Cubic Cell Calculator

What Is a Cubic Unit Cell?

A cubic unit cell is the smallest repeating cube that builds a crystal lattice. In the cubic crystal family, three Bravais lattices appear: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). Each type places atoms differently, so the relationship between atomic radius $r$ and lattice constant $a$ changes.

This calculator converts atomic radius into lattice constant for monoatomic SC, BCC, and FCC crystals. It also reports atoms per cell, packing fraction, and cell volume. For related chemistry tools, see the Molar Mass Calculator and Combustion Analysis Calculator.

Lattice Constant Formulas

For hard-sphere packing along the closest contact direction:

Simple cubic: $a = 2r$
Body-centered cubic: $a = 4r / \sqrt{3}$
Face-centered cubic: $a = 4r / \sqrt{2}$

SC has 1 atom per cell, BCC has 2, and FCC has 4. FCC (and HCP) are closest-packed structures with the highest packing efficiency among common monoatomic metals.

Worked Examples

Aluminum crystallizes as FCC with $r = 1.43\,\text{Å}$:

$$a = 4 \times \frac{1.43}{\sqrt{2}} = 4.045\,\text{Å}$$

Iron in a BCC lattice with $r = 1.24\,\text{Å}$:

$$a = 4 \times \frac{1.24}{\sqrt{3}} = 2.864\,\text{Å}$$

Polonium is a rare simple cubic metal. With $r = 1.67\,\text{Å}$, $a = 2r = 3.34\,\text{Å}$ (literature values are near $3.35\,\text{Å}$).

How to Use This Calculator

Choose SC, BCC, or FCC.
Enter the atomic radius and select Å, pm, or nm.
Read the lattice constant instantly, plus packing fraction and cell volume.

Why Lattice Constants Matter

Lattice parameters connect atomic size to macroscopic crystal geometry. They appear in density estimates, diffraction indexing, and alloy design. Experimental lattice constants often come from X-ray diffraction, while these formulas give the ideal hard-sphere geometric limit for monoatomic crystals.

Frequently Asked Questions

What are the three cubic lattice types?

Simple cubic places atoms only at cube corners. Body-centered cubic adds an atom at the cube center. Face-centered cubic adds an atom at the center of each face.

How do I calculate the lattice constant from atomic radius?

Use a = 2r for simple cubic, a = 4r/√3 for BCC, and a = 4r/√2 for FCC, where r is the atomic radius.

Which cubic structure packs atoms most efficiently?

Face-centered cubic is a closest-packed structure and packs more efficiently than simple cubic or BCC for equal hard spheres.

What unit should I use for atomic radius?

Ångströms (Å) are traditional in crystallography. Picometers and nanometers are also fine; 1 Å = 100 pm = 0.1 nm.

Do these formulas work for ionic compounds?

They are for monoatomic hard-sphere models. Ionic and multi-atom bases need different geometry based on ion radii and coordination, often measured experimentally.

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