V fcc = 4V rh
The reverse relationships can be found by determining the inverse of the matrix
in Eq. 5.26, that is,
1
1 À1
1 À1 1
À1 1
1
0
@
1
A
À1
¼
1
2
1 1 0
1 0 1
0 1 1
0
@
1
A
Hence the inverse relationship (i.e., transformation from fcc to primitive) is
h
k
l
0
@
1
A ¼
1
2
1 1 0
1 0 1
0 1 1
0
@
1
A
H
K
L
0
@
1
A
ð5:27Þ
These relationships are valid for all other face-centered and their corresponding
primitive lattices.
The Eq. 5.27 can be written as
h ¼
H
2
þ
K
2
þ 0 L
k ¼
H
2
þ 0 K þ
L
2
ð5:28Þ
Let us check the validity of Eqs. 5.25 and 5.28 by taking some examples.
Example 6 Find the equivalent fcc planes corresponding to its primitive lattice
planes: (100), (110) and (111).
Solution: Given: Planes of fcc primitive: (100), (110) and (111).
Let us take them one by one.
Case I: Plane of fcc primitive: (hkl) (100).
Substituting the values of h, k and l in Eq. 5.25, we obtain (HKL) ð11 1Þ for fcc
plane. Again, substituting the values of H, K and L in Eq. 5.28, we obtain (hkl)
(100) for primitive fcc plane, this is the same indices with which we started. This
confirms the validity of Eqs. 5.25 and 5.28.
Case II: Plane of fcc primitive: (hkl) (110).
A similar operation with the given h, k and l values in Eq. 5.25 will provide us
(HKL) (200) for fcc plane. Again, substituting the values of H, K and L in
Eq. 5.28, we obtain (hkl) (110) for primitive bcc plane. This confirms the validity
of Eqs. 5.25 and 5.28.
192
5 Unit Cell Transformations