International Tables for Crystallography
Volume H: Powder diffraction

First online edition (2019)   ISBN: 978-1-118-41628-0   doi: 10.1107/97809553602060000115  

Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

Contents

• Preface  (p. xxviii) | html | pdf |
  C. J. Gilmore, J. A. Kaduk and H. Schenk

• Introduction
  • 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 r_j can be decomposed into a vector R_s from the origin of the crystal to the origin of the unit cell containing the jth atom, and the vector u_t 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 f_t 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 |

• Instrumentation and sample preparation
  • 2.1. Instrumentation for laboratory X-ray scattering techniques  (pp. 26-50) | html | pdf | chapter contents |
    A. Kern
    • 2.1.1. Introduction  (p. 26) | html | pdf |
    • 2.1.2. Scope and terminology  (p. 26) | html | pdf |
    • 2.1.3. Historical overview  (pp. 26-28) | html | pdf |
      • 2.1.3.1. From film cameras to diffractometers  (pp. 26-28) | html | pdf |
        • 2.1.3.1.1. Film cameras  (pp. 26-27) | html | pdf |
        • 2.1.3.1.2. Diffractometers  (pp. 27-28) | html | pdf |
      • 2.1.3.2. Recent years  (p. 28) | html | pdf |
    • 2.1.4. The platform concept – fitting the instrument to the need  (pp. 28-32) | html | pdf |
      • 2.1.4.1. Basic design principles and instrument geometry considerations  (pp. 28-30) | html | pdf |
      • 2.1.4.2. Range of hardware  (pp. 30-31) | html | pdf |
      • 2.1.4.3. Range of applications  (pp. 31-32) | html | pdf |
    • 2.1.5. Goniometer designs  (pp. 32-36) | html | pdf |
      • 2.1.5.1. Geometrical conventions and scan modes  (pp. 32-34) | html | pdf |
        • 2.1.5.1.1. Goniometer base  (pp. 32-34) | html | pdf |
        • 2.1.5.1.2. Specimen stage  (p. 34) | html | pdf |
      • 2.1.5.2. Accuracy and precision  (pp. 34-35) | html | pdf |
      • 2.1.5.3. Hybrid beam-path systems  (pp. 35-36) | html | pdf |
        • 2.1.5.3.1. Multiple-beam-path systems  (p. 35) | html | pdf |
        • 2.1.5.3.2. Non-coplanar beam-path systems  (pp. 35-36) | html | pdf |
    • 2.1.6. X-ray sources and optics  (pp. 36-45) | html | pdf |
      • 2.1.6.1. X-ray beam quality measures  (pp. 36-37) | html | pdf |
      • 2.1.6.2. X-ray sources  (pp. 37-39) | html | pdf |
        • 2.1.6.2.1. Generation of X-rays and the X-ray spectrum  (p. 37) | html | pdf |
        • 2.1.6.2.2. Types of X-ray sources  (pp. 37-39) | html | pdf |
          • 2.1.6.2.2.1. Fixed-target X-ray sources  (pp. 38-39) | html | pdf |
          • 2.1.6.2.2.2. Moving-target X-ray sources  (p. 39) | html | pdf |
        • 2.1.6.2.3. Performance of X-ray sources  (p. 39) | html | pdf |
      • 2.1.6.3. X-ray optics  (pp. 39-45) | html | pdf |
        • 2.1.6.3.1. Absorptive X-ray optics  (pp. 40-41) | html | pdf |
          • 2.1.6.3.1.1. Apertures  (p. 40) | html | pdf |
          • 2.1.6.3.1.2. Metal filters  (pp. 40-41) | html | pdf |
        • 2.1.6.3.2. Diffractive X-ray optics  (pp. 41-43) | html | pdf |
          • 2.1.6.3.2.1. Single-reflection monochromators  (pp. 41-42) | html | pdf |
          • 2.1.6.3.2.2. Multiple-reflection monochromators  (pp. 42-43) | html | pdf |
        • 2.1.6.3.3. Reflective X-ray optics  (pp. 43-44) | html | pdf |
          • 2.1.6.3.3.1. Multilayer mirrors  (pp. 43-44) | html | pdf |
          • 2.1.6.3.3.2. Capillaries  (p. 44) | html | pdf |
        • 2.1.6.3.4. Combi-optics  (pp. 44-45) | html | pdf |
    • 2.1.7. X-ray detectors  (pp. 45-49) | html | pdf |
      • 2.1.7.1. Detector parameters  (pp. 45-46) | html | pdf |
      • 2.1.7.2. Detector types  (pp. 46-48) | html | pdf |
        • 2.1.7.2.1. Scintillation counters  (pp. 46-47) | html | pdf |
        • 2.1.7.2.2. Gas-ionization detectors  (pp. 47-48) | html | pdf |
          • 2.1.7.2.2.1. Wire-based proportional counters  (p. 47) | html | pdf |
          • 2.1.7.2.2.2. Micro-gap detectors  (pp. 47-48) | html | pdf |
        • 2.1.7.2.3. Semiconductor detectors  (p. 48) | html | pdf |
          • 2.1.7.2.3.1. The Si(Li) detector  (p. 48) | html | pdf |
          • 2.1.7.2.3.2. Silicon micro-strip and silicon pixel detectors  (p. 48) | html | pdf |
          • 2.1.7.2.3.3. CCD and CMOS detectors  (p. 48) | html | pdf |
      • 2.1.7.3. Position sensitivity and associated scanning modes  (pp. 48-49) | html | pdf |
        • 2.1.7.3.1. Pixel size, spatial resolution and angular resolution  (p. 49) | html | pdf |
        • 2.1.7.3.2. Dimensionality  (p. 49) | html | pdf |
        • 2.1.7.3.3. Size and shape  (p. 49) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.1.1. Diffraction of X-rays by (a) a rotating single crystal and (b) an ideal powder  (p. 28) | html | pdf |
      • Fig. 2.1.2. The basic design principle of modern diffractometers  (p. 29) | html | pdf |
      • Fig. 2.1.3. Transformation between the Bragg–Brentano and Debye–Scherrer geometries using a incident-beam X-ray optical bench  (p. 29) | html | pdf |
      • Fig. 2.1.4. Bragg–Brentano geometry  (p. 30) | html | pdf |
      • Fig. 2.1.5. Bragg–Brentano geometry  (p. 30) | html | pdf |
      • Fig. 2.1.6. Laboratory coordinates and geometric definition of the coaxial goniometer axes ω and 2θ  (p. 33) | html | pdf |
      • Fig. 2.1.7. Goniometer base configurations and scan modes suitable for both Bragg–Brentano or Debye–Scherrer geometry  (p. 33) | html | pdf |
      • Fig. 2.1.8. Goniometer base configurations and scan modes suitable for the Debye–Scherrer geometry only  (p. 33) | html | pdf |
      • Fig. 2.1.9. Geometric definition of the Eulerian and kappa geometries with identical specimen orientation in space  (p. 34) | html | pdf |
      • Fig. 2.1.10. Example of counterbalancing of a vertical θ–θ goniometer  (p. 35) | html | pdf |
      • Fig. 2.1.11. Illustration of coplanar and in-plane diffraction  (p. 35) | html | pdf |
      • Fig. 2.1.12. Sophisticated IP-GID implementation by placing two goniometers vertically with respect to each other, allowing simultaneous coplanar and in-plane measurements using two independent scattered-beam optical X-ray benches (compare with Fig  (p. 36) | html | pdf |
      • Fig. 2.1.13. Illustration of the working principle of laboratory X-ray sources: (a) fixed target, (b) rotating target, (c) liquid-metal jet  (p. 38) | html | pdf |
      • Fig. 2.1.14. Apertures used for beam collimation  (p. 40) | html | pdf |
      • Fig. 2.1.15. Motorized switchable (a) and rotating (b) absorbers  (p. 41) | html | pdf |
      • Fig. 2.1.16. Illustration of flat single-reflection monochromators  (p. 41) | html | pdf |
      • Fig. 2.1.17. Illustration of curved and ground single-reflection monochromators  (p. 42) | html | pdf |
      • Fig. 2.1.18. Illustration of multiple-reflection monochromators  (p. 42) | html | pdf |
      • Fig. 2.1.19. Schematic of graded multilayer mirrors  (p. 43) | html | pdf |
      • Fig. 2.1.20. Examples for orthogonally positioned curved mirrors for beam conditioning  (p. 43) | html | pdf |
      • Fig. 2.1.21. Schematic of monocapillary optics  (p. 44) | html | pdf |
      • Fig. 2.1.22. Schematic of polycapillary optics  (p. 44) | html | pdf |
      • Fig. 2.1.23. Incident and diffracted beam combi-optics for switching between (a) the Bragg–Brentano geometry and (b) the parallel-beam geometry  (p. 45) | html | pdf |
      • Fig. 2.1.24. Example of the use of highly sophisticated incident- and diffracted-beam combi-optics in combination with a rotatable X-ray source and a double detector arm  (p. 45) | html | pdf |
    • Tables
      • Table 2.1.1. Types of beam-path components available in laboratory X-ray powder diffraction  (p. 31) | html | pdf |
      • Table 2.1.2. X-ray applications for with modern X-ray diffractometers  (p. 32) | html | pdf |
      • Table 2.1.3. Characteristic wavelengths and absorption edges of metal filters in common use  (p. 37) | html | pdf |
      • Table 2.1.4. Maximum target loading and specific loading for some selected fixed- and moving-target X-ray sources  (p. 39) | html | pdf |
      • Table 2.1.5. Comparison of divergence and intensity for several types of germanium channel-cut monochromators  (p. 43) | html | pdf |
      • Table 2.1.6. Important detector properties at 8 keV as reported by various vendors  (p. 47) | html | pdf |
  • 2.2. Synchrotron radiation and powder diffraction  (pp. 51-65) | html | pdf | chapter contents |
    A. Fitch
    • 2.2.1. Introduction  (p. 51) | html | pdf |
    • 2.2.2. Production of synchrotron radiation  (pp. 51-54) | html | pdf |
      • 2.2.2.1. Bending magnets  (pp. 52-53) | html | pdf |
      • 2.2.2.2. Insertion devices  (pp. 53-54) | html | pdf |
        • 2.2.2.2.1. Wigglers  (p. 53) | html | pdf |
        • 2.2.2.2.2. Undulators  (pp. 53-54) | html | pdf |
        • 2.2.2.2.3. Tuning  (p. 54) | html | pdf |
    • 2.2.3. Optics  (pp. 54-56) | html | pdf |
      • 2.2.3.1. Monochromator  (pp. 54-55) | html | pdf |
      • 2.2.3.2. Mirror  (pp. 55-56) | html | pdf |
      • 2.2.3.3. Compound refractive lens  (p. 56) | html | pdf |
    • 2.2.4. Diffractometers  (pp. 56-60) | html | pdf |
      • 2.2.4.1. Parallel-beam instruments  (pp. 56-58) | html | pdf |
        • 2.2.4.1.1. Angular resolution  (pp. 57-58) | html | pdf |
        • 2.2.4.1.2. Hart–Parrish design  (p. 58) | html | pdf |
      • 2.2.4.2. Debye–Scherrer instruments  (pp. 58-59) | html | pdf |
      • 2.2.4.3. Energy-dispersive instruments  (pp. 59-60) | html | pdf |
    • 2.2.5. Considerations for powder-diffraction experiments  (pp. 60-63) | html | pdf |
      • 2.2.5.1. Polarization  (pp. 60-61) | html | pdf |
      • 2.2.5.2. Radiation damage  (p. 61) | html | pdf |
      • 2.2.5.3. Beam heating  (p. 61) | html | pdf |
      • 2.2.5.4. Choice of wavelength  (pp. 61-62) | html | pdf |
      • 2.2.5.5. Angular resolution  (p. 62) | html | pdf |
      • 2.2.5.6. Spatial resolution  (p. 62) | html | pdf |
      • 2.2.5.7. Time resolution  (p. 63) | html | pdf |
        • 2.2.5.7.1. Using fast detectors  (p. 63) | html | pdf |
        • 2.2.5.7.2. Using the pulse structure  (p. 63) | html | pdf |
      • 2.2.5.8. Beamline evolution  (p. 63) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.2.1. Schematic representation of a synchrotron storage ring with beamlines radiating tangentially from the bending magnets and in line with the straight sections  (p. 51) | html | pdf |
      • Fig. 2.2.2. Emission of a fan of radiation by the electron beam as it curves in a bending magnet from one straight section of the ring to the next  (p. 52) | html | pdf |
      • Fig. 2.2.3. Synchrotron radiation is emitted in a cone of opening angle of the order of 1/γ tangential to the electrons as they follow a curved trajectory through the bending magnet  (p. 52) | html | pdf |
      • Fig. 2.2.4. Spectrum of a bending magnet (B = 0.85 T) at the ESRF with an electron energy of 6 GeV (γ = 11 742), shown as flux per horizontal mrad for a 0.1% energy bandwidth at a storage-ring current of 200 mA  (p. 53) | html | pdf |
      • Fig. 2.2.5. Schematic illustration of a wiggler (upper) and an undulator (lower)  (p. 53) | html | pdf |
      • Fig. 2.2.6. Photon flux versus energy through a 1-mm² aperture 30 m from the source, 0.1% bandwidth, for an ESRF u35 undulator (magnetic periodicity 35 mm, 1.6 m long, magnetic gap of 11 mm, peak magnetic field B₀ = 0.71 T, electron energy 6 GeV, K = 2.31, storage-ring current 200 mA)  (p. 54) | html | pdf |
      • Fig. 2.2.7. Double-crystal monochromator arrangement  (p. 54) | html | pdf |
      • Fig. 2.2.8. Curved mirror set to collimate the beam  (p. 56) | html | pdf |
      • Fig. 2.2.9. Schematic diagram of a set of refractive lenses  (p. 56) | html | pdf |
      • Fig. 2.2.10. Multianalyser stage, nine channels separated by 2°, devised by Hodeau et al  (p. 57) | html | pdf |
      • Fig. 2.2.11. Δ(2θ) calculated from equation (2.2.2) for a beamline with a double-crystal Si(111) monochromator, an Si(111) analyser (Δ_m = Δ_a and θ_m = θ_a) and an FWHM vertical divergence of 25 µrad at λ = 0.4 Å (solid line: Δ_m ≃ 8.3 µrad, θ_m = 3.6571°), λ = 0.8 Å (dashed line: Δ_m ≃ 16.6 µrad, θ_m = 7.3292°) and λ = 1.2 Å (dotted line: Δ_m ≃ 25.2 µrad, θ_m = 11.0319°)  (p. 58) | html | pdf |
      • Fig. 2.2.12. Schematic representation of a parallel-beam diffractometer of the Hart–Parrish design  (p. 58) | html | pdf |
      • Fig. 2.2.13. (a) 120° Mythen detector box, containing helium, mounted on the powder diffractometer of the materials science beamline at the Swiss Light Source  (p. 59) | html | pdf |
      • Fig. 2.2.14. Schematic representation of an energy-dispersive diffraction arrangement  (p. 59) | html | pdf |
      • Fig. 2.2.15. Variation of Δf′ and Δf′′ with photon energy for Sn (solid line) and Sb (dotted line) in the vicinity of their K absorption edges (from the tables of Sasaki, 1989)  (p. 62) | html | pdf |
  • 2.3. Neutron powder diffraction  (pp. 66-101) | html | pdf | chapter contents |
    C. J. Howard and E. H. Kisi
    • 2.3.1. Introduction to the diffraction of thermal neutrons  (pp. 66-67) | html | pdf |
    • 2.3.2. Neutrons and neutron diffraction – pertinent details  (pp. 67-70) | html | pdf |
      • 2.3.2.1. Properties of the neutron  (p. 67) | html | pdf |
      • 2.3.2.2. Neutron scattering lengths  (pp. 67-68) | html | pdf |
      • 2.3.2.3. Refractive index for neutrons  (pp. 68-69) | html | pdf |
      • 2.3.2.4. Neutron attenuation  (p. 69) | html | pdf |
      • 2.3.2.5. Magnetic form factors and magnetic scattering lengths  (pp. 69-70) | html | pdf |
      • 2.3.2.6. Structure factors  (p. 70) | html | pdf |
    • 2.3.3. Neutron sources  (pp. 70-80) | html | pdf |
      • 2.3.3.1. The earliest neutron sources  (pp. 70-72) | html | pdf |
      • 2.3.3.2. Fission reactors for neutron-beam research  (pp. 72-73) | html | pdf |
      • 2.3.3.3. Spallation neutron sources  (pp. 73-77) | html | pdf |
      • 2.3.3.4. Neutron beam tubes and guides  (pp. 77-80) | html | pdf |
    • 2.3.4. Diffractometers  (pp. 80-92) | html | pdf |
      • 2.3.4.1. Constant-wavelength neutron diffractometers  (pp. 80-84) | html | pdf |
        • 2.3.4.1.1. Collimation  (pp. 80-81) | html | pdf |
        • 2.3.4.1.2. Monochromators  (pp. 81-82) | html | pdf |
        • 2.3.4.1.3. Neutron detectors  (pp. 82-84) | html | pdf |
        • 2.3.4.1.4. Resolution and intensity  (p. 84) | html | pdf |
      • 2.3.4.2. Time-of-flight (TOF) diffractometers  (pp. 84-88) | html | pdf |
        • 2.3.4.2.1. Instrument resolution and design  (p. 86) | html | pdf |
        • 2.3.4.2.2. Detection  (pp. 86-88) | html | pdf |
      • 2.3.4.3. Variations on a theme  (pp. 88-91) | html | pdf |
      • 2.3.4.4. Comparison of CW and TOF diffractometers  (pp. 91-92) | html | pdf |
    • 2.3.5. Experimental considerations  (pp. 92-98) | html | pdf |
      • 2.3.5.1. Preliminary considerations  (pp. 92-94) | html | pdf |
      • 2.3.5.2. Sample-related factors  (pp. 94-96) | html | pdf |
      • 2.3.5.3. Sample environment and in situ experiments  (pp. 96-98) | html | pdf |
        • 2.3.5.3.1. Sample containers  (p. 96) | html | pdf |
        • 2.3.5.3.2. Non-ambient temperature  (pp. 96-98) | html | pdf |
        • 2.3.5.3.3. Uniaxial stress  (p. 98) | html | pdf |
    • 2.3.6. Concluding remarks  (p. 98) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.3.1. Representations of the scattering of X-rays and neutrons by selected elements  (p. 66) | html | pdf |
      • Fig. 2.3.2. Comparison of X-ray and neutron powder-diffraction patterns from rutile, TiO₂  (p. 67) | html | pdf |
      • Fig. 2.3.3. The magnetic form factor for Mn²⁺ compared with the normalized X-ray form factor and the normalized neutron nuclear scattering length  (p. 67) | html | pdf |
      • Fig. 2.3.4. Magnetic structure for MnO proposed by Shull et al  (p. 68) | html | pdf |
      • Fig. 2.3.5. The Maxwellian distribution of neutron wavelengths produced within moderators at different temperatures  (p. 72) | html | pdf |
      • Fig. 2.3.6. The NBSR at the National Institute of Standards and Technology Center for Neutron Research  (p. 73) | html | pdf |
      • Fig. 2.3.7. Schematic diagram of the HFR operated by the Institut Laue–Langevin in Grenoble, France  (p. 73) | html | pdf |
      • Fig. 2.3.8. (a) The neutron energy distribution (flux) of the J-PARC neutron source for coupled, decoupled and poisoned decoupled moderators  (p. 75) | html | pdf |
      • Fig. 2.3.9. One of the disc choppers in use at the ISIS neutron facility  (p. 75) | html | pdf |
      • Fig. 2.3.10. Layout of the ISIS spallation neutron source  (p. 76) | html | pdf |
      • Fig. 2.3.11. Layout at the SINQ neutron source  (p. 78) | html | pdf |
      • Fig. 2.3.12. Schematic diagram of the ESS facility  (p. 79) | html | pdf |
      • Fig. 2.3.13. Plan of a curved neutron guide, indicating different possible neutron paths, labelled `garland' and `zig-zag'  (p. 79) | html | pdf |
      • Fig. 2.3.14. Schematic diagrams of (a) a multilayer monochromator and (b) a neutron supermirror  (p. 79) | html | pdf |
      • Fig. 2.3.15. A constant-wavelength neutron powder diffractometer  (p. 81) | html | pdf |
      • Fig. 2.3.16. Commercially available compact Soller collimators  (p. 82) | html | pdf |
      • Fig. 2.3.17. The vertically focusing monochromator constructed at the Brookhaven National Laboratory (Vogt et al  (p. 82) | html | pdf |
      • Fig. 2.3.18. (a) Schematic cross section of the POLARIS diffractometer at the ISIS facility, UK, and (b) a three-dimensional solid model of the detector chamber  (p. 85) | html | pdf |
      • Fig. 2.3.19. Raw neutron diffraction patterns from Y₃Al₅O₁₂ (YAG)  (p. 87) | html | pdf |
      • Fig. 2.3.20. Parts of the very high resolution neutron powder-diffraction patterns recorded by the backscattering detector bank on the instrument HRPD at ISIS from SrZrO₃ at (a) 1403, (b) 1153, (c) 1053, (d) 933 and (e) 293 K  (p. 89) | html | pdf |
      • Fig. 2.3.21. Neutron powder-diffraction patterns during combustion synthesis of Ti₃SiC₂ recorded in 400 ms each on the diffractometer D20 at ILL (Riley et al  (p. 90) | html | pdf |
      • Fig. 2.3.22. Illustrating (a) a CW engineering diffractometer and (b) the formation of a gauge volume at the intersection of the incident and diffracted beams  (p. 90) | html | pdf |
      • Fig. 2.3.23. Stress distribution for four stress components in an iron powder compacted within a convergent die (see also Zhang et al  (p. 91) | html | pdf |
      • Fig. 2.3.24. The engineering diffractometer ENGIN-X at ISIS  (p. 91) | html | pdf |
      • Fig. 2.3.25. Schematic showing regions of intensity–resolution space in which different diffractometer types typically operate  (p. 92) | html | pdf |
      • Fig. 2.3.26. (a) Exterior and (b) interior of the standard ILL liquid-helium cryostat for cooling samples in the range 1.8–295 K  (p. 97) | html | pdf |
      • Fig. 2.3.27. Elements of a typical mechanical testing machine used for applying uniaxial stress (pressure) to samples on an engineering neutron diffractometer  (p. 98) | html | pdf |
      • Fig. 2.3.28. (a) Cross section and (b) exterior of a self-loading die for the study of stresses in granular materials  (p. 98) | html | pdf |
    • Tables
      • Table 2.3.1. Properties of the neutron (adapted from Kisi & Howard, 2008)  (p. 68) | html | pdf |
      • Table 2.3.2. Coherent scattering lengths and absorption cross sections (for 25 meV neutrons) for selected isotopes  (p. 69) | html | pdf |
      • Table 2.3.3. Details on selected research reactors  (p. 74) | html | pdf |
      • Table 2.3.4. Details of selected spallation neutron sources  (p. 79) | html | pdf |
      • Table 2.3.5. Advantages of CW and TOF instruments (modified from Kisi & Howard, 2008)  (p. 92) | html | pdf |
      • Table 2.3.6. Suitability of problems to high-resolution or high-intensity diffractometers  (p. 93) | html | pdf |
      • Table 2.3.7. Guidance on choice of wavelength/detector bank  (p. 93) | html | pdf |
  • 2.4. Electron powder diffraction  (pp. 102-117) | html | pdf | chapter contents |
    J.-M. Zuo, J. L. Lábár, J. Zhang, T. E. Gorelik and U. Kolb
    • 2.4.1. Introduction  (pp. 102-103) | html | pdf |
    • 2.4.2. Electron powder diffraction pattern geometry and intensity  (pp. 103-104) | html | pdf |
      J.-M. Zuo and J. L. Lábár
    • 2.4.3. Electron powder diffraction techniques  (pp. 105-108) | html | pdf |
      J.-M. Zuo and J. Zhang
      • 2.4.3.1. Selected-area electron diffraction (SAED)  (p. 105) | html | pdf |
      • 2.4.3.2. Nano-area electron diffraction (NAED)  (pp. 105-106) | html | pdf |
      • 2.4.3.3. Sample preparation  (pp. 106-107) | html | pdf |
      • 2.4.3.4. Diffraction data collection, processing and calibration  (pp. 107-108) | html | pdf |
    • 2.4.4. Phase identification and phase analysis  (pp. 108-110) | html | pdf |
      J. L. Lábár
    • 2.4.5. Texture analysis  (pp. 110-111) | html | pdf |
      J. L. Lábár
    • 2.4.6. Rietveld refinement with electron diffraction data  (pp. 111-113) | html | pdf |
      T. E. Gorelik and U. Kolb
    • 2.4.7. The pair distribution function from electron diffraction data  (pp. 113-114) | html | pdf |
      T. E. Gorelik and U. Kolb
    • 2.4.8. Summary  (p. 114) | html | pdf |
    • Appendix 2.4.1. Computer programs for electron powder diffraction  (pp. 114-115) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.4.1. An electron powder diffraction pattern recorded on an imaging plate from a polycrystalline Al thin film using selected-area electron diffraction geometry with 200 kV electrons  (p. 102) | html | pdf |
      • Fig. 2.4.2. Schematic diagram of the Ewald sphere construction and the geometry for recording electron diffraction patterns  (p. 103) | html | pdf |
      • Fig. 2.4.3. Schematic illustration of selected-area electron diffraction in conventional TEM  (p. 105) | html | pdf |
      • Fig. 2.4.4. Schematic illustration of electron nanoprobe formation using a combination of condenser lenses (II and III) and the objective lens  (p. 106) | html | pdf |
      • Fig. 2.4.5. Sample preparation and lift-out using a focused ion beam (FIB)  (p. 106) | html | pdf |
      • Fig. 2.4.6. An example of electron powder diffraction recording for nanodiamonds  (p. 108) | html | pdf |
      • Fig. 2.4.7. Rietveld analysis result with powder electron diffraction data of hydroxyapatite  (p. 111) | html | pdf |
      • Fig. 2.4.8. Powder electron diffraction pattern of nanocrystalline gold demonstrating non-symmetrical background features  (p. 111) | html | pdf |
