14.12 How to Refine Peak Profiles for a Sample

There are three usual cases for how peak profiles might be fit with GSAS-II:

1. The preferred case, where the instrument is well characterized and only sample terms need to be refined.
2. Where one must work with a dataset without known instrumental broadening terms.
3. Where an instrument is being characterized to determine the instrumental broadening terms.

The third case has been discussed previously in the “Determining Instrumental Profile Terms” section, §14.10. The second case is to be avoided wherever possible, but will be discussed further, below. So, I will first consider the process for fitting the sample broadening terms when the instrumental terms have already been determined for the instrument configuration used for data collection, which is what I would prefer to see people do. Good fits for peak profiles are only possible when the peak locations (from lattice parameters and sample displacement parameters), background and reflection intensities are all well fit. Thus, there are four interlocking aspects of Rietveld fitting. Please be sure to read the chapters on peak position (Chapter 12), background (Chapter 11), and peak intensity fitting (Chapter 13) before attempting to fit sample broadening.

14.12.1 Refinement of sample terms only

The optimal way that GSAS-II profiles will be fit will be to use previously determined profile terms that accurately describe the instrumental contribution to peak profiles. When this is the case then a good fit for the observed peak profiles should be obtained with refinement of the sample terms and the instrumental broadening terms do not need to be fit for samples. This greatly simplifies the Rietveld fitting process. In very simple cases, it will be possible to refine the histogram scale factor, background, lattice parameters and sample broadening all at the same time, but when any of those parameters start fairly far from their optimum values, the refinement may move to unreasonable values for some parameters and once that happens further optimization may not be possible. For this reason, I recommend a more systematic approach where parameters are added to the model in a more deliberate sequence.

Note that for complex crystallographic studies, where the starting model will not fit the diffraction intensities well (since improving an inaccurate model is the point of the study), it will not be possible to fit peak profiles well with the starting model; but it may also be impossible to fit the crystallographic structure with a poorly-fit peak shape. One approach to this is to cycle between fitting the background, then the peak shape, then the structure, (while also optimizing the unit cell parameters) and keep returning to each of these and possibly adding new parameters, such as sample displacement and preferred orientation, in turn until an optimum fit has been obtained. To remove intensity fitting from the combined tasks for fitting all aspects of the data, some researchers perform an initial profile fitting using a Le Bail or Pawley fit (see §20.4), where peak intensities are optimized directly, rather than have them determined from atom positions, preferred orientation etc. I recommend this approach. The steps used for this can vary, but here is a good recipe to follow:

1. Refine the scale factor alone, or optionally with the background terms before starting the Le Bail/Pawley fit. The reason for this is that later in the refinement, when you turn off the Le Bail/Pawley fit, if the scale factor is very far from correct, other parameters (particularly peak shape and lattice) may refine to quite incorrect values prior to the scale factor converging. These parameters should return to their optimal values in subsequent refinement cycles, but this does not always happen.
2. Fit the background. How this is best done depends on the complexity of the background. It may be sufficient to refine a few Chebyshev terms, or one may need to define fixed background points and fit background peaks to them. This will depend on the sample. See the Background Fitting chapter (Chapter 11) for more details on different approaches to fitting background.
3. Once the background is well fit, fit the lattice parameters. Before doing this, confirm that the structural parameters place computed peaks in positions that overlap with the observed. If they don’t the likelihood is that the refinement will not progress. If using Pawley fitting, refine only the intensities of the peaks for a few cycles before refining lattice parameters. With Le Bail, this should be done for you when you perform your first refinement after setting turning the Le Bail setting on. (One can also fit the Le Bail intensities alone by running a refinement after setting the number of least-squares cycles to zero.) Don’t forget to include the refinement of the appropriate sample displacement parameter(s) for the type of instrument and sample you are using. See the Lattice Parameter Fitting chapter (Chapter 12) for more details
4. Confirm that the background and lattice are well fit: At this point the difference curve should be relatively flat and close to zero, except in the regions where peaks are found. [Using the “w” option to view (obs-calc)/sigma can help considerably in seeing this is true.] Likewise, the computed peaks should be well-centered with the observed peaks. Look for the “derivative shape” in the difference pattern when observed and computed peaks are not aligned.

