Fourier transform (see Cowley Sect. 2.2)

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Fourier transform (see Cowley Sect. 2.2)
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Fourier transform (see Cowley Sect. 2.2)
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Fourier transform (see Cowley Sect. 2.2)
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Fourier transform (see Cowley Sect. 2.2)
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Fourier transform
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Fourier transform
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Fourier transform
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Fourier transform
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Scattering of x-rays by single electron (Thomson)
(see Cowley sect. 4.1)
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Scattering of x-rays by single electron (Thomson)
o
(see Cowley sect. 4.1)
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Scattering of x-rays by single electron (Thomson)
o
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Scattering of x-rays by single electron (Thomson)
o
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Scattering of x-rays by single atom
For n electrons in an atom, time-averaged
electron density is
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Scattering of x-rays by single atom
For n electrons in an atom, time-averaged
electron density is
Can define an atomic scattering factor
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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
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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)
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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
i1
ai, bi tabulated for all elements in, e.g., De
Graef McHenry Structure of Materials, p. 299
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Dispersion - anomalous scattering
Have assumed radiation frequency gtgt
resonant frequency of electrons in atom
frequently not true
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Dispersion - anomalous scattering
Have assumed radiation frequency gtgt
resonant frequency of electrons in atom
frequently not true Need to correct
scattering factors f fo f' i f"
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Dispersion - anomalous scattering
Need to correct scattering factors f fo f'
i f"
5
f"
???K
1
2
f'
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Neutron scattering lengths
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Atom assemblies
(see Cowley sect. 5.1)

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Atom assemblies
(see Cowley sect. 5.1)

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

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Atom assemblies

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Atom assemblies
For single slit, width a g(x) 1

If scatterer is a box a, b, c
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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
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Atom assemblies

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

This requires ua h, an integer. Then
Finally
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Friedel's law

Inversion doesn't change intensities
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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
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Friedel's law

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Friedel's law

Inversion doesn't change intensities Generalizing
phase info is lost in intensity measurement
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Generalized Patterson
Suppose, for a distribution of atoms over a
finite volume

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

Then, in reciprocal space
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Generalized Patterson

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Generalized Patterson

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Generalized Patterson

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Generalized Patterson

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Source considerations

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Source considerations
Sources not strictly monochromatic - changes
Ewald construction
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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)
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Lorentz-polarization factor