crystallography ( 晶体学 )

1. crystallography (晶体学)

2. Structure is important Type of structure we discussed called crystal structure(晶体结构) In crystals, atom groups (unit cells) are repeated to form a solid material

3. Structure is important "X-ray diffraction" is important method for study of materials X-ray diffraction works because of the repeating nature of crystals To understand X-ray diffraction, must first understand repetition in crystals The study of repetition in crystals is called crystallography

4. Repetition = symmetry (对称性)Types of repetition: Rotation (旋转) Translation (平移)

5. RotationWhat is rotational symmetry?

6. Imagine that this object will be rotated (maybe)

7. Imagine that this object will be rotated (maybe)

8. Imagine that this object will be rotated (maybe)

9. Imagine that this object will be rotated (maybe)

10. Was it?

11. The object is obviously symmetric…it has symmetry

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

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

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

15. What kind of symmetry does this object have?

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

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

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

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

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

21. Another example:

22. Another example: 6 6mm m

23. And another:

24. And another: 2 2

25. What about translation?Same as rotation

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

27. 2D translationsLots of common examples

28. Each block is represented by a point

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

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

31. Not a lattice:

32. Not a lattice:

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

34. Another type of lattice - with a different symmetry rectangular

35. Another type of lattice - with a different symmetrysquare

36. Another type of lattice - with a different symmetry hexagonal

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

38. 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

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

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

41. 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

42. 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

43. We abstracted points from the shape:

44. We abstracted points from the shape: Now we abstract further:

45. Now we abstract further: This is a UNIT CELL

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

47. b a T Basis vectors and unit cells T = t a + t b b a a and b are the basis vectors for the lattice

48. c b a a, b, and c are the basis vectors for the lattice In 3-D:

49. c b T a T = t a + t b + t c a c b a, b, and c are the basis vectors for the lattice In 3-D: –––> [221] direction

50. Lattice parameters (晶格参数) : c a b b g a need 3 lengths - |a|, |b|, |c| & 3 angles - , ,  to get cell shape