About the Unit Cell Density Calculator
This unit cell calculator connects the three quantities that crystal-structure problems revolve around: the edge length of a cubic unit cell, the radius of the atoms in it, and the density of the solid. Choose simple cubic, body-centred cubic (BCC), face-centred cubic (FCC) or diamond cubic, enter the molar mass, and then give whichever one you know — edge length, atomic radius or density. The calculator returns the other two, plus the number of atoms per cell, the coordination number, the cell volume and the packing efficiency.
It is built for general chemistry and materials science students working solid-state problems, such as finding the density of copper from X-ray diffraction data, confirming that iron is BCC, or estimating Avogadro’s number from a crystal. The table compares all four lattices for your element so you can see which one matches a measured density.
The calculator assumes a single element in a perfect cubic lattice with atoms touching along the cell’s close-packed direction (edge, body diagonal or face diagonal). Real metals have small defects and thermal expansion, so calculated densities usually match handbook values within about 1%.
How to use the unit cell density calculator
- 1Choose the lattice type: FCC, BCC, simple cubic or diamond.
- 2Enter the molar mass of the element.
- 3Choose whether you know the edge length, atomic radius or density, and enter it.
- 4Read the answer, then check packing efficiency and coordination number.
- 5Use the table to see which lattice best matches a measured density.
Formula and method
Density is the mass of the atoms that belong to one unit cell divided by the cell volume. Each cubic lattice contains a fixed number of atoms per cell — 1 for simple cubic, 2 for BCC, 4 for FCC and 8 for diamond — so the mass is Z × M ÷ N_A and the volume is a³. The edge length is converted from picometres to centimetres (1 pm = 10⁻¹⁰ cm) so the density comes out in g/cm³.
Atoms touch along a different direction in each lattice, which fixes the ratio between edge length and atomic radius. Packing efficiency is the volume of Z spheres of radius r divided by a³: 52.36% for SC, 68.02% for BCC, 74.05% for FCC and 34.01% for diamond. Rearranging the density equation gives a = ∛(Z·M ÷ (N_A·ρ)).
- ρ
- Density (g/cm³)
- Z
- Atoms per unit cell
- M
- Molar mass (g/mol)
- N_A
- Avogadro constant, 6.02214076 × 10²³ mol⁻¹
- a
- Cubic cell edge length
- r
- Atomic (metallic or covalent) radius
Worked examples
Density of copper (FCC) from its edge length
Copper is FCC with a = 361.49 pm = 3.6149 × 10⁻⁸ cm. ρ = 4 × 63.546 ÷ (6.02214 × 10²³ × 4.7237 × 10⁻²³ cm³) ≈ 8.94 g/cm³, close to the handbook 8.96. r = a√2 ÷ 4 ≈ 127.8 pm.
Density of iron (BCC)
α-iron is BCC with 2 atoms per cell. ρ = 2 × 55.845 ÷ (N_A × (2.8665 × 10⁻⁸ cm)³) ≈ 7.87 g/cm³, and r = √3·a ÷ 4 ≈ 124.1 pm.
Silicon edge length from its density
Silicon has the diamond structure with 8 atoms per cell. a = ∛(8 × 28.0855 ÷ (N_A × 2.329)) = 5.431 × 10⁻⁸ cm = 543.1 pm, matching X-ray measurements. The covalent radius is √3·a ÷ 8 ≈ 117.6 pm.
Frequently asked questions
How do you calculate the density of a unit cell?+
Multiply the number of atoms per cell by the molar mass, divide by Avogadro’s number to get the mass of one cell, then divide by the cell volume a³. Remember to convert the edge length to centimetres for g/cm³.
How many atoms are in FCC, BCC and simple cubic cells?+
Simple cubic has 1 (8 corners × ⅛), BCC has 2 (corners plus one body centre) and FCC has 4 (corners plus 6 face centres × ½). The diamond cubic cell has 8.
Which cubic structure packs most efficiently?+
Face-centred cubic, at 74.05%, the maximum possible for equal spheres (shared with hexagonal close packing). BCC fills 68.02%, simple cubic 52.36% and diamond only 34.01%.
How do I tell whether a metal is FCC or BCC?+
If you know the density, molar mass and edge length, compute Z = ρ·N_A·a³ ÷ M. A result near 4 indicates FCC and near 2 indicates BCC. The table here shows the density each lattice would give.
What is the relationship between edge length and atomic radius?+
It depends on where atoms touch: a = 2r in simple cubic (along the edge), a = 4r/√3 in BCC (body diagonal) and a = 2√2·r in FCC (face diagonal).