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Non-Ideal DataDiffraction in the Real WorldScott A Speakman, Ph.D.

speakman@mit.eduhttp://prism.mit.edu/xrayThe calculated diffraction pattern represents the ideal X-ray powder sampleThe ideal powder sampleMillions of grainsRandomly oriented grainsFlatSmooth surfaceDensely packedHomogeneousSmall grain size (less than 10 microns)Infinitely thickPreferred Orientation (aka Texture): non-random orientation of crystallitesIf the crystallites in a powder sample have plate or needle like shapes it can be very difficult to get them to adopt random orientationstop-loading, where you press the powder into a holder, can cause problems with preferred orientationin samples such as metal sheets or wires there is almost always preferred orientation due to the manufacturing processfor samples with systematic orientation, XRD can be used to quantify the texture in the specimenPreferred orientation causes a systematic error in peak intensities

Phase ID of samples with Preferred OrientationWhen executing a Search & Match for phase ID, you can no longer use peak intensities to help identify the phases that are presentUncheck Match Intensity and Demote unmatched strong

Preferred Orientation causes a systematic error in peak intensitiesIt becomes more necessary that you know the chemistry and origin of the sample that you are analyzingPractice Phase IDOpen Steel_Original_C1.xrdml Fit background and search peaksRun Search & MatchSearch with Match Intensity and Demote unmatched strong; do not constrain chemistrySearch with restrictions: EDXRF showed that Fe was the majority elment (above Mg), along with small amounts of Cr. Search with restrictions and with Match Intensity and Demote Unmatched Strong unchecked

Figure out what crystallographic directions are preferredAccept Fe as a matchFerrite and austeniteDetermine the texture component for Austenite firstTo see peak (hkl)sIn the Accepted Pattern List, right-clickSelect Analyze Pattern LinesThis table makes it easy to see what major peaks are missing peaks of the preferred orientation

Figure out what crystallographic directions are preferredTo label the peak markersSelect menu Customize > Document SettingsGo to the Legends & Grids tab Click on the button Pattern View LegendCheck Line hklClick OKCheck Label pattern lines with pattern view legendClick OK

The table shows us the (hkl) of peaks that are missing; the main graphics shows us the (hkl) of peaks that are stronger than expected(022) peak of austenite is stronger than expectedSuggests [011] texture

To Refine the Preferred OrientationAdd the reference pattern to the Refinement ControlGo to the Pattern ListRight-click on the both Fe entries and select Convert pattern to phaseTo set the preferred orientationIn the Refinement Control list, click on the second Fe phaseMake sure the symmetry is FCCChange the title to AusteniteIn the Object Inspector, find the Preferred Orientation sectionSet the Direction h, k, and lThe Parameter (March/Dollase) is the amount to preferred orientation1 indicates random orientation1 indicates preferred needle orientation

We will use Automatic RefinementThe Preferred orientation setting is turned onIt contains a Toggle Directions option so that the computer will try to figure out what the preferred orientation isWe made an educated guess, so we turn this off

The austenite is well refined, so we need to determine the orientation of the ferrite

Refining 21 parameters to get a 6.3% weighted R profile

The residual is good, but the fit still has areas of mismatch

Finishing steelHow I finished the refinement of steelRefine all background parametersRefine specimen displacementRefine W, then V, then U one at a timeTurn off specimen displacement, turn on lattice parameters (cells)Turn on specimen displacementRefine WRefine Peak Shape 1Weighted R Profile= 6.3%Another preferred orientation exampleOpen snail_texture-small section.rawFit background and search peaksRun Search & MatchSearch with Match Intensity and Demote unmatched strong; do not constrain chemistrySearch with restrictions: EDXRF showed that Ca was the only element above Mg that was presentSearch with restrictions and with Match Intensity and Demote Unmatched Strong unchecked

Figure out what crystallographic directions are preferredAccept calcite as a matchTo see peak (hkl)sIn the Accepted Pattern List, right-clickSelect Analyze Pattern LinesThis table makes it easy to see what major peaks are missing peaks of the preferred orientation

The table shows us the (hkl) of peaks that are missing; the main graphics shows us the (hkl) of peaks that are stronger than expected(hh0) and (hhk) peaks tend to be stronger and (00l) peaks weakerSuggests (110) texture

To Refine the Preferred OrientationAdd the reference pattern to the Refinement ControlGo to the Pattern ListRight-click on the calcite entry and select Convert pattern to phaseTo set the preferred orientationIn the Refinement Control list, click on the Calcite phaseIn the Object Inspector, find the Preferred Orientation sectionSet the Direction h, k, and lThe Parameter (March/Dollase) is the amount to preferred orientation1 indicates random orientation1 indicates avoided orientation

