Question

Calculate the packing efficiency of body centred cubic (b.c.c.) unit cell.

Hint:

BCC packing efficiency (68%) is less than FCC (74%) but greater than simple cubic (52%).

Answer 1

Step 1: Relation between atomic radius and edge length in BCC.
In a body centred cubic structure, the body diagonal passes through 2 radii from corner atoms and 1 radius from the central atom. \[ \sqrt{3}a = 4r \] \[ a = \frac{4r}{\sqrt{3}} \] Step 2: Volume of unit cell.
\[ V_{cell} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \] Step 3: Number of atoms in BCC unit cell.
In BCC: - 8 corner atoms contribute = \( \tfrac{1}{8} \times 8 = 1\) atom
- 1 body-centred atom = 1 atom
Total = 2 atoms per unit cell. Step 4: Volume occupied by atoms.
\[ V_{atoms} = 2 \times \frac{4}{3}\pi r^3 = \frac{8}{3}\pi r^3 \] Step 5: Packing efficiency.
\[ \text{Packing Efficiency} = \frac{V_{atoms}}{V_{cell}} \times 100 \] \[ = \frac{\tfrac{8}{3}\pi r^3}{\tfrac{64r^3}{3\sqrt{3}}} \times 100 \] \[ = \frac{\pi \sqrt{3}}{8} \times 100 = 68% \] Conclusion:
The packing efficiency of a BCC unit cell is \(\boxed{68%}\).