Unit Cells: Counting Atoms and Calculating Density

A tiled bathroom floor might contain thousands of tiles, but you only need to see one tile, and know how it repeats, to describe the whole floor. Crystals work the same way. A grain of salt holds something like 10¹⁸ ions, yet its entire structure can be captured by one small box of atoms that is copied, without rotation, in three directions. That box is the unit cell.

Once you understand unit cells, a surprising amount of solid-state chemistry falls into place. You can count how many atoms belong to one cell, relate the cell’s size to the size of the atoms, predict how efficiently space is filled and, with a little arithmetic, calculate the density of a solid from X-ray measurements alone.

What a unit cell is (and isn’t)

A crystal lattice is an infinite, regular array of points, each with an identical environment. A unit cell is a parallelepiped (a slanted box) that, when stacked face to face with copies of itself, fills space and rebuilds the lattice.

Three points trip people up:

• The cell is a repeating unit, not a molecule. The atoms at its corners are shared with neighbouring cells.
• You can draw more than one valid cell. Chemists usually choose the one that shows the symmetry most clearly, called the conventional cell, even if a smaller cell exists.
• Cells are described by six numbers: three edge lengths (a, b, c) and three angles (α, β, γ). For a cube, a = b = c and all angles are 90°, so one number, the lattice parameter a, is enough.

This post focuses on cubic cells, because they are the ones most exams ask you to calculate with, and because many metals and simple ionic solids are cubic.

Counting atoms: sharing the pieces

Imagine cutting a crystal into cube-shaped cells with a very sharp knife. An atom sitting exactly at a corner of a cell gets sliced into pieces, and each piece goes to a different cell. The fraction that belongs to one particular cell depends on where the atom sits:

Position of atom	Cells sharing it	Fraction in one cell
Corner	8	⅛
Edge	4	¼
Face	2	½
Body (fully inside)	1	1

A helpful analogy is a pizza at a table: the pizza in the middle of four people is one pizza, not four, however many plates it touches. An atom at a corner is shared by eight cells, so each cell really owns an eighth of it.

Apply this to the three cubic cells:

• Simple cubic (SC): 8 corners × ⅛ = 1 atom
• Body-centred cubic (BCC): 8 × ⅛ + 1 body = 2 atoms
• Face-centred cubic (FCC): 8 × ⅛ + 6 faces × ½ = 4 atoms

The same rules apply to ionic crystals, where you count each type of ion separately. In the sodium chloride cell, the chloride ions sit at the corners and face centres (4 in total), and the sodium ions sit at the 12 edge midpoints and the body centre: 12 × ¼ + 1 = 4. Four of each gives the formula NaCl, which is a good check that you’ve counted correctly. More on that in ionic crystal structures.

Linking cell size to atom size

Treating atoms as hard spheres, you can relate the edge length a to the atomic radius r by finding the direction in which neighbouring atoms actually touch.

• SC: atoms touch along the edge, so a = 2r.
• BCC: atoms touch along the body diagonal (length a√3), which spans 4r. So a = 4r/√3.
• FCC: atoms touch along the face diagonal (length a√2), which spans 4r. So a = 2√2 r.

Getting the touching direction right is most of the battle. A quick sketch of one face (for FCC) or a slice through the body diagonal (for BCC) makes it obvious.

Packing efficiency

The packing efficiency is the fraction of the cell’s volume actually occupied by atoms:

packing efficiency = (number of atoms × volume of one atom) ÷ volume of cell

For simple cubic: one atom of volume ⁴⁄₃πr³ in a cube of side 2r.

(⁴⁄₃πr³) ÷ (2r)³ = π/6 ≈ 0.524, so 52 %.

For BCC: two atoms in a cube of side 4r/√3.

2 × ⁴⁄₃πr³ ÷ (4r/√3)³ = π√3/8 ≈ 0.680, so 68 %.

For FCC: four atoms in a cube of side 2√2 r.

4 × ⁴⁄₃πr³ ÷ (2√2 r)³ = π/(3√2) ≈ 0.740, so 74 %.

Notice that r cancels every time. Packing efficiency depends only on the geometry, not on which element you use. Simple cubic is so wasteful that only one element, polonium, adopts it under ordinary conditions. FCC reaches the maximum possible for equal spheres, which it shares with hexagonal close packing (see metal crystal structures).

Calculating density from a unit cell

Density is mass divided by volume. For a crystal you can use one unit cell as the sample:

ρ = (Z × M) ÷ (N_A × a³)

where

• Z = number of atoms (or formula units) per cell,
• M = molar mass in g/mol,
• N_A = the Avogadro constant, 6.02214076 × 10²³ mol⁻¹ (see Avogadro’s number),
• a = edge length, converted to centimetres if you want g/cm³.

