International Tables for Crystallography (2019). Vol. H. ch. 1.1, pp. 2-23
https://doi.org/10.1107/97809553602060000935

Chapter 1.1. Overview and principles of powder diffraction

Contents

  • 1.1. Overview and principles of powder diffraction  (pp. 2-23) | html | pdf | chapter contents |
    R. E. Dinnebier and S. J. L. Billinge
    • 1.1.1. Information content of a powder pattern  (p. 2) | html | pdf |
    • 1.1.2. The peak position  (pp. 2-9) | html | pdf |
      • 1.1.2.1. The Bragg equation derived  (pp. 2-4) | html | pdf |
      • 1.1.2.2. The Bragg equation from the reciprocal lattice  (pp. 4-6) | html | pdf |
      • 1.1.2.3. The Bragg equation from the Laue equation  (pp. 6-7) | html | pdf |
      • 1.1.2.4. The Ewald construction and Debye–Scherrer cones  (pp. 7-9) | html | pdf |
    • 1.1.3. The peak intensity  (pp. 9-11) | html | pdf |
      • 1.1.3.1. Adding phase-shifted amplitudes  (pp. 9-11) | html | pdf |
    • 1.1.4. The peak profile  (pp. 11-15) | html | pdf |
      • 1.1.4.1. Sample contributions to the peak profile  (pp. 13-15) | html | pdf |
        • 1.1.4.1.1. Crystallite size  (pp. 13-14) | html | pdf |
        • 1.1.4.1.2. Microstrain  (pp. 14-15) | html | pdf |
    • 1.1.5. The background  (pp. 15-19) | html | pdf |
      • 1.1.5.1. Information content in the background  (p. 15) | html | pdf |
      • 1.1.5.2. Background from extraneous sources  (pp. 15-16) | html | pdf |
      • 1.1.5.3. Sources of background from the sample  (pp. 16-19) | html | pdf |
        • 1.1.5.3.1. Elastic coherent diffuse scattering  (p. 16) | html | pdf |
        • 1.1.5.3.2. Total-scattering and atomic pair distribution function analysis  (pp. 16-18) | html | pdf |
        • 1.1.5.3.3. Inelastic coherent diffuse scattering  (pp. 18-19) | html | pdf |
        • 1.1.5.3.4. Incoherent scattering  (p. 19) | html | pdf |
    • 1.1.6. Local and global optimization of crystal structures from powder diffraction data  (pp. 19-22) | html | pdf |
      • 1.1.6.1. Rietveld refinement  (pp. 19-22) | html | pdf |
      • 1.1.6.2. Local structure refinement  (p. 22) | html | pdf |
      • 1.1.6.3. Parametric Rietveld refinement  (p. 22) | html | pdf |
    • 1.1.7. Outlook  (pp. 22-23) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 1.1.1. Schematic picture of the information content of a powder pattern  (p. 2) | html | pdf |
      • Fig. 1.1.2. Schematic drawing of a set of parallel lattice planes (111) passing through all points of the cubic lattice  (p. 3) | html | pdf |
      • Fig. 1.1.3. Illustration of the geometry used for the simplified derivation of Bragg's law  (p. 3) | html | pdf |
      • Fig. 1.1.4. Illustration of the geometry in the general case where scattering takes place at the position of atoms in consecutive planes  (p. 3) | html | pdf |
      • Fig. 1.1.5. A two-dimensional monoclinic lattice and its corresponding reciprocal lattice  (p. 4) | html | pdf |
      • Fig. 1.1.6. Illustration of the important wave and scattering vectors in the case of elastic Bragg scattering  (p. 5) | html | pdf |
      • Fig. 1.1.7. Geometrical description of a lattice plane in terms of real-space basis vectors  (p. 5) | html | pdf |
      • Fig. 1.1.8. Scattering from an object consisting of two scatterers separated by r  (p. 6) | html | pdf |
      • Fig. 1.1.9. Simplified representation of the Ewald-sphere construction as a circle in two dimensions  (p. 7) | html | pdf |
      • Fig. 1.1.10. Illustration of the reciprocal lattice associated with a single-crystal lattice (left) and a large number of randomly oriented crystallites (right)  (p. 7) | html | pdf |
      • Fig. 1.1.11. Simplified representation of the Ewald-sphere construction as a circle in two dimensions  (p. 8) | html | pdf |
      • Fig. 1.1.12. Comparison between the scattered beams originating from a single crystal (top) and a powder (bottom)  (p. 8) | html | pdf |
      • Fig. 1.1.13. Left: Debye–Scherrer rings from an ideal fine-grained powder sample of a protein (courtesy Bob Von Dreele)  (p. 9) | html | pdf |
      • Fig. 1.1.14. Graphical illustration of the phase shift between two sine waves of equal amplitude  (p. 9) | html | pdf |
      • Fig. 1.1.15. The position vector of the jth atom rj can be decomposed into a vector Rs from the origin of the crystal to the origin of the unit cell containing the jth atom, and the vector ut from the unit cell origin to the jth atom  (p. 10) | html | pdf |
      • Fig. 1.1.16. Graphical illustration of the summation of scattered wave amplitudes ft in the complex plane, accounting for the phase shifts coming from the different positions of the atoms in the unit cell  (p. 11) | html | pdf |
      • Fig. 1.1.17. Schematic illustration of the projection of the reciprocal a*c* plane (representing the three-dimensional reciprocal-lattice space) into the one-dimensional powder pattern  (p. 11) | html | pdf |
      • Fig. 1.1.18. Normalized peak-shape functions  (p. 12) | html | pdf |
      • Fig. 1.1.19. Peak fits of three selected reflections for an LaB6 standard measured with Mo Kα1 radiation (λ = 0.7093 Å) from a Ge(220) monochromator in Debye–Scherrer geometry using the fundamental-parameter approach  (p. 13) | html | pdf |
      • Fig. 1.1.20. Normalized intensity from a finite lattice with n = 3 (solid curve) and n = 8 (dashed line), demonstrating the sharpening of peaks with increasing number of unit cells n  (p. 14) | html | pdf |
      • Fig. 1.1.21. Path-length difference of the scattered ray versus the depth of the lattice plane in the crystal  (p. 14) | html | pdf |
      • Fig. 1.1.22. Comparison of raw data and the normalized reduced total-scattering structure function F(Q) = Q[S(Q) − 1]  (p. 17) | html | pdf |
      • Fig. 1.1.23. PDFs in the form of G(r) from bulk CdSe and from a series of CdSe nanoparticles  (p. 17) | html | pdf |
      • Fig. 1.1.24. Spectrum from an energy-resolving detector that shows the elastic and Compton signals as a function of scattering vector Q  (p. 19) | html | pdf |
      • Fig. 1.1.25. Flow diagram of a simulated-annealing procedure used for structure determination from powder diffraction data (from Mittemeijer & Welzel, 2012)  (p. 20) | html | pdf |
      • Fig. 1.1.26. χ2 (cost function) and `temperature' dependence of the number of moves during a simulated-annealing run  (p. 21) | html | pdf |
      • Fig. 1.1.27. Screen shot (TOPAS 4.1; Bruker-AXS, 2007) of a simulated-annealing run on Pb3O4 measured with a D8 advance diffractometer in Bragg–Brentano geometry  (p. 21) | html | pdf |
    • Tables
      • Table 1.1.1. Types of scattering from a sample  (p. 15) | html | pdf |