packing fraction of a unit cell is drfined as the fraction of the total volume of the unit cell occupied by the atom(s).
`P.E=("Volume of the atoms(s) present in a unit cell")/("Volume of unit cell")=(Zxx(4)/(3)pir^(3))/(a^(3))`
and `%` of empty space = `100- P.F.xx100 `
where Z= effective number of stoms in s cube .
r= radius of a an atoms
a = edge lenght of the cube
Packing fraction in face centered cubic unit cell is :

Explanation

Packing efficiency in simple cubic lattice. In a simple cubic lattice, the atoms are present only on the corners of the cube. The atoms touch each other along the edge. Thus edge length, a and radius of atom, r are related as
a = 2r
Volume of cubic unit cell `=a^(3)=(2r)^(3)=8r^(3)`. A simple cube unit cell contains only 1 atom.
`therefore` The volume of the occupied space `=4/(3)pir^(3)`
`therefore"Packing efficiency"=("Volume of one atom"xx100)/("Volume of the unit cell")`
`=(4/(3)pir^(3))/(8r^(3))xx100=(pixx100)/(6)`
`=52.36=52.4%` (approx.)