International Tables for Crystallography
Volume B: Reciprocal space

Second online edition (2010)   ISBN: 978-1-4020-8205-4   doi: 10.1107/97809553602060000108     | 1 | 2 |

Edited by U. Shmueli

Contents

• Preface to the third edition  (p. xiv) | 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. 3-5) | 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. 10-23) | 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. 11-12) | html | pdf |
    • 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion  (pp. 12-15) | html | pdf |
    • 1.2.8. Fourier transform of orbital products  (pp. 15-17) | html | pdf |
    • 1.2.9. The atomic temperature factor  (p. 17) | html | pdf |
    • 1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation  (pp. 17-18) | html | pdf |
    • 1.2.11. Rigid-body analysis  (pp. 18-20) | html | pdf |
    • 1.2.12. Treatment of anharmonicity  (pp. 20-22) | html | pdf |
    • 1.2.13. The generalized structure factor  (p. 22) | html | pdf |
    • 1.2.14. Conclusion  (p. 22) | html | pdf |
    • References | html | pdf |
  • 1.3. Fourier transforms in crystallography: theory, algorithms and applications  (pp. 24-113) | html | pdf | chapter contents |
    G. Bricogne
    • 1.3.1. General introduction  (p. 24) | html | pdf |
    • 1.3.2. The mathematical theory of the Fourier transformation  (pp. 24-52) | html | pdf |
    • 1.3.3. Numerical computation of the discrete Fourier transform  (pp. 52-62) | html | pdf |
    • 1.3.4. Crystallographic applications of Fourier transforms  (pp. 62-106) | html | pdf |
    • References | html | pdf |
  • 1.4. Symmetry in reciprocal space  (pp. 114-174) | html | pdf | chapter contents |
    U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve
    • 1.4.1. Introduction  (p. 114) | html | pdf |
      U. Shmueli
    • 1.4.2. Effects of symmetry on the Fourier image of the crystal  (pp. 114-117) | html | pdf |
      U. Shmueli
    • 1.4.3. Structure-factor tables  (pp. 117-119) | html | pdf |
      U. Shmueli
    • 1.4.4. Symmetry in reciprocal space: space-group tables  (pp. 119-122) | html | pdf |
      U. Shmueli
    • Appendix 1.4.1. Comments on the preparation and usage of the tables  (p. 122) | html | pdf |
      U. Shmueli
    • Appendix 1.4.2. Space-group symbols for numeric and symbolic computations  (pp. 122-134) | html | pdf |
    • Appendix 1.4.3. Structure-factor tables  (pp. 135-161) | html | pdf |
      U. Shmueli
    • Appendix 1.4.4. Crystallographic space groups in reciprocal space  (pp. 162-173) | html | pdf |
      U. Shmueli
    • References | html | pdf |
  • 1.5. Crystallographic viewpoints in the classification of space-group representations  (pp. 175-192) | html | pdf | chapter contents |
    M. I. Aroyo and H. Wondratschek
    • 1.5.1. List of abbreviations and symbols  (p. 175) | html | pdf |
    • 1.5.2. Introduction  (pp. 175-176) | html | pdf |
    • 1.5.3. Basic concepts  (pp. 176-179) | html | pdf |
    • 1.5.4. Conventions in the classification of space-group irreps  (pp. 179-181) | html | pdf |
    • 1.5.5. Examples and discussion  (pp. 181-191) | html | pdf |
    • 1.5.6. Conclusions  (p. 191) | html | pdf |
    • Appendix 1.5.1. Reciprocal-space groups   (p. 192) | html | pdf |
    • References | html | pdf |

• Reciprocal space in crystal-structure determination
  • 2.1. Statistical properties of the weighted reciprocal lattice  (pp. 195-214) | html | pdf | chapter contents |
    U. Shmueli and A. J. C. Wilson
    • 2.1.1. Introduction  (p. 195) | html | pdf |
