Question

Calculate the packing efficiency in a body centered cubic (BCC) lattice.

Answer 1

Step 1: Find the total volume of the unit cell

For Body-Centered Cubic (BCC) lattice, the relationship between the edge length $\mathrm{a}$ and the radius $\mathrm{r}$ of the unit cell is $\mathrm{a}=\frac{4\mathrm{r}}{\sqrt{3}}$

The volume of the unit cell is

$$\begin{gathered} {\mathrm{a}}^3={\left(\frac{4\mathrm{r}}{\sqrt{3}}\right)}^3 \\ \Rightarrow {\mathrm{a}}^3=\frac{64{\mathrm{r}}^3}{3\sqrt{3}} \end{gathered}$$

Step 2: Find the volume occupied by atoms in a unit cell

The volume occupied by one atom is $\frac{4}{3}{\mathrm{\pi r}}^3$

A BCC unit cell has two atoms per unit cell.

The volume occupied by 2 atoms is
$2\times \frac{4}{3}{\mathrm{\pi r}}^3=\frac{8}{3}{\mathrm{\pi r}}^3$
Step 3: Find packing efficiency

The packing efficiency$=\frac{\text{Volume occupied by atoms in a unit cell}}{\text{total volume of unit cell}}\times 100$

The packing efficiency of BCC lattice

$$\begin{gathered} =\frac{\frac{8}{3}{\mathrm{\pi r}}^3}{\frac{64{\mathrm{r}}^3}{3\sqrt{3}}}\times 100 \\ =68.04\% \end{gathered}$$

Thus, the packing efficiency in a body-centered cubic (BCC) lattice is $68.04\%$