Space Lattice are non-coplanar vectors in space forming a basis {  }
One dimensional lattice Two dimensional lattice
Three dimensional lattice
Lattice vectors and parameters
Indices of directions
Miller indices for planes
Miller indices and plane spacing
Two-dimensional lattice showing that lines of lowest indices have the greatest spacing and greatest density of lattice poi...
Reciprocal lattice
Illustration of crystal lattices and corresponding reciprocal lattices for a cubic system
Illustration of crystal lattices and corresponding reciprocal lattices for a a hexagonal system
If then and perpendicular to (hkl) plane
If then and perpendicular to (hkl) plane Proof: H • ( a 1 /h- a 2 /k) = H • ( a 1 /h- a 3 /l)=0 a 1 /h• H /| H |=[1/h 0 0]...
Symmetry (a) mirror   plane (b)rotation (c)inversion (d)roto-  inversion
Symmetry operation
Crystal system
The 14 Bravais lattices
The fourteen Bravais  lattices  Simple cubic lattices nitrogen - simple cubic copper - face centered cubic body centered c...
Tetragonal lattices a1 = a2 ≠ a3 α = β = γ = 90  simple tetragonal   Body centered Tetragonal
Orthorhombic lattices  a1 ≠ a2 ≠ a3  α = β = γ = 90  simple orthorhombic   Base centered orthorhombic Body centered ortho...
Monoclinic lattices  a1 ≠ a2 ≠ a3  α = γ = 90   ≠ β (2nd setting) α = β = 90   ≠ γ (1st setting)      Simple monoclinic ...
Triclinic lattice a1 ≠ a2 ≠ a3 α ≠ β ≠ γ  Simple triclinic
Hexagonal lattice a1 = a2 ≠ a3 α = β = 90   , γ = 120  lanthanum - hexagonal
Trigonal (Rhombohedral) lattice a1 = a2 = a3 α = β = γ ≠ 90  mercury - trigonal
Relation between rhombo-hedral and hexagonal lattices
Relation of tetragonal C lattice to tetragonal P lattice
Extension of lattice points through space by the unit cell vectors  a, b, c
Symmetry elements
Primitive and non-primitive cells Face-centerd cubic point lattice referred to cubic and rhombo-hedral cells
All shaded planes in the cubic lattice shown are planes of the zone{001}
Zone axis [uvw] Zone plane (hkl) then hu+kv+wl=0 Two zone planes (h 1 k 1 l 1 ) and (h 2 k 2 l 2 ) then zone axis [uvw]=
Plane spacing
Indexing the hexagonal system
Indexing the hexagonal system
Crystal structure  -Fe, Cr, Mo, V  -Fe, Cu, Pb, Ni
Hexagonal close-packed Zn, Mg, Be,   -Ti
FCC and HCP
 -Uranium, base-centered orthorhombic (C-centered) y=0.105±0.005
 
 
 
 
AuBe: Simple cubic u = 0.100 w = 0.406
Structure of solid solution (a) Mo in Cr (substitutional) (b) C in   -Fe (interstitial)
Atom sizes (d) and coordination
Change in coordination 12  8 12  6 12  4 size contraction, percent 3 3 12
A: Octahedral site,  B: Tetrahedral site
Twin
(a) (b) FCC annealing (c) HCP deformation twins
Twin band in FCC lattice, Plane of main drawing is (1 ī 0)
Homework assignment Problem 2-6  Problem 2-8  Problem 2-9  Problem 2-10
Stereographic  projection *Any plane passing the center of the reference sphere intersects the sphere in a trace called gr...
Stereographic  projection
 
Pole on upper sphere can also be projected to the horizontal (equatorial) plane
Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O.
Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O...
<ul><li>A great circle representing a plane is divided to two half circles, one in upper reference sphere, the other in lo...
N S E W
The position of pole P can be defined by two angles    and  
The position of projection P’ can be obtained by r = R tan(  /2)
The trace of each semi-great circle hinged along NS projects on WNES plane as a meridian
As the semi-great circle swings along NS, the end point of each radius draws on the upper sphere a curve which projects on...
The weaving of meridians and parallels makes the Wulff net
Two projected poles can always be rotated along the net normal to a same meridian (not parallel) such that their intersect...
P : a pole at (  1 ,  1 )  NMS : its trace
The projection of a plane trace and pole can be found from each other by rotating the projection along net normal to the f...
Zone circle and zone pole
If P2’ is the projection of a zone axis, then all poles of the corresponding zone planes lie on the trace of P2’
Rotation of a poles about NS axis by a fixed angle: the corresponding poles moving along a parallel *Pole A1 move to pole ...
m: mirror plane F1: face 1 F2: face 2 N1: normal of F1 N2: normal of N2 N1, N2 lie on a plane which is  丄 to m
 