      • Fig. 2.4.9. Electron powder diffraction profiles of gold nanoparticles (range 2–6 nm⁻¹) recorded at different electron diffraction camera lengths  (p. 112) | html | pdf |
  • 2.5. Two-dimensional powder diffraction  (pp. 118-149) | html | pdf | chapter contents |
    B. B. He
    • 2.5.1. Introduction  (pp. 118-119) | html | pdf |
      • 2.5.1.1. The diffraction pattern measured by an area detector  (p. 118) | html | pdf |
      • 2.5.1.2. Comparison between 2D-XRD and conventional XRD  (p. 118) | html | pdf |
      • 2.5.1.3. Advantages of two-dimensional X-ray diffraction  (pp. 118-119) | html | pdf |
    • 2.5.2. Fundamentals  (pp. 119-123) | html | pdf |
      • 2.5.2.1. Diffraction space and laboratory coordinates  (pp. 119-121) | html | pdf |
        • 2.5.2.1.1. Diffraction cones in laboratory coordinates  (pp. 119-120) | html | pdf |
        • 2.5.2.1.2. Diffraction-vector cones in laboratory coordinates  (pp. 120-121) | html | pdf |
      • 2.5.2.2. Detector space and pixel position  (pp. 121-122) | html | pdf |
        • 2.5.2.2.1. Detector position in the laboratory system  (p. 121) | html | pdf |
        • 2.5.2.2.2. Pixel position in diffraction space for a flat detector  (pp. 121-122) | html | pdf |
        • 2.5.2.2.3. Pixel position in diffraction space for a curved detector  (p. 122) | html | pdf |
      • 2.5.2.3. Sample space and goniometer geometry  (pp. 122-123) | html | pdf |
        • 2.5.2.3.1. Sample rotations and translations in Eulerian geometry  (pp. 122-123) | html | pdf |
      • 2.5.2.4. Diffraction-vector transformation  (p. 123) | html | pdf |
        • 2.5.2.4.1. Diffraction unit vector in diffraction space and sample space  (p. 123) | html | pdf |
        • 2.5.2.4.2. Transformation from diffraction space to sample space  (p. 123) | html | pdf |
        • 2.5.2.4.3. Transformation from detector space to reciprocal space  (p. 123) | html | pdf |
    • 2.5.3. Instrumentation  (pp. 123-132) | html | pdf |
      • 2.5.3.1. X-ray source and optics  (pp. 124-125) | html | pdf |
        • 2.5.3.1.1. Beam path in a diffractometer equipped with a 2D detector  (p. 124) | html | pdf |
        • 2.5.3.1.2. Liouville's theorem  (pp. 124-125) | html | pdf |
        • 2.5.3.1.3. X-ray source  (p. 125) | html | pdf |
        • 2.5.3.1.4. X-ray optics  (p. 125) | html | pdf |
      • 2.5.3.2. 2D detector  (pp. 125-128) | html | pdf |
        • 2.5.3.2.1. Active area and pixel size  (pp. 126-127) | html | pdf |
        • 2.5.3.2.2. Spatial resolution of area detectors  (p. 127) | html | pdf |
        • 2.5.3.2.3. Detective quantum efficiency and energy range  (pp. 127-128) | html | pdf |
        • 2.5.3.2.4. Detection limit and dynamic range  (p. 128) | html | pdf |
        • 2.5.3.2.5. Types of 2D detectors  (p. 128) | html | pdf |
      • 2.5.3.3. Data corrections and integration  (pp. 128-132) | html | pdf |
        • 2.5.3.3.1. Nonuniform response correction  (pp. 128-129) | html | pdf |
        • 2.5.3.3.2. Spatial correction  (pp. 129-130) | html | pdf |
        • 2.5.3.3.3. Frame integration  (p. 130) | html | pdf |
        • 2.5.3.3.4. Lorentz, polarization and absorption corrections  (pp. 130-131) | html | pdf |
        • 2.5.3.3.5. Air scatter  (pp. 131-132) | html | pdf |
        • 2.5.3.3.6. Sample absorption  (p. 132) | html | pdf |
    • 2.5.4. Applications  (pp. 132-147) | html | pdf |
      • 2.5.4.1. Phase identification  (pp. 133-136) | html | pdf |
        • 2.5.4.1.1. Relative intensity  (p. 133) | html | pdf |
        • 2.5.4.1.2. Detector distance and resolution  (pp. 133-134) | html | pdf |
        • 2.5.4.1.3. Defocusing effect  (p. 134) | html | pdf |
        • 2.5.4.1.4. Sampling statistics  (pp. 134-136) | html | pdf |
      • 2.5.4.2. Texture analysis  (pp. 136-140) | html | pdf |
        • 2.5.4.2.1. Pole density and pole figures  (p. 136) | html | pdf |
        • 2.5.4.2.2. Fundamental equations  (pp. 136-138) | html | pdf |
        • 2.5.4.2.3. Data-collection strategy  (p. 138) | html | pdf |
        • 2.5.4.2.4. Texture-data processing  (pp. 138-139) | html | pdf |
        • 2.5.4.2.5. Pole-figure interpolation and use of symmetry  (p. 139) | html | pdf |
        • 2.5.4.2.6. Orientation relationship  (pp. 139-140) | html | pdf |
      • 2.5.4.3. Stress measurement  (pp. 140-145) | html | pdf |
        • 2.5.4.3.1. Stress and strain relation  (pp. 140-141) | html | pdf |
        • 2.5.4.3.2. Fundamental equations  (pp. 141-142) | html | pdf |
        • 2.5.4.3.3. Equations for various stress states  (p. 142) | html | pdf |
        • 2.5.4.3.4. Data-collection strategy  (p. 143) | html | pdf |
        • 2.5.4.3.5. Data integration and peak evaluation  (pp. 143-144) | html | pdf |
        • 2.5.4.3.6. Stress calculation  (p. 144) | html | pdf |
        • 2.5.4.3.7. Comparison between the 2D method and the con­ventional method  (pp. 144-145) | html | pdf |
      • 2.5.4.4. Quantitative analysis  (pp. 145-147) | html | pdf |
        • 2.5.4.4.1. Crystallinity  (p. 145) | html | pdf |
        • 2.5.4.4.2. Crystallite size  (pp. 145-147) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.5.1. Diffraction patterns in 3D space from a powder sample and the diffractometer plane  (p. 118) | html | pdf |
      • Fig. 2.5.2. Diffraction pattern from a battery component containing multiple layers  (p. 119) | html | pdf |
      • Fig. 2.5.3. The diffraction cone and the corresponding diffraction-vector cone  (p. 120) | html | pdf |
      • Fig. 2.5.4. Detector positions in the laboratory-system coordinates  (p. 121) | html | pdf |
      • Fig. 2.5.5. Relationship between a pixel P and detector position in the laboratory coordinates  (p. 121) | html | pdf |
      • Fig. 2.5.6. Cylinder-shaped detector in vertical direction: (a) detector position in the laboratory coordinates; (b) pixel position in the flattened image  (p. 122) | html | pdf |
      • Fig. 2.5.7. Sample rotation and translation  (p. 122) | html | pdf |
      • Fig. 2.5.8. Unit diffraction vector in (a) the laboratory coordinates and (b) the sample coordinates  (p. 123) | html | pdf |
      • Fig. 2.5.9. X-ray beam path in a two-dimensional X-ray diffraction system  (p. 124) | html | pdf |
      • Fig. 2.5.10. Detector dimensions and maximum measurable 2θ  (p. 126) | html | pdf |
      • Fig. 2.5.11. Solid angle covered by each pixel and its location on the detector  (p. 126) | html | pdf |
      • Fig. 2.5.12. (a) Point-spread function (PSF) from a parallel point beam; (b) line-spread function (LSF) from a sharp line beam  (p. 127) | html | pdf |
      • Fig. 2.5.13. A 2D frame showing γ integration  (p. 130) | html | pdf |
      • Fig. 2.5.14. Geometric relationship between the monochromator and detector in the laboratory coordinates  (p. 131) | html | pdf |
      • Fig. 2.5.15. Absorption correction for a flat slab: (a) reflection; (b) transmission  (p. 132) | html | pdf |
      • Fig. 2.5.16. Diffraction pattern merged from four 2D frames collected from a battery material  (p. 134) | html | pdf |
      • Fig. 2.5.17. Defocusing effects: (a) cylindrical detector; (b) flat detector at various incident angles and detector swing angles; (c) comparison of defocusing factors  (p. 135) | html | pdf |
      • Fig. 2.5.18. Diffraction frame collected from a Cu film on an Si substrate showing intensity variation along γ due to texture  (p. 136) | html | pdf |
      • Fig. 2.5.19. (a) Definition of pole direction angles α and β; (b) stereographic projection in a pole figure  (p. 137) | html | pdf |
      • Fig. 2.5.20. Data-collection strategy: (a) 2D detector with D = 7 cm; (b) 2D detector with D = 10 cm; (c) point detector  (p. 138) | html | pdf |
      • Fig. 2.5.21. Pole-figure data processing: (a) a frame with the 2θ integration ranges for the (220) ring; (b) 2θ profile showing the background and peak; (c) integrated intensity distribution as a function of γ  (p. 139) | html | pdf |
      • Fig. 2.5.22. Pole-figure processing: (a) I(γ) mapped to the pole figure; (b) Pole figure after interpolation and symmetry processing  (p. 139) | html | pdf |
      • Fig. 2.5.23. Combined pole figure of a Cu (111) film on an Si (400) substrate: (a) regular 2D projection; (b) 3D surface plot  (p. 140) | html | pdf |
      • Fig. 2.5.24. Diffraction-cone distortion due to stresses  (p. 141) | html | pdf |
      • Fig. 2.5.25. Data-collection strategy schemes: (a) ω + φ scan; (b) ψ + φ scan  (p. 143) | html | pdf |
      • Fig. 2.5.26. Data integration for stress measurement  (p. 143) | html | pdf |
      • Fig. 2.5.27. Stress calculation with the 2D method and the method: (a) nine data points (abbreviated as pts) on the diffraction ring; (b) measured stress and standard deviation by different methods  (p. 145) | html | pdf |
      • Fig. 2.5.28. 2D diffraction pattern from an oriented polycrystalline polymer sample  (p. 146) | html | pdf |
      • Fig. 2.5.29. Crystallite-size analysis: (a) 2θ profile of a gold nanoparticle (grey) and regular gold metal (black); (b) γ profile of LaB₆; (c) measurement range  (p. 146) | html | pdf |
  • 2.6. Non-ambient-temperature powder diffraction  (pp. 150-155) | html | pdf | chapter contents |
    C. A. Reiss
    • 2.6.1. Introduction  (p. 150) | html | pdf |
    • 2.6.2. In situ powder diffraction  (p. 150) | html | pdf |
    • 2.6.3. Processes of interest  (p. 150) | html | pdf |
    • 2.6.4. General system setup of non-ambient chambers  (pp. 150-151) | html | pdf |
      • 2.6.4.1. Sample stage  (p. 150) | html | pdf |
      • 2.6.4.2. Temperature-control unit, process controller  (pp. 150-151) | html | pdf |
      • 2.6.4.3. Vacuum equipment, gas supply  (p. 151) | html | pdf |
      • 2.6.4.4. Water cooling  (p. 151) | html | pdf |
      • 2.6.4.5. Diffractometer and height-compensation mechanism  (p. 151) | html | pdf |
    • 2.6.5. Specimen properties  (p. 151) | html | pdf |
    • 2.6.6. High-temperature sample stages  (pp. 151-153) | html | pdf |
      • 2.6.6.1. Direct heating: strip heaters  (pp. 151-152) | html | pdf |
      • 2.6.6.2. Environmental heating: the oven  (p. 152) | html | pdf |
      • 2.6.6.3. Environmental heating: lamp furnace  (pp. 152-153) | html | pdf |
      • 2.6.6.4. Domed hot stage  (p. 153) | html | pdf |
    • 2.6.7. Low-temperature sample stages  (pp. 153-154) | html | pdf |
      • 2.6.7.1. Cryogenic cooling stages/cryostat  (pp. 153-154) | html | pdf |
      • 2.6.7.2. Cryogenic cooling stages/cryostream  (p. 154) | html | pdf |
    • 2.6.8. Temperature accuracy  (pp. 154-155) | html | pdf |
    • 2.6.9. Future  (p. 155) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.6.1. Typical hardware setup for a non-ambient X-ray diffraction experiment as described in Section 2.6.4; non-ambient chamber, temperature/process-control unit, vacuum/gas equipment, cooling water and goniometer with height-alignment stage connected to a PC  (p. 151) | html | pdf |
      • Fig. 2.6.2. An Anton Paar HTK 1200N high-temperature oven chamber on a PANalytical Empyrean system equipped with a PIXcel3D detector  (p. 151) | html | pdf |
      • Fig. 2.6.3. The interior of a typical strip-heater sample stage (Anton Paar HTK 2000N) with heating strip (A), mechanics to compensate strip expansion (B), thermocouple wires (C), heat shield (D) and water-cooled base plate (E)  (p. 152) | html | pdf |
      • Fig. 2.6.4. A typical furnace heater (Anton Paar HTK 1200N) consisting of sample holder (A), heater (B), thermal insulation (C), water-cooled housing (D), thermocouple (E) and X-ray window (F)  (p. 152) | html | pdf |
      • Fig. 2.6.5. (a) Upon heating, CaCO₃ (the peak at about 29.3° in 2θ) reacts with SiO₂ (amorphous); at 853 K the new phase α′_L-Ca₂SiO₄ is formed (the peak between 33 and 32° in 2θ)  (p. 153) | html | pdf |
      • Fig. 2.6.6. Sample-heating stage (Anton Paar DHS 1100) with lightweight, air-cooled housing (A), dome-shaped X-ray window (B) and heating plate with sample fixation (C)  (p. 153) | html | pdf |
      • Fig. 2.6.7. Monitoring of layer thickness and roughness by X-ray reflectivity measurements during annealing at 823 K  (p. 153) | html | pdf |
      • Fig. 2.6.8. Schematic drawing of the Oxford Cryosystems cryostream setup  (p. 154) | html | pdf |
      • Fig. 2.6.9. Oxford Cryostream (A) mounted on a PANalytical diffractometer for cooling a capillary (B)  (p. 154) | html | pdf |
      • Fig. 2.6.10. Atomic pair distribution function of C₆₀ at room temperature (red) and at 100 K (blue)  (p. 154) | html | pdf |
  • 2.7. High-pressure devices  (pp. 156-173) | html | pdf | chapter contents |
    A. Katrusiak
    • 2.7.1. Introduction  (p. 156) | html | pdf |
    • 2.7.2. Historical perspective  (pp. 156-157) | html | pdf |
    • 2.7.3. Main types of high-pressure environments  (pp. 157-159) | html | pdf |
    • 2.7.4. The diamond-anvil cell (DAC)  (pp. 159-160) | html | pdf |
    • 2.7.5. Variable-temperature high-pressure devices  (pp. 160-161) | html | pdf |
    • 2.7.6. Soft and biomaterials under pressure  (p. 161) | html | pdf |
    • 2.7.7. Completeness of data  (pp. 161-162) | html | pdf |
    • 2.7.8. Single-crystal data collection  (pp. 162-163) | html | pdf |
    • 2.7.9. Powder diffraction with the DAC  (pp. 163-164) | html | pdf |
    • 2.7.10. Sample preparation  (p. 164) | html | pdf |
    • 2.7.11. Hydrostatic conditions  (pp. 164-165) | html | pdf |
    • 2.7.12. High-pressure chamber and gasket in the DAC  (pp. 165-166) | html | pdf |
    • 2.7.13. High-pressure neutron diffraction  (p. 166) | html | pdf |
    • 2.7.14. Pressure determination  (pp. 166-167) | html | pdf |
    • 2.7.15. High-pressure diffraction data corrections  (pp. 167-168) | html | pdf |
    • 2.7.16. Final remarks  (p. 168) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.7.1. A cross section through a piston-and-cylinder device with a shrink-fitted double cylinder  (p. 158) | html | pdf |
      • Fig. 2.7.2. Cross sections of (a) the girdle anvil, (b) the belt anvil, (c) the Bridgman anvil and (d) the toroid anvil  (p. 158) | html | pdf |
      • Fig. 2.7.3. A schematic view of the opposed-sapphire anvil of the Kurchatov–LBB cell designed for neutron diffraction on magnetic materials (Goncharenko, 2004)  (p. 159) | html | pdf |
      • Fig. 2.7.4. Cross sections of three types of diamond anvil used in high-pressure cells: the brilliant design (supported either on the table or on the crown rim), the Drukker design (supported on the table and crown) and the Boehler–Almax design (supported on the crown)  (p. 159) | html | pdf |
      • Fig. 2.7.5. A cross section through the central part of a diamond-anvil cell, schematically showing the main elements applied in various designs  (p. 160) | html | pdf |
      • Fig. 2.7.6. A diamond-anvil cell, showing the 40° half-angle opening of the conical windows and the reciprocal space accessed for a single-crystal sample and Mo Kα or Ag Kα radiation  (p. 161) | html | pdf |
      • Fig. 2.7.7. A schematic illustration of the reciprocal space associated with a sample enclosed in a diamond-anvil cell (DAC)  (p. 162) | html | pdf |
    • Tables
      • Table 2.7.1. The (pseudo)hydrostatic limits of selected media at 296 K (Holzapfel, 1997; Miletich et al., 2000)  (p. 165) | html | pdf |
      • Table 2.7.2. Luminescence pressure sensors, their electronic transition types (s = singlet, d = doublet) and rates of spectral shifts (after Holzapfel, 1997)  (p. 167) | html | pdf |
      • Table 2.7.3. Parameters recommended for pressure determination by EOS measurements  (p. 167) | html | pdf |
      • Table 2.7.4. Pressure fixed points at ambient temperature (after Holzapfel, 1997; Hall, 1980)  (p. 167) | html | pdf |
  • 2.8. Powder diffraction in external electric and magnetic fields  (pp. 174-188) | html | pdf | chapter contents |
    H. Ehrenberg, M. Hinterstein, A. Senyshyn and H. Fuess
    • 2.8.1. Introduction  (p. 174) | html | pdf |
    • 2.8.2. Experimental conditions  (pp. 174-177) | html | pdf |
      • 2.8.2.1. Detectors  (pp. 175-176) | html | pdf |
      • 2.8.2.2. Absorption  (pp. 176-177) | html | pdf |
      • 2.8.2.3. Sample fluorescence and incoherent neutron scattering  (p. 177) | html | pdf |
    • 2.8.3. Examples  (pp. 177-186) | html | pdf |
      • 2.8.3.1. In situ studies of ferroelectrics in an external electric field  (pp. 177-181) | html | pdf |
      • 2.8.3.2. In situ studies of electrode materials and in operando investigations of Li-ion batteries  (pp. 181-183) | html | pdf |
      • 2.8.3.3. Diffraction under a magnetic field  (pp. 183-186) | html | pdf |
        • 2.8.3.3.1. General remarks  (pp. 183-184) | html | pdf |
        • 2.8.3.3.2. Frustrated magnetic systems  (pp. 184-185) | html | pdf |
          • 2.8.3.3.2.1. Kagomé staircase systems  (pp. 184-185) | html | pdf |
          • 2.8.3.3.2.2. Manganite systems  (p. 185) | html | pdf |
        • 2.8.3.3.3. Additional systems and scattering techniques  (pp. 185-186) | html | pdf |
        • 2.8.3.3.4. Concluding remarks  (p. 186) | html | pdf |
    • 2.8.4. Summary  (p. 186) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.8.1. Sample geometries for in situ experiments with an applied electric field  (p. 177) | html | pdf |
      • Fig. 2.8.2. (a), (b) High-resolution synchrotron X-ray powder diffraction patterns and TEM imaging of PZT with a varying Zr/Ti ratio  (p. 178) | html | pdf |
      • Fig. 2.8.3. Diffraction patterns of the tetragonal 101_T /110_T reflection pairs of PZT 52/48, PZT 53/47, PZT 54/46, PZT 55/45 and PZT 56/44 recorded in situ under an electric field for the first poling cycle of up to 4 kV mm⁻¹  (p. 179) | html | pdf |
      • Fig. 2.8.4. In situ transmission geometry developed by Schönau, Schmitt et al  (p. 179) | html | pdf |
      • Fig. 2.8.5. 111_C and 200_C reflections of the unpoled, remanent and applied field state of PIC 151 at ω = 0°  (p. 179) | html | pdf |
      • Fig. 2.8.6. 111_C and 200_C reflections of bipolar fatigued PIC 151 (50 Hz, 10⁷ cycles) in the remanent state (0 kV mm⁻¹ at ω = 0°, 15°, 30° and 45°)  (p. 180) | html | pdf |
      • Fig. 2.8.7. 111_C and 200_C reflections of bipolar fatigued PIC 151 (50 Hz, 10⁷ cycles) at ω = 45° with 0.0, 0.72, 1.41 and 2.0 kV mm⁻¹  (p. 180) | html | pdf |
      • Fig. 2.8.8. Diffracted intensities of the pseudo-cubic 002 reflections as a function of 2θ and time during application of a square, bipolar electric field waveform of frequency 1 Hz and amplitude of plus or minus half the coercive field  (p. 180) | html | pdf |
      • Fig. 2.8.9. Pump–probe measurements of the 200_C reflection at ω = 45°  (p. 180) | html | pdf |
      • Fig. 2.8.10. The pseudo-cubic 002 reflection of (a) 0.93BNT–0.07BT end member, (b) 0.938BNT–0.053BT–0.009KNN, (c) 0.932BNT–0.045BT–0.023KNN and (d) 0.86BNT–0.14KNN end member as a function of electric field from the initial zero-field state (top) to an applied field of 5.5 kV mm⁻¹ (bottom)  (p. 181) | html | pdf |
      • Fig. 2.8.11. Rietveld refinement based on different patterns of 0.92Bi_0.5Na_0.5TiO₃–0.06BaTiO₃–0.02K_0.5NbO₃ (a) in the unpoled and (b) in the applied field state at 6 kV mm⁻¹  (p. 181) | html | pdf |
      • Fig. 2.8.12. Li_1+δNiO₂ during charge and discharge  (p. 183) | html | pdf |
      • Fig. 2.8.13. The High Magnetic Field Facility for Neutron Scattering at the Helmholtz-Zentrum Berlin has two main components: the High Field Magnet (HFM) and the Extreme Environment Diffractometer (EXED)  (p. 183) | html | pdf |
      • Fig. 2.8.14. Neutron powder diffraction data for Co₃V₂O₈ at 9 K (left) and 2 K (right) under magnetic fields of 0, 0.5, 1.5, 2.5, 4.0 and 8.0 T  (p. 184) | html | pdf |
      • Fig. 2.8.15. The observed Bragg reflection 100 (open circles) under an applied field of (a) 0 T, (b) 2 T and (c) 4 T at 4.2 K and (d) 0 T, (e) 2 T and (f) 4 T at 16 K (taken from Yusuf et al  (p. 185) | html | pdf |
      • Fig. 2.8.16. Time-of-flight diffraction patterns of YMn₂O₅ at 1.6 K under magnetic fields between 0 and 8 T (taken from Radaelli & Chapon, 2008 )  (p. 185) | html | pdf |
      • Fig. 2.8.17. Polarized neutron diffraction patterns for Tb₂Sn₂O₇ at 2 K and 1 T (a) and 100 K and 5 T (b)  (p. 186) | html | pdf |
  • 2.9. Cells for in situ powder-diffraction investigation of chemical reactions  (pp. 189-199) | html | pdf | chapter contents |
    W. van Beek and P. Pattison
    • 2.9.1. Introduction  (p. 189) | html | pdf |
    • 2.9.2. Historical perspective  (p. 189) | html | pdf |
    • 2.9.3. Main types of reaction cells  (pp. 189-197) | html | pdf |
      • 2.9.3.1. Introduction  (p. 189) | html | pdf |
      • 2.9.3.2. Capillary cells  (pp. 189-192) | html | pdf |
      • 2.9.3.3. Reactions requiring specialist cells  (pp. 192-195) | html | pdf |
        • 2.9.3.3.1. Cells for electrochemistry  (pp. 192-193) | html | pdf |
        • 2.9.3.3.2. Cells with humidity control  (pp. 193-194) | html | pdf |
        • 2.9.3.3.3. Large-volume cells for energy-dispersive diffraction  (p. 194) | html | pdf |
        • 2.9.3.3.4. Large-volume cells for angular-dispersive diffraction  (pp. 194-195) | html | pdf |
      • 2.9.3.4. Cells specifically for neutrons  (pp. 195-197) | html | pdf |
        • 2.9.3.4.1. Introduction  (pp. 195-196) | html | pdf |
        • 2.9.3.4.2. Solid–gas reactions  (p. 196) | html | pdf |
        • 2.9.3.4.3. Electrochemistry using neutron diffraction  (p. 196) | html | pdf |
        • 2.9.3.4.4. Hydrothermal reaction cells  (pp. 196-197) | html | pdf |
    • 2.9.4. Complementary methods and future developments  (p. 197) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.9.1. (a) Swagelock VCR gland with an epoxy-glued capillary (red ellipse)  (p. 190) | html | pdf |
      • Fig. 2.9.2. (a) An exploded representation of the flow-cell/furnace components, indicating how they fit together  (p. 190) | html | pdf |
      • Fig. 2.9.3. Sketch of a typical experimental setup and a three-dimensional drawing of the in situ flow cell  (p. 191) | html | pdf |
      • Fig. 2.9.4. (a) In situ three-dimensional stacked plot of the intercalation of ibuprofen into an LDH  (p. 191) | html | pdf |
      • Fig. 2.9.5. (a) In situ synchrotron diffraction patterns (selected region) of an LiCoO₂/Li cell collected during cell charging  (p. 192) | html | pdf |
      • Fig. 2.9.6. Schematic drawing of the humidity-control system: (1) mass-flow controller, (2) adsorption dryer, (3) pressure regulator, (4) heated bubbler, (5) peristaltic pump, (6) water reservoir, (7) thermostat, (8) condensation trap, (9) mixing chamber and (10) thermostat  (p. 193) | html | pdf |
      • Fig. 2.9.7. Sequence of XRD measurements between 21 and 27° 2θ  (p. 193) | html | pdf |
      • Fig. 2.9.8. A schematic of the Oxford/Daresbury hydrothermal autoclave used for energy-dispersive X-ray diffraction studies  (p. 194) | html | pdf |
