Which of the following lattices has the highest packing efficiency (i) simple cubic (ii) body−centred cubic and (iii) hexagonal close−packed lattice?

Q 1.17: Which of the following lattices has the highest packing efficiency (i) simple cubic (ii) body−centred cubic and (iii) hexagonal close−packed lattice?

Solution:

(i)

 

In a simple cubic lattice the atoms are located only on the corners of the cube.

Let take edge length or side of the cube = a,

Let take radius of each particles               = r

The relation between radius and edge a

                                                               a     = 2r

The volume of the cubic unit cell         = side³

= a³

                                                                 = (2r)³

                                                                 = 8r³

Number of atoms in unit cell  = 8 × 1 /8

                                                   = 1

The volume of the occupied space = (4/3)πr³

(ii) In body centered cubic two atoms diagonally

 

Let take edge length or side of the cube = a,

Let take radius of each particles               = r

The diagonal of a cube is always a√3

The relation between radius and edge a will

 a√3 = 4r

divide by root 3 we get

                                                a = 4r/√3

total number of atoms in body centered cubic

number of atoms at the corner = 8 × 1/8 = 1

number of atoms at the center                  = 1

total number of atoms                               = 2

The volume of the cubic unit cell            = side³

  = a³

     = (4r/√3)³

The volume of the occupied space = (4/3)πr³

.

(iii) hexagonal close−packed lattice

Let take base of hexagonal is a and height is c  

Each angle in hexagonal will 60 degree at base

Hexagonal close−packed lattice has the highest packing efficiency of 74%.