Cubic Unit Cell Calculator: Edge Length, Atomic Radius, Packing Fraction and Density (SC, BCC, FCC, Diamond)

How it works

SC: a = 2r, Z = 1, APF = π/6 = 52.36% · BCC: a = 4r/√3, Z = 2, APF = π√3/8 = 68.02% · FCC: a = 4r/√2 = 2√2 r, Z = 4, APF = π√2/6 = 74.05% · diamond: a = 8r/√3, Z = 8, APF = π√3/16 = 34.01% · density ρ = Z × M ÷ (Nᴀ × a³) · a = ∛(Z × M ÷ (ρ × Nᴀ))

In a cubic metal the atoms are treated as hard spheres that touch along one direction of the cell. In simple cubic they touch along the edge (a = 2r), in body-centered cubic along the body diagonal (a√3 = 4r), in face-centered cubic along the face diagonal (a√2 = 4r), and in diamond cubic along a quarter of the body diagonal (a√3 = 8r). Each geometry fixes how many atoms belong to one cell (1, 2, 4 or 8, counting ⅛ of each corner atom, ½ of each face atom and all of the interior ones) and what fraction of the cell they fill. Enter the radius to get the edge length, the edge length to get the radius, or the measured density to work back to the edge length with a = ∛(Z M ÷ ρ Nᴀ). With the molar mass the calculator gives the density, cell volume, atoms per cm³, molar volume, nearest-neighbor distance and the radius of the largest interstitial void, plus a table that puts the same atoms in all four cells so you can compare packing. No other calculator on the site covers unit cells; for diffraction from these cells use the Miller indices and Bragg’s law calculators. Results assume a perfect, single-element crystal of rigid spheres.

Worked example

Copper is FCC with an atomic radius of 1.28 Å and a molar mass of 63.546 g/mol. Atoms touch along the face diagonal, so a = 2√2 × 1.28 = 3.6204 Å. The cell volume is 3.6204³ = 47.453 ų = 4.7453 × 10⁻²³ cm³. With 4 atoms per cell, ρ = 4 × 63.546 ÷ (6.02214 × 10²³ × 4.7453 × 10⁻²³) = 8.895 g/cm³, close to copper’s measured 8.96 g/cm³. The atoms fill 74.05% of the cell and each one has 12 nearest neighbors 2.56 Å away.

Sources: Omni Calculator: Cubic cell calculator (Al, Fe and Po examples) · Vedantu: Density of a unit cell (silver FCC example) · NIST: Periodic Table of the Elements, 2024 (atomic weights)

Frequently asked questions