      • Fig. 2.9.9. (a) Schematic of a sodium-halide cell in an in situ synchrotron EDXRD experimental setup  (p. 195) | html | pdf |
  • 2.10. Specimen preparation  (pp. 200-222) | html | pdf | chapter contents |
    P. S. Whitfield, A. Huq and J. A. Kaduk
    • 2.10.1. X-ray powder diffraction  (pp. 200-218) | html | pdf |
      • 2.10.1.1. Powders and particle statistics (granularity)  (pp. 201-206) | html | pdf |
      • 2.10.1.2. Preferred orientation  (pp. 206-209) | html | pdf |
      • 2.10.1.3. Absorption (surface roughness), microabsorption and extinction  (pp. 209-211) | html | pdf |
        • 2.10.1.3.1. Absorption (surface roughness)  (pp. 209-210) | html | pdf |
        • 2.10.1.3.2. Microabsorption  (p. 210) | html | pdf |
        • 2.10.1.3.3. Extinction  (pp. 210-211) | html | pdf |
      • 2.10.1.4. Holders  (pp. 211-218) | html | pdf |
        • 2.10.1.4.1. Reflection sample holders  (pp. 211-213) | html | pdf |
        • 2.10.1.4.2. Transmission sample holders  (pp. 213-218) | html | pdf |
          • 2.10.1.4.2.1. Flat foils  (pp. 213-215) | html | pdf |
          • 2.10.1.4.2.2. Capillaries  (pp. 215-218) | html | pdf |
    • 2.10.2. Neutron powder diffraction  (pp. 218-221) | html | pdf |
      • 2.10.2.1. Specimen form  (p. 218) | html | pdf |
      • 2.10.2.2. Sample size  (p. 218) | html | pdf |
      • 2.10.2.3. Specimen containment  (pp. 218-219) | html | pdf |
      • 2.10.2.4. Isotopes, absorption and activation  (pp. 219-221) | html | pdf |
    • 2.10.3. Conclusions  (p. 221) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.10.1. Calculated corundum powder patterns using the structure of Lewis et al  (p. 200) | html | pdf |
      • Fig. 2.10.2. 2D images of the spotty Debye rings of a coarse (~35 µm) cement powder using a Co Kα radiation 1 mm point source  (p. 201) | html | pdf |
      • Fig. 2.10.3. 2D image from the same sample after reducing the crystallites down to a few µm, together with the properly averaged integrated data  (p. 201) | html | pdf |
      • Fig. 2.10.4. 2D image showing the Debye rings when the unmilled sample from Fig  (p. 201) | html | pdf |
      • Fig. 2.10.5. The appearance of specimen granularity in a hand-ground specimen of a railroad tank car corrosion deposit  (p. 202) | html | pdf |
      • Fig. 2.10.6. An extreme example of granularity  (p. 202) | html | pdf |
      • Fig. 2.10.7. Optical microscope images of the surfaces of the hand-ground and micronized specimens of Scott's Moss Control Granules  (p. 203) | html | pdf |
      • Fig. 2.10.8. Rietveld refinement plot for micronized Scott's Moss Control Granules  (p. 203) | html | pdf |
      • Fig. 2.10.9. A rocking curve (φ scan) of (Ba_0.7Sr_1.3)TiO₄, with the detector fixed at 9.647° 2θ, the top of a strong peak in the synchrotron pattern  (p. 203) | html | pdf |
      • Fig. 2.10.10. Rietveld plot of a mixture of β-17α-estradiol hemihydrate and α-17α-estradiol  (p. 204) | html | pdf |
      • Fig. 2.10.11. The main 101 reflection in data collected from a very coarse (~100 µm) highly crystalline quartz  (p. 204) | html | pdf |
      • Fig. 2.10.12. Comparison between capillary (0.3 mm, 0.8265 Å) and rocking flat-plate (strip heater, 1.2386 Å, ω ±2°) data from the Australian Synchrotron  (p. 204) | html | pdf |
      • Fig. 2.10.13. φ scans of the five fingers of quartz for (a) <38 µm, (b) <15 µm and (c) micronized samples  (p. 205) | html | pdf |
      • Fig. 2.10.14. Optical micrographs of (a) -400 mesh quartz at 100× magnification and (b) quartz milled in a McCrone micronizer for 15 min in isopropyl alcohol at 150× magnification  (p. 206) | html | pdf |
      • Fig. 2.10.15. Diagram showing the source of improved particle statistics in reflection geometry using a 1D position sensitive detector (PSD) versus a point detector  (p. 206) | html | pdf |
      • Fig. 2.10.16. Top: diffraction pattern of top-loaded 400 mesh phlogopite mica  (p. 207) | html | pdf |
      • Fig. 2.10.17. Diffraction pattern of top-loaded miconized phlogopite mica  (p. 207) | html | pdf |
      • Fig. 2.10.18. Diffraction pattern of miconized phlogopite mica when back-loaded onto a smooth surface  (p. 207) | html | pdf |
      • Fig. 2.10.19. 20× optical micrograph of a cross section of the 400-grit carborundum paper used for back-loaded mica  (p. 207) | html | pdf |
      • Fig. 2.10.20. Diffraction pattern of micronized phlogopite mica when back-loaded onto 400-grit carborundum paper  (p. 208) | html | pdf |
      • Fig. 2.10.21. Diffraction pattern of top-loaded spray-dried phlogopite mica  (p. 208) | html | pdf |
      • Fig. 2.10.22. SEM micrograph of spray-dried micronized phlogopite mica (courtesy of M  (p. 208) | html | pdf |
      • Fig. 2.10.23. View through the alignment scope of the spherical spray-dried mica inside a 0.5 mm capillary  (p. 208) | html | pdf |
      • Fig. 2.10.24. Plot of the ratio of the integrated intensities of the 001/200 reflections of the mica using different sample-preparation techniques  (p. 208) | html | pdf |
      • Fig. 2.10.25. SEM micrograph of wollastonite needles  (p. 208) | html | pdf |
      • Fig. 2.10.26. Effect of preferential orientation on data from top-loaded wollastonite compared with the calculated pattern from the literature wollastonite-1A structure (Ohashi, 1984)  (p. 209) | html | pdf |
      • Fig. 2.10.27. Rietveld refinement fit to the literature wollastonite-1A structure (Ohashi, 1984) with data from a 0.3 mm capillary with no orientation corrections  (p. 209) | html | pdf |
      • Fig. 2.10.28. Rietveld refinement fit to the literature wollastonite-1A structure (Ohashi, 1984) with data from a 0.2 mm capillary with no orientation corrections  (p. 209) | html | pdf |
      • Fig. 2.10.29. The effect of surface roughness on the intensity compared to that of a bulk copper specimen  (p. 209) | html | pdf |
      • Fig. 2.10.30. Powder patterns of a commercial cobalt silicate sample, measured from a (rough) slurry-mounted specimen (red) and from a (flat) conventional front-packed specimen (green)  (p. 210) | html | pdf |
      • Fig. 2.10.31. Derivation of the equation relating peak displacement to sample displacement (s) in parafocusing geometry  (p. 211) | html | pdf |
      • Fig. 2.10.32. A home-made top-loading zero-background silicon holder with a 0.5 mm deep cavity  (p. 211) | html | pdf |
      • Fig. 2.10.33. Commercial holder for air-sensitive samples  (p. 212) | html | pdf |
      • Fig. 2.10.34. Filling a commercial back-loading sample holder  (p. 212) | html | pdf |
      • Fig. 2.10.35. Data from powdered sucrose on a Bragg–Brentano instrument, with the peak intensities normalized to the first reflection  (p. 212) | html | pdf |
      • Fig. 2.10.36. Diffraction pattern from a silicon-wafer zero-background holder, smears of Vaseline and Corning high-vacuum grease, and the surface treated with hairspray  (p. 213) | html | pdf |
      • Fig. 2.10.37. Parts prior to assembly of a transmission foil sample in the holder  (p. 214) | html | pdf |
      • Fig. 2.10.38. Transmission data from double layers (as used for powder samples) of different polymer substrate films  (p. 214) | html | pdf |
      • Fig. 2.10.39. Diffraction pattern from loose SRM640c powder between two 50 µm Kapton foils  (p. 214) | html | pdf |
      • Fig. 2.10.40. Transmitted light view of a micronized quartz sample through 50 µm Kapton foils  (p. 214) | html | pdf |
      • Fig. 2.10.41. Comparison of data from micronized 40S mica taken in reflection and transmission geometry, and spray-dried material in reflection geometry  (p. 214) | html | pdf |
      • Fig. 2.10.42. Comparison of the diffraction patterns of pure SnO₂ from a 0.3 mm quartz capillary in transmission and reflection geometries with Cu Kα radiation  (p. 216) | html | pdf |
      • Fig. 2.10.43. Raw diffraction data from 0.3 mm capillaries of SnO₂ diluted with 8000 grit diamond powder and carbon black  (p. 216) | html | pdf |
      • Fig. 2.10.44. Rietveld refinement of the diamond-diluted data with the SnO₂ cassiterite structure  (p. 216) | html | pdf |
      • Fig. 2.10.45. Comparison of data from SnO₂ when diluted with diamond inside a 0.3 mm capillary and pure SnO₂ coated on the outside of a 0.3 mm capillary  (p. 217) | html | pdf |
      • Fig. 2.10.46. Comparison of the background from four different 0.5 mm-diameter capillaries  (p. 217) | html | pdf |
      • Fig. 2.10.47. Platform and pin mounts for capillary samples  (p. 217) | html | pdf |
      • Fig. 2.10.48. A 0.5mm capillary secured into a standard brass capillary pin using dental wax at both ends of the pin  (p. 217) | html | pdf |
      • Fig. 2.10.49. Goniometer head position in relation to the goniometer-mounted alignment scope  (p. 218) | html | pdf |
      • Fig. 2.10.50. Two examples of sample holders used in neutron powder diffraction  (p. 218) | html | pdf |
      • Fig. 2.10.51. Two sample holders used for high-temperature studies  (p. 219) | html | pdf |
      • Fig. 2.10.52. Aerodynamic levitation system to suspend melts at temperatures to 2773 K and beyond for neutron diffraction measurements  (p. 219) | html | pdf |
      • Fig. 2.10.53. Calculation of the penetration of neutrons into Li₃N using the online tool at https://www.ncnr.nist.gov/resources/activation/   (p. 220) | html | pdf |
    • Tables
      • Table 2.10.1. Intensity (counts) and mean deviation in intensity of the main quartz 101 reflection with a stationary sample of -325 mesh quartz powder  (p. 205) | html | pdf |
      • Table 2.10.2. Theoretical behaviour of different crystallite sizes of quartz in a volume of 20 mm³  (p. 205) | html | pdf |
      • Table 2.10.3. Absorption and physical characteristics of the capillaries whose data are shown in Fig. 2.10.46  (p. 215) | html | pdf |

• Methodology
  • 3.1. The optics and alignment of the divergent-beam laboratory X-ray powder diffractometer and its calibration using NIST standard reference materials  (pp. 224-251) | html | pdf | chapter contents |
    J. P. Cline, M. H. Mendenhall, D. Black, D. Windover and A. Henins
    • 3.1.1. Introduction  (p. 224) | html | pdf |
    • 3.1.2. The instrument profile function  (pp. 224-230) | html | pdf |
    • 3.1.3. Instrument alignment  (pp. 230-235) | html | pdf |
    • 3.1.4. SRMs, instrumentation and data-collection procedures  (pp. 235-238) | html | pdf |
    • 3.1.5. Data-analysis methods  (pp. 238-241) | html | pdf |
    • 3.1.6. Instrument calibration  (pp. 241-250) | html | pdf |
    • 3.1.7. Conclusions  (p. 250) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.1.1. A schematic diagram illustrating the operation and optical components of a Bragg–Brentano X-ray powder diffractometer  (p. 224) | html | pdf |
      • Fig. 3.1.2. A schematic diagram illustrating the operation and optical components of a Bragg–Brentano X-ray diffractometer equipped with a position-sensitive detector  (p. 225) | html | pdf |
      • Fig. 3.1.3. A schematic diagram illustrating the geometry of a Johansson incident-beam monochromator  (p. 225) | html | pdf |
      • Fig. 3.1.4. Diagrammatic representations of convolutions leading to the observed XRPD profile  (p. 226) | html | pdf |
      • Fig. 3.1.5. The flat specimen error aberration profile as a function of incident-slit size (R = 217.5 mm)  (p. 227) | html | pdf |
      • Fig. 3.1.6. The flat specimen error aberration profiles for a 1° incident slit as a function 2θ (R = 217.5 mm)  (p. 227) | html | pdf |
      • Fig. 3.1.7. The PSD defocusing error aberration profiles for a silicon strip PSD as a function of window width (R = 217.5 mm, incident slit = 1° and strip width = 75 µm)  (p. 227) | html | pdf |
      • Fig. 3.1.8. Axial divergence aberration profiles shown for several levels of axial divergence  (p. 228) | html | pdf |
      • Fig. 3.1.9. Axial divergence aberration profiles for primary and secondary Soller slits of 2.3° as a function of 2θ (R = 217.5 mm)  (p. 228) | html | pdf |
      • Fig. 3.1.10. Linear attenuation aberration profiles that would roughly correspond to SRMs 676a (50 cm⁻¹), 640e and 1976b (100 cm⁻¹), and 660c (800 cm⁻¹) at 90° 2θ, where the transparency effect is at a maximum (R = 217.5 mm)  (p. 228) | html | pdf |
      • Fig. 3.1.11. The emission spectrum of Cu Kα radiation as provided by Hölzer et al  (p. 229) | html | pdf |
      • Fig. 3.1.12. Illustration of the Kα₃ lines and tube-tails contributions to an observed profile on a log scale, shown with two fits: the fundamental-parameters approach, which includes these features, and the split pseudo-Voigt PSF, which does not  (p. 229) | html | pdf |
      • Fig. 3.1.13. Illustration of the effect of the Johansson optic on the Cu Kα emission spectrum  (p. 229) | html | pdf |
      • Fig. 3.1.14. Diagrammatic explanation of the conditions necessary to realize a properly aligned X-ray powder diffractometer  (p. 230) | html | pdf |
      • Fig. 3.1.15. Diagrammatic view illustrating the use of a knife edge to determine the 2θ zero angle  (p. 231) | html | pdf |
      • Fig. 3.1.16. Diagrammatic view of the glass tunnel for determination of θ and 2θ zero angles  (p. 232) | html | pdf |
      • Fig. 3.1.17. Design of experiments using a pinhole optic to align the X-ray source with the receiving slit  (p. 232) | html | pdf |
      • Fig. 3.1.18. Successful results from the pinhole experiment showing variation in profile shape with successive adjustment of tube tilt; the central peak of highest intensity indicates the state of parallelism between the source and the receiving slit  (p. 232) | html | pdf |
      • Fig. 3.1.19. Results from 2θ scans at successive θ angles using the glass tunnel to determine the θ and 2θ zero angles  (p. 233) | html | pdf |
      • Fig. 3.1.20. Diagram of hypothetical results from two zero-angle measurements (Fig  (p. 233) | html | pdf |
      • Fig. 3.1.21. Final results from a θ–2θ scan using the glass tunnel, indicating the correct determination of θ and 2θ zero angles  (p. 233) | html | pdf |
      • Fig. 3.1.22. Figures found within the instructions for a Siemens D500 incident-beam monochromator in a Huber 611 monochromator housing, illustrating image formation and movement for correct and incorrect settings of tilt and azimuth angles (reproduced with verbal permission from Huber)  (p. 234) | html | pdf |
      • Fig. 3.1.23. The X-ray powder diffractometer designed and fabricated at NIST, in conventional divergent-beam format  (p. 237) | html | pdf |
      • Fig. 3.1.24. The NIST-built powder diffractometer configured with the Johansson incident-beam monochromator and a position-sensitive detector  (p. 237) | html | pdf |
      • Fig. 3.1.25. Diagrammatic representation of a powder-diffraction line profile, illus­trating the metrics Δ(2θ) and half-width-at-half-maximum (HWHM)  (p. 238) | html | pdf |
      • Fig. 3.1.26. Δ(2θ) curve using SRM 660b illustrating the peak-position shifts as function of 2θ  (p. 239) | html | pdf |
      • Fig. 3.1.27. Simulated and actual FWHM data from SRM 660b using the two Voigt PSFs with (`with Caglioti') and without constraints  (p. 239) | html | pdf |
      • Fig. 3.1.28. Left and right HWHM data from SRM 660b using the split pseudo-Voigt PSF fitted with uniform weighting  (p. 239) | html | pdf |
      • Fig. 3.1.29. Comparison of Δ(2θ) curves determined with profile fitting of SRM 660b data without the use of any constraints, as a function of 2θ  (p. 242) | html | pdf |
      • Fig. 3.1.30. Δ(2θ) curves from SRM 660b determined with profile fitting using the Caglioti function and the unconstrained split pseudo-Voigt PSF with uniform weighting  (p. 242) | html | pdf |
      • Fig. 3.1.31. FWHM data from SRM 660b using various split PSFs fitted without constraints  (p. 242) | html | pdf |
      • Fig. 3.1.32. Fits of the split pseudo-Voigt PSF to the low-angle 100, mid-angle 310 and high-angle 510 lines from SRM 660b illustrating the erroneous peak position and FWHM value reported for the 100 and 510 lines, respectively  (p. 243) | html | pdf |
      • Fig. 3.1.33. FWHM data from fits of the split pseudo-Voigt and split Pearson VII PSFs to simulated low- and high-resolution data  (p. 244) | html | pdf |
      • Fig. 3.1.34. Fits of a split Pearson VII PSF to data from SRM 660b collected using a Johansson IBM  (p. 245) | html | pdf |
      • Fig. 3.1.35. Δ(2θ) curves from the NIST machine configured with a Johansson IBM, illustrating a comparison of results from second-derivative and various profile-fitting methods  (p. 245) | html | pdf |
      • Fig. 3.1.36. FWHM data from SRM 660b collected using the NIST machine configured with a Johansson IBM, illustrating a comparison of results from various profile-fitting and data-collection methods  (p. 245) | html | pdf |
      • Fig. 3.1.37. FWHM data from SRM 660b collected using the NIST machine configured with a Johansson IBM and PSD, illustrating the contribution to defocusing at low angles with increasing window width  (p. 246) | html | pdf |
      • Fig. 3.1.38. FWHM data from SRMs 640e, 1976b and 660c collected with the IBM and PSD (4 mm window) and fitted using the split Pearson VII PSF with uniform weighting  (p. 246) | html | pdf |
      • Fig. 3.1.39. Fits of three SRM 660b lines obtained with a Rietveld analysis using the Thompson, Cox and Hastings formalism of the pseudo-Voigt PSF and the Finger model for asymmetry  (p. 247) | html | pdf |
      • Fig. 3.1.40. Fits of SRM 676a obtained from a Rietveld analysis using GSAS with the Thompson, Cox and Hastings formalism of the pseudo-Voigt PSF and the Finger model for asymmetry  (p. 248) | html | pdf |
      • Fig. 3.1.41. Fit quality realized with a fundamental-parameters-approach analysis of SRM 660b peak-scan data using TOPAS  (p. 248) | html | pdf |
      • Fig. 3.1.42. Δ(2θ) data from the 20 data sets collected for the certification of SRMs 660c and 640e, determined via FPA analyses using TOPAS  (p. 249) | html | pdf |
      • Fig. 3.1.43. Qualification of a machine using SRM 1976b  (p. 250) | html | pdf |
    • Tables
      • Table 3.1.1. Aberrations comprising the geometric component of the IPF  (p. 226) | html | pdf |
      • Table 3.1.2. Run-time parameters used for collection of the data used for certification of SRM 660b  (p. 238) | html | pdf |
  • 3.2. The physics of diffraction from powders  (pp. 252-262) | html | pdf | chapter contents |
    P. W. Stephens
    • 3.2.1. Introduction  (p. 252) | html | pdf |
    • 3.2.2. Idealized diffraction from powders  (pp. 252-257) | html | pdf |
      • 3.2.2.1. Peak positions  (pp. 252-253) | html | pdf |
      • 3.2.2.2. Diffraction peak intensities  (pp. 253-255) | html | pdf |
        • 3.2.2.2.1. X-rays  (pp. 253-254) | html | pdf |
        • 3.2.2.2.2. Neutrons and nuclear scattering  (p. 254) | html | pdf |
        • 3.2.2.2.3. Neutrons and magnetic scattering  (pp. 254-255) | html | pdf |
      • 3.2.2.3. Peak shapes  (pp. 255-257) | html | pdf |
        • 3.2.2.3.1. Domain size  (pp. 255-256) | html | pdf |
        • 3.2.2.3.2. Strain  (p. 256) | html | pdf |
        • 3.2.2.3.3. Instrumental contributions  (pp. 256-257) | html | pdf |
    • 3.2.3. Complications due to non-ideal sample or instrument properties  (pp. 257-261) | html | pdf |
      • 3.2.3.1. Absorption within a homogeneous sample  (pp. 257-258) | html | pdf |
      • 3.2.3.2. Absorption with multiphase samples  (p. 258) | html | pdf |
      • 3.2.3.3. Granularity, microabsorption and surface roughness  (pp. 258-259) | html | pdf |
      • 3.2.3.4. Anisotropic strain broadening  (pp. 259-260) | html | pdf |
      • 3.2.3.5. Preferred orientation (texture)  (p. 260) | html | pdf |
      • 3.2.3.6. Extinction  (pp. 260-261) | html | pdf |
    • 3.2.4. The Debye scattering equation  (p. 261) | html | pdf |
    • 3.2.5. Summary  (p. 261) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.2.1. Computed powder line shape from an ensemble of spherical particles of diameter 100a, including comparison to Gaussian and Lorentzian line shapes of equal FWHM  (p. 255) | html | pdf |
      • Fig. 3.2.2. Sketch illustrating symmetrical diffraction from a flat sample  (p. 258) | html | pdf |
      • Fig. 3.2.3. Sketch illustrating an X-ray propagating through a mixture of two kinds of particles  (p. 259) | html | pdf |
      • Fig. 3.2.4. Weight fractions that would be measured from a mixture of equal parts of three phases with Cu Kα radiation  (p. 259) | html | pdf |
      • Fig. 3.2.5. Diffraction peak width versus diffraction angle from a powder pattern of face-centred cubic Rb₃C₆₀ (Stephens, 1999)  (p. 259) | html | pdf |
      • Fig. 3.2.6. Extinction correction for spherical particles according to Thorkildsen & Larsen (1998)  (p. 261) | html | pdf |
    • Tables
      • Table 3.2.1. Restrictions and reflections of anisotropic strain parameters for the various Laue classes  (p. 260) | html | pdf |
  • 3.3. Powder diffraction peak profiles  (pp. 263-269) | html | pdf | chapter contents |
    R. B. Von Dreele
    • 3.3.1. Introduction  (p. 263) | html | pdf |
    • 3.3.2. Peak profiles for constant-wavelength radiation (X-rays and neutrons)  (pp. 263-265) | html | pdf |
      • 3.3.2.1. Introduction – symmetric peak profiles  (pp. 263-264) | html | pdf |
      • 3.3.2.2. Constant-wavelength powder profile asymmetry  (p. 264) | html | pdf |
      • 3.3.2.3. Peak-displacement effects  (pp. 264-265) | html | pdf |
      • 3.3.2.4. Fundamental parameters profile modelling  (p. 265) | html | pdf |
    • 3.3.3. Peak profiles for neutron time-of-flight experiments  (pp. 265-266) | html | pdf |
      • 3.3.3.1. The experiment  (p. 265) | html | pdf |
      • 3.3.3.2. The neutron pulse shape  (pp. 265-266) | html | pdf |
      • 3.3.3.3. The neutron TOF powder peak profile  (p. 266) | html | pdf |
    • 3.3.4. Peak profiles for X-ray energy-dispersive experiments  (pp. 266-267) | html | pdf |
    • 3.3.5. Sample broadening  (pp. 267-268) | html | pdf |
      • 3.3.5.1. Crystallite size broadening  (p. 267) | html | pdf |
      • 3.3.5.2. Microstrain broadening  (pp. 267-268) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.3.1. The band of intensity diffracted by a sample with height 2S, as seen by a detector with opening 2H and a detector angle 2φ moving in the detector cylinder  (p. 264) | html | pdf |
      • Fig. 3.3.2. Low-angle synchrotron powder diffraction line (2θ ≃ 4.1°) fitted by the Finger et al  (p. 264) | html | pdf |
      • Fig. 3.3.3. The observed and calculated Ni 222 diffraction line profile from the Back Scattering Spectrometer, Harwell Laboratory, Chilton, UK  (p. 266) | html | pdf |
