6.2 Matrix Representation of Symmetry Operations
Whenever a symmetry operation is applied to a unit cell, the coordinate axes
associated with it change. The resulting change can be conveniently described by
the following linear equations.
x
0
1 ¼ c 11 x 1 þ c 12 x 2 þ c 13 x 3
x
0
2 ¼ c 21 x 1 þ c 22 x 2 þ c 23 x 3
x
0
3 ¼ c 31 x 1 þ c 32 x 2 þ c 33 x 3
This can be written in the matrix form as
x
0
1
x
0
2
x
0
3
0
@
1
A ¼
c 11 c 12 c 13
c 21 c 22 c 23
c 31 c 32 c 33
0
@
1
A
x 1
x 2
x 3
0
@
1
A
ð6:1Þ
The matrix elements are given by
C ij ¼
c 11 c 12 c 13
c 21 c 22 c 23
c 31 c 32 c 33
0
@
1
A
The first script in the symbol C ij refers to the new axes and the second to the old
axes. Therefore, the matrix elements may be conveniently recognized as the
direction cosines between the new and old axes in the right-handed coordinate
system taken in the order old ! new as shown in the Fig. 6.14. Accordingly, the
matrix elements in terms of direction cosines are given as
C ij ¼
cos x 1 x
0
1
À
Á cos x 1 x
0
2
À
Á cos x 1 x
0
3
À
Á
cos x 2 x
0
1
À
Á cos x 2 x
0
2
À
Á cos x 2 x
0
3
À
Á
cos x 3 x
0
1
À
Á cos x 3 x
0
2
À
Á cos x 3 x
0
3
À
Á
0
@
1
A
ð6:2Þ
Fig. 6.14 Directions cosines
6.2 Matrix Representation of Symmetry Operations
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