This tutorial will show how to perform the calibration of the instrumental function of a generic powder diffractometer using a monochromatic Cu k-alpha source; the procedure can be applied, with minor changes, to other laboratory equipment using different radiation and diffraction geometries or even to synchrotron instruments.
The main goal of the calibration is to characterize the intrinsic peak broadening due to the machine optical setup, as well as to quantify possible goniometric misalignments. This step is of paramount importance when performing size/strain analysis, as the actual broadening observed in a diffraction profile can be considered as the convolution of intrinsic instrumental broadening and the actual sample broadening due to microstructural imperfections. In general, it is advisable to perform the instrument calibration every time the optical setup (slits, monochromators, etc) of the diffractometer is changed, as well of course after extraordinary maintenance.
For this tutorial we will refer to diffraction data acquired during the Size-Strain Line-Broadening Analysis of the Ceria Round-Robin Sample, published in Journal of Applied Crystallography 37 (2004) 911-924. In particular, the instrument calibration will be performed on the diffraction pattern acquired on the annealed CeO2 sample, which exhibits no intrinsic broadening.
Start ReX as usual or, if already started, select the “New project” command from the main toolbar. The default analysis will be setup, with an empty dataset and sample model and a standard powder diffractometer model which can be used as a starting point for the calibration. Since we will need to do a lot of parameter selection, it is advisable to make the parameter view always visible by dragging it in a convenient location in the GUI layout or, better, switch to the “Rietveld” perspective by selecting the “Rietveld Analysis” option in the “View->Open perspective” command menu.
Now, download the ReX examples package (if you haven’t already) and decompress in a folder of your choice; load the “lebailsh.xy” diffraction pattern located in the “calibration” folder by selecting the Dataset node in the project tree and selecting the “load data” command from the main toolbar (or the context menu). The pattern will be added to the default dataset and displayed in the dataset plot window; if not already enabled, select the “square root” option in the Y scale command in the window menu to better visualize high-angle features.
Now, we need to load the CeO2 crystallographic information file file from the database; to do so, select the “Sample” node in the project tree and select the “Load Object” command from the menu in the main toolbar. Load the CeO2.cif file from the calibration folder; the phase should appear as a sub-node of Sample in the project tree.
We can now start the actual Rietveld refinement procedure. First of all, select the instrument node; the corresponding parameter tree will appear in the parameter view. Check the “Scale factor”, “2-theta offset” and all the coefficients in the “Polynomial” background node as refinable parameters:
To improve the preliminary refinement, we now select the CeO2 phase in the project tree and check the cell parameter and isotropic U factors parameters:
Please note that when using a certified standard (like e.g. NIST 640e), accurate cell parameters values are usually provided in the standard certificate and don’t have to be refined; also atomic displacement factors can be left out from the refinement when reliable literature values are available.
After running the analysis again, the fit should improve significantly, especially for what concerns diffracted intensities along the whole 2-theta range. Peak shape fitting, on the other hand, will still be quite unsatisfactory, especially at high 2-theta angles:
This is due to the fact that we are using a default instrumental broadening, with a constant peak shape parameterization along the whole 2-theta range. To begin with the actual instrument profile function calibration, select the Instrument node in the project tree; in the Properties view, make sure “Pseudo Voigt” is selected in the Profile function combo panel, as well as no asymmetry correction:
Then, go to the parameter view and enable the W parameter (corresponding to the first Caglioti coefficient) under the Pseudo-Voigt node in the parameter tree:
Run the refinement with the selected parameters; the fit should improve a bit, giving a final value of W around 0.003 which corresponds to a better average FWHM of the diffraction peaks. To improve the fit significantly, however, we need to refine the other parameters defining the FWHM and gaussianity coefficients of the peak shape as a function of 2-theta; to do so, it is mandatory to proceed one step at a time, to avoid numerical instabilities in the optimization. In the next refinement step, select the V parameter (second Caglioti coefficient) and run the refinement again; the fit should improve considerably, with a Rwp around 16%.
Proceed enabling the other Pseudo-Voigt parameters step by step, running the refinement again for each newly enabled parameter; a suggested order is W, V, U, then Eta 0 and Eta 1 (Eta 2 can be left out in most cases); the final Rwp should be around 13%, with peak shapes giving a satisfactory fit around the whole diffraction range. Finally, the K polarization factor and the K-alpha2 ratio factor (both located under the Cu-Ka node in the instrument) should be refined, giving a final Rwp around 11.5% and parameter values similar to the ones reported below:
To conclude this tutorial, it is worth spending a final note about using the calibration file. The analysis can be kept as it is (with the final parameters set enabled) as a calibration template when doing routine calibrations of the same instrument, with approximately the same optical setup; after modifying significant aspects of the instrumentation (alignment, slits, etc), however, it is suggested to run the analysis from scratch, enabling the parameters one at a time in the sequence described before.
On the other hand, when analyzing real samples, in particular when dealing with size/strain related effects, it is convenient to save the calibration analysis file with a different name, after fixing all the instrument related parameters and removing both the CeO2 phase and the calibration pattern; this file can then be used as a starting base for all the subsequent analyses performed under the same instrumental setup.