      • Fig. 3.3.4. Microstrain surface for sodium parahydroxybenzoate multiplied by 10⁶  (p. 268) | html | pdf |
  • 3.4. Indexing a powder diffraction pattern  (pp. 270-281) | html | pdf | chapter contents |
    A. Altomare, C. Cuocci, A. Moliterni and R. Rizzi
    • 3.4.1. Introduction  (p. 270) | html | pdf |
    • 3.4.2. The basic concepts of indexing  (pp. 270-272) | html | pdf |
      • 3.4.2.1. Figures of merit  (pp. 271-272) | html | pdf |
      • 3.4.2.2. Geometrical ambiguities  (p. 272) | html | pdf |
    • 3.4.3. Indexing methods  (pp. 272-275) | html | pdf |
      • 3.4.3.1. Traditional indexing methods  (pp. 273-275) | html | pdf |
        • 3.4.3.1.1. Zone-indexing strategy  (pp. 273-274) | html | pdf |
        • 3.4.3.1.2. Shirley–Ishida–Watanabe (SIW) heuristic strategy  (p. 274) | html | pdf |
        • 3.4.3.1.3. Index-heuristics  (p. 274) | html | pdf |
        • 3.4.3.1.4. Index-permutation strategy  (p. 274) | html | pdf |
        • 3.4.3.1.5. Successive-dichotomy search method  (pp. 274-275) | html | pdf |
      • 3.4.3.2. Non-traditional indexing methods  (p. 275) | html | pdf |
        • 3.4.3.2.1. The topographs method  (p. 275) | html | pdf |
        • 3.4.3.2.2. Global-optimization methods  (p. 275) | html | pdf |
          • 3.4.3.2.2.1. Genetic-algorithm search method  (p. 275) | html | pdf |
          • 3.4.3.2.2.2. Monte Carlo search method  (p. 275) | html | pdf |
          • 3.4.3.2.2.3. Grid-search method  (p. 275) | html | pdf |
    • 3.4.4. Software packages for indexing and examples of their use  (pp. 275-280) | html | pdf |
      • 3.4.4.1. Traditional indexing programs  (p. 276) | html | pdf |
        • 3.4.4.1.1. ITO (Visser, 1969)  (p. 276) | html | pdf |
        • 3.4.4.1.2. TREOR90 (Werner et al., 1985)  (p. 276) | html | pdf |
        • 3.4.4.1.3. DICVOL91 (Boultif & Louër, 1991)  (p. 276) | html | pdf |
      • 3.4.4.2. Evolved indexing programs  (pp. 276-277) | html | pdf |
        • 3.4.4.2.1. N-TREOR09 (Altomare et al., 2009)  (pp. 276-277) | html | pdf |
        • 3.4.4.2.2. DICVOL06 (Louër & Boultif, 2006, 2007) and DICVOL14 (Louër & Boultif, 2014)  (p. 277) | html | pdf |
      • 3.4.4.3. Non-traditional indexing programs  (pp. 277-278) | html | pdf |
        • 3.4.4.3.1. Conograph: indexing via the topographs method  (p. 277) | html | pdf |
        • 3.4.4.3.2. GAIN: indexing via a genetic-algorithm search method  (pp. 277-278) | html | pdf |
        • 3.4.4.3.3. McMaille: indexing via a Monte Carlo search method  (p. 278) | html | pdf |
      • 3.4.4.4. Crysfire: a suite of indexing programs  (p. 278) | html | pdf |
      • 3.4.4.5. Two commercial programs  (p. 278) | html | pdf |
        • 3.4.4.5.1. SVD-Index  (p. 278) | html | pdf |
        • 3.4.4.5.2. X-CELL  (p. 278) | html | pdf |
      • 3.4.4.6. Examples of applications of indexing programs  (pp. 278-280) | html | pdf |
        • 3.4.4.6.1. Indexing using DICVOL06  (pp. 278-279) | html | pdf |
        • 3.4.4.6.2. Indexing using N-TREOR09  (pp. 279-280) | html | pdf |
    • 3.4.5. Conclusion  (p. 280) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.4.1. The list of possible cells for the decafluoroquarterphenyl structure automatically found using N-TREOR09  (p. 279) | html | pdf |
    • Tables
      • Table 3.4.1. Expressions for Q(hkl) for different types of symmetry  (p. 271) | html | pdf |
      • Table 3.4.2. Examples of lattices leading to geometrical ambiguities  (p. 273) | html | pdf |
      • Table 3.4.3. Classification of indexing methods  (p. 276) | html | pdf |
  • 3.5. Data reduction to |F_hkl| values  (pp. 282-287) | html | pdf | chapter contents |
    A. Le Bail
    • 3.5.1. Introduction  (p. 282) | html | pdf |
    • 3.5.2. Algorithms  (pp. 282-284) | html | pdf |
      • 3.5.2.1. Unrestrained cell  (p. 282) | html | pdf |
      • 3.5.2.2. Restrained cell  (p. 282) | html | pdf |
        • 3.5.2.2.1. Pawley method  (p. 283) | html | pdf |
        • 3.5.2.2.2. Le Bail method  (pp. 283-284) | html | pdf |
    • 3.5.3. Pitfalls in the extraction of accurate |F_hkl| values using the Pawley and Le Bail methods  (pp. 284-285) | html | pdf |
      • 3.5.3.1. Consequences of (exact or accidental) overlap  (p. 284) | html | pdf |
      • 3.5.3.2. Preferred-orientation effects  (p. 284) | html | pdf |
      • 3.5.3.3. Background-estimation effects  (pp. 284-285) | html | pdf |
    • 3.5.4. Applications and by-products  (pp. 285-286) | html | pdf |
      • 3.5.4.1. Supporting indexing and space-group determination  (p. 286) | html | pdf |
      • 3.5.4.2. Structure solution  (p. 286) | html | pdf |
    • 3.5.5. Conclusion  (p. 286) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.5.1. Data reduction to |F_hkl| values for the C₆F₁₀ Pawley (1981) test case by the Le Bail method using FULLPROF  (p. 285) | html | pdf |
      • Fig. 3.5.2. The C₆F₁₀ Monte Carlo molecule positioning by the real-space ESPOIR program produces that best fit (R_p = 13.6%) of the pseudo powder pattern built from the previously extracted |F_hkl| values (Fig  (p. 285) | html | pdf |
      • Fig. 3.5.3. Projection along the b axis of the C₆F₁₀ structure model in P2₁/n before Rietveld refinement  (p. 286) | html | pdf |
  • 3.6. Whole powder pattern modelling: microstructure determination from powder diffraction data  (pp. 288-303) | html | pdf | chapter contents |
    M. Leoni
    • 3.6.1. Introduction  (pp. 288-289) | html | pdf |
    • 3.6.2. Fourier methods  (pp. 289-298) | html | pdf |
      • 3.6.2.1. Definitions  (p. 289) | html | pdf |
      • 3.6.2.2. Peak profile and the convolution theorem  (p. 289) | html | pdf |
      • 3.6.2.3. The Warren–Averbach method and its variations  (pp. 289-290) | html | pdf |
      • 3.6.2.4. Beyond the Warren–Averbach method  (p. 290) | html | pdf |
      • 3.6.2.5. Whole powder pattern modelling (WPPM)  (pp. 290-291) | html | pdf |
      • 3.6.2.6. Broadening components  (pp. 291-296) | html | pdf |
        • 3.6.2.6.1. Instrument  (p. 291) | html | pdf |
        • 3.6.2.6.2. Source emission profile  (p. 291) | html | pdf |
        • 3.6.2.6.3. Optical elements  (pp. 291-292) | html | pdf |
        • 3.6.2.6.4. Domain size and shape  (pp. 292-293) | html | pdf |
        • 3.6.2.6.5. Strain broadening (lattice distortions)  (p. 293) | html | pdf |
        • 3.6.2.6.6. Dislocations  (pp. 293-295) | html | pdf |
        • 3.6.2.6.7. Twin and deformation faults  (pp. 295-296) | html | pdf |
        • 3.6.2.6.8. Antiphase domain boundaries  (p. 296) | html | pdf |
      • 3.6.2.7. Assembling the equations into a peak and modelling the data  (pp. 296-298) | html | pdf |
        • 3.6.2.7.1. Alternative approaches  (pp. 297-298) | html | pdf |
    • 3.6.3. Examples of WPPM analysis  (pp. 298-301) | html | pdf |
      • 3.6.3.1. Nanocrystalline ceria  (pp. 298-299) | html | pdf |
      • 3.6.3.2. Copper oxide  (pp. 299-301) | html | pdf |
    • Appendix 3.6.1. Functions for profile shapes  (p. 301) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.6.1. X-ray powder diffraction pattern of nanocrystalline ceria calcined at 673 K  (p. 298) | html | pdf |
      • Fig. 3.6.2. Parameterization of the instrumental resolution function using a pseudo-Voigt and the relationship of Caglioti et al  (p. 298) | html | pdf |
      • Fig. 3.6.3. Size distribution of the ceria powder: WPPM (line) and TEM (histogram)  (p. 299) | html | pdf |
      • Fig. 3.6.4. TEM micrograph of the calcined ceria powder  (p. 299) | html | pdf |
      • Fig. 3.6.5. Result of WPPM of ball-milled Cu₂O: raw data (dots), model (line) and difference (lower line)  (p. 300) | html | pdf |
      • Fig. 3.6.6. Domain-size distribution of cuprite: (a) WPPM result and (b) distribution multiplied by D³  (p. 300) | html | pdf |
    • Tables
      • Table 3.6.1. Scherrer constants (K_w and K_β) for various domain shapes (Langford & Wilson, 1978)  (p. 288) | html | pdf |
      • Table 3.6.2. Models for antiphase domain boundaries for the Cu₃Au case  (p. 296) | html | pdf |
  • 3.7. Crystallographic databases and powder diffraction  (pp. 304-324) | html | pdf | chapter contents |
    J. A. Kaduk
    • 3.7.1. Introduction  (pp. 304-305) | html | pdf |
      • 3.7.1.1. History of the PDF/ICDD  (p. 304) | html | pdf |
      • 3.7.1.2. Search/match  (pp. 304-305) | html | pdf |
    • 3.7.2. Powder Diffraction File (PDF)  (pp. 305-313) | html | pdf |
      • 3.7.2.1. Sources and formats of the PDF  (p. 306) | html | pdf |
      • 3.7.2.2. Quality marks in the PDF  (pp. 306-307) | html | pdf |
      • 3.7.2.3. Features of the PDF  (pp. 307-309) | html | pdf |
      • 3.7.2.4. Boolean logic in phase identification  (pp. 309-313) | html | pdf |
        • 3.7.2.4.1. Water-still deposit  (pp. 309-310) | html | pdf |
        • 3.7.2.4.2. Vanadium phosphate butane-oxidation catalyst  (p. 310) | html | pdf |
        • 3.7.2.4.3. Valve deposit from a piston aviation engine  (p. 310) | html | pdf |
        • 3.7.2.4.4. Isocracker sludge  (pp. 310-311) | html | pdf |
        • 3.7.2.4.5. Amoxicillin  (pp. 311-312) | html | pdf |
        • 3.7.2.4.6. Pseudoephedrine  (p. 312) | html | pdf |
        • 3.7.2.4.7. Commercial multivitamin: Centrum A to Zn  (pp. 312-313) | html | pdf |
    • 3.7.3. Cambridge Structural Database (CSD)  (pp. 313-314) | html | pdf |
      • 3.7.3.1. Mercury  (p. 314) | html | pdf |
    • 3.7.4. Inorganic Crystal Structure Database (ICSD)  (pp. 314-316) | html | pdf |
      • 3.7.4.1. General features of the ICSD  (p. 315) | html | pdf |
      • 3.7.4.2. Features particularly useful for powder crystallography  (pp. 315-316) | html | pdf |
    • 3.7.5. Pearson's Crystal Data (PCD/LPF) (with Pierre Villars and Karen Cenzual)  (pp. 316-318) | html | pdf |
      • 3.7.5.1. General information  (p. 316) | html | pdf |
      • 3.7.5.2. Evaluation procedure  (p. 316) | html | pdf |
      • 3.7.5.3. Standardized crystallographic data  (pp. 316-317) | html | pdf |
      • 3.7.5.4. Consequent prototype assignment  (p. 317) | html | pdf |
      • 3.7.5.5. Assigned atom coordinates  (p. 317) | html | pdf |
      • 3.7.5.6. External links  (p. 317) | html | pdf |
      • 3.7.5.7. Retrievable database fields  (pp. 317-318) | html | pdf |
      • 3.7.5.8. Particular software features  (p. 318) | html | pdf |
    • 3.7.6. Metals data file (CRYSTMET)  (p. 318) | html | pdf |
    • 3.7.7. Protein Data Bank (PDB)  (pp. 318-319) | html | pdf |
      • 3.7.7.1. Powder diffraction by proteins  (pp. 318-319) | html | pdf |
      • 3.7.7.2. Calculation of protein powder patterns (with Kenny Ståhl)  (p. 319) | html | pdf |
    • 3.7.8. Crystallography Open Database (COD) (with Saulius Gražulis)  (pp. 319-321) | html | pdf |
    • 3.7.9. Other internet databases  (p. 321) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.7.1. A plot generated by the Results/Graph Fields function in the Powder Diffraction File  (p. 309) | html | pdf |
      • Fig. 3.7.2. The result of applying a commercial search/match program (Jade 9.5; Materials Data, 2012) to the powder pattern of a water-still scale  (p. 309) | html | pdf |
      • Fig. 3.7.3. Variation of the unit-cell volume with the magnesium content in magnesian calcites in the Powder Diffraction File  (p. 310) | html | pdf |
      • Fig. 3.7.4. The results of applying a commercial search/match program (Jade 9.5; Materials Data, 2012) to the (background-subtracted, Kα₂-stripped) powder pattern of a butane-oxidation catalyst  (p. 311) | html | pdf |
      • Fig. 3.7.5. Comparison of the low-quality experimental PDF entry 00–047–0967 with the high-quality calculated pattern 01–074–2749 located by searching the experimental pattern against the rest of the PDF  (p. 311) | html | pdf |
      • Fig. 3.7.6. The four crystalline phases identified in a butane-oxidation catalyst  (p. 311) | html | pdf |
      • Fig. 3.7.7. The four phases identified in a valve deposit from an aircraft engine by automated search/match methods and guessing based on the appearance of the sample  (p. 312) | html | pdf |
      • Fig. 3.7.8. The seven phases identified in the valve deposit from an aircraft engine  (p. 312) | html | pdf |
      • Fig. 3.7.9. The phases identified in a deposit from a refinery isocracker  (p. 312) | html | pdf |
      • Fig. 3.7.10. The final Rietveld plot from refinement of the isocracker deposit  (p. 313) | html | pdf |
      • Fig. 3.7.11. Phases identified in amoxicillin powder from a commercial capsule  (p. 313) | html | pdf |
      • Fig. 3.7.12. Phases identified by automated search/match in a Centrum A to Zn multivitamin tablet  (p. 313) | html | pdf |
      • Fig. 3.7.13. Overview of the trends from the different corrections  (p. 319) | html | pdf |
      • Fig. 3.7.14. Calculated and experimental powder patterns for (a) lysozyme, (b) trigonal insulin and (c) cubic insulin  (p. 320) | html | pdf |
      • Fig. 3.7.15. (a) The website and search interface of the Crystallography Open Database (COD) permits searches of crystallographic data by a range of parameters and unrestricted retrieval of the found data  (p. 320) | html | pdf |
    • Tables
      • Table 3.7.1. Criteria for the assignment of quality marks to calculated patterns in the Powder Diffraction File  (p. 308) | html | pdf |
  • 3.8. Clustering and visualization of powder-diffraction data  (pp. 325-343) | html | pdf | chapter contents |
    C. J. Gilmore, G. Barr and W. Dong
    • 3.8.1. Introduction  (p. 325) | html | pdf |
    • 3.8.2. Comparing 1D diffraction patterns  (pp. 325-327) | html | pdf |
      • 3.8.2.1. Spearman's rank order coefficient  (p. 325) | html | pdf |
      • 3.8.2.2. Pearson's r coefficient  (p. 325) | html | pdf |
      • 3.8.2.3. Combining the correlation coefficients  (p. 325) | html | pdf |
      • 3.8.2.4. Full-profile qualitative pattern matching  (p. 326) | html | pdf |
      • 3.8.2.5. Generation of the correlation and distance matrices  (pp. 326-327) | html | pdf |
    • 3.8.3. Cluster analysis  (pp. 327-329) | html | pdf |
      • 3.8.3.1. Dendrograms  (p. 327) | html | pdf |
      • 3.8.3.2. Estimating the number of clusters  (pp. 327-328) | html | pdf |
      • 3.8.3.3. Metric multidimensional scaling  (pp. 328-329) | html | pdf |
      • 3.8.3.4. Principal-component analysis  (p. 329) | html | pdf |
      • 3.8.3.5. Choice of clustering method  (p. 329) | html | pdf |
      • 3.8.3.6. The most representative sample  (p. 329) | html | pdf |
      • 3.8.3.7. Amorphous samples  (p. 329) | html | pdf |
    • 3.8.4. Data visualization  (pp. 329-331) | html | pdf |
      • 3.8.4.1. Primary data visualization  (pp. 329-330) | html | pdf |
      • 3.8.4.2. Secondary visualization using parallel coordinates, the grand tour and minimum spanning trees  (pp. 330-331) | html | pdf |
        • 3.8.4.2.1. Parallel-coordinates plots  (pp. 330-331) | html | pdf |
        • 3.8.4.2.2. The grand tour  (p. 331) | html | pdf |
        • 3.8.4.2.3. Powder data as a tree: the minimum spanning trees  (p. 331) | html | pdf |
    • 3.8.5. Further validating and visualizing clusters: silhouettes and fuzzy clustering  (pp. 331-333) | html | pdf |
      • 3.8.5.1. Silhouettes  (pp. 331-333) | html | pdf |
      • 3.8.5.2. Fuzzy clustering  (p. 333) | html | pdf |
      • 3.8.5.3. The PolySNAP program and DIFFRAC.EVA  (p. 333) | html | pdf |
    • 3.8.6. Examples  (pp. 333-337) | html | pdf |
      • 3.8.6.1. Aspirin data  (pp. 333-335) | html | pdf |
        • 3.8.6.1.1. Aspirin data with amorphous samples included  (p. 335) | html | pdf |
      • 3.8.6.2. Phase transitions in ammonium nitrate  (pp. 335-337) | html | pdf |
    • 3.8.7. Quantitative analysis with high-throughput PXRD data without Rietveld refinement  (pp. 337-339) | html | pdf |
      • 3.8.7.1. Example: inorganic mixtures  (pp. 338-339) | html | pdf |
    • 3.8.8. Using spectroscopic data  (pp. 339-340) | html | pdf |
    • 3.8.9. Combining data types: the INDSCAL method  (pp. 340-342) | html | pdf |
      • 3.8.9.1. An example combining PXRD and Raman data  (p. 342) | html | pdf |
    • 3.8.10. Quality control  (p. 342) | html | pdf |
    • 3.8.11. Computer software  (p. 342) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.8.1. The use of the Pearson (r) and Spearman (R) correlation coefficients to quantitatively match powder patterns: (a) r = 0.93, R = 0.68; (b) r = 0.79, R = 0.90; (c) r = 0.66, R = 0.22  (p. 326) | html | pdf |
      • Fig. 3.8.2. Four different methods of estimating the number of clusters present in a set of 23 powder patterns for the drug doxazosin  (p. 327) | html | pdf |
      • Fig. 3.8.3. Example of a parallel-coordinates plot in six dimensions, with axes labeled X1, X2, …, X6, for a set of 80 organic PXRD samples partitioned into four clusters  (p. 330) | html | pdf |
      • Fig. 3.8.4. Flowchart for the cluster-analysis and data-visualization procedure described in this chapter  (p. 330) | html | pdf |
      • Fig. 3.8.5. Powder patterns for 13 commercial aspirin samples partitioned into five sets  (p. 331) | html | pdf |
      • Fig. 3.8.6. (a) The initial default dendrogram using the centroid clustering method on 13 PXRD patterns from 13 commercial aspirin samples  (p. 332) | html | pdf |
      • Fig. 3.8.7. The use of minimum spanning trees (MSTs)  (p. 333) | html | pdf |
      • Fig. 3.8.8. The use of silhouettes in defining the details of the clustering  (p. 334) | html | pdf |
      • Fig. 3.8.9. The complete cluster analysis for the aspirin samples  (pp. 335-336) | html | pdf |
      • Fig. 3.8.10. The aspirin data including data from five amorphous samples  (p. 337) | html | pdf |
      • Fig. 3.8.11. Ammonium nitrate phase transitions  (p. 338) | html | pdf |
      • Fig. 3.8.12. Identifying mixtures using lanthanum strontium copper oxide and caesium thiocyanate diffraction data taken from the ICDD Clay Minerals database  (p. 338) | html | pdf |
      • Fig. 3.8.13. (a) The dendrogram generated from 74 Raman spectra without background corrections applied  (p. 339) | html | pdf |
      • Fig. 3.8.14. Clustering the 74 Raman spectra without background corrections applied using first-derivative data  (p. 339) | html | pdf |
      • Fig. 3.8.15. A flowchart for the INDSCAL method using Raman and PXRD data  (p. 340) | html | pdf |
      • Fig. 3.8.16. Clustering 48 PXRD spectra with background corrections applied for three polymorphs of sulfathiazole  (p. 341) | html | pdf |
      • Fig. 3.8.17. Visualization tools for quality-control procedures using a modified MMDS plot  (p. 342) | html | pdf |
    • Tables
      • Table 3.8.1. Six commonly used clustering methods  (p. 327) | html | pdf |
      • Table 3.8.2. Estimate of the number of clusters for the 23 sample data set for doxazosin  (p. 328) | html | pdf |
  • 3.9. Quantitative phase analysis  (pp. 344-373) | html | pdf | chapter contents |
    I. C. Madsen, N. V. Y. Scarlett, R. Kleeberg and K. Knorr
    • 3.9.1. Introduction  (p. 344) | html | pdf |
    • 3.9.2. Phase analysis  (pp. 344-345) | html | pdf |
    • 3.9.3. QPA methodology  (pp. 345-350) | html | pdf |
      • 3.9.3.1. Absorption–diffraction method  (pp. 345-346) | html | pdf |
      • 3.9.3.2. Internal standard method  (pp. 346-347) | html | pdf |
        • 3.9.3.2.1. Selection of an internal standard  (pp. 346-347) | html | pdf |
      • 3.9.3.3. Reference intensity ratio methods  (p. 347) | html | pdf |
      • 3.9.3.4. Matrix-flushing method  (pp. 347-348) | html | pdf |
      • 3.9.3.5. Full-pattern fitting methods  (p. 348) | html | pdf |
      • 3.9.3.6. Rietveld-based QPA  (pp. 348-350) | html | pdf |
    • 3.9.4. Demonstration of methods  (pp. 350-353) | html | pdf |
      • 3.9.4.1. Absorption–diffraction method  (p. 350) | html | pdf |
      • 3.9.4.2. Internal standard method  (pp. 350-351) | html | pdf |
      • 3.9.4.3. Reference intensity ratio  (p. 351) | html | pdf |
      • 3.9.4.4. Matrix flushing  (pp. 351-352) | html | pdf |
      • 3.9.4.5. Rietveld-based methods  (pp. 352-353) | html | pdf |
    • 3.9.5. Alternative methods for determination of calibration constants  (pp. 353-356) | html | pdf |
      • 3.9.5.1. Standardless determination of the phase constant C  (pp. 353-354) | html | pdf |
      • 3.9.5.2. Demonstration of the Zevin approach  (pp. 354-355) | html | pdf |
      • 3.9.5.3. Experiment constant – a whole-sample approach  (pp. 355-356) | html | pdf |
    • 3.9.6. Quantification of phases with partial or no known crystal structures  (pp. 356-360) | html | pdf |
      • 3.9.6.1. Use of calibrated models  (pp. 356-358) | html | pdf |
        • 3.9.6.1.1. Generation of calibrated PONKCS models  (p. 357) | html | pdf |
        • 3.9.6.1.2. Application of the model  (pp. 357-358) | html | pdf |
      • 3.9.6.2. Modelling of structural disorder  (pp. 358-359) | html | pdf |
      • 3.9.6.3. Quantitative determination of amorphous material  (pp. 360-361) | html | pdf |
    • 3.9.7. QPA from in situ experimentation  (pp. 361-362) | html | pdf |
      • 3.9.7.1. Data analysis  (pp. 361-362) | html | pdf |
    • 3.9.8. QPA using neutron diffraction data  (p. 362) | html | pdf |
    • 3.9.9. QPA using energy-dispersive diffraction data  (pp. 362-364) | html | pdf |
    • 3.9.10. Improving accuracy  (pp. 364-370) | html | pdf |
      • 3.9.10.1. Standard deviations and error estimates  (p. 364) | html | pdf |
      • 3.9.10.2. Minimizing systematic errors  (pp. 364-365) | html | pdf |
      • 3.9.10.3. Minimizing sample-related errors  (pp. 365-370) | html | pdf |
        • 3.9.10.3.1. Crystallite-size issues  (p. 365) | html | pdf |
        • 3.9.10.3.2. Preferred orientation  (pp. 365-366) | html | pdf |
        • 3.9.10.3.3. Microabsorption  (pp. 366-368) | html | pdf |
        • 3.9.10.3.4. Whole-pattern-refinement effects  (p. 368) | html | pdf |
        • 3.9.10.3.5. Element analytical standards  (pp. 368-369) | html | pdf |
        • 3.9.10.3.6. Phase-specific methods: diffraction SRMs, round-robin samples and synthetic mixtures  (pp. 369-370) | html | pdf |
    • 3.9.11. Summary  (p. 370) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.9.1. Plot of the analysed concentration (black diamonds – left axis) and the bias (open triangles – right axis) expressed as wt% for fluorite using the absorption–diffraction method  (p. 351) | html | pdf |
      • Fig. 3.9.2. Plot of the analysed concentration (black diamonds – left axis) and the bias (open triangles – right axis) expressed as wt% for fluorite using the internal standard method with zincite designated as the internal standard  (p. 351) | html | pdf |
      • Fig. 3.9.3. Plot of the 24 determined RIR values for fluorite (black diamonds) and zincite (open triangles) as a function of corundum concentration  (p. 351) | html | pdf |
