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III. Analytical Aspects

Hand-Outs: 13. III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2. Single Crystal Diffraction CuAu. (hk0). (hk1). Hand-Outs: 13. III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2.

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III. Analytical Aspects

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  1. Hand-Outs: 13 III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2 Single Crystal Diffraction CuAu (hk0) (hk1)

  2. Hand-Outs: 13 III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2 Powder Diffraction CuAu Single Crystal Diffraction CuAu (hk0) (hk1) • One-dimensional • Difficult to “solve” structures • Fingerprinting; phase analysis, accurate lattice constants

  3. Hand-Outs: 13 III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2 Powder Diffraction CuAu • One-dimensional • Difficult to “solve” structures • Fingerprinting; phase analysis, accurate lattice constants • Detection: • Film – intersect cones to produce “lines”; • Debye-Scherrer; Guinier; • (2) PSDs – X-ray phosphors • (3) Time-of-Flight – detectors at specific angles; • continuous kinetic energy; variable  (NPD)

  4. Hand-Outs: 13 III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2 Structure Solution by Powder Diffraction CuAu (Cu K) Rietveld Refinement (1) Fit peak profiles (peak shapes vs. 2) (2) Fit peak intensities (integrated – structure) NPD: Peaks are Gaussian shape; XPD: Peaks are combination of Gaussian and Lorentzian (“pseudo-Voigt” functions)

  5. Hand-Outs: 14 III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2 Each crystal grain gives a diffraction pattern Ihkl(n) = |Fhkl(n)|2

  6. Hand-Outs: 14 III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2 Several grains gives a superposition Ihkl = n|Fhkl(n)|2 (n|Fhkl(n)|)2 Each crystal grain gives a diffraction pattern Ihkl(n) = |Fhkl(n)|2

  7. Hand-Outs: 14 III. Analytical Aspects Diffraction: Powder Diffraction Cheetham & Day, Chapter 2 Several grains gives a superposition Ihkl = n|Fhkl(n)|2 (n|Fhkl(n)|)2 Each crystal grain gives a diffraction pattern Ihkl(n) = |Fhkl(n)|2 40 Grains 200 Grains ITOT = nIn  = 2dhkl sin  Preferred Orientation: e.g., MoS2

  8. Hand-Outs: 15 III. Analytical Aspects Diffraction: Multiplicities (Symmetry) Cheetham & Day, Chapter 2 Laue: 4/m 4001(hkl) = (khl) m001(hkl) = (hkl) Multiplicity = 8

  9. Hand-Outs: 15 III. Analytical Aspects Diffraction: Multiplicities (Symmetry) Cheetham & Day, Chapter 2 Laue: 4/mmm 4001(hkl) = (khl) m001(hkl) = (hkl) m100(hkl) = (hkl) m110(hkl) = (khl) Multiplicity = 16

  10. Hand-Outs: 16 III. Analytical Aspects Diffraction: d-Spacings Cheetham & Day, Chapter 2  = 2 dhkl sin  1/dhkl2 Expressions (= 4 sin2  / ) d330 = d411

  11. Hand-Outs: 16 III. Analytical Aspects Diffraction: d-Spacings Cheetham & Day, Chapter 2 CuAu: P4/mmma = 3.966 Å, c = 3.673 Å {111} = (111), (111), (111), (111), + inverses  = 2 dhkl sin  {020} (001) {110} (002) {021} {112} {022} {131}

  12. Hand-Outs: 16 III. Analytical Aspects Diffraction: d-Spacings Cheetham & Day, Chapter 2  = 2 dhkl sin  CuAu Fm3m a = 3.868 Å {111} h+k, h+l, k+l = even # {020} Longer axis (d002) in real space; Smaller scattering angle (2) {111} CuAu P4/mmm a = 3.966 Å c = 3.673 Å {020} (001) {110}

  13. Hand-Outs: 17 III. Analytical Aspects Diffraction: Diffuse Scattering (Background from “disorder” in the sample) • Types of Disorder: • Thermal • Occupational • Displacement • Stacking Faults Fhkl= Fh = fe2ihu + (ff)e2ihu + fe2ih(uu) + (ff)e2ih(uu) Bragg Peaks Occupational Disorder Thermal/Displacement Disorder Combination

  14. Hand-Outs: 17 III. Analytical Aspects Diffraction: Diffuse Scattering (Background from “disorder” in the sample) • Types of Disorder: • Thermal • Occupational • Displacement • Stacking Faults Fhkl= Fh = fe2ihu + (ff)e2ihu + fe2ih(uu) + (ff)e2ih(uu) Bragg Peaks Occupational Disorder Thermal/Displacement Disorder Combination Thermal/Displacement Disorder (large h diffuse intensity) ff = 0 uu small; significant phase effects for large h; u: nonrandom – satellites or systematic extinctions.

