Three cubic arrangements host most elemental crystals: simple cubic, body-centered cubic and face-centered cubic. This Cubic Cell Calculator takes the structure, the atomic radius and the molar mass, and returns the atoms per unit cell, the lattice edge length, the atomic packing factor and the theoretical density — the standard quartet of materials-science calculations from one geometry.
Worked Example: FCC Copper, r = 128 pm
In FCC, atoms touch along the face diagonal, so 4r = a√2 and the edge is a = 2√2 r = 2 × 1.4142 × 128 = 362.0 pm — against a measured copper lattice constant of 361.5 pm, agreement within a fraction of a percent. The cell owns 4 atoms (8 corners × ⅛ + 6 faces × ½), the packing factor is the close-packed π/(3√2) = 0.740, and the theoretical density follows: mass per cell = 4 × 63.546 / 6.022×10²³ = 4.220×10⁻²² g over a volume of (3.620×10⁻⁸ cm)³ = 4.744×10⁻²³ cm³, giving 8.89 g/cm³ versus copper’s measured 8.96 — the small shortfall is the hard-sphere idealization (real atoms are not billiard balls and the radius is itself a fitted quantity), and agreement at the 1% level is what this model promises. Change only the structure selector and the same radius tells the other geometries’ stories: BCC touches along the body diagonal (a = 4r/√3, 2 atoms, APF 0.680), simple cubic along the edge (a = 2r, 1 atom, APF 0.524) — one radius, three densities, which is why allotropy changes a metal’s weight without changing its atoms.
Counting Atoms and the Geometry of Touching
The census is bookkeeping about sharing: a corner atom is divided among 8 cells (⅛ each), a face atom between 2 (½), a body-center atom belongs wholly to its cell. Hence SC = 8 × ⅛ = 1; BCC = 1 + 1 = 2; FCC = 1 + 6 × ½ = 4. The edge formulas encode which line the spheres touch along — edge for SC, face diagonal for FCC, body diagonal for BCC — and the touching direction also sets the coordination number: 6, 12 and 8 respectively, FCC’s 12 being the maximum possible for equal spheres, shared only with hexagonal close packing (same APF 0.740, the Kepler ceiling). Density inverts just as cleanly: measured density plus structure yields the lattice constant, and X-ray crystallography lives on exactly that — diffraction gives a, this page’s algebra gives the rest, including a route to Avogadro’s number itself when density and molar mass are known (the silicon-sphere experiments that redefined the kilogram ran this calculation at metrological precision).
Which Metals Choose Which Cell
FCC hosts copper, aluminum, nickel, lead, gold and austenitic steels — ductile, because close-packed planes slide easily. BCC hosts iron at room temperature, chromium, tungsten and ferritic steels — stronger, less forgiving, and in iron’s case temporary: at 912 °C iron rearranges to FCC (austenite), the transformation on which all of steel heat treatment stands, with the density change of this page’s algebra (BCC iron’s model density from its 126 pm radius computes near 7.5 g/cm³ in the hard-sphere idealization) driving measurable volume changes during hardening. Simple cubic claims almost no elemental metals — polonium is the textbook exception — but the SC lattice underlies compound structures like CsCl (two interpenetrating SC sublattices) and perovskites, where the counting rules here still apply ion by ion. Whatever the system, the workflow is the one this page automates: census the cell, relate a to r along the touching direction, and let density fall out of mass over volume.
Common Mistakes
Counting corner atoms as whole atoms — the ⅛ sharing is the entire census game. Using the radius relation of the wrong structure (a = 2√2 r is FCC only; forcing it onto a BCC metal corrupts both edge and density). Slipping units in the density step: the edge must be converted from picometres to centimetres (×10⁻¹⁰) before cubing, or the density is wrong by 10³⁰. Quoting APF 0.74 for BCC — that is FCC/HCP’s ceiling; BCC packs 0.68. And treating the hard-sphere density as exact: thermal expansion, vacancies and the fitted nature of metallic radii keep honest agreement near the 1% level, as the copper example shows.
And the Non-Cubic Cousin
Zinc, magnesium and titanium crystallize hexagonal close-packed instead — a different cell, but the same 0.740 packing factor and coordination 12, because the ceiling belongs to sphere geometry rather than to the cube. The counting habit transfers intact: establish how atoms are shared, find the touching direction, and the density algebra of this page follows.
Related Tools
Continue with the Diffusion Coefficient Calculator, the Langmuir Isotherm Calculator, the Molar Mass Calculator, or browse the full Chemistry Calculators hub.
Frequently Asked Questions
How many atoms are in each cubic unit cell?
Simple cubic: 1 (8 corners × ⅛). BCC: 2 (corners plus the whole body-center atom). FCC: 4 (corners plus 6 faces × ½). Sharing fractions are the whole calculation.
What is the edge length formula for FCC?
a = 2√2 r, from atoms touching along the face diagonal (4r = a√2). For copper at r = 128 pm that gives 362.0 pm — matching the measured lattice constant of 361.5 pm closely.
What is the atomic packing factor?
The fraction of cell volume the atoms occupy: SC 0.524, BCC 0.680, FCC 0.740 — the last being the maximum possible for equal spheres, shared with hexagonal close packing.
How is theoretical density computed?
Density = (atoms per cell × molar mass / Avogadro’s number) ÷ edge³. FCC copper: 4 × 63.546 / 6.022×10²³ g in a (3.620×10⁻⁸ cm)³ cell = 8.89 g/cm³, about 1% under the measured 8.96.
Why does iron change structure on heating?
BCC ferrite rearranges to FCC austenite at 912 °C — the basis of steel heat treatment. The repacking changes density per this page’s algebra, producing the volume shifts hardening exploits and engineers must accommodate.
Is simple cubic used by any real element?
Essentially one: polonium. But the simple cubic lattice underlies important compound structures — CsCl and perovskites are built from interpenetrating simple-cubic sublattices.