3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)An X-ray diffraction image for the protein myoglobin. Source: Wikipedia. 3.012 Fund of Mat Sci: Structure – Lecture 18X-RAYS AT WORKModel of helical domains in myoglobin. Image courtesy of Magnus Manske3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Wed Nov 23• Prof Wuensch Lecture Notes• http://capsicum.me.utexas.edu/ChE386K/for many details (Lect 19 onwards, but note different 2π convention)• Buy turkey3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. X-rays generation: undulators and wigglers in synchrotrons, bremsstrahlung and core excitations (e.g. Kα) in X-ray tubes2. Reciprocal lattice3. Diffraction gratings – Huygens construction4. Laue diffraction from periodicarrays in 1-d, 2-d, 3-dGrooveIncident Plane Wave (Lambda = 2/11 * Grating Pitch)Diffraction Grating0th Order1st Order2nd Order3rd Order(Left-pointing 1st, 2nd, and3rd order, and all higher orderbeams not shown.)Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Reciprocal lattice (IV)() () ()()12 323 3112222123123 123 123d is distance between two planes of Millewith , , integers,, , are the reciprocal-lattice vectors hklaa aaaabb baaa aaa aaaG bb bhi j hijGhijπππ∗×××===⋅× ⋅× ⋅×=++=rr rrrrrr rrr r rr r rr rrrrrrhkl= r indices h k l in the reciprocal lattice 2dπFirst and second Laue conditions()()0000coscoscos cosnnxaS aaS aaaSSnααααλ⋅=⋅=−=⋅−=rrrrrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Soa.Soa.SaSFigure by MIT OCW.Figure by MIT OCW.a2a13.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)All three Laue conditions()()()102030integer multiple of integer multiple of integer multiple of aSSaSSaSSλλλ⋅− =⋅− =⋅− =rrrrrrrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Ewald constructionFigure by MIT OCW.hklDiffracted beamhklPd*hkld*hkl= S-S0λS02πλ2πλ2π2π/λOrigin of the reciprocal latticeReflecting sphereIncident beamθθSCrystal at the center of the sphereOCθ3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Laue condition needs “white” spectrum202201 Reflected BeamIncident Beam200 Reflected Beam1022010011011002π /λmax2π /λmin200001002101201002102202201Reflected BeamReflecting Sphere forSmallest WavelengthReflecting Sphere forLargest WavelengthFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Alternate geometrical view002001O100200001002Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Bragg Law2sinhklndλθ=OPP'P''ddO'A(hkl)(hkl)(hkl)BDdCEθλθθFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Equivalence to Laue condition02222cos 2sinhklhklSSddππππνθλλ λ∗⎛⎞−====⎜⎟⎝⎠rrFigure by MIT OCW.θθθνν(hkl)2π/λ2π S/λDiffracted BeamReciprocal-lattice OriginEwald SphereIncident Beam2π S0/λλdhkl2πS-S0=((*3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Powder diffraction (I)202201 Reflected BeamIncident Beam200 Reflected Beam1022010011011002π /λmax2π /λmin200000001002101201002102202201Reflected BeamReflecting Sphere forSmallest WavelengthReflecting Sphere forLargest WavelengthFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Powder diffraction (II)Image removed for copyright reasons. Please see the diagram at http://capsicum.me.utexas.edu/ChE386K/html/powder_diffraction_3.htm.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Powder diffraction (III)Diagrams of the Powder Method removed for copyright reasons.See the images at http://www.matter.org.uk/diffraction/x-ray/powder_method.htm3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)X-ray filtersImage removed for copyright reasons. Please see the graph at http://capsicum.me.utexas.edu/ChE386K/html/absorption_edge.htm.Image removed for copyright reasons. Please see the diagrams at http://capsicum.me.utexas.edu/ChE386K/html/filter.htm.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Debye-Scherrer cameraPhotographs of a Debye-Scherrer camera removed for copyright reasons.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Interplanar spacings22222Cubic: hkladhkl=++Debye-Scherrer camera2sinhkldλθ=22222Cubic: hkladhkl=++3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Figure by MIT OCW.Tables removed for copyright reasons. See http://www.matter.org.uk/diffraction/x-ray/indexing_powder_pattern.htm3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Systematic absencesImage removed for copyright reasons.Please see the table at http://capsicum.me.utexas.edu/ChE386K/html/systematic_absences.htm.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Effects of symmetry on diffractionImages removed for copyright reasons.Please see the images at http://capsicum.me.utexas.edu/ChE386K/html/diffraction_symmetry1.htm.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Structure FactorImage removed for copyright reasons.Please see the graph at http://capsicum.me.utexas.edu/ChE386K/html/scattering_factor_curve.htm.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Friedel’s law• The diffraction pattern is always centrosymmetric, even if the crystal is not centrosymmetric3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Point symmetry + inversion = LaueImage removed for copyright reasons. Please see the table at http://capsicum.me.utexas.edu/ChE386K/html/diffraction_symmetry2.htm.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Back-reflection and transmission LaueDiagrams of the Laue Method removed for copyright reasons. See the images at
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