6. FCC and Primitive (For All Lattices)
The axial relationships between the translation vectors of fcc and its primitive unit
cells are shown in Fig. 5.6. Let us consider a crystal plane which is referred to as
(hkl) in the primitive system of axes a 1 , b 1 , c 1 and (HKL) in fcc system of axes
a 2 , b 2 , c 2 . The relationships between the two sets of axes are:
a 2 ¼ a 1 þ b 1 À c 1
b 2 ¼ a 1 À b 1 + c 1
c 2 ¼ À a 1 + b 1 + c 1
Now, using these equations, we can write the following transformation equations
H ¼ 1h þ 1k À 1l
K ¼ 1h À 1k þ 1l
L ¼ À1h þ 1k þ 1l
ð5:25Þ
The matrix form of these equations is
H
K
L
0
@
1
A =
1
1 À1
1 À1 1
À1 1
1
0
@
1
A
h
k
l
0
@
1
A
ð5:26Þ
The determinant of the matrix in Eq. 5.26, D ¼ 4: Therefore using Eq. 5.3, we
can obtain the volume relationship between the two unit cells
Fig. 5.6 FCC and its
primitive unit cell axes
5.2 Transformation of Indices of Crystal Planes (Unit Cell)
191