Question

A body-centered cubic lattice is made up of hollow spheres of $B$. Spheres of solid $A$ are present in hollow spheres of $B$. Radius $A$ is half of radius of $B$. What is the ratio of total volume of spheres of $B$ unoccupied by $A$ in a unit cell and volume of unit cell?

• A. $\frac{7 \sqrt{3 \pi}}{64}$
• B. $\frac{7 \sqrt{3}}{128}$
• C. $\frac{7 . \pi}{24}$
• D. $\frac{7 \pi}{64 \sqrt{3}}$

Answer 1

Answer: (D)

Number of atoms of $B$ in unit cell $=2$

Total volume of B unoccupied by A in a unit cell

$=2 \times \frac{4}{3}\left( R ^{3}- r ^{3}\right) \times \pi=\frac{7 \pi R^{3}}{3}$

Vol. of unit cell $=a^{3}=\frac{64}{3 \sqrt{3}} R^{3}$

For bcc, $\sqrt{3} a=4 R$

$\therefore \rho_{\text {ratio }}=\frac{7 \pi R^{3} / 3}{\frac{64}{3 \sqrt{3}} R^{3}}=\frac{7 \pi}{64 \sqrt{3}}$