crystal structures

• Crystalline


  Atoms are situated in a repeating or periodic array over large atomic distances. Crystalline solids have short range order as well as long range order (E.g. metals & ceramics)

  
• Amorphous


  Atoms are not packed in definite orderly arrangement. Amorphous solids have short range order but there is no long range order. (E.g. Glass & Polymers)
• Above their melting point, metals are liquids, and their atoms are randomly arranged and relatively free to move. However, when cooled below their melting point, metals rearrange to form ordered, crystalline structures.

  
• Amorphous solids


  • Tangled mass of long chained molecules (Polymers)
  • Three dimensional network of atoms with no long range order (glass)
• Single Crystals


  For a crystalline solid when the periodic and repeated arrangement of atoms extends throughout the entire specimen it results a single crystal
• Polycrystalline Materials


  A solid can be composed of many crystalline grains, not aligned with each other. Where they meet is called a grain boundary.
• Anisotropy


  Different directions in the crystal have a different packing. This causes anisotropy in the properties of crystals; for instance, the deformation depends on the direction in which a stress is applied.
• Lattice


  A three dimensional array of points coinciding with atom positions
• Unit cell


  The smallest structure that repeats itself by translation through the crystal
• Types of unit cells

  • Face-centered cubic (FCC)
  • Body-centered cubic (BCC)
  • Hexagonal close-packed (HCP)
• Body-centered cubic (BCC) unit cell


  • Atoms at each of the eight corners of a cube plus one atom in the center of the cube
• Face-centered cubic (FCC) unit cell


  • Eight atoms at corners of the unit cell and one atom centered in each of the faces
• Hexagonal close-packed (HCP) unit cell


  • Crystal arrangement of close-packed layers of particles where three layers of particles alternate positions
• Important properties of unit cells

  • Type of atoms and their radii
  • Cell dimensions
  • Number of atoms per unit cell
  • Coordination number
  • Atomic packing factor
• Density computations


  Divide mass of atoms in unit cell by volume of unit cell
• Polymorphism is when a material may exist in more than one crystal structure. If the material is an elemental solid, it is called allotropy.
• Close-packed crystal structures


  FCC and HCP are built by packing spheres on top of each other, in the hollow sites. The packing is alternate between two types of sites, ABABAB.. in the HCP structure, and alternates between three types of positions, ABCABC… in the FCC crystals.
• Packing factor (packing Density)


  Fraction of volumes of the unit cell occupied by the atoms. FCC & HCP ≈ 0.74, BCC ≈ 0.68
• Coordination Number (CN)


  Number of equidistant nearest neighboring atoms for each atom. BCC → CN= 8, HCP & FCC → CN= 12
• Lattice Parameters (constants)


  Height, width, Breadth of a unit cell
• ABCABC… positions


  Types of positions in FCC crystals
• Building 3D packing pattern

  1. Pack atoms two dimensionally in atomic planes

  2. Stack these planes on top of one another to give crystal
• Weight of atoms = atoms mole x atoms unitcell x g mole / 4 / 02 10.6 / 5.63 23 = 4.2192 x 10^23 g/unit cell
• Volume of unit cell = a^3 = (3.61 x 10^8)^3 cm^3
• Density = 4.2192x10^22 / (3.61x10^8)^3 = 8.96 g/cm^3
• Packing factor (packing Density)


  Fraction of volumes of the unit cell occupied by the atoms
• Packing factor = volume of atoms in unit cell / volume of the unit cell, FCC & HCP ≈ 0.74, BCC ≈ 0.68
• Coordination Number (CN)


  Number of equidistant nearest neighboring atoms for each atom
• BCC → CN= 8, HCP & FCC → CN= 12
• Relationship between lattice parameters (a, b, c) and the atomic radius (r)


  FCC: atoms contact in all the face diagonals, a = 2√2r

  BCC: Atoms contact along the body diagonal, a = 4√3r

  HCP: 2r = √3a, 4r = 3a
• Close Packed Planes (CPP) and Close Packed Directions (CPD)


  • Parallel planes and directions considered to be similar
• Packing factor


  True Volume / Bulk Volume

  True Volume = Volume occupied by atoms

  Bulk Volume = Volume occupied by unit cell
• Miller Indices


  Set of three numbers used to specify any plane or direction in a crystal
• Miller indices for a Plane

  • (100)
  • (010)
• Miller indices for a Direction

  • [106]
• Planar Density


  Number of atoms per unit area within a plane
• For a simple cubic structure, planar density of (100) > (110) > (111), and the close-packed plane is (100)
• For a FCC structure, planar density of (111) > (100) > (110), and the close-packed plane is (111)
• For a BCC structure, planar density of (110) > (100) > (111), and the close-packed plane is (110)
• Linear Density


  Number of atoms per unit length in a particular direction of the unit cell