Q 10 Calculate the efficiency of packing in case of a metal crystal for (i) simple cubic (ii) body–centred cubic (iii) face–centred cubic (with the assumptions that atoms are touching each other).

Q 10 Calculate the efficiency of packing in case of a metal crystal for

(i) simple cubic

(ii) body–centred cubic

(iii) face–centred cubic (with the assumptions that atoms are touching each other).

(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 x 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 x 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)

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

Let take radius of each particles               = r

The diagonal of a square  is always a√2

The relation between radius and edge a will

 a√2 = 4r

divide by root 3 we get

                        a          = 4r/√2

total number of atoms in body centered cubic

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

number of atoms at the face    = 6 x1/2    = 3

total number of atoms                                 = 4

The volume of the cubic unit cell             = side³

                                                                       = a3

                                                                       = (4r/√2)³

                                                                        = (2√2 r)³

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

.