The neutron and X-ray diffraction of powder samples results in a pattern characterised by reflections (peaks in intensity) at certain positions.
This terminology will be used here although the technique is equally applicable to alternative scales such as x-ray energy or neutron time-of-flight.
The relation is The most common powder X-ray diffraction (XRD) refinement technique used today is based on the method proposed in the 1960s by Hugo Rietveld.
[2] The Rietveld method fits a calculated profile (including all structural and instrumental parameters) to experimental data.
It employs the non-linear least squares method, and requires the reasonable initial approximation of many free parameters, including peak shape, unit cell dimensions and coordinates of all atoms in the crystal structure.
The Rietveld method is an incredibly powerful technique which began a remarkable era for powder XRD and materials science in general.
It is possible to determine the accuracy of a crystal structure model by fitting a profile to a 1D plot of observed intensity vs angle.
However, it can be used to find structural details missing from a partial or complete ab initio structure solution, such as unit cell dimensions, phase quantities, crystallite sizes/shapes, atomic coordinates/bond lengths, micro strain in crystal lattice, texture, and vacancies.
A typical diffraction pattern can be described by the positions, shapes, and intensities of multiple Bragg reflections.
Next, peak positions should be indexed and used to determine unit cell parameters, symmetry, and content.
To do this successfully, there is a requirement for excellent data which means good resolution, low background, and a large angular range.
as an integral: The instrumental function depends on the location and geometry of the source, monochromator, and sample.
Most commonly though, is the pseudo-Voigt function, a weighted sum of the former two (the full Voigt profile is a convolution of the two, but is computationally more demanding).
This can be problematic for non-ideal powder XRD data, such as those collected at synchrotron radiation sources, which generally exhibit asymmetry due to the use of multiple focusing optics.
The Finger–Cox–Jephcoat function is similar to the pseudo-Voigt, but has better handling of asymmetry, which is treated in terms of axial divergence.
The function is a convolution of pseudo-Voigt with the intersection of the diffraction cone and a finite receiving slit length using two geometrical parameters,
In the case of monochromatic neutron sources the convolution of the various effects has been found to result in a reflex almost exactly Gaussian in shape.
At very low diffraction angles the reflections may acquire an asymmetry due to the vertical divergence of the beam.
The integrated intensity depends on multiple factors, and can be expressed as the following product: where: The width of the diffraction peaks are found to broaden at higher Bragg angles.
In solid polycrystalline samples the production of the material may result in greater volume fraction of certain crystal orientations (commonly referred to as texture).
First, non-linear least squares fitting has an iterative nature for which convergence may be difficult to achieve if the initial approximation is too far from correct, or when the minimized function is poorly defined.
The latter occurs when correlated parameters are being refined at the same time, which may result in divergence and instability of the minimization.
This iterative nature also means that convergence to a solution does not occur immediately for the method is not exact.
For a low background, the functions are defined by contributions from the integrated intensities and peak shape parameters.
Mathematically they are easily accounted for, but practically, due to the finite accuracy and limited resolution of experimental data, each new phase can lower the quality and stability of the refinement.
Again, background is treated as a Chebyshev polynomial of the first kind ("Handbook of Mathematical Functions", M. Abramowitz and IA.
Now, given the considerations of background, peak shape functions, integrated intensity, and non-linear least squares minimization, the parameters used in the Rietveld refinement which put these things together can be introduced.
Given the complexity of Rietveld refinement it is important to have a clear grasp of the system being studied (sample, and instrumentation) to ensure that results are accurate, realistic, and meaningful.
range, and a good model – to serve as the initial approximation in the least squares fitting – are necessary for a successful, reliable, and meaningful Rietveld refinement.
There are some concerns about the reliability of these figures, as well there is no threshold or accepted value which dictates what represents a good fit.