3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Structure – Lecture 17X-RAY DIFFRACTIONImage of a spiral sea shell (left) and Rosalyn Franklin's original picture of a DNA Alpha Helix (right). Images removed for copyright reasons.A beautiful spiral, and … an even more beautiful one3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Wed Nov 9• Read: Prof Wuensch Lecture Notes3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: 1. Glide planes, screw axes2. Space groups3. Bravais lattices: sc, bcc, fcc (also, lattice with a basis)4. Primitive, conventional, and Wigner-Seitz cells5. Miller indices6. Diamond, zincblend, perovskites, rocksalt, CsCl3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Probing with radiation• Wavelength need to be smaller than typical interatomic distances• Beams of photon (X-rays), electrons, neutrons• We look at coherent (all same atoms behave in the same way), elastic (no energy is lost) scattering• Elastic: diffraction. Inelastic: spectroscopies• We “interrogate” long-range order with coherent elastic scatteringExamples: http://imagers.gsfc.nasa.gov/ems/ems.html3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Energy of an accelerated electron3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)How do we generate soft X-rays (~1000 eV)?• Relativistic effects: every time a charge is accelerated or decelerated: wigglers and undulators in a synchrotronDiagram explaining the mechanics of a synchrotron removed for copyright reasons. See http://geographyfieldwork.com/SynchrotronWorks.htmImage Copyright ©EPSIM 3D/JF Santarelli, Synchotron Soleil3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)How do we generate soft X-rays?• In the lab: beam of electrons striking a metal target– Electrons are decelerated, and they emit radiation on a broad spectrum of frequencies. This is called Bremsstrahlung– In addition, we excite core electrons, that decay back emitting radiation at K, L, M lines3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Moseley law8 10 12 14 16 18 20 22Re 75Yb 70Tb 65Nd 60Cs 55Sn 50Rh 45Zr 40Br 35Zn 30Mn 25Ca 20P 15LαKαKβL seriesK seriesFrequency (Hz)1016Moseley Plot of Characteristic X-Raysn=5n=4n=3n=2n=1KLMNLγLβLαMαMβKβKαKγKδFigure by MIT OCW.Figure by MIT OCW.hυLα = 13.6eV (Z - 7.4)2122132hυKα = 13.6eV (Z - 1)2112122=3413.6 (Z -1)2 eV3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)How do we generate X-rays?Intensity IWavelength λλminIntensity IWavelength λλminIncrease of acceleratingvoltageIncrease of heating voltageX Ray Emission Spectrum.02 .04 .06 .08 .10 .12Relative Intensity0123Wavelength (nm)BrehmsstrahlungcontinuumX-rays from a molybdenumtarget at 35 kVCharacteristic x-raysKβKαFigure by MIT OCW.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)The Laue experimentX-ray photograph of zinc blende from the Laue experiment removed for copyright reasons. See http://capsicum.me.utexas.edu/ChE386K/html/laue_experiment.htm.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)How does a crystal diffract ?Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Diffraction (wave-like instead of particle-like)Figure by MIT OCW.Source: WikipediaFigure by MIT OCW.Diffraction from a gratingGrooveIncident 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.Reciprocal lattice (I)• Let’s start with a Bravais lattice, defined in terms of its primitive lattice vectors…()123, , integer numbers,,Rl m nlmnRlmnaaa=+ +=rrrrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)1ar3arar2Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Reciprocal lattice (II)• …and then let’s take a plane wave( ) exp[ ( )]rA ikrΨ=⋅rrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Reciprocal lattice (III)• What are the wavevectors for which our plane wave has the same amplitude at all lattice points ?3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Reciprocal lattice (IV)() () ()()12 323 3112222123123 123 1232 n integer is satisfied bywith , , integers,provided , , are the reci procal-lattice vector aa aaaabb baaa aaa aaakR nGh i j hijGhbb bijππππ×××===⋅× ⋅× ⋅×⋅==++=rr rrrrrr rrrr rrr rrrrr rrrrrs3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Examples of reciprocal lattices()2311232aabaaaπ×=⋅×rrrrrrDirect lattice Reciprocal latticeSimple cubic Simple cubicFCC BCCBCC FCCOrthorhombic Orthorhombic3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)First Laue condition()()0cos cosnxABCD a nααλ−= − =BACDαoαnaFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)First Laue condition (vector form)()()0000coscoscos cosnnxaS aaS aaaSSnααααλ⋅=⋅=−=⋅−=rrrrrrrFigure by MIT OCW.S0a.S0a.SaS3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Second Laue conditionFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Third Laue condition3.012 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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