A plane not passing through the center of the reference sphere intersects the sphere on a small circle which also projects...
Projection of a small circle centered at Y
Rotation of a pole A1 along an inclined axis B1: B1  B3     B2    B2     B3     B1 A1  A1     A2     A3     A4  ...
Rotation of a pole A1 along an inclined axis B1:
A 1  rotate about B 1  forming a small circle in the reference sphere, the small circle projects along A 1 , A 4 , D, arc ...
Rotation of 3 directions along b axis
Rotation of 3 directions along b axis
Rotation of 3 directions along b axis
Standard coordinates for crystal axes
Standard coordinates for crystal axes
Standard coordinates for crystal axes
Standard coordinates for crystal axes
Projection of a monoclinic crystal +C -b +b -a + a x x 011 0-1-1 01-1 0-11 -110 -1-10 110 1-10
Projection of a monoclinic crystal
Projection of a monoclinic crystal
Projection of a monoclinic crystal
(a) Zone plane (stippled) (b) zone circle with zone axis ā, note [100] • [0xx]=0
Location of axes  for a triclinic crystal: the circle on net has a radius of    along WE axis of the net
 
Zone circles corresponding to a, b, c axes of a triclinic crystal
Standard projections of cubic crystals on (a) (001), (b) (011)
d/(a/h)=cos  , d/(b/k)=cos  , d/(c/l)=cos  h:k:l=a  cos  : b  cos  : c  cos  measure 3 angles to calculate hkl
The face poles of six faces related by -3 axis that is (a) perpendicular (b) oblique to the plane of projection
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972 B3102005 Cullity Chapter 2

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972 B3102005 Cullity Chapter 2

