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 ${}^{\prime }{a}^{\prime }.$ 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 $=1-\frac{8}{3}\pi \frac{r^3}{a^3}$