Cubic Cell Calculator

Turn an atomic radius into a cubic lattice constant, and back.

Simple, body centered and face centered cells in six length units.

Updated October 19, 2026

CreatorFrank Zhao

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Introduction

A cubic cell is the repeating box of a crystal: a cube whose edge is called the lattice constant, $a$ , with atoms arranged inside it in one of three standard ways. This page converts between that edge and the radius of the atom it is built from, in either direction.

There is no single answer, because the three packings touch along different directions. In a simple cubic cell the atoms touch along the cube edge; in a body-centred cell along the body diagonal; in a face-centred cell along the face diagonal. Each direction is a different multiple of the radius, so the same atom needs a wider cell as the packing gets denser.

That makes the page useful from both ends. If you have a radius from a table and want to know how large the cell should be, fill in the radius. If you have a cell edge from a paper, a phase diagram or a diffraction pattern and want the radius behind it, fill in the edge. And running the same atom through all three packings is the quickest way to see why a measured edge points to one structure rather than another.

3 packings

Simple, body-centred and face-centred cubic

2 to 2.83

Cell edge per radius, packing by packing

Both directions

Radius in gives an edge; edge in gives a radius

Quick start

Choose the packing

The three options are the three cubic cells: simple cubic has one atom at each corner, body-centred cubic adds one at the centre of the cell, and face-centred cubic adds one at the centre of each face. Changing the choice keeps the radius you typed and re-solves the cell edge for the new packing, so you can compare them without retyping anything.

Type the radius

Put the radius into the radius box. The examples on this page use ångströms, the scale crystallographers usually quote for atoms; a value such as $1.67\ \mathrm{Å}$ is a fair starting point.

Or start from a cell edge

When the edge is the number you already have, type it into the lattice-constant box instead. The radius box is solved from it by the same relation read backwards, so either box can be the starting point.

Read the cell edge

With simple cubic selected, a radius of $1.67\ \mathrm{Å}$ gives a cell edge of twice that:

a_{\mathrm{SC}}

=

2 \times 1.67\ \mathrm{Å}

=

3.34\ \mathrm{Å}

The same atom in a body-centred cell would need $3.8567\ \mathrm{Å}$ , and in a face-centred cell $4.72347\ \mathrm{Å}$ . The packing, not the radius, is what moves the answer by that much.

Which way the atoms touch

All three formulas answer one question: which straight line through the cell runs from atom centre to atom centre, and how many radii fit along it?

Simple cubic: the edge is the contact line

Corner atoms touch their neighbours along the cube edge, so the edge is exactly two radii long. One atom per cell, six nearest neighbours, and the shortest possible relation between radius and edge:

a_{\mathrm{SC}} = 2r

Body-centred cubic: the body diagonal

The corner atoms touch the atom in the centre along the body diagonal. That diagonal is $\sqrt{3}$ times the edge, and a corner-to-corner run through the centre carries four radii:

\sqrt{3}\,a_{\mathrm{BCC}}

=

4r

\;\Rightarrow\;

a_{\mathrm{BCC}}

=

\frac{4r}{\sqrt{3}}

Face-centred cubic: the face diagonal

On every face, the corner atoms touch the atom at the centre of that face along the face diagonal. A face diagonal is $\sqrt{2}$ times the edge, and again it carries four radii:

\sqrt{2}\,a_{\mathrm{FCC}}

=

4r

\;\Rightarrow\;

a_{\mathrm{FCC}}

=

\frac{4r}{\sqrt{2}} = 2\sqrt{2}\,r

Reading the same relation backwards

Each formula is one multiplication, so it inverts in one step: divide the edge by its coefficient to recover the radius.

r = \frac{a_{\mathrm{SC}}}{2}

\qquad

r = \frac{\sqrt{3}\,a_{\mathrm{BCC}}}{4}

\qquad

r = \frac{\sqrt{2}\,a_{\mathrm{FCC}}}{4}

Put the three side by side and the geometry becomes a single number per packing: how many radii make one cell edge.

Packing	Contact line	Edge per radius	Atoms per cell	Nearest neighbours	Space filled
Simple cubic	Cube edge	2	1	6	52.36%
Body-centred cubic	Body diagonal	2.3094	2	8	68.02%
Face-centred cubic	Face diagonal	2.8284	4	12	74.05%

For the same atom, the face-centred cell is 41.4% wider than the simple cubic one, and the body-centred cell 15.5% wider. That extra width is what buys the denser packing: 74.05% of the cell filled instead of 52.36%, once the atoms are treated as touching spheres.

