Question

The packing fraction of face centered cubic (fcc) lattice is:

• A. \(0.68\)
• B. \(0.74\)
• C. \(0.76\)
• D. \(0.32\)

Hint:

Packing fraction of FCC is \(0.74\), which is higher than BCC and simple cubic.

Answer 1

The Correct Option is B

Concept:
Packing fraction is the fraction of the total unit cell volume occupied by atoms.

Step 1: Number of atoms in FCC unit cell.
In FCC lattice: \[ N=4 \]

Step 2: Relation between lattice parameter and atomic radius.
For FCC: \[ a=2\sqrt{2}r \]

Step 3: Packing fraction formula. \[ \text{Packing fraction}=\frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} \] \[ =\frac{4\left(\frac{4}{3}\pi r^3\right)}{a^3} \] Substituting \(a=2\sqrt{2}r\), \[ \text{Packing fraction}=\frac{4\left(\frac{4}{3}\pi r^3\right)}{(2\sqrt{2}r)^3} \] \[ =\frac{\pi}{3\sqrt{2}} \] \[ =0.74 \] \[ \therefore \text{Correct Answer is (B)} \]