Le Bail Intensity Extraction in GSAS-II - Sucrose

Exercise files are found here. The data files are also used in the 3rd section of the Fit Peaks/Autoindexing tutorial.

Background

In a Rietveld fit, a number of different items are fit, potentially at the same time. These include the lattice parameters, atomic positions and displacement (aka “temperature”, U_iso and U_ij) parameters for one or more phases, as well as peak shapes parameters either for broadening from either the sample or the instrument (avoid both together), background fitting terms and one or two correction terms needed to describe peak shifts due to apparent sample placement. In addition, there could be other terms needed to describe additional experimental conditions such as for absorption, preferred orientation or extinction. These parameters can be divided into four categories: experimental (peak shapes, peak shifts and background), lattice parameters, structural parameters (atom positions and U values) and intensity correction terms (absorption, preferred orientation and extinction).

In Rietveld, the reflection intensities are generated from the structural and intensity correction parameters, but there are two approaches to full-pattern fitting that treat reflection intensities as arbitrary: (1) Pawley fitting, developed by Stuart Pawley [Pawley, G. S. (1981). "Unit-Cell Refinement from Powder Diffraction Scans." Journal of Applied Crystallography 14: 357-361.], where peak intensities are treated as least-squares variables. (2) Le Bail fitting, from Armel Le Bail [Le Bail, A. (2005). "Whole powder pattern decomposition methods and applications: A retrospection." Powder Diffraction 20(4): 316-326.] The Le Bail approach, demonstrated by this tutorial, takes advantage of the fact that Hugo Rietveld developed a rather ingenious approach to estimating reflection intensities: the idea behind this is that for every point in a pattern where one or more reflections contribute, one can estimate the relative contribution from each reflection from the structure factor magnitude, F_calc, from the approximate crystal structure model, adjusted for reflection multiplicity (m) and preferred orientation. Simply apportioning the observed powder pattern intensities point by point to each reflection provides an estimate for the observed structure factors, F_obs. The quality of this estimate improves as the crystal structure model approaches reality. Le Bail’s method cleverly appropriates Rietveld’s intensity extraction, but assuming the crystal structure model is unknown, sets all F_calc values to 1. After the intensity extraction is performed, a set of F_obs values is obtained. These initially treat every F_obs/m ratio as equivalent and will not be very accurate, but they are much better reflection estimates than the initial F_calc values (all unity). The F_calc values are set to the extracted F_obs and the Le Bail fit can be repeated, giving even better intensity estimates. This process is equivalent to a steepest descents minimization and it will slowly converge to a self-consistent set of intensities that will give best possible agreement to the observed pattern. Note that the apportionment of intensity for reflections that are very closely overlapped will tend to make their F_calc values approximately equal. (Note that the Pawley approach will tend make m*F_obs values equal for complete reflection overlap; neither is likely to be accurate.)

Note that when either Pawley or Le Bail intensity extraction are used, all structural and intensity correction parameters no longer affect the intensity computations. Such parameters should not be included in the fit. If included inadvertently, GSAS-II will usually remove them from the fit along with providing annoying warning messages, but it is possible that this parameter removal process could fail and the refinement may not run properly.

GSAS-II provides two ways to update Le Bail reflection intensities. The Calculate/"Le Bail fit" menu command will perform three intensity extraction cycles, but will not change any GSAS-II parameters. When the Calculate/Refine menu command is used, least-squares minimization of experimental and lattice parameters is alternated with Le Bail fitting. Note that when first initiating a Le Bail fit, the reflection intensities are very small and fitting any lattice or peak width parameters is likely to diverge and fail. Even background parameters may be affected by poor reflection intensity estimates. For this reason, it is best to allow the reflection intensities to come near convergence using "Le Bail fit" before refining any parameters with "Refine". Usually, it is only necessary to do this at the beginning of a fit or if the Le Bail intensity values are reset (do this by turning the extraction flag off and then on again), but if parameters change significantly and the Le Bail intensities are far from optimal, it is an excellent idea to reconverge them with "Le Bail fit" again.

There are many reasons to use Le Bail fitting. The technique was developed to obtain a set of reflection intensities for structure solution. It can also be used to treat an impurity, where lattice parameters are known but the structural details are not or for fitting lattice parameters where likewise the structure is not known. Le Bail fitting is also useful prior to a structural fit. It allows determination of reasonable starting values for the background, peak shapes and unit cell parameters so they can be fixed until the late stages of the refinement. Another use is to see when artifacts in the background or peak shapes limit how well the pattern can be fit. A Le Bail or Pawley fit provides an estimate for the best possible R_wp and reduced χ² values that can be expected for a dataset. For complex patterns, where it is difficult to discern where allowed reflections are present, comparison of Le Bail or Pawley fits can provide insight on extinction conditions. A lower symmetry space group with fewer extinctions should provide a better fit to warrant its consideration.

