(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%.