Presentation on theme: "X-Ray Diffraction Background and Fundamentals"— Presentation transcript:
1 X-Ray Diffraction Background and Fundamentals Prof. Thomas KeySchool of Materials Engineering
2 Crystalline materials are characterized by the orderly periodic arrangements of atoms. The (200) planes of atoms in NaClThe (220) planes of atoms in NaClThe unit cell is the basic repeating unit that defines a crystal.Parallel planes of atoms intersecting the unit cell are used to define directions and distances in the crystal.These crystallographic planes are identified by Miller indices.
3 Bragg’s law is a simplistic model to understand what conditions are required for diffraction. dhklFor parallel planes of atoms, with a space dhkl between the planes, constructive interference only occurs when Bragg’s law is satisfied.In our diffractometers, the X-ray wavelength l is fixed.Consequently, a family of planes produces a diffraction peak only at a specific angle q.Additionally, the plane normal must be parallel to the diffraction vectorPlane normal: the direction perpendicular to a plane of atomsDiffraction vector: the vector that bisects the angle between the incident and diffracted beamThe space between diffracting planes of atoms determines peak positions.The peak intensity is determined by what atoms are in the diffracting plane.draw the diffraction vector on this slide, or make a second slide explicitly illustrating the diffraction vector
4 Our powder diffractometers typically use the Bragg-Brentano geometry. DetectorX-ray tubeΦwq2qAnglesThe 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)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.
5 Coupled 2θ Measurements DetectorMotorized Source SlitsX-ray tubeΦwq2qIn “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.AnglesThe 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)
6 The X-ray Shutter is the most important safety device on a diffractometer X-rays exit the tube through X-ray transparent Be windows.X-Ray safety shutters contain the beam so that you may work in the diffractometer without being exposed to the X-rays.Being aware of the status of the shutters is the most important factor in working safely with X rays.
7 The wavelength of X rays is determined by the anode of the X-ray source. Electrons from the filament strike the target anode, producing characteristic radiation via the photoelectric effect.The anode material determines the wavelengths of characteristic radiation.While we would prefer a monochromatic source, the X-ray beam actually consists of several characteristic wavelengths of X rays.KLM
8 Why does this sample second set of peaks at higher 2θ values? Hints:It’s AluminaCu sourceDetector has a single channel analyzer006113Kα1Kα2
9 Diffraction Pattern Collected Where A Ni Filter Is Used To Remove Kβ Ka1Ka2What could this be?W La1Due to tungsten contaminationK alpha 1 and K alpha 2 overlap heavily at low angles and are easier to discriminate at high angles.Kb9
10 Wavelengths for X-Radiation are Sometimes Updated CopperAnodesBearden(1967)Holzer et al.(1997)CobaltCu Ka1ÅÅCo Ka1ÅÅCu Ka2ÅÅCo Ka2ÅÅCu KbÅÅCo KbÅÅMolybdenumChromiumMo Ka1ÅÅCr Ka1ÅÅMo Ka2ÅÅCr Ka2ÅÅMo KbÅÅCr KbÅÅOften quoted values from Cullity (1956) and Bearden, Rev. Mod. Phys. 39 (1967) are incorrect.Values from Bearden (1967) are reprinted in international Tables for X-Ray Crystallography and most XRD textbooks.Most recent values are from Hölzer et al. Phys. Rev. A 56 (1997)
19 Not all Planes Produce Peaks Peak IntensityStructure FactorsOnly if [Fhkl]2 ≠0 does a peak appearwhereIo = Intensity of the incident X-ray beamp = Multiplicity factor (a function of the crystallography of the material)C = Experimental constant (related to temperature, absorption, fluorescence, and crystal imperfection). Temperature factor=e-2M; Absorption factor = A(θ).LP = Lorentz-Polarization factor.ƒn = atomic scattering factor of atom ‘n’ is a measure of the scattering efficiencyu,v,w are the atomic positions in the unit cellh,k,l are the Miller indices of the reflection.N is number of atoms in the unit cellThe summation is performed over all atoms in the unit cell.These calculations are easily doable for simple structures
20 Structure Factors: Useful Knowledge (1)Atomic scattering factors vary as a function of atomic number (Z) and diffraction angle (θ)Values can be looked up in tablesLinear extrapolations are used for calculating the values between those listed.(2)
24 X-Ray Diffraction Patterns 111200220311222BCC or FCC?Relative intensities determined by:
25 A random polycrystalline sample that contains thousands of crystallites should exhibit all possible diffraction peaks2002201112223112q2q2qFor every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract (the plane perpendicular bisects the incident and diffracted beams).Basic assumptions of powder diffraction are that for every set of planes there is an equal number of crystallites that will diffract and that there is a statistically relevant number of crystallites, not just one or two.
26 Why are peaks missing? The sample is a cut piece of Morton’s Salt 111200220311222JCPDF#The sample is a cut piece of Morton’s SaltJCPDF# is supposed to fit it (Sodium Chloride Halite)
27 It’s a single crystal (a big piece of rock salt) 1112002203112222qThe (200) planes would diffract at °2q; however, they are not properly aligned to produce a diffraction peakThe (222) planes are parallel to the (111) planes.At °2q, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.
36 Is that enough information? Example 5Radiation from a copper source -Is that enough information?“Professor my peaks split!”
