rietveld-method part ii - cefet-mg€¦ · dr. peter g. weidler rietveld-method ii further criteria...

Name of Institute, Faculty, Department1 10/31/17KIT – The Research University in the Helmholtz Association www.kit.edu

Dr. Peter G. Weidler Institute of Functional Interfaces IFG

Rietveld-Methodpart II

IFG, KIT Campus North - 2 - November 2017Dr. Peter G. Weidler Rietveld-Method II

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


Profile Parameter

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

Profile Parameter Fitting Strategy


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.


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...


Profile Parameter common errors

truncation effect:

calculation range too short


Profile Parameter intensity errors

correct

too high

too low


Profile Parameter position errors

too high

too low


Profile Parameter line width errors

too broad too small


Profile Parameter asymmetry errors

profile shape function symmetric

profile asymmetric


Profile Parameter combined errors

FWHM too small

profile asymmetry too small


Profile Parameter combined errors

FWHM too small

intensity too small


Further criteriaFurther 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.


Further criteriaFurther 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 0.10


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).


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.


Estimated Standard Deviations

CeO: a (Å) = 5.41301+/- 0.00002 <--- 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


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


R-values

contain information about the fit of model to data

large R-values indicate errors in the modele.g. wrong positions, occupancies, temp factors


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.


R-values

also known as χ²

if data `over-collected' , i.e. errors no longer dominated by counting statisticsR

exp will be very small and S >> 1

if data `under-collected' (trash) or operator has done something wrongS 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


R-values

A statistic strongly recommended.optimum value = 2.0;

indicating mismatch of observed and calculated profileeven if area match

usually dwd

is small at begin, and increases towards the end of the refinement


R-values

R values are useful indicators for the evaluation of arefinement, especially in the case of small improvementsto the model, but they should not be overinterpreted.

The most important criteria for judging the quality of aRietveld refinement are

(i) the fit of the calculated pattern to the observed dataevaluated 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.


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



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.



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



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



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 identicaldisplacement 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?



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?


QXRD and RietveldQXRD 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. 409-426

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. 383-400.

Scarlett, N.V.Y. and Madsen I.C. (2006), “Quantification of phases with partial or no known crystal structures” Powder Diffraction, 21(4), 278-284.


QXRD and Rietveld

“Quantitative phase analysis by X-ray diffraction (QXRD) is the onlyanalytical technique that is truly phase sensitive”

Diffraction data derived directly from the crystal structure of each phaseResults are not inferred via indirect measurementCapable 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 !


QXRD and Rietveld: internal standardMaterial 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


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


QXRD and Rietveld and realityIUCr 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



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 educationof users of diffraction methodology and Rietveld-based software


QXRD and Rietveld and realityPrecision, 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 refinementas 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



Corundum Magnetite ZirconMean Analysed wt% (n=3) 56.52 17.06 26.42Mean Rietveld error 0.15 0.11 0.11S.D. of Analysed wt% 0.63 0.41 0.35XRF 50.4(2) 19.6(1) 29.5(1)Weighed 50.46 19.46 29.90Mean of bias 6.06 -2.58 -3.48

Results often quoted as Rietveld wt% ± Rietveld errorCorundum 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 knownCorundum 56(6) Magnetite 17(3) Zircon 26(3)

Ian Madsen - CSIRO Process Science & Engineering - 9th Topas User Meeting, Germany November 2011



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 ofmagnitude 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


Summary

short overview

get informed read publications, workshops ( even "foreign" code) wikis etc....

try outboth Rietveld and XRD measurement

be critical !!!

... but always have fun

Obregado !!


Literature

R.A. Young The Rietveld MethodIUCr, Oxford University Press, 1993, pp.299 40€ 95BRL

D.L. Bish & J.E. Post (Eds) Modern Powder DiffractionReviews in Mineralogy Vol 20Mineralogical Society of America, 1989, pp.369 30€ 70BRL

W.I.F. David, K. Shankland, L.B. McCusker, Ch. BaerlocherStructure Determination from Powder Diffraction DataIUCr, Oxford Science Publications, 2002 (2011 reprint), pp. 337

68€ 160BRL

International Union of Crystallographywww.iucr.com


CEFET

Bruker AXS do Brasil and Bruker AXS Germany, Knielingen

UMFG

Acknowledgment

INCT-Acqua

Ian Madsen, CSIRO (QXRD)

rietveld-method part ii - cefet-mg€¦ · dr. peter g. weidler rietveld-method ii further criteria...

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