crystallography ( 晶体学 )

1. crystallography (晶体学)

2. Most materials we use are crystalline, partially crystalline, or can be made crystalline or partially crystalline Crystalline means atom arrangement is periodic (周期性的) - repeats throughout space

3. Most materials we use are crystalline, partially crystalline, or can be made crystalline or partially crystalline metals

4. Most materials we use are crystalline, partially crystalline, or can be made crystalline or partially crystalline spinels (ferrites - 铁电的)

5. Most materials we use are crystalline, partially crystalline, or can be made crystalline or partially crystalline BaTiO3 (钡钛氧3)

6. Most materials we use are crystalline, partially crystalline, or can be made crystalline or partially crystalline zeolite (沸石)

7. Most materials we use are crystalline, partially crystalline, or can be made crystalline or partially crystalline carbon nanotubes (纳米管) (CNTs)

8. Most materials we use are crystalline, partially crystalline, or can be made crystalline or partially crystalline polyethylene (聚乙烯) polymer

9. This type of structure is called crystal structure(晶体结构) In crystals, atom groups (unit cells) repeated to form a solid material The study of this repetition in crystals is called crystallography

10. Repetition = symmetry (对称性)Types of repetition: rotation (旋转) translation (平移)

11. RotationWhat is rotational symmetry?

12. Imagine this object will be rotated (maybe)

13. Imagine this object will be rotated (maybe)

14. Imagine this object will be rotated (maybe)

15. Imagine this object will be rotated (maybe)

16. Was it?

17. The object is obviously symmetric…it has symmetry

18. The object is obviously symmetric…it has symmetryCan be rotated 90° w/o detection

19. …………so symmetry is really doing nothing

20. Symmetry is doing nothing -or at least doing something so that it looks like nothing was done!

21. What kind of symmetry does this object have?

22. What kind of symmetry does this object have? 4 (旋转轴)

23. What kind of symmetry does this object have? 4 m (镜)

24. What kind of symmetry does this object have? 4 m

25. What kind of symmetry does this object have? 4 m

26. What kind of symmetry does this object have? 4 4mm (点群) m

27. Another example:

28. Another example: 6 6mm m

29. And another:

30. And another: 2 2

31. What about translation?Same as rotation

32. What about translation?Same as rotationEx: one dimensional array of points Translations are restricted to only certain values to get symmetry (periodicity)

33. 2D translationsLots of common examples

34. Each block is represented by a point

35. This array of points is a LATTICE (晶格)

36. Lattice - infinite(无限的),perfectly periodic (周期性的) array of points in a space

37. Not lattice:

38. Not lattice:

39. Not lattice - ….some kind of STRUCTURE because not just points

40. Another type of lattice - with different symmetry rectangular

41. Another type of lattice - with different symmetrysquare

42. Another type of lattice - with different symmetry hexagonal (六角)

43. Back to rotation - This lattice exhibits 6-fold symmetry hexagonal

44. overlap not allowed Periodicity and rotational symmetryWhat types of rotational symmetry allowed? object with 4-fold symmetry translates OK object with 5-fold symmetry doesn't translate

45. a a Periodicity and rotational symmetrySuppose periodic row of points is rotated through ± :

46. S a t t a vector S = an integer x basis translation t Periodicity and rotational symmetryTo maintain periodicity:

47. S a t t a vector S = an integer x basis translation t t cos  = S/2 = mt/2 m cos  axis 2 1 0 2 π 1 1 1/2 π/3 5π/3 6 0 0 π/2 3π/2 4 -1 -1/2 2π/3 4π/3 3 -2 -1 - ππ2

48. m cos  axis 2 1 0 2 π 1 1 1/2 π/3 5π/3 6 0 0 π/2 3π/2 4 -1 -1/2 2π/3 4π/3 3 -2 -1 - ππ2 Only rotation axes consistent with lattice periodicity in 2-D or 3-D

49. Now we abstract further: this is UNIT CELL We abstracted points from the shape:

50. Now we abstract further: This is a UNIT CELL Represented by two lengths and an angle …….or, alternatively, by two vectors