Question

How would you calculate the number of particles in a unit cell?

Answer 1

The number of particles per unit cell in a crystal lattice is calculated based on the locations of the atoms or ions. These locations can be at the corners, on the faces, along the edges, or at the centre of the unit cell. Each location contributes a certain fraction of an atom to the unit cell.

Here’s how each location contributes to the unit cell:

• Corner atoms: An atom at a corner of a unit cell is shared by 8 unit cells. Therefore, a corner atom contributes `1/8`th of an atom to one unit cell.
• Edge atoms: An atom on an edge is shared by 4 unit cells. Therefore, an edge atom contributes `1/4`th of an atom to one unit cell.
• Face atoms: An atom on a face is shared by 2 unit cells. Therefore, a face atom contributes `1/2` of an atom to one unit cell.
• Centre atom: A particle at the center of the unit cell belongs entirely to the unit cell, contributing 1 particle.

The total number of atoms in a unit cell is the sum of the contributions from the corner atoms, edge atoms, face atoms, and center atoms.

Apply to different unit cells:

1. Simple-Centred Cubic (scc):

8 corners = `(8 xx 1/8)` = 1

∴ In simple-centred cubic, 1 particle per unit cell.

2. Body-Centred Cubic (bcc): 

8 corners = `(8 xx 1/8)` = 1

1 center paticle = 1

Total = 1 (corners) + 1 (center) = 2

∴ In body-centred cubic, 2 particles per unit cell.

3. Face-Centred Cubic (fcc):

8 corners = `(8 xx 1/8) = 1`

6 face particles = `(6 xx 1/2) = 3`

Total = 1 (corners) + 3 (center) = 4

∴ In face-centred cubic, 4 particles per unit cell.