Question

Packing factors for a body centred cubic and simple cubic structure are, respectively :

• A. $0.68$ and $0.524$
• B. $0.524$ and $0.68$
• C. $0.48$ and $0.52$
• D. $0.52$ and $0.48$

Answer 1

The correct option is A $0.68$ and $0.524$

$\text{Body centred cubic}:$

Body centred cubic lattice has radius $=r=\frac{\sqrt{3}a}{4}$

The unit cell of body centred cubic has $2$ atoms on the average.

$\therefore$ Volume of atoms $=2\times \frac{4}{3}\pi r^3$ $=2\times \frac{4}{3}\times \pi \times {\left[\begin{array}{c}\frac{\sqrt{3}a}{4}\end{array}\right]}^3=\frac{\sqrt{3}}{8}\pi a^3$

The volume of unit cell $=a^3$

$\therefore$ Packing factor $=\frac{\sqrt{3}}{8}\frac{\pi a^3}{a^3}=0.68$

$\text{Simple Cubic}:$

Simple cubic has radius $=r=\frac{a}{2}$

The unit cell of sc has one atom.

$\therefore$ Volume of atom $=\frac{4}{3}\pi {\left(\begin{array}{c}\frac{a}{2}\end{array}\right)}^3=\frac{4\pi a^3}{24}$

The volume of unit cell $=a^3$

$\therefore$ Packing factor $=\frac{4}{24}\frac{\pi a^3}{a^3}=0.524$

Hence, the correct option is $\text{A}$