Crystallography for 2nd Year Students

Crystallography

• Crystallography is a branch of chemistry for second-year students.
• Interstitial sites are locations in a crystal where atoms or ions different from the "normal" atoms can be placed.

Interstitial Sites

• Cubic site: An interstitial site with a coordination number of eight. An atom/ion in this site touches eight other atoms/ions.
• Octahedral site: An interstitial site with a coordination number of six. An atom/ion in this site touches six other atoms/ions.
• Tetrahedral site: An interstitial site with a coordination number of four. An atom/ion in this site touches four other atoms/ions.

Crystals Having Filled Interstitial Sites

• FCC lattices have:
  • 3 octahedral (Oh) sites at edge centers and 1 at body center.
  • 8 tetrahedral (Th) sites at 1/2, 1/4, 1/4 positions.
• Interstitial sites are important because they allow for the derivation of various crystal structures (FCC, BCC, HCP) with slightly different arrangements.

Density Calculations

• Density (ρ) of a crystalline material is calculated as:
  • ρ = (number of atoms in the unit cell, n) × (mass of atom, m) / (volume of unit cell, v)
• Number of atoms in unit cells:
  • BCC: 2
  • FCC: 4
  • HCP: 6
• Mass of atom (m) is the atomic weight (a) in g/mol or amu.
• To convert from amu to grams: (atomic weight in amu) / (Avogadro's number).
• Avogadro's number (NA): 6.023 × 10²³ atoms/mol
• Volume of the unit cell (Vc):
  • SC: a³
  • BCC: a³√3/4
  • FCC: a³√2/4

Density Calculation Example (Copper)

• Radius (Rcu) of copper = 0.128 nm
• Crystal structure of copper = FCC
• Atomic weight (Acu) of copper = 63.5 g/mol
• Number of atoms per unit cell (n) = 4
• Volume of unit cell (Vc) = a³ = (2R√2)³ = 16√2R³
• Calculate Density (ρ) using the formula above
• Results in density of approximately 8.89 g/cm³.

Planar Density

• Planar Density (PD) represents the density of atomic packing on a particular crystal plane.
• PD = (Number of atoms on a plane) / (Area of the plane)
• Example: In an FCC lattice, the {110} planes have 2 atoms per unit area.

Linear Density

• Linear Density (LD) refers to the number of atoms per unit length along a specific direction.
• LD = (Number of atoms along the direction) / (Length of the direction)
• Example: The <110> directions in an FCC lattice have a linear density of 2 atoms/√2a.

Lattice Constants and Atomic Radii

• Table provided that gives lattice constant (a,c) and atomic radius (R) for various metals with different crystal structures (BCC, FCC, HCP).

X-Ray Diffraction

• Bragg's law:
  • 2d sin θ = mλ
  • m: order of diffraction
  • d: spacing between crystal planes
  • θ: diffraction angle
  • λ: wavelength of X-rays
• Diffraction patterns help identify the crystal structure of a material.