What is the volume occupied by particles in BCC structure if ' $a$ ' is edge length of unit cell?

What is the volume occupied by particles in BCC structure if ' $a$ ' is edge length of unit cell?
  1. $\frac{\sqrt{3} \pi \mathrm{a}^3}{8}$
  2. $\frac{\pi \mathrm{a}^3}{3 \sqrt{2}}$
  3. $\frac{\pi \mathrm{a}^3}{12 \sqrt{2}}$
  4. $\frac{\sqrt{3} \pi \mathrm{a}^3}{16}$

Solution

Number of particles in BCC unit cell $(Z)=2$ Relation between edge length and radius, $\begin{aligned} & \sqrt{3} \mathrm{a}=4 \mathrm{r} \\ & \mathrm{r}=\frac{\sqrt{3} \mathrm{a}}{4} \end{aligned}$ Volume occupied by particles in BCC $=2 \times$ Volume of one atom $\begin{aligned} & =2 \times \frac{4}{3} \pi \mathrm{r}^3 \\ & =2 \times \frac{4}{3} \pi\left(\frac{\sqrt{3} \mathrm{a}}{4}\right)^3 \\ & =\frac{2 \times 4}{3} \times \pi \times \frac{3 \sqrt{3} \mathrm{a}^3}{64} \\ & =\frac{\sqrt{3} \pi \mathrm{a}^3}{8} \end{aligned}$

Asked in: MHT CET 2021 (23 Sep Shift 1)

Practice more Solid State questions on Aicharya