i. Simple or primitive cubic lattice:
a. In simple cubic unit cell, eight constituent particles (spheres) are present at eight corners of unit cell.
b. Unit cells repeat in three dimensions to form the complete crystal. The constituent particles at the corners are shared by neighbouring unit cells.
c. A constituent particle present at a corner is shared by eight neighbouring unit cells. Its contribution to a unit cell is only 1/8.
d. Thus, the number of atoms present in each unit cell = 8 corner atoms \(\times\) \(\cfrac{1}{8}\) atom per unit cell = 1
Thus, simple cubic lattice has one atom per unit cell.
ii. Body-centred cubic lattice (bcc):
a. In body-centred cubic unit cell, eight constituent particles (spheres) are present at eight corners of unit cell. One constituent particle is present at the centre of the body of the unit cell.
b. A constituent particle present at a corner is shared by eight neighbouring unit cells. Its contribution to a unit cell is only 1/8.
Thus, the number of atoms present at the corners per unit cell
= 8 corner atoms \(\times\cfrac{1}{8}\) atom per unit cell = 1
c. The constituent particle present at the centre of the body of the unit cell is not shared by any other unit cell. Its contribution to a unit cell is l.
Thus, the number of atoms present at the centre of the cube = 1
d. The total number of atoms present in the unit cell = 1 + 1 = 2
Thus, body centred cubic has 2 atoms per unit cell.
Note: When layers of constituent particles are superimposed on one another, a simple cubic structure is formed. To form a body centred cubic structure, second layer of constituent particles is fitted on the depressions of the first layer and the third layer is fitted on the depressions of the second layer. Each sphere is in contact with eight spheres. Four spheres are present in the layer above and four spheres are present in the layer below. Thus, the coordination number of the constituent particle in body-centred cubic lattice is eight.
iii. Face-centred cubic lattice (fcc):
a. In face-centred cubic unit cell, eight constituent particles (spheres) are present at eight corners of unit cell. Six constituent particles (spheres) are present at centres of six faces.
b. A constituent particle present at a corner is shared by eight neighbouring unit cells. Its contribution to a unit cell is only 1/8.
Thus, the number of atoms present at corners per unit cell
= 8 corner atoms \(\times\cfrac{1}{8}\)atom per unit cell = 1
c. A constituent particle present at the centre of a face is shared by two neighbouring unit cells. Its contribution to a unit cell is only 1/2.
The number of atoms present at faces per unit cell = 6 atoms at the faces \(\times\cfrac{1}{2}\) atom per unit cell = 3
d. The total number of atoms per unit cell = 1 + 3 = 4
Thus, a face centred cubic unit cell has 4 atoms per unit cell.
Note: The coordination number of the constituent particle in a cubic close-packed structure or face centred cubic lattice is twelve i.e., four spheres are present in the layer above, four spheres in the layer below and four spheres in the layer of the constituent particle.
