Structure Refinement Rietveld Profile Analysis

In powder diffraction, there is commonly a problem arising from the overlap of peaks in the diffraction pattern, especially at high scattering angles. Accurate intensities, and hence structure factors, for these reflections cannot be refined. A

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technique developed by Rietveld ’ known as profile analysis is used to overcome this problem. The Rietveld method involves the refinement o f a structural model to obtain the best fit between the observed and calculated diffraction profiles, rather than individual reflections, and relies on the ability to describe the peak shape of individual Bragg reflections using a parameterised analytical function. This method was originally developed for neutron diffraction data owing to the accurately

Gaussian peak shapes obtained using monochromatised neutron beams from reactor sources. However, due to the narrow, well-defined Bragg peaks obtained from synchrotron radiation and the low background, the Rietveld method can also be applied to synchrotron powder data. The technique proceeds as follows.

At each point, I, in the diffraction pattern there is an intensity, yi. A function (Equation 5) comprising of the observed and calculated intensity at each step in the diffraction pattern is minimised using a least squares procedure.

S = ' Z w ,\y , - (5)

where: wi = weighting function where w =l/yi yi = observed intensity at the f ^ step yic = calculated intensity at the step and the sum is over all data points.

In the Rietveld method, no effort is made initially to allocate observed intensity to particular Bragg reflections, or to resolve overlapped reflections. Therefore a reasonably good starting model is required. To obtain a starting model, one can use the structure solution technique as detailed in the previous section, an isostructural material with a known structure or computer modelling techniques'^.

Typically, many Bragg reflections contribute to the observed intensity yi, at any point I in the pattern. The calculated intensity yic depends on the structure factor |FkP values calculated from the structural model. Other contributions are parameters defining the background, a function to describe the shape o f the Bragg peaks and the zero-point error, which corrects for any shift in the peak position by detector errors. The full expression for the calculated intensity is given in Equation 6.

K

s is the scale factor, to place the calculated data on the same scale as the

experimental; K represents the Miller indices hkl for a Bragg reflection; Lk contains

the Lorentz, polarisation and multiplicity factors; (j) corresponds to the chosen analytical function to represent the shape o f the peaks; is the preferred orientation function, used in samples where there is a strong tendency for the crystallites in a sample to be orientated more in one way than the others; A is an absorption factor, which differs with instrument geometry; yyi is the background intensity at the I* step and 20i-20a^ is the zero-point corrected peak position.

The peak shapes observed are a function o f both the sample and the instrument and they vary as a function o f 20. The most widely used peak shape for x-ray data is the pseudo-Voigt approximation of the Voigt fimction, and is a linear combination of Lorentzian and Gaussian components.

To assess the quality o f the model, there are several R-factors that are commonly used to calculate the goodness o f fit, in particular:

R-profile R = —

L V i

y 1 /

- / I

z ^ly.

{N - P)

Where yi and yic are the observed and calculated intensities at the I* step; Iko and Ikc are the observed and calculated intensities assigned to the Bragg reflection at the end of the refinement cycles; wj is a weighting factor; N is the number of

observations (i.e. the number of y\ used) and P is the number o f parameters adjusted. Ideally Rwp should approach the statistically expected R-value, Re, which reflects the quality o f the data. Therefore, the goodness-of-fit which is the ratio o f the two, should approach 1 towards the end o f the refinement.

Although these R-factors provide guidance as to the quality of the structural model, a difference profile plot, which displays the difference between the observed and calculated profiles, is the best way o f following and guiding a Rietveld refinement. Also, chemical intuition in terms o f the bond lengths and bond angles obtained must be applied. Constraints can be included in the refinement to ensure the geometry of the structural model remains sensible.

In a typical refinement, first the lattice parameters, the zero-point error and the background are refined, which is followed by the refinement o f the peak shape

variables. Once the profile has reached convergence, the atom coordinates can be entered and the structural model may be refined, i.e. the coordinates and the

temperature factors (isotropic or anisotropic). Soft constraints are often included in the refinement o f the structural model to help maintain realistic bond angles and lengths between the atoms; these can be removed once the model starts converging. At each stage o f the refinement, the R-factors and the Goodness o f fit, are calculated too, so the quality o f the fit can be assessed. Once convergence is achieved, the parameters should be checked to ensure they are meaningful.

In this work, the program GSAS^^ was used in the refinement of the high-resolution x-ray powder diffraction data collected.