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

Explanation

The number of atoms per unit cell in FCC structures is 4. (Z =4)
Each atom is considered as one sphere.
Let the edge length of the unit cell= a.
Radius of the sphere =r
Length of the face diagonal = b
From the`DeltaABC`,

`AC^2 = BC^(2) + AB^(2)`
`b^(2) = a^(2) + a^(2)`
` b = sqrt(2)a | b =4r`
`:. sqrt(2)a = 4r`
` a= ( 4r)/(sqrt(2)) = 2 sqrt(2)r `
Volume of one atom `= 4 xx (4)/(3) pi r^(3) = (16)/(3) pi r^(3)`
The total volume of the unit cell `=a^(3) = (2sqrt(2)r)^(3) = 16 sqrt(2) r^(3)`
Packing efficiency `=("Volume of an atom " xx 4)/( "Total volume of the unit cell ") xx 100`
= `((16)/(3) pi r^(3))/(16 sqrt(2) r^(3)) xx 100 = 74 %`