FROM DIFFRACTION TO STRUCTURE
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1 3.012 Fund of Mat Sci: Structure Lecture 19 FROM DIFFRACTION TO STRUCTURE Images removed for copyright reasons. 3-fold symmetry in silicon along the [111] direction. Forward (left) and backward (right) scattering.
2 Homework for Mon Nov 28 Study: 3.3 Allen-Thomas (Symmetry constraints) Read all of Chapter 1 Allen-Thomas
3 Last time: 1. Laue condition in 3 dimensions 2. Ewald construction 3. Bragg law, and equivalence to Laue condition 4. Powder diffraction, Debye-Scherrer
4 Atoms as spherical scatterers Images removed for copyright reasons. Figure by MIT OCW.
5 Huygens construction (Left-pointing 1st, 2nd, and 3rd order, and all higher order beams not shown.) 0th Order 1st Order 2nd Order 3rd Order Groove Incident Plane Wave (Lambda = 2/11 * Grating Pitch) Diffraction Grating 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 S 0 a. S 0 a. S a S a r 2 a r 1 Figure by MIT OCW. Figure by MIT OCW.
7 ( Ewald construction 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.
8 ( Equivalence to Laue condition nλ = d hkl 2sinθ ( ) 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 r r S S π π cosν d π π = = hkl = = 2sinθ λ λ dhkl λ Figure by MIT OCW.
9 Laue condition needs white spectrum 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.
10 Powder diffraction Image removed for copyright reasons. Please see the diagram at
11 Interplanar spacings 2 a 2π Cubic: d 2 = d = hkl hkl h + k + l dhkl
12 Debye-Scherrer camera nλ = d hkl 2sinθ 2 Cubic: dhkl = 2 a h + k + l Figure by MIT OCW sinθ h + k + l = a nλ 2
13 Tables removed for copyright reasons. See
14 Systematic absences Image removed for copyright reasons. Please see the table at
15 Effects of symmetry on diffraction Images removed for copyright reasons. Please see the images at
16 Structure Factor Images removed for copyright reasons. Please see the graph at
17 Friedel s law The diffraction pattern is always centrosymmetric, even if the crystal is not centrosymmetric
18 Point symmetry + inversion = Laue Image removed for copyright reasons. Please see the table at
19 X-ray powder diffraction images removed for copyright reasons. X-ray powder diffraction for silver, aluminum, gold, and copper
20 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 for these lines can yield very accurate values for the crystal unit cell size.
21 Rocksalt X-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.
22 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)
23 Scalar, vector, tensor properties Mass (0), polarization (1), strain (2) Z Deformed Undeformed X P P' u Y Z u = R' - R P P' o R R' Y X Figure by MIT OCW.
24 Transformation of a vector
25 Orthogonal Matrices
26 Transformation of a tensor
27 Neumann s principle the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal
28 Tensor properties of materials
29 Symmetry constraints
30 Curie s Principle a crystal under an external influence will exhibit only those symmetry elements that are common to the crystal without the influence and the influence without the crystal
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