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

Instrument Settings Bruker D8 Focus Source Slits Type of measurement Cu Kα 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

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

Bragg’s law and Peak Positions. For parallel planes of atoms, with a space dhkl 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 l are: Cu Kα1=1.540598 Å and Cu Kα2=1.544426 Å Or Cu Kα(avg)=1.54278 Å dhkl is dependent on the lattice parameter (atomic/ionic radii) and the crystal structure Ihkl=IopCLP[Fhkl]2 determines the intensity of the peak draw the diffraction vector on this slide, or make a second slide explicitly illustrating the diffraction vector

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

It is important that your sample be at the correct height Z-Displacements Detector Tetragonal PZT a=4.0215Å b=4.1100Å R 011 110 111 002 200 θ Disp 2θ It is important that your sample be at the correct height

Z-Displacements vs. Change in Lattice Parameter Change In Lattice Parameter Strain/Composition? Lattice Parameters a=4.0215 Å c=4.1100 Å a=4.07A c=4.16A 101/110 Tetragonal PZT 002/200 111 Z-Displaced Fit Disp.=1.5mm D= 1.5 mm a [A] 4.07 c [A] 4.16 Disp Shifts due to z-displacements are systematically different and differentiable from changes in lattice parameter 7

Crystal Orientation Matters Sample Preparation Crystal Orientation Matters

Orientations Matter in Single Crystals (a big piece of rock salt) 111 200 220 311 222 2q The (200) planes would diffract at 31.82 °2q; however, they are not properly aligned to produce a diffraction peak The (222) planes are parallel to the (111) planes. At 27.42 °2q, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.

For phase identification you want a random powder (polycrystalline) sample 200 220 111 222 311 2q 2q 2q 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

Inducing Texture In A Powder Sample Sample Preparation Inducing Texture In A Powder Sample

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 <10mm 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

An Examination of Table Salt 200 111 220 311 222 Salt Sprinkled on double stick tape What has Changed? NaCl <100> With Randomly Oriented Crystals Hint Typical Shape Of Crystals It’s the same sample sprinkled on double stick tape but after sliding a glass slide across the sample

Texture in Samples Common Occurrences How to Prevent 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.

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% po is p of a sample with a random crystallographic orientation. Where for (00l) Ihkl is the integrated intensity of the (hkl) reflection 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

One of the most important uses of XRD Phase Identification One of the most important uses of XRD

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

Identifying Non-Cubic Phases

ICCD: JCPDS Files

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

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

Good for alloys with continuous solid solutions 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

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

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

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

Peak Shifts and Peak Broadening Strain Effects Peak Shifts and Peak Broadening

Other Factors contributing to contribute to the observed peak profile

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 contributions

Crystallite Size Broadening Peak Width B(2q) 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) p102-113. Remember: Instrument contributions must be subtracted

Methods used to Define Peak Width 46.7 46.8 46.9 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9 2 q (deg.) Intensity (a.u.) 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 46.7 46.8 46.9 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9 2 q (deg.) Intensity (a.u.)

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

Hint: Why are the intensities different? Which of these diffraction patterns comes from a nanocrystalline material? 66 67 68 69 70 71 72 73 74 2 q (deg.) Intensity (a.u.) Hint: Why are the intensities different? These 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 broadening 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 35

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 TEM images collected by Jane Howe at Oak Ridge National Laboratory.

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

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.

Reporting Data

Diffraction patterns are best reported using dhkl and relative intensity rather than 2q and absolute intensity. The peak position as 2q depends on instrumental characteristics such as wavelength. The peak position as dhkl is an intrinsic, instrument-independent, material property. Bragg’s Law is used to convert observed 2q positions to dhkl. 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.

Powder diffraction data consists of a record of photon intensity versus detector angle 2q. Diffraction data can be reduced to a list of peak positions and intensities Each dhkl corresponds to a family of atomic planes {hkl} individual planes cannot be resolved- this is a limitation of powder diffraction versus single crystal diffraction Raw Data Reduced dI list Position [°2q] Intensity [cts] 25.2000 372.0000 25.2400 460.0000 25.2800 576.0000 25.3200 752.0000 25.3600 1088.0000 25.4000 1488.0000 25.4400 1892.0000 25.4800 2104.0000 25.5200 1720.0000 25.5600 1216.0000 25.6000 732.0000 25.6400 456.0000 25.6800 380.0000 25.7200 328.0000 hkl dhkl (Å) Relative Intensity (%) {012} 3.4935 49.8 {104} 2.5583 85.8 {110} 2.3852 36.1 {006} 2.1701 1.9 {113} 2.0903 100.0 {202} 1.9680 1.4

Extra Examples Crystal Structure vs. Chemistry

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. Assume that they are both random powder samples SrTiO3 and CaTiO3 Ca Z=20; Sr Z=38 Zr Z=40; Y Z=39 Plays into why having an un-textured sample is important 200 210 211 2θ (Deg.) 43

Two samples of Yttria stabilized Zirconia Why might the two patterns differ? 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 ZrO2 50% Y in ZrO2 45 50 55 60 65 2θ (Deg) Intensity(Counts) Ca Z=20; Sr Z=38 Zr Z=40; Y Z=39 Plays into why having an un-textured sample is important R(Y3+) = 0.104Å R(Zr4+) = 0.079Å 44

Questions

Supplimental Information

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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. Lorentzian (Cauchy) Gaussian Integral Breadth (PV)