1. Crystallography and Diffraction Techniques Myoglobin

2. Types of diffraction - X-ray diffraction - Electron diffraction - Neutron diffraction Enhanced visibility of hydrogen atoms by neutron crystallography on fully deuterated myoglobin Myoglobin diffraction pattern 1962 Nobel Prize by Max Perutz and Sir John Cowdery KendrewMax PerutzSir John Cowdery Kendrew

3. X-ray Diffraction

4. Water

5. Light

6. Electron

7. Constructive

8. Destructive

9. Diffraction from atoms

10. Continue

11. 1 A About 1 Å

12. Wave of mater

13. Wave of electrons The electrons are accelerated in an electric potential U to the desired velocity:

14. Crystal diffraction

15. Gas, liquid, powder diffraction

17. Surface diffraction

18. Diffraction by diffractometer

19. Example of spots by diffractometer

20. X-ray Crystallography

21. Electron density

22. Deformation Electron Density

23. Macromolecule X-ray Crystallography

24. Generation of X-rays

25. What is K  and K  (for Cu) ? K  : 2p  1s K  : 3p  1s

26. X-ray tube

28. An optical grating and diffraction of light

29. Lattice planes

30. Lattice planes => reflection

31. Lattice planes review

32. Bragg ’ s Law

35. 2dsin(theta)=n lumda

36. Bragg ’ s Law

37. Atomic scattering factor

39. intensity

40. Phase and intensity

41. Electron density

42. Diffraction of one hole

43. Diffraction of two holes

44. Diffraction of 5 holes

45. 2D four holes

46. From real lattice to reciprocal lattice Real holesReflection pattern Crystal lattice is a real lattice, while its reflection pattern is its corresponding reciprocal lattice.

47. TEM image of Si? or Diamond? Real lattice viewed from (110) direction. Si Diamond

48. Electron Diffraction

49. Conversion of Real Lattice to Reciprocal Lattice PPP PPP PPP PPP PPP PPP PPP PPP PPP PPP

50. Ewald Sphere and Diffraction Pattern The Ewald sphere is a geometric construct used in X-ray crystallography which neatly demonstrates the relationship between: the wavelength of the incident and diffracted x-ray beams, the diffraction angle for a given reflection, the reciprocal lattice of the crystal Paul Peter Ewald (1888~1985)

51. Ewald Sphere

52. A vector of reciprocal lattice represents a set of parallel planes in a crystal lattice 2d sin  = n (1/d hkl )/(2/ ) = sin  (hkl)  

53. Reciprocal Lattice and Ewald Sphere

54. Detector, Reciprocal Lattice and Ewald Sphere

55. 3D View of Ewald Sphere and Reciprocal Sphere

56. Techniques of X-ray diffraction Single Crystal and Powder X-ray Diffractions many many many very small single crystals

57. Diffractometers for Single Crystal and Powder X-ray Diffractions

58. Single Crystal and Powder X- ray Diffraction Patterns

59. The powder XRD method

60. Formation of a cone of diffracted radiation

61. XRPD on film electron diffraction of powder sample

62. Finger Print Identification Finger Print Identification for Known Compounds by comparing experimental XRPD to those in PDF database

63. Some peaks may not be observed due to preferred orientation For example, layered structure such as graphite.

64. X-ray powder diffraction patterns of crystalline and amorphous sample

65. Scherrer Formula t = thickness of crystal in Å B = width in radians, at an intensity equal to half the maximum intensity However, this type of peak broadening is negligible when the crystallite size is larger than 200 nm. B is often calculated relative to a reference solid (with crystallite size >500 nm) added to the sample: B 2 =Bs 2 -Br 2.

