Unit 1

• Crystal


  Mineral formed underground from three-dimensional repeating patterns of atoms
• Crystals used in early civilizations


  • Quartz
  • Garnet
  • Diamonds
• Synthetic crystals used in industries


  • Diamond bits
  • Synthetic quartz, ruby and sapphire
  • Ruby laser
• Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids

  
• Crystal structure


  Ordered arrangement of atoms, ions or molecules in a crystalline material

  
• Atom


  Smallest constituent of ordinary matter that has the properties of a chemical element
• Ion


  Atom or molecule that has a non zero net electrical charge
• Cation


  Positively charged ion
• Anion


  Negatively charged ion
• Molecule


  Electrically neutral group of two or more atoms held together by chemical bonds
• Crystal/Crystalline solid


  Solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions

  
• Unit cell


  Smallest group of particles in the material that constitutes the repeating pattern
• Crystalline lattice


  Regular arrangement of atoms within a crystalline solid

  
• Coordination number


  Number of atoms with which a given atom can strongly interact
• Packing efficiency


  Percentage of the volume of the unit cell occupied by the spheres
• Simple cubic lattice


  • One atom at each corner
  • Atoms touch along each edge
  • Packing efficiency = 52%
• Body-centered cubic lattice


  • One atom in the center and one at each corner
  • Atoms touch along a diagonal through the center of the cube
  • Packing efficiency = 68%
• Face-centered cubic lattice


  • One atom at each corner and one atom in the center of each face
  • Atoms touch along the diagonal face
  • Packing efficiency = 74%

  
• Relating density to crystal structure


  Example problem: Calculating density of solid crystalline chromium
• Chromium


  Crystallizes with a body-centered cubic unit cell
• Chromium atom radius


  125 pm
• Calculating density of solid crystalline chromium

  1. Volume (V) = (4/3)πr^3

  2. Mass (m) = (Avogadro's number * atomic weight) / moles

  3. Density (ρ) = m/V
• Density of solid crystalline chromium is 7.18 g/cm^3
• Simple Cubic

  • Number of atoms per unit cell: 1
  • Relation between side of cell (l) and atomic radius (r): l = 2r
  • Packing Efficiency: 52.4%
  • Empty Space: 47.6%
• Body-Centered Cubic


  • Number of atoms per unit cell: 2
  • Relation between side of cell (l) and atomic radius (r): l = √(3/2)r
  • Packing Efficiency: 68%
  • Empty Space: 32%
• Face-Centered Cubic

  • Number of atoms per unit cell: 4
  • Relation between side of cell (l) and atomic radius (r): l = 2r√2
  • Packing Efficiency: 74%
  • Empty Space: 26%

  
• Closest-Packed Structures


  • Simple cubic structure has a lot of empty spaces
  • More efficient packing is achieved by offsetting the second layer by 1/2 atom so that the atoms sit in the indentations formed by the atoms in the layer below
• Hexagonal Closest Packing


  • Third layer aligned with first layer
  • ABAB pattern
  • Coordination number = 12
  • Packing efficiency = 74%
  • Unit cell is hexagonal
• Cubic Closest Packing


  • Third layer offset from first layer
  • ABCABC pattern
  • Coordination number = 12
  • Packing efficiency = 74%
  • Identical to face-centred cubic unit cell structure
• Metal Crystal Structures


  • Aluminum: FCC, Atomic Radius: 0.1431 nm
  • Cadmium: HCP, Atomic Radius: 0.1490 nm
  • Chromium: BCC, Atomic Radius: 0.1249 nm
  • Cobalt: HCP, Atomic Radius: 0.1253 nm
  • Copper: FCC, Atomic Radius: 0.1278 nm
  • Gold: FCC, Atomic Radius: 0.1442 nm
  • Iron: BCC, Atomic Radius: 0.1241 nm
  • Lead: FCC, Atomic Radius: 0.1750 nm
  • Magnesium: HCP, Atomic Radius: 0.1599 nm
  • Molybdenum: BCC, Atomic Radius: 0.1363 nm
  • Nickel: FCC, Atomic Radius: 0.1246 nm
  • Platinum: FCC, Atomic Radius: 0.1387 nm
  • Silver: FCC, Atomic Radius: 0.1445 nm
  • Tantalum: BCC, Atomic Radius: 0.1430 nm
  • Titanium: HCP, Atomic Radius: 0.1445 nm
  • Tungsten: BCC, Atomic Radius: 0.1371 nm
  • Zinc: HCP, Atomic Radius: 0.1332 nm
• Translational Symmetry


  Periodic repetition of a structural feature across a length or through an area or volume
• Point Symmetry


  Periodic repetition of a structural feature around a point. Includes reflection, rotation, and inversion.

  
• Reflection Symmetry


  Structural features on one side of a plane passing through the center of a crystal are the mirror image of the structural features on the other side
• Rotational Symmetry


  Structural element is rotated a fixed number of degrees about a central point and then repeated

  
• Inversion Symmetry


  Any line drawn through the origin at the center of the crystal will connect two identical features on opposite sides
• Rotoinversion Symmetry


  Combination of rotation and inversion. 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold rotoinversion operations exist.
• There are 32 different possible combinations of symmetry elements, corresponding to 32 crystal classes
• Crystal Systems


  • Isometric (Cubic)
  • Hexagonal
  • Tetragonal
  • Orthorhombic
  • Monoclinic
  • Triclinic

  
• Isometric (Cubic) Crystal System

  • 4 3-fold axes of symmetry
  • Up to 3 4-fold axes of rotational symmetry
  • Up to 6 2-fold axes of symmetry
  • Up to 9 mirror planes

  
• Tetragonal Crystal System


  • 1 4-fold symmetry axis
  • Up to 4 2-fold axes of rotation
  • Center of inversion
  • Up to 5 mirror planes