  15. Hand-Outs: 17 III. Analytical Aspects Diffraction: Diffuse Scattering (Background from “disorder” in the sample) • Types of Disorder: • Thermal • Occupational • Displacement • Stacking Faults Fhkl= Fh = fe2ihu + (ff)e2ihu + fe2ih(uu) + (ff)e2ih(uu) Bragg Peaks Occupational Disorder Thermal/Displacement Disorder Combination Thermal/Displacement Disorder (large h diffuse intensity) Occupational Disorder (small h diffuse intensity) uu= 0 fflargest near small h;

  16. Hand-Outs: 17 III. Analytical Aspects Diffraction: Diffuse Scattering (Background from “disorder” in the sample) • Types of Disorder: • Thermal • Occupational • Displacement • Stacking Faults Fhkl= Fh = fe2ihu + (ff)e2ihu + fe2ih(uu) + (ff)e2ih(uu) Bragg Peaks Occupational Disorder Thermal/Displacement Disorder Combination Stacking Faults (specific streaking of diffuse scattering) CCP = “ABC” “Disordered” HCP = “AB” Disordered, Streaking Bragg Peaks only; Streaking from finite size effects Bragg Peaks only; Streaking from finite size effects Ordered, CP Planes

  17. Hand-Outs: 18 III. Analytical Aspects Diffraction: Modulated Structures http://cryst.iphy.ac.cn/VEC/Tutorials/mD_direct_methods/what's.html References De Wolff, Acta Cryst. A30, 777-785 (1974) Van Smaalen, Crystallogr. Rev. 4, 79-202 (1995)

  18. c* 16 16 9 9 Hand-Outs: 19 III. Analytical Aspects Diffraction: Modulated Structures http://cryst.iphy.ac.cn/VEC/Tutorials/mD_direct_methods/what's.html 9 16 c Reciprocal Space Sr9/8TiS3 (2 subsystems with 2 different periodicities) Real Space

  19. Hand-Outs: 20 III. Analytical Aspects Diffraction: Pair Distribution Functions (PDFs) Bragg Peaks: Constructive Interference; long-range, average structure (e.g. Rietveld) Diffuse Scattering: deviations in average structure – short-range structure Pair-Distribution Function (PDF) = Fourier transform of normalized scattering intensity, S(q) Experimental: Pair Density Average # Density Theoretical: bi = scattering length/factor for atom i rij = separation between atoms i and j G(r), Gcalc(r) = “Bond-length” distribution between all pairs of atoms up to some maximum distance, weighted by scattering power of two atoms Need to measure up to q = ; terminated at qmax  termination ripples

  20. Hand-Outs: 20 III. Analytical Aspects Diffraction: Pair Distribution Functions (PDFs) http://www.pa.msu.edu/cmp/billinge-group Structure function Raw data PDF

  21. Hand-Outs: 21 III. Analytical Aspects Diffraction: Pair Distribution Functions (PDFs) http://www.pa.msu.edu/cmp/billinge-group (a) A C60 molecule. (b) C60(s) (c) The scattering result (d) PDF Sit on a C atom and look at neighborhood Nearest neighbor distance = 1.4 Å; 2nd n.n. distance = 2.2 Å, and so on. G(r) has sharp peaks at these positions. (Structural information in the PDF) There are no sharp peaks beyond 7.1 Å, the diameter of the ball, because the balls are spinning with respect to each other.

  22. Hand-Outs: 22 III. Analytical Aspects Diffraction: Pair Distribution Functions (PDFs) http://www.pa.msu.edu/cmp/billinge-group LaMnO3 PDF Red line: from structure model Blue line: from experimental data Green line: difference Proffen et al., Phys. Rev. B60, 9973 (1999).

  23. r1 << x r1 r2 r2 ~ x/2 Intra-domain structure Inter-domain structure Hand-Outs: 22 III. Analytical Aspects Diffraction: Pair Distribution Functions (PDFs) http://www.pa.msu.edu/cmp/billinge-group Observing Domains in the PDF Nice (Brief) Review: S.J.L. Billinge, Z. Kristallogr. 219, 117-121 (2004) Future Developments: Using PDFs to solve “nanoscale” structure problems…

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