Diffraction as a Fourier Transform

Detlef Smilgies


In the kinematic approximation (or Born approximation), the diffraction pattern is given by the Fourier transform of the
electron density ρ(r):

S(q) = ∫ ρ(r) exp(iqr) dr
I(q) ~ |S(q)|2
where the scattering intensity I(q) is essentially given by the squared modulus of the structur factor S(q) and q is the scattering vector. The goal of this section is to make use of some fundamental properties of the Fourier transform, in order to shed some light on the structure of the diffraction pattern.

The electron density of a material can be approximated by the overlap of electron densities of individual atoms.

ρ(r) = ∑nρat(rn)
S(q) = ∑n fat,n(q) exp(iqrn)
where the index n runs over each atom. The atomic form factor fat,n(q) is the Fourier transform of the electron density of atom n. If we approximate the atomic electron density by a simple Gaussian, fat,n(q) would again be a Gaussian. Tabulated values for all atoms can be found in the International Tables of Crystallography. A particularly useful form is the parametrized form of fat,n(q), where the calculated values were fitted with a superposition of Gaussians.

We speak of diffraction, if the material studied has some regular crystalline structure, i.e. a lattice. A point lattices
is given by lattice vectors a, b, c, so that each point of the lattice can be written as

rmnp = m a + n b + p c
with m,n,p integer numbers. We can associate a density
latt(r) = ∑mnpδ(r-rmnp)
to such a point lattice. Its Fourier transform F(q) is again a point lattice, in this time in reciprocal space:
ghkl=h a* + k b* + l c*
Latt(q) = ∑hklδ(q-ghkl)
where the reciprocal lattice vectors are derived from a, b, c by
a* = 2π b×c / (a×b)c
and cyclic permutation. F(q) describes the diffraction pattern of the point lattice {rmnp}
 

We obtain the simplest kind of material, a monoatomic lattice, if we put an atom rat(r) at each lattice point,
which can be described by the convolution

ρ(r) = ρat(r) * latt(r)
where the convolution of two functions f(r) and g(r) is defined as
{f * g}(r) = f(r) * g(r) = ∫ f(r') g(r-r') dr'
In words, we smear out the delta function at each lattice point with the atomic charge density rat(r).
 

Why do we go through this effort, writing down a simple thing in a complicated way? Well, we want to use
some powerful properties of the Fourier Transform, to save us a lot of calculations in the following.
The convolution theorem states that the Fourier transform of a the convolution of two functions f and g
is simply the product of the Fourier transforms of the two functions:

FT{f * g} = FT{f} FT{g}
Application to our point lattice yields thus without any further calculation
S(q) = fat(q) Latt(q)
i.e. the intensities of the diffraction peaks (hkl) fall off like |fat(ghkl)|2 .
 

If there is more than one atom per unit cell, we have a more complicated electron density:

ρ uc(r) = ∑ nrat,n(r-rn)
The index n now runs over all atoms within the unit cell, and rn are the positions of the atoms within the unit
cell. We have already developed all the tools previously for evaluating the structure factor
ρ (r) = ρ uc(r) * latt(r)
S(q) = Suc(q) * Latt(q)
with the structure factor of the unit cell
Suc(q) = ∑ n fat,n(q) exp(iqrn)
Here we made use of another property of the Fourier transform
FT{ f(r-r0) } = FT{ f(r) } * exp(iqr0).
in words: a shift in real space r0 yields a phase factor exp(iqr0) in reciprocal space.
 

This final form of the scattering factor is what we need, in order to use the kinematic approximation for
structure analysis of materials. The whole structural information is contained in the smooth function Suc(q). However, a single unit cell would not yield a measurable scattering signal. What happens in a crystal? We will only sample
Suc(q) at q=ghkl. However, since r(r) is a periodic function, the Fourier coefficients S(ghkl) already contain the whole information.
 

In fact, we actually only measure

I(ghkl)  = | S(ghkl) |2
i.e. we lose the phase information contained in S(q), which is generally a complex number. Now what happens, if we Fourier backtransform I(q)? Applying the convolution theorem backwards we get
FT-1{ I(q) } = FT-1{ S(q) S*(q) } = FT-1{ S(q) } * FT-1{ S*(q) } = ρ(r) * ρ(r) = P(r)
The Patterson function P(r) is the autocorrelation function of the electron density. This means, we have lost the information about the absolute positions of each atom within the unit cell, but we still can derive interatomic distances from the peaks of P(r). The Patterson function is the classic tool of crystallography, which derived schemes on how to reconstruct the original lattice structure from this information. In following steps the structure will be further refined by fitting structure models to the measured intensities.

This scheme is works for about 100 atoms in the unit cell. Nowadays the classic approach has been replaced by so-called direct methods that try to guess phases subject to constraints that only models with charge densities resembling atomic densities are further refined. Direct methods cover a much larger parameter space that the classic refinement.

For very large unit cells as in protein crystals direct methods still fail. In such cases schemes have been devised, where
phase information can be recovered by comparing scattering intensities from the native material and heavy atom doped material. Instead of doping proteins with heavy atoms - which is not a trivial matter - recently methods have been devised, where phase information can be obtained, by comparing the diffraction intensities at an absorption edge and off-resonant (MAD). At the absorption edge the scattering power of the excited atom is reduced fom the off-resonant scattering power proportional to Z, so that it "looks" like a different atom. Hence the data set can be taken twice, once away from the edge and once right at the edge. Differences in the intensities due to f ' and f '' can be used to regain information on the scattering phases. These differences are only a few percent of the reflection intensity, hence high-quality data are needed.