Calculate the packing efficiency in Face Centred Cubic (FCC) structure.

Explanation

The number of atoms per unit cell in FCC structures is four. Each atom is considered as one sphere.
Let the edge length of the unit cell = a
Radius of the sphere = r
Radius of the face diagonal = b
In ABC,
`AC^2 = EC^2 + AB^2`
`b^2 = a^2 + a^2`
`b= sqrt(2) a`
But `b=4r`
`therefore sqrt(2) . a= 4r`
`a = (4r)/( sqrt2) =2 sqrt(2 r)`
Volume of one sphere `= (4)/(3) pi r^3`
Since FCC lattice contains 4 atoms (spheres) per unit cell,
The volume of four spheres in fee `= 4 xx (4)/(3) pir^3 = (16)/(3) pir^3`
The total volume of the unit cell `=a^(3) (2 sqrt2 r)^3`
Packing efficiency `=("Volume of four spheres in unit cell")/("Total volume of the unit cell")xx 100`
Packing efficiency `= ((16)/(3) pir^3)/( (2 sqrt(2r))^^ 3)xx 100 = 74%`