1. According to rule 1, the identity element E forms a class of its own.
2. According to rule 2, C 2 belongs to a separate class.
3. According to rule 2, both C 4 and its inverse C
3
4 belong to one class (due to the
presence of a vertical C 2 axis).
4. According to rule 4, the first two mirror symmetries r x and r y belong to one
class while the other mirror symmetries r 1 and r 2 belong to another class.
Hence, in all there are five separate classes. All eight symmetry elements can be
written in five classes as: E, C 2 , 2C 4 , 2r x and 2r 1 .
Example 13 Determine the number of classes corresponding to the point group
D 2d ( 42m) whose symmetry elements are E, S 4 , S
2
3 , S
3
4 , C 2x , C 2y , r 1 and r 2 .
Solution: Given: Point group D 2d ( 42m), symmetry elements are: E, S 4, S
2
3 , S
3
4 , C 2x ,
C 2y , r 1 and r 2 .
No. of classes = ?
We know that E and C 2 are inverses of their own. Similarly, S 4 and S
3
4 are the
inverses of each other. C 2x , C 2 , r 1 , and r 2 are inverses of their own.
Let us obtain the number of classes using the above-mentioned rules.
1. According to rule 1, the identity element E forms a class of its own.
2. According to rule 2, C 2 belongs to a separate class.
3. According to rule 2, S 4 and S
3
4 belong to one class (due to vertical C 2 axis).
4. According to rule 3, C 2x and C 2y belong to another separate class (due to mirror
plane r 1 or r 2 which interchanges points on the two axes).
5. According to rule 4, r 1 and r 2 belong to the same class (due to S 4 axis which
interchanges points on the two mirrors) .
Hence, in all there are five separate classes. All eight symmetry elements can be
written in five classes as: E, C 2, 2S 4 , 2C 2x , 2r 1 .
Multiple Choice Questions (MCQ)
1. Permissible rotational symmetries in a crystalline solid are limited to:
(a) 5
(b) 4
(c) 3
(d) 2
2. Two mirrors at an angle h together produce a rotation of:
(a) h
(b) 2h
(c) 3h
(d) 4h
6.3 Group (Point) Representation of Symmetry Operations
251
2. According to rule 2, C 2 belongs to a separate class.
3. According to rule 2, both C 4 and its inverse C
3
4 belong to one class (due to the
presence of a vertical C 2 axis).
4. According to rule 4, the first two mirror symmetries r x and r y belong to one
class while the other mirror symmetries r 1 and r 2 belong to another class.
Hence, in all there are five separate classes. All eight symmetry elements can be
written in five classes as: E, C 2 , 2C 4 , 2r x and 2r 1 .
Example 13 Determine the number of classes corresponding to the point group
D 2d ( 42m) whose symmetry elements are E, S 4 , S
2
3 , S
3
4 , C 2x , C 2y , r 1 and r 2 .
Solution: Given: Point group D 2d ( 42m), symmetry elements are: E, S 4, S
2
3 , S
3
4 , C 2x ,
C 2y , r 1 and r 2 .
No. of classes = ?
We know that E and C 2 are inverses of their own. Similarly, S 4 and S
3
4 are the
inverses of each other. C 2x , C 2 , r 1 , and r 2 are inverses of their own.
Let us obtain the number of classes using the above-mentioned rules.
1. According to rule 1, the identity element E forms a class of its own.
2. According to rule 2, C 2 belongs to a separate class.
3. According to rule 2, S 4 and S
3
4 belong to one class (due to vertical C 2 axis).
4. According to rule 3, C 2x and C 2y belong to another separate class (due to mirror
plane r 1 or r 2 which interchanges points on the two axes).
5. According to rule 4, r 1 and r 2 belong to the same class (due to S 4 axis which
interchanges points on the two mirrors) .
Hence, in all there are five separate classes. All eight symmetry elements can be
written in five classes as: E, C 2, 2S 4 , 2C 2x , 2r 1 .
Multiple Choice Questions (MCQ)
1. Permissible rotational symmetries in a crystalline solid are limited to:
(a) 5
(b) 4
(c) 3
(d) 2
2. Two mirrors at an angle h together produce a rotation of:
(a) h
(b) 2h
(c) 3h
(d) 4h
6.3 Group (Point) Representation of Symmetry Operations
251