Crystal Lattice Density Calculator

Crystal Lattice Density The theoretical (X-ray) density of a crystal is calculated from its unit cell geometry and composition. ρ = (Z × M) / (Nₐ × V) Where: Z = number of formula units per unit cell M = molar mass of the formula unit (g/mol) Nₐ = Avogadro’s number (6.022 × 10²³ mol⁻¹) V = unit cell volume (cm³)

Cubic Crystal Systems Simple cubic: Z = 1, a = b = c, V = a³ Body-centered cubic (BCC): Z = 2, V = a³ Face-centered cubic (FCC): Z = 4, V = a³ Diamond cubic: Z = 8, V = a³

Other Crystal Systems Tetragonal: V = a² × c Orthorhombic: V = a × b × c Hexagonal: V = a² × c × sin(60°) = a² × c × √3/2

Packing Efficiency Simple cubic: 52.4%, atoms touch along the cube edge BCC: 68.0%, atoms touch along the body diagonal FCC: 74.0%, the highest packing for equal spheres (HCP ties it) Diamond: 34.0%, a very open structure held together by directional covalent bonds

Atomic radius from the lattice constant

The same touching geometry that sets the packing efficiency also gives you the atomic radius, which is where most published metallic radii come from in the first place:

Copper at a = 3.615 Å gives r = 1.278 Å, and iron at a = 2.8665 Å gives r = 1.241 Å. Both match the tabulated metallic radii to three figures, which is a useful way to check you picked the right structure: get the structure wrong and the radius comes out visibly off.

Converting Lattice Constants Lattice constants are given in ångströms (Å) or pm. 1 Å = 10⁻⁸ cm = 100 pm. Use cm³ in the density formula. Typical metal lattice constants: 2.5 to 6 Å. Typical ionic crystal constants: 4 to 12 Å.

Common Examples Iron (BCC): a = 2.8665 Å, M = 55.845, Z = 2 → ρ = 7.874 g/cm³ Copper (FCC): a = 3.615 Å, M = 63.55, Z = 4 → ρ = 8.935 g/cm³ (the measured value is 8.96; the small gap is normal, since X-ray density assumes a perfect defect-free crystal) NaCl (FCC-type): a = 5.64 Å, M = 58.44, Z = 4 → ρ = 2.164 g/cm³ Diamond (diamond cubic): a = 3.567 Å, M = 12.01, Z = 8 → ρ = 3.515 g/cm³

Note the iron figure. Round the lattice constant to 2.87 Å, as plenty of textbooks do, and the density drops to 7.847, because a cubed lattice constant magnifies any rounding threefold. If your answer misses a published density by a few tenths of a percent, the lattice constant is almost always the culprit rather than the arithmetic.

Theoretical against measured density

X-ray density assumes every lattice site is occupied. Real crystals have vacancies, dislocations and grain boundaries, so a measured density normally comes in slightly lower. A gap under about 0.5% is ordinary. A larger one usually means one of three things: the structure assumed here is not the structure present, Z is wrong for the compound, or the sample has real porosity, which is the common case with sintered ceramics and powder-metallurgy parts. Enter a measured density and the calculator reports the gap for you.