For example, the independent symmetry elements of the point group mm2 are
two mirror planes m, m and the number of equivalent points associated with each
mirror plane is 2. Therefore, the number of equivalent positions for the point group
mm2 is 2 Â 2 = 4.
A list of symmetry matrix and fractional coordinates of equivalent positions is
provided for 2-D point groups in Table 3.1.
Making use of the concept of equivalent points, crystal structures can be entirely
specified with fractional coordinates.
Solved Examples
Example 1 In a unit cell of plane hexagonal lattice with a = b = 10 Å and
c = 120°, an atom has its fractional coordinates x = 0.5 and y = 0.5. Draw four unit
cells on its edges and write their fractional coordinates.
Solution: Given: a = b = 10 Å, c = 120°, fractional coordinates of an atom in the
given unit cell: x = 0.5 and y = 0.5.
Table. 3.1 Symmetry matrix and fractional coordinates of equivalent points in 2-D
S.
No
Symmetry
operation
No. of
equivalent
points
Symmetry
matrix
Fractional coordinates
1
Identity
1
1 0
0 1
(x, y)
2
2-fold
rotation
(jj to z-axis)
2
1 0
0 1
(x, y), (x, y)
3
Mirror line
(jj to y or
x-axis)
2
1 0
0 1
(x, y), (x, y) or (x, y), (x, y)
4
4-fold
rotation
4
0 1
1 0
(x, y), (y, x), (x, y), (y, x)
5
3-fold
rotation
3
0 1
1 1
(x, y), (y, x-y), (y-x, x)
6
6-fold
rotation
6
1 1
1 0
(x, y), (x-y, x), (y, x-y),
(x, y), (y-x, x), (y, y-x)
7
mm2
4
1 0
0 1
(x, y), (x, y), (x, y), (x, y)
8
4 mm
8
0 1
1 0
(x, y), (y, x), (x, y), (y, x), (x, y),
(x, y), (y, x), (y, x)
9
3 m (3m1)
6
1 0
1 1
(x, y), (y, x-y), (y-x, x), (y, x),
(y-x, x), (x, x-y)
10
6 mm
12
1 1
0 1
(x, y), (x-y, x), (y, x-y), (x, y),
(y-x, x), (y, y-x),
(x-y, y), (x, y-x), (y, x), (y, x),
(x, x-y), (y-x, y)
3.1 Fractional Coordinates
97