crystal size analysis

Estimating Crystallite SizeUsing XRDScott A Speakman, [email protected]

http://prism.mit.edu/xrayMIT Center for Materials Science and Engineering

Center for Materials Science and Engineering

WarningThese slides have not been extensively proof-read, and therefore may contain errors.While I have tried to cite all references, I may have missed some these slides were prepared for an informal lecture and not for publication.If you note a mistake or a missing citation, please let me know and I will correct it.

I hope to add commentary in the notes section of these slides, offering additional details. However, these notes are incomplete so far.


Goals of Todays LectureProvide a quick overview of the theory behind peak profile analysisDiscuss practical considerations for analysisDemonstrate the use of lab software for analysisempirical peak fitting using MDI JadeRietveld refinement using HighScore PlusDiscuss other software for peak profile analysisBriefly mention other peak profile analysis methodsWarren Averbach Variance methodMixed peak profilingwhole pattern Discuss other ways to evaluate crystallite size

Assumptions: you understand the basics of crystallography, X-ray diffraction, and the operation of a Bragg-Brentano diffractometer


A Brief History of XRD1895- Rntgen publishes the discovery of X-rays1912- Laue observes diffraction of X-rays from a crystal

when did Scherrer use X-rays to estimate the crystallite size of nanophase materials?


The Scherrer Equation was published in 1918Peak width (B) is inversely proportional to crystallite size (L)

P. Scherrer, Bestimmung der Grsse und der inneren Struktur von Kolloidteilchen mittels Rntgenstrahlen, Nachr. Ges. Wiss. Gttingen 26 (1918) pp 98-100.

J.I. Langford and A.J.C. Wilson, Scherrer after Sixty Years: A Survey and Some New Results in the Determination of Crystallite Size, J. Appl. Cryst. 11 (1978) pp 102-113.


The Laue Equations describe the intensity of a diffracted peak from a single parallelopipeden crystalN1, N2, and N3 are the number of unit cells along the a1, a2, and a3 directionsWhen N is small, the diffraction peaks become broaderThe peak area remains constant independent of N


Which of these diffraction patterns comes from a nanocrystalline material? These diffraction patterns were produced from the exact same sample Two different diffractometers, with different optical configurations, were used The apparent peak broadening is due solely to the instrumentation


Many factors may contribute tothe observed peak profileInstrumental Peak ProfileCrystallite SizeMicrostrainNon-uniform Lattice DistortionsFaultingDislocationsAntiphase Domain BoundariesGrain Surface RelaxationSolid Solution InhomogeneityTemperature Factors

The peak profile is a convolution of the profiles from all of these contributions


Instrument and Sample Contributions to the Peak Profile must be DeconvolutedIn order to analyze crystallite size, we must deconvolute:Instrumental Broadening FW(I)also referred to as the Instrumental Profile, Instrumental FWHM Curve, Instrumental Peak ProfileSpecimen Broadening FW(S)also referred to as the Sample Profile, Specimen Profile

We must then separate the different contributions to specimen broadeningCrystallite size and microstrain broadening of diffraction peaks


Contributions to Peak ProfilePeak broadening due to crystallite sizePeak broadening due to the instrumental profileWhich instrument to use for nanophase analysisPeak broadening due to microstrainthe different types of microstrainPeak broadening due to solid solution inhomogeneity and due to temperature factors


Crystallite Size BroadeningPeak Width due to crystallite size varies inversely with crystallite sizeas the crystallite size gets smaller, the peak gets broaderThe peak width varies with 2q as cos qThe crystallite size broadening is most pronounced at large angles 2ThetaHowever, the instrumental profile width and microstrain broadening are also largest at large angles 2thetapeak intensity is usually weakest at larger angles 2thetaIf using a single peak, often get better results from using diffraction peaks between 30 and 50 deg 2thetabelow 30deg 2theta, peak asymmetry compromises profile analysis


The Scherrer Constant, KThe constant of proportionality, K (the Scherrer constant) depends on the how the width is determined, the shape of the crystal, and the size distributionthe most common values for K are:0.94 for FWHM of spherical crystals with cubic symmetry0.89 for integral breadth of spherical crystals w/ cubic symmetry1, because 0.94 and 0.89 both round up to 1 K actually varies from 0.62 to 2.08For an excellent discussion of K, refer to JI Langford and AJC Wilson, Scherrer after sixty years: A survey and some new results in the determination of crystallite size, J. Appl. Cryst. 11 (1978) p102-113.


