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(A) In a $bcc$ (body-centered cubic) unit cell,the number of atoms per unit cell is $Z = 2$. The relationship between the edge length $a$ and the atom

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$68$

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(A) In a $bcc$ (body-centered cubic) unit cell,the number of atoms per unit cell is $Z = 2$. The relationship between the edge length $a$ and the atomic radius $r$ is $4r = \sqrt{3}a$,or $a = \frac{4r}{\sqrt{3}}$. The volume of the unit cell is $V_{cell} = a^3 = (\frac{4r}{\sqrt{3}})^3 = \frac{64r^3}{3\sqrt{3}}$. The volume occupied by $2$ atoms is $V_{atoms} = 2 	imes \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3$. The packing efficiency is $\frac{V_{atoms}}{V_{cell}} 	imes 100 = \frac{(8/3) \pi r^3}{(64/3\sqrt{3}) r^3} 	imes 100 = \frac{\sqrt{3} \pi}{8} 	imes 100 \approx 68 \%$.

What is the total volume occupied by atoms in a $bcc$ unit cell (in $\%$)?

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