Question

The unit cell in a body centred cubic lattice is given in the figure. Each sphere has a radius $r$ and the cube has side $'a'.$ What fraction of the total cube volume is empty?

• A. $1-\frac{8}{3} \frac{\pi r^{3}}{a^{3}}$
• B. $\frac{4}{3} \frac{\pi r^{3}}{a^{3}}$
• C. $\frac{r}{a}$
• D. $2-\frac{4}{3} \frac{\pi r^{3}}{a^{3}}$

Answer 1

Answer: (A)

$Z _{ eff }$ of $BCC =2$
Volume occupied by atoms $=2 \times \frac{4}{3} \pi r^{3}$
Volume of unit cell $=a^{3}$
Packing fraction $=\frac{8}{3} \pi \frac{r^{3}}{a^{3}}$
Fraction of empty unit cell $=1-\frac{8}{3} \pi \frac{r^{3}}{a^{3}}$