Metals and many simple solids crystallise in cubic lattices, where the repeating unit – the unit cell – is a cube of edge length a. In a simple cubic (SC) cell the atoms sit only at the eight corners and touch along the edges. A body-centred cubic (BCC) cell adds one atom in the middle, and the atoms touch along the body diagonal. A face-centred cubic (FCC) cell puts an atom in the centre of each face, and they touch along the face diagonals; FCC is the same arrangement as cubic close packing.
Because corner atoms are shared by eight cells and face atoms by two, an SC cell contains 1 atom, BCC 2 and FCC 4. The diamond structure (carbon, silicon, germanium) is an FCC lattice with a second, shifted set of atoms, giving 8 per cell. The geometry of touching spheres fixes the relation between a and the atomic radius r, and from it the atomic packing factor – the fraction of space filled by atoms: 52.4% (SC), 68.0% (BCC), 74.0% (FCC) and only 34.0% for diamond.
Once the cell size is known, the density follows directly: the mass of Z atoms divided by the volume a³. Turned around, a measured density and molar mass give the lattice constant and radius – a classic way to check X-ray data or estimate Avogadro’s number.
How to use the Cubic Unit Cell Calculator
- Lattice type
- I know the
- Atomic radius (r)
- Edge length (a)
- Density (ρ)
- Molar mass of the element (M)
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose the lattice and what you know: the atomic radius, the edge length, or the density. The calculator applies the touching-sphere relation for that lattice, then computes the cell volume, packing efficiency and density with ρ = Z·M/(N_A·a³), with a converted to centimetres.
Worked example
Aluminium is FCC with a metallic radius of 1.43 Å. The edge length is a = 2√2 × 1.43 = 4.0447 Å (the measured value is 4.0495 Å), and the cell volume is 66.17 ų = 6.617 × 10⁻²³ cm³. With 4 atoms per cell and M = 26.982 g/mol, ρ = 4 × 26.982 ÷ (6.02214 × 10²³ × 6.617 × 10⁻²³) = 2.709 g/cm³, close to the handbook density of 2.70 g/cm³. For BCC iron with r = 1.241 Å, a = 4 × 1.241 ÷ √3 = 2.866 Å and ρ = 2 × 55.845 ÷ (6.02214 × 10²³ × (2.866 × 10⁻⁸)³) ≈ 7.88 g/cm³.
Cubic lattices compared
| Lattice | Atoms per cell | Coordination | a in terms of r | Packing efficiency | Examples |
|---|---|---|---|---|---|
| Simple cubic | 1 | 6 | 2r | 52.36% | α-Po |
| Body-centred cubic | 2 | 8 | 4r/√3 ≈ 2.309r | 68.02% | Fe, Cr, W, Mo, Na |
| Face-centred cubic | 4 | 12 | 2√2·r ≈ 2.828r | 74.05% | Al, Cu, Ni, Ag, Au, Pb |
| Diamond cubic | 8 | 4 | 8r/√3 ≈ 4.619r | 34.01% | C, Si, Ge |
Inputs explained
| Field | What to enter |
|---|---|
| Lattice type | Choose one: Simple cubic (SC), Body-centred cubic (BCC), Face-centred cubic (FCC / ccp), Diamond cubic |
| I know the | Choose one: Atomic radius r, Edge length (lattice constant) a, Density ρ |
| Atomic radius (r) | Units: Å, nm, pm, m |
| Edge length (a) | Units: Å, nm, pm, m |
| Density (ρ) | Units: g/mL, g/cm³, kg/m³, kg/L, lb/ft³ |
| Molar mass of the element (M) | Units: g/mol, kg/mol, Da (u), kDa |
What each result means
- Edge length (lattice constant) a = [edge]
- Atomic radius r = [radius]
- Atomic radius r = [radius]
- Edge–radius relation = [Lattice type] – 1
- Packing efficiency (atomic packing factor) = ([Atoms per unit cell (Z)]×4÷3×pi×[radius]^3÷[edge]^3)×100
- Empty space in the cell = (1 – [Atoms per unit cell (Z)]×4÷3×pi×[radius]^3÷[edge]^3)×100
- Nearest-neighbour distance (2r) = 2×[radius]
- Unit cell volume (a³) = [edge]^3
- Theoretical density = [Atoms per unit cell (Z)]×[Molar mass of the element (M)]÷(6.02214076e23×([edge]×100)^3)
- Atoms per cm³ (× 10²²) = [Atoms per unit cell (Z)]÷([edge]×100)^3÷1e22
Formula
SC: a = 2r, Z = 1, APF = π/6 · BCC: a = 4r/√3, Z = 2, APF = π√3/8 · FCC: a = 2√2·r, Z = 4, APF = π/(3√2) · diamond: a = 8r/√3, Z = 8 · ρ = Z·M ÷ (N_A·a³)
Worked example
Inputs — Lattice type: Face-centred cubic (FCC / ccp); I know the: Atomic radius r; Atomic radius (r): 1.43 Å; Molar mass of the element (M): 26.982 g/mol.
Results — Atoms per unit cell (Z) = 4; Edge length (lattice constant) a = 4.04465 Å; Edge–radius relation = a = 2√2·r – atoms touch along the face diagonal; Coordination number = 12; Packing efficiency (atomic packing factor) = 74.05 %; Empty space in the cell = 25.95 %; Nearest-neighbour distance (2r) = 2.86 Å; Unit cell volume (a³) = 66.1673 ų; Theoretical density = 2.7086 g/mL; Atoms per cm³ (× 10²²) = 6.0453.
Key terms
- Unit cell
- The smallest repeating block whose translations rebuild the whole crystal.
- Lattice constant (a)
- The edge length of the cubic unit cell.
- Atomic packing factor
- The fraction of the cell volume occupied by atoms modelled as hard spheres.
- Coordination number
- The number of nearest neighbours of each atom.
- Close packing
- Arrangement of equal spheres that fills 74% of space (FCC and HCP).
Frequently asked questions
How many atoms are in SC, BCC and FCC unit cells?
1, 2 and 4. Corner atoms count ⅛ (8 × ⅛ = 1), body-centre atoms count 1, and face atoms count ½ (6 × ½ = 3).
What is the packing efficiency of FCC?
π/(3√2) = 74.05%, the densest possible packing of equal spheres – the same as hexagonal close packing. BCC fills 68.02% and simple cubic 52.36%.
How do I get the edge length from the atomic radius?
a = 2r for SC, a = 4r/√3 ≈ 2.309r for BCC and a = 2√2·r ≈ 2.828r for FCC.
Which metals are FCC and which BCC?
FCC: aluminium, copper, nickel, silver, gold, platinum, lead. BCC: α-iron, chromium, tungsten, molybdenum, sodium, potassium. Only α-polonium is simple cubic.
How do I calculate density from the unit cell?
ρ = Z·M/(N_A·a³). For copper (FCC, a = 3.615 Å, M = 63.546 g/mol): 4 × 63.546 ÷ (6.022 × 10²³ × 4.724 × 10⁻²³ cm³) ≈ 8.93 g/cm³.
Why is my calculated density slightly off?
Metallic radii are rounded and depend on coordination number; real crystals also contain vacancies and impurities, and lattice constants change with temperature.
What is the coordination number?
The number of nearest neighbours touching each atom: 6 in SC, 8 in BCC, 12 in FCC and 4 in diamond.