Give the No. of lattice points in one unit cell & percentage
efficiency of packing in the case of the following crystals:
i) Simple cubic ii) body centred cubic iii) face centered cubic
Answers
Answer:
(i) The efficiency of packing in case of simple cubic unit cell is given below:
A simple cubic unit cell contains one atom per unit cell.
Also, a=2r, where a is the edge length and r is the radius of atom.
Total volume of unit cell =a
3
.
Packing efficiency =
Total volume of unit cell
Volume of one sphere
×100
Packing efficiency =
8r
3
3
4
πr
3
×100=52.4%
(ii) The efficiency of packing in case of body-centred cubic unit cell is given below:
A body-centred cubic unit cell contains two atoms per unit cell.
Also,
3
a=4r, where a is the edge length and r is the radius of atom.
Total volume of unit cell =a
3
.
Packing efficiency =
Total volume of unit cell
Volume of two spheres
×100
Packing efficiency =
3
3
64
×r
3
3
8
πr
3
×100=68%
(iii) The efficiency of packing in case of face-centered cubic unit cell (with the assumptions that atoms are touching each other) is given below:
A face-centered cubic unit cell contains four atoms per unit cell.
Also, a=2
2
r, where a is the edge length and r is the radius of atom.
Total volume of unit cell =a
3
Packing efficiency =
Total volume of unit cell
Volume of four spheres
×100
Packing efficiency =
16
2
r
3
3
16
πr
3
×100=74%