Rietveld-Method part II

Transcription

1 Rietveld-Method part II Dr. Peter G. Weidler Institute of Functional Interfaces IFG 1 10/31/17 KIT The Research University in the Helmholtz Association Name of Institute, Faculty, Department

2 closer look on some details......continued data collection background contribution, peak-shape function, refinement of profile parameters, refinement of structural parameters, use of geometric restraints, calculation of e.s.d.'s, interpretation of R values some common problems and possible solution QA with Rietveld November 2017

3 Profile Parameter Fitting Strategy Before the refinement can be started, the positions of the observed and calculated peaks must match well. That is, the unit-cell parameters and the 2θcorrection (including both the zero offset and the displacement error) need to be refined first. (--> ev. quick Pawley-fit) These parameters can be highly correlated; work around: using standard material (e.g. NIST SRM materials) with the sample If an internal standard cannot be used for some reason (e.g. insufficient sample, sensitive sample), at least the 2θ calibration of the instrument should be checked carefully using an external standard November 2017

4 Profile Parameter Fitting Strategy good match of peak positions achieved --> refinement of profile parameters first peak width -- then peak asymmetry as a function of 2θ Variation of the FWHM of the peak shape as a function of 2θ is usually modelled with the equation derived by Caglioti al. (1958) While these functions are simple and usually work well, they cannot be used to model anisotropic line broadening (i.e. hkl-dependent line broadening) and may not describe the 2θ dependence very well November 2017

5 Checking the refinement development The profile fit is best seen in a plot of the observed, calculated patterns and the difference curve. profile plots are much more informative than R parameters Difference plots indicate profile parameter problem deficiency in the structural model Examples follow November 2017

6 Profile Parameter common errors truncation effect: calculation range too short November 2017

7 Profile Parameter intensity errors correct too high too low November 2017

8 Profile Parameter position errors too high too low November 2017

9 Profile Parameter line width errors too broad too small November 2017

10 Profile Parameter asymmetry errors profile shape function symmetric profile asymmetric November 2017

11 Profile Parameter combined errors FWHM too small profile asymmetry too small November 2017

12 Profile Parameter combined errors FWHM too small intensity too small November 2017

13 Further criteria Changes in positional parameters cause changes in structure-factor magnitudes and therefore in relative peak intensities Atomic displacement (thermal) parameters have the effect of emphasizing the high-angle region (smaller thermal parameters) or deemphasizing it (larger thermal parameters). Advisable to start refinement of structural parameters with the positions of heavier atoms and then try lighter atoms If the latter refinement converges, all atomic positions in the model can then be refined simultaneously. At this point, the refinement of the somewhat trickier parameters can be attempted November 2017

14 Further criteria IMPORTANT The scale, the occupancy parameters and the thermal parameters are highly correlated with one another, and are more sensitive to the background correction than are the positional parameters The structure should be refined to convergence. --> maximum shift/e.s.d. in the final cycle of refinement should be no more than November 2017

15 Constraints powder diffraction data are a one-dimensional projection of three-dimensional data, --> inherent loss of information. One way to compensate for this loss, at least in part, is to supplement the diffraction data with information from another source. geometric information (typical bond distances and angles) lattice parameters --> Vegard rule... --> reduce the number of parameters (e.g. a rigid body) November 2017

16 Number of Observations (and number of parameters The Rietveld algorithm will allow many more parameters to be refined than the data can actually support (because mathematically the number of observations is the number of steps in the profile), so the user has to intervene with common sense. vs November 2017

17 Estimated Standard Deviations CeO: a (Å) = / <--- correct or just a number various Rietveld refinement programs calculate e.s.d.'s differently! same data and the same structural model, one program will not necessarily produce the same e.s.d.'s for the structural parameters different interpretations of how the errors are best estimated. Several of the methods used to calculate the e.s.d.'s deviate from this strict statistical logic in an attempt to allow for systematic errors in the structural model. A powder diffractionist needs to know which method is used to calculate the e.s.d.'s in the Rietveld program he/she is using and to be aware that opinions vary regarding the most appropriate method; for most purposes that is sufficient. In any publication, method used to calculate the e.s.d.'s should be stated November 2017

18 R-values Although a difference profile plot is probably the best way of following and guiding a Rietveld refinement, the fit of the calculated pattern to the observed data can also be given numerically. This is usually done in terms of agreement indices or R- values November 2017

19 R-values contain information about the fit of model to data large R-values indicate errors in the model e.g. wrong positions, occupancies, temp factors November 2017

20 R-values if background subtracted, yi(obs) is net intensity after subtraction, but if the background is refined, yi(obs) [and yi(calc)] is likely to include the background contribution. In the latter case, a high background will automatically produce a low Rwp value, because a significant part of the intensity is accounted for by the background function. Thus, the comparison of profile R values from different kinds of powder diffraction experiments can be extremely misleading November 2017

