Which of the following lattices has the highest packing efficiency
(i) simple cubic
(ii) body-centred cubic and
(iii) Hexagonal close-packed lattice?
(i)
In a simple cubic lattice the atoms are located only at the corners of the cube.
Let us assume the edge length or the side of the cube = a
And the radius of each particle = r
The relation between radius and edge a
Can be given as a = 2r
The volume of the cubic unit cell = side3 = a3
= (2r3)
= 8r3
Number of atoms in unit cell 
The volume of the occupied space 
And we know that, the packing efficiency 
= 52.36%
(ii) 
Let us assume the edge length or the side of the cube = a
And the radius of each particle = r
The diagonal of a cube is always a 
The relation between radius and the edge will be 
= 4r
Divide by root 3 we get A 
Total number of atoms in body centred cubic
Number of atoms at the corner
Number of atoms at the centre = 1
Total number of atoms = 2
The volume of the cubic unit cell = side3
= a3
= (4r/a√3)3
The volume of the occupied space 
Packing efficiency 
= 68%
(iii) Let the base of the hexagon is a and the height is c
Each angle in hexagonal will be 60 degree at the base
Packing efficiency of
Hexagonal close- packed lattice 
a = 2r c 
= 1.633a
= 74%
Thus, hexagonal close- packed lattice has the highest packing efficiency of 74%.