Crystalography

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Crystalography

  1. 1. Complied by: Prof. Vijaya Agarwala BE, MTech, PhD Professor and Head, Center of Excellence Nanotechnology & Professor, Metallurgical and Materials Engineering and IIT Roorkee MT201B: Materials Science L-3, T-1, P-0 4 credits: CWS-25%, MTE-25%, ETE-50%
  2. 2. Materials Science 2 lntroduction to Crystallography: Crystal defects: point defects, line defects, dislocations surface defects and volume defects; Principles of Alloy Formation : primary and intermediate phases, their formation, solid solutions, Hume Rothery rules, Binary Equilibria: Binary phase diagrams involving isomorphous, eutectic, peritectic and eutectoid reactions. phase rule, lever rule, effect of non- equilibrium cooling on structure and distribution of phases. Some common binary phase diagrams viz : Cu-Ni, Al-Si, Pb-Sn, Cu-Zn, Cu-Sn and Fe-C and important alloys belonging to these systems;
  3. 3. The shell model of the atom in which electrons are confined to live within certain shells and in subshells within shells
  4. 4. Fig 1.3 Materials Science Force is considered the change in potential energy, E, over a change in position. F = dE/dr
  5. 5. Fig 1.8 The formation of ionic bond between Na and Cl atoms in NaCl. The attraction Is due to coulombic forces. Materials Science
  6. 6. Fig 1.10 Sketch of the potential energy per ion-pair in solid NaCl. Zero energy corresponds to neutral Na and Cl atoms infinitely separated. Materials Science
  7. 7. Materials Science Fig 1.12 The origin of van der Walls bonding between water molecules. (a) The H2O molecule is polar and has a net permanent dipole moment (b) Attractions between the various dipole moments in water gives rise to van der Walls bonding
  8. 8. Materials Science 8 Covalent bonding -sharing of electron -strong bond, so high MP -directional, low electrical conductivity Metallic Bonding -random movements of electron, electron cloud -high electrical conductivity
  9. 9. Crystal Systems • Most solids are crystalline with their atoms arranged in a regular manner. • Long-range order : the regularity can extend throughout the crystal. • Short-range order : the regularity does not persist over appreciable distances. eg. amorphous materials such as glass and wax. • Liquids have short-range order, but lack long-range order. • Gases lack both long-range and short-range order Ref: http://me.kaist.ac.kr/upload/course/MAE800C/chapter2-1.pdf 9Materials Science
  10. 10. Crystal Structures (Contd…) • Five regular arrangements of lattice points that can occur in two dimensions. (a) square; (b) primitive rectangular; (c) centered rectangular; (d) hexagonal; (e) oblique. 10Materials Science
  11. 11. Point lattice 11Materials Science
  12. 12. Unit cell Lattice parameters: a, b, c, α, β and γ 12Materials Science
  13. 13. Crystal systems and Bravais lattice 13Materials Science
  14. 14. Number of lattice points per cell Where, Ni = number of interior points, Nf = number of points on faces, Nc = number of points on corners. 14Materials Science
  15. 15. base-centered arrangement of points is not a new lattice 15Materials Science
  16. 16. Any of the fourteen Bravais lattices may be referred to a combinatin of primitive unit cells. Face centered cubic lattice shown may be referred to the primitive cubic cell and rhombohedral cell (indicated by dashed lines, its axial angle between a is 600, and each of its side is √2 a, where a is the lattice parameter of cubic cell. 16Materials Science
  17. 17. FCC 17Materials Science 000, ½ ½ 0, ½ 0 ½ , 0 ½ ½ ¼ ¼ ¼ , ¾ ¾ ¼, ¾ ¼ ¾, ¼ ¾ ¾
  18. 18. 18Materials Science
  19. 19. 19Materials Science
  20. 20. 20Materials Science BCC
  21. 21. BCC 21Materials Science
  22. 22. HCP 22Materials Science
  23. 23. DC 23Materials Science
  24. 24. Materials Science 24 A C G H D F I J G H I J x Z 000, ½ ½ 0, ½ 0 ½ , 0 ½ ½ ¼ ¼ ¼ , ¾ ¾ ¼, ¾ ¼ ¾, ¼ ¾ ¾
  25. 25. ZnS 25Materials Science 3a 3b 5a 5b
  26. 26. SiO2 26Materials Science
  27. 27. Graphite 27Materials Science
  28. 28. C60 28Materials Science
  29. 29. Fig 1.43 Three allotropes of carbon Materials Science
  30. 30. CNT 30Materials Science
  31. 31. 31 NaCl Materials Science
  32. 32. Coordination number Number of nearest neighbors of an atom in the crystal lattice 32Materials Science
  33. 33. 5 • Rare due to poor packing (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson) SIMPLE CUBIC STRUCTURE (SC) 33Materials Science Polonium is a chemical element with the symbol Po and atomic number 84, discovered in 1898 by Marie and Pierre Curie. A rare and highly radioactive element ...
