1. Objectives • Written and graphic symbols of symmetry elements
• Basic symmetry elements and symmetry operations
• Written and graphic symbols of symmetry elements
• Crystal systems and Miller indices
• Lattices and unit cell

2. Periodic array in a crystal: Example 1
STM (Scanning Tunneling Microscope) image of a platinum surface
IBM Research Almaden Research Center

3. Periodic array in a crystal: Example 2
TopoMetrix
Corporation 
Interconnected 6-membered rings of graphite
and the triangular geometry about each carbon atom.

4. Symmetrical crystal forms

5. Non-isometric forms
Fig. 5.38, Klein pg. 205

6. Isometric forms
Fig. 5.38, Klein pg. 206

7. Rotation
A Symmetrical Pattern 6 6

8. 6 6 Two-fold rotation 360˚/2 Motif Element Operation The axis
The plane (perpendicular to the axis)
The terms: motif, symmetry element, symmetry operation, the pattern
2-fold, 360/2=180
The symbal 6
Operation

9. Three-fold rotation
360o/3 6
step 1 6
step 3 6
step 2

10. d 9 n-fold Rotation a Z t 6 6 6 6 6 1-fold 2-fold 3-fold 4-fold 6-fold
identity

11. 6 6 Inversion In 2D, inversion = 2-fold rotation
Role play: difference between
In 2D, inversion = 2-fold rotation
In 3D, inversion ≠ 2-fold rotation

12. Rotation + Inversion 3

13. Rotation + Inversion 3 1

14. Rotation + Inversion 3

15. Rotation + Inversion 3 1 2

16. Rotation + Inversion 3

17. Rotation + Inversion 3

18. Rotation + Inversion 3 1 2 3

19. Rotation + Inversion 3 1 2 3 4

20. Rotation + Inversion 3 1 2 5

21. Rotation + Inversion 3 3 5 1 4 2 6

22. Crystal systems: length/angle relations
Klein Fig. 5.27, pg. 196

23. Crystal System - Symmetry Characteristics

24. Crystal system - Symmetry characteristics
Klein Fig. 5.25, pg. 193

25. Lattice, lattice point, unit cell

26. Escher Print: Equivalent points

27. Escher Print: Which Unit Cell?

28. Plane lattices (nets): 5 unique types
Fig. 5.50, Klein, pg 218

29. Bravais lattices (14 unique types)
Triclinic
¹ b
¹ g c c c
Fig. 5.63
Klein, pg 232
Table 5.9
Klein, pg 233 b b P
I = C a a
Monoclinic g o
a =
= 90
¹ b a b c c b a P C F I
Orthorhombic o a b c b g
a = =
= 90

30. Bravais lattices (14 unique types)2 1 P I
Tetragonal 2 a 1
= a c
a = b = g
= 90 o a 3 a 2
Fig. 5.63
Klein, pg 232
Table 5.9
Klein, pg 233 a 1 P F I
Isometric 1 2 a
= a 3
a = b = g
= 90 o