21 R-values also known as χ² if data `over-collected', i.e. errors no longer dominated by counting statistics R exp will be very small and S >> 1 if data `under-collected' (trash) or operator has done something wrong S will be less than 1 Strange S values can also arise from data for which the e.s.d.'s of the counts have been incorrectly calculated (e.g. counts given as counts per second are assumed to be the absolute counts). The most important criteria for judging the quality of a Rietveld refinement are (i) the fit of the calculated pattern to the observed data and (ii) the chemical sense of the structural model November 2017

22 R-values A statistic strongly recommended. optimum value = 2.0; indicating mismatch of observed and calculated profile even if area match usually d wd is small at begin, and increases towards the end of the refinement November 2017

23 R-values R values are useful indicators for the evaluation of a refinement, especially in the case of small improvements to the model, but they should not be overinterpreted. The most important criteria for judging the quality of a Rietveld refinement are (i) the fit of the calculated pattern to the observed data evaluated on the basis of the final profile plot (using the complete range of data collected) (ii) the chemical sense of the structural model evaluated on a careful examination of the final atomic parameters November 2017

24 Some common problems and maybe solutions The background is not well fitted Try a different function, background subtraction, or a combination of the two The peak shapes are poorly described (i) Check the difference plot --> characteristic difference profiles --> indicating that a specific profile parameter should be reset or further refined. (ii) Try a different peak-shape function. (iii) Check that there is an asymmetry correction in the peak-shape function. (iv) Check to see if the peak widths are dependent and require a more sophisticated function to describe the 2θdependence November 2017

25 Some common problems and maybe solutions There is a mismatch between the peak positions in the calculated and observed patterns (i) Determine the unit-cell parameters with an independent measurement using an internal standard. (ii) Check that the 2θ correction (zero offset and sample displacement) function used is appropriate for the diffractometer geometry. The tails of the peaks in the calculated pattern seem to be cut off prematurely Try increasing the peak range used in the calculation November 2017

26 Some common problems and maybe solutions The relative intensities of a few reflections are too high but none is too low Check the sample used for data collection. This may indicate a problem with poor particle statistics (i.e. `rocks in the dust'). The only solution is to recollect the data after regrinding/sieving (and spinning the sample during data collection). There are small unindexed peaks in the diffraction pattern If other preparations of the same material have these peaks with similar relative intensities, they are probably not due to an impurity November 2017

27 Some common problems and maybe solutions The refinement does not converge Do the peak positions match? Is the background correction sensible? Is the scale factor correct? Is the structural model complete? Check for oscillations in the parameter shifts and apply damping factors as needed. Examine the covariance matrix for correlations between parameters. If high correlation is present between two variables, it may not be sensible to refine both November 2017

28 Some common problems and maybe solutions The refinement does not converge Try refining fewer parameters initially. Try adding geometric restraints If geometric restraints are already in use, are they correct? Set thermal (atomic displacement) parameters at sensible values and hold them fixed (or constrain similar atoms to have identical displacement parameters). Try a different space group. Is there something fundamentally wrong with the model? Do the data support the number of parameters being re ned? November 2017

29 Some common problems and maybe solutions Refinement converged, but there is an angle dependent intensity mismatch and/or unreasonable thermal parameters Check the Lorentz-polarization correction. Should an absorption correction be applied? Is a surface-roughness correction indicated? Have the atoms been identified correctly? Check the scattering factors used (especially if they have been input by hand). Is there preferred orientation in the sample? November 2017

30 QXRD and Rietveld Madsen, I.C., N.V.Y. Scarlett, L.M.D. Cranswick, and T. Lwin (2001) Outcomes of the International Union of Crystallography Commission on Powder Diffraction Round Robin on Quantitative Phase Analysis: samples 1a to 1h. Journal of Applied Crystallography, 34(4): pp Scarlett, N.V.Y., I.C. Madsen, L.M.D. Cranswick, T. Lwin, E. Groleau, G. Stephenson, M. Aylmore, and N. Agron-Olshina (2002) Outcomes of the International Union of Crystallography Commission on Powder Diffraction Round Robin on Quantitative Phase Analysis: samples 2, 3, 4, synthetic bauxite, natural granodiorite and pharmaceuticals. Journal of Applied Crystallography, 35(4): pp Scarlett, N.V.Y. and Madsen I.C. (2006), Quantification of phases with partial or no known crystal structures Powder Diffraction, 21(4), November 2017

31 QXRD and Rietveld Quantitative phase analysis by X-ray diffraction (QXRD) is the only analytical technique that is truly phase sensitive Diffraction data derived directly from the crystal structure of each phase Results are not inferred via indirect measurement Capable of analysing polymorphs Mathematical basis of QPA is well established, but Limitations on accuracy are mostly experimental Many sources of error Instrument configuration Particle statistics Counting error Preferred orientation Micro-absorption Operator error! November 2017