  34. 34. 6 • APF for a simple cubic structure = 0.52 Adapted from Fig. 3.19, Callister 6e. ATOMIC PACKING FACTOR 34Materials Science
  35. 35. • Coordination # = 8 7 Adapted from Fig. 3.2, Callister 6e. (Courtesy P.M. Anderson) • Close packed directions are cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. BODY CENTERED CUBIC STRUCTURE (BCC) 35Materials Science
  36. 36. a R 8 • APF for a body-centered cubic structure = 0.68 Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell Adapted from Fig. 3.2, Callister 6e. ATOMIC PACKING FACTOR: BCC 36Materials Science
  37. 37. 9 • Coordination # = 12 Adapted from Fig. 3.1(a), Callister 6e. (Courtesy P.M. Anderson) • Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. FACE CENTERED CUBIC STRUCTURE (FCC) 37Materials Science
  38. 38. Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a 10 • APF for a body-centered cubic structure = 0.74 Adapted from Fig. 3.1(a), Callister 6e. ATOMIC PACKING FACTOR: FCC 38Materials Science
  39. 39. 14 Example: Copper Data from Table inside front cover of Callister (see next slide): • crystal structure = FCC: 4 atoms/unit cell • atomic weight = 63.55 g/mol (1 amu = 1 g/mol) • atomic radius R = 0.128 nm (1 nm = 10 cm)-7 Compare to actual: Cu = 8.94 g/cm3 Result: theoretical Cu = 8.89 g/cm3 THEORETICAL DENSITY, 39Materials Science
  40. 40. 15 Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008 Atomic radius (nm) 0.143 ------ 0.217 0.114 ------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0.125 0.128 ------ 0.122 0.122 0.144 ------ ------ Density (g/cm3) 2.71 ------ 3.5 1.85 2.34 ------ 8.65 1.55 2.25 1.87 ------ 7.19 8.9 8.94 ------ 5.90 5.32 19.32 ------ ------ Adapted from Table, "Charac- teristics of Selected Elements", inside front cover, Callister 6e. Characteristics of Selected Elements at 20C 40Materials Science
  41. 41. metals• ceramics• polymers 16 Metals have... • close-packing (metallic bonding) • large atomic mass Ceramics have... • less dense packing (covalent bonding) • often lighter elements Polymers have... • poor packing (often amorphous) • lighter elements (C,H,O) Composites have... • intermediate values Data from Table B1, Callister 6e. DENSITIES OF MATERIAL CLASSES 41Materials Science
  42. 42. Materials Science 42 Physical Properties •Acoustical properties •Atomic properties •Chemical properties •Electrical properties •Environmental properties •Magnetic properties •Optical properties •Density Mechanical properties •Compressive strength •Ductility •Fatigue limit •Flexural modulus •Flexural strength •Fracture toughness •Hardness •Poisson's ratio •Shear modulus •Shear strain •Shear strength •Softness •Specific modulus •Specific weight •Tensile strength •Yield strength •Young's modulus
  43. 43. 18 • Most engineering materials are polycrystals. • Nb-Hf-W plate with an electron beam weld. • Each "grain" is a single crystal. • If crystals are randomly oriented, overall component properties are not directional. • Crystal sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers). Adapted from Fig. K, color inset pages of Callister 6e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) 1 mm POLYCRYSTALS 43Materials Science
  44. 44. 19 • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: • Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic. 200 m Data from Table 3.3, Callister 6e. (Source of data is R.W. Hertzberg, Deformatio n and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.) Adapted from Fig. 4.12(b), Callister 6e. (Fig. 4.12(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].) SINGLE VS POLYCRYSTALS 44Materials Science
  45. 45. Face-Centered Cubic Nanoparticles • Figure (a) shows the 12 neighbors that surround an atom (darkened circle) located in the center of a cube for a FCC lattice. • Figure (b) presents another perspective of the 12 nearest neighbors. These 13 atoms constitute the smallest theoretical nanoparticle for an FCC lattice. • Figure (c) shows the 14-sided polyhedron, called a dekatessarahedron, that is generated by connecting the atoms with planer faces 45Materials Science
  46. 46. If another layer of 42 atoms is layed around the 13-atom nanoparticle, one obtains a 55-atom nanoparticle with the same dekatessarahedron shape. Lager nanoparticles with the same polyhedral shape are obtained by adding more layers, and the sequence of numbers in the resulting particles, N N=1, 13, 55, 147,.., which are called structural magic numbers. 46Materials Science
  47. 47. Atoms in nano clusters • For n layers, the number of atoms N and the number of atoms on the surface Nsurf in this FCC nanoparticle is given by the formula, N = 1/3(10 n3 −15 n2 +11 n −3) Nsurf =10n2 − 20n +12 47Materials Science
