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

Explanation

The edge length `=.a.`
Radius of sphere = r
Face diagonal AC = b = 4r
In `DeltaABC , AC^(2) = b^(2) =AB^(2) + BC^(2)`
`=a^(2) + a^(2) = 2a^(2)`
`:. b=sqrt(2)a " But " b = 4r = sqrt(2) a `
` a= (4r)/(sqrt(2)) = 2 sqrt(2) r `
Volume of sphere `= ((4)/(3))pir^(3)`
Volume of cube `=a^(3) = (2sqrt(2)r)^(3)`
Number of particle per unit cell in FCC = Z = 4 spheres .
Packing efficiency `=("Volume occupied by four spheres in the unit cell " xx 100)/("Total volume of the unit cell ") or `
Packing efficiency `=(ZV_("sphere") xx 100)/(a^(3))`
`=(4 xx (4)/(3) pir^(3) xx 100)/((2sqrt(2)r)^(3))=(16 pi r^(3) xx 100)/( 48 sqrt(2) r^(3))`
` (pi xx 100)/(3 sqrt(2)) = 74%`