Page 19 - Solid State

1.7.2 Efficiency of From Fig. 1.21, it is clear that the
Packing in atom at the centre will be in touch
Body- with the other two atoms diagonally
Centred arranged.
Cubic In Δ EFD,
Structures b = a + a = 2a 2
2
2
2
b= 2a
Now in Δ AFD
2
2
2
2
2
c = a + b = a + 2a = 3a 2
c= 3a
The length of the body diagonal
Fig. 1.21: Body-centred cubic unit
c is equal to 4r, where r is the radius
cell (sphere along the
of the sphere (atom), as all the three
body diagonal are shown
spheres along the diagonal touch with solid boundaries).
each other.
Therefore, 3a = 4r
4r
a=
3
3
Also we can write, r = a
4
In this type of structure, total number of atoms is 2 and their volume
( ) 3
4
is 2 × 3 π r .
⎛ 4 ⎞ 3 ⎛ 4 ⎞ 3
3
3
Volume of the cube, a will be equal to ⎜ r ⎟ or a = ⎜ r ⎟ .
⎝ 3 ⎠ ⎝ 3 ⎠
Therefore,
×
Volumeoccupied by twospheres in the unit cell 100
Packing efficiency = %
Total volumeof the unit cell
2 × (4/3 ) rπ 3 × 100
= %
⎡ ( ) 4/ 3 r ⎤ 3
⎣ ⎦

(8/3 ) rπ 3 × 100
= % = 68%
)
( 64/ 3 3 r 3
1.7.3 Packing In a simple cubic lattice the atoms are located only on the corners of the
Efficiency in cube. The particles touch each other along the edge (Fig. 1.22).
Simple Cubic Thus, the edge length or side of the cube ‘a’, and the radius of each particle,
Lattice r are related as
a = 2r
3
3
The volume of the cubic unit cell = a = (2r) = 8r 3
Since a simple cubic unit cell contains only 1 atom
4 3
The volume of the occupied space = r π
3
19 The Solid State