Consider a face$-$centered cubic $($FCC$)$ crystal structure with a lattice constant $a=4A$

$($a$)$ Calculate the number of tetrahedral and octahedral voids per unit cell in an FCC structure. Also, determine the fractional coordinates of one tetrahedral void and one octahedral void within the unit cell.

$[5]$

$($b$)$ If atoms with radius $r=0\mathrm{.}414R($where R is the radius of the host atoms$)$ occupy all the octahedral voids in the FCC structure, determine the new atomic packing factor $($APF$)$ for the combined structure. Assume all voids are filled without changing the host lattice arrangement.

$5$