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Q.
The packing efficiency in a body centered cubic (bcc) structure is closest to
KVPYKVPY 2017The Solid State
Solution:
For bcc lattice,
In $\Delta E F D$
$b^{2}=a^{2}+a^{2}=2 a^{2}$
$b=\sqrt{2 a}$
Now, in $\triangle A F D$
$c^{2}=a^{2}+b^{2}$
$=a^{2}+2 a^{2}=3 a^{2}$
$c=\sqrt{3} a$
Also, $\sqrt{3} a=4 r$
$\frac{\sqrt{3}}{4} a=r$
In bec, $Z=2$
$\therefore $ Volume of cube $=a^{3}=\left(\frac{4}{\sqrt{3}} r\right)^{3}$
Packing effeciency
Volume occupied by two
$=\frac{\text { spheres in a unit cell }}{\text { Total volume of unit cell }} \times 100$
$=\frac{2 \times \frac{4}{3} \pi r^{3} \times 100}{[4 / \sqrt{3} r]^{3}}=68 \%$