      • Fig. 3.9.4. Plot of the analysed concentration (black diamonds – left axis) and the bias (open triangles – right axis) expressed as wt% for fluorite using the matrix-flushing method with RIRs of 1.0, 3.617 and 4.856 for corundum, fluorite and zincite, respectively  (p. 352) | html | pdf |
      • Fig. 3.9.5. The results of QPA of the in situ XRD data collected during the seeding experiments of Webster et al  (p. 354) | html | pdf |
      • Fig. 3.9.6. The results of QPA of the in situ XRD data collected during the seeding experiments of Webster et al  (p. 354) | html | pdf |
      • Fig. 3.9.7. The results of QPA of the in situ XRD data collected during the seeding experiments of Webster et al  (p. 354) | html | pdf |
      • Fig. 3.9.8. SEM image of Al(OH)₃ (grey hexagon) which has crystallized on goethite seed (light grey needles) (Webster et al  (p. 355) | html | pdf |
      • Fig. 3.9.9. Plot of the bias (known − determined) in the analysed phase abundances using the Zevin & Kimmel (1995) approach for corundum (black diamonds), fluorite (open triangles) and zincite (crosses)  (p. 356) | html | pdf |
      • Fig. 3.9.10. Plot of the experiment constant K as a function of known phase concentration for corundum (closed diamonds), fluorite (open triangles) and zincite (crosses) using the phase-specific method  (p. 356) | html | pdf |
      • Fig. 3.9.11. Turbostratic disorder, illustrated by the stacking of two hexagonal layers rotated by 7°  (p. 358) | html | pdf |
      • Fig. 3.9.12. Section of the reciprocal lattice of a turbostratically disordered pseudo-hexagonal C-centred structure  (p. 359) | html | pdf |
      • Fig. 3.9.13. Output of Rietveld refinement of XRD data (Cu Kα radiation) for a synthetic sample containing a mixture crystalline and amorphous phases  (p. 360) | html | pdf |
      • Fig. 3.9.14. Raw in situ XRD data (Co Kα radiation) collected during the synthesis of the iron-ore sinter bonding phase SFCA-I (Webster et al  (p. 362) | html | pdf |
      • Fig. 3.9.15. Results of Rietveld-based QPA of the in situ data sequence shown in Fig  (p. 363) | html | pdf |
      • Fig. 3.9.16. Basic experimental arrangement for energy-dispersive diffraction  (p. 363) | html | pdf |
      • Fig. 3.9.17. Variation of the magnetite (filled diamonds) and quartz (open squares) concentration of an iron-ore sample with grinding time  (p. 365) | html | pdf |
      • Fig. 3.9.18. Increase of the March–Dollase (Dollase, 1986) parameter and related decrease of the degree of preferred orientation with grinding time for the two amphibole species actinolite (filled diamonds) and grunerite (open squares) in an iron ore  (p. 366) | html | pdf |
      • Fig. 3.9.19. Backscattered-electron SEM image of a mixture of approximately equal amounts of corundum (dark grey), magnetite and zircon (lighter grey)  (p. 367) | html | pdf |
      • Fig. 3.9.20. Bias as a function of phase concentration for industrial salt samples for (i) structure-based QPA (filled symbols) and (ii) calibrated hkl_phase (open symbols) for halite (circles) and sylvite (squares)  (p. 368) | html | pdf |
      • Fig. 3.9.21. Profile fit of anatase and rutile (a) without and (b) with a Kβ filter absorption-edge correction  (p. 369) | html | pdf |
      • Fig. 3.9.22. Output of Rietveld refinement and results of QPA for the iron-ore certified reference material SX 11–14 from Dillinger Hütte  (p. 370) | html | pdf |
    • Tables
      • Table 3.9.1. Weighed composition (weight fraction) of the eight mixtures comprising sample 1 in the IUCr CPD round robin on QPA (Madsen et al., 2001)  (p. 350) | html | pdf |
      • Table 3.9.2. Average values (n = 3) of net peak intensity derived using profile fitting for the strongest peaks of corundum (113), fluorite (022) and zincite (011)  (p. 350) | html | pdf |
      • Table 3.9.3. Phase calibration constants for corundum, fluorite and zincite determined using the Zevin (Zevin & Kimmel, 1995) and Knudsen (Knudsen, 1981) method  (p. 355) | html | pdf |
      • Table 3.9.4. Comparison of errors generated during the analysis of XRD data (Cu Kα radiation) from three sub-samples of sample 4 from the IUCr CPD round robin on QPA (Scarlett et al., 2002)  (p. 364) | html | pdf |
      • Table 3.9.5. Calculated values of μD (where μ is the linear absorption coefficient and D is the particle diameter) for Cu Kα X-rays for corundum, magnetite and zircon with a range of particle sizes  (p. 367) | html | pdf |
      • Table 3.9.6. Compositional analysis of the Dillinger Hütte iron-ore certified reference material SX 11–14, (i) derived from QPA results, taking into account the nominal stoichiometry of the phases (XRD) and (ii) the certified analyses (Cert) (Knorr & Bornefeld, 2013)  (p. 369) | html | pdf |
  • 3.10. Accuracy in Rietveld quantitative phase analysis with strictly monochromatic Mo and Cu radiations  (pp. 374-384) | html | pdf | chapter contents |
    L. León-Reina, A. Cuesta, M. García-Maté, G. Álvarez-Pinazo, I. Santacruz, O. Vallcorba, A. G. De la Torre and M. A. G. Aranda
    • 3.10.1. Introduction  (pp. 374-375) | html | pdf |
    • 3.10.2. Compounds and series  (pp. 375-376) | html | pdf |
      • 3.10.2.1. Single phases  (p. 375) | html | pdf |
      • 3.10.2.2. Crystalline inorganic series  (p. 375) | html | pdf |
      • 3.10.2.3. Crystalline organic series  (p. 375) | html | pdf |
      • 3.10.2.4. Variable amorphous content series  (pp. 375-376) | html | pdf |
    • 3.10.3. Analytical techniques  (pp. 376-377) | html | pdf |
      • 3.10.3.1. Mo Kα₁ laboratory X-ray powder diffraction (LXRPD)  (p. 376) | html | pdf |
      • 3.10.3.2. Cu Kα₁ laboratory X-ray powder diffraction (LXRPD)  (p. 376) | html | pdf |
      • 3.10.3.3. Transmission synchrotron X-ray powder diffraction (SXRPD)  (pp. 376-377) | html | pdf |
    • 3.10.4. Powder-diffraction data analysis  (p. 377) | html | pdf |
    • 3.10.5. Crystalline single phases  (pp. 377-379) | html | pdf |
    • 3.10.6. Limits of detection and quantification  (pp. 379-381) | html | pdf |
    • 3.10.7. Increasing inorganic crystalline phase content series  (p. 381) | html | pdf |
    • 3.10.8. Increasing crystalline organic phase content series  (pp. 381-382) | html | pdf |
    • 3.10.9. Increasing amorphous content series within an inorganic crystalline phase matrix  (pp. 382-383) | html | pdf |
    • 3.10.10. Conclusions  (pp. 383-384) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 3.10.1. Irradiated volume for a flat sample holder in transmission mode using (a) Mo radiation and (b) Cu radiation, and (c) reflection mode using Cu radiation  (p. 374) | html | pdf |
      • Fig. 3.10.2. Scanning electron microscopy micrographs for the studied phases (×1000)  (p. 376) | html | pdf |
      • Fig. 3.10.3. (a) Raw Mo Kα₁ powder patterns for the inorganic series composed of a constant matrix of calcite, gypsum and quartz, and increasing amounts of insoluble anhydrite (peaks highlighted with a solid square)  (p. 377) | html | pdf |
      • Fig. 3.10.4. Selected region of the powder patterns showing the main diffraction peak of insoluble anhydrite for the low-content samples to investigate the limit of detection  (p. 378) | html | pdf |
      • Fig. 3.10.5. (a) Raw Mo Kα₁ powder patterns for the organic series composed of a constant matrix of glucose, fructose and lactose, and increasing amounts of xylose (peaks highlighted with an asterisk)  (p. 379) | html | pdf |
      • Fig. 3.10.6. Selected range of the Rietveld plots for CGpQ_4.0A: (a) Mo Kα₁ and (b) Cu Kα₁ patterns  (p. 380) | html | pdf |
      • Fig. 3.10.7. Rietveld quantification results for (a) the insoluble anhydrite series (within an inorganic crystalline matrix), (b) the xylose series (within an organic crystalline matrix) and (c) the ground-glass series (within an inorganic crystalline matrix) as a function of the weighed amount of each phase  (p. 381) | html | pdf |
      • Fig. 3.10.8. Raw powder patterns for the amorphous-material-containing series composed of a constant matrix of calcite and zincite, and increasing amounts of ground glass  (p. 383) | html | pdf |
    • Tables
      • Table 3.10.1. Cambridge Structural Database (CSD)/Inorganic Crystal Structure Database (ICSD) reference codes for all phases used for Rietveld refinements in this work and the linear absorption coefficients for the wavelengths used  (p. 375) | html | pdf |
      • Table 3.10.2. Rietveld quantitative phase analyses for the crystalline inorganic mixtures measured with Cu Kα₁ and Mo Kα₁ radiations  (p. 380) | html | pdf |
      • Table 3.10.3. RQPA for the crystalline organic mixtures measured with Cu Kα₁ and Mo Kα₁ radiations  (p. 382) | html | pdf |
      • Table 3.10.4. Rietveld quantitative phase analyses of the CQZ_xGl mixture, where quartz (Q) is the internal standard, to derive amorphous content (am), obtained from SXRPD, Mo Kα₁ and Cu Kα₁ patterns  (p. 382) | html | pdf |

• Structure determination
  • 4.1. An overview of currently used structure determination methods for powder diffraction data  (pp. 386-394) | html | pdf | chapter contents |
    K. Shankland
    • 4.1.1. Introduction  (p. 386) | html | pdf |
    • 4.1.2. Methods used in SDPD  (pp. 386-387) | html | pdf |
    • 4.1.3. Conventional direct methods of structure determination  (pp. 387-388) | html | pdf |
    • 4.1.4. Modified direct methods of structure determination  (p. 388) | html | pdf |
    • 4.1.5. The direct-methods sum function  (p. 388) | html | pdf |
    • 4.1.6. The Patterson function  (p. 388) | html | pdf |
    • 4.1.7. Resonant (anomalous) scattering  (pp. 388-389) | html | pdf |
    • 4.1.8. Isomorphous replacement  (p. 389) | html | pdf |
    • 4.1.9. Maximum-entropy methods  (p. 389) | html | pdf |
    • 4.1.10. Charge flipping  (pp. 389-390) | html | pdf |
    • 4.1.11. Molecular envelopes  (p. 390) | html | pdf |
    • 4.1.12. Model building  (p. 390) | html | pdf |
    • 4.1.13. Molecular replacement  (p. 390) | html | pdf |
    • 4.1.14. Global optimization  (pp. 390-391) | html | pdf |
    • 4.1.15. Maximum-likelihood methods  (p. 391) | html | pdf |
    • 4.1.16. Local minimization  (p. 391) | html | pdf |
    • 4.1.17. Active use of prior information for particular structural classes  (pp. 391-392) | html | pdf |
    • 4.1.18. Combined figures of merit  (p. 392) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.1.1. A generalized scheme for structure determination from powder diffraction data  (p. 386) | html | pdf |
      • Fig. 4.1.2. A set of structure factors is Fourier transformed to yield a density map  (p. 389) | html | pdf |
    • Tables
      • Table 4.1.1. Some representative examples of structures solved using the methods described in this chapter  (p. 387) | html | pdf |
  • 4.2. Solving crystal structures using reciprocal-space methods  (pp. 395-413) | html | pdf | chapter contents |
    A. Altomare, C. Cuocci, A. Moliterni and R. Rizzi
    • 4.2.1. Introduction  (p. 395) | html | pdf |
    • 4.2.2. Intensity extraction for RS methods  (p. 395) | html | pdf |
    • 4.2.3. Direct methods  (pp. 395-399) | html | pdf |
      • 4.2.3.1. Normalized structure factors  (pp. 396-397) | html | pdf |
      • 4.2.3.2. Structure invariants  (p. 397) | html | pdf |
      • 4.2.3.3. Triplet invariants  (p. 397) | html | pdf |
      • 4.2.3.4. How direct methods work  (pp. 397-399) | html | pdf |
    • 4.2.4. Improving data normalization  (pp. 399-400) | html | pdf |
      • 4.2.4.1. Pseudotranslational symmetry  (p. 399) | html | pdf |
      • 4.2.4.2. Preferred orientation  (pp. 399-400) | html | pdf |
      • 4.2.4.3. Systematic decomposition  (p. 400) | html | pdf |
    • 4.2.5. Patterson methods  (pp. 400-401) | html | pdf |
      • 4.2.5.1. The use of the Patterson function for estimating integrated intensities  (p. 401) | html | pdf |
    • 4.2.6. Charge flipping  (pp. 401-403) | html | pdf |
    • 4.2.7. Maximum-entropy methods  (pp. 403-404) | html | pdf |
    • 4.2.8. Optimization of the structure model  (pp. 404-406) | html | pdf |
      • 4.2.8.1. Fourier recycling (FR)  (p. 404) | html | pdf |
      • 4.2.8.2. Weighted least-squares (WLSQ) refinement  (pp. 404-405) | html | pdf |
      • 4.2.8.3. Resolution bias modification (RBM)  (pp. 405-406) | html | pdf |
    • 4.2.9. Software packages for powder solution  (pp. 406-410) | html | pdf |
      • 4.2.9.1. Example of structure solution by XLENS  (p. 406) | html | pdf |
      • 4.2.9.2. Example of structure solution by SUPERFLIP  (p. 407) | html | pdf |
      • 4.2.9.3. The input file for a default run of EXPO  (p. 407) | html | pdf |
      • 4.2.9.4. An example of model optimization by EXPO for an inorganic compound  (pp. 407-408) | html | pdf |
      • 4.2.9.5. Example of model optimization by EXPO for an organic compound  (pp. 408-410) | html | pdf |
      • 4.2.9.6. The ALLTRIALS tool in EXPO: exploring all the phase sets  (p. 410) | html | pdf |
      • 4.2.9.7. Graphical tools in EXPO  (p. 410) | html | pdf |
    • 4.2.10. Conclusions  (p. 410) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.2.1. Flow chart for the DM procedure  (p. 398) | html | pdf |
      • Fig. 4.2.2. Crystal structure of (S)-(+)-ibuprofen solved by XLENS  (p. 406) | html | pdf |
      • Fig. 4.2.3. Crystal structure of the zeolite ZrPOF-EA solved by SUPERFLIP  (p. 407) | html | pdf |
      • Fig. 4.2.4. Default input file for EXPO for the structure of clomipramine  (p. 407) | html | pdf |
      • Fig. 4.2.5. Structure solution for chromium chromate tetrachromate, Cr₈O₂₁: (a) the powder pattern; (b) the DM structure model; (c) the model optimized by the automatic EXPO WLSQ-FR procedure  (p. 408) | html | pdf |
      • Fig. 4.2.6. Structure solution for clomipramine hydrochloride, C₁₉H₂₃ClN₂·HCl: (a) the powder pattern; (b) the DM structure model; (c) the model optimized by the automatic EXPO RBM procedure  (p. 409) | html | pdf |
      • Fig. 4.2.7. The ALLTRIALS procedure in EXPO for solving 1,4-bis-(2-phenethyloxy-ethanesulfonyl)-piperazine, C₂₄H₃₄N₂O₆S₂: (a) the phasing sets ranked according to CFOM; (b) the phasing sets ranked according to R_F; (c) the correct solution is automatically detected as set No  (p. 409) | html | pdf |
  • 4.3. Real-space methods for structure solution from powder-diffraction data: application to molecular structures  (pp. 414-432) | html | pdf | chapter contents |
    W. I. F. David
    • 4.3.1. Introduction  (pp. 414-416) | html | pdf |
    • 4.3.2. Real-space structure determination: a global-optimization extension of the Rietveld method  (p. 416) | html | pdf |
    • 4.3.3. Optimizing the process of real-space structure determination from powder-diffraction data  (pp. 416-418) | html | pdf |
      • 4.3.3.1. Introduction  (pp. 416-417) | html | pdf |
      • 4.3.3.2. Parameterization of the molecular structure  (pp. 417-418) | html | pdf |
    • 4.3.4. The nature and magnitude of the global-optimization challenge  (pp. 418-424) | html | pdf |
      • 4.3.4.1. Interpretation of the least-squares cost function  (pp. 419-421) | html | pdf |
      • 4.3.4.2. The chi-squared hypersurface  (pp. 421-424) | html | pdf |
    • 4.3.5. Global-optimization algorithms  (pp. 424-430) | html | pdf |
      • 4.3.5.1. Stochastic search algorithms  (pp. 425-429) | html | pdf |
        • 4.3.5.1.1. Introduction  (p. 426) | html | pdf |
        • 4.3.5.1.2. Monte Carlo methods  (p. 426) | html | pdf |
        • 4.3.5.1.3. Simulated-annealing methods  (pp. 426-427) | html | pdf |
        • 4.3.5.1.4. Parallel tempering  (pp. 427-428) | html | pdf |
        • 4.3.5.1.5. Genetic algorithms  (pp. 428-429) | html | pdf |
        • 4.3.5.1.6. Differential evolution  (p. 429) | html | pdf |
      • 4.3.5.2. Deterministic search algorithms  (pp. 429-430) | html | pdf |
        • 4.3.5.2.1. Introduction  (p. 429) | html | pdf |
        • 4.3.5.2.2. The hybrid Monte Carlo algorithm  (pp. 429-430) | html | pdf |
    • 4.3.6. Using maximum-likelihood techniques to tackle incomplete structural models  (pp. 430-431) | html | pdf |
      • 4.3.6.1. Introduction  (p. 430) | html | pdf |
      • 4.3.6.2. Working with incomplete structural models: maximum-likelihood methods  (pp. 430-431) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.3.1. (a) The maze of strategies associated with the determination of crystal structures from powder-diffraction data (after Baerlocher & McCusker, 2002) and (b) the modified `global-optimization' maze showing the double start point and simplification of the principal maze  (p. 415) | html | pdf |
      • Fig. 4.3.2. (a) The Cartesian molecular coordinate system for hydrochlorothiazide  (p. 417) | html | pdf |
      • Fig. 4.3.3. The reported values for both and in (a) the final stages of the structure-solution process for promazine hydrochloride  (p. 420) | html | pdf |
      • Fig. 4.3.4. (a) The molecular structure and (b) the b-axis projection of the crystal structure of the polymorphic form B of famotidine  (p. 421) | html | pdf |
      • Fig. 4.3.5. The frequency distribution for famotidine (form B)  (p. 421) | html | pdf |
      • Fig. 4.3.6. Two-dimensional section of the hypersurface showing the variation in as function of x and z translations  (p. 422) | html | pdf |
      • Fig. 4.3.7. Comparison of the crystal structure shifted by z = 1/4 with the correct crystal structure of famotidine (form B)  (p. 422) | html | pdf |
      • Fig. 4.3.8. (a) The log-normal distribution of local minima obtained from steepest-descents minimization of 20 000 randomly selected famotidine form B crystal structures  (p. 423) | html | pdf |
      • Fig. 4.3.9. Three good solutions for the crystal structure determination of famotidine form B  (p. 423) | html | pdf |
      • Fig. 4.3.10. The crystal structure of famotidine form B associated with a Δx = 1/4 shift from the correct solution  (p. 424) | html | pdf |
      • Fig. 4.3.11. The move-by-move change in observed during a real-space structure-solution run for famotidine form B  (p. 424) | html | pdf |
      • Fig. 4.3.12. (a) A simple periodic potential-energy function constructed over a square cell  (p. 425) | html | pdf |
      • Fig. 4.3.13. Molecular structures of 1,4-bis(2-phenethyloxyethanesulfonyl)piperazine, famotidine, 2-{[3-(2-phenylethoxy)propyl]sulfonyl}ethyl benzoate and verapamil hydrochloride  (p. 426) | html | pdf |
      • Fig. 4.3.14. The molecular structure and crystal structure of the malaria pigment B-haematin  (p. 427) | html | pdf |
      • Fig. 4.3.15. A plot of the best χ<sup>2</sup> values versus the number of χ<sup>2</sup> evaluations for the 20 simulated-annealing structure-solution attempts for capsaicin (after Shankland et al  (p. 427) | html | pdf |
      • Fig. 4.3.16. A single-point molecular crossover for cimetidine  (p. 428) | html | pdf |
      • Fig. 4.3.17. Mutation of a single gene influencing the orientation of a cimetidine molecule within the unit cell  (p. 428) | html | pdf |
      • Fig. 4.3.18. A plot of the best χ<sup>2</sup> values versus the number of χ<sup>2</sup> evaluations for the 20 hybrid Monte Carlo structure-solution attempts for capsaicin (after Shankland et al  (p. 429) | html | pdf |
      • Fig. 4.3.19. (a) The correct crystal structure of remacemide nitrate  (p. 430) | html | pdf |
    • Tables
      • Table 4.3.1. The symbolic representation of the initial bonds, angles and dihedral angles described in the Z-matrix of hydrochlorothiazide  (p. 418) | html | pdf |
      • Table 4.3.2. Z-matrix for hydrochlorothiazide  (p. 418) | html | pdf |
      • Table 4.3.3. Estimated minimum d-spacing range extent (in Å) for structure solution for different complexities of molecular structures using low (5 × 10<sup>−3</sup>), medium (2 × 10<sup>−3</sup>) and high (2 × 10<sup>−4</sup>) resolution instrumentation  (p. 421) | html | pdf |
  • 4.4. The use of supplementary information to solve crystal structures from powder diffraction  (pp. 433-441) | html | pdf | chapter contents |
    A. J. Florence
    • 4.4.1. General remarks  (p. 433) | html | pdf |
    • 4.4.2. Molecular models  (pp. 433-439) | html | pdf |
      • 4.4.2.1. Molecular volume  (p. 434) | html | pdf |
      • 4.4.2.2. Fragment model selection: bond length and angles  (pp. 434-435) | html | pdf |
      • 4.4.2.3. Flexible ring conformations  (pp. 435-436) | html | pdf |
      • 4.4.2.4. Torsion-angle constraints  (pp. 436-438) | html | pdf |
        • 4.4.2.4.1. Crystal-structure-derived information  (pp. 436-437) | html | pdf |
        • 4.4.2.4.2. Solid-state-NMR-derived information  (pp. 437-438) | html | pdf |
      • 4.4.2.5. Intermolecular distance constraints  (p. 438) | html | pdf |
      • 4.4.2.6. H-atom location in structures solved from X-ray powder data  (pp. 438-439) | html | pdf |
      • 4.4.2.7. Crystal-structure prediction and crystal-structure solution  (p. 439) | html | pdf |
    • 4.4.3. Concluding remarks  (p. 439) | html | pdf |
      • 4.4.3.1. Test data for SDPD  (p. 439) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.4.1. Key steps in crystal-structure determination from powder diffraction data (after Florence, 2009)  (p. 433) | html | pdf |
      • Fig. 4.4.2. Automatic generation of reliable flexible-ring conformations in Corina  (p. 435) | html | pdf |
      • Fig. 4.4.3. Determining ring conformation during global optimization using ring opening  (p. 435) | html | pdf |
      • Fig. 4.4.4. The structure of ibuprofen with flexible torsions highlighted (arrows)  (p. 436) | html | pdf |
      • Fig. 4.4.5. A trimodal torsion-angle distribution obtained from the CSD using Mogul for the C—C—C—C torsion in the query fragment (shown)  (p. 436) | html | pdf |
      • Fig. 4.4.6. Top: structure of cimetidine molecule <sup>13</sup>C enriched at C2 and C16  (p. 437) | html | pdf |
      • Fig. 4.4.7. (a) Structure of hydrochlorothiazide (HCT) with H1 and H2 labelled; (b) overlay of HCT form-II SDPD (black) and single-crystal (grey) structures showing the discrepancy in N—H orientation around the N—S bond; (c) CASTEP-optimized SDPD (grey) and single-crystal (black) structures  (p. 438) | html | pdf |
    • Tables
      • Table 4.4.1. Comparison of molecular volumes for tetradecane-1,14-diol and carbamazepine from three different sources  (p. 434) | html | pdf |
      • Table 4.4.2. Allowed ranges for the flexible torsion angles (τ1–13) identified in the verapamil cation using Mogul (from Florence et al., 2009)  (p. 437) | html | pdf |
  • 4.5. Solving and refining inorganic structures  (pp. 442-451) | html | pdf | chapter contents |
    R. Černý and V. Favre-Nicolin
    • 4.5.1. The molecular/non-molecular and the organic/inorganic boundaries  (p. 442) | html | pdf |
    • 4.5.2. Modelling of non-molecular compounds  (p. 442) | html | pdf |
