$($a$)$ Prove that the packing fraction for the f$.$c$.$c$.$ structures is $0\mathrm{.}74.$

$($b$)$ Determine the radius of a spherical atom $($in terms of the radius of the host atom$)$

that can just fit into the tetrahedral and octahedral voids in the fcc structure.

$($c$)$ Determine the packing fraction in the following structures: $($i$)$ Simple Cubic; $($ii$)$

Body Centered Cubic; $($iii$)$ Diamond Cubic.

$($d$)$ Given that the radius of the $Z{n}^{2+}$ and $S^{2-}$ ions are $0\mathrm{.}083nm$ and $0\mathrm{.}174nm,$

respectively, calculate the packing fraction of the Zinc Blende Structure

$($e$)$ Draw the following planes and directions in a tetragonal unit cell: $(001),(011),$

$(113),[110],[201],$ and $[101].$ Show cell axes.

$($f$)$ Show by means of a $(110)$ sectional drawing that $[111]$ is perpendicular to $(111)$ in

the cubic system but not, in general, in the tetragonal system.