The packing efficiency in a body centered cub | vedclass

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(C) The correct option is $(C)$.For a $bcc$ lattice,the body diagonal is $c = \sqrt{3} a$.Also,the atoms touch along the body diagonal,so $c = 4r$,whe

Vedclass · Accepted Answer

$68$

Vedclass · Answer

(C) The correct option is $(C)$.For a $bcc$ lattice,the body diagonal is $c = \sqrt{3} a$.Also,the atoms touch along the body diagonal,so $c = 4r$,where $r$ is the radius of the sphere.Therefore,$\sqrt{3} a = 4r$,which gives $a = \frac{4r}{\sqrt{3}}$.In a $bcc$ unit cell,the number of atoms per unit cell $(Z)$ is $2$.The volume of the two spheres is $2 	imes \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3$.The total volume of the unit cell is $a^3 = (\frac{4r}{\sqrt{3}})^3 = \frac{64r^3}{3\sqrt{3}}$.Packing efficiency = $\frac{	ext{Volume of two spheres}}{	ext{Total 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 \, \%$

The packing efficiency in a body centered cubic $(bcc)$ structure is closest to $.... \, \%$

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