      • 4.5.2.1. Principles  (p. 442) | html | pdf |
      • 4.5.2.2. Building units  (pp. 442-443) | html | pdf |
      • 4.5.2.3. Special positions and the sharing of atoms between building units  (pp. 443-444) | html | pdf |
      • 4.5.2.4. Is there a need for precise unit-cell contents?  (p. 444) | html | pdf |
      • 4.5.2.5. Building units and the convergence of the DSM algorithm  (p. 444) | html | pdf |
      • 4.5.2.6. Random versus non-random DSM algorithms  (pp. 444-445) | html | pdf |
      • 4.5.2.7. The choice of the cost function in DSM algorithms  (p. 445) | html | pdf |
    • 4.5.3. Solving, refining and completing inorganic structures  (pp. 445-448) | html | pdf |
      • 4.5.3.1. A general problem: poor quality powder data  (p. 445) | html | pdf |
      • 4.5.3.2. Indexing of multiphase samples  (p. 445) | html | pdf |
      • 4.5.3.3. X-rays versus neutrons  (pp. 445-446) | html | pdf |
      • 4.5.3.4. Chemical and positional disorder  (pp. 446-447) | html | pdf |
      • 4.5.3.5. Structure validation – theoretical issues  (pp. 447-448) | html | pdf |
    • 4.5.4. Examples  (pp. 448-450) | html | pdf |
      • 4.5.4.1. Polyhedral compounds  (pp. 448-449) | html | pdf |
      • 4.5.4.2. Hybrid compounds  (p. 449) | html | pdf |
      • 4.5.4.3. Close-packed compounds  (p. 450) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.5.1. A non-molecular crystal can be described by primary building units (PBUs), which can be (a) isolated atoms, (b) coordination polyhedra, (c) molecules, or a more complex and flexible arrangement of secondary building units (SBUs)  (p. 442) | html | pdf |
      • Fig. 4.5.2. Ab initio structure-solution algorithms working in direct space must be able to automatically handle (a) atoms that are shared between individual building units and (b) atoms that are on special positions  (p. 443) | html | pdf |
      • Fig. 4.5.3. Modelling of the aluminium methylphosphonate structure using: (left) three H<sub>3</sub>C-PO<sub>3</sub> blocks MP1–MP3 and two Al atoms, and (right) one AlO<sub>5</sub> trigonal bipyramid for Al1, one AlO<sub>4</sub> tetrahedron for Al2 and three H<sub>3</sub>C-P groups MP1–MP3  (p. 444) | html | pdf |
      • Fig. 4.5.4. In situ synchrotron-radiation X-ray powder diffraction data as a function of temperature for a ball-milled sample of LiBH<sub>4</sub>:CsBH<sub>4</sub> (2:1) (room temperature to 428 K, λ = 0.8210 Å)  (p. 446) | html | pdf |
      • Fig. 4.5.5. Yttrium borohydride appears to show positional disorder on the BH<sub>4</sub> groups when only XPD data are used (left: Ravnsbaek, Filinchuk, Černý, Ley et al  (p. 446) | html | pdf |
      • Fig. 4.5.6. XPD (Cu Kα<sub>1</sub>, left) and NPD (λ = 1.5558 Å, right) patterns for the high-temperature modification of yttrium borohydride  (p. 447) | html | pdf |
      • Fig. 4.5.7. Structural model of Li<sub>4</sub>Al<sub>3</sub>(BH<sub>4</sub>)<sub>13</sub> as solved from synchrotron XPD (left) and as corrected by DFT optimization (right)  (p. 447) | html | pdf |
      • Fig. 4.5.8. Structural fragment of a nanocrystalline inorganic–organic hybrid compound VO(C<sub>6</sub>H<sub>5</sub>COO)<sub>2</sub> (Djerdj et al  (p. 448) | html | pdf |
      • Fig. 4.5.9. Structural model of Mg(BH<sub>4</sub>)<sub>2</sub> as solved from XPD using a DSM (Černý et al  (p. 448) | html | pdf |
      • Fig. 4.5.10. The crystal structure of MOF-525 (Morris et al  (p. 449) | html | pdf |
      • Fig. 4.5.11. The crystal structure of the close-packed intermetallic phase MgIr (Černý et al  (p. 450) | html | pdf |
  • 4.6. Zeolites  (pp. 452-464) | html | pdf | chapter contents |
    L. B. McCusker and Ch. Baerlocher
    • 4.6.1. Introduction  (p. 452) | html | pdf |
    • 4.6.2. Framework structure determination  (pp. 452-458) | html | pdf |
      • 4.6.2.1. Preliminary steps  (p. 454) | html | pdf |
      • 4.6.2.2. Model building  (pp. 454-455) | html | pdf |
      • 4.6.2.3. Focus  (pp. 455-456) | html | pdf |
      • 4.6.2.4. Direct methods  (p. 456) | html | pdf |
      • 4.6.2.5. Charge flipping  (pp. 456-457) | html | pdf |
      • 4.6.2.6. Using additional information  (pp. 457-458) | html | pdf |
    • 4.6.3. Structure refinement  (pp. 458-462) | html | pdf |
      • 4.6.3.1. Space-group ambiguity  (p. 458) | html | pdf |
      • 4.6.3.2. Geometric restraints  (p. 458) | html | pdf |
      • 4.6.3.3. Disorder  (pp. 458-459) | html | pdf |
      • 4.6.3.4. Locating non-framework species  (pp. 459-460) | html | pdf |
      • 4.6.3.5. Rietveld refinement of SSZ-74  (pp. 460-462) | html | pdf |
      • 4.6.3.6. Locating isomorphously substituted T atoms  (p. 462) | html | pdf |
    • 4.6.4. Conclusions  (pp. 462-463) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.6.1. The 231 framework type codes assigned by March 2016 and how their structures were solved  (p. 452) | html | pdf |
      • Fig. 4.6.2. The evolution of methods used to solve zeolite framework structures  (p. 453) | html | pdf |
      • Fig. 4.6.3. A histogram of the number of T atoms in the asymmetric unit for the 231 zeolite framework types, showing the methods used to solve their structures  (p. 453) | html | pdf |
      • Fig. 4.6.4. The framework structure of zeolite A (LTA) showing the 8-rings on the surfaces of the cubic unit cell and the linkages between them to form the lta cavity  (p. 455) | html | pdf |
      • Fig. 4.6.5. The `butterfly' unit and projections of the three 10- and 12-ring framework types that have this unit, one-dimensional channels and a 5 Å repeat along the channel  (p. 455) | html | pdf |
      • Fig. 4.6.6. Flow chart of the Focus program  (p. 456) | html | pdf |
      • Fig. 4.6.7. Flow chart of the charge-flipping algorithm implemented in Superflip  (p. 457) | html | pdf |
      • Fig. 4.6.8. Structure envelope for SSZ-74  (p. 458) | html | pdf |
      • Fig. 4.6.9. The disorder of the Na⁺ ion (green) in the 8-ring of zeolite A (LTA)  (p. 459) | html | pdf |
      • Fig. 4.6.10. (a) The measured powder diffraction pattern for the silicoalumino­phos­phate SAPO-40 (AFR)  (p. 459) | html | pdf |
      • Fig. 4.6.11. Difference electron-density maps (blue) calculated for SAPO-40 (AFR) along the [001] (left) and [010] (right) directions using (a) the scale factor calculated for the full diffraction pattern (Fig  (p. 460) | html | pdf |
      • Fig. 4.6.12. The structure completion and Rietveld refinement of SSZ-74 (-SVR)  (p. 461) | html | pdf |
      • Fig. 4.6.13. The structure-directing agent (SDA) used in the synthesis of SSZ-74 and the difference electron-density map (orange) generated from the pattern and model shown in Fig  (p. 462) | html | pdf |
  • 4.7. Rietveld refinement  (pp. 465-472) | html | pdf | chapter contents |
    B. H. Toby
    • 4.7.1. History  (pp. 465-466) | html | pdf |
    • 4.7.2. Rietveld versus single-crystal refinements  (p. 466) | html | pdf |
    • 4.7.3. Multiphase and multiple data set fitting  (pp. 466-467) | html | pdf |
    • 4.7.4. The mechanism of Rietveld fitting  (p. 467) | html | pdf |
    • 4.7.5. Types of profile functions  (p. 467) | html | pdf |
    • 4.7.6. Observed structure-factor estimates  (pp. 467-468) | html | pdf |
    • 4.7.7. Estimating observed structure factors without a structure  (p. 468) | html | pdf |
    • 4.7.8. Agreement factors  (pp. 468-469) | html | pdf |
    • 4.7.9. Restraints and constraints in Rietveld refinement  (pp. 469-470) | html | pdf |
    • 4.7.10. The order in which to introduce parameters in a fit  (pp. 470-471) | html | pdf |
    • 4.7.11. Examples  (p. 471) | html | pdf |
    • 4.7.12. Conclusions  (p. 471) | html | pdf |
    • References | html | pdf |
  • 4.8. Application of the maximum-entropy method to powder-diffraction data  (pp. 473-488) | html | pdf | chapter contents |
    O. V. Magdysyuk, S. van Smaalen and R. E. Dinnebier
    • 4.8.1. Introduction  (pp. 473-474) | html | pdf |
    • 4.8.2. The principle of maximum entropy  (pp. 474-475) | html | pdf |
    • 4.8.3. Prior densities in MEM calculations  (pp. 475-476) | html | pdf |
    • 4.8.4. Crystallographic MEM equations  (p. 476) | html | pdf |
    • 4.8.5. MEM algorithms  (pp. 476-477) | html | pdf |
      • 4.8.5.1. Exponential modelling algorithm (Gull & Daniell, 1978; Collins, 1982; Collins & Mahar, 1983)  (pp. 476-477) | html | pdf |
      • 4.8.5.2. Sakata–Sato algorithm (Sakata & Sato, 1990; Sakata et al., 1990, 1992, 1993; Kumazawa et al., 1995)  (p. 477) | html | pdf |
      • 4.8.5.3. Cambridge algorithm (Bryan & Skilling, 1980, 1986; Burch et al., 1983; Skilling & Bryan, 1984; Gull & Skilling, 1987, 1999)  (p. 477) | html | pdf |
      • 4.8.5.4. Two-channel MaxEnt method (Papoular & Gillon, 1990; Zheludev et al., 1995; Papoular et al., 1996)  (p. 477) | html | pdf |
      • 4.8.5.5. Valence-only MaxEnt method (Roversi et al., 1998)  (p. 477) | html | pdf |
    • 4.8.6. Constraints in the MEM based on the measured intensities from powder-diffraction data  (pp. 477-480) | html | pdf |
      • 4.8.6.1. F and G constraints  (pp. 478-479) | html | pdf |
      • 4.8.6.2. Constraints using `partly phased' reflections for anomalous-scattering X-ray powder diffraction  (pp. 479-480) | html | pdf |
      • 4.8.6.3. Prior-derived F constraints  (p. 480) | html | pdf |
      • 4.8.6.4. Observed structure factors  (p. 480) | html | pdf |
    • 4.8.7. Analysis of the electron-density distribution from powder-diffraction data  (pp. 480-485) | html | pdf |
      • 4.8.7.1. Types of MEM-reconstructed electron-density distribution  (pp. 480-482) | html | pdf |
      • 4.8.7.2. MEM-based improvement of the structure model from charge flipping  (p. 482) | html | pdf |
      • 4.8.7.3. Combination of MEM and Rietveld refinement (REMEDY cycle)  (pp. 482-483) | html | pdf |
      • 4.8.7.4. MEM-assisted structure solution by charge flipping  (pp. 483-484) | html | pdf |
      • 4.8.7.5. Disorder and anharmonic atomic displacements  (p. 484) | html | pdf |
      • 4.8.7.6. Accurate electron densities  (pp. 484-485) | html | pdf |
    • 4.8.8. Extensions of the MEM  (pp. 485-486) | html | pdf |
      • 4.8.8.1. Distribution of normalized residuals of the structure factors  (p. 485) | html | pdf |
      • 4.8.8.2. Stopping criterion  (pp. 485-486) | html | pdf |
      • 4.8.8.3. Generalization of MaxEnt algorithm to n dimensions (n > 3) and application to powder-diffraction data of aperiodic structures  (p. 486) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.8.1. Single-crystal (left) and powder (right) X-ray diffraction data for Pb<sub>3</sub>O<sub>4</sub>  (p. 473) | html | pdf |
      • Fig. 4.8.2. Electron-density distribution for threefold disordered pentamethyl­cyclopentadienyllithium (Li[C<sub>10</sub>H<sub>15</sub>], LiCp*)  (p. 474) | html | pdf |
      • Fig. 4.8.3. Electron-density distribution in the (110) plane of Si, calculated from powder X-ray diffraction data  (p. 474) | html | pdf |
      • Fig. 4.8.4. Dividing of the total intensity of the overlapping reflections into the contributions of individual reflections by Rietveld refinement (with a structural model) and by Le Bail fitting (without a structural model) for powder-diffraction data for K<sub>2</sub>C<sub>2</sub>O<sub>4</sub> (adapted from Samy et al  (p. 478) | html | pdf |
      • Fig. 4.8.5. Observed intensity (points) as a function of the scattering angle, together with lines representing (a) the calculated intensity of the Rietveld refinement and (b) the intensities after decomposition according to equation (4.8.22) (from Takata, 2008)  (p. 479) | html | pdf |
      • Fig. 4.8.6. Flow chart of the procedure for reconstruction of the different MEM maps from powder-diffraction data  (p. 481) | html | pdf |
      • Fig. 4.8.7. Structural model from Rietveld refinement and MEM electron densities of disordered α-Rb<sub>2</sub>C<sub>2</sub>O<sub>4</sub> (left) and α-Rb<sub>2</sub>CO<sub>3</sub> (right)  (p. 482) | html | pdf |
      • Fig. 4.8.8. Flow chart of MEM/Rietveld method with structure refinement and MEM imaging of the charge density (adapted from Takata, 2008)  (p. 483) | html | pdf |
      • Fig. 4.8.9. Acetylene adsorption isotherm and crystal structures of CPL-1 with adsorption of acetylene corresponding to the amount of adsorbed gas  (p. 483) | html | pdf |
      • Fig. 4.8.10. The top panel shows the (110) plane of the crystal structure of SiAl<sub>4</sub>O<sub>2</sub>N<sub>4</sub>; red and yellow indicate fractional occupancies by oxygen and nitrogen, respectively; dark blue indicates the fractional occupancy of the (Si,Al)2A and (Si,Al)2B sites  (p. 484) | html | pdf |
      • Fig. 4.8.11. Crystal structure and MEM-reconstructed negative nuclear-density map of one unit cell of Li<sub>1.3</sub>Al<sub>0.3</sub>Ti<sub>1.7</sub>(PO<sub>4</sub>)<sub>3</sub>, and the section of this density  (p. 484) | html | pdf |
  • 4.9. Structure validation  (pp. 489-514) | html | pdf | chapter contents |
    J. A. Kaduk
    • 4.9.1. Introduction  (p. 489) | html | pdf |
    • 4.9.2. Statistical measures (with contributions from B. H. Toby)  (pp. 489-490) | html | pdf |
    • 4.9.3. Graphical measures (with contributions from B. H. Toby and J. K. Stalick)  (pp. 490-496) | html | pdf |
    • 4.9.4. Chemical reasonableness  (pp. 496-508) | html | pdf |
      • 4.9.4.1. Organic compounds  (pp. 496-503) | html | pdf |
        • 4.9.4.1.1. Calcium tartrate tetrahydrate  (pp. 497-500) | html | pdf |
        • 4.9.4.1.2. Guaifenesin  (pp. 500-502) | html | pdf |
        • 4.9.4.1.3. Cobalt(II) acetate tetrahydrate, Co(C<sub>2</sub>H<sub>3</sub>O<sub>2</sub>)<sub>2</sub>(H<sub>2</sub>O)<sub>4</sub>  (pp. 502-503) | html | pdf |
      • 4.9.4.2. Inorganic compounds  (pp. 503-508) | html | pdf |
        • 4.9.4.2.1. The bond-valence method  (p. 503) | html | pdf |
        • 4.9.4.2.2. Hexaaquairon(II) tetrafluoroborate, [Fe(H<sub>2</sub>O)<sub>6</sub>](BF<sub>4</sub>)<sub>2</sub>  (pp. 503-505) | html | pdf |
        • 4.9.4.2.3. (Ba<sub>1.5</sub>Sr<sub>0.5</sub>)TiO<sub>4</sub>  (pp. 505-506) | html | pdf |
        • 4.9.4.2.4. (Ba<sub>1.25</sub>Sr<sub>0.75</sub>)TiO<sub>4</sub>  (pp. 506-507) | html | pdf |
        • 4.9.4.2.5. Mullite, Al<sub>4.85</sub>Si<sub>1.18</sub>O<sub>9.77</sub>  (p. 507) | html | pdf |
        • 4.9.4.2.6. 6H perovskite Ba<sub>3</sub>CaSb<sub>2</sub>O<sub>9</sub>  (pp. 507-508) | html | pdf |
        • 4.9.4.2.7. Quartz  (p. 508) | html | pdf |
    • 4.9.5. Working by analogy  (p. 508) | html | pdf |
      • 4.9.5.1.   (p. 508) | html | pdf |
    • 4.9.6. Structure validation in general  (pp. 508-509) | html | pdf |
    • 4.9.7. CheckCIF/PLATON  (pp. 509-512) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.9.1. A `Rietveld' plot from the refinement of calcium tartrate tetrahydrate  (p. 491) | html | pdf |
      • Fig. 4.9.2. The calcium tartrate tetrahydrate Rietveld plot of Fig  (p. 491) | html | pdf |
      • Fig. 4.9.3. The calcium tartrate tetrahydrate Rietveld plot of Fig  (p. 491) | html | pdf |
      • Fig. 4.9.4. A normalized Rietveld plot for calcium tartrate tetrahydrate  (p. 492) | html | pdf |
      • Fig. 4.9.5. A weighted Rietveld plot for calcium tartrate tetrahydrate  (p. 492) | html | pdf |
      • Fig. 4.9.6. (a) A Rietveld plot from a refinement of NIST SRM 660a, LaB₆  (p. 492) | html | pdf |
      • Fig. 4.9.7. The calcium tartrate tetrahydrate Rietveld plot of Fig  (p. 492) | html | pdf |
      • Fig. 4.9.8. A normal probability plot from the refinement of calcium tartrate tetrahydrate  (p. 493) | html | pdf |
      • Fig. 4.9.9. A Rietveld plot of calcium tartrate tetrahydrate after refinement of a phase fraction and a three-term shifted Chebyshev background function  (p. 493) | html | pdf |
      • Fig. 4.9.10. A weighted Rietveld plot of calcium tartrate tetrahydrate after refinement of a phase fraction and a three-term shifted Chebyshev background function  (p. 493) | html | pdf |
      • Fig. 4.9.11. A small region of the calcium tartrate tetrahydrate Rietveld plot early in the refinement  (p. 494) | html | pdf |
      • Fig. 4.9.12. The same 34.5–36° region of the pattern of calcium tartrate tetrahydrate after refinement of the cell parameters and specimen displacement  (p. 494) | html | pdf |
      • Fig. 4.9.13. (a) The 34.5–36° region of the calcium tartrate tetrahydrate pattern after refinement of profile coefficients  (p. 494) | html | pdf |
      • Fig. 4.9.14. A full-scale Rietveld plot from the refinement of Ba₃CaSb₂O₉ (Rowda et al  (p. 495) | html | pdf |
      • Fig. 4.9.15. The plot of Fig  (p. 495) | html | pdf |
      • Fig. 4.9.16. Full scale Rietveld plot of (Ba_1.25Sr_0.75)TiO₄  (p. 495) | html | pdf |
      • Fig. 4.9.17. A portion of the plot of Fig  (p. 495) | html | pdf |
      • Fig. 4.9.18. A portion of the Rietveld plot of (Ba_1.25Sr_0.75)TiO₄ after the correct unit cell and structure had been identified  (p. 495) | html | pdf |
      • Fig. 4.9.19. A Rietveld plot of (Ba_1.5Sr_0.5)TiO₄ early in the refinement, with an incorrect unit cell for the major phase  (p. 496) | html | pdf |
      • Fig. 4.9.20. The low-angle portion of the Rietveld plot of (Ba_1.5Sr_0.5)TiO₄, showing that the unit cell predicts many peaks which are not observed  (p. 496) | html | pdf |
      • Fig. 4.9.21. The structure of a tartrate anion  (p. 497) | html | pdf |
      • Fig. 4.9.22. A Mercury plot of the distribution of the lengths of the central C3—C4 bond in tartrate anions  (p. 497) | html | pdf |
      • Fig. 4.9.23. A Mercury histogram of the O8—C3—C4 bond angle in tartrate anions  (p. 497) | html | pdf |
      • Fig. 4.9.24. A Mercury polar histogram of the O8—C3—C4—O9 torsion angles in tartrate anions  (p. 497) | html | pdf |
      • Fig. 4.9.25. Refined molecular structure of guaifenesin  (p. 500) | html | pdf |
      • Fig. 4.9.26. A Mogul Geometry Check histogram of the `unusual' C5—O16 distance in guaifenesin, showing its position in the histogram of equivalent distances  (p. 501) | html | pdf |
      • Fig. 4.9.27. A Mogul Geometry Check histogram of the `unusual' C6—O11 distance in guaifenesin, showing its position in the histogram of equivalent distances  (p. 501) | html | pdf |
      • Fig. 4.9.28. A Mogul Geometry Check histogram of the `unusual' C12—O11—C6 angle in guaifenesin, showing its position in the histogram of equivalent angles  (p. 501) | html | pdf |
      • Fig. 4.9.29. A Mogul Geometry Check histogram of the `unusual' C17—O16—C5 angle in guaifenesin, showing its position in the histogram of equivalent angles  (p. 501) | html | pdf |
      • Fig. 4.9.30. A Mogul Geometry Check histogram of the O20—C19—C18—C17 torsion angle in guaifenesin, showing its position in the histogram of equivalent angles  (p. 501) | html | pdf |
      • Fig. 4.9.31. Comparison of the isotropic displacement coefficients for the powder (PANKUL01) and two single-crystal (PANKUL and PANKUL02) structures of guaifenesin  (p. 502) | html | pdf |
      • Fig. 4.9.32. Mercury overlay of the two single-crystal structures (PANKUL and PANKUL02) of guaifenesin, showing the differences in the positions of the hydroxyl protons  (p. 502) | html | pdf |
      • Fig. 4.9.33. Comparison of the experimental powder pattern of [Fe(H₂O)₆](BF₄)₂ and the PDF entry 00–021-0427 reported for this compound  (p. 505) | html | pdf |
      • Fig. 4.9.34. The observed pattern of (Ba_1.5Sr_0.5)TiO₄ compared with the experimental and calculated PDF entries for Ba₂TiO₄  (p. 505) | html | pdf |
      • Fig. 4.9.35. A Rietveld plot from an incorrect refinement of the structure of (Ba_1.5Sr_0.5)TiO₄  (p. 506) | html | pdf |
    • Tables
      • Table 4.9.1. Average bond distances in a tartrate anion  (p. 497) | html | pdf |
      • Table 4.9.2. Average bond angles in a tartrate anion  (p. 497) | html | pdf |
      • Table 4.9.3. Bond distances (Å) and angles (°) in cobalt(II) acetate tetrahydrate (Kaduk & Partenheimer, 1997)  (p. 503) | html | pdf |
      • Table 4.9.4. Root-mean-square density-weighted displacements (Å) of atoms in various refinements of cobalt(II) acetate tetrahydrate (Kaduk & Partenheimer, 1997)  (p. 503) | html | pdf |
      • Table 4.9.5. Displacement coefficients (Ų) from various refinements of cobalt(II) acetate tetrahydrate  (p. 504) | html | pdf |
      • Table 4.9.6. Average bond distances and angles from 26 experimental determinations of α-quartz compared with average values from 20 determinations in the ICSD  (p. 508) | html | pdf |
      • Table 4.9.7. R.m.s. differences Δ (Å) between the non-H atoms in experimental and DFT-optimized crystal structures of the citrate group in group-1 citrate salts  (p. 509) | html | pdf |
  • 4.10. Use of CIF for powder diffraction data  (pp. 515-521) | html | pdf | chapter contents |
    B. H. Toby
    • 4.10.1. Background  (p. 515) | html | pdf |
    • 4.10.2. A brief introduction to CIF  (pp. 515-516) | html | pdf |
    • 4.10.3. CIF usage for powder diffraction  (pp. 516-517) | html | pdf |
    • 4.10.4. Creation of pdCIF data files  (pp. 517-518) | html | pdf |
    • 4.10.5. Creating readers for pdCIF data files  (p. 518) | html | pdf |
      • 4.10.5.1. Ordinate  (p. 518) | html | pdf |
      • 4.10.5.2. Observed intensities  (p. 518) | html | pdf |
      • 4.10.5.3. Background intensities  (p. 518) | html | pdf |
      • 4.10.5.4. Computed intensities  (p. 518) | html | pdf |
    • 4.10.6. CIF software for powder diffraction  (pp. 518-521) | html | pdf |
      • 4.10.6.1. enCIFer  (p. 519) | html | pdf |
      • 4.10.6.2. The IUCr publCIF application  (p. 519) | html | pdf |
      • 4.10.6.3. The IUCr checkCIF web application  (p. 519) | html | pdf |
      • 4.10.6.4. The IUCr pdCIFplot application  (pp. 519-520) | html | pdf |
      • 4.10.6.5. International Centre for Diffraction Data (ICDD) Genie application  (p. 520) | html | pdf |
      • 4.10.6.6. pdCIF creation in GSAS-II  (pp. 520-521) | html | pdf |
    • 4.10.7. Conclusions  (p. 521) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 4.10.1. An example of an IUCr Journals web page where powder diffraction data have been submitted with the publication  (p. 519) | html | pdf |
      • Fig. 4.10.2. An example web-displayed pdCIFplot graphic for powder diffraction data from the article shown in Fig  (p. 519) | html | pdf |
      • Fig. 4.10.3. The publication-quality version of Fig  (p. 520) | html | pdf |
      • Fig. 4.10.4. An example dialogue window from GSAS-II used in preparation of a CIF file  (p. 520) | html | pdf |
      • Fig. 4.10.5. An example dialogue window from GSAS-II used in the preparation of a CIF file, to edit the bond distances and angles for one phase  (p. 520) | html | pdf |
