Identify FCC, BCC and HCP structures and compute a metal's density from its unit cell.
Why it matters
Most metals and many ceramics are crystalline: their atoms sit in a regular, repeating 3D pattern. The pattern matters. Whether a metal is easily formed or tends to crack in the cold, how densely its atoms pack, and how it responds to heat treatment all depend on its crystal structure. Three structures cover most engineering metals, and learning them is the first step to reading the microstructure of anything metallic.
Lattices and unit cells
A crystal can be described by its unit cell: the smallest box of atoms that, repeated in every direction, builds the whole crystal. For cubic cells the box is a cube with edge length a, the lattice parameter, usually a few tenths of a nanometre. Treat atoms as hard spheres of radius R touching their nearest neighbours.
Four numbers describe each structure:
- n, the number of atoms belonging to one unit cell (corner atoms are shared by 8 cells, face atoms by 2);
- the coordination number (CN), how many nearest neighbours each atom touches;
- the relation between a and R, from where the atoms touch;
- the atomic packing factor (APF), the fraction of the cell volume filled by atoms.
The three common metal structures
Body-centred cubic (BCC). An atom at each corner and one in the centre. Atoms touch along the body diagonal, so 4R = a·√3.
n = 8 × 1/8 + 1 = 2 CN = 8 a = 4R/√3 APF = 0.68
Examples: iron at room temperature (ferrite), chromium, tungsten, molybdenum.
Face-centred cubic (FCC). An atom at each corner and one in the centre of each face. Atoms touch along the face diagonal, so 4R = a·√2.
n = 8 × 1/8 + 6 × 1/2 = 4 CN = 12 a = 2R·√2 APF = 0.74
Examples: aluminium, copper, nickel, gold, silver, and iron above 912 °C (austenite).
Hexagonal close-packed (HCP). Layers of close-packed atoms stacked in an ABAB pattern. Like FCC it is as dense as identical spheres can be:
n = 6 per hexagonal cell CN = 12 APF = 0.74 ideal c/a = 1.633
Examples: titanium, magnesium, zinc, cobalt.
FCC and HCP are both close packed; they differ only in the stacking of the close-packed layers (ABCABC for FCC, ABAB for HCP).
Why structure changes behaviour
Metals deform plastically by planes of atoms slipping past each other. FCC metals have many close-packed slip planes and directions, so they are typically ductile even at low temperature: aluminium and copper can be drawn into fine wire. BCC metals, including ordinary steels, can become brittle when cold (the ductile-to-brittle transition, covered later). HCP metals have fewer easy slip systems and are often less formable at room temperature.
Some materials change crystal structure with temperature. This is polymorphism (called allotropy for pure elements). Iron is BCC below 912 °C, FCC from 912 to 1,394 °C, then BCC again up to its melting point. That change is the basis of steel heat treatment.
Worked example: the density of copper
The theoretical density of a crystal is the mass of the atoms in one unit cell divided by the cell's volume:
ρ = n · A / (Vc · NA)
where A is the atomic mass (g/mol), Vc = a³ for a cubic cell, and NA = 6.022 × 10²³ atoms/mol.
Copper is FCC with R = 0.128 nm and A = 63.55 g/mol.
a = 2 · 0.128 · √2 = 0.3620 nm = 3.620 × 10⁻⁸ cm
Vc = a³ = 4.745 × 10⁻²³ cm³
ρ = 4 × 63.55 / (4.745 × 10⁻²³ × 6.022 × 10²³) = 254.2 / 28.57 = 8.90 g/cm³
The measured density of copper is 8.96 g/cm³: within 1 %. Doing the same for BCC iron (R = 0.124 nm, A = 55.85 g/mol) gives a = 0.2864 nm and ρ ≈ 7.90 g/cm³, against a measured 7.87 g/cm³.
Watch the units: convert nanometres to centimetres (1 nm = 10⁻⁷ cm) so the answer comes out in g/cm³.
Polycrystals and non-crystalline materials
A real metal part is polycrystalline: millions of small crystals (grains), each with the same structure but a different orientation, meeting at grain boundaries. Glasses and many polymers are amorphous: their atoms or chains have no long-range order. That is why glass softens gradually instead of melting at one temperature.
Common mistakes
- Counting corner atoms as whole atoms. Each corner atom is shared by eight cells.
- Using the wrong diagonal. BCC atoms touch along the body diagonal; FCC atoms touch along the face diagonal.
- Unit slips between nm, Å (1 Å = 0.1 nm) and cm in density calculations.
Recap
- BCC: 2 atoms, CN 8, APF 0.68. FCC: 4 atoms, CN 12, APF 0.74. HCP: CN 12, APF 0.74.
- ρ = n·A / (a³·NA) predicts density within about 1 % for pure metals.
- Iron changes from BCC to FCC on heating, which makes steel heat treatment possible.
Try it
On the Do tab, finish the density function and compute the densities of copper and iron from their crystal structures.