Number of Atoms in Unit Cell — Definition

Definition

Imagine a crystal as a giant, repeating pattern of building blocks. Each of these smallest, repeating building blocks is called a 'unit cell'. Just like how bricks make up a wall, unit cells make up a crystal.

Now, these unit cells contain atoms, ions, or molecules at specific positions. When we talk about the 'number of atoms in a unit cell', we're not just counting every atom that *touches* the unit cell, but rather the *effective* number of atoms that *belong exclusively* to that single unit cell.

Think of it this way: if an atom is sitting exactly at a corner of a cube, it's actually being shared by eight different cubes (unit cells) that meet at that corner. So, for any one of those cubes, only one-eighth of that atom truly 'belongs' to it.

Similarly, an atom exactly in the center of a face of a cube is shared by two unit cells (the one it's on and the adjacent one), so it contributes half of itself to each. An atom sitting exactly in the middle of an edge is shared by four unit cells, contributing one-fourth to each.

And finally, an atom located entirely within the body of the unit cell, not touching any face or edge or corner, belongs completely (100%) to that unit cell.

So, to find the total effective number of atoms in a unit cell, we sum up these fractional contributions. For example, in a simple cubic (primitive) unit cell, atoms are only at the 8 corners. Since each corner atom contributes $1/8$, the total number of atoms is $8 \times (1/8) = 1$.

In a body-centered cubic (BCC) unit cell, there are atoms at the 8 corners and one atom at the body center. So, it's $8 \times (1/8) + 1 \times 1 = 1 + 1 = 2$ atoms. For a face-centered cubic (FCC) unit cell, there are atoms at the 8 corners and at the center of each of the 6 faces.

This gives $8 \times (1/8) + 6 \times (1/2) = 1 + 3 = 4$ atoms. This effective number, 'Z', is a crucial parameter in solid-state chemistry.