(International Tables) Reciprocal space

International Tables for Crystallography
Volume B: Reciprocal space

First online edition (2006) ISBN: 978-0-7923-6592-1 doi: 10.1107/97809553602060000102 | 1 | 2 |

Edited by U. Shmueli

Preface to the second edition (pp. xxv-xxvi) | html | pdf |
U. Shmueli

General relationships and techniques
- 1.1. Reciprocal space in crystallography (pp. 2-9) | html | pdf | chapter contents |
  U. Shmueli
  - 1.1.1. Introduction (p. 2) | html | pdf |
  - 1.1.2. Reciprocal lattice in crystallography (pp. 2-3) | html | pdf |
  - 1.1.3. Fundamental relationships (pp. 3-4) | html | pdf |
  - 1.1.4. Tensor-algebraic formulation (pp. 5-7) | html | pdf |
  - 1.1.5. Transformations (pp. 7-8) | html | pdf |
  - 1.1.6. Some analytical aspects of the reciprocal space (pp. 8-9) | html | pdf |
  - References | html | pdf |
- 1.2. The structure factor (pp. 10-24) | html | pdf | chapter contents |
  P. Coppens
  - 1.2.1. Introduction (p. 10) | html | pdf |
  - 1.2.2. General scattering expression for X-rays (p. 10) | html | pdf |
  - 1.2.3. Scattering by a crystal: definition of a structure factor (p. 10) | html | pdf |
  - 1.2.4. The isolated-atom approximation in X-ray diffraction (pp. 10-11) | html | pdf |
  - 1.2.5. Scattering of thermal neutrons (p. 11) | html | pdf |
  - 1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism (pp. 11-13) | html | pdf |
  - 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion (pp. 14-15) | html | pdf |
  - 1.2.8. Fourier transform of orbital products (pp. 17-18) | html | pdf |
  - 1.2.9. The atomic temperature factor (p. 18) | html | pdf |
  - 1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation (pp. 18-19) | html | pdf |
  - 1.2.11. Rigid-body analysis (pp. 19-22) | html | pdf |
  - 1.2.12. Treatment of anharmonicity (pp. 22-23) | html | pdf |
  - 1.2.13. The generalized structure factor (pp. 23-24) | html | pdf |
  - 1.2.14. Conclusion (p. 24) | html | pdf |
  - References | html | pdf |
- 1.3. Fourier transforms in crystallography: theory, algorithms and applications (pp. 25-98) | html | pdf | chapter contents |
  G. Bricogne
  - 1.3.1. General introduction (p. 25) | html | pdf |
  - 1.3.2. The mathematical theory of the Fourier transformation (pp. 25-49) | html | pdf |
  - 1.3.3. Numerical computation of the discrete Fourier transform (pp. 49-58) | html | pdf |
  - 1.3.4. Crystallographic applications of Fourier transforms (pp. 58-98) | html | pdf |
  - References | html | pdf |
- 1.4. Symmetry in reciprocal space (pp. 99-161) | html | pdf | chapter contents |
  U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve
  - 1.4.1. Introduction (p. 99) | html | pdf |
    U. Shmueli
  - 1.4.2. Effects of symmetry on the Fourier image of the crystal (pp. 99-102) | html | pdf |
    U. Shmueli
  - 1.4.3. Structure-factor tables (pp. 102-104) | html | pdf |
    U. Shmueli
  - 1.4.4. Symmetry in reciprocal space: space-group tables (pp. 104-106) | html | pdf |
    U. Shmueli
  - Appendix 1.4.1. Comments on the preparation and usage of the tables (pp. 106-107) | html | pdf |
    U. Shmueli
  - Appendix 1.4.2. Space-group symbols for numeric and symbolic computations (pp. 107-119) | html | pdf |
  - Appendix 1.4.3. Structure-factor tables (pp. 120-149) | html | pdf |
    U. Shmueli
  - Appendix 1.4.4. Crystallographic space groups in reciprocal space (pp. 150-161) | html | pdf |
    U. Shmueli
  - References | html | pdf |
- 1.5. Crystallographic viewpoints in the classification of space-group representations (pp. 162-188) | html | pdf | chapter contents |
  M. I. Aroyo and H. Wondratschek
  - 1.5.1. List of symbols (p. 162) | html | pdf |
  - 1.5.2. Introduction (p. 162) | html | pdf |
  - 1.5.3. Basic concepts (pp. 162-165) | html | pdf |
  - 1.5.4. Conventions in the classification of space-group irreps (pp. 165-168) | html | pdf |
  - 1.5.5. Examples and conclusions (pp. 168-176) | html | pdf |
  - Appendix 1.5.1. Reciprocal-space groups $[{\cal G}^{*}]$ (p. 176) | html | pdf |
  - References | html | pdf |