The unit conversion is where most marks are lost: 1 pm = 10⁻¹⁰ cm, so a cell edge in picometres must be multiplied by 10⁻¹⁰ before you cube it.

Worked example 1: copper (FCC)

Copper is FCC with a lattice parameter of about 361.5 pm. Its molar mass is 63.546 g/mol.

1. Z = 4.
2. a = 361.5 pm = 3.615 × 10⁻⁸ cm.
3. a³ = (3.615 × 10⁻⁸ cm)³ = 4.724 × 10⁻²³ cm³.
4. Mass of the cell = 4 × 63.546 ÷ (6.02214076 × 10²³) = 4.221 × 10⁻²² g.
5. ρ = 4.221 × 10⁻²² g ÷ 4.724 × 10⁻²³ cm³ = 8.93 g/cm³.

The measured density of copper listed on its element page is 8.933 g/cm³, so the agreement is excellent. Small differences, when they show up, have real causes: lattice parameters change slightly with temperature (a cell measured warmer is a little larger), real samples contain vacancies and grain boundaries, and a rounded lattice parameter carries rounding error that gets tripled when you cube it. A 0.1 % error in a becomes roughly a 0.3 % error in ρ.

We can also recover copper’s radius: r = a√2 ⁄ 4 = 361.5 × 1.4142 ÷ 4 ≈ 128 pm.

Worked example 2: iron (BCC)

At room temperature iron is BCC (α-iron) with a ≈ 286.7 pm and M = 55.845 g/mol.

1. Z = 2.
2. a³ = (2.867 × 10⁻⁸ cm)³ = 2.357 × 10⁻²³ cm³.
3. Mass of the cell = 2 × 55.845 ÷ (6.02214076 × 10²³) = 1.855 × 10⁻²² g.
4. ρ = 1.855 × 10⁻²² ÷ 2.357 × 10⁻²³ = 7.87 g/cm³.

That matches the tabulated 7.874 g/cm³ very closely. The radius from a = 4r/√3 is r = √3 × 286.7 ÷ 4 ≈ 124 pm.

Worked example 3: sodium chloride (rock salt)

For ionic solids, Z counts formula units. NaCl has 4 formula units per cell, a ≈ 564 pm and M = 22.990 + 35.45 = 58.44 g/mol.

1. a³ = (5.64 × 10⁻⁸ cm)³ = 1.794 × 10⁻²² cm³.
2. Mass of the cell = 4 × 58.44 ÷ (6.02214076 × 10²³) = 3.882 × 10⁻²² g.
3. ρ = 3.882 × 10⁻²² ÷ 1.794 × 10⁻²² = 2.16 g/cm³.

The same equation run backwards is historically important: if you know the density and the cell edge very precisely, you can solve for N_A. Measurements of this kind on nearly perfect silicon crystals helped pin down the Avogadro constant before its value was fixed exactly in 2019.

To check the final step of any of these, plug the mass and volume into the density calculator.

Common mistakes

• Counting every atom you can see. An FCC drawing shows 14 atoms, but the cell owns only 4. Always apply the sharing fractions.
• Forgetting to convert pm to cm before cubing. Cube 361.5 without converting and the answer is off by a factor of 10³⁰.
• Using the wrong touching direction. In BCC, corner atoms don’t touch each other, so a ≠ 2r.
• Using Z = 1 for NaCl because the formula has one Na. Z is the number of formula units in the cell, which is 4 for rock salt.
• Thinking a larger packing efficiency always means a denser material. Packing efficiency ignores atomic mass. It tells you about space, not weight.
• Treating the conventional cell as the smallest cell. For FCC, a smaller (rhombohedral) primitive cell holding one atom exists. The cubic cell is used because it shows the symmetry.

Key takeaways

• A unit cell is the repeating box that builds a crystal; corner, edge and face atoms are shared (⅛, ¼, ½).
• Atoms per cubic cell: SC 1, BCC 2, FCC 4. For ionic cells, count each ion and check the formula.
• Radius links: SC a = 2r, BCC a = 4r/√3, FCC a = 2√2 r.
• Packing efficiency: SC 52 %, BCC 68 %, FCC 74 %, independent of the element.
• Density from a cell: ρ = ZM ÷ (N_A a³), with a in cm. Copper gives 8.93 g/cm³, iron 7.87 g/cm³ and NaCl 2.16 g/cm³.