Download presentation

Presentation is loading. Please wait.

Published byDamon Washington Modified over 2 years ago

2
The 10 two-dimensional crystallographic point groups Interactive exercise Eugen Libowitzky Institute of Mineralogy and Crystallography 2012

3
For teachers… (1) "Copy and paste" can be used to extend the exercise pages or to change the graphics objects. (2) Symmetry elements of a group and most graphics objects have been aligned at the page center. (3) The interactive fields and the symmetry elements of each point group are grouped. (4) The group m is available with vertical and horizontal m.

4
Basics In two-dimensional lattices* and patterns the following local symmetries (i.e. without repetition by shift / translation; centered around one point) may occur: (1) Rotation about a 1-, 2-, 3-, 4-, 6-fold rotation axis (RA) (2) Reflection(s) through one (or more) mirror plane(s)** m * Remarks: Lattices have nothing to do with point groups! However, they constrain the possible rotation axes to those five given above! ** In two dimensions the term "mirror line" also makes sense.

5
Sym. elementSym. operation Printed symbol Graphical symbol 1-fold RA360° rotation1none 2-fold RA180° rotation2 3-fold RA120° rotation3 4-fold RA90° rotation4 6-fold RA60° rotation6 mirror planereflectionm |

6
Point groups… In two dimensions these 5 rotation axes alone or in combination with the mirror plane(s) result in exactly 10 possible combinations or arrangements of symmetry elements that intersect in one point: the 10 two-dimensional crystallographic* point groups * Remark: 5-, 7- and higher-fold rotation axes would result in an infinite number of two-dimensional point groups. However, these are not compatible with lattices. – See note (*) before!

7
Symmetry An object is "symmetric" in a geometric sense, if it appears identical (in shape and orientation) after a symmetry operation (except "1" - equal to identity). Example: 2-fold rotation

8
The 10 two-dim. cryst. point groups 2mm 3m3m4mm 6mm m (1m) 2 34 6 1

9
The point group symbol (according to Hermann-Mauguin) In cases where only one symmetry element is present the symbol is just 1, 2, 3, 4, 6, m (the latter also 1m). In all other cases the rotation axis is written first, each set of symmetrically equivalent mirror planes behind. 2mm2mm 3m3m4mm4mm6mm6mm

10
! Caution - Pitfall (1) ! The absolute orientation of symmetry elements in space has no impact on the point group assignment! Example: All three objects belong to point group m! )

11
! Caution - Pitfall (2) ! "Two-dimensional" means that an object extends only in the very two dimensions of a plane. Even if we observe the object from the third dimension in space, there is no way for a symmetry operation between an apparent "front-" and "backside". Example: The two-dim. "Ampelmännchen" CANNOT be rotated about a 2-fold RA within the plane to the "backside"!

12
! Caution - Pitfall (3) ! Symmetry is always complete, i.e. the operations must be repeatable until the starting point is reached again, and all parts of the object and even the symmetry elements must be included! A few examples: ) One detail "too much" prevents m! One "missing" detail prevents the complete 6-fold rotation and alle but one vertical m!

13
! Caution - Pitfall (4) ! Quite frequently an object is assigned a certain symmetry that is well present but which underestimates the true symmetry and thus results in the wrong point group. An extreme example: graph of a snowflake (6mm) Symmetry well present, but underestimated / incomplete!

14
! Caution - Pitfall (5) ! Symmetry elements or operations do not necessarily result in reproduction / repetition of object details. This is only true for general sites (remote from the symmetry elements). Special sites (exactly on a symmetry element) are repeated in themselves and therefore appear unchanged! A few examples: VZH

15
Interactive exercise: (1) Assign the symmetry to the object in the page center. (2) Mouse click (left) on one of the 10 point group labels. (3) The symmetry elements appear in the object. RIGHT: Bright chimes sound. Green check symbol. Disappears after repeated mouse click. WRONG: Dull "error" sound. Disappears after three seconds or after repeated mouse click. CONTINUE: Return key or mouse click to the background.

16
m134622mm4mm6mm3m

17
m134622mm4mm6mm3m

18
m134622mm4mm6mm3m

19
m134622mm4mm6mm3m

20
m134622mm4mm6mm3m

21
m134622mm4mm6mm3m

22
m134622mm4mm6mm3m

23
m134622mm4mm6mm3m

24
m134622mm4mm6mm3m

25
m134622mm4mm6mm3m

26
W m134622mm4mm6mm3m

27
96 m134622mm4mm6mm3m

28
m134622mm4mm6mm3m

29
m134622mm4mm6mm3m

30
m134622mm4mm6mm3m

31
m134622mm4mm6mm3m

32
HEXE m134622mm4mm6mm3m

33
m134622mm4mm6mm3m

34
m134622mm4mm6mm3m

35
m134622mm4mm6mm3m

36
m134622mm4mm6mm3m

37
m134622mm4mm6mm3m

38
m134622mm4mm6mm3m

39
m134622mm4mm6mm3m

40
m134622mm4mm6mm3m

41
m134622mm4mm6mm3m

42
m134622mm4mm6mm3m

43
m134622mm4mm6mm3m

44
SOS m134622mm4mm6mm3m

45
m134622mm4mm6mm3m

46
m134622mm4mm6mm3m

47
m134622mm4mm6mm3m

48
m134622mm4mm6mm3m

49
m134622mm4mm6mm3m

50
m134622mm4mm6mm3m

51
m134622mm4mm6mm3m

52
m134622mm4mm6mm3m

53
m134622mm4mm6mm3m

54
m134622mm4mm6mm3m

55
m134622mm4mm6mm3m

56
m134622mm4mm6mm3m

57
m134622mm4mm6mm3m

58
m134622mm4mm6mm3m

59
m134622mm4mm6mm3m

60
m134622mm4mm6mm3m

61
m134622mm4mm6mm3m

62
6009 m134622mm4mm6mm3m

63
m134622mm4mm6mm3m

64
m134622mm4mm6mm3m

65
m134622mm4mm6mm3m

66
m134622mm4mm6mm3m

67
OHO m134622mm4mm6mm3m

68
m134622mm4mm6mm3m

69
m134622mm4mm6mm3m

70
m134622mm4mm6mm3m

71
m134622mm4mm6mm3m

72
m134622mm4mm6mm3m

73
m134622mm4mm6mm3m

74
m134622mm4mm6mm3m

75
m134622mm4mm6mm3m

76
m134622mm4mm6mm3m

77
m134622mm4mm6mm3m

78
m134622mm4mm6mm3m

79
m134622mm4mm6mm3m

80
m134622mm4mm6mm3m

81
m134622mm4mm6mm3m

82
m134622mm4mm6mm3m

83
m134622mm4mm6mm3m

84
m134622mm4mm6mm3m

85
m134622mm4mm6mm3m

86
m134622mm4mm6mm3m

Similar presentations

OK

Symmetry Translations (Lattices) A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary.

Symmetry Translations (Lattices) A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on rational numbers for class 8 Ppt on main distribution frame Ppt on nelson mandela and mahatma gandhi Download small ppt on global warming Download ppt on transportation in animals and plants Ppt on product specification file Ppt on bgp routing protocol Ppt on waves tides and ocean currents definition Ppt on tourism in pakistan Ppt on volatility of stock market