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General theory of diffraction
X-rays scatter off the charge density (r), neutrons scatter
off the spin density. Coherent scattering (diffraction) creates
the Fourier transform of (r) from real to reciprocal space:
Ã(k) =  (r) ei (kk0) r d3r
à = |Ã|  e i 
I = |Ã|2 =
measured
intensity
kk0 = scattering vector = Ghkl for periodic structures
k0
k
(r)
Real space
Fourier transform from r to k:
Inverse FT from k to r:
I(k)
Reciprocal space
Ã(k) =  A(r) ei k r d3r
A(k) = (2)3  Ã(k) e+i k r d3k
Structure determination by diffraction
The diffraction pattern is determined by three considerations:
1) The Bragg condition (= energy and momentum conservation)
determines the position of the diffraction spots in k-space.
It originates from the whole crystal lattice.
2) The structure factor describes the intensity modulation of
the diffraction spots by the atoms inside the unit cell.
3) The atomic scattering factor describes the diffraction at an
individual atom.
This is in tune with the mantra that large objects in real space
correspond to small objects in k-space. 1) represents the largest
possible object in real space (the infinite lattice), and it becomes
the smallest possible object in k-space (a point). 2) represents a
medium-sized object (the unit cell), and 3) the smallest object
(an atom). They affect increasingly larger portions of k-space.
Structure factor
The structure factor Shkl is given by:
Shkl =  f exp[-i Ghkl r]
=  f exp[-i 2 (h u+k v+l w)]
where r is the position of atom number  inside the unit cell
and f its atomic scattering factor. r can be expressed in
terms of the real space lattice vectors by the indices u,v,w
just like G is expressed by the Miller indices h,k,l.
The structure factor leads to the extinction of certain spots,
i.e. those for which there is destructive interference between
two equal atoms in the unit cell. For example, the spot for G100
vanishes for the fcc lattice, since the (100) planes through the
corner atoms interfere destructively with those through the
face-centered atoms. G200 is the first Bragg spot on the x-axis.
Atomic scattering factor
The atomic scattering factor f is given by:
f =  (r) · exp[-i Ghkl r] d3r
where  is the charge density of a single atom inside the
unit cell. The integral over the charge density of an atom
is proportional to the number of electrons, i.e. the atomic
number Z. Furthermore, the diffraction intensity is given
by the square of the structure factor. That leads to a
strong increase of the diffraction intensity for heavy
atoms with high Z .
Experimental methods for structure analysis
Energy and momentum conservation impose four constraints in
three-dimensional diffraction. They cannot all be fulfilled by
adjusting the three k components of the diffracted wave (for
an arbitrary incident wave). Something else has to give. Either
the energy or the direction of the incident wave needs to be
flexible. This can be accomplished in several ways:
1) Incident x-rays with a continuous energy spectrum (Laue).
2) Rotate the crystal (popular with protein crystallography).
3) Use polycrystalline samples (powder diffraction, Debye-Scherrer).
Laue diffraction pattern
Laue diffraction pattern of
NaCl taken with neutrons.
See a projection of k-space.
Powder diffraction pattern
Observe rings around the
incoming and outgoing beam.
(Cylindrical film unfolded.)
Extra diffraction rings visible
for the ordered Cu3Au alloy.
Horizontal scan across the rings for Si powder. The (100), (200) reflections are forbidden in the
diamond structure, since their structure factor vanishes.
The phase problem
Mathematically, the structure (= charge density) in real space can be
obtained from the amplitude of the diffracted wave in k-space by an
inverse Fourier transform from k to r . However, the amplitude is a
complex number of the form A = |A|· ei , which contains the phase .
Only the intensity I=|A|2 is measured, not the phase .
Crystallographers have developed many tricks to retrieve the phase.
For example, sulfur can be replaced by selenium in proteins, which is
chemically similar. But selenium diffracts X-rays much more when the
X-ray energy is tuned to be in resonance with an inner shell excitation
(“anomalous scattering”). The difference between diffraction patterns
on- and off-resonance provides the phase information.
Simple crystal structures can be solved by calculating the diffraction
pattern for trial structures containing adjustable parameters. Those
are obtained by a least square fit to the diffraction intensities.
Reconstruction of a single nano-object (ptychography)
With the recent advent of laser-like X-ray sources there has been
great interest in Fourier-transforming the diffraction pattern of a
single object, such as a protein molecule or a virus (see next slide).
There is a theorem that allows the reconstruction of the phase if the
object is located in a well-defined finite aperture, with no diffracting
objects outside.
The idea is as follows: 1) Start with arbitrary phase in k-space and
perform an inverse Fourier transform from k to r. A phase error will
produce a finite amplitude outside the aperture. 2) Correct the error
by setting the amplitude outside the aperture to zero. 3) Perform a
Fourier transform from r to k. If the resulting diffraction intensity
disagrees with the data, adjust the amplitude |A| in k-space and go
back to 1). This loop needs to be iterated many times, but it converges
eventually to the correct amplitude and phase in both r- and k-space.
Such a method allows (in principle) lensless imaging with atomic
resolution, which is limited only by the wavelength of the X-rays.
Diffraction from a single object
X-ray diffraction pattern
of a single Mimivirus
particle imaged at the
LCLS at Stanford, which
produces laser-like X-rays.
The X-ray pulse stripped
most of the electrons
from the atoms, leading
to a Coulomb explosion.
But it was so short (< 50
femtoseconds) that the
atoms did not have time
to move until after this
image was obtained
(“diffract and destroy”).
Combining thousands of
such images with various
orientations of the virus
(tomography) provides a
three-dimensional image.