    • 2.1.2. The average intensity of general reflections  (pp. 195-196) | html | pdf |
    • 2.1.3. The average intensity of zones and rows  (pp. 196-197) | html | pdf |
    • 2.1.4. Probability density distributions – mathematical preliminaries  (pp. 197-200) | html | pdf |
    • 2.1.5. Ideal probability density distributions  (pp. 200-202) | html | pdf |
    • 2.1.6. Distributions of sums, averages and ratios  (pp. 202-203) | html | pdf |
    • 2.1.7. Non-ideal distributions: the correction-factor approach  (pp. 203-207) | html | pdf |
    • 2.1.8. Non-ideal distributions: the Fourier method  (pp. 207-212) | html | pdf |
    • References | html | pdf |
  • 2.2. Direct methods  (pp. 215-243) | html | pdf | chapter contents |
    C. Giacovazzo
    • 2.2.1. List of symbols and abbreviations  (p. 215) | html | pdf |
    • 2.2.2. Introduction  (p. 215) | html | pdf |
    • 2.2.3. Origin specification  (pp. 215-216) | html | pdf |
    • 2.2.4. Normalized structure factors  (pp. 216-221) | html | pdf |
    • 2.2.5. Phase-determining formulae  (pp. 221-230) | html | pdf |
    • 2.2.6. Direct methods in real and reciprocal space: Sayre's equation  (pp. 230-231) | html | pdf |
    • 2.2.7. Scheme of procedure for phase determination: the small-molecule case  (pp. 231-232) | html | pdf |
    • 2.2.8. Other multisolution methods applied to small molecules  (pp. 232-234) | html | pdf |
    • 2.2.9. Some references to direct-methods packages: the small-molecule case  (pp. 234-235) | html | pdf |
    • 2.2.10. Direct methods in macromolecular crystallography  (pp. 235-239) | html | pdf |
    • References | html | pdf |
  • 2.3. Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing  (pp. 244-281) | html | pdf | chapter contents |
    L. Tong, M. G. Rossmann and E. Arnold
    • 2.3.1. Introduction  (pp. 244-247) | html | pdf |
    • 2.3.2. Interpretation of Patterson maps  (pp. 247-251) | html | pdf |
    • 2.3.3. Isomorphous replacement difference Pattersons  (pp. 251-255) | html | pdf |
    • 2.3.4. Anomalous dispersion  (pp. 255-257) | html | pdf |
    • 2.3.5. Noncrystallographic symmetry  (pp. 258-260) | html | pdf |
    • 2.3.6. Rotation functions  (pp. 260-269) | html | pdf |
    • 2.3.7. Translation functions  (pp. 269-272) | html | pdf |
    • 2.3.8. Molecular replacement  (pp. 272-275) | html | pdf |
    • 2.3.9. Conclusions  (p. 275) | html | pdf |
    • References | html | pdf |
  • 2.4. Isomorphous replacement and anomalous scattering  (pp. 282-296) | html | pdf | chapter contents |
    M. Vijayan and S. Ramaseshan
    • 2.4.1. Introduction  (p. 282) | html | pdf |
    • 2.4.2. Isomorphous replacement method  (pp. 282-283) | html | pdf |
    • 2.4.3. Anomalous-scattering method  (pp. 284-287) | html | pdf |
    • 2.4.4. Isomorphous replacement and anomalous scattering in protein crystallography  (pp. 287-293) | html | pdf |
    • 2.4.5. Anomalous scattering of neutrons and synchrotron radiation. The multiwavelength method  (pp. 293-294) | html | pdf |
    • References | html | pdf |
  • 2.5. Electron diffraction and electron microscopy in structure determination  (pp. 297-402) | html | pdf | chapter contents |
    J. M. Cowley, J. C. H. Spence, M. Tanaka, B. K. Vainshtein, B. B. Zvyagin, P. A. Penczek and D. L. Dorset
    • 2.5.1. Foreword  (pp. 297-299) | html | pdf |
      J. M. Cowley and J. C. H. Spence
    • 2.5.2. Electron diffraction and electron microscopy  (pp. 299-307) | html | pdf |
      J. M. Cowley