Factors that affect K and crystallite size analysishow the peak width is defined how crystallite size is definedthe shape of the crystal the size distribution


Methods used in Jade to Define Peak WidthFull Width at Half Maximum (FWHM)the width of the diffraction peak, in radians, at a height half-way between background and the peak maximum

Integral Breadththe total area under the peak divided by the peak heightthe width of a rectangle having the same area and the same height as the peakrequires very careful evaluation of the tails of the peak and the backgroundFWHM


Integral Breadth Warren suggests that the Stokes and Wilson method of using integral breadths gives an evaluation that is independent of the distribution in size and shapeL is a volume average of the crystal thickness in the direction normal to the reflecting planesThe Scherrer constant K can be assumed to be 1

Langford and Wilson suggest that even when using the integral breadth, there is a Scherrer constant K that varies with the shape of the crystallites


Other methods used to determine peak widthThese methods are used in more the variance methods, such as Warren-Averbach analysisMost often used for dislocation and defect density analysis of metalsCan also be used to determine the crystallite size distributionRequires no overlap between neighboring diffraction peaks

Variance-slopethe slope of the variance of the line profile as a function of the range of integration

Variance-interceptnegative initial slope of the Fourier transform of the normalized line profile


How is Crystallite Size DefinedUsually taken as the cube root of the volume of a crystalliteassumes that all crystallites have the same size and shapeFor a distribution of sizes, the mean size can be defined asthe mean value of the cube roots of the individual crystallite volumesthe cube root of the mean value of the volumes of the individual crystallites

Scherrer method (using FWHM) gives the ratio of the root-mean-fourth-power to the root-mean-square value of the thicknessStokes and Wilson method (using integral breadth) determines the volume average of the thickness of the crystallites measured perpendicular to the reflecting planeThe variance methods give the ratio of the total volume of the crystallites to the total area of their projection on a plane parallel to the reflecting planes


Remember, Crystallite Size is Different than Particle SizeA particle may be made up of several different crystallitesCrystallite size often matches grain size, but there are exceptions


Crystallite ShapeThough the shape of crystallites is usually irregular, we can often approximate them as:sphere, cube, tetrahedra, or octahedraparallelepipeds such as needles or platesprisms or cylindersMost applications of Scherrer analysis assume spherical crystallite shapesIf we know the average crystallite shape from another analysis, we can select the proper value for the Scherrer constant K

Anistropic peak shapes can be identified by anistropic peak broadeningif the dimensions of a crystallite are 2x * 2y * 200z, then (h00) and (0k0) peaks will be more broadened then (00l) peaks.


Anistropic Size BroadeningThe broadening of a single diffraction peak is the product of the crystallite dimensions in the direction perpendicular to the planes that produced the diffraction peak.


Crystallite Size Distributionis the crystallite size narrowly or broadly distributed?is the crystallite size unimodal?

XRD is poorly designed to facilitate the analysis of crystallites with a broad or multimodal size distribution

Variance methods, such as Warren-Averbach, can be used to quantify a unimodal size distributionOtherwise, we try to accommodate the size distribution in the Scherrer constantUsing integral breadth instead of FWHM may reduce the effect of crystallite size distribution on the Scherrer constant K and therefore the crystallite size analysis


Instrumental Peak ProfileA large crystallite size, defect-free powder specimen will still produce diffraction peaks with a finite widthThe peak widths from the instrument peak profile are a convolution of:X-ray Source ProfileWavelength widths of Ka1 and Ka2 linesSize of the X-ray sourceSuperposition of Ka1 and Ka2 peaksGoniometer OpticsDivergence and Receiving Slit widthsImperfect focusingBeam sizePenetration into the samplePatterns collected from the same sample with different instruments and configurations at MIT


What Instrument to Use?The instrumental profile determines the upper limit of crystallite size that can be evaluatedif the Instrumental peak width is much larger than the broadening due to crystallite size, then we cannot accurately determine crystallite sizeFor analyzing larger nanocrystallites, it is important to use the instrument with the smallest instrumental peak width

Very small nanocrystallites produce weak signalsthe specimen broadening will be significantly larger than the instrumental broadeningthe signal:noise ratio is more important than the instrumental profile


Comparison of Peak Widths at 47 2q for Instruments and Crystallite SizesRigaku XRPD is better for very small nanocrystallites, Fit Background) and use Fixed Background for very complicated patternsmore complex background functions will usually fail when fitting nanocrystalline materialsThis linear background fit modeled the background too low. A level fit would not work, so the fixed background must be used.


