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

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