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# 972 B3102005 Xray3

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### 972 B3102005 Xray3

1. 1. Diffraction theory: diffraction by crystals
2. 2. Fig. 3.1 Diffraction of a parallel primary beam by a small crystal.
3. 3. (3.1) … .structure factor (3.2) After summation over all atoms:
4. 4. (3.3)
5. 5. (3.4)
6. 6. Let
7. 7. (3.6) (3.5)
8. 8. Maximum I p happens only when h’, k’ and l’ are integers.
9. 9. Fig. 3.2 The function (sin 2 Nx)/sin 2 x for N=20. The function peaks at values of x which are integral multiples of π, and it is essentially zero everywhere else.
10. 13. (3.10) Fig. 3.3 Representation of the basis vector r n in terms of the fractional coordinates x n , y n , and z n . Structure factor =
11. 14. For face-centered crystals: (3.11) hkl unmixed ： hkl mixed ：
12. 15. For body-centered crystals: h+k+l = even ： h+k+l = odd ：
13. 16. For rock salt, NaCl hkl unmixed ： hkl mixed ： hkl all even ： hkl all odd ： hkl mixed ：
14. 17. In the Zinc blende form of ZnS hkl unmixed ： h+k+l = 4n ： h+k+l =(2n+1)2 ： hkl all odd ： hkl mixed ：
15. 18. For copper hkl unmixed ： hkl mixed ： For Tungsten h+k+l = even ： h+k+l = odd ：
16. 19. For Zn (hcp): For hexagonal close-packed structure h+2k =3n , l =even ： h+2k =3n  1 , l =odd : h+2k =3n  1 , l =even ： h+2k =3n , l =odd ：
17. 20. Hexagonal close-packed Zn, Mg, Be,  -Ti
18. 21. Thermal diffuse scattering: Let
19. 24. For single kind of atom: Thermal diffuse scattering: Debye-Waller factor:
20. 25.  = Debye Temperature  Appendix 13 h = Planck’s constant k = Boltzmann’s constant m = mass of the vibration atom x =  /T  (x)  Appendix 13 For Cubic element:
21. 26. For iron at 20  C
22. 27. Diffraction method variable fixed powder variable (in part) fixed Rotating- crystal fixed variable Laue   method
23. 28. -S 0 /  S/  [hkl]* (hkl)   P Ewald sphere construction for a crystal
24. 29. For powder sample: Each reciprocal lattice point circulates around the origin to form a sphere, which intersects the Ewald sphere in a Debye ring, see next page.
25. 30. A Debye ring of a powder sample
26. 31. Debye ring of a powder sample: intersection of two spheres
27. 38. Diffractometer
28. 39. ＊ Detector travels along the measuring circle ＊ Detector intersects each Debye ring in one arc
29. 40. [2h 2k 2l]* Increasing  Increasing  A crystal in a diffractometer
30. 41. A powder sample in a diffractometer
31. 43. Pinhole camera
32. 45. Laue camera
33. 47. (a) Transmission (b) back-reflection Laue W radiation, 30kV, 19mA
34. 49. Rotating crystal method
35. 51. Rotating crystal method
36. 52. Peak broadening <ul><li>Instrument broadening </li></ul><ul><li>Sample broadening </li></ul>§ Particle-size broadening § Non-uniform strain § Stacking fault Scherrer’s formula B = grain size