Caleulate the packing efficiency in a unit cell of Cubic Close Packing(CCP) structure. 

Explanation

The number of atoms per unit cell in fee structures is four. Each atom is considered as one sphere.
so, the volume of four atoms (four spheres) `=4xx4/3pir^3=(16)/(3)pir^3`
from this figure it is clear that there are three spheres ,touching each other along the face diagonal. Let .a. be the edge length of the cube and .r. be the radius or the sphere The face diagonal AC=b then b=4r
In ABC, `AC^2=BC^2+AB^2`
`b^2=a^2+a^2" "b^2=2a^2`
`:.b=sqrt(2a)=4r" "b=sqrt(2a)=4r`
`a=(4r)/(sqrt2)=2sqrt(2r)`
The volume of the cubic unit cell `=a^3=(2sqrtr)^3`
Therefore, Packing efficiency=
`=("Volume of four atoms (four spheres)in unit cell")/("Volume of the unit cell")xx100%`

Packing efficiency`=((16)/3pir^3)/((2sqrt2r)^3)xx100%`
Packing efficiency`=((16)/(3)xx(22)/(7)r^3)/(8(sqrt2)^3r^3)xx100%`
acking efficiency`=((16)/(3)xx(22)/(7))/(8(sqrt2)^3)xx100%=74%`
Therefore 74% of unit cell is occupied by atoms and the rest 26% is empty space in ice structures.
The packing fraction in fcc structures (ccp)=0.74
The fraction of empty space in fcc structure (ccp) 0.26