Question

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

• A. $\frac{\sqrt{3}\pi a^3}{8}$
• B. $\frac{\pi a^3}{3\sqrt{2}}$
• C. $\frac{\pi a^3}{12\sqrt{2}}$
• D. $\frac{\sqrt{3}\pi a^3}{16}$

Hint:

To verify your answers quickly during exams, remember that the Packing Efficiency of a lattice is defined as $\frac{\text{Occupied Volume}}{\text{Total Cell Volume } (a^3)}$. Since the packing efficiency of a $\mathrm{BCC}$ lattice is roughly $68\%$, checking which option yields $\approx 0.68a^3$ leads you directly to the correct option: $\frac{\sqrt{3}\pi}{8} \approx \frac{1.732 \times 3.1416}{8} \approx 0.68$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the total volume occupied exclusively by the metallic spherical particles contained inside a single body-centered cubic ($\mathrm{BCC}$) unit cell, expressed as a function of the unit cell edge length $a$.

Step 2: Key Formula or Approach:
The total volume occupied by the particles in a unit cell is given by: $$\text{Occupied Volume} = Z \times \text{Volume of one spherical particle} = Z \times \frac{4}{3}\pi r^3$$ Where:

• $Z$ is the effective number of atoms in the unit cell ($Z = 2$ for $\mathrm{BCC}$).

• $r$ is the atomic radius of the particle.
For a body-centered cubic lattice, the particles touch along the body diagonal, establishing the geometric relationship: $$\sqrt{3}a = 4r \implies r = \frac{\sqrt{3}a}{4}$$

Step 3: Detailed Explanation:
Substitute the expression for radius $r$ into the volume formula for a single sphere: $$\text{Volume of one sphere} = \frac{4}{3}\pi \left(\frac{\sqrt{3}a}{4}\right)^3 = \frac{4}{3}\pi \left(\frac{3\sqrt{3}a^3}{64}\right) = \frac{\sqrt{3}\pi a^3}{16}$$ Since a body-centered cubic unit cell contains exactly $Z = 2$ effective atoms, multiply the volume of a single sphere by 2 to find the total occupied volume: $$\text{Total Occupied Volume} = 2 \times \left(\frac{\sqrt{3}\pi a^3}{16}\right) = \frac{\sqrt{3}\pi a^3}{8}$$

Step 4: Final Answer:
The volume occupied by the particles in a $\mathrm{BCC}$ structure is $\frac{\sqrt{3}\pi a^3}{8}$, which corresponds to option (A).