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)

$\frac{7\pi}{64\sqrt{3}}$

Answer 2

No. 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 = a³ = $\frac{64}{3\sqrt{3}}R^{3}$

for BCC $\sqrt{3}a$ = 4R

\ r_ratio = $\frac{7\pi R^{3}/3}{\frac{64}{3\sqrt{3}}R^{3}}$= $\frac{7\pi}{64\sqrt{3}}$