What this calculator does
Three cubic lattices pack atoms with very different efficiency. Simple cubic manages 52.4%, body-centred cubic 68.0% and face-centred cubic 74.0%, which is the densest packing possible for identical spheres.
Those percentages are pure geometry and do not depend on the atomic radius at all. What the radius changes is the edge length: a 128 pm atom gives a 362.04 pm cell in FCC and only 256.00 pm in simple cubic, because in FCC the atoms touch along the face diagonal rather than along the edge.
The formula
Each lattice has a fixed relationship between atomic radius and edge length, set by which direction the atoms touch along. Simple cubic touches along the edge, BCC along the body diagonal and FCC along the face diagonal. Atoms per cell counts the fractions contributed by corner, face and body positions, and packing efficiency is their combined volume over the cell volume.
| Term | Meaning |
|---|---|
| Simple cubic | Atoms at corners only. One atom per cell, 52.4% efficient. Rare in nature. |
| Body-centred cubic | Corners plus one in the centre. Two per cell, 68.0% efficient. |
| Face-centred cubic | Corners plus one on each face. Four per cell, 74.0% efficient. |
| Coordination number | Nearest neighbours: 6 for SC, 8 for BCC, 12 for FCC. |
The inputs explained
| Field | What to enter |
|---|---|
| Lattice type | Lattice type. Most metals are FCC or BCC; polonium is the only element that is simple cubic at room temperature. |
| Atomic radius (pm) | Atomic radius in picometres. Copper is 128, iron 126. |
When to use it
Determining density from structure
Atoms per cell and cell volume give the theoretical density, which can be compared with a measurement.
Interpreting diffraction data
An X-ray measured lattice constant converts back to an atomic radius through these relationships.
Comparing metal structures
Why iron changes structure on heating, and what that does to its density, follows from these figures.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the three lattices compare?
The same atom in each of the three cubic lattices.
| Lattice type | Edge length a | Atoms per unit cell | Packing efficiency |
|---|---|---|---|
| Simple cubic | 256.00 pm | 1 | 52.4% |
| Body-centred cubic | 295.60 pm | 2 | 68.0% |
| Face-centred cubic | 362.04 pm | 4 | 74.0% |
Questions
Why is FCC the most efficient?
Because each atom touches twelve neighbours rather than eight or six, filling space more completely. Its 74.0% is the theoretical maximum for identical spheres, a result conjectured by Kepler in 1611 and not rigorously proved until 1998.
Does packing efficiency depend on atom size?
No. It is the ratio of atom volume to cell volume, and both scale with the cube of the radius, so the radius cancels entirely. A lattice of large atoms and one of small atoms have identical efficiency if they share a structure.
Why do metals adopt different structures?
A balance of electronic structure and temperature. Iron is BCC at room temperature, FCC when heated past 912 °C and BCC again above 1,394 °C. That FCC phase is denser, which is why iron contracts slightly when it transforms on heating.
How do I get density from this?
Multiply atoms per cell by the atomic mass, divide by the Avogadro constant, then divide by the cell volume. The answer comes out in grams per cubic centimetre once the units are reconciled, and comparing it with a measured density tests the assumed structure.
For crystal planes and spacings, see the Miller indices calculator. For ionic crystal energetics, see the lattice energy calculator.