Simple Cubic, BCC and FCC Structures: Crystal Lattices

A crystal is an orderly arrangement of atoms that repeats in all directions, and the small repeating box is called the unit cell. The three simplest cubic unit cells are named by where their atoms sit. In a simple cubic (SC) cell, atoms sit only at the eight corners of the cube. In a body-centred cubic (BCC) cell, there is one extra atom at the very centre of the cube. In a face-centred cubic (FCC) cell, there is one extra atom at the centre of each of the six square faces.

This matters because the way atoms pack together decides the density, strength and even the malleability of a material. Iron's BCC form makes steel strong and hard, while aluminium's FCC structure makes it soft and easy to roll into foil.

Key Concepts

• Unit cell: the smallest repeating box of the crystal; its edge length is the lattice constant a.
• Coordination number: how many nearest neighbours each atom touches. SC = 6, BCC = 8, FCC = 12.
• Nearest-neighbour distance: distance between centres of two touching atoms. SC = a, BCC = √3·a/2, FCC = a/√2.
• Atomic packing factor (APF): fraction of the cell volume occupied by atoms. SC = 0.52, BCC = 0.68, FCC = 0.74.
• Atoms per cell: corners count 1/8, faces 1/2, body centre 1 — giving 1 (SC), 2 (BCC) and 4 (FCC).

Formulas

• Atomic radius from touching atoms: SC: 2r = a; BCC: 4r = √3·a (atoms touch along the body diagonal); FCC: 4r = √2·a (atoms touch along the face diagonal). SI unit: metres.
• APF = (atoms per cell × 4/3·π·r³) / a³. This gives π/6 ≈ 0.52 for SC, π√3/8 ≈ 0.68 for BCC and π/(3√2) ≈ 0.74 for FCC. The formulas assume hard, identical, touching spheres.
• Density: ρ = n·M / (N_A·a³), where n is atoms per cell, M the molar mass in kg/mol, N_A = 6.022×10²³ mol⁻¹ and a the lattice constant in metres. Result in kg/m³.

Worked Example

Iron at room temperature is BCC with a = 2.87 Å (1 Å = 10⁻¹⁰ m). Find its density.

1. Atoms per cell for BCC: n = 1 + 8×(1/8) = 2.
2. Cell edge in centimetres: a = 2.87×10⁻⁸ cm. Cell volume a³ = (2.87×10⁻⁸)³ = 2.36×10⁻²³ cm³.
3. Mass in one cell: 2 atoms × 55.85 g/mol ÷ 6.022×10²³ atoms/mol = 1.855×10⁻²² g.
4. Density ρ = 1.855×10⁻²² / 2.36×10⁻²³ = 7.86 g/cm³, matching the measured 7.87 g/cm³ and confirming the BCC model.

Common Misconceptions

• "The pictures show real atom sizes." Textbook drawings exaggerate atom size so you can see the positions. Real atoms are much smaller relative to the cell, and there is empty space between them.
• "FCC and BCC are different materials." They are different arrangements, and one element can switch: iron is BCC below 912 °C and FCC above it.
• "Simple cubic is common." Almost no element uses it at normal pressure; only polonium does. BCC and FCC dominate because they pack more tightly.

Real-World Applications

• Steelmaking: the BCC-to-FCC change in iron on heating is what makes heat treatment and quenching of steel possible.
• Copper wiring and aluminium foil: both are FCC, which gives the 12 slip directions that make these metals ductile and easy to draw into wires.
• X-ray diffraction: each structure produces its own fingerprint of diffraction peaks, so unknown crystals are identified by matching the pattern to SC, BCC or FCC.

SC, BCC and FCC hold 1, 2 and 4 atoms per cell with packing fractions 0.52, 0.68 and 0.74. Their geometry fixes density, strength and deformation behaviour — the foundation of materials science.