      • Fig. 4.10.6. An example dialogue window from GSAS-II for editing the contents of a CIF template  (p. 521) | html | pdf |

• Defects, texture and microstructure
  • 5.1. Domain size and domain-size distributions  (pp. 524-537) | html | pdf | chapter contents |
    M. Leoni
    • 5.1.1. Introduction  (p. 524) | html | pdf |
    • 5.1.2. The Scherrer formula and integral-breadth methods  (p. 524) | html | pdf |
    • 5.1.3. Fourier methods  (p. 526) | html | pdf |
    • 5.1.4. The peak profile and the common volume function (`ghost')  (pp. 526-528) | html | pdf |
    • 5.1.5. Common volume function for simple shapes  (pp. 528-530) | html | pdf |
    • 5.1.6. Calculation of the intensity profile for simple shapes  (pp. 530-531) | html | pdf |
    • 5.1.7. Meaning of the size Fourier coefficients: mean size and column-length distribution  (p. 531) | html | pdf |
    • 5.1.8. Extension of the Fourier approach to a size distribution  (pp. 531-533) | html | pdf |
    • 5.1.9. Analytical distributions  (p. 533) | html | pdf |
    • 5.1.10. Histogram (arbitrary) distributions  (pp. 533-534) | html | pdf |
    • 5.1.11. Example  (pp. 534-536) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 5.1.1. Geometry employed to derive the Scherrer equation  (p. 525) | html | pdf |
      • Fig. 5.1.2. A domain and its ghost translated by a quantity L along hkl  (p. 527) | html | pdf |
      • Fig. 5.1.3. The intersection of two circles  (p. 528) | html | pdf |
      • Fig. 5.1.4. Size Fourier term for a sphere of 10 nm and tangent to the origin (initial slope)  (p. 531) | html | pdf |
      • Fig. 5.1.5. Nanocrystalline ceria pattern; the indices for some reflections are indicated  (p. 534) | html | pdf |
      • Fig. 5.1.6. Analysis of the 111 and 200 reflections of ceria: (a) calculation using the FWHM, (b) fit using Voigt functions: the single profiles are shown, (c) modelling as a single sphere, (d) modelling as a lognormal distribution of spheres, (e) comparison of the instrumental profile with the measured profile and modelling using a lognormal distribution of spheres, taking the instrumental contribution into account, and (f) cosine Fourier transform of the left side of the 111 peak  (p. 535) | html | pdf |
    • Tables
      • Table 5.1.1. Coefficients for the calculation of the common volume function for a sphere, cube, tetrahedron and octahedron  (p. 529) | html | pdf |
  • 5.2. Stress and strain  (pp. 538-554) | html | pdf | chapter contents |
    N. C. Popa
    • 5.2.1. The importance of determining stress and the diffraction method  (p. 538) | html | pdf |
    • 5.2.2. Strain and stress in single crystals, elastic constants and transformations  (p. 538) | html | pdf |
    • 5.2.3. Strain and stress in polycrystalline samples  (pp. 540-542) | html | pdf |
      • 5.2.3.1. Types of strains and stresses, and the strain/stress orientation distribution function  (pp. 540-541) | html | pdf |
      • 5.2.3.2. The mean and variance of the observable strain, the peak shift and broadening  (pp. 541-542) | html | pdf |
    • 5.2.4. Determining the strain/stress by diffraction  (pp. 542-543) | html | pdf |
      • 5.2.4.1. Determining the macrostrain/stress  (pp. 542-543) | html | pdf |
      • 5.2.4.2. Determining the microstrain  (p. 543) | html | pdf |
    • 5.2.5. Macrostrain/stress in isotropic samples: classical approximations  (pp. 543-546) | html | pdf |
      • 5.2.5.1. The Voigt model  (p. 544) | html | pdf |
      • 5.2.5.2. The Reuss model  (pp. 544-545) | html | pdf |
      • 5.2.5.3. The Hill average  (p. 545) | html | pdf |
      • 5.2.5.4. The Kroner model  (p. 545) | html | pdf |
      • 5.2.5.5. The `sin²Ψ' method  (pp. 545-546) | html | pdf |
      • 5.2.5.6. Determining the single-crystal elastic constants  (p. 546) | html | pdf |
    • 5.2.6. The hydrostatic state in isotropic polycrystals  (pp. 546-547) | html | pdf |
    • 5.2.7. Calculating the macroscopic strain/stress using spherical harmonics  (pp. 547-552) | html | pdf |
      • 5.2.7.1. Strain expansion in generalized spherical harmonics: the Popa and Balzar approach  (pp. 547-548) | html | pdf |
      • 5.2.7.2. The selection rules for all Laue classes  (p. 548) | html | pdf |
      • 5.2.7.3. Generalized spherical harmonics of real type, and the WSODF index  (pp. 548-549) | html | pdf |
      • 5.2.7.4. Determining the macrostrain/stress state of the sample  (pp. 549-550) | html | pdf |
      • 5.2.7.5. Simplified (`short') harmonics representation of the peak shift, and the `mixed' representation  (p. 550) | html | pdf |
      • 5.2.7.6. Implementation in Rietveld codes  (pp. 550-551) | html | pdf |
      • 5.2.7.7. An application: determining the averaged macroscopic strain and stress tensors in a rolled uranium plate from time-of-flight neutron diffraction data  (pp. 551-552) | html | pdf |
      • 5.2.7.8. Limitations of the spherical-harmonics approach and possible further developments  (p. 552) | html | pdf |
    • 5.2.8. The spherical-harmonics approach to strain broadening  (pp. 552-553) | html | pdf |
      • 5.2.8.1. Ignoring the macrostrain variance  (p. 552) | html | pdf |
      • 5.2.8.2. The double-dependent anisotropic strain breadth (DDASB)  (pp. 552-553) | html | pdf |
      • 5.2.8.3. The `classical' limit of the DDASB  (p. 553) | html | pdf |
    • References | html | pdf |
    • Tables
      • Table 5.2.1. The matrix C for all Laue groups represented by specific constraints  (p. 540) | html | pdf |
      • Table 5.2.2. Quadratic and quartic forms for symmetries higher than triclinic  (p. 545) | html | pdf |
      • Table 5.2.3. Assignment of functions to the coefficients   (p. 549) | html | pdf |
      • Table 5.2.4. The real generalized spherical harmonics  (p. 549) | html | pdf |
      • Table 5.2.5. The monomials j<sub>μl</sub>(a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>) for l = 2, 4, 6  (p. 550) | html | pdf |
      • Table 5.2.6. The complete set of polynomials J<sub>μl</sub> for l = 2, 4, 6 for all Laue groups  (p. 551) | html | pdf |
  • 5.3. Quantitative texture analysis and combined analysis  (pp. 555-580) | html | pdf | chapter contents |
    D. Chateigner, L. Lutterotti and M. Morales
    • 5.3.1. Introduction  (p. 555) | html | pdf |
    • 5.3.2. Crystallographic quantitative texture analysis (QTA)  (pp. 555-567) | html | pdf |
      • 5.3.2.1. Orientation distribution (OD)  (pp. 555-558) | html | pdf |
        • 5.3.2.1.1. The orientation space H  (pp. 555-556) | html | pdf |
        • 5.3.2.1.2. The orientation distribution (OD) or orientation distribution function (ODF)  (p. 556) | html | pdf |
        • 5.3.2.1.3. Pole figures  (pp. 556-558) | html | pdf |
          • 5.3.2.1.3.1. Mathematical expression  (pp. 556-557) | html | pdf |
          • 5.3.2.1.3.2. Diffraction pole figures and orientation of planes  (p. 557) | html | pdf |
          • 5.3.2.1.3.3. From diffraction measurements to pole figures and ODs  (p. 557) | html | pdf |
          • 5.3.2.1.3.4. Pole-figure normalization  (pp. 557-558) | html | pdf |
      • 5.3.2.2. The fundamental equation of quantitative texture analysis  (pp. 558-559) | html | pdf |
      • 5.3.2.3. Resolution of the fundamental equation  (pp. 559-562) | html | pdf |
        • 5.3.2.3.1. Generalized spherical-harmonics expansion  (pp. 559-560) | html | pdf |
        • 5.3.2.3.2. Vector method  (p. 560) | html | pdf |
        • 5.3.2.3.3. WIMV method  (p. 560) | html | pdf |
        • 5.3.2.3.4. Arbitrarily defined cells (ADC) method  (p. 560) | html | pdf |
        • 5.3.2.3.5. Entropy-maximization method  (p. 560) | html | pdf |
        • 5.3.2.3.6. EWIMV method  (p. 560) | html | pdf |
        • 5.3.2.3.7. Component method  (pp. 560-561) | html | pdf |
        • 5.3.2.3.8. Positivity and exponential harmonics  (pp. 561-562) | html | pdf |
        • 5.3.2.3.9. Radon transform and Fourier analysis  (p. 562) | html | pdf |
      • 5.3.2.4. Inverse pole figures  (pp. 562-563) | html | pdf |
      • 5.3.2.5. OD refinement reliability estimators  (p. 563) | html | pdf |
      • 5.3.2.6. Texture-strength factors  (pp. 563-564) | html | pdf |
        • 5.3.2.6.1. Texture index  (p. 564) | html | pdf |
          • 5.3.2.6.1.1. OD texture index  (p. 564) | html | pdf |
          • 5.3.2.6.1.2. Pole-figure texture index  (p. 564) | html | pdf |
        • 5.3.2.6.2. Texture entropy  (p. 564) | html | pdf |
        • 5.3.2.6.3. Pole-figure and ODF strengths  (p. 564) | html | pdf |
        • 5.3.2.6.4. Correlation between F<sup>2</sup> and S  (p. 564) | html | pdf |
      • 5.3.2.7. Texture types  (pp. 564-566) | html | pdf |
        • 5.3.2.7.1. Random texture  (pp. 564-565) | html | pdf |
        • 5.3.2.7.2. Planar textures  (p. 565) | html | pdf |
        • 5.3.2.7.3. Fibre textures  (p. 565) | html | pdf |
        • 5.3.2.7.4. Three-dimensional textures  (pp. 565-566) | html | pdf |
        • 5.3.2.7.5. Typical OD components  (p. 566) | html | pdf |
      • 5.3.2.8. Reciprocal-space mapping  (pp. 566-567) | html | pdf |
    • 5.3.3. Magnetic quantitative texture analysis (MQTA)  (pp. 567-569) | html | pdf |
      • 5.3.3.1. Magnetization curves and magnetic moment distributions  (p. 567) | html | pdf |
      • 5.3.3.2. Magnetic pole figures and magnetic ODs  (pp. 567-568) | html | pdf |
        • 5.3.3.2.1. Magnetic pole figures and ODs  (p. 567) | html | pdf |
        • 5.3.3.2.2. Fundamental equations of MQTA  (pp. 567-568) | html | pdf |
      • 5.3.3.3. An example  (pp. 568-569) | html | pdf |
    • 5.3.4. Modelling of preferred orientation in the Rietveld method  (pp. 569-571) | html | pdf |
      • 5.3.4.1. Rietveld and March approaches  (p. 569) | html | pdf |
      • 5.3.4.2. March–Dollase approach  (pp. 569-570) | html | pdf |
      • 5.3.4.3. Modified March–Dollase models  (p. 570) | html | pdf |
      • 5.3.4.4. Donnet–Jouanneaux function  (p. 570) | html | pdf |
      • 5.3.4.5. Modelling by spherical harmonics (and exponential)  (p. 570) | html | pdf |
      • 5.3.4.6. The use of standard functions (or texture components)  (p. 570) | html | pdf |
      • 5.3.4.7. Remarks  (pp. 570-571) | html | pdf |
    • 5.3.5. Combined analysis: structure, texture, microstructure, stress, phase, layering and reflectivity analyses in a single approach  (pp. 571-578) | html | pdf |
      • 5.3.5.1. Problems  (pp. 571-572) | html | pdf |
      • 5.3.5.2. Intensity of a pattern and general scheme  (p. 572) | html | pdf |
      • 5.3.5.3. Minimum experimental requirements  (p. 572) | html | pdf |
      • 5.3.5.4. Theoretical implementation  (pp. 572-577) | html | pdf |
        • 5.3.5.4.1. Instrumental broadening calibration  (pp. 572-573) | html | pdf |
        • 5.3.5.4.2. Peak-displacement errors  (p. 573) | html | pdf |
        • 5.3.5.4.3. Background fitting  (pp. 573-574) | html | pdf |
        • 5.3.5.4.4. Reflection intensities  (pp. 574-575) | html | pdf |
        • 5.3.5.4.5. Line profiles and sample broadening  (p. 575) | html | pdf |
        • 5.3.5.4.6. Texture computation  (p. 575) | html | pdf |
        • 5.3.5.4.7. Residual strains/stresses and evaluation of macroscopic tensors  (pp. 575-576) | html | pdf |
        • 5.3.5.4.8. Absorption and layers  (pp. 576-577) | html | pdf |
      • 5.3.5.5. Implementation  (p. 577) | html | pdf |
      • 5.3.5.6. Examination of a solution  (pp. 577-578) | html | pdf |
    • 5.3.6. Conclusions  (p. 578) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 5.3.1. Crystal and sample reference frames K<sub>B</sub> = (x<sub>B</sub>, y<sub>B</sub>, z<sub>B</sub>) and K<sub>A</sub> = (x<sub>A</sub>, y<sub>A</sub>, z<sub>A</sub>)  (p. 555) | html | pdf |
      • Fig. 5.3.2. Definition of the three Euler angles α<sub>c</sub>, β<sub>c</sub> and γ<sub>c</sub> that define the orientation of one crystallite frame K<sub>B</sub> = (a, b, c) of an orthogonal crystal cell in the sample coordinate system K<sub>A</sub>  (p. 555) | html | pdf |
      • Fig. 5.3.3. (a) P<sub>h</sub>(y) diffraction pole figure for one crystallite  (p. 556) | html | pdf |
      • Fig. 5.3.4. Relationship between the three-dimensional object f(g) and the pole figures P<sub>h</sub>(y)  (p. 557) | html | pdf |
      • Fig. 5.3.5. (a) An OD as γ<sub>c</sub> sections and (b) as {100} and {001} pole figures  (p. 558) | html | pdf |
      • Fig. 5.3.6. The inverse pole-figure sectors as a function of the crystal symmetry  (p. 562) | html | pdf |
      • Fig. 5.3.7. Visual examination of the OD refinement reliability using experimental and recalculated normalized {104}, {110}, {113}, {202}, {116}, {211}, {125} and {300} pole figures (in successive order)  (p. 563) | html | pdf |
      • Fig. 5.3.8. Entropy variation with texture index for modelled f(g) functions  (p. 564) | html | pdf |
      • Fig. 5.3.9. {100}, {001} and {110} pole figures of a planar texture in an orthorhombic crystal system  (p. 565) | html | pdf |
      • Fig. 5.3.10. {100}, {001} and {110} pole figures of a cyclic-planar texture in an orthorhombic crystal system  (p. 565) | html | pdf |
      • Fig. 5.3.11. {100}, {001} and {110} pole figures of a fibre texture in an orthorhombic crystal system  (p. 565) | html | pdf |
      • Fig. 5.3.12. {100}, {001} and {110} pole figures of a cyclic-fibre texture in an orthorhombic crystal system  (p. 565) | html | pdf |
      • Fig. 5.3.13. {100}, {001} and {110} pole figures of a three-dimensional texture in an orthorhombic crystal system, with 〈001〉* directions along Z<sub>A</sub> and 〈100〉* directions along X<sub>A</sub>  (p. 565) | html | pdf |
      • Fig. 5.3.14. The difference between a perfect single crystal and a polycrystal with perfect three-dimensional crystallite orientations  (p. 566) | html | pdf |
      • Fig. 5.3.15. Isodensity surfaces representing isolated OD components in a Cartesian coordinate system in H space  (p. 566) | html | pdf |
      • Fig. 5.3.16. Several isodensity surfaces representing a cyclic-fibre OD (a) and a cyclic-planar OD (b) in H space  (p. 566) | html | pdf |
      • Fig. 5.3.17. Illustration of a case for which a magnetization measurement in two perpendicular directions cannot reveal the macroscopic magnetic anisotropy  (p. 567) | html | pdf |
      • Fig. 5.3.18. {110} pole figures at zero field (a), under 0.3 T (b) and the difference (c)  (p. 568) | html | pdf |
      • Fig. 5.3.19. Illustration of the effects of defocusing and misadjustment on peak shapes and diffractometer resolution function  (p. 573) | html | pdf |
      • Fig. 5.3.20. (a) Randomly selected diagrams (at increasing χ values and broadening due to defocusing from bottom to top) illustrating the quality of the fit, together with the textures of AlN and Pt  (p. 574) | html | pdf |
      • Fig. 5.3.21. Low-index recalculated pole figures and Z<sub>A</sub> inverse pole figures for the three textured phases of the stack  (p. 577) | html | pdf |
    • Tables
      • Table 5.3.1. Preferred-orientation (PO) modelling methods implemented in some Rietveld packages, and their capability to perform a full texture analysis (QTA) with determination of the OD from several patterns  (p. 571) | html | pdf |
      • Table 5.3.2. Results obtained from combined analysis on an AlN/Pt/TiO<sub>x</sub>/Al<sub>2</sub>O<sub>3</sub>/Ni(Co–Cr–Al–Y) stack  (p. 577) | html | pdf |
  • 5.4. Thin films and multilayers  (pp. 581-600) | html | pdf | chapter contents |
    M. Birkholz
    • 5.4.1. Introduction  (p. 581) | html | pdf |
    • 5.4.2. Effects of absorption  (p. 581) | html | pdf |
      • 5.4.2.1. Absorption factor  (pp. 581-582) | html | pdf |
      • 5.4.2.2. Chemical phase identification  (pp. 582-583) | html | pdf |
      • 5.4.2.3. Determination and interpretation of the product μt  (pp. 583-584) | html | pdf |
      • 5.4.2.4. Phase inhomogeneities  (p. 584) | html | pdf |
    • 5.4.3. Grazing-incidence configurations  (pp. 584-590) | html | pdf |
      • 5.4.3.1. Grazing-incidence X-ray diffraction (GIXRD)  (pp. 584-587) | html | pdf |
      • 5.4.3.2. Penetration depth and information depth  (p. 587) | html | pdf |
      • 5.4.3.3. Depth-dependent properties  (pp. 587-588) | html | pdf |
      • 5.4.3.4. Index of refraction for X-rays  (p. 588) | html | pdf |
      • 5.4.3.5. Total external reflection and critical angle  (pp. 588-590) | html | pdf |
      • 5.4.3.6. Use of synchrotron radiation  (p. 590) | html | pdf |
    • 5.4.4. Thin-film textures and their depth dependence  (pp. 590-593) | html | pdf |
      • 5.4.4.1. Texture factors  (p. 590) | html | pdf |
      • 5.4.4.2. Pole-figure scans and analysis  (pp. 590-592) | html | pdf |
      • 5.4.4.3. Texture gradients  (pp. 592-593) | html | pdf |
    • 5.4.5. Stress and strain analysis  (pp. 593-594) | html | pdf |
    • 5.4.6. X-ray reflectivity (XRR)  (pp. 594-597) | html | pdf |
      • 5.4.6.1. Reflectivity from a substrate  (pp. 594-595) | html | pdf |
      • 5.4.6.2. Reflectivity of a single layer  (pp. 595-596) | html | pdf |
      • 5.4.6.3. Reflectivity of multilayers and superlattices  (pp. 596-597) | html | pdf |
    • 5.4.7. Grazing-incidence X-ray scattering (GIXS)  (pp. 597-598) | html | pdf |
    • 5.4.8. Conclusions and perspective  (p. 598) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 5.4.1. Schematic representation of absorption in a thin-film sample during a θ/2θ scan  (p. 581) | html | pdf |
      • Fig. 5.4.2. Absorption factor A for 500 nm and 1 µm Al and Nb films as function of 2θ  (p. 582) | html | pdf |
      • Fig. 5.4.3. Schematics of absorption in a multilayer system  (p. 583) | html | pdf |
      • Fig. 5.4.4. Visualization of geometrical thickness (left) and projected thickness (right)  (p. 584) | html | pdf |
      • Fig. 5.4.5. The geometry in grazing-incidence diffraction is characterized by a small incidence angle α that is kept constant during measurement  (p. 585) | html | pdf |
      • Fig. 5.4.6. The two diffractometer configurations most frequently applied in GIXRD  (p. 585) | html | pdf |
      • Fig. 5.4.7. Schematics of the illuminated sample depth for the derivation of the absorption factor A<sub>α</sub> in GIXRD  (p. 586) | html | pdf |
      • Fig. 5.4.8. The absorption factor A<sub>α</sub> in GIXRD geometry for various incidence angles α and values of the product μt  (p. 586) | html | pdf |
      • Fig. 5.4.9. (a) GIXRD, α = 1.8°, and (b) θ/2θ diffraction pattern from the same TiO<sub>2</sub>/Ti layer system on steel  (p. 587) | html | pdf |
      • Fig. 5.4.11. Penetration depth τ<sub>1/e</sub> as function of incident angle α for TiO<sub>2</sub> and Cu Kα radiation according to (5.4.45) (straight line) and (5.4.23) (dashed line)  (p. 589) | html | pdf |
      • Fig. 5.4.12. Schematic drawing for deriving the Bragg peak position shift in GIXRD due to the refraction of the incoming beam  (p. 590) | html | pdf |
      • Fig. 5.4.13. Average information depth as a function of tilt angle ψ for a 2θ = 32° reflection from thin BaTiO<sub>3</sub> films  (p. 591) | html | pdf |
      • Fig. 5.4.14. Thickness dependence of twist Δφ for yttria-stabilized zirconia (YSZ) and Gd<sub>2</sub>Zr<sub>2</sub>O<sub>7</sub> films  (p. 592) | html | pdf |
      • Fig. 5.4.15. Resistivity of thin ZnO:Al films versus reciprocal texture index 1/J (Birkholz et al  (p. 592) | html | pdf |
      • Fig. 5.4.16. Variation of the products μt of representative thin-film materials having high, average and low electron densities in the X-ray energy range between 5 and 15 keV  (p. 593) | html | pdf |
      • Fig. 5.4.17. Simulation of an X-ray reflectivity scan from a quartz glass substrate of varying surface roughness σ probed by Cu Kα radiation  (p. 595) | html | pdf |
      • Fig. 5.4.18. Scheme of reflected and refracted beams for deriving the phase difference Δ  (p. 595) | html | pdf |
      • Fig. 5.4.19. Simulated X-ray reflectivity of a Ta<sub>2</sub>O<sub>5</sub> thin film, t = 25 nm, on a fused silica substrate recorded with Cu Kα radiation  (p. 596) | html | pdf |
      • Fig. 5.4.20. Notion of participating wavevectors of XRR of a multilayer system  (p. 596) | html | pdf |
      • Fig. 5.4.21. Simulated X-ray reflectivity patterns of barrier coating SnO/NiCrO<sub>x</sub>/Ag/ZnO/SnO on glass recorded with Cu Kα radiation  (p. 597) | html | pdf |
      • Fig. 5.4.22. Simulation of X-ray reflectivity of a (TiO<sub>2</sub>/C) superlattice on an Si wafer  (p. 597) | html | pdf |
    • Tables
      • Table 5.4.1. Mass densities ρ, according to Weast (1986), electron densities ρ<sub>e</sub> and critical angles α<sub>c</sub> of some oxides used as optical coatings  (p. 588) | html | pdf |
  • 5.5. Multigrain crystallography and three-dimensional grain mapping  (pp. 601-616) | html | pdf | chapter contents |
    H. F. Poulsen and G. B. M. Vaughan
    • 5.5.1. List of symbols  (p. 601) | html | pdf |
    • 5.5.2. Introduction  (pp. 601-602) | html | pdf |
    • 5.5.3. Experimental setup  (pp. 602-603) | html | pdf |
    • 5.5.4. Diffraction geometry  (pp. 603-605) | html | pdf |
      • 5.5.4.1. The laboratory and rotated coordinate systems  (p. 603) | html | pdf |
      • 5.5.4.2. Detector coordinate systems  (pp. 603-604) | html | pdf |
      • 5.5.4.3. Diffraction  (p. 604) | html | pdf |
      • 5.5.4.4. Forward projection, no detector tilt  (pp. 604-605) | html | pdf |
      • 5.5.4.5. Forward projection with detector tilt and non-centred sample  (p. 605) | html | pdf |
        • 5.5.4.5.2. Detector tilt specification II  (p. 605) | html | pdf |
    • 5.5.5. Indexing  (pp. 605-606) | html | pdf |
      • 5.5.5.1. Detector tilt specification I  (p. 605) | html | pdf |
    • 5.5.6. Multigrain crystallography  (pp. 606-607) | html | pdf |
      • 5.5.6.1. Monophase materials  (pp. 606-607) | html | pdf |
      • 5.5.6.2. Multiphase materials containing unknown phases  (p. 607) | html | pdf |
    • 5.5.7. Centre-of-mass and stress mapping  (pp. 607-608) | html | pdf |
    • 5.5.8. 3D grain and orientation mapping  (pp. 608-609) | html | pdf |
      • 5.5.8.1. Approach 1: Grain-by-grain volumetric mapping  (p. 608) | html | pdf |
      • 5.5.8.2. Approach 2: Orientation mapping by Monte Carlo optimization  (pp. 608-609) | html | pdf |
    • 5.5.9. Representation of crystallographic orientation  (pp. 609-613) | html | pdf |
      • 5.5.9.1. Euler angles (Bunge definition)  (pp. 609-610) | html | pdf |
      • 5.5.9.2. Rodrigues vectors  (pp. 610-611) | html | pdf |
      • 5.5.9.3. Unit quaternions  (pp. 611-612) | html | pdf |
        • 5.5.9.3.1. Quaternion basics  (p. 611) | html | pdf |
        • 5.5.9.3.2. Relation between unit quaternions and orientations  (pp. 611-612) | html | pdf |
        • 5.5.9.3.3. Distance  (p. 612) | html | pdf |
      • 5.5.9.4. Bounding cubes  (pp. 612-613) | html | pdf |