    • 2.5.3. Point-group and space-group determination by convergent-beam electron diffraction  (pp. 307-356) | html | pdf |
      M. Tanaka
    • 2.5.4. Electron-diffraction structure analysis (EDSA)  (pp. 356-361) | html | pdf |
      B. K. Vainshtein and B. B. Zvyagin
    • 2.5.5. Image reconstruction  (pp. 361-366) | html | pdf |
      B. K. Vainshtein
    • 2.5.6. Three-dimensional reconstruction  (pp. 366-375) | html | pdf |
      B. K. Vainshtein and P. A. Penczek
    • 2.5.7. Single-particle reconstruction  (pp. 375-388) | html | pdf |
      P. A. Penczek
    • 2.5.8. Direct phase determination in electron crystallography  (pp. 388-394) | html | pdf |
      D. L. Dorset
    • References | html | pdf |

• Dual bases in crystallographic computing
  • 3.1. Distances, angles, and their standard uncertainties  (pp. 404-409) | html | pdf | chapter contents |
    D. E. Sands
    • 3.1.1. Introduction  (p. 404) | html | pdf |
    • 3.1.2. Scalar product  (p. 404) | html | pdf |
    • 3.1.3. Length of a vector  (p. 404) | html | pdf |
    • 3.1.4. Angle between two vectors  (pp. 404-405) | html | pdf |
    • 3.1.5. Vector product  (p. 405) | html | pdf |
    • 3.1.6. Permutation tensors  (p. 405) | html | pdf |
    • 3.1.7. Components of vector product  (p. 405) | html | pdf |
    • 3.1.8. Some vector relationships  (pp. 405-406) | html | pdf |
    • 3.1.9. Planes  (p. 406) | html | pdf |
    • 3.1.10. Variance–covariance matrices  (pp. 406-408) | html | pdf |
    • 3.1.11. Mean values  (p. 408) | html | pdf |
    • 3.1.12. Computation  (pp. 408-409) | html | pdf |
    • References | html | pdf |
  • 3.2. The least-squares plane  (pp. 410-417) | html | pdf | chapter contents |
    R. E. Marsh and V. Schomaker
    • 3.2.1. Introduction  (p. 410) | html | pdf |
    • 3.2.2. Least-squares plane based on uncorrelated, isotropic weights  (pp. 410-413) | html | pdf |
    • 3.2.3. The proper least-squares plane, with Gaussian weights  (pp. 413-416) | html | pdf |
    • Appendix 3.2.1.   (pp. 416-417) | html | pdf |
    • References | html | pdf |
  • 3.3. Molecular modelling and graphics  (pp. 418-448) | html | pdf | chapter contents |
    R. Diamond and L. M. D. Cranswick
    • 3.3.1. Graphics  (pp. 418-434) | html | pdf |
      R. Diamond
    • 3.3.2. Molecular modelling, problems and approaches  (pp. 434-438) | html | pdf |
      R. Diamond
    • 3.3.3. Implementations  (pp. 438-442) | html | pdf |
      R. Diamond
    • 3.3.4. Graphics software for the display of small and medium-sized molecules  (pp. 442-445) | html | pdf |
      L. M. D. Cranswick
    • References | html | pdf |
  • 3.4. Accelerated convergence treatment of R⁻ⁿ lattice sums  (pp. 449-457) | html | pdf | chapter contents |
    D. E. Williams
    • 3.4.1. Introduction  (p. 449) | html | pdf |
    • 3.4.2. Definition and behaviour of the direct-space sum  (p. 449) | html | pdf |
    • 3.4.3. Preliminary description of the method  (pp. 449-450) | html | pdf |
    • 3.4.4. Preliminary derivation to obtain a formula which accelerates the convergence of an R⁻ⁿ sum over lattice points X(d)  (pp. 450-452) | html | pdf |
    • 3.4.5. Extension of the method to a composite lattice  (pp. 452-453) | html | pdf |
    • 3.4.6. The case of n = 1 (Coulombic lattice energy)  (pp. 453-454) | html | pdf |
    • 3.4.7. The cases of n = 2 and n = 3  (p. 454) | html | pdf |
    • 3.4.8. Derivation of the accelerated convergence formula via the Patterson function  (p. 454) | html | pdf |
    • 3.4.9. Evaluation of the incomplete gamma function  (pp. 454-455) | html | pdf |