6. Initial Peak Width7. Initial Peak LocationThese setting determine the way that Jade calculates the initial peak profile, before refinementInitial Widthif the peak is not significantly broadened by size or strain, then use the FWHM curveif the peak is significantly broadened, you might have more success if you Specify a starting FWHMInitial Locationusing PDF overlays is always the preferred optionif no PDF reference card is available, and the peak is significantly broadened, then you will want to manually insert peaks- the Peak Search will not workResult of auto insertion using peak search and FWHM curve on a nanocrystalline broadened peak. Manual peak insertion should be used instead.


8. Display OptionsCheck the options for what visual components you want displayed during the profile fittingTypically use:Overall ProfileIndividual ProfilesBackground CurveLine MarkerSometimes use:Difference PatternPaint Individuals


9. Fitting ResultsThis area displays the results for profile fit peaksNumbers in () are estimated standard deviations (ESD)if the ESD is marked with (?), then that peak profile function has not yet been refinedClick once on a row, and the Main Display Area of Jade will move to show you that peak, and a blinking cursor will highlight that peakYou can sort the peak fits by any column by clicking on the column header


Other buttons of interestExecuteRefinementAutofitAll PeaksSee OtherOptionsHelpSave Text Fileof Results


Clicking Other OptionsUnify Variables: force all peaks to be fit using the same profile parameterUse FWHM or Integral Breadth for Crystallite Size AnalysisSelect What Columns to Show in the Results Area


Procedure for Profile Fitting a Diffraction PatternOpen the diffraction patternOverlay the PDF referenceZoom in on first peak(s) to analyzeOpen the profile fitting dialogue to configure optionsRefine the profile fit for the first peak(s) Review the quality of profile fitMove to next peak(s) and profile fitContinue until entire pattern is fit


Procedure for Profile Fitting1. Open the XRD pattern2. Overlay PDF reference for the sample


Procedure for Profile Fitting3. Zoom in on First Peak to Analyzetry to zoom in on only one peakbe sure to include some background on either side of the peak


Procedure for Profile Fittingwhen you open the profile fitting dialogue, an initial peak profile curve will be generatedif the initial profile is not good, because initial width and location parameters were not yet set, then delete ithighlight the peak in the fitting resultspress the delete key on your keyboard4. Open profile fitting dialogue to configure parameter

5. Once parameters are configured properly, click on the blue triangle to execute Profile Fitting you may have to execute the refinement multiple times if the initial refinement stops before the peak is sufficiently fit


Procedure for Profile Fitting6. Review Quality of Profile FitThe least-squares fitting residual, R, will be listed in upper right corner of screenthe residual R should be less than 10%The ESD for parameters such as 2-Theta and FWHM should be small, in the last significant figure


Procedure for Profile Fitting7. Move to Next Peak(s) In this example, peaks are too close together to refine individuallyTherefore, profile fit the group of peaks togetherProfile fitting, if done well, can help to separate overlapping peaks


Procedure for Profile Fitting8. Continue until the entire pattern is fitThe results window will list a residual R for the fitting of the entire diffraction patternThe difference plot will highlight any major discrepancies


Instrumental FWHM Calibration CurveThe instrument itself contributes to the peak profileBefore profile fitting the nanocrystalline phase(s) of interest profile fit a calibration standard to determine the instrumental profile

Important factors for producing a calibration curveUse the exact same instrumental conditionssame optical configuration of diffractometersame sample preparation geometrycalibration curve should cover the 2theta range of interest for the specimen diffraction patterndo not extrapolate the calibration curve


Instrumental FWHM Calibration CurveStandard should share characteristics with the nanocrystalline specimensimilar mass absorption coefficientsimilar atomic weightsimilar packing densityThe standard should not contribute to the diffraction peak profilemacrocrystalline: crystallite size larger than 500 nmparticle size less than 10 micronsdefect and strain freeThere are several calibration techniquesInternal StandardExternal Standard of same compositionExternal Standard of different composition


Internal Standard Method for CalibrationMix a standard in with your nanocrystalline specimena NIST certified standard is preferreduse a standard with similar mass absorption coefficientNIST 640c SiNIST 660a LaB6NIST 674b CeO2 NIST 675 Micastandard should have few, and preferably no, overlapping peaks with the specimenoverlapping peaks will greatly compromise accuracy of analysis


