1. Point Groups
Roya Majidi
1393

2. Point Groups Space Groups 7 Crystal Systems 32 point groups in 3D
How symmetry operators without translation combine
Point Groups
7 Crystal Systems
32 point groups in 3D
Lattices have only 7 distinct point groups
How symmetry operators with translation combine
Space Groups
14 Bravais Lattices
Lattices have only 14 distinct space groups
230 space groups in 3D

3. Classification of Symmetry Operators
Dimension of the Operator
Takes an object to its mirror form or not
Based on
If the operator acts at a point or moves a point
(i.e. outside a unit cell)
If it plays a role in the shape of a crystal or not (Macroscopic/Microscopic)

4. Rotation Reflection Inversion Translation Screw Rotation
Basic Symmetry
Operations
Inversion
Translation
Screw Rotation
Compound Symmetry
Operations
Glide Reflection
Rotoinversion

5. Operator Symmetry Inversion Rotation Mirror Central (I, O, C)
Classification based on the dimension invariant entity of the symmetry operator
Operator
Symmetry
Inversion
Central (I, O, C)
Rotation
Axial (A)
Mirror
Planar (m, P)

6. Mirror m

10. Vertical Mirror y x z
(x y z) (-x y z)

11. Vertical Mirror y x z
(x y z) (x -y z)

12. Horizontals Mirror y x z
(x y z) (x y -z)

13. Inversion 1

15. 6 6

16. (x y z) (-x –y –z)

17. n
Rotation Axis
If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where:
Crystals can only have 1, 2, 3, 4 or 6 fold symmetry

18. 1, 2, 3, 4, 6
A1, A2, A3, A4, or A6
C1, C2, C3, C4, or C6

19. Two-fold rotation = 360o/2 rotation

26.  = 360
n = 1
1-fold rotation axis 1

27. Symbol for 2-fold axis
 = 180
n = 2
2-fold rotation axis 2

28. Symbol for 3-fold axis
 = 120
n = 3
3-fold rotation axis 3

29.  = 90
n = 4
4-fold rotation axis 4

30.  = 60
n = 6
6-fold rotation axis 6

31. Roto-inversion operations
A roto-inversion operator rotates a point/object and then inverts it (inversion operation) in one go.
Roto-inversion operations

38. Try combining a 2-fold rotation axis with a mirror
2-D Symmetry
Try combining a 2-fold rotation axis with a mirror

39. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
(could do either step first)

40. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)

41. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
Is that all??

42. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
No! A second mirror is required

43. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror
The result is Point Group 2mm
“2mm” indicates 2 mirrors
The mirrors are different
(not equivalent by symmetry)

44. Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror

45. Now try combining a 4-fold rotation axis with a mirror Step 1: reflect
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect

46. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 1

47. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 2

48. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 3

49. Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?

50. Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors

51. Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
Point group name??

52. Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
Point group name??
4mm

53. Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
Point group name??
4mm

54. 3-fold rotation axis with a mirror creates point group 3m
2-D Symmetry
3-fold rotation axis with a mirror creates point group 3m

55. 6-fold rotation axis with a mirror creates point group 6mm
2-D Symmetry
6-fold rotation axis with a mirror creates point group 6mm

56. Example
4mm
622

57. Example mm m

58. Example
4mm m

59. Example
No symmetry 4
This is Amorphous!!

60. We now have 10 unique 3-D symmetry operations: 1 2 3 4 6 i m 3 4 6
Combinations of these elements are also possible
A complete analysis of symmetry about a point in space requires that we try all possible combinations of these symmetry elements

61. Rotoinversion a. 1-fold rotoinversion ( 1 )
3-D Symmetry
Rotoinversion
a. 1-fold rotoinversion ( 1 )

62. 3-D Symmetry Rotoinversion a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)

63. 3-D Symmetry Rotoinversion a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)
Step 2: invert
This is the same as i, so not
a new operation

64. 3-D Symmetry Rotoinversion
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Note: this is a temporary step, the intermediate motif element does not exist in the final pattern

