ch3

• Crystalline material


  Atoms are situated in a repeating or periodic array over large atomic distances - long-range order exists
• Noncrystalline or amorphous material


  Long-range atomic order is absent
• Crystal structure


  • Properties of some materials are directly related to their crystal structures
  • Significant property differences exist between crystalline and noncrystalline materials having the same composition
• Atomic hard-sphere model


  Atoms (or ions) are thought of as being solid spheres having well-defined diameters
• Lattice


  A three-dimensional array of points coinciding with atom positions (or sphere centers)
• Unit cell


  Small repeat entities used to describe crystal structures, wherein all the atom positions in the crystal may be generated by translations of the unit cell integral distances along each of its edges
• Metallic bonding is nondirectional in nature, leading to relatively large numbers of nearest neighbors and dense atomic packings for most metallic crystal structures
• Face-centered cubic (FCC) crystal structure

  Unit cell has cubic geometry, with atoms located at each of the corners and the centers of all the cube faces
• Body-centered cubic (BCC) crystal structure


  Unit cell has cubic geometry, with atoms located at all eight corners and a single atom at the cube center
• Hexagonal close-packed (HCP) crystal structure

  Unit cell has a hexagonal base and a parallelepiped shape
• Single crystal


  Material in which a continuous periodic arrangement of atoms extends throughout the entire sample
• Polycrystalline material


  Material composed of many small crystallites or grains
• Isotropy


  Material property is the same in all directions
• Anisotropy


  Material property varies with direction
• Body-centered cubic (BCC)


  Crystal structure where atoms are located at the corners and center of a cube
• Body-centered cubic (BCC) crystal structure


  • Center and corner atoms touch one another along cube diagonals
  • Unit cell length a and atomic radius R are related through a = 4R/√3
• Metals with BCC structure

  • Chromium
  • Iron
  • Tungsten
• Simple cubic (SC) crystal structure


  Unit cell with atoms only at the corners of a cube
• None of the metallic elements have the simple cubic crystal structure
• Hexagonal close-packed (HCP)


  Metallic crystal structure with a hexagonal unit cell
• Hexagonal close-packed (HCP) crystal structure


  • Top and bottom faces consist of 6 atoms forming regular hexagons
  • Atoms in midplane have nearest neighbors in adjacent planes
• Coordination number and atomic packing factor for HCP

  12 and 0.74
• Metals with HCP structure

  • Cadmium
  • Magnesium
  • Titanium
  • Zinc
• Determining FCC unit cell volume

  1. Use unit cell edge length a = 2R√2

  2. Volume VC = a^3 = 16R^3√2
• Calculating atomic packing factor for FCC

  1. Total atom volume VS = 16/3 πR^3

  2. Unit cell volume VC = 16R^3√2

  3. APF = VS/VC = 0.74
• Determining HCP unit cell volume

  1. Volume VC = 3a^2c√3/2

  2. In terms of atomic radius R: VC = 6R^2c√3
• Theoretical density for metals

  ρ = nA/(VCNA), where n = atoms per unit cell, A = atomic weight, VC = unit cell volume, NA = Avogadro's number
• Polymorphism


  Phenomenon where a material can have more than one crystal structure
• Allotropy


  Polymorphism in elemental solids
• Unit cell


  Parallelepiped that defines the geometry of a crystal structure
• Unit cell geometry


  • Defined by 3 edge lengths (a, b, c) and 3 interaxial angles (α, β, γ)
  • Lattice parameters
• Crystal systems


  • Cubic
  • Tetragonal
  • Hexagonal
  • Orthorhombic
  • Rhombohedral
  • Monoclinic
  • Triclinic
• Cubic system has the greatest degree of symmetry
• Triclinic system has the least symmetry
• FCC and BCC structures belong to the cubic crystal system, HCP falls within the hexagonal system
• Lattice position coordinates (Px, Py, Pz)

  Defined by fractional multiples (q, r, s) of the unit cell edge lengths (a, b, c)
• Determining crystallographic direction indices [uvw]

  1. Construct right-handed x-y-z coordinate system

  2. Determine coordinates of vector tail and head points

  3. Subtract tail from head coordinates

  4. Normalize coordinate differences by respective lattice parameters

  5. Reduce to smallest integer values
• Common crystallographic directions


  • [100]
  • [110]
  • [111]
• Negative indices are represented by a bar over the index
• Determination of directional indices

  1. Take note of vector tail and head coordinates

  2. Compute coordinate differences

  3. Use equations to calculate u, v, and w

  4. Enclose u, v, and w indices in brackets