Internal Standard Method for CalibrationAdvantages:know that standard and specimen patterns were collected under identical circumstances for both instrumental conditions and sample preparation conditionsthe linear absorption coefficient of the mixture is the same for standard and specimen Disadvantages: difficult to avoid overlapping peaks between standard and broadened peaks from very nanocrystalline materialsthe specimen is contaminatedonly works with a powder specimen


External Standard Method for CalibrationIf internal calibration is not an option, then use external calibrationRun calibration standard separately from specimen, keeping as many parameters identical as is possible

The best external standard is a macrocrystalline specimen of the same phase as your nanocrystalline specimenHow can you be sure that macrocrystalline specimen does not contribute to peak broadening?


Qualifying your Macrocrystalline Standardselect powder for your potential macrocrystalline standardif not already done, possibly anneal it to allow crystallites to grow and to allow defects to healuse internal calibration to validate that macrocrystalline specimen is an appropriate standardmix macrocrystalline standard with appropriate NIST SRMcompare FWHM curves for macrocrystalline specimen and NIST standardif the macrocrystalline FWHM curve is similar to that from the NIST standard, than the macrocrystalline specimen is suitablecollect the XRD pattern from pure sample of you macrocrystalline specimendo not use the FHWM curve from the mixture with the NIST SRM


Disadvantages/Advantages of External Calibration with a Standard of the Same CompositionAdvantages:will produce better calibration curve because mass absorption coefficient, density, molecular weight are the same as your specimen of interestcan duplicate a mixture in your nanocrystalline specimenmight be able to make a macrocrystalline standard for thin film samplesDisadvantages: time consumingdesire a different calibration standard for every different nanocrystalline phase and mixturemacrocrystalline standard may be hard/impossible to producecalibration curve will not compensate for discrepancies in instrumental conditions or sample preparation conditions between the standard and the specimen


External Standard Method of Calibration using a NIST standardAs a last resort, use an external standard of a composition that is different than your nanocrystalline specimenThis is actually the most common method usedAlso the least accurate methodUse a certified NIST standard to produce instrumental FWHM calibration curve


Advantages and Disadvantages of using NIST standard for External CalibrationAdvantagesonly need to build one calibration curve for each instrumental configurationI have NIST standard diffraction patterns for each instrument and configuration available for download from http://prism.mit.edu/xray/standards.htmknow that the standard is high quality if from NISTneither standard nor specimen are contaminatedDisadvantagesThe standard may behave significantly different in diffractometer than your specimendifferent mass absorption coefficientdifferent depth of penetration of X-raysNIST standards are expensivecannot duplicate exact conditions for thin films


Consider- when is good calibration most essential?For a very small crystallite size, the specimen broadening dominates over instrumental broadeningOnly need the most exacting calibration when the specimen broadening is small because the specimen is not highly nanocrystallineFWHM of Instrumental Profileat 48 2q

0.061 degBroadening Due to Nanocrystalline Size


Steps for Producing an Instrumental ProfileCollect data from calibration standardProfile fit peaks from the calibration standardProduce FWHM curveSave FWHM curveSet software preferences to use FHWH curve as Instrumental Profile


Steps for Producing an Instrumental ProfileCollect XRD pattern from standard over a long rangeProfile fit all peaks of the standards XRD patternuse the profile function (Pearson VII or pseudo-Voigt) that you will use to fit your specimen patternindicate if you want to use FWHM or Integral Breadth when analyzing specimen patternProduce a FWHM curvego to Analyze > FWHM Curve Plot


Steps for Producing an Instrumental Profile4. Save the FWHM curve

go to File > Save > FWHM Curve of Peaksgive the FWHM curve a name that you will be able to find againthe FWHM curve is saved in a database on the local computeryou need to produce the FWHM curve on each computer that you useeverybody elses FHWM curves will also be visible


Steps for Producing an Instrumental Profile5. Set preferences to use the FWHM curve as the instrumental profile Go to Edit > PreferencesSelect the Instrument tabSelect your FWHM curve from the drop-down menu on the bottom of the dialogueAlso enter Goniometer RadiusRigaku Right-Hand Side: 185mmRigaku Left-Hand Side: 250mmPANalytical XPert Pro: 240mm


Other Software Preferences That You Should Be Aware OfReport TabCheck to calculate Crystallite Size from FWHMset Scherrer constant