  1. 1. Space Lattice are non-coplanar vectors in space forming a basis { }
  2. 2. One dimensional lattice Two dimensional lattice
  3. 3. Three dimensional lattice
  4. 4. Lattice vectors and parameters
  5. 5. Indices of directions
  6. 6. Miller indices for planes
  7. 7. Miller indices and plane spacing
  8. 8. Two-dimensional lattice showing that lines of lowest indices have the greatest spacing and greatest density of lattice points
  9. 9. Reciprocal lattice
  10. 10. Illustration of crystal lattices and corresponding reciprocal lattices for a cubic system
  11. 11. Illustration of crystal lattices and corresponding reciprocal lattices for a a hexagonal system
  12. 12. If then and perpendicular to (hkl) plane
  13. 13. If then and perpendicular to (hkl) plane Proof: H • ( a 1 /h- a 2 /k) = H • ( a 1 /h- a 3 /l)=0 a 1 /h• H /| H |=[1/h 0 0] • [hkl]*/| H |=1/| H |=d hkl H a 1 a 2 a 3
  14. 14. Symmetry (a) mirror plane (b)rotation (c)inversion (d)roto- inversion
  15. 15. Symmetry operation
  16. 16. Crystal system
  17. 17. The 14 Bravais lattices
  18. 18. The fourteen Bravais lattices Simple cubic lattices nitrogen - simple cubic copper - face centered cubic body centered cubic Cubic lattices a 1 = a 2 = a 3 α = β = γ = 90 o
  19. 19. Tetragonal lattices a1 = a2 ≠ a3 α = β = γ = 90  simple tetragonal Body centered Tetragonal
  20. 20. Orthorhombic lattices a1 ≠ a2 ≠ a3 α = β = γ = 90  simple orthorhombic Base centered orthorhombic Body centered orthorhombic Face centered orthorhombic
  21. 21. Monoclinic lattices a1 ≠ a2 ≠ a3 α = γ = 90  ≠ β (2nd setting) α = β = 90  ≠ γ (1st setting) Simple monoclinic Base centered monoclinic
  22. 22. Triclinic lattice a1 ≠ a2 ≠ a3 α ≠ β ≠ γ Simple triclinic
  23. 23. Hexagonal lattice a1 = a2 ≠ a3 α = β = 90  , γ = 120  lanthanum - hexagonal
  24. 24. Trigonal (Rhombohedral) lattice a1 = a2 = a3 α = β = γ ≠ 90  mercury - trigonal
  25. 25. Relation between rhombo-hedral and hexagonal lattices
  26. 26. Relation of tetragonal C lattice to tetragonal P lattice
  27. 27. Extension of lattice points through space by the unit cell vectors a, b, c
  28. 28. Symmetry elements
  29. 29. Primitive and non-primitive cells Face-centerd cubic point lattice referred to cubic and rhombo-hedral cells
  30. 30. All shaded planes in the cubic lattice shown are planes of the zone{001}
  31. 31. Zone axis [uvw] Zone plane (hkl) then hu+kv+wl=0 Two zone planes (h 1 k 1 l 1 ) and (h 2 k 2 l 2 ) then zone axis [uvw]=
  32. 32. Plane spacing
  33. 33. Indexing the hexagonal system
  34. 34. Indexing the hexagonal system
  35. 35. Crystal structure  -Fe, Cr, Mo, V  -Fe, Cu, Pb, Ni
  36. 36. Hexagonal close-packed Zn, Mg, Be,  -Ti
  37. 37. FCC and HCP
  38. 38.  -Uranium, base-centered orthorhombic (C-centered) y=0.105±0.005
  39. 43. AuBe: Simple cubic u = 0.100 w = 0.406
  40. 44. Structure of solid solution (a) Mo in Cr (substitutional) (b) C in  -Fe (interstitial)
  41. 45. Atom sizes (d) and coordination
  42. 46. Change in coordination 12  8 12  6 12  4 size contraction, percent 3 3 12
  43. 47. A: Octahedral site, B: Tetrahedral site
  44. 48. Twin
  45. 49. (a) (b) FCC annealing (c) HCP deformation twins
  46. 50. Twin band in FCC lattice, Plane of main drawing is (1 ī 0)
  47. 51. Homework assignment Problem 2-6 Problem 2-8 Problem 2-9 Problem 2-10
  48. 52. Stereographic projection *Any plane passing the center of the reference sphere intersects the sphere in a trace called great circle * A plane can be represented by its great circle or pole, which is the intersection of its plane normal with the reference sphere
  49. 53. Stereographic projection
  50. 55. Pole on upper sphere can also be projected to the horizontal (equatorial) plane
  51. 56. Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O.
  52. 57. Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O. U L P P’ P P’ X O O
  53. 58. <ul><li>A great circle representing a plane is divided to two half circles, one in upper reference sphere, the other in lower sphere </li></ul><ul><li>Each half circle is projected as a trace on the equatorial plane </li></ul><ul><li>The two traces are symmetrical with respect to their associated common diameter </li></ul>
  54. 59. N S E W
  55. 60. The position of pole P can be defined by two angles  and 
  56. 61. The position of projection P’ can be obtained by r = R tan(  /2)
  57. 62. The trace of each semi-great circle hinged along NS projects on WNES plane as a meridian
  58. 63. As the semi-great circle swings along NS, the end point of each radius draws on the upper sphere a curve which projects on WNES plane as a parallel
  59. 64. The weaving of meridians and parallels makes the Wulff net
  60. 65. Two projected poles can always be rotated along the net normal to a same meridian (not parallel) such that their intersecting angle can be counted from the net
  61. 66. P : a pole at (  1 ,  1 ) NMS : its trace
  62. 67. The projection of a plane trace and pole can be found from each other by rotating the projection along net normal to the following position
  63. 68. Zone circle and zone pole
  64. 69. If P2’ is the projection of a zone axis, then all poles of the corresponding zone planes lie on the trace of P2’
  65. 70. Rotation of a poles about NS axis by a fixed angle: the corresponding poles moving along a parallel *Pole A1 move to pole A2 *Pole B1 moves 40 ° to the net end then another 20 ° along the same parallel to B1’ corresponding to a movement on the lower half reference sphere, pole corresponding to B1’ on upper half sphere is B2
  66. 71. m: mirror plane F1: face 1 F2: face 2 N1: normal of F1 N2: normal of N2 N1, N2 lie on a plane which is 丄 to m
  67. 73. A plane not passing through the center of the reference sphere intersects the sphere on a small circle which also projects as a circle, but the center of the former circle does not project as the center of the latter.
  68. 74. Projection of a small circle centered at Y
  69. 75. Rotation of a pole A1 along an inclined axis B1: B1  B3  B2  B2  B3  B1 A1  A1  A2  A3  A4  A4 A plane not passing through the center of the reference sphere intersects the sphere on a small circle which also projects as a circle .
  70. 76. Rotation of a pole A1 along an inclined axis B1:
  71. 77. A 1 rotate about B 1 forming a small circle in the reference sphere, the small circle projects along A 1 , A 4 , D, arc A 1 , A 4 , D centers around C (not B 1 ) in the projection plane
  72. 78. Rotation of 3 directions along b axis
  73. 79. Rotation of 3 directions along b axis
  74. 80. Rotation of 3 directions along b axis
  75. 81. Standard coordinates for crystal axes
  76. 82. Standard coordinates for crystal axes
  77. 83. Standard coordinates for crystal axes
  78. 84. Standard coordinates for crystal axes
  79. 85. Projection of a monoclinic crystal +C -b +b -a + a x x 011 0-1-1 01-1 0-11 -110 -1-10 110 1-10
  80. 86. Projection of a monoclinic crystal
  81. 87. Projection of a monoclinic crystal
  82. 88. Projection of a monoclinic crystal
  83. 89. (a) Zone plane (stippled) (b) zone circle with zone axis ā, note [100] • [0xx]=0
  84. 90. Location of axes for a triclinic crystal: the circle on net has a radius of  along WE axis of the net
  85. 92. Zone circles corresponding to a, b, c axes of a triclinic crystal
  86. 93. Standard projections of cubic crystals on (a) (001), (b) (011)
  87. 94. d/(a/h)=cos  , d/(b/k)=cos  , d/(c/l)=cos  h:k:l=a  cos  : b  cos  : c  cos  measure 3 angles to calculate hkl
  88. 95. The face poles of six faces related by -3 axis that is (a) perpendicular (b) oblique to the plane of projection
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