3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Structure – Lecture 19FROM DIFFRACTION TO STRUCTUREImages removed for copyright reasons.3-fold symmetry in silicon along the [111] direction. Forward (left) and backward (right) scattering.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Mon Nov 28• Study: 3.3 Allen-Thomas (Symmetry constraints) • Read all of Chapter 1 Allen-Thomas3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Laue condition in 3 dimensions2. Ewald construction3. Bragg law, and equivalence to Laue condition4. Powder diffraction, Debye-Scherrer3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Atoms as spherical scatterersImages removed for copyright reasons.Figure by MIT OCW.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Huygens constructionGrooveIncident 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.)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)S0a.S0a.SaS2ar1arFigure by MIT OCW.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Ewald constructionθθθνν(hkl)2π/λ2π S/λDiffracted BeamReciprocal-lattice OriginEwald SphereIncident Beam2π S0/λλdhkl2πS-S0=((*Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Equivalence to Laue conditionθθθνν(hkl)2π/λ2π S/λDiffracted BeamReciprocal-lattice OriginEwald SphereIncident Beam2π S0/λλdhkl2πS-S0=((*02222cos 2sinhklhklSSddππππνθλλ λ∗⎛⎞−====⎜⎟⎝⎠rr()2sinhklndλθ=Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Laue condition needs “white” spectrum202201 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 diffractionImage 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)Interplanar spacings222222Cubic: = hkl hklhkladdhkl dπ∗⎛⎞=⎜⎟++⎝⎠3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Debye-Scherrer camera2sinhklndλθ=22222Cubic: hkladhkl=++2222 22sinhkl anθλ⎛⎞++=⎜⎟⎝⎠Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)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)X-ray powder diffraction images removed for copyright reasons.• X-ray powder diffraction for silver, aluminum, gold, and copper3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)X-ray powder diffraction images removed for copyright reasons.• Detail, back-scattering direction, showing the line splitting that takes place due to the presence of the K-alpha-1 and K-alpha-2 lines of the copper spectrum which the x-ray machine produced. Measurements of the diffraction angles forthese lines can yield very accurate values for the crystal unit cell size.RocksaltX-ray powder diffraction images removed for copyright reasons.Source: Wikipedia• Detail of the small-angle end of the film, showing that the NaCl and KCl patterns do not look the same, and that the crystal sizes are different. The KCl appears almost exactly as if it were a simple cubic, due to the fact that the K and Cl ions, very close to each other on the periodic table, are almost exactly alike. The Na and Cl ions are not so close on the periodic table, thus their ions are not the same, thus the powder diffraction pattern does not look like the pattern from a simple cubic crystal. 3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Physical properties and their relation to symmetry• Density (mass, from a certain volume)• Pyroelectricity (polarization from temperature)• Conductivity (current, from electric field)• Piezoelectricity (polarization, from stress)• Stiffness (strain, from stress)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Scalar, vector, tensor properties• Mass (0), polarization (1), strain (2)YUndeformedXDeformedZXPP'uRPP'R'u = R' - RoZYFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Transformation of a vector3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Orthogonal Matrices3.012
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