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

Answer 1

The correct option is A $1-\frac{8}{3}\frac{\pi r^3}{a^3}$

Total volume of the cube$=a^3$
For the body centred cubic lattice, effective number of atoms, $z=2$
Volume of one atom$=\frac{4}{3}\pi r^3$
So, total occupied volume $=2\times \frac{4}{3}\pi r^3=\frac{8}{3}\pi r^3$
Hence, fraction of occupied space=$\frac{\text{Ocuppied volume}}{\text{Total volume}}=\frac{8}{3}\frac{\pi r^3}{a^3}$

Fraction of empty space $1-\frac{8}{3}\frac{\pi r^3}{a^3}.$
Hence, (a) is correct.