This refinement may have failed, so undo and try againCorrelation matrix shows correlation between: Specimen displacement and lattice parametersPeak width and preferred orientation

Undo and Try AgainChange the Automatic Rietveld stepsDo not refine Lattice Parameters and W (Halfwidth)Refine Preferred orientation before refining peak positions (specimen displacement)After scale factor adjusts, there is not enough intensity for different peaks to refined the peak positions

The low angle data fits well with this preferred orientation model

This is an example of strong preferred orientationThe March-Dollase function can only work for limited cases of preferred orientationA program with spherical harmonics can model more pronounced types of preferred orientationIn a case like this, the texture needs to be studied using pole figure analysis; Rietveld can provide only limited results

More preferred orientation with J Kaduk Dealing with Difficult SamplesOther complications: fluorescenceWe are studying Fe-bearing steel with a diffractometer that used Cu wavelength radiationFe and Co absorb the Cu wavelength radiation, and then fluoresce that energy as their characteristic X-raysThese fluoresced X-rays become background noiseThe absorption of X-rays also decreases the depth of penetration of the X-rays, which limits the irradiated volume of the sample and makes it more probable that you will have problems with particle statisticsHow to calculate the depth of penetrationt= 0.5L sinqt is the thickness of the sample contributing to 99% of the diffracted intensity L is the path length calculated based on the formula:IL=IO x exp (m/r)rLabut all that math is hard ... so let Thomas Degen do it for yougo to menu Tools > MAC CalculatorPenetration depth is 6 microns with Cu wavelength radiation and 31 microns with Co wavelength radiation

Problems with FluorescenceIncreases background noiseDecreases irradiated volumeDecreases the diffraction signalDecreases the fraction of X-rays scattered by the sampleEvery X-ray that is absorbed is an X-ray that was not diffractedBest solution is to change the X-ray anodeThis is not always practicalA diffracted-beam monochromator can eliminate the fluoresced X-rays, decreasing the background noiseThis also decreases the overall signal

Peak Ht:BkgRatioPeak AreaWithout monochromator8446:42501.981565With monochromator1157:4525.7251Anistropic Peak BroadeningSmall crytallite sizes produce peak broadeningIf the nanocrystals in a sample have an anistropic shape, then different peaks will be broadened differentlyExample: nanorod in which the axial direction of the rod corresponds to the c-axis of the crystalThe crystal dimension in the c direction is much larger than the direction in the a or b directionsThe (00l) peaks, which correspond to planes stacked along the c-axis, will be sharper corresponding to the larger dimensionThe (h00), (0k0), and (hk0) peaks, which correspond to planes stacked along the diameter of the nanorod, will be broader due to the smaller dimension.cAnistropic broadening can also change peak heights, giving the appearance of preferred orientation

NormalNanorod along the c-axisThe anistropic broadening model in HSP can deal with many simple cases

More complex cases may require using multiple models of the phaseIn the Refinement Control, right-click on phase and select Duplicate PhaseOne phase may model anistropic broadening in the [100] direction and the second phase may model anistropic broadening in the [110] directionThis approach also works for modeling more complex types of preferred orientation- create two phases with different preferred orientation functions

Anistropic Peak Broadening may also be due to complex defect structures in layered materialsIn this case, (h0h) peaks are broadened and asymmetricthis is due to a lack of correlation between well-formed (h0h) planesThis cannot be modeled in current Rietveld codes

Dealing with Peak AsymmetryPeak asymmetry is produced by:Axial divergenceSample transparencyAxial divergence can be reduced by using Soller slits

Sample Transparency ErrorX Rays penetrate into your samplethe depth of penetration depends on:the mass absorption coefficient of your samplethe incident angle of the X-ray beamThis produces errors because not all X rays are diffracting from the same location Angular errors and peak asymmetryGreatest for organic and low absorbing (low atomic number) samplesCan be eliminated by using parallel-beam opticsCan be reduced by using a thin sample

m is the linear mass absorption coefficient for a specific sampleModeling peak asymmetry in HighScore PlusThe asymmetry correction that works with the pseudo-Voigt function only offers limited adjustment

No asymmetry correctionFull asymmetry correctionThe Pseudo Voigt 3(FJC Asymmetry) correction can be manually adjusted to model more pronounced asymmetryS/L Asymmetry and D/L Asymmetry cannot be refinedProcedure: run a simple low angle standard such as mica or silver behenateManually adjust S/L and D/L to fit your dataUse those values the refinement of your sample of interestThis only works if your sample does not contribute additional asymmetry due to transparency

