International Tables for Crystallography Volume B: Reciprocal space [B, 4 ed.] 9781402082054

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INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY

Volume B RECIPROCAL SPACE

Edited by U. SHMUELI

Contributing authors E. Arnold: CABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA. [2.3] M. I. Aroyo: Departamento de Fisı´ca de la Materia Condensada, Facultad de Cienca y Technologı´a, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain. [1.5] A. Authier: Institut de Mine´ralogie et de la Physique des Milieux Condense´s, Baˆtiment 7, 140 rue de Lourmel, 75015 Paris, France. [5.1] H. Boysen: Department fu¨r Geo- und Umweltwissenschaften, Sektion Kristallographie, LudwigMaximilians Universita¨t, Theresienstrasse 41, 80333 Mu¨nchen, Germany. [4.2] G. Bricogne: Global Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Baˆtiment 209D, Universite´ Paris-Sud, 91405 Orsay, France. [1.3] P. Coppens: Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 142603000, USA. [1.2] J. M. Cowley:† Arizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287-1504, USA. [2.5.1, 2.5.2, 4.3, 5.2] L. M. D. Cranswick: Neutron Program for Materials Research, National Research Council Canada, Building 459, Chalk River Laboratories, Chalk River, Ontario, Canada K0J 1J0. [3.3.4] T. A. Darden: Laboratory of Structural Biology, National Institute of Environmental Health Sciences, 111 T. W. Alexander Drive, Research Triangle Park, NC 27709, USA. [3.5] R. Diamond: MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England. [3.3.1, 3.3.2, 3.3.3] D. L. Dorset: ExxonMobil Research and Engineering Co., 1545 Route 22 East, Clinton Township, Annandale, New Jersey 08801, USA. [2.5.8, 4.5.1, 4.5.3] F. Frey: Department fu¨r Geo- und Umweltwissenschaften, Sektion Kristallographie, LudwigMaximilians Universita¨t, Theresienstrasse 41, 80333 Mu¨nchen, Germany. [4.2] C. Giacovazzo: Dipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 Bari, Italy. [2.2] J. K. Gjønnes: Institute of Physics, University of Oslo, PO Box 1048, N-0316 Oslo 3, Norway. [4.3]

P. Goodman†: School of Physics, University of Melbourne, Parkville, Australia. [5.2] R. W. Grosse-Kunstleve: Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Mailstop 4-230, Berkeley, CA 94720, USA. [1.4] J.-P. Guigay: European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France. [5.3] T. Haibach: Laboratory of Crystallography, Department of Materials, ETH Ho¨nggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland. [4.6] S. R. Hall: Crystallography Centre, University of Western Australia, Nedlands 6907, WA, Australia. [1.4] H. Jagodzinski: Department fu¨r Geo- und Umweltwissenschaften, Sektion Kristallographie, LudwigMaximilians Universita¨t, Theresienstrasse 41, 80333 Mu¨nchen, Germany. [4.2] R. E. Marsh: The Beckman Institute–139–74, California Institute of Technology, 1201 East California Blvd, Pasadena, California 91125, USA. [3.2] R. P. Millane: Department of Electrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. [4.5.1, 4.5.2] A. F. Moodie: Department of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia. [5.2] P. A. Penczek: The University of Texas – Houston Medical School, Department of Biochemistry and Molecular Biology, 6431 Fannin, MSB 6.218, Houston, TX 77030, USA. [2.5.6, 2.5.7] P. S. Pershan: Division of Engineering and Applied Science and The Physics Department, Harvard University, Cambridge, MA 02138, USA. [4.4] S. Ramaseshan†: Raman Research Institute, Bangalore 560 080, India. [2.4] M. G. Rossmann: Department of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA. [2.3] D. E. Sands: Department of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA. [3.1] M. Schlenker: Laboratoire Louis Ne´el du CNRS, BP 166, F-38042 Grenoble Cedex 9, France. [5.3] V. Schomaker†: Department of Chemistry, University of Washington, Seattle, Washington 98195, USA. [3.2] U. Shmueli: School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel. [1.1, 1.4, 2.1]

† Deceased.

† Deceased.