Reciprocal space in crystal-structure determination
- 2.1. Statistical properties of the weighted reciprocal lattice (pp. 190-209) | html | pdf | chapter contents |
  U. Shmueli and A. J. C. Wilson
  - 2.1.1. Introduction (p. 190) | html | pdf |
  - 2.1.2. The average intensity of general reflections (pp. 190-191) | html | pdf |
  - 2.1.3. The average intensity of zones and rows (pp. 191-192) | html | pdf |
  - 2.1.4. Probability density distributions – mathematical preliminaries (pp. 192-195) | html | pdf |
  - 2.1.5. Ideal probability density distributions (pp. 195-197) | html | pdf |
  - 2.1.6. Distributions of sums, averages and ratios (pp. 197-199) | html | pdf |
  - 2.1.7. Non-ideal distributions: the correction-factor approach (pp. 199-203) | html | pdf |
  - 2.1.8. Non-ideal distributions: the Fourier method (pp. 203-209) | html | pdf |
  - References | html | pdf |
- 2.2. Direct methods (pp. 210-234) | html | pdf | chapter contents |
  C. Giacovazzo
  - 2.2.1. List of symbols and abbreviations (p. 210) | html | pdf |
  - 2.2.2. Introduction (p. 210) | html | pdf |
  - 2.2.3. Origin specification (pp. 210-215) | html | pdf |
  - 2.2.4. Normalized structure factors (pp. 215-217) | html | pdf |
  - 2.2.5. Phase-determining formulae (pp. 217-225) | html | pdf |
  - 2.2.6. Direct methods in real and reciprocal space: Sayre's equation (pp. 225-227) | html | pdf |
  - 2.2.7. Scheme of procedure for phase determination (pp. 227-228) | html | pdf |
  - 2.2.8. Other multisolution methods applied to small molecules (pp. 228-230) | html | pdf |
  - 2.2.9. Some references to direct-methods packages (p. 230) | html | pdf |
  - 2.2.10. Direct methods in macromolecular crystallography (pp. 231-234) | html | pdf |
  - References | html | pdf |
- 2.3. Patterson and molecular-replacement techniques (pp. 235-263) | html | pdf | chapter contents |
  M. G. Rossmann and E. Arnold
  - 2.3.1. Introduction (pp. 235-238) | html | pdf |
  - 2.3.2. Interpretation of Patterson maps (pp. 238-242) | html | pdf |
  - 2.3.3. Isomorphous replacement difference Pattersons (pp. 242-246) | html | pdf |
  - 2.3.4. Anomalous dispersion (pp. 246-248) | html | pdf |
  - 2.3.5. Noncrystallographic symmetry (pp. 248-250) | html | pdf |
  - 2.3.6. Rotation functions (pp. 250-258) | html | pdf |
  - 2.3.7. Translation functions (pp. 258-260) | html | pdf |
  - 2.3.8. Molecular replacement (pp. 260-262) | html | pdf |
  - 2.3.9. Conclusions (pp. 262-263) | html | pdf |
  - References | html | pdf |
- 2.4. Isomorphous replacement and anomalous scattering (pp. 264-275) | html | pdf | chapter contents |
  M. Vijayan and S. Ramaseshan
  - 2.4.1. Introduction (p. 264) | html | pdf |
  - 2.4.2. Isomorphous replacement method (pp. 264-265) | html | pdf |
  - 2.4.3. Anomalous-scattering method (pp. 265-269) | html | pdf |
  - 2.4.4. Isomorphous replacement and anomalous scattering in protein crystallography (pp. 269-274) | html | pdf |
  - 2.4.5. Anomalous scattering of neutrons and synchrotron radiation. The multiwavelength method (pp. 274-275) | html | pdf |
  - References | html | pdf |
- 2.5. Electron diffraction and electron microscopy in structure determination (pp. 276-345) | html | pdf | chapter contents |
  J. M. Cowley, P. Goodman, B. K. Vainshtein, B. B. Zvyagin and D. L. Dorset
  - 2.5.1. Foreword (pp. 276-277) | html | pdf |
    J. M. Cowley
  - 2.5.2. Electron diffraction and electron microscopy (pp. 277-285) | html | pdf |
    J. M. Cowley
  - 2.5.3. Space-group determination by convergent-beam electron diffraction (pp. 285-306) | html | pdf |
    P. Goodman
  - 2.5.4. Electron-diffraction structure analysis (EDSA) (pp. 306-310) | html | pdf |
    B. K. Vainshtein and B. B. Zvyagin
  - 2.5.5. Image reconstruction (pp. 310-315) | html | pdf |
    B. K. Vainshtein
  - 2.5.6. Three-dimensional reconstruction (pp. 316-320) | html | pdf |
    B. K. Vainshtein
  - 2.5.7. Direct phase determination in electron crystallography (pp. 320-326) | html | pdf |
    D. L. Dorset
  - References | html | pdf |