Display tabCheck the last option to have crystallite sizes reported in nanometersDo not check last option to have crystallite sizes reported in Angstroms


Using the Scherrer Method in Jade to Estimate Crystallite Sizeload specimen dataload PDF reference patternProfile fit as many peaks of your data that you can


Scherrer Analysis Calculates Crystallite Size based on each Individual Peak ProfileCrystallite Size varies from 22 to 30 over the range of 28.5 to 95.4 2qAverage size: 25 Standard Deviation: 3.4 Pretty good analysisNot much indicator of crystallite strainWe might use a single peak in future analyses, rather than all 8


FWHM vs Integral BreadthUsing FWHM: 25.1 (3.4)Using Breadth: 22.5 (3.7)Breadth not as accurate because there is a lot of overlap between peaks- cannot determine where tail intensity ends and background begins


Analysis Using Different Values of KFor the typical values of 0.81 < K < 1.03the crystallite size varies between 22 and 29 The precision of XRD analysis is never better than 1 nmThe size is reproducibly calculated as 2-3 nm


For Size & Strain Analysis using Williamson-Hull type Plot in Jadeafter profile fitting all peaks, click size-strain button or in main menus, go to Analyze > Size&Strain Plot


Williamson Hull Ploty-interceptslope


Manipulating Options in the Size-Strain Plot of JadeSelect Mode of AnalysisFit Size/StrainFit SizeFit StrainSelect Instrument Profile CurveShow OriginDeconvolution ParameterResultsResiduals for Evaluation of FitExport or Save1234567


Analysis Mode: Fit Size Onlyslope= 0= strain


Analysis Mode: Fit Strain Onlyy-intercept= 0 size=


Analysis Mode: Fit Size/Strain


Comparing ResultsIntegral BreadthFWHM


Manually Inserting Peak ProfilesClick on the Profile Edit Cursor buttonLeft click to insert a peak profileRight click to delete a peak profileDouble-click on the Profile Edit Cursor button to refine the peak


ExamplesRead Y2O3 on ZBH Fast Scan.savmake sure instrument profile is IAP XPert FineOptics ZBHNote scatter of dataNote larger average crystallite size requiring good calibrationdata took 1.5 hrs to collect over range 15 to 146 2q could only profile fit data up to 90 2q; intensities were too low after thatRead Y2O3 on ZBH long scan.savmake sure instrument profile is IAP XPert FineOptics ZBHcompare Scherrer and Size-Strain PlotNote scatter of data in Size-Strain Plotdata took 14 hrs to collect over range of 15 to 130 2qsize is 56 nm, strain is 0.39%

by comparison, CeO2 with crystallite size of 3 nm took 41min to collect data from 20 to 100 2q for high quality analysis


ExamplesLoad CeO2/BN*.xrdmlOverlay PDF card 34-0394shift in peak position because of thermal expansionmake sure instrument profile is IAP XPert FineOptics ZBHlook at patterns in 3D viewScans collected every 1min as sample annealed in situ at 500Cmanually insert peak profileuse batch mode to fit peakin minutes have record of crystallite size vs time


ExamplesSize analysis of Si core in SiO2 shellread Si_nodule.savmake sure instrument profile is IAP Rigaku RHSshow how we can link peaks to specific phasesshow how Si broadening is due completely to microstrainZnO is a NIST SRM, for which we know the crystallite size is between 201 nmwe estimate 179 nm- shows error at large crystallite sizes


We can empirically calculate nanocrystalline diffraction pattern using JadeLoad PDF reference cardgo to Analyze > Simulate PatternIn Pattern Simulation dialogue boxset instrumental profile curveset crystallite size & lattice straincheck fold (convolute) with instrument profileClick on Clear Existing Display and Create New Pattern or Click on Overlay Simulated Pattern

demonstrate with card 46-1212observe peak overlap at 36 2q as peak broaden

Whole Pattern Fitting


Emperical Profile Fitting is sometimes difficultoverlapping peaksa mixture of nanocrystalline phasesa mixture of nanocrystalline and macrocrystalline phase


Or we want to learn more information about samplequantitative phase analysishow much of each phase is present in a mixturelattice parameter refinementnanophase materials often have different lattice parameters from their bulk counterpartsatomic occupancy refinement


