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 a side, $a$.
What fraction of the total cube volume of empty?

• A. $1-\frac{8}{3}\pi \frac{r^3}{a^3}$
• B. $\frac{4}{3}\cdot \pi \frac{r^3}{a^3}$
• C. $\frac{r}{a}$
• D. $2-\frac{4}{3}\pi \frac{r^3}{a^3}$

Answer 1

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

$Z_{eff}$ 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= $1-\frac{8}{3}\pi \frac{r^3}{a^3}$

Answer: (A) $1-\frac{8}{3}\pi \frac{r^3}{a^3}$