Cubic Systems: Simple Cubic, BCC and FCC Explained

Cubic systems are the simplest and most symmetric of the seven crystal systems: the unit cell is a cube with all edges equal (a = b = c) and all angles 90°. Only three arrangements of lattice points fit in a cube — simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC) — and together they describe the structures of most common metals and many ionic compounds.

They matter because the cubic arrangement decides how densely atoms pack and therefore how a metal behaves: FCC metals like copper and aluminium are soft and ductile (easy to draw into wire), while BCC iron is harder, and the packing fraction even decides how much carbon steel can absorb.

Key Concepts

• Simple cubic (SC): atoms only at the 8 corners → 1 atom per unit cell. Coordination number 6. Very few elements use it (polonium is the classic example).
• Body-centred cubic (BCC): atoms at corners plus one at the body centre → 2 atoms per unit cell. Coordination number 8. Examples: iron (at room temperature), chromium, tungsten, sodium.
• Face-centred cubic (FCC): atoms at corners plus six face centres → 4 atoms per unit cell. Coordination number 12. Examples: copper, aluminium, nickel, silver, gold, lead.
• Packing efficiency: the fraction of the cell volume actually filled by atoms — 52% for SC, 68% for BCC, 74% for FCC (the maximum for equal spheres).

Formulas

• SC: atoms touch along the edge, so a = 2r (r = atomic radius). Atoms per cell = 1; packing efficiency = (π/6) ≈ 52.4%.
• BCC: atoms touch along the body diagonal, so √3 a = 4r, i.e. a = 4r/√3. Atoms per cell = 2; packing efficiency = (π√3/8) ≈ 68.0%.
• FCC: atoms touch along the face diagonal, so √2 a = 4r, i.e. a = 4r/√2. Atoms per cell = 4; packing efficiency = (π/(3√2)) ≈ 74.0%.

Worked Example

Copper crystallises in FCC with atomic radius r = 128 pm. Find its unit-cell edge length and density (molar mass of Cu = 63.55 g/mol).

1. FCC relation: a = 4r/√2 = 4 × 128 / 1.4142 = 362 pm = 3.62 × 10⁻⁸ cm.
2. Cell volume = a³ = (3.62 × 10⁻⁸)³ = 4.745 × 10⁻²³ cm³.
3. Mass per cell: 4 atoms × 63.55 g/mol ÷ (6.022 × 10²³ atoms/mol) = 4.221 × 10⁻²² g.
4. Density = mass/volume = 4.221 × 10⁻²² / 4.745 × 10⁻²³ = 8.90 g/cm³, matching copper's known density (~8.96 g/cm³).

Common Misconceptions

• "FCC and cubic close packing are different structures." They are the same arrangement viewed differently — FCC is the unit-cell description, cubic close packing is the layer-stacking description (ABCABC…). Both give 74% packing.
• "BCC is close-packed." No — BCC at 68% is not a close packing of spheres. Only FCC (cubic close packing) and HCP reach the 74% maximum.
• "The unit cell edge equals twice the atomic radius in all cubics." Only in simple cubic. In BCC and FCC the atoms touch along the body and face diagonals, giving a = 4r/√3 and a = 4r/√2 respectively.

Real-World Applications

• Metallurgy: FCC metals (Cu, Al, Ni) are ductile and used for wires and foils; BCC metals (W, Cr) stay strong at high temperature for turbine blades and tools.
• Steel hardening: heating iron converts BCC ferrite to FCC austenite, which dissolves far more carbon — quenching then traps the carbon, hardening the steel.

In short, the three cubic systems — SC (1 atom, 52%), BCC (2 atoms, 68%) and FCC (4 atoms, 74%) — describe most metals.