    • 5.5.10. Representation of elastic strain  (p. 613) | html | pdf |
      • 5.5.10.1. Definition of strain  (p. 613) | html | pdf |
      • 5.5.10.2. Strain-to-stress conversion  (p. 613) | html | pdf |
    • 5.5.11. Crystal symmetry in relation to multigrain samples  (pp. 613-614) | html | pdf |
      • 5.5.11.1. Fundamental zone  (p. 614) | html | pdf |
      • 5.5.11.2. Determining the orientation in the fundamental zone  (p. 614) | html | pdf |
      • 5.5.11.3. Use of symmetry-equivalent orientations for strain and stress characterization  (p. 614) | html | pdf |
    • 5.5.12. Discussion  (p. 614) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 5.5.1. Sketch of the 3DXRD principle  (p. 602) | html | pdf |
      • Fig. 5.5.2. Typical 3DXRD data from a coarse-grained undeformed polycrystal  (p. 602) | html | pdf |
      • Fig. 5.5.3. Definition of the detector pixel coordinate system (see text)  (p. 604) | html | pdf |
      • Fig. 5.5.4. Grain areas reconstructed by Voronoi tessellation based on experimentally measured centres of mass (the points)  (p. 607) | html | pdf |
      • Fig. 5.5.5. Stress map of 1750 grains around a notch in an Mg AZ321 sample during tensile deformation  (p. 608) | html | pdf |
      • Fig. 5.5.6. Rendition of the 3D grain structure in a cylindrical β-Ti specimen containing 1008 grains, as obtained by the DCT algorithm  (p. 608) | html | pdf |
      • Fig. 5.5.7. Map of one layer from a cylinder of high-purity nickel  (p. 609) | html | pdf |
      • Fig. 5.5.8. Monte Carlo based reconstruction of a test case representing 20% deformed aluminium  (p. 609) | html | pdf |
      • Fig. 5.5.9. Definition of the Euler angles (φ<sub>1</sub>, ϕ, φ<sub>2</sub>) according to Bunge (1969)  (p. 610) | html | pdf |
      • Fig. 5.5.10. Definition of the Rodrigues vector r  (p. 610) | html | pdf |
      • Fig. 5.5.11. Symmetry of projection lines through point r<sup>0</sup> in Rodrigues space  (p. 611) | html | pdf |
      • Fig. 5.5.12. Three-dimensional illustration of the concepts of bounding squares  (p. 612) | html | pdf |
      • Fig. 5.5.13. The TEM equivalent of 3DXRD orientation mapping  (p. 614) | html | pdf |
    • Tables
      • Table 5.5.1. Irreducible part of Euler space for the seven Bravais classes  (p. 614) | html | pdf |
  • 5.6. X-ray diffraction from non-crystalline materials: the Debye model  (pp. 617-648) | html | pdf | chapter contents |
    S. Bates
    • 5.6.1. Outline  (p. 617) | html | pdf |
    • 5.6.2. Crystalline and non-crystalline: an introduction to the Debye model  (pp. 617-619) | html | pdf |
    • 5.6.3. Application of the Debye equation to a single molecule  (pp. 619-621) | html | pdf |
    • 5.6.4. The Debye–Menke equation  (pp. 621-622) | html | pdf |
    • 5.6.5. Verification of the Debye–Menke intensity function  (p. 622) | html | pdf |
    • 5.6.6. Background removal, intensity normalization and choice of X-ray optics  (pp. 622-625) | html | pdf |
    • 5.6.7. Application of the Debye normalization to semi-quantitative analysis  (pp. 625-627) | html | pdf |
    • 5.6.8. Correction for the instrumental intensity response  (pp. 627-631) | html | pdf |
    • 5.6.9. The full Debye normalization procedure  (pp. 631-632) | html | pdf |
    • 5.6.10. Application of the Debye normalization procedure  (pp. 632-635) | html | pdf |
    • 5.6.11. Universal appearance of non-crystalline powder patterns  (pp. 635-638) | html | pdf |
    • 5.6.12. Steps towards an effective lattice model of high-density randomly packed materials  (pp. 638-642) | html | pdf |
    • 5.6.13. Practical application of the effective lattice function determination  (pp. 642-645) | html | pdf |
    • 5.6.14. Debye diffraction models for larger molecular ensembles  (pp. 645-646) | html | pdf |
    • 5.6.15. Conclusions  (pp. 646-647) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 5.6.1. A chemical structure diagram for mannitol  (p. 619) | html | pdf |
      • Fig. 5.6.2. Calculated Debye diffraction response for a single mannitol molecule displayed with the coherent and incoherent diffraction responses  (p. 620) | html | pdf |
      • Fig. 5.6.3. Zoomed-in view of Fig  (p. 620) | html | pdf |
      • Fig. 5.6.4. The Menke molecular scattering function displayed along with the Debye diffraction response and Guinier small-angle scattering response for a single mannitol molecule  (p. 621) | html | pdf |
      • Fig. 5.6.5. The diffraction response from a single mannitol molecule as calculated by the Debye–Menke equation displayed with the independent coherent scattering function  (p. 621) | html | pdf |
      • Fig. 5.6.6. Overlay of the Debye and Debye–Menke response for mannitol with a weighted histogram of atom–atom pair distances within mannitol  (p. 622) | html | pdf |
      • Fig. 5.6.7. Calculated X-ray scattering responses for 100 mannitol molecules packed into a sphere of radius 1.71 nm  (p. 622) | html | pdf |
      • Fig. 5.6.8. Measured Bragg–Brentano data for SRM 640d depicting the diffuse background and calculated Bremsstrahlung contributions  (p. 624) | html | pdf |
      • Fig. 5.6.9. Measured Bragg–Brentano data for Si SRM 640d displayed with the calculated full background response including the instrumental contribution, Bremsstrahlung, thermal diffuse scattering and Compton scattering  (p. 624) | html | pdf |
      • Fig. 5.6.10. Measured non-crystalline powder pattern for a dry sucrose lyophilizate compared with the scaled blank background and resulting analytical signal  (p. 626) | html | pdf |
      • Fig. 5.6.11. Normalized analytical data for dry and partially dry sucrose lyophilizates  (p. 626) | html | pdf |
      • Fig. 5.6.12. Square root of the full data ensemble variance from the mean display with two orthogonal principal variance components (expressed as real positive components)  (p. 626) | html | pdf |
      • Fig. 5.6.13. Scale factors for the variance principal components displayed in Fig  (p. 627) | html | pdf |
      • Fig. 5.6.14. Normalized non-crystalline X-ray powder patterns for `dry' sucrose, free water and a hypothetical `wet' sucrose reference (variable component)  (p. 627) | html | pdf |
      • Fig. 5.6.15. Measured powder patterns for an aligned hexatriacontane intensity reference and the silicon powder standard SRM 640d  (p. 628) | html | pdf |
      • Fig. 5.6.16. Reduced integrated peak intensities for hexatriacontane and Si SRM 640d plotted on a log scale with a derived analytical instrumental intensity response function using the core response function with m = 0.0 and n = 2.1  (p. 630) | html | pdf |
      • Fig. 5.6.17. Calculated powder patterns for a ∼3 nm silicon nanocube using both Debye and Bragg diffraction equations  (p. 630) | html | pdf |
      • Fig. 5.6.18. Non-crystalline powder pattern for glassy indomethacin after correction for the Bragg–Brentano instrument intensity response  (p. 632) | html | pdf |
      • Fig. 5.6.19. A single indomethacin molecule with the five significant torsion degrees of freedom indicated in red  (p. 632) | html | pdf |
      • Fig. 5.6.20. Non-crystalline indomethacin data after intensity correction and removal of the Compton response displayed with the Debye–Menke curve for a single indomethacin molecule  (p. 632) | html | pdf |
      • Fig. 5.6.21. Non-crystalline indomethacin data after intensity correction and removal of the Compton response displayed with the Debye–Menke curves for different indomethacin molecular conformations  (p. 633) | html | pdf |
      • Fig. 5.6.22. Non-crystalline indomethacin data collected for two sample-preparation methods after intensity correction (LP) and removal of Compton scattering  (p. 634) | html | pdf |
      • Fig. 5.6.23. Non-crystalline indomethacin data collected for glasses prepared from alpha and gamma starting materials  (p. 634) | html | pdf |
      • Fig. 5.6.24. A chemical structure diagram for ibuprofen  (p. 634) | html | pdf |
      • Fig. 5.6.25. Non-crystalline ibuprofen data collected for glasses prepared from R,S and S crystalline starting materials  (p. 634) | html | pdf |
      • Fig. 5.6.26. Normalized non-crystalline powder patterns for melt–quench and cryo-ground simvastatin displayed with the calculated Debye–Menke response for a single molecule of simvastatin  (p. 635) | html | pdf |
      • Fig. 5.6.27. A chemical structure diagram for simvastatin  (p. 635) | html | pdf |
      • Fig. 5.6.28. Normalized observed reflection powder patterns for simvastatin, indomethacin and water after full Debye normalization  (p. 635) | html | pdf |
      • Fig. 5.6.29. Normalized observed non-crystalline powder patterns (LP and Compton corrected) plotted with primary halo responses using a microstrain model with a microstrain variable of about 22%  (p. 636) | html | pdf |
      • Fig. 5.6.30. Cumulative disorder in the relative position of a building block as a function of steps away from the starting block  (p. 637) | html | pdf |
      • Fig. 5.6.31. Normalized observed non-crystalline powder patterns (LP and Compton corrected) plotted with primary halo responses using a random-walk size model with = 1  (p. 638) | html | pdf |
      • Fig. 5.6.32. Debye–Menke response for mannitol compared with the mean simulated powder pattern for high-density randomly packed mannitol based upon multiple 100 molecule clusters generated using PACKMOL  (p. 638) | html | pdf |
      • Fig. 5.6.33. Schematic representation of a spherical exclusion zone centred on each basic unit  (p. 640) | html | pdf |
      • Fig. 5.6.34. Effect on the calculated Debye–Menke single-molecule response for mannitol of a hard-sphere exclusion zone  (p. 640) | html | pdf |
      • Fig. 5.6.35. Extracted lattice function for high-density randomly packed mannitol compared with the theoretical lattice function derived from a spherical exclusion zone with a radius of 2.6 Å  (p. 640) | html | pdf |
      • Fig. 5.6.36. Extracted lattice function for low-density randomly packed mannitol compared with the theoretical lattice function derived from a spherical exclusion zone of 3.0 Å  (p. 641) | html | pdf |
      • Fig. 5.6.37. Schematic illustration of a spherical core–shell-type model to describe more complex exclusion zones and packing models for basic units  (p. 641) | html | pdf |
      • Fig. 5.6.38. Extracted lattice function for high-density randomly packed mannitol compared with the theoretical lattice function derived from a three-sphere concentric exclusion zone  (p. 641) | html | pdf |
      • Fig. 5.6.39. Molecular centre pair distribution extracted from high-density mannitol packing compared with a three-sphere effective lattice model  (p. 641) | html | pdf |
      • Fig. 5.6.40. Normalized atom pair distribution extracted from high-density mannitol packing compared with a three-sphere effective lattice model  (p. 642) | html | pdf |
      • Fig. 5.6.41. Effective lattice function for a single indomethacin molecule compared with the `best fit' core–shell model (three-shell model) (inset)  (p. 642) | html | pdf |
      • Fig. 5.6.42. Fully normalized glassy indomethacin data (measured on the melt–quench alpha form) displayed with simulated data sets derived from single-molecule and dimer basic units  (p. 643) | html | pdf |
      • Fig. 5.6.43. Fully normalized glassy indomethacin data (measured on the melt–quench alpha form) displayed with a simulated data set derived from a single-molecule basic unit and the corresponding Debye–Menke response  (p. 643) | html | pdf |
      • Fig. 5.6.44. Effective lattice functions extracted from non-crystalline powder patterns measured on melt–quench (MQ) and cryo-ground (CG) simvastatin  (p. 643) | html | pdf |
      • Fig. 5.6.45. Core–shell lattice function fit to the effective lattice function for cryo-ground simvastatin  (p. 644) | html | pdf |
      • Fig. 5.6.46. Core–shell lattice functions for cryo-ground and melt–quench simvastatin  (p. 644) | html | pdf |
      • Fig. 5.6.47. Fully normalized glassy nifedipine data (measured on a melt–quench glassy form) displayed with a single-molecule Debye–Menke response  (p. 644) | html | pdf |
      • Fig. 5.6.48. A chemical structure diagram for nifedipine  (p. 644) | html | pdf |
      • Fig. 5.6.49. Effective lattice function for nifedipine extracted from measured glassy data plotted with the `best fit' core–shell lattice model  (p. 645) | html | pdf |
      • Fig. 5.6.50. Fully normalized non-crystalline nifedipine diffraction response displayed with a simulated data set derived from a single-molecule basic unit with a core–shell lattice and the corresponding Debye–Menke response  (p. 645) | html | pdf |
      • Fig. 5.6.51. Schematic illustration of the multiple bubble Debye calculation approach using each atom for a single molecule of mannitol (without hydrogen atoms) as a unique bubble centre  (p. 645) | html | pdf |
      • Fig. 5.6.52. Total diffraction response for the Debye bubble calculation applied to the nifedipine crystal structure BICCIZ  (p. 646) | html | pdf |
      • Fig. 5.6.53. Total diffraction response for the Debye bubble calculation applied to the nifedipine crystal structure BICCIZ01  (p. 646) | html | pdf |
    • Tables
      • Table 5.6.1. Analytical peaks for hexatriacontane (top) and silicon (bottom) listed by Miller index (hkl)  (p. 629) | html | pdf |
  • 5.7. Nanometre-scale structure from powder diffraction: total scattering and atomic pair distribution function analysis  (pp. 649-672) | html | pdf | chapter contents |
    S. J. L. Billinge
    • 5.7.1. Introduction  (p. 649) | html | pdf |
      • 5.7.1.1. Types of nanostructure  (p. 649) | html | pdf |
    • 5.7.2. Scattering from nanostructures  (pp. 649-659) | html | pdf |
      • 5.7.2.1. The Debye equation  (pp. 650-652) | html | pdf |
      • 5.7.2.2. Correlation functions  (pp. 652-654) | html | pdf |
      • 5.7.2.3. Low-angle scattering intensity  (pp. 654-655) | html | pdf |
      • 5.7.2.4. Calculating ) from models  (pp. 655-657) | html | pdf |
        • 5.7.2.4.1. Calculating in real space for bulk crystals  (p. 655) | html | pdf |
        • 5.7.2.4.2. Calculating in real space for nanoparticles modelled as attenuated bulk crystals  (pp. 655-656) | html | pdf |
        • 5.7.2.4.3. Calculating as the Fourier transform of the properly normalized Debye function  (p. 656) | html | pdf |
        • 5.7.2.4.4. Calculating in real space from discrete nanoparticle models  (pp. 656-657) | html | pdf |
      • 5.7.2.5. The extent of small-angle scattering  (p. 657) | html | pdf |
      • 5.7.2.6. PDFs from structures with multiple elements  (pp. 657-658) | html | pdf |
      • 5.7.2.7. Differential structure functions  (pp. 658-659) | html | pdf |
    • 5.7.3. Collecting data for total-scattering and PDF measurements  (pp. 659-663) | html | pdf |
      • 5.7.3.1. Neutrons  (pp. 659-660) | html | pdf |
      • 5.7.3.2. X-rays  (pp. 660-661) | html | pdf |
      • 5.7.3.3. Electrons  (pp. 661-662) | html | pdf |
      • 5.7.3.4. Q ranges and Q resolutions  (pp. 662-663) | html | pdf |
    • 5.7.4. Data reduction  (pp. 663-664) | html | pdf |
    • 5.7.5. Extracting structural information  (pp. 664-669) | html | pdf |
      • 5.7.5.1. Obtaining the PDF from a model  (p. 664) | html | pdf |
      • 5.7.5.2. PDFs from multi-element material  (p. 664) | html | pdf |
      • 5.7.5.3. Model-independent information from the PDF  (pp. 664-665) | html | pdf |
      • 5.7.5.4. Modelling approaches  (pp. 665-669) | html | pdf |
        • 5.7.5.4.1. Small-box modelling  (pp. 665-666) | html | pdf |
        • 5.7.5.4.2. Big-box modelling  (pp. 666-668) | html | pdf |
        • 5.7.5.4.3. Ab initio nanostructure solution from PDF data  (pp. 668-669) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 5.7.1. TEM images illustrating the variety in nanostructured materials: (a) Nanostructuring in a bulk material, (b) intercalation in a mesoporous host and (c) nanoparticles  (p. 649) | html | pdf |
      • Fig. 5.7.2. Scattering intensity calculated using the Debye function [equation (5.7.13)] for a number of discrete particles: (a) benzene, (b) fullerene C₆₀ and (c) a BaTiO₃ nanoparticle  (p. 651) | html | pdf |
      • Fig. 5.7.3. Schematic showing how the PDF is calculated  (p. 652) | html | pdf |
      • Fig. 5.7.4. Atomic form factors and the Morningstar–Warren approximation for CdSe  (p. 653) | html | pdf |
      • Fig. 5.7.5. An illustration of the annulus used to calculate the coordination number  (p. 653) | html | pdf |
      • Fig. 5.7.6. The nanoparticle form factors for a range of particle shapes  (p. 654) | html | pdf |
      • Fig. 5.7.7. (a) The calculated diffraction pattern from a faceted 2 nm gold nanoparticle  (p. 656) | html | pdf |
      • Fig. 5.7.8. Schematic of the 2D detector and sample placement for a typical RAPDF measurement  (p. 660) | html | pdf |
      • Fig. 5.7.9. An example of the better powder average (reduced spottiness of the diffraction pattern) obtained when a sample with a poor powder average is spun  (p. 661) | html | pdf |
      • Fig. 5.7.10. (a) TEM image of 100 nm Au nanoparticles (black dots) used for ePDF measurement  (p. 662) | html | pdf |
      • Fig. 5.7.11. A schematic of the buildup of the PDF from the structural model for face-centred-cubic nickel  (p. 664) | html | pdf |
      • Fig. 5.7.12. PDF fit of crystalline nickel  (p. 665) | html | pdf |
      • Fig. 5.7.13. A schematic of three different motional correlations of atoms and their corresponding PDFs  (p. 666) | html | pdf |
      • Fig. 5.7.14. An example of an RMC refinement to the neutron total-scattering data of ZrW₂O₈  (p. 667) | html | pdf |
      • Fig. 5.7.15. Total-scattering structure functions, with low-angle scattering (inset) from surfactant micelles of CTAB with different substitutions of hydrogen/deuterium  (p. 668) | html | pdf |
      • Fig. 5.7.16. Structure solution of C₆₀ from neutron PDF data using the LIGA algorithm (Juhás et al  (p. 669) | html | pdf |
    • Tables
      • Table 5.7.1. List of symbols and abbreviations used in this chapter  (p. 650) | html | pdf |
  • 5.8. Scattering methods for disordered heterogeneous materials  (pp. 673-696) | html | pdf | chapter contents |
    A. J. Allen
    • 5.8.1. Introduction and overview  (pp. 673-675) | html | pdf |
      • 5.8.1.1. Amorphous and non-crystalline materials  (p. 673) | html | pdf |
      • 5.8.1.2. Disordered and heterogeneous (and multi-component) materials  (pp. 673-674) | html | pdf |
      • 5.8.1.3. Small-angle and wide-angle scattering tools  (p. 674) | html | pdf |
      • 5.8.1.4. Tabulated summary of quantitative information obtainable  (pp. 674-675) | html | pdf |
      • 5.8.1.5. Different notations  (p. 675) | html | pdf |
    • 5.8.2. Recommended measurement tools  (pp. 675-683) | html | pdf |
      • 5.8.2.1. Small-angle scattering using a position-sensitive detector  (pp. 675-677) | html | pdf |
      • 5.8.2.2. Ultra-small-angle scattering using crystal diffraction optics  (pp. 677-678) | html | pdf |
      • 5.8.2.3. Data reduction and calibration of small-angle scattering data  (pp. 678-680) | html | pdf |
        • 5.8.2.3.1. Reduction and calibration of 2D SAS data from a position-sensitive detector  (p. 679) | html | pdf |
        • 5.8.2.3.2. Reduction and calibration of 1D USAXS or USANS data  (p. 679) | html | pdf |
        • 5.8.2.3.3. Corrections for multiple scattering and flat background  (pp. 679-680) | html | pdf |
      • 5.8.2.4. Reflectivity, grazing-incidence small-angle scattering and diffraction  (pp. 680-681) | html | pdf |
      • 5.8.2.5. Wide-angle scattering and other methods for disordered structures  (pp. 681-683) | html | pdf |
    • 5.8.3. Quantitative analysis of disordered heterogeneous materials  (pp. 683-692) | html | pdf |
      • 5.8.3.1. Interpretative models for analysis of SAS data  (pp. 683-691) | html | pdf |
        • 5.8.3.1.1. Guinier approximation  (p. 683) | html | pdf |
        • 5.8.3.1.2. Porod scattering regime  (p. 683) | html | pdf |
        • 5.8.3.1.3. Scattering invariant  (p. 683) | html | pdf |
        • 5.8.3.1.4. Debye–Bueche model  (p. 683) | html | pdf |
        • 5.8.3.1.5. Shape effects in the form factor  (pp. 683-685) | html | pdf |
        • 5.8.3.1.6. Particle pair density distribution function (PDDF)  (pp. 685-686) | html | pdf |
        • 5.8.3.1.7. Size distribution analysis  (pp. 686-687) | html | pdf |
        • 5.8.3.1.8. Particle structure factor and interparticle interference effects  (p. 687) | html | pdf |
        • 5.8.3.1.9. Fractal models  (pp. 687-688) | html | pdf |
        • 5.8.3.1.10. Anomalous SAXS and contrast-variation SANS  (pp. 688-689) | html | pdf |
        • 5.8.3.1.11. Magnetic SANS analysis  (pp. 689-690) | html | pdf |
        • 5.8.3.1.12. Further analysis of X-ray reflectivity and GI-SAXS  (pp. 690-691) | html | pdf |
      • 5.8.3.2. Small-angle scattering effects on wide-angle scattering analysis  (p. 691) | html | pdf |
      • 5.8.3.3. Combining information from different methods  (pp. 691-692) | html | pdf |
    • 5.8.4. Prospects for future development and recommended further reading  (pp. 692-693) | html | pdf |
      • 5.8.4.1. Developments at X-ray synchrotron facilities  (p. 692) | html | pdf |
      • 5.8.4.2. Developments at steady-state and pulsed neutron sources  (pp. 692-693) | html | pdf |
      • 5.8.4.3. Future prospects  (p. 693) | html | pdf |
      • 5.8.4.4. Further reading  (p. 693) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 5.8.1. Examples of scattering methods applied to disordered, heterogeneous materials, together with the scale ranges of interest and the applicable scattering-based methods  (p. 673) | html | pdf |
      • Fig. 5.8.2. (a) Schematic of the basic measurement geometry for SAXS or SANS using a 2D PSD  (p. 675) | html | pdf |
      • Fig. 5.8.3. (a) Schematic of a typical USAXS instrument with regular 1D (slit-smeared) measurement configuration  (p. 677) | html | pdf |
      • Fig. 5.8.4. Schematic of typical USANS configuration at a reactor-based neutron source: s = sample, PG = pyrolitic graphite  (p. 678) | html | pdf |
      • Fig. 5.8.5. (a) Schematic of basic reflectivity measurement geometry for XRR or NR  (p. 680) | html | pdf |
      • Fig. 5.8.6. (a) Schematic of typical total-scattering [S(Q) − 1] data versus Q for model Au core–shell nanoparticles  (p. 682) | html | pdf |
      • Fig. 5.8.7. Characteristic SAS intensity profiles of ensembles of scattering features, both monodispersed and polydispersed in size  (p. 684) | html | pdf |
    • Tables
      • Table 5.8.1. Information on disordered or heterogeneous material systems using scattering methods  (p. 674) | html | pdf |