66. 2d sin  = Some equations to calculate cell parameters (d-spacings)

67. X-ray powder diffraction patterns for potassium halides

68. Structure Factor, Intensity and Electron Density R 1 =  ||F o | - |F c ||/  |F o | F calc F obs

69. Electron density maps by X-ray diffraction

70. Scattering of X-rays by a crystal-systematic absences

71. Systematic Absences

72. Systematic absence for C-center: (x,y,z) ≣ (x+1/2, y+1/2, z) F hkl = (1/V)  f j exp[2  i(hx j +ky j +lz j )] = (1/V)  f j [cos2  (hx j +ky j +lz j )+isin2  (hx j +ky j +lz j )] = (1/V)  f j {cos2  (hx j +ky j +lz j )+cos2  [h(x j +1/2) +k(y j +1/2)+lz j )]}+i{sin2  (hx j +ky j +lz j ) +sin2  [h(x j +1/2)+k(y j +1/2)+lz j )]} j=1 N N/2

73. let 2  (hx j +ky j +lz j )=  j cos(A+B)=cosAcosB-sinAsinB sin(A+B)=sinAcosB+cosAsinB (1/V)  f j  cos2  (hx j +ky j +lz j )+cos2  h(x j +1/2)+k(y j +1/2)+lz j )]} +i  sin2  (hx j +ky j +lz j )+sin2  h(x j +1/2)+k(y j +1/2)+lz j )]} =(1/V)  f j  cos  j +cos  j +  h+k))+i[sin  j +sin  j +  h+k))]} =(1/V)  f j  cos  j +cos  j cos  h+k)]+i  sin  j +sin  j cos  h+k)]} ={[cos  h+k) + 1]}/V  f j  cos  j + isin  j ] So when cos  h+k) = -1 that is when h+k = 2n+1, F hkl = 0 Condition for systematic absences caused by C-center: For all (hkl), when h+k = 2n+1, I hkl = 0

74. F hkl =(1/V)  f j  cos2  (hx j +ky j +lz j )+isin2  (hx j +ky j +lz j )] =(1/V)  f j {  cos2  (hx j +ky j +lz j )+cos2  (-hx j +k(y j +1/2)-lz j )] +i  sin2  (hx j +ky j +lz j )+ sin2  (-hx j +k(y j +1/2)-lz j )]} For reflections at (0 k 0) F hkl = (1/V)  f j {[cos(2  ky j )+ cos(2  ky j )cos(k  )] + i[sin(2  ky j )+ sin(2  ky j )cos(k  )]} =[(cos(k  )+1)/v]  f j [cos(2  ky j )+ i[sin(2  ky j )] Systematic absences for 2 1 //b where (x,y,z) ≣ (-x,y+1/2,-z) So the conditions for 2 1 //b screw axis: For all reflections at (0 k 0), when k = 2n+1, I hkl =0

75. Conditions of Systematic Absences I-center: for all (hkl), h+k+l = 2n+1, I hkl = 0 F-center: for all (hkl), h+k = 2n+1, h+l = 2n+1 k+l = 2n+1, I hkl = 0 (or h, k, l not all even or all odd) c-glide (b-axis), for all (h0l), l = 2n+1, I hkl = 0 n-glide (b-axis), for all (h0l), h+l = 2n+1, I hkl = 0 d-glide (b-axis), for all (h0l), h+l = 4n+1, 2 or 3, I hkl = 0 3 1 //b screw axis, for all (0k0), k = 3n+1, 3n+2, I hkl = 0 其他類推

76. Setup of Conventional Single Crystal X-ray Diffractometer

77. Electron diffraction Electron diffraction e -   0.04 Å Can see crystal structure of very small area Associated with TEM f much larger than that of X-ray: can see superlattice Ni–Mo alloy (18 % Mo) with fcc structure. Weak spots result from superlattice of Mo arrangement.

78. Secondary diffraction of electron diffraction Extra reflections may appear in the diffraction pattern The intensities of diffracted beam are unreliable

80. Neutron diffraction

82. Antiferromagnetic superstructure in MnO, FeO and NiO MnO Fe 3 O 4 The most famous anti-ferromagnetic, manganese oxide (MnO) helped earn the Nobel prize for C. Shull, who showed how such magnetic structures could be obtained by neutron diffraction (but not with the more common X-ray diffraction).

83. Schematic neutron and X-ray diffraction patterns for MnO