Download (overview of) data analysis

(very short) theory
Based on total scattering data, which include Bragg diffraction and the diffuse
contribution arising from disordered features, PDF methods are particularly valuable in the
structural analysis of nanoscale and disordered (crystalline and non-crystalline) systems, for
example, glass, liquid, amorphous, nanocrystalline, heterogeneous and crystalline materials. The
PDF technique recovers structural information in the form of a distribution of atomic distances
and thus does not rely on the assumption of translational invariance that traditional
crystallographic methods apply.
A diffraction experiment is designed to measure the total scattering as a function of
scattering vector Q:
where
and
are the wave-vectors of the incident and scattered particles. A key experimental
criterion for obtaining quantitative PDFs is to be able to measure the structure function
to
large values of Q with sufficient statistics. Unfortunately, it’s not directly accessible from the
scattering experiment, but can be deduced from the experimental patterns after suitable
corrections for polarization, absorption, inelastic scattering, multiple scattering, static
approximation, and after normalization to absolute units. The term
is related to the elastic
part of the total diffracted intensity by the following expression:
where
,
and
are respectively the atomic concentration, average (over isotopes and spin
states) of the neutron-nucleus scattering length, and mean square scattering length of element a.
In the x-ray case
a, which is
with
in the equation is replaced by
, the atomic scattering factor of element
dependent. The pair distribution function is given by the Fourier transform of
total atomic number density of the sample:
,
Note that the
integration is defined over all the -space (0 to ∞). In practice, the diffraction
measurements are carried out over finite
-range. The Fourier transform between 0 and
leads to broader peaks and non-physical oscillations in
. This effect of truncation can be
reduced by convoluting structure factor with a modification function
at
which decreases to 0
. The most known are Lorch function and the square step function.
In the case of multi-component system the structure factor can be generalized as a sum of
weighted independent contributions arisen from the different correlations, where
and
are the
atomic species:
and
are the concentrations,
and
are the scattering length values, as described
previously. In the case of x-ray diffraction, the scattering lengths are replaced by the -dependent
form factors. These weighting factors
determine the extent to which different pairs
contributes to neutron/x-ray structure factors. Moreover the comparison of x-ray and neutron
measurements provides the additional information about the origin of the diffraction peaks. If we
take the example of Ge-S glass with 35.71 at. % of Ge. The x-ray weighting factors for the GeGe, Ge-S and S-S are 0.2768, 0.4986 and 0.2245 (as evaluated from the form factor values at
=
0), while those with neutrons are 0.3781, 0.4736 and 0.1483, respectively. From the definitions
given earlier, it follows that the correlation involving two Ge atoms emphasized in
to
relative
, while S-S correlations are deemphasized. In the same way the pair distribution function
in the real space can also be written as a weighted sum of partial distribution functions with the
same weighting factors.
Each partial radial distribution function can be integrated allowing the definition of the
coordination number:
where
is always the total atomic number density. We note that the number is symmetric
, since by definition pair distribution function describes the probability to find a
atom at a distance r from an
atom (and vice versa).
(overview of) data analysis
The coherent scattering per atom is deduced from the experimental scattering data. A
number of different corrections must be applied depending on the radiation and geometry used in
the experiment.
The data from neutron experiment is corrected for background and container scattering,
self-attenuation, multiple scattering, and inelasticity effects to give the total structure factor
. To obtain a sample’s cross section it is necessary to subtract the intensities from the
sample container, the sample environment and the background. Therefore additional
measurements of container and/or environment and the background are needed. In the next step
one should take in account the attenuation and multiple scattering. The multiple scattering
contributes to the background and can be calculated analytically. By the calculation of the
attenuation one should pay attention to geometry of the sample. Furthermore, the intensities must
be corrected with so-called Placzek corrections, for the departures from the static approximation.
Finally the normalization to the absolute units can be done by using the measurements from a
vanadium sample of the same configuration, because
In the case of x-rays one should consider various processes:

Elastic scattering

Compton scattering (with usual simplification as mono energetic)

Fluorescence resonant Raman scattering (that is negligible, until one treat anomalous
scattering)

Multiple scattering
The steps needed in analyzing the data are similar to the ones used for neutrons. The corrections
for container/background scattering, sample self-attenuation and multiple scattering are done in
the same manner as in the case with the neutrons. One should just replace the total attenuation
factors of neutrons with x-ray absorptions ones. In addition, when using the x-ray as a radiation
probe, one must correct the intensities with the polarization factor. The further differences lie in
atomic form factors that are Q-dependent, inelasticity corrections (Compton scattering) and the
normalization procedure. The values for atomic scattering factors and the Compton scattering
cross sections as a function of Q can be found in several tables. Finally the normalization to the
absolute units is done directly from the scattering pattern. Measured intensities at high-Q values
show very small modulations about the value
the normalization constant.
. So from this behavior one can determine