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Practical X-Ray Diffraction Prof. Thomas Key School of Materials Engineering

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Instrument Settings Source –Cu K α Slits –Less than 3.0 Type of measurement –Coupled 2θ –Detector scan –Etc. Angle Range –Increment –Rate (deg/min) Detector –LynxEye (1D) Bruker D8 Focus

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Coupled 2θ Measurements X-ray tube Detector Φ In “Coupled 2θ” Measurements: –The incident angle is always ½ of the detector angle 2 . –The x-ray source is fixed, the sample rotates at °/min and the detector rotates at 2 °/min. Angles –The incident angle (ω) is between the X-ray source and the sample. –The diffracted angle (2 ) is between the incident beam and the detector. –In plane rotation angle (Φ) Motorized Source Slits

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Bragg’s law and Peak Positions. For parallel planes of atoms, with a space d hkl between the planes, constructive interference only occurs when Bragg’s law is satisfied. –First, the plane normal must be parallel to the diffraction vector Plane normal: the direction perpendicular to a plane of atoms Diffraction vector: the vector that bisects the angle between the incident and diffracted beam –X-ray wavelengths are: Cu K α 1 = Å and Cu K α 2 = Å Or Cu K α (avg) = Å –d hkl is dependent on the lattice parameter (atomic/ionic radii) and the crystal structure –I hkl =I o pCL P [F hkl ] 2 determines the intensity of the peak

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Sample Preparation ( Common Mistakes and Their Problems ) Z-Displacements –Sample height matters –Causes peaks to shift Sample orientation of single crystals –Affects which peaks are observed Inducing texture in powder samples –Causes peak integrated intensities to vary

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Z-Displacements R Tetragonal PZT –a=4.0215Å –b=4.1100Å Disp 2θ2θ θ It is important that your sample be at the correct height Detector

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Z-Displacements vs. Change in Lattice Parameter Lattice Parameters –a= Å –c= Å Z-Displaced Fit Disp.=1.5mm Change In Lattice Parameter Strain/Composition? 101/ /200 Disp a=4.07A c=4.16A Tetragonal PZT Shifts due to z-displacements are systematically different and differentiable from changes in lattice parameter

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Sample Preparation Crystal Orientation Matters

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Orientations Matter in Single Crystals (a big piece of rock salt) 22 At °2 , Bragg’s law fulfilled for the (111) planes, producing a diffraction peak. The (200) planes would diffract at °2 ; however, they are not properly aligned to produce a diffraction peak The (222) planes are parallel to the (111) planes

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For phase identification you want a random powder (polycrystalline) sample 22 22 22 When thousands of crystallites are sampled, for every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract All possible diffraction peaks should be exhibited Their intensities should match the powder diffraction file

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Sample Preparation Inducing Texture In A Powder Sample

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Preparing a powder specimen An ideal powder sample should have many crystallites in random orientations –the distribution of orientations should be smooth and equally distributed amongst all orientations If the crystallites in a sample are very large, there will not be a smooth distribution of crystal orientations. You will not get a powder average diffraction pattern. –crystallites should be <10 m in size to get good powder statistics Large crystallite sizes and non-random crystallite orientations both lead to peak intensity variation –the measured diffraction pattern will not agree with that expected from an ideal powder –the measured diffraction pattern will not agree with reference patterns in the Powder Diffraction File (PDF) database

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Salt Sprinkled on double stick tape What has Changed? NaCl An Examination of Table Salt It’s the same sample sprinkled on double stick tape but after sliding a glass slide across the sample Hint Typical Shape Of Crystals With Randomly Oriented Crystals

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Texture in Samples Common Occurrences –Plastically deformed metals (cold rolled) –Powders with particle shapes related to their crystal structure Particular planes form the faces Elongated in particular directions (Plates, needles, acicular,cubes, etc.) How to Prevent –Grind samples into fine powders –Unfortunately you can’t or don’t want to do this to many samples.

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A Simple Means of Quantifying Texture Lotgering degree of orientation (ƒ) –A comparison of the relative intensities of a particular family of (hkl) reflections to all observed reflections in a coupled 2θ powder x-ray diffraction (XRD) Spectrum –ƒ is specifically considered a measure of the “degree of orientation” and ranges from 0% to 100% –p o is p of a sample with a random crystallographic orientation. Jacob L. Jones, Elliott B. Slamovich, and Keith J. Bowman, “Critical evaluation of the Lotgering degree of orientation texture indicator,” J. Mater. Res., Vol. 19, No. 11, Nov 2004 Where for (00l) –I hkl is the integrated intensity of the (hkl) reflection

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Phase Identification One of the most important uses of XRD

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For cubic structures it is often possible to distinguish crystal structures by considering the periodicity of the observed reflections.