    • 3.4.10. Summation over the asymmetric unit and elimination of intramolecular energy terms  (p. 455) | html | pdf |
    • 3.4.11. Reference formulae for particular values of n  (pp. 455-456) | html | pdf |
    • 3.4.12. Numerical illustrations  (pp. 456-457) | html | pdf |
    • References | html | pdf |
  • 3.5. Extensions of the Ewald method for Coulomb interactions in crystals  (pp. 458-481) | html | pdf | chapter contents |
    T. A. Darden
    • 3.5.1. Introduction  (pp. 458-460) | html | pdf |
    • 3.5.2. Lattice sums of point charges  (pp. 460-471) | html | pdf |
    • 3.5.3. Generalization to Gaussian- and Hermite-based continuous charge distributions  (pp. 471-474) | html | pdf |
    • 3.5.4. Computational efficiency  (pp. 474-480) | html | pdf |
    • References | html | pdf |

• Diffuse scattering and related topics
  • 4.1. Thermal diffuse scattering of X-rays and neutrons  (pp. 484-491) | html | pdf | chapter contents |
    B. T. M. Willis
    • 4.1.1. Introduction  (p. 484) | html | pdf |
    • 4.1.2. Dynamics of three-dimensional crystals  (pp. 484-487) | html | pdf |
    • 4.1.3. Scattering of X-rays by thermal vibrations  (pp. 487-488) | html | pdf |
    • 4.1.4. Scattering of neutrons by thermal vibrations  (pp. 488-489) | html | pdf |
    • 4.1.5. Phonon dispersion relations  (pp. 489-490) | html | pdf |
    • 4.1.6. Measurement of elastic constants  (pp. 490-491) | html | pdf |
    • References | html | pdf |
  • 4.2. Disorder diffuse scattering of X-rays and neutrons  (pp. 492-539) | html | pdf | chapter contents |
    F. Frey, H. Boysen and H. Jagodzinski
    • 4.2.1. Introduction  (pp. 492-493) | html | pdf |
    • 4.2.2. Basic scattering theory  (pp. 493-495) | html | pdf |
    • 4.2.3. Qualitative treatment of structural disorder  (pp. 495-507) | html | pdf |
    • 4.2.4. General guidelines for analysing a disorder problem  (pp. 507-509) | html | pdf |
    • 4.2.5. Quantitative interpretation  (pp. 509-526) | html | pdf |
    • 4.2.6. Disorder diffuse scattering from aperiodic crystals  (pp. 526-528) | html | pdf |
    • 4.2.7. Computer simulations and modelling  (pp. 528-530) | html | pdf |
    • 4.2.8. Experimental techniques and data evaluation  (pp. 530-535) | html | pdf |
    • References | html | pdf |
  • 4.3. Diffuse scattering in electron diffraction  (pp. 540-546) | html | pdf | chapter contents |
    J. M. Cowley and J. K. Gjønnes
    • 4.3.1. Introduction  (pp. 540-541) | html | pdf |
    • 4.3.2. Inelastic scattering  (pp. 541-542) | html | pdf |
    • 4.3.3. Kinematical and pseudo-kinematical scattering  (p. 542) | html | pdf |
    • 4.3.4. Dynamical scattering: Bragg scattering effects  (pp. 542-544) | html | pdf |
    • 4.3.5. Multislice calculations for diffraction and imaging  (p. 544) | html | pdf |
    • 4.3.6. Qualitative interpretation of diffuse scattering of electrons  (pp. 544-545) | html | pdf |
    • References | html | pdf |
  • 4.4. Scattering from mesomorphic structures  (pp. 547-566) | html | pdf | chapter contents |
    P. S. Pershan
    • 4.4.1. Introduction  (pp. 547-549) | html | pdf |
    • 4.4.2. The nematic phase  (pp. 549-551) | html | pdf |
    • 4.4.3. Smectic-A and smectic-C phases  (pp. 551-554) | html | pdf |
    • 4.4.4. Phases with in-plane order  (pp. 554-561) | html | pdf |
    • 4.4.5. Discotic phases  (p. 561) | html | pdf |
    • 4.4.6. Other phases  (pp. 561-562) | html | pdf |
    • References | html | pdf |
  • 4.5. Polymer crystallography  (pp. 567-589) | html | pdf | chapter contents |
