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)
(see Cowley sect. 4.1)

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

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

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. 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) Z

17. Scattering of x-rays by single atom
Soln for f() looks like this
(in electron scattering units)
Curve-fitting fcn:
f = Z x sin2 /2 x  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. For this electron density, there is a Fourier transform
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 -  C,D = o + 1 - 2
Coster, Knol, & Prins (1930) expt:
Used AuL1 (1.274 Å) & AuL2 (1.285 Å) ZnKedge = 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