Calculate packing efficiency in a CCP or HCP or FCC crystal lattice.

Explanation

Let 'a' be the edge length of the cube.
'b' be the face diagonal
'r': is the radius of the sphere b=4.

In `triangleABC`
`b^(2) =a^(2) + a^(2)`
`b^(2) =2a^(2)`
`b= sqrt(2) xx a`
`4r = sqrt(2) xx a`
`a = 4/sqrt(2) r`, `a=2 sqrt(2)r`
Volume of the cube `=a^(3) =(2sqrt(2)r)^(3)`.
Each unit cell of CCP structure contain 4 spheres.
Volume of four spheres =`4 xx 4/3 pi r^(3)`
Packing efficiency `=("Volume of 4 atoms in the unit cell")/("Total volume of the unit cell") xx 100`
`=(4 xx 4/3 pi r^(3))/(2sqrt(2)r)^(3) xx 100`
Packing efficiency = 74%