Fig. 4.1. The position coordinates of the eight corner atoms will remain the same
for the three cubic cases.
The position coordinates of the central atom in a bcc unit cell are
1
2
1
2
1
2
 Ã
Â
Ã
as
shown in Fig. 4.4. For simplicity, sometimes only two position coordinates, for
example, [[000]] and
1
2
1
2
1
2
 Ã
Â
Ã
are specified to represent a bcc unit cell while another
position coordinates (for corner atoms) are assumed to be understood.
Similarly, the position coordinates of six face-centered atoms in an fcc unit cell
are:
1
2
1
2 0
Â
Ã
Â
Ã
; 0
1
2
1
2
Â
Ã
Â
à ;
1
2 0
1
2
Â
Ã
Â
à ;
1
2
1
2 1
Â
Ã
Â
Ã
; 1
1
2
1
2
Â
Ã
Â
Ã
and
1
2 1
1
2
Â
Ã
Â
à :
They are shown in Fig. 4.5. Like bcc, only four position coordinates, for
example, [[000]].
1
2
1
2 0
Â
Ã
Â
Ã
; 0
1
2
1
2
Â
Ã
Â
Ã
and
1
2 0
1
2
Â
Ã
Â
Ã
are specified to represent an fcc unit
cell while other position coordinates are assumed to be understood.
Example 2 Find the Miller indices of a plane that makes intercepts of 2a, 3b and
4c along the three crystallographic axes, where a, b, c are primitive translation
vectors of the lattice.
Solution: Following the above said procedure, we have
(i)
Intercepts
2a
3b
4c
(ii)
Division by unit translation
2a
a ¼ 2
3b
b ¼ 3
4c
c ¼ 4
(iii)
Reciprocals
1
2
1
3
1
4
(iv)
After clearing fraction
6
4
3
⟹ The required Miller indices of the plane are (643).
Example 3 Find the Miller indices of a plane that makes intercepts as 3a: 4b on the
x and y axes, and is parallel to z-axis.
Fig. 4.4 The position
coordinates of the central
atom and corner atoms in a
bcc unit cell
140
4 Unit Cell Representations of Miller Indices