37 X-radiation for diffraction measurements is produced by a sealed tube or rotating anode. Sealed X-ray tubes tend to operate at 1.8 to 3 kW.Rotating anode X-ray tubes produce much more flux because they operate at 9 to 18 kW.A rotating anode spins the anode at 6000 rpm, helping to distribute heat over a larger area and therefore allowing the tube to be run at higher power without melting the target.Both sources generate X rays by striking the anode target wth an electron beam from a tungsten filament.The target must be water cooled.The target and filament must be contained in a vacuum.
38 Spectral Contamination in Diffraction Patterns Ka1Ka1Ka2Ka1Ka2Ka2K alpha 1 and K alpha 2 overlap heavily at low angles and are easier to discriminate at high angles.The Ka1 & Ka2 doublet will almost always be presentVery expensive optics can remove the Ka2 lineKa1 & Ka2 overlap heavily at low angles and are more separated at high anglesW lines form as the tube ages: the W filament contaminates the target anode and becomes a new X-ray sourceW and Kb lines can be removed with opticsW La1Kb
39 Divergence slits are used to limit the divergence of the incident X-ray beam. The slits block X-rays that have too great a divergence.The size of the divergence slit influences peak intensity and peak shapes.Narrow divergence slits:reduce the intensity of the X-ray beamreduce the length of the X-ray beam hitting the sampleproduce sharper peaksthe instrumental resolution is improved so that closely spaced peaks can be resolved.
40 Varying Irradiated area of the sample the area of your sample that is illuminated by the X-ray beam varies as a function of:incident angle of X raysdivergence angle of the X raysat low angles, the beam might be wider than your sample“beam spill-off”
41 The constant volume assumption In a polycrystalline sample of ‘infinite’ thickness, the change in the irradiated area as the incident angle varies is compensated for by the change in the penetration depthThese two factors result in a constant irradiated volume(as area decreases, depth increase; and vice versa)This assumption is important for many aspects of XRPDMatching intensities to those in the PDF reference databaseCrystal structure refinementsQuantitative phase analysisThis assumption is not necessarily valid for thin films or small quantities of sample on a ZBH
42 One by-product of the beam divergence is that the length of the beam illuminating the sample becomes smaller as the incident angle becomes larger.The length of the incident beam is determined by the divergence slit, goniometer radius, and incident angle.This should be considered when choosing a divergence slits size:if the divergence slit is too large, the beam may be significantly longer than your sample at low anglesif the slit is too small, you may not get enough intensity from your sample at higher anglesAppendix A in the SOP contains a guide to help you choose a slit size.The width of the beam is constant: 12mm for the Rigaku RU300.
44 Detectors point detectors position sensitive detectors observe one point of space at a timeslow, but compatible with most/all opticsscintillation and gas proportional detectors count all photons, within an energy window, that hit themSi(Li) detectors can electronically analyze or filter wavelengthsposition sensitive detectorslinear PSDs observe all photons scattered along a line from 2 to 10° long2D area detectors observe all photons scattered along a conic sectiongas proportional (gas on wire; microgap anodes)limited resolution, issues with deadtime and saturationCCDlimited in size, expensivesolid state real-time multiple semiconductor stripshigh speed with high resolution, robust
45 Sources of Error in XRD Data Sample Displacementoccurs when the sample is not on the focusing circle (or in the center of the goniometer circle)The greatest source of error in most dataA systematic error:S is the amount of displacement, R is the goniometer radius.at 28.4° 2theta, s=0.006” will result in a peak shift of 0.08°Can be minimized by using a zero background sample holderCan be corrected by using an internal calibration standardCan be analyzed and compensated for with many data analysis algorithmsFor sample ID, simply remember that your peak positions may be shifted a little bitCan be eliminated by using parallel-beam optics
46 Other sources of error Axial divergence Flat specimen error Due to divergence of the X-ray beam in plane with the samplecreates asymmetric broadening of the peak toward low 2theta anglesCreates peak shift: negative below 90° 2theta and positive above 90°Reduced by Soller slits and/or capillary lensesFlat specimen errorThe entire surface of a flat specimen cannot lie on the focusing circleCreates asymmetric broadening toward low 2theta anglesReduced by small divergence slits; eliminated by parallel-beam opticsPoor counting statisticsThe sample is not made up of thousands of randomly oriented crystallites, as assumed by most analysis techniquesThe sample might be textured or have preferred orientationCreates a systematic error in peak intensitiesSome peaks might be entirely absentThe sample might have large grain sizesProduces ‘random’ peak intensities and/or spotty diffraction peaks
47 sample transparency error X 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 locationAngular errors and peak asymmetryGreatest for organic and low absorbing (low atomic number) samplesCan be eliminated by using parallel-beam optics or reduced by using a thin samplem is the linear mass absorption coefficient for a specific sample
48 Techniques in the XRD SEF X-ray Powder Diffraction (XRPD)Single Crystal Diffraction (SCD)Back-reflection Laue Diffraction (no acronym)Grazing Incidence Angle Diffraction (GIXD)X-ray Reflectivity (XRR)Small Angle X-ray Scattering (SAXS)
49 Available Free Software GSAS- Rietveld refinement of crystal structuresFullProf- Rietveld refinement of crystal structuresRietan- Rietveld refinement of crystal structuresPowderCell- crystal visualization and simulated diffraction patternsJCryst- stereograms