Question

What is the packing efficiency in fcc lattice?

• A. 74%
• B. 52%
• C. 78%
• D. 68%

Hint:

For fcc lattices, remember: number of atoms per unit cell = 4, edge length \(a = 2\sqrt{2}r\), and packing efficiency = 74%.

Answer 1

The Correct Option is A

Step 1: Understand the fcc structure.
Face-centered cubic (fcc) lattice has atoms at all corners and the centers of each face of the cube. There are 8 corner atoms and 6 face atoms.

Step 2: Calculate number of atoms per unit cell.
- Each corner atom contributes 1/8th to the unit cell: \(8 \times \frac{1}{8} = 1\) atom.
- Each face-centered atom contributes 1/2 to the unit cell: \(6 \times \frac{1}{2} = 3\) atoms.
- Total atoms per fcc unit cell = 4 atoms.

Step 3: Relate radius and unit cell length.
In fcc, the face diagonal \(= 4r\), where \(r\) is atomic radius.
Edge length of cube \(a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r\).

Step 4: Calculate volume occupied by atoms.
Volume of 4 atoms \(= 4 \times \frac{4}{3}\pi r^3 = \frac{16}{3}\pi r^3\).

Step 5: Calculate unit cell volume.
Unit cell volume \(= a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2} r^3\).

Step 6: Determine packing efficiency.
Packing efficiency \(= \frac{\text{volume occupied by atoms}}{\text{volume of unit cell}} \times 100\% = \frac{\frac{16}{3}\pi r^3}{16\sqrt{2} r^3} \times 100\% = 74\%\).

Step 7: Conclusion.
The packing efficiency of fcc lattice is 74%, which is the highest among cubic lattices.