a 2 ¼ a 1 + b 1 À c 1
b 2 ¼ a 1 À b 1 + c 1
c 2 ¼ À a 1 þ b 1 þ c 1
Now, using these equations, we can write the following transformation equations
H ¼ 1 h þ 1 k À 1 l
K ¼ 1 h À 1 k þ 1 l
L ¼ À1 h þ 1 k þ 1 l
ð5:4Þ
The matrix form of these equations is
H
K
L
0
@
1
A ¼
1
1 À1
1 À1 1
À1 1
1
0
@
1
A
h
k
l
0
@
1
A
ð5:5Þ
The determinant of the matrix in Eq. 5.5, D ¼ 4: Therefore using Eq. 5.3, we
can obtain the volume relationship between the two unit cells
V FCC ¼ 4V RH
The reverse relationships can be found by determining the inverse of the matrix
in Eq. 5.5, 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 rhombohedral) 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
The same can be written as
h ¼
H
2
þ
K
2
þ 0 L
k ¼
H
2
þ 0 K þ
L
2
l ¼ 0 H þ
K
2
þ
L
2
ð5:6Þ
Let us check the validity of Eqs. 5.4 and 5.6 by taking some examples.
5.2 Transformation of Indices of Crystal Planes (Unit Cell)
181