Question

Packing efficiency of body centred unit cell is:

• A. 85%
• B. 68%
• C. 33.33%
• D. 45%

Answer 1

The correct option is B 68%

Total number of atoms per unit cell in bcc= $2$

From $\Delta EFD$,

$\Rightarrow b^2=a^2+a^2$

$\Rightarrow b=\sqrt{2}a$

$\Delta AFD$,

$\Rightarrow c^2=a^2+b^2=a^2+2a^2=3a^2$

$\Rightarrow c=\sqrt{3}a$

But $c=4r$ ($r$= radius of sphere)

$\therefore \sqrt{3}a=4r$

$\Rightarrow a=\frac{4r}{\sqrt{3}}$

Total number of atoms per unit cell in bcc= $2$

Packing efficiency= $\frac{\text{Volume occupied by 2 spheres in unit cell}\times 100}{\text{Total volume of unit cell}}$

$=\frac{2\times \frac{4}{3}\Pi r^3\times 100}{a^3}$

$=\frac{2\times \frac{4}{3}\Pi r^3\times 100}{{\left[\frac{4}{\sqrt{3}}r\right]}^3}=68$ %

$\therefore$ Packing efficiency of bcc $=$ $68$ %