H
K
L
0
@
1
A ¼
1
0 1/3
0
1 1/3
À1 À1 1/3
0
@
1
A
h
k
l
0
@
1
A
ð5:17Þ
The determinant of the matrix in Eq. 5.17, D ¼ 1: Therefore using Eq. 5.3, we
can obtain the volume relationship between the two unit cells
V TR ¼ V SH
Now, combining the hexagonal indices algebraically, we obtain
H þ K þ L
ð
Þ ¼ 1
ð5:18Þ
This is an important consequence which tells us that when the planes in a simple
hexagonal lattice are referred to as trigonal axes, only certain combinations of
trigonal indices given by Eq. 5.18 are allowed. This limits the possible X-ray
reflections in the study of such crystals.
The reverse relationships can be found by determining the inverse of the matrix
in Eq. 5.17, that is,
Fig. 5.4 Simple hexagon and
trigonal axes
5.2 Transformation of Indices of Crystal Planes (Unit Cell)
187
K
L
0
@
1
A ¼
1
0 1/3
0
1 1/3
À1 À1 1/3
0
@
1
A
h
k
l
0
@
1
A
ð5:17Þ
The determinant of the matrix in Eq. 5.17, D ¼ 1: Therefore using Eq. 5.3, we
can obtain the volume relationship between the two unit cells
V TR ¼ V SH
Now, combining the hexagonal indices algebraically, we obtain
H þ K þ L
ð
Þ ¼ 1
ð5:18Þ
This is an important consequence which tells us that when the planes in a simple
hexagonal lattice are referred to as trigonal axes, only certain combinations of
trigonal indices given by Eq. 5.18 are allowed. This limits the possible X-ray
reflections in the study of such crystals.
The reverse relationships can be found by determining the inverse of the matrix
in Eq. 5.17, that is,
Fig. 5.4 Simple hexagon and
trigonal axes
5.2 Transformation of Indices of Crystal Planes (Unit Cell)
187