The unit cell calculator above handles the three cubic lattices that most introductory and intermediate solid-state chemistry rests on: simple cubic, body-centred cubic and face-centred cubic. Give it an atomic radius and it returns the edge length of the cell; give it an edge length and it returns the radius. Either way it reports how many atoms the cell contains, what fraction of the cell volume those atoms actually occupy, and the theoretical density that follows from packing that mass into that volume.
Arb Digital publishes this alongside a wider set of free chemistry tools. The arithmetic here is short but easy to get wrong, because each structure has its own geometric relationship between the edge and the radius, and the density calculation involves a unit conversion from picometres to centimetres that is cubed — so an error of a factor of ten in the length becomes a factor of a thousand in the density. Having the conversion done consistently removes the most common source of a wildly wrong answer.
What This Unit Cell Calculator Does
A unit cell is the smallest repeating block that, stacked in three dimensions without gaps or rotations, reproduces the whole crystal. Every property of the crystal that depends on how tightly the atoms sit — density above all — can be derived from a single cell, which is what makes the cell such a useful object. This page covers the cubic family, where all three edges are equal and all angles are ninety degrees, so a single number describes the cell.
For each structure the tool applies three fixed geometric facts. The atoms per cell come from sharing: a corner atom is split between eight neighbouring cells and contributes one eighth, a face atom is split between two cells and contributes one half, and a body-centre atom belongs entirely to its own cell. The edge-to-radius relationship comes from asking along which direction the atoms actually touch. The packing efficiency then follows by dividing the total sphere volume in the cell by the cell volume.
Two boundaries are worth stating. This page describes the metric geometry of the cell — how big it is and what is in it. If you want to index a crystallographic plane or direction, the companion Miller indices calculator is the right page, and if you want the electrostatic energy released when an ionic lattice forms, that is the lattice energy calculator. This tool neither indexes planes nor computes energies.
How to Use It
- Choose the structure. Simple cubic is rare among elements; body-centred cubic is common among alkali metals and early transition metals; face-centred cubic is the closest-packed cubic arrangement and is very common among the later transition metals.
- Choose the direction. Work from a metallic radius when you have one tabulated, or from a lattice parameter when you have a diffraction result to work back from.
- Enter the length in picometres. Multiply an ångström figure by 100 and a nanometre figure by 1,000 before entering it.
- Enter the molar mass of whatever sits at each lattice point. Our molar mass calculator will build it from a formula if you need it.
- Compare the theoretical density with the measured one. A gap between the two is informative, and the section below explains what it usually means.
The Formulas and How It's Calculated
Start with the atom count. In simple cubic there are eight corners at one eighth each, so Z = 1. Body-centred cubic adds a whole atom at the centre, giving Z = 2. Face-centred cubic has eight corners at one eighth plus six faces at one half, giving Z = 4.
Next, where the atoms touch. In simple cubic they touch along the cell edge, so a = 2r. In body-centred cubic they touch along the body diagonal, whose length is a√3 and which spans four radii, so a = 4r/√3. In face-centred cubic they touch along the face diagonal, of length a√2, again spanning four radii, so a = 4r/√2 = 2√2 r.
Packing efficiency is then Z × (4/3)πr³ divided by a³. Substituting the three relationships gives the standard values: about 52.4% for simple cubic, 68.0% for body-centred cubic and 74.0% for face-centred cubic, the last being the maximum achievable for equal spheres. Chemistry LibreTexts sets out the same derivation in its section on cubic lattices and close packing.
Finally, density. Mass in the cell is Z × M / NA, where NA is the Avogadro constant, defined exactly as 6.022 140 76 × 1023 mol−1 in the NIST CODATA value for the Avogadro constant. Divide by a³ expressed in cubic centimetres and you have grams per cubic centimetre.
A Worked Example With Copper
Copper is face-centred cubic with a metallic radius near 128 pm and a molar mass of 63.55 g/mol. The edge length is 2√2 × 128 = 362.0 pm, which is 3.620 × 10−8 cm. Cubing that gives 4.746 × 10−23 cm³. The cell holds four atoms, so its mass is 4 × 63.55 / 6.022 × 1023 = 4.222 × 10−22 g. Dividing mass by volume gives about 8.89 g/cm³, against a measured value near 8.96. That is the level of agreement the model should give for a well-behaved metal, and the small shortfall is exactly the kind of discrepancy the next section is about.
Run the same numbers through a body-centred cubic assumption and the answer collapses. The edge would be 4 × 128 / √3 = 295.6 pm, the cell would hold two atoms, and the density would come out near 8.16 g/cm³. Nothing about the arithmetic warns you; only knowing that copper is face-centred does. Structure is an input, not something the calculation can infer.
When Theoretical and Measured Density Disagree
A theoretical density is what a perfect, defect-free crystal of that structure would weigh. Real materials are lighter, and usually for one of three reasons. Vacancies are the first: every crystal above absolute zero has some fraction of empty lattice sites, and the equilibrium concentration climbs steeply with temperature. Grain boundaries and dislocations are the second, since the material at a boundary is less densely packed than the bulk. Porosity is the third and by far the largest in a sintered or cast sample, where voids on a scale far above the atomic can remove several percent of the mass.
