Lesson 1: XRD and Rietveld Refinement

1 Lesson 1 XRD and Rietveld RefinementNicola D belinRMS Foundation, Bettlach, SwitzerlandMarch 2, 2015, Lyon, FrancePowder Diffraction2n = 2 d sin( )d 2 (120)(100)(010)Powder sampleX-ray Diffractometer3 SampleX-ray TubeDetectorStartScanEnd2 angleDigital Diffractometers4 Transmission GeometryGlass CapillaryFoilFluid CellCapillaries are ideal for: Light atoms (Polymers, Pharmaceuticals) Small amounts Hazardous materials Air-sensitive materialsUse characteristic radiation with low absorption coefficientFlat powder sampleReflective GeometryReflective Geometry is ideal for: Absorbing materials (Ceramics, Metals) Thin films Texture analysisUse characteristic radiation with high absorption coefficientDiffraction Pattern510203040506002004006008001000 Intensity [counts] diffraction Angle [ 2 ]Features.

2 Peak Positions ( 2theta)Peak Intensities (counts)Peak Width ( 2theta)Phases areidentifiedfrompeak positionsonly6 Search-Match software:-Extract peak positions-Compare with databaseDatabases:-PDF-2 (commercial)-PDF-4+ (commercial)-COD (free online resource) Rietveld Refinement7 For more than just identification: Rietveld refinementProf. Hugo RietveldExtracts much more information frompowder XRD data:-Unit cell dimensions-Phase quantities-Crystallite sizes / shapes-Atomic coordinates / Bond lengths-Micro-strain in crystal lattice-Texture effects-Substitutions / VacanciesNo phase identification!

3 Identify your phases first(unknown phase no Rietveld Refinement )Needs excellent data quality!No structure solution(just structure Refinement ) Rietveld Refinement8 Known structuremodel10203040506001000200030004 000 Intensity [counts] diffraction Angle [ 2 ]Calculate theoreticaldiffraction patternCompare withmeasured patternOptimize structure model, repeat calculation10203040506001000200030004000 Intensity [counts] diffraction Angle [ 2 ]Minimize differences between calculated and observedpattern by least-squares methodRietveld Refinement9102030405060-1000010002000300 040005000 Intensity [cts] diffraction Angle [ 2theta] Measured Pattern (Iobs)

4 Calculated Pattern (Icalc) Difference (Iobs-Icalc)Beginning of the Refinement :- Phase was identified correctly (peaksat the right position)- But differences exist:- Peak width- Peak positions slightly shifted- IntensitiesRietveld Refinement101020304050600100020003000400 05000 Intensity [cts] diffraction Angle [ 2theta] Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)After the Refinement :- Straight difference curve (only noise)Modelling the Peak [cts] diffraction Angle [ 2 ]K 1K 2 AsymmetryMathematical modelfor peak shape requiredModelling the Peak Profile12 Traditional ( Rietveld ) Approach.

5 Pseudo Voigt curves for K 1, K 2and K VP(x)= n * L(x)+ (1-n) * G(x)Gaussian curveLorentzian curveLorentzian ( = )Gaussian ( = )Pseudo-Voigt (n = )L(x) = 11+( )2x-x0 G(x) = exp[-ln(2) ( )2]x-x0 Pseudo-Voigt Curves13n = = = = = = = = = Curves14K 1K 2K Fitting n, to peaksof a reference [cts] diffraction Angle [ 2 ]Pseudo-Voigt: Problems15 AsymmetryPeaks at low 2 anglesare curvesare [cts] diffraction Angle [ 2 ] [ 2 ]Alternatives to Pseudo-Voigt Function16 Alternatives to PV function:- Pearson VII- Thompson-Cox-Hastings PV- Split PV- PV with axial divergence(Finger-Cox-Jephcoat PV)Fundamental Parameters Approach FPA17 Calculate the peak profile fromthe device configurationTake into account the contributions of.

6 - Source emission profile (X-ray wavelength distributionfromTube)- Every optical element in the beam path (position, size, etc.)- Sample contributions (peak broadening due to crystallite size & strain) ConfigurationSampleFundamental Parameters Parameters [ ] 2 2 120121122 2 If done properly:Very good description of the peak profileSummary: Rietveld Basics20- Calculate XRD pattern frommodel structure- Minimize differences between calculated and measured pattern- Accurate mathematical description of peak profile required:- Classical Rietveld approach.

7 Fit a peak shape function(PVor similar) to reference pattern- Fundamental Parameters Approach: Calculate peakprofile fromdevice configurationRefinement Strategies21102030405060-100001000200030 0040005000 Intensity [cts] diffraction Angle [ 2theta] Measured Pattern (Iobs) Calculated Pattern (Icalc) Difference (Iobs-Icalc)RelationPattern Features Structural FeaturesRefinement Strategy: Mismatches22- Peak Position- Absolute Intensities- Relative Intensities- Peak WidthHow to fix this? Refinement Strategies23 Wrong peak positions:- Unit cell dimensions-Sample height displacement-Zero-shift (instrument misalignment) Refinement Strategies24 Refined unit cell dimensions:Peak positions matched!

8 Refinement Strategies25 Wrong absolute intensities:- Weight fraction (scaling) Refinement Strategies26 Refined scale factor:Intensities improved (but not fixed)! Refinement Strategies27 Wrong relative intensities:- Preferred orientation- Graininess- Atomic species- Atomic coordinates- Site occupancies- Thermal displacementparametersLet s try this firstRefinement Strategies28 Refined texture:Intensities fixed! Refinement Strategies29 Wrong peak width:- Crystallite size- Micro-strain in crystal structure-Surface roughnessRefinement Strategies30 Refined crystallite sizes and micro-strain:Peak shape fixed!

9 Refined Crystal Structure31 Phase composition: 100% Al2O3 CorundumStarting ModelRefinedParameterValueUnit + nmUnit + nmCrystallite Size1267 +- 138 nmAtomic Coordinates Al / / Coordinates O / / nmUnit nmCrystallite Coordinates Al / / Coordinates O / / : Refinement Strategy32 Effect in diffraction pattern Origin in crystal structure modelWrong peak positionsUnit cell dimensionsSample height displacementZero-shiftWrong absolute intensities Weight fraction (scaling)

10 Wrong relative intensities Preferred orientationGrainy sampleAtomic species / Substitutions / VacanciesAtomic coordinatesSite occupanciesThermal displacement parametersWrong peak widthCrystallite sizeMicro-strainSurface roughnessTransparencyRietveld Refinement33 Known structuremodel10203040506001000200030004 000 Intensity [counts] diffraction Angle [ 2 ]Calculate theoreticaldiffraction patternCompare withmeasured patternOptimize structure model, repeat calculation10203040506001000200030004000 Intensity [counts] diffraction Angle [ 2 ]Minimize differences between calculated and observedpattern by least-squares methodRietveld Software Packages34 Academic Software:- Fullprof- GSAS- BGMN- Maud- Brass.