Crystal Structure and Unit Cells

• Metals are solids
• Structures of metals can be determined using X-ray, neutron, and electron diffraction
• Diffraction gives uniquely precise structural detail on solids

Crystals

• Crystals exhibit long range order
• Crystals act as diffraction gratings for radiation
• Spacings in crystals are on the order of Å
• Radiation must have a similar λ to this (X-rays)
• X-ray diffraction is like a Young's slit experiment
• Crystallography knows the wavelength, but not the spacing in the crystal

Unit Cells

• Long range order in a crystal is characterised by a unit cell
• Unit cell is the basic building block of the crystal
• A section through the (3D) crystal structure of formamide

Unit Cell Dimensions

• The repeat unit is the unit cell
• The unit cell is a box in 3D
• The lengths of the sides are labelled a, b, and c
• The angles are labelled α, β, and γ
• The angles in the box do not need to be 90°
• Examples of Unit Cell Dimensions:
  • NaCl: a/Å 5.66, b/Å 5.66, c/Å 5.66, α/° 90, β/° 90, v/° 90
  • Formamide: a/Å 3.60, b/Å 9.04, c/Å 6.99, α/° 90, β/° 100.5, v/° 90
  • Mamba venom: a/Å 104.60, b/Å 38.27, c/Å 39.42, α/° 90, β/° 90, v/° 90

Crystal Systems

• Seven crystal systems exist
• Triclinic: a≠b≠c, α≠β ≠ γ
• Monoclinic: a≠b≠c, α= y = 90°, β ≠ 90°
• Orthorhombic: a≠ b≠c, α = β = γ = 90°
• Tetragonal: a = b ≠ c, α = β = γ = 90°
• Rhombohedral: a = b = c, α = β = γ ≠ 90°
• Hexagonal: a = b≠c, α = β = 90°, γ = 120°
• Cubic: a = b = c, α = β = γ = 90°

Centring

• Cells are primitive with one lattice point on each vertex
• There is one lattice point per cell.
• This is the simplest cell that could be drawn
• It is always possible to describe a crystal structure in this way
• Sometimes it is useful to take a larger cell to make the symmetry of the crystal more obvious
• An example has a=b=c and α=β=γ=109.5°
• This can be described with a body centred cubic cell which makes the symmetry of the lattice more obvious

Face-centred cubic unit cells

• A face-centred cubic unit cell if the primitive cell has a=b=c and α=β=γ=60°

Metals: Close Packing

• Metals can be thought of as been composed as hard spheres
• The first A-Layer has a unit cell with a=b and y=120°
• The third dimension is in the next A layer up
• This defines a hexagonal unit cell: Hexagonal Close Packing
• The unit cell of the ABCABC... stack is rhombohedral, but a=b=c and α=β=γ=60°, so this can be transformed into a face-centred cubic unit cell
• This is cubic close packing

Non-close Packed Metals Structures:

• The coordination number is HCP and CCP is 12
• HCP and CCP are the most efficient hard sphere packing
• Two other simple metallic structures also exist
• Body centred cubic (BCC): Z = 2.
  • Only slightly less efficient
  • Formally 8-coordinate, but 6 more atoms are only 15% further away
• Primitive cubic
  • Very rare and much less efficient
  • Z = 1
  • Only Po has this structure
  • More important in ionic salts

Summary of Lattice Structures

• HCP
  • Sequence: ABABAB...
  • Coordination: 12
  • Atoms/Cell (Z): 2
  • Examples: Mg, Co, Zn
  • Efficiency: 0.74
• CCP
  • Sequence: ABCABC...
  • Coordination: 12
  • Atoms/Cell (Z): 4
  • Examples: Ni, Cu
  • Efficiency: 0.74
• BCC
  • Sequence: Not close packed
  • Coordination: 8+6
  • Atoms/Cell (Z): 2
  • Examples: Na, Cr, Fe
  • Efficiency: 0.68
• PC
  • Sequence: Not close packed
  • Coordination: 6
  • Atoms/Cell (Z): 1
  • Examples: Po
  • Efficiency: 0.52

Polymorphism

• Metals may exhibit different structures under different conditions.
• This is called polymorphism
  • Fe is BCC at RT
  • CCP at 1179 K
  • BCC again at 1674 K to mpt
  • HCP at high pressure

Bonding in Metals:

• High coordination numbers of metals implies delocalised bonding.
• A simple picture of metal bonding is of ions in a delocalised 'electron sea'.

MO Theory (Band Theory)

• In band theory, a bulk metal is treated like a huge molecule
• Can be illustrated by thinking of a 1D metal built of a long row of metal orbitals

Sublimation Energies

• Maxima occur at group 5 (half-filled d-band)
• 4th period > 3rd period: better overlap of d-orbitals
• The dip after V in the 3rd period is associated with magnetism
• Examples of rmax(nd) / Å:
  • Co: 0.40
  • Rh: 0.68
  • Ir: 0.77