The geometry is exact; the radius is the approximation

The coefficients 2, 2.3094 and 2.8284 come from the cube itself and carry no experimental uncertainty. Everything that can be wrong in the answer arrives through the radius: which definition a table used, at what temperature it was measured, and how closely the real atoms behave like touching spheres. Choosing that one number well is the subject of the next section.

Worked examples

One atom, three cells

Situation: A table gives an atomic radius of 1.43 Å and you do not yet know which cubic packing the material adopts. Run the same radius through all three packings before hunting for the structure.

Inputs

Atomic radius

1.43\ \mathrm{Å}

Packings triedSimple, body-centred and face-centred

Result

Simple cubic

2.86\ \mathrm{Å}

Body-centred

3.30244\ \mathrm{Å}

Face-centred

4.04465\ \mathrm{Å}

In practice: Whichever edge is closest to a measured value identifies the packing. A mismatch across all three suggests the radius you entered, rather than the structure, is the problem.

Backwards from a measured edge

Situation: A body-centred metal has a measured cell edge of 3.000 Å and you want the radius behind it, for comparison with a table of atomic radii.

Inputs

PackingBody-centred cubic

Cell edge

3.000\ \mathrm{Å}

Result

r

=

\frac{\sqrt{3} \times 3.000\ \mathrm{Å}}{4}

=

1.29904\ \mathrm{Å}

In practice: Quote 1.29904 Å (129.904 pm) as the touching-sphere radius. It is half the nearest-neighbour distance, which for a body-centred cell is $\sqrt{3}\,a/2 = 2.598\ \mathrm{Å}$ .

Getting the radius right

Use the touching-sphere radius, not just any radius

Tables list several radii for the same element and they are not interchangeable. Iron is the clean example: WebElements gives it a measured cell edge of 286.65 pm and an Fe–Fe distance of 248.2 pm in the metal, so the radius this page needs is 124.1 pm — half that distance. Enter 124.1 pm in body-centred mode and the page returns 2.86597 Å, within 0.02% of the measured edge.

Enter instead the 140 pm that the radius table calls iron's empirical atomic radius, and you get 3.23316 Å — about 13% too wide. Its van der Waals radius of 244 pm is larger again. Halving the nearest-neighbour distance is the check that keeps you out of that trap.

Temperature is part of the measurement

A metal expands as it warms, so the cell edge measured at room temperature is slightly longer than one measured near absolute zero. For careful work, take the radius and the edge from the same temperature rather than mixing a room-temperature edge with a low-temperature radius.

Keep more digits than you report

If the edge feeds another calculation, use the unrounded value. Rounding 4.04465 Å to 4.04 Å before the next step already costs about 0.1%, and that error is carried into everything that follows.

Record the packing with the number

A cell edge alone does not say which packing produced it. Write the packing, the radius and the unit together, and the number stays checkable months later. An edge quoted without its packing cannot be compared with a measured value in any meaningful way.

Questions

Does this work for ionic crystals such as table salt?

Not directly. Sodium chloride has four formula units in its cubic cell, and its edge follows from the sodium–chlorine distance rather than from one atomic radius. The same is true of caesium chloride and most other compounds: their cells have a basis, so the edge is measured by diffraction and then used, not predicted from a single radius. For those solids the next step is usually the lattice energy calculator, which takes the cation–anion distance and the structure.

Is the lattice constant the same as the distance between neighbouring atoms?

Only in simple cubic, where the edge itself is the contact line. $a = 2r$ there. In a body-centred cell the nearest neighbours are $\sqrt{3}\,a/2$ apart, which is $2r$ , and in a face-centred cell $\sqrt{2}\,a/2$ . The cell edge is longer than the nearest-neighbour distance in both of those packings, so a bond length should be halved before it goes into the radius box.

Why does a radius above 3 Å get a warning?

It is a sanity check, not a hard physical limit. The page keeps the number you entered and asks you to look again, because a touching-sphere radius that large almost always means a different kind of radius went into the box: an ionic radius, a van der Waals radius, or a whole cell edge. The warning clears as soon as the value is back in the usual range.

How far to trust it

The model is identical touching spheres. The three relations assume every atom is the same hard sphere and that neighbours touch exactly. Real electron clouds, bond character and directional bonding make the effective radius depend on the direction and on the coordination.
One atom per lattice point. Metals and single-element crystals fit the model. Compounds whose cell has a basis need a measured edge, and alloys can have a slightly different edge from either pure metal.
Radius tables disagree with each other. Empirical, calculated, covalent, metallic and van der Waals radii for the same element can differ by tens of percent. Use a metallic or touching-sphere value, and say which one you used when you report an edge.
For specifications, measure rather than infer. Thermal expansion, defects, pressure and alloying all move a real cell edge. When the number matters for a material specification or a manufacturing decision, use diffraction data measured under the conditions you care about.

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