  48. 48. Atomic packing • In two dimensions the most efficient way to pack identical circles is equilateral triangle arrangement shown in figure (a). • A second hexagonal layer of spheres can be placed on top of the first to form the most efficient packing of two layers, as shown in figure (b). • For efficient packing, the third layer can be placed either above the first layer with an atom at the location indicated by T or in the third possible arrangement with an atom above the position marked by X on the figure. • In the first case a hexagonal lattice with a hexagonal close packed (HCP) structure is generated, and in the second case a face-centered cubic lattice results. 48Materials Science
  49. 49. Voids X on figure is called an octahedral site The radius(aoct) of octahedral site is = 0.41421ao where ao is the radius of the spheres. There are also smaller sites, called tetrahedral sites, labeled T This is a smaller site since its radius aT= 0.2247ao 49Materials Science
  50. 50. Void types 50Materials Science
  51. 51. Stacking sequences: FCC & HCP 51Materials Science
  52. 52. 52Materials Science
  53. 53. 53Materials Science
  54. 54. HCP structure 54 Stacking sequence Materials Science
  55. 55. 55Materials Science
  56. 56. Fig 1.40 Materials Science
  57. 57. Lattice directions- MI The direction of any line in a lattice may be described by first drawing a line through the origin parallel to the given line and then giving the coordinates of any point on the line through the origin. -smallest integer value - Negative directions are shown by bars eg. 0,0,0 - 57Materials Science
  58. 58. Plane designation by Miller indices -Miller indices are always cleared of fractions - If a plane is parallel to a given axis, its fractional intercept on that axis is taken as infinity, Miller index is zero - If a plane cuts a negative axis, the corresponding index is negative and is written with a bar over it. -Planes whose indices are the negatives of one another are parallel and lie on opposite sides of the origin, e.g., (210) and (-2ī0). -- Planes belonging to the same family is denoted by curly bracket , {hkl} 58Materials Science
  59. 59. Fig 1.41 Labeling of crystal planes and typical examples in the cubic lattice Materials Science
  60. 60. Miller indices of lattice planes 60Materials Science
  61. 61. Miller Index 61Materials Science
  62. 62. The hexagonal unit cell : Miller –Bravais indices of planes and directions 63Materials Science
  63. 63. Zone= zonal planes + zonal axis -Zone axis and (hkl) the zonal plane All shaded planes belong to the same zone i.e parallel to an axis called zone axsis 64Materials Science u v w h1 k1 l1 h2 k2 l2
  64. 64. 68Materials Science
  65. 65. Crystal defects 69 1.Point defect- Vacancy, Impurity atoms ( substitutional and interstitial) Frankel and Schottky defect ( ionic solids & nonstochiometric) 2. Line defect- Edge dislocation Screw dislocation, Mixed dislocation 3. Surface defects- Grain boundaries Twin boundary Surfaces, stacking faults Interphases Materials Science
  66. 66. 70Materials Science
  67. 67. 71Materials Science
  68. 68. 72Materials Science
  69. 69. Frankel and Schottky defect 73Materials Science
  70. 70. 74Materials Science
  71. 71. Non stochiometry 75 Conduction in ionic crystal ZnO crystal containing extra Zn2+ Crystal is electronically neutral, (i.e. 2+ & 2- ) Zn2+ O2- Materials Science
  72. 72. Dislocation line and b are perpendicular to each other 76Materials Science
  73. 73. Movement of edge dislocation 77Materials Science
  74. 74. 78Materials Science
  75. 75. Cause of slip 79Materials Science
  76. 76. Elastic stress field responsible for electron scattering and increase in electrical resistivity lattice strain around dislocation 80Materials Science
  77. 77. 81Materials Science
  78. 78. The closest packed plane and the closest packed direction of FCC The plane and directions for the dislocation movement 82Materials Science
  79. 79. Tensile specimen - breaks How does the dislocation affect the failure? 83Materials Science
  80. 80. Dislocation line and b are parallel to each other 84Materials Science
  81. 81. By resolving, the contribution from both types of dislocations can be determined 85Materials Science
  82. 82. TEM -dislocaions 86Materials Science
  83. 83. 3. Surface defects 87Materials Science
  84. 84. Low angle GB 88Materials Science
  85. 85. 89Materials Science
  86. 86. 91Materials Science
  87. 87. Stacking fault -occurs when there is a flaw in the stacking sequence 93Materials Science
  88. 88. Interfaces of phases Coherent semi-coherent incoherent Al-Cu system 94Materials Science
  89. 89. Materials Science 95