The interesting case is the opposite direction. If a measured density is higher than the theoretical value, the usual explanation is not a denser crystal but a wrong assumption — most often an interstitial impurity squeezed into the holes between the host atoms, or an incorrect structure assignment. Comparing your figure against a bulk measurement made with our density calculator is a quick way to catch a mis-assigned lattice before it propagates through a longer calculation.
Why Radius Values Differ Between Tables
An atom has no sharp edge, so every atomic radius is an operational definition rather than a measurement of a boundary. The metallic radius is half the distance between nearest neighbours in the metal, and it is the one this calculator implicitly assumes. The covalent radius is half the bond length in a homonuclear covalent bond and is generally smaller. The van der Waals radius, taken from the closest approach of non-bonded atoms, is considerably larger. Ionic radii differ again, and depend on charge and on coordination number.
The practical consequence is that mixing radius types produces answers that are wrong by ten or twenty percent while looking entirely plausible. If you take a radius from one table and a lattice parameter from another and they disagree, suspect the definition before you suspect the crystal. When you have diffraction data, the safest route is to work backwards: measure the lattice parameter, use our Bragg's law calculator to relate diffraction angle and spacing, and let the radius fall out of the geometry.
Beyond the Three Cubic Cases
Cubic cells are the easy end of crystallography. The full set of Bravais lattices runs to fourteen, spanning tetragonal, orthorhombic, monoclinic, triclinic, hexagonal and rhombohedral systems, each with its own edge lengths and angles. Hexagonal close packing deserves a special mention: it fills exactly the same 74% of space as face-centred cubic and has the same coordination number of twelve, differing only in the stacking sequence of the close-packed layers — ABAB rather than ABCABC. Chemistry LibreTexts covers that comparison in its treatment of unit cells and basic structures.
Ionic solids introduce a further complication this page does not model: two different sphere sizes. In rock salt, zinc blende and fluorite the anions form a close-packed array and the cations occupy octahedral or tetrahedral holes, so the touching condition involves the sum of two radii rather than twice one. The atom-count logic still works, and the density formula still works with the formula mass in place of the atomic mass, but the edge-to-radius relationships shown here do not apply.
Arb Digital publishes hundreds of free, no-signup tools covering chemistry, physics, biology, finance and everyday maths — no accounts, no stored data.
Browse All Free Tools Contact Arb DigitalCommon Mistakes to Avoid
- Assuming the atoms touch along the edge in every structure. They touch along the edge only in simple cubic, along the face diagonal in FCC and along the body diagonal in BCC.
- Counting a corner atom as a whole atom. Corners contribute one eighth, faces one half, edges one quarter, and only a body-centre atom counts as one.
- Forgetting to cube the unit conversion. Picometres to centimetres is a factor of 1010, so the volume factor is 1030.
- Mixing radius definitions — metallic, covalent, ionic and van der Waals radii are different quantities and are not interchangeable.
- Expecting exact agreement with a measured density. Vacancies, grain boundaries and porosity all make real samples lighter than the ideal cell predicts.
Related Free Tools From Arb Digital
For the crystallographic side, the Miller indices calculator handles plane and direction indexing and the Bragg's law calculator links diffraction angle to interplanar spacing. For the chemistry, the molar mass calculator and the average atomic mass calculator supply the mass term, while the moles to grams calculator handles the stoichiometric conversions. The lattice energy calculator covers ionic lattice energetics and the density calculator handles bulk mass over volume. Browse the full free online tools hub for more.
Frequently Asked Questions
Four. Eight corner atoms each contribute one eighth, giving one atom, and six face atoms each contribute one half, giving three more. Simple cubic contains one atom per cell and body-centred cubic contains two.
Simple cubic fills about 52.4% of the cell volume, body-centred cubic about 68.0%, and face-centred cubic about 74.0%. The face-centred value is the theoretical maximum for identical spheres and is shared with hexagonal close packing.
It depends on where the atoms touch. In simple cubic the edge equals twice the radius. In body-centred cubic the edge equals four radii divided by the square root of three. In face-centred cubic the edge equals four radii divided by the square root of two.
Multiply the atoms per cell by the molar mass, divide by the Avogadro constant to get the mass in the cell, then divide by the cell volume. Convert the edge length to centimetres before cubing it so the result comes out in grams per cubic centimetre.
Because the calculation assumes a perfect crystal. Real samples contain vacancies, grain boundaries, dislocations and, in cast or sintered material, pores. All of these remove mass without changing the lattice, so measured densities are usually slightly lower.
Six in simple cubic, eight in body-centred cubic and twelve in face-centred cubic. The coordination number counts nearest neighbours touching a given atom, and it rises with packing efficiency.
Only partly. The atom-counting rules and the density formula still apply if you use the formula mass, but the edge-to-radius relationships assume identical spheres. Ionic structures involve a large anion array with smaller cations in the holes, so the touching condition uses the sum of two different radii.
This calculator applies the standard geometric relationships for cubic unit cells and is provided for educational purposes. Theoretical densities describe idealised, defect-free crystals and will differ from measurements made on real material.