Dual bases in crystallographic computing
- 3.1. Distances, angles, and their standard uncertainties (pp. 348-352) | html | pdf | chapter contents |
  D. E. Sands
  - 3.1.1. Introduction (p. 348) | html | pdf |
  - 3.1.2. Scalar product (p. 348) | html | pdf |
  - 3.1.3. Length of a vector (p. 348) | html | pdf |
  - 3.1.4. Angle between two vectors (p. 348) | html | pdf |
  - 3.1.5. Vector product (p. 349) | html | pdf |
  - 3.1.6. Permutation tensors (p. 349) | html | pdf |
  - 3.1.7. Components of vector product (p. 349) | html | pdf |
  - 3.1.8. Some vector relationships (p. 349) | html | pdf |
  - 3.1.9. Planes (pp. 349-350) | html | pdf |
  - 3.1.10. Variance–covariance matrices (pp. 350-351) | html | pdf |
  - 3.1.11. Mean values (pp. 351-352) | html | pdf |
  - 3.1.12. Computation (p. 352) | html | pdf |
  - References | html | pdf |
- 3.2. The least-squares plane (pp. 353-359) | html | pdf | chapter contents |
  R. E. Marsh and V. Schomaker
  - 3.2.1. Introduction (p. 353) | html | pdf |
  - 3.2.2. Least-squares plane based on uncorrelated, isotropic weights (pp. 353-355) | html | pdf |
  - 3.2.3. The proper least-squares plane, with Gaussian weights (pp. 355-358) | html | pdf |
  - Appendix 3.2.1. (pp. 358-359) | html | pdf |
  - References | html | pdf |
- 3.3. Molecular modelling and graphics (pp. 360-384) | html | pdf | chapter contents |
  R. Diamond
  - 3.3.1. Graphics (pp. 360-376) | html | pdf |
  - 3.3.2. Molecular modelling, problems and approaches (pp. 377-380) | html | pdf |
  - 3.3.3. Implementations (pp. 380-384) | html | pdf |
  - References | html | pdf |
- 3.4. Accelerated convergence treatment of R⁻ⁿ lattice sums (pp. 385-397) | html | pdf | chapter contents |
  D. E. Williams
  - 3.4.1. Introduction (p. 385) | html | pdf |
  - 3.4.2. Definition and behaviour of the direct-space sum (p. 385) | html | pdf |
  - 3.4.3. Preliminary description of the method (pp. 385-386) | html | pdf |
  - 3.4.4. Preliminary derivation to obtain a formula which accelerates the convergence of an $[{\bi R}^{\bf -{\bi n}}]$ sum over lattice points X(d) (pp. 386-388) | html | pdf |
  - 3.4.5. Extension of the method to a composite lattice (pp. 388-389) | html | pdf |
  - 3.4.6. The case of $[{\bf{\bi n} = 1}]$ (Coulombic lattice energy) (p. 389) | html | pdf |
  - 3.4.7. The cases of $[{\bf{\bi n} = 2}]$ and $[{\bf{\bi n} = 3}]$ (p. 389) | html | pdf |
  - 3.4.8. Derivation of the accelerated convergence formula via the Patterson function (pp. 389-390) | html | pdf |
  - 3.4.9. Evaluation of the incomplete gamma function (p. 390) | html | pdf |
  - 3.4.10. Summation over the asymmetric unit and elimination of intramolecular energy terms (p. 390) | html | pdf |
  - 3.4.11. Reference formulae for particular values of n (pp. 390-391) | html | pdf |
  - 3.4.12. Numerical illustrations (pp. 391-392) | html | pdf |
  - References | html | pdf |