Exercise: modeling asymmetryOpen the file Silver Behenate.hpfWe have modeled the silver behenate as an orthorhombic unit cellThe most important parameter is the c 58.06We are using a LeBail fit because the crystal structure of silver behenate is not knownChange the peak profile to Pseudo Voigt 3(FJC Asymmetry)This is found in the Refinement section of the Object Inspector for Global ParametersStart a refinementBegin to manually vary S/L and D/L Asymmetry in the Profile Variables for the phase, running the refinement again each timeYou may be able to try refining S/L or D/L with no other variablesS/L=0.025D/L=0.025

J Kaduk follow up comments on transparencyJ Kaduk slides on Absorption, Surface Roughness, and Particle Statistics Skip to Speakman constant volume assumption, thin samples, inhomogeneous samplesParticle StatisticsXRPD Methods are based on the irradiation of millions of crystallites in a polycrystalline sampleIf there are not enough crystallites contributing to the diffraction pattern, then the observed peak intensities will be erroneousPoor particle statistics may be created if:The average grain size is too largeThe irradiated volume is too smallEither because the X-ray beam is too small or because there is not enough material in the sampleThe collimation of the X-ray beam is too tightThese factors interrelate- the grain size that is acceptable for a loosely collimated X-ray beam might be too large for a tightly collimated X-ray beamOnly a small fraction of the crystallites irradiated contribute to the diffraction pattern

A diffraction peak is observed when the crystallographic direction is parallel to the diffraction vectorThe crystallographic direction is the vector normal to the family of atomic planes that produce the diffraction peakThe diffraction vector is the vector that bisects the angle between the incident and scattered X-ray beam2q2qThe crystallographic direction (black arrow) is parallel to the diffraction vector (blue arrow), so the illustrated planes will diffract.The crystallite is now tilted so that the crystallographic direction (black arrow) is NOT parallel to the diffraction vector (blue arrow), so the illustrated planes will NOT diffract.Only a small fraction of the crystallites irradiated contribute to the diffraction pattern

A small fraction of crystallites will be properly oriented to diffract for each observable 2q2qA small fraction of grains (shaded blue) in this sample are properly oriented to produce the (100) diffraction peak A different fraction of grains (shaded blue) are properly oriented to produce the (110) diffraction peak Some grains (shaded blue) are oriented in such a way that they do not contribute to any diffraction peakParticle Statistics are determined byThe number of crystallites that are irradiatedThe irradiated volumeThe irradiated area (width and length of the X-ray beam)The depth of penetration of the X-raysThe average crystallite sizeThe particle packing factor (porosity)The fraction of irradiated crystallites that contribute to the diffraction peakVertical divergence of the X-ray beamDetector size and aperture (receiving slit)

The Number of Irradiated CrystallitesThe Irradiated Volume will be discussed in the next section (the constant volume assumption)The X-ray beam width and length are determined by the instrument configurationThe depth of penetration depends on , the linear mass absorption coefficient, of the specimenFor now, the irradiated volume will be treated as a value Vi

The number of irradiated crystallites (Ni) is:a is the average crystallite sizeAssumes a cubic crystallite shape where the length of the side of the cube is aEmpirical testing has shown that a Correct Beam Overflow option in HSPthrow away (clip or exclude) low angle data where beam was larger than sampleuse automatic divergence slitsDeviations from the constant volume assumption: Automatic Divergence Slits Automatic Divergence Slits (ADS) is possible with computer-controlled variable divergence slitsthe divergence slit is changed during the scan to maintain a constant X-ray beam lengthadvantage: get better intensity at high angles without risking beam overflow at low anglesdisadvantage:at higher angles, the background is noisier and the peaks are broader because of the larger divergence slitconstant volume assumption not preservedas penetration depth increases, the X-ray beam length does not shortenhave more irradiated volume at higher anglesDeviations from the constant volume assumption: Automatic Divergence SlitsHow to correct for increasing irradiated volume due to automatic divergence slitsCorrect with HSPuse Treatment > Corrections > Convert Divergence Slit

a round-robin led by Armel LeBail showed that converted ADS data works as well as data collected with fixed divergence slitsDeviations from the constant volume assumption: Thin SamplesIf the thickness of a sample is not greater than the maximum penetration depth of the X-ray beam, then the constant volume assumption will not be preservedwhen the penetration depth exceeds the thickness of the sample, intensity will begin to decrease as 1/sin q A sample may be thin because ofpreparation as a monolayer of powder on a ZBHa thin film on a substrateit is just a wafer thin sample

