Results: Rietveld Refinement Results

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Results: Rietveld Refinement Results

It is essential that evidence of the quality of a Rietveld refinement be presented along with the final parameter values.
The article should therefore include the following:-

1. Final Plot
   A plot of observed and final calculated intensities versus 2q (or d spacing for time-of-flight or energy dispersive studies) is required. A plot of the difference in the two profiles (observed intensity minus calculated intensity) should also appear on the same figure, as well as markers showing the positions of all reflections allowed by the space group for the structure. Examples of such plots were shown earlier (e.g. the best fit to the PbSO₄ data in which the observed intensities are represented by the black dots, the calculated intensities by the red line, the difference plot by the pink line and the reflection markers are shown in green).
2. R Factors
   The values and definitions of all R factors should be provided, these include R_wp, R_exp, R_I, and c²(=(R_wp/R_exp)².
3. Final Values
   The final values of all refined parameters, with their estimated standard deviations should be provided. These are normally provided in tabular form. Examples of such tables are:

Interpretation

In any article reporting the results of a Rietveld refinement, the final plot, the R factors and the refined parameters table all contain important information relating to the quality of the fit of calculated to observed data, and so they should be studied carefully.

The plot of the difference between observed and calculated intensities should be examined with particular note. There are several features to look for:

1. the quality of the fit of the background (the difference plot should be zero in regions of background only).
2. look for any reflections whose observed and calculated intensities don't agree, the larger the difference and number of such reflections, the worse the quality of the final structure. Differences between the observed and calculated intensities indicate that the refined structure is not entirely consistent with the observed diffraction intensities. This means that there might be atoms incorrectly assigned or missing, or that additional corrections need to be applied to give a better fit.
3. if any particular classes of reflection don't match well this might indicate that preferred orientation or peak broadening effects have not been considered fully.
4. if there is a systematic shape to every reflection in the difference fit then the zero point, lattice parameters or the peak shape might not be correctly fitted.

The R factors reported for the refinement should be as low as possible and c² should be close to one. If the value of c² < 1, then the refinement has been over-parameterised or the peak-to-background ratio of the data is low. If the values of R_wp and R_p are high then you should not immediately class the fit as poor, as these values are partly dependent on how the factors are defined by the Rietveld program, and by the intrinsic quality of the diffraction data. If any R factors are not quoted in an article, then look more closely at the values of the refined parameters and the final profile plot. In summary, consider all the R factors together when assessing the quality of fit.

The tables of final refined parameter values provide much information on the quality of the Rietveld refinement presented and how the refinement was done. You should look closely at the values of the parameters and the errors (esd's). The smaller these are, the more precisely the corresponding parameters have been determined, and the more precise any derived values will be. For example, the lower the error on atomic positions the more precise will be the values of the calculated bond distances and angles. When the value of the refined parameter is equal to or less than approximately 3 times the esd then the parameter value is not more significant than its error and must be treated with caution, especially if the value is used to justify an argument later in the discussion section of the article, for instance, on the presence of an atom in the structure with a low site occupancy.

Certain points concerning the information presented in the table of atomic parameters should be scrutinised carefully. These include the following:

1. Atomic coordinates - look to see if any general position sites are not refined and why (maybe, for instance, the space group is polar), that atomic coordinates on any symmetry elements are not refined, and whether any atoms are constrained to have the same coordinates or to have the same shift in their direction vector.
2. Temperature factors - check that atoms of the same element have similar temperature factors if they are in similar coordination environments and that heavier atoms have the lower temperature factor values (for X-ray data). If one atom has a much higher temperature factor than others of the same type check to see if there is an explanation provided in the discussion section; for example, is that atom not tightly bound in the structure or is there disorder?
   If anisotropic temperature factors have been refined, check that the values for the temperature factor along the three axes of the thermal ellipsoid are all positive (may need a program to do this easily) and again, if the magnitude along one direction is larger than any other, see if any explanation is proposed.
   Also check whether the temperature factors are fixed, constrained or refined freely.
3. Site Occupancies - see if they are fixed or refined, and whether the values, when multiplied by the site multiplicity, are as predicted by the molecular formula.

Examples of several deductions that can be made about the refinement process from the above table of atomic coordinates, isotropic temperature factors and occupancies are:-

1. Si(1) and Al(1) have been constrained to have the same position, isotropic temperature factor (all of which have been refined) and a total occupancy of 100 % (for which the values have been fixed at 66.7 % of Si and 33.3 % of Al).
2. The isotropic temperature factors and and atomic coordinates of each of the oxygen atoms have been refined independently, but their occupancies have been fixed at 100 %.
3. The Pd atom's position and site occupancy have been refined but its temperature factor has been fixed at a pre-defined value.

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