    R. P. Millane and D. L. Dorset
    • 4.5.1. Overview  (p. 567) | html | pdf |
      R. P. Millane and D. L. Dorset
    • 4.5.2. X-ray fibre diffraction analysis  (pp. 568-583) | html | pdf |
      R. P. Millane
    • 4.5.3. Electron crystallography of polymers  (pp. 583-586) | html | pdf |
      D. L. Dorset
    • References | html | pdf |
  • 4.6. Reciprocal-space images of aperiodic crystals  (pp. 590-624) | html | pdf | chapter contents |
    W. Steurer and T. Haibach
    • 4.6.1. Introduction  (pp. 590-591) | html | pdf |
    • 4.6.2. The n-dimensional description of aperiodic crystals  (pp. 591-598) | html | pdf |
    • 4.6.3. Reciprocal-space images  (pp. 598-621) | html | pdf |
    • 4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals  (pp. 621-623) | html | pdf |
    • References | html | pdf |

• Dynamical theory and its applications
  • 5.1. Dynamical theory of X-ray diffraction  (pp. 626-646) | html | pdf | chapter contents |
    A. Authier
    • 5.1.1. Introduction  (p. 626) | html | pdf |
    • 5.1.2. Fundamentals of plane-wave dynamical theory  (pp. 626-630) | html | pdf |
    • 5.1.3. Solutions of plane-wave dynamical theory  (pp. 630-633) | html | pdf |
    • 5.1.4. Standing waves  (p. 633) | html | pdf |
    • 5.1.5. Anomalous absorption  (pp. 633-634) | html | pdf |
    • 5.1.6. Intensities of plane waves in transmission geometry  (pp. 634-638) | html | pdf |
    • 5.1.7. Intensity of plane waves in reflection geometry  (pp. 638-640) | html | pdf |
    • 5.1.8. Real waves  (pp. 640-642) | html | pdf |
    • Appendix 5.1.1. Basic equations  (pp. 642-644) | html | pdf |
    • References | html | pdf |
  • 5.2. Dynamical theory of electron diffraction  (pp. 647-653) | html | pdf | chapter contents |
    A. F. Moodie, J. M. Cowley and P. Goodman
    • 5.2.1. Introduction  (p. 647) | html | pdf |
    • 5.2.2. The defining equations  (p. 647) | html | pdf |
    • 5.2.3. Forward scattering  (pp. 647-648) | html | pdf |
    • 5.2.4. Evolution operator  (p. 648) | html | pdf |
    • 5.2.5. Projection approximation – real-space solution  (p. 648) | html | pdf |
    • 5.2.6. Semi-reciprocal space  (pp. 648-649) | html | pdf |
    • 5.2.7. Two-beam approximation  (p. 649) | html | pdf |
    • 5.2.8. Eigenvalue approach  (pp. 649-650) | html | pdf |
    • 5.2.9. Translational invariance  (p. 650) | html | pdf |
    • 5.2.10. Bloch-wave formulations  (p. 650) | html | pdf |
    • 5.2.11. Dispersion surfaces  (p. 650) | html | pdf |
    • 5.2.12. Multislice  (p. 651) | html | pdf |
    • 5.2.13. Born series  (pp. 651-652) | html | pdf |
    • 5.2.14. Approximations  (p. 652) | html | pdf |
    • References | html | pdf |
  • 5.3. Dynamical theory of neutron diffraction  (pp. 654-664) | html | pdf | chapter contents |
    M. Schlenker and J.-P. Guigay
    • 5.3.1. Introduction  (p. 654) | html | pdf |
    • 5.3.2. Comparison between X-rays and neutrons with spin neglected  (pp. 654-655) | html | pdf |
    • 5.3.3. Neutron spin, and diffraction by perfect magnetic crystals  (pp. 655-658) | html | pdf |
    • 5.3.4. Extinction in neutron diffraction (nonmagnetic case)  (pp. 658-659) | html | pdf |
    • 5.3.5. Effect of external fields on neutron scattering by perfect crystals  (p. 659) | html | pdf |
    • 5.3.6. Experimental tests of the dynamical theory of neutron scattering  (pp. 659-660) | html | pdf |
    • 5.3.7. Applications of the dynamical theory of neutron scattering  (pp. 660-661) | html | pdf |
    • References | html | pdf |