3.012 Fund of Mat Sci: Structure Lecture 18
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1 3.012 Fund of Mat Sci: Structure Lecture 18 X-RAYS AT WORK An X-ray diffraction image for the protein myoglobin. Source: Wikipedia. Model of helical domains in myoglobin. Image courtesy of Magnus Manske
2 Homework for Wed Nov 23 Prof Wuensch Lecture Notes for many details (Lect 19 onwards, but note different 2π convention) Buy turkey
3 Last time: 1. X-rays generation: undulators and wigglers in synchrotrons, bremsstrahlung and core excitations (e.g. K α ) in X-ray tubes 2. Reciprocal lattice 3. Diffraction gratings Huygens construction 4. Laue diffraction from periodic arrays in 1-d, 2-d, 3-d (Left-pointing 1st, 2nd, and 3rd order, and all higher order beams not shown.) 0th Order 1st Order 2nd Order 3rd Order Groove Figure by MIT OCW. Incident Plane Wave (Lambda = 2/11 * Grating Pitch) Diffraction Grating
4 Reciprocal lattice (IV) r r r r G = hb + ib + jb with h, i, j integers, r r r r r r r a 2 a 3 r a 3 a 1 r a 1 a 2 b 1 = 2π r r r b 2 = 2π r r r b 3 = 2π r r r a 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 a 3 r G= ( h i j) ( ) ( ) ( ),, are the reciprocal-lattice vectors d is distance between two planes of Miller indices h k l hkl in the reciprocal lattice = 2π d hkl
5 First and second Laue conditions S o a a. S o r r a S = acosαn r r a S0 = acosα0 r r r a α α = a S S = n λ ( cos ) ( ) n cos 0 0 x a. S S Figure by MIT OCW. a 2 a 1 Figure by MIT OCW.
6 All three Laue conditions r r r a ( ) 1 S S0 = r r r a ( ) 2 S S0 = r r r a S S = ( ) 3 0 integer multiple of λ integer multiple of λ integer multiple of λ
7 Ewald construction Diffracted beam hkl hkl P Incident beam 2π/λ d* 2π hkl S λ d* hkl = S-S 0 2π λ θ θ C 2π S λ 0 Crystal at the center of the sphere θ O Origin of the reciprocal lattice Reflecting sphere Figure by MIT OCW.
8 Laue condition needs white spectrum 201 Reflected Beam Incident Beam 200 Reflected Beam π /λ max 2π /λ min Reflected Beam Reflecting Sphere for Smallest Wavelength Reflecting Sphere for Largest Wavelength Figure by MIT OCW.
9 Alternate geometrical view O Figure by MIT OCW.
10 Bragg Law P O λ P' O' P'' (hkl) (hkl) (hkl) θ A B E θ d C D θ d d Figure by MIT OCW. nλ = d hkl 2sin θ
11 ( Equivalence to Laue condition Diffracted Beam 2π S/λ ν * d hkl (= S-S 0 λ 2π 2π/λ θ Incident θ ν Beam θ 2π S 0 /λ (hkl) Reciprocallattice Origin Ewald Sphere Figure by MIT OCW. r r S S π π cosν d π π = = hkl = = 2sinθ λ λ dhkl λ
12 Powder diffraction (I) 201 Reflected Beam Incident Beam 200 Reflected Beam 201 Reflected Beam 2π /λ max π /λ min Reflecting Sphere for Smallest Wavelength Reflecting Sphere for Largest Wavelength Figure by MIT OCW.
13 Powder diffraction (II) Image removed for copyright reasons. Please see the diagram at
14 Powder diffraction (III) Diagrams of the Powder Method removed for copyright reasons. See the images at
15 X-ray filters Image removed for copyright reasons. Please see the graph at Image removed for copyright reasons. Please see the diagrams at
16 Debye-Scherrer camera Photographs of a Debye-Scherrer camera removed for copyright reasons. Figure by MIT OCW.
17 Interplanar spacings Cubic: d 2 hkl = a h + k + l
18 Debye-Scherrer camera λ = d hkl 2sinθ 2 Cubic: dhkl = 2 a h + k + l Figure by MIT OCW.
19 Tables removed for copyright reasons. See
20 Systematic absences Image removed for copyright reasons. Please see the table at
21 Effects of symmetry on diffraction Images removed for copyright reasons. Please see the images at
22 Structure Factor Image removed for copyright reasons. Please see the graph at
23 Friedel s law The diffraction pattern is always centrosymmetric, even if the crystal is not centrosymmetric
24 Point symmetry + inversion = Laue Image removed for copyright reasons. Please see the table at
25 Back-reflection and transmission Laue Diagrams of the Laue Method removed for copyright reasons. See the images at
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