVvuU =+−=−=−= );();2( 3 1 );2( 3 1 Require that: 0=++ TVU Vectorially 0aaa 21 =++ 3 caa caaa wvu WTVU ++= +++ 21 321 a1 a2 a2
  70. 70. 70 a1 a1 -a2 -a3 ]0112[ a2 a3 x1 x2 x3 wWvuTuvVvuU =+−=−=−= );();2( 3 1 );2( 3 1 [ ] ]0112[0]100[ 3 1 3 1 3 2 ≡−−⇒ [ ] ]0121[0]010[ 3 1 3 2 3 1 ≡−−⇒ [ ] ]2011[0]011[ 3 2 3 1 3 1 ≡−−⇒ x1: x2: x3: Slip directions in hcp 0112 Miller-Bravais indices of slip directions in hcp crystal:
  71. 71. 71 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 h U + k V + i T +l W = 0
  72. 72. 72 CUBIC CRYSTALS [hkl] ⊥⊥⊥⊥ (hkl) Angle between two directions [h1k1l1] and [h2k2l2]: C [111] (111) 2 2 2 2 2 2 2 1 2 1 2 1 212121 cos lkhlkh llkkhh ++++ ++ =θ
  73. 73. 73 dhkl Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell 222 lkh acubic hkld ++ = O x (100) ad =100 B O x z E 2011 a d =
  74. 74. 74 [uvw] Miller indices of a direction (i.e. a set of parallel directions) <uvw> Miller indices of a family of symmetry related directions (hkl) Miller Indices of a plane (i.e. a set of parallel planes) {hkl} Miller indices of a family of symmetry related planes [uvtw] Miller-Bravais indices of a direction, (hkil) plane in a hexagonal system Summary of Notation convention for Indices
  75. 75. 75 Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices Reciprocal lattice
  76. 76. 76 Reciprocal Lattice A lattice can be defined in terms of three vectors a, b and c along the edges of the unit cell We now define another triplet of vectors a*, b* and c* satisfying the following property: The triplet a*, b* and c* define another unit cell and thus a lattice called the reciprocal lattice. The original lattice in relation to the reciprocal lattice is called the direct lattice. VVV b)(a c a)(c b c)(b a × = × = × = ∗∗∗ ;; V= volume of the unit cell
  77. 77. 77 Every crystal has a unique Direct Lattice and a corresponding Reciprocal Lattice. The reciprocal lattice vector ghkl has the following interesting and useful properties: A RECIPROCAL LATTICE VECTOR ghkl is a vector with integer components h, k, l in the reciprocal space: ∗∗∗ ++= cbag lkhhkl Reciprocal Lattice (contd.)
  78. 78. 78 hkl hkl d 1 =g 1. The reciprocal lattice vector ghkl with integer components h, k, l is perpendicular to the direct lattice plane with Miller indices (hkl): )(hklhkl ⊥g Properties of Reciprocal Lattice Vector: 2. The length of the reciprocal lattice vector ghkl with integer components h, k, l is equal to the reciprocal of the interplanar spacing of direct lattice plane (hkl):
  79. 79. 79 Full significance of the concept of the reciprocal lattice will be appreciated in the discussion of the x- ray and electron diffraction.
  80. 80. 80 Thank You
  81. 81. 81 Example: Monatomic Body-Centred Cubic (BCC) crystal e.g., Fe, Ti, Cr, W, Mo etc. Atomic content of a unit cell: fractional coordinates: Fractional coordinates of the two atoms 000; ½ ½ ½ Unit cell description : 8 Effective no. of atoms in the unit cell 218 8 1 =+× corners Body centre 000 ½ ½ ½

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