For Whole Pattern Fitting, Usually use Rietveld Refinementmodel diffraction pattern from calculationsWith an appropriate crystal structure we can precisely calculate peak positions and intensitiesthis is much better than empirically fitting peaks, especially when they are highly overlappingWe also model and compensate for experimental errors such as specimen displacement and zero offsetmodel peak shape and width using empirical functionswe can correlate these functions to crystallite size and strain

we then refine the model until the calculated pattern matches the experimentally observed pattern

for crystallite size and microstrain analysis, we still need an internal or external standard


Peak Width Analysis in Rietveld RefinementHighScore Plus can use pseudo-Voigt, Pearson VII, or Voigt profile functionsFor pseudo-Voigt and Pearson VII functionsPeak shape is modeled using the pseudo-Voigt or Pearson VII functionsThe FWHM term, HK, is a component of both functionsThe FWHM is correlated to crystallite size and microstrainThe FWHM is modeled using the Cagliotti EquationU is the parameter most strongly associated with strain broadeningcrystallite size can be calculated from U and WU can be separated into (hkl) dependent components for anisotropic broadening


Using pseudo-Voigt and Pears VIII functions in HighScore PlusRefine the size-strain standard to determine U, V, and W for the instrumental profilealso refine profile function shape parameters, asymmetry parameters, etcRefine the nanocrystalline specimen dataImport or enter the U, V, and W standard parametersIn the settings for the nanocrystalline phase, you can specify the type of size and strain analysis you would like to executeDuring refinement, U, V, and W will be constrained as necessary for the analysisSize and Strain: Refine U and WStrain Only: Refine USize Only: Refine U and W, U=W


ExampleOpen ZnO Start.hpfShow crystal structure parametersnote that this is hexagonal polymorphCalculate Starting StructureEnter U, V, and W standardU standard= 0.012364V standard= -0.002971W standard= 0.015460Set Size-Strain Analysis Optionstart with Size OnlyThen change to Size and StrainRefine using Size-Strain Analysis Automatic Refinement


The Voigt profile function is applicable mostly to neutron diffraction dataUsing the Voigt profile function may tries to fit the Gaussian and Lorentzian components separately, and then convolutes themcorrelate the Gaussian component to microstrainuse a Cagliotti function to model the FWHM profile of the Gaussian component of the profile functioncorrelate the Lorentzian component to crystallite sizeuse a separate function to model the FWHM profile of the Lorentzian component of the profile functionThis refinement mode is slower, less stable, and typically applies to neutron diffraction data onlythe instrumental profile in neutron diffraction is almost purely Gaussian


HighScore Plus WorkshopJan 29 and 30 (next Tues and Wed)from 1 to 5 pm both days

Space is limited: register by tomorrow (Jan 25)preferable if you have your own laptop

Must be a trained independent user of the X-Ray SEF, familiar with XRD theory, basic crystallography, and basic XRD data analysis


Free Software Empirical Peak FittingXFitWinFitcouples with Fourya for Line Profile Fourier AnalysisShadowcouples with Breadth for Integral Breadth AnalysisPowderXFITsucceeded by PROFILEWhole Pattern FittingGSASFullprofReitanAll of these are available to download from http://www.ccp14.ac.uk


Other Ways of XRD AnalysisMost alternative XRD crystallite size analyses use the Fourier transform of the diffraction patternVariance MethodWarren Averbach analysis- Fourier transform of raw dataConvolution Profile Fitting Method- Fourier transform of Voigt profile functionWhole Pattern Fitting in Fourier SpaceWhole Powder Pattern Modeling- Matteo Leoni and Paolo ScardiDirectly model all of the contributions to the diffraction patterneach peak is synthesized in reciprocal space from it Fourier transformfor any broadening source, the corresponding Fourier transform can be calculatedFundamental Parameters Profile Fittingcombine with profile fitting, variance, or whole pattern fitting techniquesinstead of deconvoluting empirically determined instrumental profile, use fundamental parameters to calculate instrumental and specimen profiles


Complementary AnalysesTEMprecise information about a small volume of samplecan discern crystallite shape as well as size

PDF (Pair Distribution Function) Analysis of X-Ray Scattering

Small Angle X-ray Scattering (SAXS)

Raman

AFM

Particle Size Analysiswhile particles may easily be larger than your crystallites, we know that the crystallites will never be larger than your particles


