Question

Packing fraction in a body-centered cubic cell of crystals is:

• A. $\frac{\sqrt{3}}{8}\pi$
• B. $\frac{\pi }{6}$
• C. $\frac{\sqrt{2}}{6}\pi$
• D. $\frac{1}{3\sqrt{2}}\pi$

Answer 1

The correct option is D $\frac{1}{3\sqrt{2}}\pi$

We know that,

$\begin{array}{c}APF=\frac{N_{atoms}{V}_{atom}}{V_{unit\_cell}}=\frac{6.\frac{4}{3}\pi r^3}{\frac{3\sqrt{3}}{2}{a}^{2}c}\\ =\frac{6.\frac{4}{3}\pi r^3}{\frac{3\sqrt{3}}{2}{\left(2r\right)}^2\sqrt{\frac{2}{3}}.4r}=\frac{6.\frac{4}{3}\pi r^3}{\frac{3\sqrt{3}}{2}.\sqrt{\frac{2}{3}}.16r^3}\\ =\frac{\pi }{\sqrt{18}}=\frac{\pi }{3\sqrt{2}}\approx 0.74048048\end{array}$

hence option (D) is correct