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Identifying Non-Cubic Phases

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ICCD: JCPDS Files

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Phase Identification One of the most important uses of XRD Typical Steps –Obtain XRD pattern –Measure d-spacings –Obtain integrated intensities –Compare data with known standards in the –JCPDS file, which are for random orientations There are more than 50,000 JCPDS cards of inorganic materials

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Measuring Changes In A Single Phase’s Composition by X-Ray Diffraction

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Vegard’s Law Good for alloys with continuous solid solutions Ex) Au-Pd To create the plot on the right Using the crystal structure of the alloy calculate “a” for each metal Draw a straight line between them as shown on the chart to the left. To calculate the composition Calculate “a” from d-spacings “a” will be an atomic weighted fraction of “a” of the two metal

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Measuring Changes In Phase Fraction Using I/Icor and Direct Comparison Method

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Phase Fractions Using I/Icorr –Where ω= weight fraction I(hkl)=Reference’s relative intensity I exp (hkl)=Experimental integrated intensity 1

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Phase Fractions Direct Comparison Method –Where v=Volume fraction V=Volume of the unit cell Because this is already a complicated method, many choose to go ahead and use Rietveld Refinement

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Strain Effects Peak Shifts and Peak Broadening

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Other Factors contributing to contribute to the observed peak profile

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Many factors may contribute to the observed peak profile Instrumental Peak Profile –Slits –Detector arm length Crystallite Size Microstrain –Non-uniform Lattice Distortions (aka non-uniform strain) –Faulting –Dislocations –Antiphase Domain Boundaries –Grain Surface Relaxation Solid Solution Inhomogeneity Temperature Factors The peak profile is a convolution of the profiles from all of these contributionsThe peak profile is a convolution of the profiles from all of these contributions

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Crystallite Size Broadening Peak Width B(2 ) varies inversely with crystallite size The constant of proportionality, K (the Scherrer constant) depends on the how the width is determined, the shape of the crystal, and the size distribution –The most common values for K are 0.94 (for FWHM of spherical crystals with cubic symmetry), 0.89 (for integral breadth of spherical crystals with cubic symmetry, and 1 (because 0.94 and 0.89 both round up to 1). –K actually varies from 0.62 to 2.08 –For an excellent discussion of K, 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) p Remember: –Instrument contributions must be subtracted

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(deg.) Intensity (a.u.) (deg.) Intensity (a.u.) Methods used to Define Peak Width Full 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 Breadth –the total area under the peak divided by the peak height –the width of a rectangle having the same area and the same height as the peak –requires very careful evaluation of the tails of the peak and the background FWHM

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Williamson-Hull Plot y-interceptslope K≈0.94 Grain size broadening Grain size and strain broadening Gausian Peak Shape Assumed

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(deg.) Intensity (a.u.) Which of these diffraction patterns comes from a nanocrystalline material? These diffraction patterns were produced from the exact same sample The apparent peak broadening is due solely to the instrumentation –0.0015° slits vs. 1° slits Hint: Why are the intensities different?

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

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Anistropic Size Broadening The 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. Use 111 and 222 peaks Use 200 and 400 peaks To determine aspect ratios

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Crystallite Shape Though the shape of crystallites is usually irregular, we can often approximate them as: –sphere, cube, tetrahedra, or octahedra –parallelepipeds such as needles or plates –prisms or cylinders Most applications of Scherrer analysis assume spherical crystallite shapes If 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 broadening –if the dimensions of a crystallite are 2x * 2y * 200z, then (h00) and (0k0) peaks will be more broadened then (00l) peaks.

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Reporting Data

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Diffraction patterns are best reported using d hkl and relative intensity rather than 2 and absolute intensity. The peak position as 2 depends on instrumental characteristics such as wavelength. –The peak position as d hkl is an intrinsic, instrument-independent, material property. Bragg’s Law is used to convert observed 2 positions to d hkl. The absolute intensity, i.e. the number of X rays observed in a given peak, can vary due to instrumental and experimental parameters. –The relative intensities of the diffraction peaks should be instrument independent. To calculate relative intensity, divide the absolute intensity of every peak by the absolute intensity of the most intense peak, and then convert to a percentage. The most intense peak of a phase is therefore always called the “100% peak”. –Peak areas are much more reliable than peak heights as a measure of intensity.

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Powder diffraction data consists of a record of photon intensity versus detector angle 2 . Diffraction data can be reduced to a list of peak positions and intensities –Each d hkl corresponds to a family of atomic planes {hkl} –individual planes cannot be resolved- this is a limitation of powder diffraction versus single crystal diffraction hkld hkl (Å)Relative Intensit y (%) {012} {104} {110} {006} {113} {202} Position [°2 ] Intensity [cts] Raw Data Reduced dI list

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Extra Examples Crystal Structure vs. Chemistry

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Two Perovskite Samples What are the differences? –Peak intensity –d-spacing Peak intensities can be strongly affected by changes in electron density due to the substitution of atoms with large differences in Z, like Ca for Sr. SrTiO 3 and CaTiO 3 2θ (Deg.) Assume that they are both random powder samples

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θ (Deg) Intensity(Counts) Two samples of Yttria stabilized Zirconia Substitutional Doping can change bond distances, reflected by a change in unit cell lattice parameters The change in peak intensity due to substitution of atoms with similar Z is much more subtle and may be insignificant 10% Y in ZrO 2 50% Y in ZrO 2 Why might the two patterns differ? R(Y 3+ ) = 0.104Å R(Zr 4+ ) = Å

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Questions

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Supplimental Information

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Free Software Empirical Peak Fitting –XFit –WinFit couples with Fourya for Line Profile Fourier Analysis –Shadow couples with Breadth for Integral Breadth Analysis –PowderX –FIT succeeded by PROFILE Whole Pattern Fitting –GSAS –Fullprof –Reitan All of these are available to download from

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Dealing With Different Integral Breadth/FWHM Contributions Contributions Lorentzian and Gaussian Peak shapes are treated differently B=FWHM or β in these equations Williamson-Hall plots are constructed from for both the Lorentzian and Gaussian peak widths. The crystallite size is extracted from the Lorentzian W-H plot and the strain is taken to be a combination of the Lorentzian and Gaussian strain terms. Gaussian Lorentzian (Cauchy) Integral Breadth (PV)

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