(iv) Nine mirror planes, 3 bisecting the parallel faces, and 6 connecting diagonal
edges. All of them are passing through the center of the cube.
(v) One center of symmetry at the center of the cube.
Thus, there are:
6-diads + 4-triads + 3-tetrads = 13 axes
3 surface planes + 6 diagonal planes = 9 planes
Center of symmetry = 1
Therefore, the total number of symmetry elements in a cube = 13 + 9 + 1 = 23.
Example 11 Determine the equivalent Schoenflies notations corresponding to the
International notations, 2; 3; 4 and 6.
Solution: Given: International notation of rotoinversion symmetries are:
2; 3; 4 and 6. Let us find their equivalent Schoenflies notations, one by one.
(a) The symbol 2 indicates that the crystal has rotoinversion axis of order 2.
Therefore,
2 ¼ C
1
2 :i ¼ C
1
2 :C
1
2 :r h ¼ Er h ¼ r mirrorplane
ð
Þ
where C
1
2 :C
1
2 ¼ E is the identity element.
(b) Similarly, the symbol 3 indicates that the crystal has rotoinversion axis of order
3. Therefore,
3 ¼ C
1
3 :i ¼ C
1
3 :C
1
2 :r h ¼ C
2
6 :C
3
6 :r h ¼ C
5
6 :r h ¼ S
5
6
This is equivalent to the rotoreflection axis of order 6.
(c) The symbol 4 indicates that the crystal has rotoinversion axis of order 4.
Therefore,
4 ¼ C
1
4 :i ¼ C
1
4 :C
1
2 :r h ¼ C
1
4 :C
2
4 :r h ¼ C
3
4 :r h ¼ S
3
4
This is equivalent to the rotoreflection axis of order 4.
(d) The symbol 6 indicates that the crystal has rotoinversion axis of order 6.
Therefore,
6 ¼ C
1
6 :i = C
1
6 :C
1
2 :r h ¼ C
1
6 :C
3
6 :r h ¼ C
4
6 :r h ¼ C
2
3 :r h ¼ C
3
3 :C
2
3 :r h ¼ C
5
3 :r h ¼ S
5
3
This is equivalent to the rotoreflection axis of order 3.
218
6 Unit Cell Symmeteries and Their Representations