Question

The packing efficiency in a body centered cubic (bcc) structure is closest to

• A. $74\%$
• B. $63\%$
• C. $68\%$
• D. $52\%$

Answer 1

Answer: (C)

For bcc lattice,

In $\Delta EFD$
$b^2=a^2+a^2=2a^2$
$b=\sqrt{2a}$
Now, in $△AFD$
$c^2=a^2+b^2$
$=a^2+2a^2=3a^2$
$c=\sqrt{3}a$
Also, $\sqrt{3}a=4r$
$\frac{\sqrt{3}}{4}a=r$
In bec, $Z=2$
$\therefore$ Volume of cube $=a^3={\left(\frac{4}{\sqrt{3}}r\right)}^3$
Packing effeciency
Volume occupied by two
$=\frac{\text{ spheres in a unit cell }}{\text{ Total volume of unit cell }}\times 100$
$=\frac{2\times \frac{4}{3}\pi r^3\times 100}{[4/\sqrt{3}r{]}^3}=68\%$