65. 3-D Symmetry Rotoinversion b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 2: invert

66. Rotoinversion b. 2-fold rotoinversion ( 2 ) The result:
3-D Symmetry
Rotoinversion
b. 2-fold rotoinversion ( 2 )
The result:

67. 3-D Symmetry Rotoinversion b. 2-fold rotoinversion ( 2 )
This is the same as m, so not a new operation

68. Rotoinversion c. 3-fold rotoinversion ( 3 )
3-D Symmetry
Rotoinversion
c. 3-fold rotoinversion ( 3 )

69. 3-D Symmetry Rotoinversion 1 c. 3-fold rotoinversion ( 3 )
Step 1: rotate 360o/3
Again, this is a temporary step, the intermediate motif element does not exist in the final pattern 1

70. 3-D Symmetry Rotoinversion c. 3-fold rotoinversion ( 3 )
Step 1: rotate 360o/3
Step 2: invert through center

71. 3-D Symmetry Rotoinversion 1 2 c. 3-fold rotoinversion ( 3 )
Completion of the first sequence 1 2

72. Rotoinversion c. 3-fold rotoinversion ( 3 ) Rotate another 360/3
3-D Symmetry
Rotoinversion
c. 3-fold rotoinversion ( 3 )
Rotate another 360/3

73. Rotoinversion c. 3-fold rotoinversion ( 3 ) Invert through center
3-D Symmetry
Rotoinversion
c. 3-fold rotoinversion ( 3 )
Invert through center

74. 3-D Symmetry Rotoinversion 3 1 2 c. 3-fold rotoinversion ( 3 )
Complete second step to create face 3 3 1 2

75. 3-D Symmetry Rotoinversion 3 1 4 2 c. 3-fold rotoinversion ( 3 )
Third step creates face 4
(3  (1)  4) 3 1 4 2

76. 3-D Symmetry Rotoinversion 1 5 2 c. 3-fold rotoinversion ( 3 )
Fourth step creates face 5 (4  (2)  5) 5 1 2

77. 3-D Symmetry Rotoinversion 5 1 6 c. 3-fold rotoinversion ( 3 )
Fifth step creates face 6
(5  (3)  6)
Sixth step returns to face 1 5 1 6

78. Rotoinversion c. 3-fold rotoinversion ( 3 ) This is unique
3-D Symmetry
Rotoinversion
c. 3-fold rotoinversion ( 3 )
This is unique 3 5 1 4 2 6

79. Rotoinversion d. 4-fold rotoinversion ( 4 )
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )

80. Rotoinversion d. 4-fold rotoinversion ( 4 )
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )

81. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4

82. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert

83. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert

84. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4

85. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert

86. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert

87. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4

88. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 6: Invert
3-D Symmetry
Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
6: Invert

89. 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 )
This is also a unique operation

90. 3-D Symmetry Rotoinversion d. 4-fold rotoinversion ( 4 )
A more fundamental representative of the pattern

91. Rotoinversion e. 6-fold rotoinversion ( 6 ) Begin with this framework:
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 )
Begin with this framework:

92. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1

93. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1

94. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 2

95. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 2

96. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 2

97. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 2

98. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 2 4

99. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 2 4

100. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 5 2 4

101. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 5 2 4

102. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 5 2 6 4

103. 3-D Symmetry Rotoinversion e. 6-fold rotoinversion ( 6 )
Note: this is the same as
a 3-fold rotation axis perpendicular to a mirror plane
Top View

104. Rotoinversion e. 6-fold rotoinversion ( 6 ) A simpler pattern
3-D Symmetry
Rotoinversion
e. 6-fold rotoinversion ( 6 )
A simpler pattern
Top View

105. 3-D Symmetry The 32 3-D Point Groups (sometime crystal classes)
Every 3-D pattern must conform to one of them.
This includes every crystal, and every point within a crystal
Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

106. 3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System
(more later when we consider translations)
Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

107. Crystal has 4mm symmetry
Example
Crystal has 4mm symmetry

108. Example
Crystal has
4/m 2/m 2/m symmetry