Dealing with Thin SamplesUse automatic divergence slitsuseful only for very thin samples, when the penetration depth of the X-ray beam exceeds the sample thickness over the entire measurement rangemaintains a constant irradiated length, and the thinness of the sample enforces a constant penetration depthconsequently, the irradiated volume is constantwhen you collect data from a thin sample using ADS, you need to lie to HSP and tell it that you used fixed slitsHSP takes fixed slits to mean that the constant volume assumption was observedselect the pattern in the Scan Listin the object inspector, find instrument settings > divergence slit typechange from automatic to fixed

Dealing with Thin SamplesIf data were collected with a fixed slit, apply a divergence slit correctionThen, lie to HSP and tell it that the data are from automatic slitsselect the pattern in the Scan Listin the object inspector, find instrument settings > divergence slit type

change from automatic to fixed use Treatment > Corrections > Convert Divergence Slitconvert data from fixed to automatic slitapplies a sin q correction to the peak intensities

This has limited effectiveness in my experiencethis approach greatly increases noise at high angles

Dealing with Thin SamplesUse grazing incidence angle X-ray diffractiononly if you are using parallel-beam opticsloss of angular resolution- peaks are broadeneduse a fixed incident angle, so that the irradiated area does not changethe incident angle is fixed at a small value to limit the depth of penetration of the X-rays, favoring scattering from the upper layers of the thin filmGIXD used to be the standard for thin film analysiswhen given the choice between GIXD with a point detector or Bragg-Brentano geometry with the XCelerator, the choice is harderoften get more intensity from thin film peaks using the more efficient detector rather than the grazing incident angleWhat if we cannot compensate for a thin sample?There will be errors in the refined model- especially thermal parametersthe model will try to compensate for the intensity being too low at higher angles of 2thetaWe can still use Rietveld refinement to quantify parameters that are independent of intensityunit cell lattice parametersnanocrystallite size and microstrainWe can do semi-quantitative phase composition analysisuse the RIR method with peaks that are close to each otherover a narrow range of 2theta, we can approximate the irradiated volume as nearly constant

Inhomogeneous SamplesThe changing size of the X-ray beam can be problematic if the sample is inhomogeneousPoorly mixed powderLayered microstructure, such as coatings on a substrateTo collect refinable data, do not allow the X-ray beam size to change during the data collectionUse GIXD technique (fixed incident beam)Use automatic divergence slits if the sample is thinThe penetration depth will change during the scan, even if the X-ray beam length is held constant so there must not be any thickness to allow the changing penetration depth.Chart1240.0030462013120.006092564980.009139253460.012186429148.015234254740.018282892834.307046791830.024383257626.694101977124.03048882821.851725791920.036600911618.501198267117.18557796316.045784120215.048849870714.169565296213.388322719412.689642430612.061140695711.492792619510.97639604610.505175310710.07348405549.67657936689.31044803738.97167143448.6573193278.36486567818.09212127947.83717942437.59837176717.37423220777.16346715016.96493085886.77760492316.60058105086.43304657946.2742722166.12360161615.98044248775.84425896645.71456505375.59091895135.47291814925.36019515645.2524137755.14926584085.0504683634.9557610074.86490387344.7776755334.6938712854.61330160784.53579077964.4611756464.38930451844.32003618524.25323902454.18879020484.1265749654.06648596374.00842269023.95229093143.89800228753.84547373173.79462720953.74538927343.69769074893.651466433.60665479973.56319777373.52104046623.48013097353.44042017543.40186155273.3644110183.32802676063.2926691023.25830036373.22488474293.19238819963.16077834963.13002436673.10009689133.07096794513.04261085253.01500016622.98811159882.96192195882.93640909012.91155181662.88732988942.86372393792.84071542392.81828659852.79642046152.77510072392.75431177182.7340386334

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Sheet1Irradiated Length (mm)1240.002120.01380.01460.01548.02640.02734.31830.02926.691024.031121.851220.041318.501417.191516.051615.051714.171813.391912.692012.062111.492210.982310.512410.07259.68269.31278.97288.66298.36308.09317.84327.60337.37347.16356.96366.78376.60386.43396.27406.12415.98425.84435.71445.59455.47465.36475.25485.15495.05504.96514.86524.78534.69544.61554.54564.46574.39584.32594.25604.19614.13624.07634.01643.95653.90663.85673.79683.75693.70703.65713.61723.56733.52743.48753.44763.40773.36783.33793.29803.26813.22823.19833.16843.13853.10863.07873.04883.02892.99902.96912.94922.91932.89942.86952.84962.82972.80982.78992.751002.73

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