Textbook ReferencesHP Klug and LE Alexander, X-Ray Diffraction Procedures for Polycrystalline and Amorphous Materials, 2nd edition, John Wiley & Sons, 1974.Chapter 9: Crystallite Size and Lattice Strains from Line BroadeningBE Warren, X-Ray Diffraction, Addison-Wesley, 1969reprinted in 1990 by Dover PublicationsChapter 13: Diffraction by Imperfect CrystalsDL Bish and JE Post (eds), Reviews in Mineralogy vol 20: Modern Powder Diffraction, Mineralogical Society of America, 1989.Chapter 6: Diffraction by Small and Disordered Crystals, by RC Reynolds, Jr.Chapter 8: Profile Fitting of Powder Diffraction Patterns, by SA Howard and KD PrestonA. Guinier, X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies, Dunod, 1956.reprinted in 1994 by Dover Publications


ArticlesD. Balzar, N. Audebrand, M. Daymond, A. Fitch, A. Hewat, J.I. Langford, A. Le Bail, D. Lour, O. Masson, C.N. McCowan, N.C. Popa, P.W. Stephens, B. Toby, Size-Strain Line-Broadening Analysis of the Ceria Round-Robin Sample, Journal of Applied Crystallography 37 (2004) 911-924 S Enzo, G Fagherazzi, A Benedetti, S Polizzi, A Profile-Fitting Procedure for Analysis of Broadened X-ray Diffraction Peaks: I. Methodology, J. Appl. Cryst. (1988) 21, 536-542.A Profile-Fitting Procedure for Analysis of Broadened X-ray Diffraction Peaks. II. Application and Discussion of the Methodology J. Appl. Cryst. (1988) 21, 543-549B Marinkovic, R de Avillez, A Saavedra, FCR Assuno, A Comparison between the Warren-Averbach Method and Alternate Methods for X-Ray Diffraction Microstructure Analysis of Polycrystalline Specimens, Materials Research 4 (2) 71-76, 2001.D Lou, N Audebrand, Profile Fitting and Diffraction Line-Broadening Analysis, Advances in X-ray Diffraction 41, 1997.A Leineweber, EJ Mittemeijer, Anisotropic microstrain broadening due to compositional inhomogeneities and its parametrisation, Z. Kristallogr. Suppl. 23 (2006) 117-122BR York, New X-ray Diffraction Line Profile Function Based on Crystallite Size and Strain Distributions Determined from Mean Field Theory and Statistical Mechanics, Advances in X-ray Diffraction 41, 1997.


Instrumental Profile Derived from different mounting of LaB6 In analysis of Y2O3 on a ZBH, using the instrumental profile from thin SRM gives a size of 60 nm; using the thick SRM gives a size of 64 nm

ZBH vs Toploaded

0.0440.053

0.0420.05

0.0450.052

0.0430.051

0.0450.052

0.0480.054

0.0520.059

0.0550.06

0.0550.06

0.0590.063

0.0680.067

0.0630.063

0.0660.069

0.0690.071

0.080.078

0.0820.083

0.0940.092

0.0880.094

0.0990.099

0.1140.119

0.1320.137

0.1540.146

0.1670.17

0.1960.197

10 micron thick

0.3 mm thick

2Theta

FWHM

LaB6 ZBH

2-ThetadHeightAreaArea%ShapeSkewFWHMBreadth

21.374.1546111853859262.50.5190.4080.0440.058

30.3952.938417752617251000.630.2040.0420.058

37.4522.399471432659843.10.6320.1230.0450.062

43.5162.078381813572220.6160.040.0430.059

48.9691.858683313126150.60.6630.0540.0450.063

541.696739761593025.80.7030.0150.0480.067

63.2271.4695123054638.80.736-0.1190.0520.074

67.5611.385433411575625.50.7420.0630.0550.079

71.7571.314322761089517.70.793-0.0610.0550.08

75.8591.25311395700211.30.7550.0490.0590.084

79.8831.199822613082.10.726-0.0370.0680.097

83.861.152778643687.10.8420.0220.0630.093

87.8061.11081544930215.10.9390.0080.0660.101

95.6821.039125916022.60.929-0.2330.0690.103

99.6551.00811045746512.10.877-0.0910.080.119

103.6790.9797860630010.20.9220.0670.0820.122

107.7640.953633027854.50.91-0.0140.0940.141

111.9450.929452241336.70.913-0.2020.0880.132

116.2620.907856782412.71-0.0530.0990.153

120.7420.886245146597.50.9610.0060.1140.173

130.4280.848524229284.71-0.0790.1320.203

135.8290.831327235935.80.80.0390.1540.221

141.7960.815210631632726.51-0.1560.1670.257

148.70.85329261150.954-0.1490.1960.291

LaB6 ZBH

Lorentzian Component

Skewness

FWHM

Breadth

2Theta (deg)

