Fourier transform (see Cowley Sect. 2.2)

Fourier transform (see Cowley Sect. 2.2)

Fourier transform (see Cowley Sect. 2.2)

Fourier transform (see Cowley Sect. 2.2)

Fourier transform

Fourier transform

Fourier transform

Fourier transform

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

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

Scattering of x-rays by single electron (Thomson)

Scattering of x-rays by single electron (Thomson)

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

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

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

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

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 /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

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

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"

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

Neutron scattering lengths

Atom assemblies (see Cowley sect. 5.1)

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

Atom assemblies (see Cowley sect. 5.1)

Atom assemblies

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

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

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

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

Friedel's law Inversion doesn't change intensities

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

Friedel's law

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

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

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

Generalized Patterson

Generalized Patterson

Generalized Patterson

Generalized Patterson

Source considerations

Source considerations Sources not strictly monochromatic - changes Ewald construction

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)

Lorentz-polarization factor