Cubic Cell
Calculate lattice constants and APF for SC, BCC, and FCC cubic cells from atomic radius. Free online crystallography calculator with comparison charts.
About This Calculator
The Cubic Cell Calculator is a chemistry and materials science tool designed for students, researchers, and professionals working with crystal structures. It computes the lattice constant (a), atomic packing factor (APF), and unit cell volume for all three cubic Bravais lattices — simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) — from a single atomic radius input. Simply enter the atomic radius of your element in ångströms (Å) and instantly compare the structural parameters across the three cubic cell types side by side.
The lattice constant formulas are derived from the geometry of closely packed spheres in each cubic arrangement. For simple cubic cells where atoms touch along the cube edge, a = 2r. For body-centered cubic cells where atoms touch along the body diagonal, a = 4r/√3. For face-centered cubic cells where atoms touch along the face diagonal, a = 4r/√2. The atomic packing factor (APF) represents the fraction of the unit cell volume occupied by atoms: π/6 (52.36%) for SC, π√3/8 (68.02%) for BCC, and π√2/6 (74.05%) for FCC — making FCC the closest-packed structure alongside hexagonal close-packed (HCP).
Regional Notes
Worldwide: Crystallography uses standardized formulas and the ångström (Å = 10⁻¹⁰ m) as the unit of atomic measurement globally. The atomic radius values for elements are consistent across international scientific literature. Common metals and their crystal structures: iron (BCC at room temperature, FCC above 912°C), aluminum (FCC), copper (FCC), tungsten (BCC), and polonium (SC — the only element with this structure under normal conditions).
Frequently Asked Questions
What are the three types of cubic unit cells?
The three types of cubic unit cells are simple cubic (SC) with one atom per unit cell, body-centered cubic (BCC) with two atoms per unit cell, and face-centered cubic (FCC) with four atoms per unit cell. FCC has the highest atomic packing factor of about 74%.
How do you calculate the lattice constant of a cubic cell?
For a simple cubic cell, a = 2r. For a body-centered cubic (BCC) cell, a = 4r/√3. For a face-centered cubic (FCC) cell, a = 4r/√2, where r is the atomic radius and a is the lattice constant.
What is the atomic packing factor of a simple cubic cell?
The atomic packing factor (APF) of a simple cubic cell is π/6 ≈ 0.5236 or about 52.36%. This means about 52% of the unit cell volume is occupied by atoms, with the remaining 48% being empty space.
What is the atomic packing factor of a BCC cell?
The atomic packing factor (APF) of a body-centered cubic (BCC) cell is π√3/8 ≈ 0.6802 or about 68%. Common BCC metals include iron, chromium, tungsten, and vanadium.
What is the atomic packing factor of an FCC cell?
The atomic packing factor (APF) of a face-centered cubic (FCC) cell is π√2/6 ≈ 0.7405 or about 74%. FCC is one of the two closest-packed crystal structures (the other being HCP). Common FCC metals include aluminum, copper, gold, and silver.
Which elements crystallize in a simple cubic structure?
Polonium (Po) is the only known element that crystallizes in a simple cubic structure under normal conditions. Most metals prefer the more closely packed BCC, FCC, or HCP structures.
How many atoms are in a BCC unit cell?
A body-centered cubic (BCC) unit cell contains 2 atoms total: 1/8 atom at each of the 8 corners (8 × 1/8 = 1) plus 1 full atom at the body center.
How many atoms are in an FCC unit cell?
A face-centered cubic (FCC) unit cell contains 4 atoms total: 1/8 atom at each of the 8 corners (8 × 1/8 = 1) plus 1/2 atom at each of the 6 face centers (6 × 1/2 = 3).