32 QXRD and Rietveld: internal standard Material should be stable & unreactive (especially for in situ studies) Simple diffraction pattern minimal overlap with sample peaks Standard MAC should be similar to sample MAC Avoid introducing micro-absorption effects Minimal sample related effect on observed intensities No preferred orientation 100% (or known) crystallinity Minimal graininess Possibilities -Al2O3 (corundum) TiO2 (rutile) ZnO (zincite) Cr2O3 (eskolaite) -Fe2O3 (hematite) CeO2 (cerianite) CaF2 (fluorite) C (diamond) Alternate approach Use an independent measure (e.g. chemical analysis) to derive the concentration of a phase already present in the sample Designate it as the internal standard November 2017

33 QXRD and Rietveld Most Rietveld programs just yield relative abundance of phases: all included phases are always 100% need additional knowledge of phase content or elemental composition (XRF) QXRD by Rietveld should ALWAYS be counter checked by XRF November 2017

34 QXRD and Rietveld and reality IUCr Round Robin on QXRD November 2017

35 QXRD and Rietveld and reality IUCr Round Robin on QXRD November 2017

36 QXRD and Rietveld and reality IUCr Round Robin on QXRD November 2017

37 QXRD and Rietveld and reality IUCr Round Robin on QXRD November 2017

38 QXRD and Rietveld and reality IUCr Round Robin on QXRD Largest & most common sources of error relate to the carbon-based bipedal life forms descended from apes # involved in the analysis The dreaded operator error Some sources of error Incorrect crystal structures: space group, atom coords, occupancies, Beq s Use of poor profile models Omission of phase(s) from the analysis Errors in phase ID Failure to refine parameters Unit cell, thermal, etc.. Refinement of parameters which are not supported by the data! Inappropriate use of correction models just because you CAN doesn t mean you SHOULD! Preferred orientation correction Microabsorption correction # Adams D. (1979) The Hitchhiker s Guide to the Galaxy November 2017

39 QXRD and Rietveld and reality IUCr Round Robin on QXRD Poor assessment of output Acceptance of physically unrealistic parameters (esp. thermal parameters) Acceptance of incomplete refinements High values of R-factors Refined parameters not checked Visual fit of model not checked Poor understanding of issues in data collection and analysis procedures Rietveld software often treated as black-box These issues can only be solved through continuing education of users of diffraction methodology and Rietveld-based software November 2017

40 QXRD and Rietveld and reality Precision, Accuracy and calculated errors The issue of precision and accuracy in QPA via XRD is a difficult one Analysts most often report Rietveld errors calculated during refinement as the errors in the final quantification These values numbers are related purely to the mathematical fit of the model They do not represent either precision or accuracy of the QPA! Determination of actual accuracy is no trivial task Needs recourse to some other measure of the sample November 2017

41 QXRD and Rietveld and reality Precision, Accuracy and calculated errors November 2017

42 QXRD and Rietveld and reality Precision, Accuracy and calculated errors Corundum Magnetite Zircon Mean Analysed wt% (n=3) Mean Rietveld error S.D. of Analysed wt% XRF 50.4(2) 19.6(1) 29.5(1) Weighed Mean of bias Results often quoted as Rietveld wt% ± Rietveld error Corundum 56.5(2) Magnetite 17.1(1) Zircon 26.4(1) If replicates are done (rare) results quoted as Rietveld wt% ± SD of mean Corundum 56.5(6) Magnetite 17.1(4) Zircon 26.4(4) 'Real result errors only available if answer already known Corundum 56(6) Magnetite 17(3) Zircon 26(3) Ian Madsen - CSIRO Process Science & Engineering - 9th Topas User Meeting, Germany November November 2017

43 QXRD and Rietveld and reality Precision, Accuracy and calculated errors Rietveld errors are not a measure of accuracy or precision They only represent the goodness of fit between calculated and observed patterns Standard deviation of replicates precision of analysis, not accuracy Many times larger than Rietveld errors Low R-factors could lead the analyst to conclude that the mean value ± SD is an adequate measure of the phase abundance and error Rietveld errors and replication errors can be at least an order of magnitude smaller than the bias (measured weighed) In this case, bias is due to severe micro-absorption Represents the true accuracy which can be achieved if no further steps are taken to identify and eliminate the cause of the bias November 2017

44 Summary short overview get informed read publications, workshops ( even "foreign" code) wikis etc... try out both Rietveld and XRD measurement be critical!!!... but always have fun Obregado!! November 2017

45 Literature R.A. Young The Rietveld Method IUCr, Oxford University Press, 1993, pp BRL D.L. Bish & J.E. Post (Eds) Modern Powder Diffraction Reviews in Mineralogy Vol 20 Mineralogical Society of America, 1989, pp BRL W.I.F. David, K. Shankland, L.B. McCusker, Ch. Baerlocher Structure Determination from Powder Diffraction Data IUCr, Oxford Science Publications, 2002 (2011 reprint), pp BRL International Union of Crystallography November 2017

46 Acknowledgment Bruker AXS do Brasil and Bruker AXS Germany, Knielingen Ian Madsen, CSIRO (QXRD) CEFET UMFG INCT-Acqua November 2017