Problem: Show that the atomic packing factor for BCC is 0.68. 

Atomic packing factor:

$\boxed{\mathbf{APF}\mathbf{=}\frac{\mathbf{vol}\mathbf{.}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{atoms}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{unit}\mathbf{ }\mathbf{cell}}{\mathbf{total}\mathbf{ }\mathbf{unit}\mathbf{ }\mathbf{cell}\mathbf{ }\mathbf{volume}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{s}}}{{\mathbf{V}}_{\mathbf{c}}}}$

A body-centered cubic (BCC) unit cell is composed of a cube with one atom at each of its corners and one atom at the center of the cube. 

• corner atoms contribute 1/8 and the center atom contribute 1

$\mathbf{\#}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{atoms}\mathbf{=}\left(\mathbf{8}\mathbf{\times }\frac{\mathbf{1}}{\mathbf{8}}\right)\mathbf{+}\mathbf{1}$

# of atoms = 2 per 1 unit cell

Volume of atoms in unit cell:

$$\begin{gathered} \boxed{{\mathbf{V}}_{\mathbf{s}}\mathbf{=}\mathbf{2}\mathbf{ }\mathbf{atoms}\mathbf{\times }{\mathbf{V}}_{\mathbf{sphere}}} \\ {\mathbf{V}}_{\mathbf{s}}\mathbf{=}\mathbf{2}\mathbf{ }\mathbf{atoms}\mathbf{\times }\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}} \\ {\mathbf{V}}_{\mathbf{s}}\mathbf{=}\frac{\mathbf{8}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}} \end{gathered}$$

Edge length, a:

$$\begin{gathered} \boxed{\mathbf{a}\mathbf{=}\frac{\mathbf{4}\mathbf{r}}{\sqrt{\mathbf{3}}}} \\ \mathbf{r}\mathbf{=}\frac{\sqrt{\mathbf{3}}}{\mathbf{4}}\mathbf{a} \end{gathered}$$

a = edge length
r = radius