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(D) Packing efficiency is defined as the fraction of the total volume of the unit cell occupied by the atoms.For a $BCC$ (Body-Centered Cubic) structu

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$68$

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(D) Packing efficiency is defined as the fraction of the total volume of the unit cell occupied by the atoms.For a $BCC$ (Body-Centered Cubic) structure:$1$. The number of atoms per unit cell $(z)$ is $2$.$2$. The relationship between the edge length $(a)$ and the atomic radius $(r)$ is $r = \frac{\sqrt{3}a}{4}$,or $a = \frac{4r}{\sqrt{3}}$.$3$. The volume of the unit cell $(V)$ is $a^3 = (\frac{4r}{\sqrt{3}})^3 = \frac{64r^3}{3\sqrt{3}}$.$4$. The volume occupied by $2$ atoms is $2 	imes \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3$.$5$. Packing efficiency = $\frac{	ext{Volume of atoms}}{	ext{Volume of unit cell}} 	imes 100 = \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} 	imes 100 = \frac{\sqrt{3} \pi}{8} 	imes 100 \approx 0.68 	imes 100 = 68 \%$.

What is the percentage efficiency of packing in $BCC$ structure (in $\%$)?

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