Diffuse scattering and related topics
- 4.1. Thermal diffuse scattering of X-rays and neutrons (pp. 400-406) | html | pdf | chapter contents |
  B. T. M. Willis
  - 4.1.1. Introduction (p. 400) | html | pdf |
  - 4.1.2. Dynamics of three-dimensional crystals (pp. 400-402) | html | pdf |
  - 4.1.3. Scattering of X-rays by thermal vibrations (pp. 403-404) | html | pdf |
  - 4.1.4. Scattering of neutrons by thermal vibrations (pp. 404-405) | html | pdf |
  - 4.1.5. Phonon dispersion relations (pp. 405-406) | html | pdf |
  - 4.1.6. Measurement of elastic constants (p. 406) | html | pdf |
  - References | html | pdf |
- 4.2. Disorder diffuse scattering of X-rays and neutrons (pp. 407-442) | html | pdf | chapter contents |
  H. Jagodzinski and F. Frey
  - 4.2.1. Scope of this chapter (pp. 407-408) | html | pdf |
  - 4.2.2. Summary of basic scattering theory (pp. 408-410) | html | pdf |
  - 4.2.3. General treatment (pp. 410-420) | html | pdf |
  - 4.2.4. Quantitative interpretation (pp. 420-438) | html | pdf |
  - 4.2.5. Measurement of diffuse scattering (pp. 438-442) | html | pdf |
  - References | html | pdf |
- 4.3. Diffuse scattering in electron diffraction (pp. 443-448) | html | pdf | chapter contents |
  J. M. Cowley and J. K. Gjønnes
  - 4.3.1. Introduction (pp. 443-444) | html | pdf |
  - 4.3.2. Inelastic scattering (p. 444) | html | pdf |
  - 4.3.3. Kinematical and pseudo-kinematical scattering (p. 445) | html | pdf |
  - 4.3.4. Dynamical scattering: Bragg scattering effects (pp. 445-447) | html | pdf |
  - 4.3.5. Multislice calculations for diffraction and imaging (p. 447) | html | pdf |
  - 4.3.6. Qualitative interpretation of diffuse scattering of electrons (pp. 447-448) | html | pdf |
  - References | html | pdf |
- 4.4. Scattering from mesomorphic structures (pp. 449-465) | html | pdf | chapter contents |
  P. S. Pershan
  - 4.4.1. Introduction (pp. 449-451) | html | pdf |
  - 4.4.2. The nematic phase (pp. 451-453) | html | pdf |
  - 4.4.3. Smectic-A and smectic-C phases (pp. 453-456) | html | pdf |
  - 4.4.4. Phases with in-plane order (pp. 456-463) | html | pdf |
  - 4.4.5. Discotic phases (p. 463) | html | pdf |
  - 4.4.6. Other phases (pp. 463-464) | html | pdf |
  - 4.4.7. Notes added in proof to first edition (pp. 464-465) | html | pdf |
  - References | html | pdf |
- 4.5. Polymer crystallography (pp. 466-485) | html | pdf | chapter contents |
  R. P. Millane and D. L. Dorset
  - 4.5.1. Overview (p. 466) | html | pdf |
    R. P. Millane and D. L. Dorset
  - 4.5.2. X-ray fibre diffraction analysis (pp. 466-481) | html | pdf |
    R. P. Millane
  - 4.5.3. Electron crystallography of polymers (pp. 481-485) | html | pdf |
    D. L. Dorset
  - References | html | pdf |
- 4.6. Reciprocal-space images of aperiodic crystals (pp. 486-532) | html | pdf | chapter contents |
  W. Steurer and T. Haibach
  - 4.6.1. Introduction (pp. 486-487) | html | pdf |
  - 4.6.2. The n-dimensional description of aperiodic crystals (pp. 487-494) | html | pdf |
  - 4.6.3. Reciprocal-space images (pp. 494-515) | html | pdf |
  - 4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals (pp. 516-517) | html | pdf |
  - References | html | pdf |