FWHM or Breadth

LaB6 Toploaded


21.4324.142723506935811000.3850.2580.0530.067

30.4572.9326187707364078.70.4810.1790.050.066

37.5122.395674893072132.80.5180.0830.0520.069

43.5762.075362612511426.80.498-0.0360.0510.067

49.0261.856690613756140.10.5460.0260.0520.069

54.0561.695142181861619.90.612-0.0220.0540.074

63.2841.4683149172107.70.628-0.0370.0590.081

67.6121.384540702009921.50.665-0.0050.060.083

71.811.313530641525116.30.676-0.0230.060.083

75.9091.2524174792799.90.7280.0690.0630.089

79.931.1992299184420.981-0.1690.0670.103

83.9071.1522112561496.60.810.0010.0630.091

87.8511.1104198512150130.859-0.070.0690.102

95.731.038750531403.40.818-0.0650.0710.104

99.6961.007815641106011.80.927-0.1390.0780.118

103.7130.9794119588809.50.927-0.140.0830.124

107.8030.953342034953.70.9270.0240.0920.139

111.9840.929269158386.20.911-0.0690.0940.141

116.2940.9069132612178130.996-0.0930.0990.153

120.7670.886156059306.30.929-0.1540.1190.177

130.4560.8484376472150.978-0.010.1370.21

135.8420.831346562996.71-0.1450.1460.226

141.8160.815117022636828.20.956-0.1330.170.259

148.7150.79998101438615.41-0.1740.1970.297

Si Disc


28.4353.1363457541922291000.7170.2420.0490.07

47.2941.92052126611174058.10.7970.1140.060.088

56.1131.6377114366671934.70.8880.0830.0650.097

69.1211.35791993151107.90.8790.1240.0860.127

76.3681.246137702775514.40.8670.10.0830.123

88.0221.108744393781719.70.9020.030.0950.142

94.9441.045321482064310.70.929-0.0130.1070.161

106.7030.96011331143387.50.957-0.1080.1190.18

114.0890.91820322601513.50.967-0.0460.1410.214

127.5380.858715512661913.81-0.1860.1880.287

136.90.8282865168938.80.947-0.0820.2170.326

Scherrer published on the use of XRD to estimate crystallite size in 1918, a mere 6 years after the first X-ray diffraction pattern was ever observed. Remember nanophase science is not quite as new or novel as we like to think. Paul Scherrer, Determination of ?? and the inner structure of colloids using X-rays, K is the Scherrer constant, lambda is the wavelength of X-radiation, theta is the diffraction anglemore detail on the Scherrer equation laterThe Scherrer equation is rooted in the peak profiles described by the Laue equationsWhen N1, N2, and N3 are large numbers, the three quotients are non-zero only when the Laue equations are closely satisfied; this produces a sharp peak (nearly a delta-function, see fig 3.2 on pg 30).When N1, N2, and N3 are not large numbers, the three quotients broaden. This is directly observed as broader peaks in the diffraction pattern.The area underneath the peak is constant, so that peak broadening is accompanied by a decrease in the maximum peak heightThese two diffraction patterns come from the exact same sample (silicon). The apparent difference in peak broadening is due to the instrument optics, not due to specimen broadeningTEM images collected by Jane Howe at Oak Ridge National Laboratory. K=0.94 for FWHM of spherical crystals with cubic symmetryK=0.89 for integral breadth of spherical crystals w/ cubic symmetryK=1, because 0.94 and 0.89 both round up to 1 K=1.03 for tetrahedral crystallites (100) peakK=0.81 for octrahedral crystallites (100) peak0.81 < K < 1.03 are the typical extremesBecause this is a complex pattern, without a lot of peak broadening, and because we had only a small quantity of sample, we were required to execute a very long scanWe collected a full range scan of CeO2 before and afterwe determined that there was no significant microstrain, and that Scherrer analysis of a single diffraction peak yielded results that were comparable to analysis of the full patternTherefore, we felt confident that this short range data collection would give us reasonably accurate resultsWe added ZnO as an internal standard to help us quantify the amount of amorphous materialwe cannot really use ZnO as a size-strain standard because we know that it is slightly nanocrystallinethe upper limit for practical crystallite size analysis on the Rigaku is 100nm; on the PANalytical is about 150 nm

crystal size analysis

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