2 001
½
ŠÂm 010
½
Š¼
À1 0 0
0 À1 0
0
0 1
0
B
@
1
C
A
1 0 0
0 À1 0
0 0 1
0
B
@
1
C
A
¼
À1 0 0
0 1 0
0 0 1
0
B
@
1
C
A = m[100]
The resulting mirror plane is perpendicular to the x axis.
Example 7 In an orthorhombic crystal system, show that two mutually perpendicular mirror planes produce a 2-fold axis.
Solution: Given: Crystal system is orthorhombic, two mutually perpendicular
mirror planes. Let us suppose that the two mutually perpendicular mirror planes are:
m[100] and m[010]. Now, writing them in the matrix form and taking their product,
we obtain
m 001
½
ŠÂm 010
½
Š ¼
À1 0 0
0 1 0
0 0 1
0
B
@
1
C
A
1 0 0
0 À1 0
0 0 1
0
B
@
1
C
A
¼
À1 0 0
0 À1 0
0
0 1
0
B
@
1
C
A ¼ 2½001Š
The resulting 2-fold axis represents the intersection of the two mirrors along
c-axis.
Example 8 In an orthorhombic crystal system, show that two mutually perpendicular 2-fold axes produce another 2-fold axis perpendicular to each of the first two
axes.
Solution: Given: Crystal system is orthorhombic, two mutually perpendicular
2-fold axes. Let us suppose that the two mutually perpendicular axes are: 2[010]
and 2[001]. Now, writing 2[010] and 2[001] in the matrix form and taking their
product, we obtain
2 001
½
ŠÂ2 010
½ Š ¼
À1 0 0
0 À1 0
0
0 1
0
B
@
1
C
A
À1 0 0
0 1 0
0 0 À1
0
B
@
1
C
A
¼
1 0
0
0 À1 0
0 0 À1
0
B
@
1
C
A ¼ 2½100Š
The resulting 2-fold axis is perpendicular to the first two axes.
6.2 Matrix Representation of Symmetry Operations
227