  90. 90. Materials Science 96 Principles of Alloy Formation : primary and intermediate phases, their formation, solid solutions, Hume Rothery rule
  91. 91. Materials Science 97
  92. 92. Materials Science 98
  93. 93. Materials Science 99
  94. 94. Materials Science 100
  95. 95. Materials Science 101
  96. 96. Materials Science 102
  97. 97. Materials Science 103
  98. 98. Materials Science 104
  99. 99. Materials Science 105
  100. 100. Materials Science 106
  101. 101. Materials Science 107
  102. 102. Materials Science 108
  103. 103. Materials Science 109
  104. 104. Materials Science 110
  105. 105. Materials Science 111
  106. 106. Materials Science 112
  107. 107. Materials Science 113
  108. 108. Materials Science 114
  109. 109. Definition of Phase: • A phase is a region of material that is chemically uniform, physically distinct, and (often) mechanically separable. • A phase is a physically separable part of the system with distinct physical and chemical properties. System - A system is that part of the universe which is under consideration. • In a system consisting of ice and water in a glass jar, the ice cubes are one phase, the water is a second phase, and the humid air over the water is a third phase. The glass of the jar is another separate phase. 115Materials Science
  110. 110. Gibbs' phase rule proposed by Josiah Willard Gibbs The phase rule is an expression of the number of variables in equation(s) that can be used to describe a system in equilibrium. Degrees of freedom, F F = C − P + 2 Where, P is the number of phases in thermodynamic equilibrium with each other C is the number of components 116Materials Science
  111. 111. Phase rule at constant pressure • Condensed systems have no gas phase. When their properties are insensitive to the (small) changes in pressure, which results in the phase rule at constant pressure as, F = C − P + 1 117Materials Science
  112. 112. Types of Phase diagram 118 1. Unary phase diagram 2. Binary phase diagrams 3. Ternary phase diagram Materials Science
  113. 113. Unary phase diagram Critical pressure Liquid phase Pressure Temperature Solid Phase gaseous phase 119Materials Science
  114. 114. Binary phase diagrams 1. Binary isomorphous systems (complete solid solubility) 2. Binary eutectic systems (limited solid solubility) 3. Binary systems with intermediate phases/compounds 120Materials Science
  115. 115. Binary phase diagram - isomorphous system 121Materials Science
  116. 116. The Lever Rule Finding the amounts of phases in a two phase region: 1. Locate composition and temperature in diagram 2. In two phase region draw the tie line or isotherm 3. Fraction of a phase is determined by taking the length of the tie line to the phase boundary for the other phase, and dividing by the total length of tie line The lever rule is a mechanical analogy to the mass balance calculation. The tie line in the two-phase region is analogous to a lever balanced on a fulcrum. 122Materials Science
  117. 117. microstrucures 123Materials Science
  118. 118. Binary phase diagram –2. limited solubility • A phase diagram for a binary system displaying an eutectic point. 124Materials Science
  119. 119. Cu-Ag system 125Materials Science
  120. 120. Sn-Bi system 126Materials Science
  121. 121. Pb-Sn system 127Materials Science
  122. 122. Pb-Sn system 128Materials Science
  123. 123. Mechanism of growth Pb-Sn system 129Materials Science
  124. 124. Fig 1.69 Materials Science The equilibrium phase diagram of the Pb-Sn alloy. The microstructure on the left show the observations at various points during the cooling of a 90% Pb-10% Sn from the melt along the dashed line (the overall alloy composition remains constant at 10% Sn). Pb-Sn system
  125. 125. Cu- Zn system 131Materials Science
  126. 126. Ternary phase diagrams MgO-Al2O3-SiO2 system at 1 atm. pressure Fe-Ni-Cr ternary alloy system 132Materials Science
  127. 127. Formation of nano crystallites/ grains Nuclei of the solid phase form and they grow to consume all the liquid at the solidus line. 13 atoms constitute to a theoretical nano- particle for a FCC lattice having two layers. 55 and 147 atoms for 3 and 4 layer clusters. If the size of the crystallites are in the nanometer range, they are called nanocrystals/grains. High temperature structure can be retained at lower temperature by quenching. 133Materials Science
  128. 128. Single crystal A single crystal solid is a material in which the crystal lattice of the entire sample is continuous no grain boundaries- grain boundaries can have significant effects on the physical and electrical properties of a material single crystals are of interest to electric device applications 135Materials Science

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