Question

The fraction of the total volume occupied by the atoms present in a face centred cubic unit cell is

• A. \(\dfrac{\pi}{6}\)
• B. \(\dfrac{\pi}{3\sqrt{2}}\)
• C. \(\dfrac{\pi\sqrt{3}}{8}\)
• D. \(2\sqrt{2}\) \bigskip

Hint:

Packing efficiencies: \[ \begin{array}{c|c} \textbf{Structure} & \textbf{Packing Efficiency}
\hline \text{Simple Cubic} & 52.4\%
\text{BCC} & 68\%
\text{FCC/HCP} & 74\% \end{array} \]

Answer 1

The Correct Option is B

Concept: FCC packing fraction = \(\frac{\pi}{3\sqrt2}\).

Step 1: FCC: Z=4, \(a=2\sqrt2 r\). Packing fraction = \(\frac{4\cdot\frac43\pi r^3}{(2\sqrt2 r)^3} = \frac{\pi}{3\sqrt2}\).

Step 2: Write the final answer. \(\boxed{\frac{\pi}{3\sqrt2}}\)