Question

The packing efficiency of a simple cubic crystal with an interstitial atom exactly fitting at the body center is:

• A. $0.48$
• B. $0.52$
• C. $0.73$
• D. $0.91$

Answer 1

Answer: (C)

$0.73$

For a simple cubic, $a=2r$
Therefore, the radius of the interstitial atom exactly fitting at the centre of the cube is $R=\frac{1}{2}\times (a\sqrt{3}-2r)$

$R=\frac{1}{2}\times (a\sqrt{3}-a)$

$R=0.366a$

Packing fraction $=\frac{\text{volume occupied by spheres}}{a^3}$
Packing fraction $=\frac{(\frac{1}{8}\times 8\times \frac{4\pi r^3}{3})+(1\times \frac{4\pi R^3}{3})}{a^3}$

Packing fraction$=\frac{4\pi ({\left(\frac{a}{2}\right)}^3+(0.366a{)}^3)}{3a^3}$

Packing fraction $=0.7285$.