What is the packing efficiency in BCC unit cell ?

In a body-centered cubic (BCC) unit cell, the atoms touch each other along the body diagonal.

Let the edge length of the unit cell be $a$ and the radius of the atom be $r$.

The length of the body diagonal is $\sqrt{3}$a$.<br><br>Since the atoms touch along the body diagonal, we have:<br><br>\sqrt{3}a = 4r \Rightarrow a = \dfrac{4r}{\sqrt{3}}$

The number of atoms per unit cell in a BCC structure is $Z = 2$.

Volume of the unit cell is $V = a^3 = \left(\dfrac{4r}{\sqrt{3}}\right)^3 = \dfrac{64r^3}{3\sqrt{3}}$

Volume occupied by the atoms is $v = 2 \times \dfrac{4}{3}\pi r^3 = \dfrac{8}{3}\pi r^3$

Packing efficiency is given by:

$\text{Packing efficiency} = \dfrac{v}{V} \times 100$

$\text{Packing efficiency} = \dfrac{\dfrac{8}{3}\pi r^3}{\dfrac{64r^3}{3\sqrt{3}}} \times 100 = \dfrac{\sqrt{3}\pi}{8} \times 100$

Substituting the values of $\pi$ and $\sqrt{3}:<br><br>\text{Packing efficiency} \approx 68\ \%$

Answer: $68\ \%$