5. Try refining the isotropic broadening parameter. Should that be the isotropic microstrain or crystallite parameter? My GSAS-II coauthor Bob’s comments on this are valuable: “…ask yourself about how the sample was made. If hand-ground, the size will not usually be small enough to cause broadening, but there will be lots of microstrain that could be complex. If the sample is a precipitate or has been annealed, then the size & microstrain may be both “ideal” & cause little broadening at all. Nanomaterials (by definition) will have size broadening but unknown (& probably difficult [impossible] to determine) microstrain because of the broad peaks. Always apply the ‘Occam’s Razor’ principle & look for the simplest model that fits the data.”

   For this reason, I suggest starting with the isotropic microstrain parameter except for nanomaterials, where the crystallite size makes more sense. The microstrain value starts with a default of 1000 and should not be refined if it drops below a value that is insignificant for the instrument that has been used. Should this happen, reset it and turn off the refinement of this parameter.

6. While it is fairly rare to have data over a wide enough Q range to refine both crystallite size and microstrain together, I will often try this to see if it is possible. Usually, I save the refinement under a new name prior to doing this, so that it is easy to discard the fit with both parameters refined together after it goes awry. Note that while small values of microstrain produce insignificant broadening, large values for the size parameter are insignificant. The default value for the size parameter, 1 micron, is insignificant for most diffractometers so a value any larger is an indication of a problem, but 11-BM can show broadening of a few microns. Again, reset and turn off the fit for a parameter that refines out of a reasonable range, but unless you have prior information that a very small-grain sample is in use, or note that a fit with size alone gives a significantly better fit than microstrain alone, microstrain broadening is more plausible than crystallite size alone and as such is a more reasonable model.
7. Try uniaxial anisotropic broadening for whichever parameter is providing the greatest broadening contribution if refinement of broadening is significantly improving the fit. While one might expect it to be fairly clear from the difference plot that some peaks are wider than others, in practice I often discover anisotropic broadening is present only because the fit improves significantly only when I try this option. The uniaxial model, which changes the number of terms from one to two is sufficient for testing for this improvement, and the default broadening axis of 001 is the best choice for most high-symmetry systems (111 for rhombohedric cells), but for orthorhombic, monoclinic or triclinic systems it may well make sense to also try other axes. If the ratio of the fit metric (R_wp or GOF) before-and-after the addition of the extra parameter only changes by a few percent or less then the two broadening values are likely quite close and anisotropic broadening is not improving the model – return to an isotropic model.
8. If uniaxial microstrain broadening improves the fit significantly, try refining with the generalized microstrain model, particularly for higher symmetry systems. I have seen good fits with monoclinic cells, but I would not attempt this with a triclinic structure with less than superb data with plenty of widely separated peaks. If this does not produce a significant improvement in the fit (the ratio of the fit metric before-and-after the addition of the extra parameters changes by less than a few percent), return to the uniaxial model. Also, before deciding to use this model, particularly if the fit improvement is small considering the number of parameters that are added, look at the plot for the microstrain surface and see if the directions where there are the greatest and least deviations in the lattice parameters and see that it makes chemical sense from the structure. If the broadening is predominantly from crystallite size, I do not suggest use of the generalized crystallite broadening mode, unless you are sufficiently knowledgeable to know which terms are consistent with the symmetry of your lattice.
9. Once the profile terms, background and lattice parameters are determined well from a Pawley or Le Bail fit, if one is faced with a complex crystallographic analysis, where the process will be more complex than just turning on the refinement flags for a few atoms, for example where interstitial or charge-balancing atoms may need to be located, it can be helpful to stop these terms while exploring structural options. When turning off the Le Bail/Pawley fit, make sure to turn on the refinement of the histogram scale factor. If the scale factor is far from correct (this can be checked by running a zero-cycle fit), it is best to refine for a cycle this before varying any other parameters. Once the crystallographic parameters have been well-fit, for a final refinement run, do return the profile terms, background and lattice parameters back into the fit.
10. For some datasets it is not possible to refine the peak shape, ADP’s (\(\rm U_{iso}\) values) and background together. This is discussed in detail in §11.5. This happens where the data has so many peaks at high Q that there are no areas far enough from reflections so that there is Bragg intensity in every data point over a wide region. When this is the case it may be possible that there is no unique minimum for the optimum \(\rm U_{iso}\) values along with the background location. As will be discussed further, if the high-Q background is placed too low, then more intensity is needed in high Q reflections, which causes \(\rm U_{iso}\) to be set to values lower than the real value. My solution for this is to fix all the U_iso values to a value somewhat larger than I would expect, based on my experience for the material type and data collection temperature, and then fit the background and sample broadening but not \(\rm U_{iso}\) parameters. I then stop fitting the background, but do fit the U_iso values again. At this point the U_iso values will typically refine to larger values than what I had before. When you publish the work, do note that this was done and why.