Prerequisite

In this exercise you we will use Le Bail fitting to extract approximate reflection intensities. This is similar to what is done in the beginning of the “Charge Flipping in GSAS-II - sucrose” tutorial, which uses Pawley fitting instead. It follows logically from the third example in the “Fit Peaks/Autoindexing in GSAS-II” tutorial, from where the sucrose unit cell was determined. Alternately, a project (start.gpx) file is provided in the exercise files to start without the previous tutorial.

Step 1: Setup for fit

From the final project file, after completing the indexing and creating the phase, make the the following changes:

· Controls: Set the “Max cycles” value to 5

· Limits: Raise the upper data limit from 8 to 24 degrees

· Background: Add a background peak at 5.5 deg with sigma=10,000, set the flag to refine the peak intensity (window shown in Step 5, below)

· Instrument parameters: Use the Operations/“Reset profile” menu command to return to the beamline-supplied terms; turn off refinement of all terms

· Sample parameters: turn off refinement of the histogram scale factor

· Phase Sucrose/Data tab: Use “Add powder histograms” to link this phase to the powder dataset; set the refine flag on for the size term and the microstrain term.

· Phase Sucrose/Atoms tab: Use “Edit Atoms”/“Append atom” menu command to add an atom to the structure. It does not matter what the atom type is or where it is located, since the phase structure will not be used in the computation, but one atom must be present for the phase to be created properly.

· Phase Sucrose/General tab: Set the “Refine the unit cell” flag.

These changes have been applied to a GSAS-II project file from the conclusion of the 3rd Example within “Fit Peaks/Autoindexing in GSAS-II” and is provided as for this tutorial as file start.gpx.

Step 2: Select Le Bail fitting

On the phase Data tab, select the “Do new Le Bail extraction” option to put the refinement into Le Bail mode.

Step 3: Perform an initial Le Bail-only fit

Initiate a fit that improves only the Le Bail intensities using the Calculate/“Le Bail fit” menu command. (Note that control-B is a shortcut for this, except on the Mac where the shortcut is ⌘B).

This will run through three cycles of Le Bail fitting. This will only optimize the intensities of reflections. No other parameter values will be optimized until Step 5, when a least squares fit is done. After one use, a fairly reasonable fit is obtained, as shown below.

Press OK to accept these results.

Step 4: Perform additional Le Bail-only fits

The Le Bail intensities can be further fit using the Calculate/“Le Bail fit” menu command again. The fit improves very slightly, as shown below.

Since the improvement is so small, there is little value in running this again (though there is also no harm in doing this). In other projects, it may be necessary to perform this step more than twice before the fit no longer improves significantly. Here, any improvement to the fit will require refinement of the the experimental and lattice parameters, which is done via least-squares fitting.

Step 5: Combined Le Bail/Least Squares fit

In the instructions for Step 1, refinement flags were set for three background terms, the intensity value for a single background peak, a sample broadening term and a microstrain broadening term plus four lattice parameters (10 parameters in total). This can be verified using the Calculate/“View LS parameters” menu command (shortcut: control-L or ⌘L).

All that is needed at this stage to perform a combined Le Bail/Least Squares fit is to use the Calculate/Refine menu command.

The refinement progresses well and a much better, though far from perfect, fit is obtained.

Step 6: Add more parameters

Plotting the (obs-cal)/sigma curve (press the w key) and enlarging it (use the crossed-arrows button and drag the right mouse button upwards inside the lower box), as shown above, shows that further background improvement might be expected with more background terms.

· Change to use 6 chebyschev-1 peaks.

· Also refine sig for the single background peak.

Repeating "Refine" improves the fit somewhat, as shown below.

Step 7: Explore peak shape terms

At this point the fit is still far from what would be expected based on counting statistics (which would yield a reduced χ² value close to 1.) Looking closely at the data and computed pattern, as shown below for example, leads one to believe that the profile is still not being well-fit.

A Le Bail fit provides a good way to explore different peak shape broadening options. Investigation of a number of models for anisotropic peak broadening finds that the R_wp can be reduced just a bit, to 11.9%, through use of the Stephens model for microstrain (refining 9 terms in the Generalized Mustrain model), with a plausible microstrain directional plot, as below. Considering that 8 more terms have been added to the refinement, this is a fairly minor improvement, which many would say is not worthwhile. A fit that uses fewer terms and is almost as good uses uniaxial crystallite domain size broadening along the 001 axis and uniaxial microstrain broadening along the 010 axis. This adds 2 only parameters.

The important result here is that this sample exhibits non-ideal broadening artifacts. This could be due to milling of the sucrose during manufacture creating defects that cannot be well-fit with a conventional peak shape. From this fit, it is now clear that best possible fit to be expected from any structural model will have a R_wp on the order of 12% and a reduced χ² value on the order of 11.8, since no crystal structure model could give a better intensity fit than a Le Bail or Pawley fit.

This completes this exercise.