Question

Figure 41-1ashows 14 atoms that represent the unit cell of copper. However, because each of these atoms is shared with one or more adjoining unit cells, only a fraction of each atom belongs to the unit cell shown. What is the number of atoms per unit cell for copper? (To answer, count up the fractional atoms belonging to a single unit cell.)

Short answer

The number of atoms per unit cell is 4.

Step-by-step solution

The given data

The unit cell of the copper shows the representation of 14 atoms.

Understanding the concept of lattice structure of copper

A lattice structure is a three-dimension open structure that is made up of more than one repeating unit cell of an atom. The cells define the connectivity of the constituent atoms in the unit cell structure. They can be of mainly four types of lattice structures: FCC, BCC, HCP, and simple cubic. Now, here the given unit cell of copper has an FCC structure. The total number of atoms per unit cell is calculated by calculating the atoms’ contributions that are placed at the corners and face constituting the structure.

Calculation of the number of atoms per unit cell

As the lattice structure copper consisting of 14 electrons is an FCC structure, thus the contribution to the structure is only from the face and corner atoms.

The contribution of each corner atom per unit cell of copper is $\frac{1}{8}$. So, the total contribution by 8 corner atoms is 1.

The contribution of each face atom per unit cell of copper is $\frac{1}{2}$. Thus, the contribution of 6 face atoms per unit cell of copper is 3.

Now, the total contribution of the FCC lattice in the copper unit cell is given by:

$\left(\frac{1}{8}\times 8\right)+\left(\frac{1}{2}\times 6\right)=4$

Hence, the number of atoms per unit cell of copper is 4.