Dynamical theory and its applications
- 5.1. Dynamical theory of X-ray diffraction (pp. 534-551) | html | pdf | chapter contents |
  A. Authier
  - 5.1.1. Introduction (p. 534) | html | pdf |
  - 5.1.2. Fundamentals of plane-wave dynamical theory (pp. 534-538) | html | pdf |
  - 5.1.3. Solutions of plane-wave dynamical theory (pp. 538-541) | html | pdf |
  - 5.1.4. Standing waves (p. 541) | html | pdf |
  - 5.1.5. Anomalous absorption (p. 541) | html | pdf |
  - 5.1.6. Intensities of plane waves in transmission geometry (pp. 541-545) | html | pdf |
  - 5.1.7. Intensity of plane waves in reflection geometry (pp. 545-548) | html | pdf |
  - 5.1.8. Real waves (pp. 548-550) | html | pdf |
  - Appendix 5.1.1. (pp. 550-551) | html | pdf |
  - References | html | pdf |
- 5.2. Dynamical theory of electron diffraction (pp. 552-556) | html | pdf | chapter contents |
  A. F. Moodie, J. M. Cowley and P. Goodman
  - 5.2.1. Introduction (p. 552) | html | pdf |
  - 5.2.2. The defining equations (p. 552) | html | pdf |
  - 5.2.3. Forward scattering (p. 552) | html | pdf |
  - 5.2.4. Evolution operator (pp. 552-553) | html | pdf |
  - 5.2.5. Projection approximation – real-space solution (p. 553) | html | pdf |
  - 5.2.6. Semi-reciprocal space (p. 553) | html | pdf |
  - 5.2.7. Two-beam approximation (pp. 553-554) | html | pdf |
  - 5.2.8. Eigenvalue approach (p. 554) | html | pdf |
  - 5.2.9. Translational invariance (p. 554) | html | pdf |
  - 5.2.10. Bloch-wave formulations (p. 555) | html | pdf |
  - 5.2.11. Dispersion surfaces (p. 555) | html | pdf |
  - 5.2.12. Multislice (p. 555) | html | pdf |
  - 5.2.13. Born series (pp. 555-556) | html | pdf |
  - 5.2.14. Approximations (p. 556) | html | pdf |
  - References | html | pdf |
- 5.3. Dynamical theory of neutron diffraction (pp. 557-569) | html | pdf | chapter contents |
  M. Schlenker and J.-P. Guigay
  - 5.3.1. Introduction (p. 557) | html | pdf |
  - 5.3.2. Comparison between X-rays and neutrons with spin neglected (pp. 557-558) | html | pdf |
  - 5.3.3. Neutron spin, and diffraction by perfect magnetic crystals (pp. 558-561) | html | pdf |
  - 5.3.4. Extinction in neutron diffraction (non-magnetic case) (pp. 561-562) | html | pdf |
  - 5.3.5. Effect of external fields on neutron scattering by perfect crystals (p. 562) | html | pdf |
  - 5.3.6. Experimental tests of the dynamical theory of neutron scattering (pp. 562-563) | html | pdf |
  - 5.3.7. Applications of the dynamical theory of neutron scattering (pp. 563-564) | html | pdf |
  - References | html | pdf |

International Tables for CrystallographyVolume B: Reciprocal space

Edited by U. Shmueli

Contents

International Tables for Crystallography
Volume B: Reciprocal space