Question

Calculate the packing efficiency in a face-centered cubic (FCC) crystal.

Hint:

FCC structures (like in Cu, Al) have highest packing efficiency among simple lattices: 74%.

Answer 1

Step 1: Use formula for packing efficiency: \[ \text{Efficiency} = \frac{\text{Volume occupied by atoms in unit cell}}{\text{Total volume of unit cell}} \times 100 \] In FCC, atoms touch along face diagonal: \[ \sqrt{2}a = 4r \Rightarrow r = \frac{\sqrt{2}}{4}a \] Volume of 4 atoms: \[ 4 \cdot \frac{4}{3}\pi r^3 = \frac{16}{3} \pi r^3 \] \[ = \frac{16}{3} \pi \left( \frac{\sqrt{2}}{4}a \right)^3 = \frac{16}{3} \pi \cdot \frac{2\sqrt{2}}{64} a^3 = \frac{\pi \sqrt{2}}{6} a^3 \] \[ \text{Efficiency} = \frac{\pi \sqrt{2}}{6} \cdot 100 \approx 74% \] Final Answer: \[74% \]