Note that while usually I recommend fitting sample broadening only after fitting the lattice and background, when the peaks in the data are much more broad than the instrumental resolution, it can be difficult to fit the lattice and background well when the peak shapes are very poorly fit. (The reverse should never occur where the computed peaks are more broad than the observed pattern!). Looking at the “Rietveld plot” which shows the observed and computed patterns and their differences, particularly for peaks where intensities are well fit will demonstrate how well the peak widths in the fit match the data. In this case, it may be useful to refine either a single size or microstrain value (which one it is not overly important) so that the peak shape is close to correct before working out the details on how to model the background. Once the background is well-fit the process above can then be followed.

When it is not clear to me which model I want to use, I will frequently set up two refinements, for example, one with a microstrain term, and one with a crystallite size term, or an isotropic model compared to the uniaxial model, to see which gives a larger drop in the GOF, reduced c² or R_wp values. I will then compare the two fits and continue with the refinement that provides the largest improvement and discard the other. If no significant drop is seen in either refinement in comparison to the default value, then sample broadening is not significant. It may show more improvement if attempted later in the final stages of the refinement when all other parameters are fit, but if then the fit does not improve, do not include sample broadening in the final fit and note when publishing that negligible sample broadening was observed. If both terms provide fairly large, but similar fit improvements then it is possible that both types of sample broadening are present, but unless the data presents discernable peaks over a wide range of Q, it is quite possible that the two terms have more-or-less equivalent effect on the fit.

As noted, it is possible to make a trial refinement where both isotropic crystallite size broadening and microstrain broadening are refined together. If this is done, it should be confirmed that two things occur before continuing refinement with both terms: (A) the fit should be significantly better with both terms than with either alone and (B) the values should make sense, in that a very small microstrain value or a very large crystallite size value means that the term is not changing the peak widths and can be removed. Continue to monitor the values as even though the refinement starts with reasonable values, as while the initial fits to these parameters may have reasonable values, but this may not persist as the refinement progresses.

14.12.2 Refinement of instrumental profile terms

GSAS-II can still be used with the “old fashioned” approach for fitting profile parameters in a CW fit, where U, V, W, X and/or Y are refined, but my recommendation would be to refine U, V & W for the histogram and microstrain (or rarely crystallite size) for each sample. If your high-school best friend offers you powder diffraction data from her newly discovered room-temperature superconductor and asks you to confirm the structure using Rietveld, you will want to use that data, no matter that you do not have profile calibration data or an instrument parameter file for her diffractometer. On the other hand, one hopes that every TOF pattern will have calibration information, as TOF data is pretty much impossible to fit without instrument parameters or data on a standard that allows their derivation.

While under usual circumstances it is not possible to refine the sample broadening at the same time as the diffractometer X and Y terms, there are occasions where this can be done. In a multi-phase fit one might refine X & Y for the histogram and then refine microstrain (or size, probably not both) for all but one phase. Note that if this is done, the refined microstrain/size values should be interpreted as being on an arbitrary scale, but it is possible to compare values obtained for the different phases. Since the refined sample parameters cannot make peaks sharper than the values dictated by X & Y, in a multiphase sample, the phase with the sharpest peaks is the one that should have the sample terms fixed. Alternately, set the sample broadening terms to have non-negligible crystallite or microstrain broadening for the phase where these terms will not be fit, so that the other phases can have sample broadening terms that can be smaller than those of the fixed phase. It is also possible to refine uniaxial microstrain or generalized microstrain, even though X & Y are being refined. Having said this, if you are in the compromised situation where data must be fit without the ability to calibrate the instrumental broadening, I think it is much simpler to leave X & Y as zero and refine only the sample broadening terms.