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9.1.1.2. Unit Cells

Table of Contents

The repeating building block of a crystal

A crystal is made from a pattern that repeats in space. The smallest repeating block of that pattern is called the unit cell. If you copy a unit cell and shift it again and again in the allowed crystal directions, you can rebuild the whole crystal.

A unit cell is not usually a single atom. It is a small region of space that contains enough information to reproduce the entire crystal structure. This includes the shape of the region, its size, and the positions of the atoms inside it.

You can think of a wallpaper pattern on a floor. One small tile, repeated many times, makes the full pattern. In the same way, one unit cell, repeated in three dimensions, makes a crystal.

Shape and edges of a unit cell

A unit cell is described by three edge lengths and three angles. The edge lengths are usually written as $a$, $b$, and $c$. The angles between them are written as $\alpha$, $\beta$, and $\gamma$.

The volume of the unit cell tells us how much space one repeating block occupies. For a simple rectangular cell, the volume is

$$
V = abc
$$

In more general cases, the volume depends on the angles too, but for beginners it is enough to first understand the rectangular case clearly.

The choice of unit cell is not always unique. The same crystal can sometimes be described by different repeating cells. Physicists often prefer the simplest one that shows the symmetry clearly.

A unit cell is a repeating region of space, not just a group of atoms.
To describe a unit cell, we need its edge lengths, its angles, and the atom arrangement inside it.

Lattice points and atoms in the cell

The corners of a unit cell are lattice points, positions that mark the repeating structure. Atoms may sit at these lattice points, but they may also appear at other positions inside the cell.

An important idea is that atoms at the boundaries are shared with neighboring cells. This means that a corner atom does not belong fully to one cell.

The usual sharing rules are shown below.

Atom positionShared by how many cellsContribution to one unit cell
Corner8$\frac{1}{8}$
Edge center4$\frac{1}{4}$
Face center2$\frac{1}{2}$
Body center1$1$

These fractions are very important when counting how many atoms belong to one unit cell.

When counting atoms in a unit cell, boundary atoms are shared.
Corner atom contribution: $\frac{1}{8}$
Face-centered atom contribution: $\frac{1}{2}$
Body-centered atom contribution: $1$

Simple examples of cubic unit cells

A very common family is the cubic unit cells. In these cells,

$$
a = b = c
$$

and

$$
\alpha = \beta = \gamma = 90^\circ
$$

The difference between the cubic types comes from where the atoms are placed.

Simple cubic

In a simple cubic cell, atoms appear only at the 8 corners. Since each corner contributes $\frac{1}{8}$, the total number of atoms per unit cell is

$$
8 \times \frac{1}{8} = 1
$$

Simple cubic unit cell

Body-centered cubic

In a body-centered cubic, or BCC, cell, there are atoms at the 8 corners and one atom in the very center of the cube.

So the total number of atoms is

$$
8 \times \frac{1}{8} + 1 = 2
$$

Body-centered cubic unit cell

Face-centered cubic

In a face-centered cubic, or FCC, cell, there are atoms at the 8 corners and one atom at the center of each of the 6 faces.

The total number of atoms is

$$
8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4
$$

Face-centered cubic unit cell

Basis and crystal structure

The full crystal structure is produced by combining a lattice with a set of atoms attached to each lattice point. That attached set is often called the basis.

So in a simple picture,

$$
\text{crystal structure} = \text{lattice} + \text{basis}
$$

This matters because two materials can have the same lattice shape but different atoms or different atom positions inside the unit cell.

For example, a unit cell may contain more than one type of atom. In that case, the cell still repeats regularly, but the repeated internal pattern is more complex.

The unit cell must include the full repeating internal arrangement of atoms.
Repeating only the lattice points is not enough to specify a real crystal.

Why unit cells matter

Unit cells are useful because many physical properties of a solid can be linked to them. If we know the unit cell dimensions and how many atoms are inside, we can find quantities like atomic density and mass density.

If a unit cell contains $n$ atoms, and each atom has mass $m$, then the mass of one unit cell is

$$
m_{\text{cell}} = nm
$$

If the unit cell volume is $V_{\text{cell}}$, then the density is

$$
\rho = \frac{m_{\text{cell}}}{V_{\text{cell}}}
$$

In chemistry and materials science, the atomic mass is often given per mole. Then the mass of one atom is

$$
m = \frac{M}{N_A}
$$

where $M$ is the molar mass and $N_A$ is Avogadro's number. So the density becomes

$$
\rho = \frac{nM}{N_A V_{\text{cell}}}
$$

This equation connects microscopic structure to a measurable macroscopic property.

For a crystal with $n$ atoms per unit cell,
$$
\rho = \frac{nM}{N_A V_{\text{cell}}}
$$
This is a key relation between unit-cell structure and material density.

Conventional and primitive cells

Sometimes the chosen unit cell is the smallest possible repeating one. This is called a primitive cell. It contains exactly one lattice point in total.

In other cases, a larger cell is chosen because its shape shows the symmetry more clearly. This is called a conventional cell.

For example, the cubic cells often shown in textbooks are conventional cells. They are easy to visualize and useful for describing symmetry, even if a smaller primitive choice may exist.

Type of cellMain idea
Primitive cellSmallest repeating cell, contains one lattice point
Conventional cellOften larger, chosen to show symmetry clearly

Visualizing repetition in space

The power of the unit cell idea is repetition by translation. If one unit cell is shifted by integer multiples of its edge vectors, the same arrangement appears again.

If the cell edges are represented by vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$, then equivalent lattice points occur at positions

$$
\vec{R} = n_1 \vec{a} + n_2 \vec{b} + n_3 \vec{c}
$$

where $n_1$, $n_2$, and $n_3$ are integers.

This equation says that the crystal is built by repeating one unit cell throughout space.

Repetition of a unit cell in two dimensions

A final picture

A unit cell is the basic repeating three-dimensional block of a crystal. It is defined by its geometry and by the arrangement of atoms inside it. By repeating this one cell through space, we generate the full solid. Understanding unit cells lets us count atoms correctly, compare crystal structures, and connect atomic arrangement to measurable properties such as density and symmetry.

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