Calculate the efficiency of packing in case of a metal crystal for face centred cubic lattice.

Explanation

We have a unit cell having edge length a and face diagonal AC = b.

`AC^2 = BC^2 + AB^2`
`[:. AC = b, BC = a, AB = a]`
`b^2 = a^2 + a^2 or b = sqrt(2a)`
If r is the radius of the sphere, then
`b =4 r = sqrt2a`
or `a = (4r)/sqrt2 = 2 sqrt(2r)` or `r = a /(2 sqrt2)`
We know that, number of spheres per unit cell in ccp structure is 4.
Therefore,total of spheres `= 4 xx 4/3 xx pi r^3`
Volume of cube `(a^3) = (2sqrt2r)^3`.
So, packing efficiency ` = ("Volume occupied by 4spheres")/("Total voluume of unit cell") xx 100`
(in prercentage)
` = (4 xx4/3 pi r^3)/((2sqrt2r)^3) xx 100 = (16/3 pi r^3)/(16 sqrt2r^3) xx 100 = 74%`