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972 B3102005 Cullity Chapter 2

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1. Space Lattice are non-coplanar vectors in space forming a basis { } 2. One dimensional lattice Two dimensional lattice 3. Three dimensional lattice 4. Lattice vectors…
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  • 1. Space Lattice are non-coplanar vectors in space forming a basis { }
  • 2. One dimensional lattice Two dimensional lattice
  • 3. Three dimensional lattice
  • 4. Lattice vectors and parameters
  • 5. Indices of directions
  • 6. Miller indices for planes
  • 7. Miller indices and plane spacing
  • 8. Two-dimensional lattice showing that lines of lowest indices have the greatest spacing and greatest density of lattice points
  • 9. Reciprocal lattice
  • 10. Illustration of crystal lattices and corresponding reciprocal lattices for a cubic system
  • 11. Illustration of crystal lattices and corresponding reciprocal lattices for a a hexagonal system
  • 12. If then and perpendicular to (hkl) plane
  • 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. Symmetry (a) mirror plane (b)rotation (c)inversion (d)roto- inversion
  • 15. Symmetry operation
  • 16. Crystal system
  • 17. The 14 Bravais lattices
  • 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. Tetragonal lattices a1 = a2 ≠ a3 α = β = γ = 90  simple tetragonal Body centered Tetragonal
  • 20. Orthorhombic lattices a1 ≠ a2 ≠ a3 α = β = γ = 90  simple orthorhombic Base centered orthorhombic Body centered orthorhombic Face centered orthorhombic
  • 21. Monoclinic lattices a1 ≠ a2 ≠ a3 α = γ = 90  ≠ β (2nd setting) α = β = 90  ≠ γ (1st setting) Simple monoclinic Base centered monoclinic
  • 22. Triclinic lattice a1 ≠ a2 ≠ a3 α ≠ β ≠ γ Simple triclinic
  • 23. Hexagonal lattice a1 = a2 ≠ a3 α = β = 90  , γ = 120  lanthanum - hexagonal
  • 24. Trigonal (Rhombohedral) lattice a1 = a2 = a3 α = β = γ ≠ 90  mercury - trigonal
  • 25. Relation between rhombo-hedral and hexagonal lattices
  • 26. Relation of tetragonal C lattice to tetragonal P lattice
  • 27. Extension of lattice points through space by the unit cell vectors a, b, c
  • 28. Symmetry elements
  • 29. Primitive and non-primitive cells Face-centerd cubic point lattice referred to cubic and rhombo-hedral cells
  • 30. All shaded planes in the cubic lattice shown are planes of the zone{001}
  • 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. Plane spacing
  • 33. Indexing the hexagonal system
  • 34. Indexing the hexagonal system
  • 35. Crystal structure  -Fe, Cr, Mo, V  -Fe, Cu, Pb, Ni
  • 36. Hexagonal close-packed Zn, Mg, Be,  -Ti
  • 37. FCC and HCP
  • 38.  -Uranium, base-centered orthorhombic (C-centered) y=0.105±0.005
  • 43. AuBe: Simple cubic u = 0.100 w = 0.406
  • 44. Structure of solid solution (a) Mo in Cr (substitutional) (b) C in  -Fe (interstitial)
  • 45. Atom sizes (d) and coordination
  • 46. Change in coordination 12  8 12  6 12  4 size contraction, percent 3 3 12
  • 47. A: Octahedral site, B: Tetrahedral site
  • 48. Twin
  • 49. (a) (b) FCC annealing (c) HCP deformation twins
  • 50. Twin band in FCC lattice, Plane of main drawing is (1 ī 0)
  • 51. Homework assignment Problem 2-6 Problem 2-8 Problem 2-9 Problem 2-10
  • 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
  • 53. Stereographic projection
  • 55. Pole on upper sphere can also be projected to the horizontal (equatorial) plane
  • 56. Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O.
  • 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
  • 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>
  • 59. N S E W
  • 60. The position of pole P can be defined by two angles  and 
  • 61. The position of projection P’ can be obtained by r = R tan(  /2)
  • 62. The trace of each semi-great circle hinged along NS projects on WNES plane as a meridian
  • 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
  • 64. The weaving of meridians and parallels makes the Wulff net
  • 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
  • 66. P : a pole at (  1 ,  1 ) NMS : its trace
  • 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
  • 68. Zone circle and zone pole
  • 69. If P2’ is the projection of a zone axis, then all poles of the corresponding zone planes lie on the trace of P2’
  • 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
  • 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
  • 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.
  • 74. Projection of a small circle centered at Y
  • 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 .
  • 76. Rotation of a pole A1 along an inclined axis B1:
  • 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
  • 78. Rotation of 3 directions along b axis
  • 79. Rotation of 3 directions along b axis
  • 80. Rotation of 3 directions along b axis
  • 81. Standard coordinates for crystal axes
  • 82. Standard coordinates for crystal axes
  • 83. Standard coordinates for crystal axes
  • 84. Standard coordinates for crystal axes
  • 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
  • 86. Projection of a monoclinic crystal
  • 87. Projection of a monoclinic crystal
  • 88. Projection of a monoclinic crystal
  • 89. (a) Zone plane (stippled) (b) zone circle with zone axis ā, note [100] • [0xx]=0
  • 90. Location of axes for a triclinic crystal: the circle on net has a radius of  along WE axis of the net
  • 92. Zone circles corresponding to a, b, c axes of a triclinic crystal
  • 93. Standard projections of cubic crystals on (a) (001), (b) (011)
  • 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
  • 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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