Rietveld Analysis Tips 2
Using Parameter Correlation Coefficients to Take the Black Box Hazard Out of Rietveld Analysis Calculations
Pre-amble
This is the second in an occasional series written for Rietveld practitioners, especially novices. This note follows Rietveld Analysis Tips I. Obtaining Sensible Starting Parameters - see September 2005 AXAA Newsletter (O'Connor, 2005). What are parameter correlations and why bother about them? These numbers comprise a powerful set of information for use:
1. when designing your refinement strategy for a given class of materials and diffractometer data set,
2. for fine-tuning the strategy as the refinement progresses, and
3. finally, in optimising the quality of the results.
By using these numbers, you can answer the often-asked questions, such as:
The apparently mysterious behaviour of some refinements can be understood, and readily managed, if the Rietvelder appreciates through an understanding of correlations why the calculation is not refining or has diverged. For example, if the initial values for the peak profiles give peak shapes which are substantially wrong, then refinement of these parameters will dramatically shift other parameters to which the peak profile parameters are strongly linked, and the refinement will likely 'go off the rails'. Looking at the Parameter Correlation Matrix It is very important to become familiar with the parameter correlation values for the model that you are refining and the diffraction pattern which you have measured for the Rietveld analysis. Your Rietveld program of choice should surely output the correlation matrix at the end of a Rietveld calculation. I would not attempt Rietveld calculations without this information being provided by the program. For example, program Rietica provides the matrix values near the end of the calculation via the VIEW OUTPUT button under the INFORMATION tab. Table 1 gives an example.
Table 1. Correlation coefficients from a Rietveld analysis of ?-Al2O3 polycrystalline sample. Refinement conducted with Bragg-Brentano XRD data and the Rietica Rietveld program. Strong parameter correlations (values 70 % - 100 %) are shown in bold font, medium strength correlations (40 %- 70 %) in regular font; weak correlations (10 %- 40 %) in smaller font; and very weak correlations (less than 10 %) are not shown. Parameter codes: SD is the specimen displacement correction for the sample; B0, B1, B2 and B3 are polynomial fit parameters for the pattern background; SC is the phase scale factor; LPa and LPc are the refinable lattice parameters for the hexagonal cell; TAl and TO are the isotropic thermal parameters for the Al and O atoms, respectively; and U, V, W, ? and As are the peak profile parameters for the pseudo-Voigt function.
Pre-amble
This is the second in an occasional series written for Rietveld practitioners, especially novices. This note follows Rietveld Analysis Tips I. Obtaining Sensible Starting Parameters - see September 2005 AXAA Newsletter (O'Connor, 2005). What are parameter correlations and why bother about them? These numbers comprise a powerful set of information for use:
1. when designing your refinement strategy for a given class of materials and diffractometer data set,
2. for fine-tuning the strategy as the refinement progresses, and
3. finally, in optimising the quality of the results.
By using these numbers, you can answer the often-asked questions, such as:
- In which order should I refine the parameters?
- Why has my refinement diverged, or at least reached a 'false minimum', and might I do to recover the situation?
- Why do some refined parameters appear to be physically unreasonable, for example the lattice parameters indicating real changes in the cell volume when this seems physically unacceptable?
The apparently mysterious behaviour of some refinements can be understood, and readily managed, if the Rietvelder appreciates through an understanding of correlations why the calculation is not refining or has diverged. For example, if the initial values for the peak profiles give peak shapes which are substantially wrong, then refinement of these parameters will dramatically shift other parameters to which the peak profile parameters are strongly linked, and the refinement will likely 'go off the rails'. Looking at the Parameter Correlation Matrix It is very important to become familiar with the parameter correlation values for the model that you are refining and the diffraction pattern which you have measured for the Rietveld analysis. Your Rietveld program of choice should surely output the correlation matrix at the end of a Rietveld calculation. I would not attempt Rietveld calculations without this information being provided by the program. For example, program Rietica provides the matrix values near the end of the calculation via the VIEW OUTPUT button under the INFORMATION tab. Table 1 gives an example.
Table 1. Correlation coefficients from a Rietveld analysis of ?-Al2O3 polycrystalline sample. Refinement conducted with Bragg-Brentano XRD data and the Rietica Rietveld program. Strong parameter correlations (values 70 % - 100 %) are shown in bold font, medium strength correlations (40 %- 70 %) in regular font; weak correlations (10 %- 40 %) in smaller font; and very weak correlations (less than 10 %) are not shown. Parameter codes: SD is the specimen displacement correction for the sample; B0, B1, B2 and B3 are polynomial fit parameters for the pattern background; SC is the phase scale factor; LPa and LPc are the refinable lattice parameters for the hexagonal cell; TAl and TO are the isotropic thermal parameters for the Al and O atoms, respectively; and U, V, W, ? and As are the peak profile parameters for the pseudo-Voigt function.
