International Tables for Crystallography (2019). Vol. H, ch. 5.6, pp. 617-648
https://doi.org/10.1107/97809553602060000971

Chapter 5.6. X-ray diffraction from non-crystalline materials: the Debye model

Contents

  • 5.6. X-ray diffraction from non-crystalline materials: the Debye model  (pp. 617-648) | html | pdf | chapter contents |
    • 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 [2^{1/2} {N_s}] = 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 |