Question

Calculate the packing efficiency in a Body Centered Cubic (BCC) lattice?

Answer 1

For Body Centered Cubic (BCC) lattice, the relationship between the edge length a and the radius r of the unit cell is $a=\frac{4r}{\sqrt{3}}$
The volume of the unit cell is $a^3=(\frac{4r}{\sqrt{3}}{)}^3=\frac{64r^3}{3\sqrt{3}}$
The volume occupied by 1 atom is $\frac{4}{3}\pi r^3$
A BCC unit cell has 2 atoms per unit cell.
The volume occupied by 2 atoms is
$2\times \frac{4}{3}\pi r^3=\frac{8}{3}\pi r^3$
The packing efficiency $=\frac{\text{Volume occupied by atoms in a unit cell}}{\text{total volume of unit cell}}\times 100$
The packing efficiency $=\frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}}\times 100=68.04$ %