Calculate density, packing efficiency, atomic radius, and cell parameters for simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) unit cells.
Crystalline solids are built from repeating unit cells — the smallest structural unit that, when stacked in three dimensions, reproduces the entire crystal. The three cubic unit cell types are Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC). Each type has a unique arrangement of atoms, packing efficiency, and geometric relationship between the lattice parameter (edge length) and atomic radius.
Understanding unit cells is fundamental to solid-state chemistry and materials science. The unit cell type determines the density of the crystal, the coordination number (nearest neighbors), and the packing efficiency (fraction of space occupied by atoms). Metals, alloys, and ionic compounds all adopt specific crystal structures that control their physical properties.
This calculator lets you choose a cell type and either enter the lattice parameter or atomic radius to compute the other, along with density, packing efficiency, void space, and coordination number. Preset metals are included for quick exploration of real crystal structures.
Essential for solid-state chemistry and materials science. Calculate crystal density, compare cubic cell types, and understand the geometric relationships that govern crystal structures. This cubic unit cell calculator helps you compare outcomes quickly and reduce avoidable mistakes when making day-to-day care decisions. Use the estimate as a planning baseline and confirm final decisions with a qualified professional when risk is high.
Simple Cubic: a = 2r, atoms/cell = 1, CN = 6, packing = 52.4% BCC: a = 4r/√3, atoms/cell = 2, CN = 8, packing = 68.0% FCC: a = 2√2·r, atoms/cell = 4, CN = 12, packing = 74.0% Density = (Z × M) / (Nₐ × a³) where Z = atoms per cell, M = molar mass, Nₐ = Avogadro's number, a = edge length
Result: Density = 8.89 g/cm³
Copper crystallizes in FCC. With r = 128 pm: a = 2√2 × 128 = 361.8 pm = 3.618 × 10⁻⁸ cm. Density = (4 × 63.546) / (6.022×10²³ × (3.618×10⁻⁸)³) = 8.89 g/cm³, matching the experimental value.
X-ray crystallography uses Bragg's law (nλ = 2d sin θ) to determine lattice parameters from diffraction patterns. The pattern of present and absent reflections identifies the cell type: SC shows all (hkl), BCC requires h+k+l = even, FCC requires h,k,l all odd or all even.
Many elements exist in multiple crystal structures. Carbon forms diamond (FCC variant), graphite, and fullerenes. Iron transitions from BCC (α-Fe) to FCC (γ-Fe) to BCC (δ-Fe) with increasing temperature. These structural changes affect hardness, conductivity, and magnetic properties.
Real crystal structures include NaCl (two interpenetrating FCC lattices), CsCl (BCC-like but two species), diamond cubic (FCC with tetrahedral basis), and wurtzite. Understanding the cubic fundamentals is essential before tackling these more complex arrangements.
Copper, aluminum, gold, silver, nickel, platinum, lead, and calcium are FCC. FCC metals tend to be ductile and good conductors.
Iron (at room temperature), tungsten, chromium, molybdenum, sodium, potassium, and barium are BCC. This keeps planning practical and lowers the chance of preventable errors.
FCC has 74.0% packing (the maximum for identical spheres) vs. 68.0% for BCC. FCC atoms touch along the face diagonal while BCC atoms touch along the body diagonal.
HCP has the same packing efficiency as FCC (74.0%) but a hexagonal unit cell. Zinc, magnesium, titanium, and cobalt are HCP.
Corner atoms are shared by 8 cells (count 1/8); edge atoms by 4 (1/4); face atoms by 2 (1/2); body-center atoms by 1. SC: 8×(1/8) = 1. BCC: 8×(1/8) + 1 = 2. FCC: 8×(1/8) + 6×(1/2) = 4.
The number of nearest neighbors touching each atom. SC = 6, BCC = 8, FCC = 12. Higher coordination generally means higher density and packing efficiency.