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X-ray Scattering Analysis: Fourier Transform and Atom Assemblies

Explore the Fourier transform in X-ray scattering analysis, including Thomson scattering, atomic scattering factors, dispersion, and correcting scattering factors. Learn about Fourier transforms for electron densities, atom assemblies, Friedel's law, and generalized Patterson analysis in X-ray crystallography. Understand the Lorentz factor and source considerations in X-ray diffraction experiments.

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X-ray Scattering Analysis: Fourier Transform and Atom Assemblies

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  1. Fourier transform (see Cowley Sect. 2.2)

  2. Fourier transform (see Cowley Sect. 2.2)

  3. Fourier transform (see Cowley Sect. 2.2)

  4. Fourier transform (see Cowley Sect. 2.2)

  5. Fourier transform

  6. Fourier transform

  7. Fourier transform

  8. Fourier transform

  9. Scattering of x-rays by single electron (Thomson) (see Cowley sect. 4.1)

  10. Scattering of x-rays by single electron (Thomson) o (see Cowley sect. 4.1)

  11. Scattering of x-rays by single electron (Thomson) o

  12. Scattering of x-rays by single electron (Thomson) o

  13. Scattering of x-rays by single atom For n electrons in an atom, time-averaged electron density is

  14. Scattering of x-rays by single atom For n electrons in an atom, time-averaged electron density is Can define an atomic scattering factor

  15. Scattering of x-rays by single atom For n electrons in an atom, time-averaged electron density is Can define an atomic scattering factor For spherical atoms

  16. Z Scattering of x-rays by single atom Need to find (r) …. A QM problem But soln for f() looks like this (in electron scattering units)

  17. Scattering of x-rays by single atom Soln for f() looks like this (in electron scattering units) Curve-fitting fcn: f = Z - 41.78214 x sin2/2x  ai e-b sin/ 3 or 4 2 2 i i=1 ai,bi tabulated for all elements in, e.g., De Graef & McHenry: Structure of Materials, p. 299

  18. Dispersion - anomalous scattering Have assumed radiation frequency >> resonant frequency of electrons in atom … frequently not true

  19. Dispersion - anomalous scattering Have assumed radiation frequency >> resonant frequency of electrons in atom … frequently not true Need to correct scattering factors f = fo + f' + i f"

  20. Dispersion - anomalous scattering Need to correct scattering factors f = fo + f' + i f" 5 f" 1 2 K f'

  21. Neutron scattering lengths

  22. Atom assemblies (see Cowley sect. 5.1)

  23. Atom assemblies (see Cowley sect. 5.1) For this electron density, there is a Fourier transform F(u) is a fcn in reciprocal space

  24. Atom assemblies (see Cowley sect. 5.1)

  25. Atom assemblies

  26. Atom assemblies For single slit, width a & g(x) = 1 If scatterer is a box a, b, c

  27. Atom assemblies For single slit, width a & g(x) = 1 If scatterer is a box a, b, c For periodic array of zero-width slits

  28. Atom assemblies This requires ua = h, an integer. Then Finally

  29. Atom assemblies This requires ua = h, an integer. Then Finally

  30. Friedel's law Inversion doesn't change intensities

  31. Friedel's law Consider ZnS - one side crystal terminated by Zn atoms, other side by S atoms Phase differences (on scattering are 1 (S) & 2 (Zn) A,B = o + 2 - 1 C,D = o + 1 - 2 Coster, Knol, & Prins (1930) expt: Used AuL1 (1.274 Å) & AuL2 (1.285 Å) ZnKedge = 1.280 Å Expect phase changes and thus intensities different for 1 from Zn side; 2 unaffected

  32. Friedel's law

  33. Friedel's law Inversion doesn't change intensities Generalizing: phase info is lost in intensity measurement

  34. Generalized Patterson Suppose, for a distribution of atoms over a finite volume

  35. Generalized Patterson Suppose, for a distribution of atoms over a finite volume Then, in reciprocal space

  36. Generalized Patterson

  37. Generalized Patterson

  38. Generalized Patterson

  39. Generalized Patterson

  40. Source considerations

  41. Source considerations Sources not strictly monochromatic - changes Ewald construction

  42. Lorentz factor Lorentz factor takes into account change in scattering volume size & scan rate as a fcn of angle for a particular diffraction geometry E.g., for powder diffraction and (unpolarized beam)

  43. Lorentz-polarization factor

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