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
  37. 53. Origin of particle-size broadening   (radians, Debye-Scherrer formulae)
  38. 56. A single (analyzing) crystal with a specific (hkl) plane//sample surface
  39. 57. ∑ A =sum of atomic weights of all atoms in a unit cell X-ray density   in g/cm 3 , A in grams, V’ in Å 3
  40. 58. Homework assignment <ul><li>Cullity 3-1 </li></ul><ul><li>Cullity 3-3 </li></ul><ul><li>Cullity 4-3 </li></ul><ul><li>Cullity 4-4 </li></ul>
  41. 59. (3.7)
  42. 60. (3.8) (3.9)

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