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CONTRIBUTING AUTHORS

J. C. H. Spence: Department of Physics, Arizona State University, Tempe, AZ 95287-1504, USA. [2.5.1]

M. Vijayan: Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India. [2.4]

W. Steurer: Laboratory of Crystallography, Department of Materials, ETH Ho¨nggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland. [4.6]

D. E. Williams†: Department of Chemistry, University of Louisville, Louisville, Kentucky 40292, USA. [3.4] B. T. M. Willis: Department of Chemistry, Chemistry Research Laboratory, University of Oxford, Mansfield Road, Oxford OX1 3TA, England. [4.1]

M. Tanaka: Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Japan. [2.5.3] L. Tong: Department of Biological Sciences, Columbia University, New York 10027, USA. [2.3]

A. J. C. Wilson†: St John’s College, Cambridge, England. [2.1] H. Wondratschek: Institut fu¨r Kristallographie, Universita¨t, D-76128 Karlsruhe, Germany. [1.5]

B. K. Vainshtein†: Institute of Crystallography, Academy of Sciences of Russia, Leninsky prospekt 59, Moscow B-117333, Russia. [2.5.4, 2.5.5, 2.5.6]

B. B. Zvyagin†: Institute of Ore Mineralogy (IGEM), Academy of Sciences of Russia, Staromonetny 35, 109017 Moscow, Russia. [2.5.4]

† Deceased.

† Deceased.

vi

Contents PAGE

Preface (U. Shmueli) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

xiii

Preface to the second edition (U. Shmueli)

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xiii

Preface to the third edition (U. Shmueli)

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xiv

PART 1. GENERAL RELATIONSHIPS AND TECHNIQUES 1.1. Reciprocal space in crystallography (U. Shmueli) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

2

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1.1.3. Fundamental relationships .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.1.1. Introduction

1.1.2. Reciprocal lattice in crystallography

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1.1.4. Tensor-algebraic formulation 1.1.5. Transformations

1.1.6. Some analytical aspects of the reciprocal space

1.2. The structure factor (P. Coppens) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.3. Scattering by a crystal: definition of a structure factor

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1.2.4. The isolated-atom approximation in X-ray diffraction

1.2.1. Introduction

1.2.2. General scattering expression for X-rays

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1.2.5. Scattering of thermal neutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.8. Fourier transform of orbital products .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation .. .. .. .. .. .. .. ..

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1.2.11. Rigid-body analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion 1.2.9. The atomic temperature factor

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1.2.13. The generalized structure factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.3. Fourier transforms in crystallography: theory, algorithms and applications (G. Bricogne) .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.4. Symmetry in reciprocal space (U. Shmueli) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.4.2. Effects of symmetry on the Fourier image of the crystal .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

114

1.2.12. Treatment of anharmonicity 1.2.14. Conclusion

1.3.1. General introduction

1.3.2. The mathematical theory of the Fourier transformation 1.3.3. Numerical computation of the discrete Fourier transform 1.3.4. Crystallographic applications of Fourier transforms

1.4.1. Introduction

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1.4.4. Symmetry in reciprocal space: space-group tables .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.4.3. Structure-factor tables

Appendix A1.4.1. Comments on the preparation and usage of the tables

Appendix A1.4.2. Space-group symbols for numeric and symbolic computations (U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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Appendix A1.4.3. Structure-factor tables

Appendix A1.4.4. Crystallographic space groups in reciprocal space

1.5. Crystallographic viewpoints in the classification of space-group representations (M. I. Aroyo and H. Wondratschek) .. .. .. ..

175

1.5.1. List of abbreviations and symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.5.2. Introduction 1.5.3. Basic concepts

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CONTENTS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.5.4. Conventions in the classification of space-group irreps 1.5.5. Examples and discussion 1.5.6. Conclusions

Appendix A1.5.1. Reciprocal-space groups ðGÞ


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192

PART 2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION .. .. .. .. .. .. .. .. .. .. .. ..

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2.1.2. The average intensity of general reflections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.1.3. The average intensity of zones and rows .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.2. Direct methods (C. Giacovazzo) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

215

2.2.1. List of symbols and abbreviations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.2.4. Normalized structure factors

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2.2.5. Phase-determining formulae

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221

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2.2.8. Other multisolution methods applied to small molecules .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

232

2.2.9. Some references to direct-methods packages: the small-molecule case .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

234

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235

2.1. Statistical properties of the weighted reciprocal lattice (U. Shmueli and A. J. C. Wilson) 2.1.1. Introduction

2.1.4. Probability density distributions – mathematical preliminaries 2.1.5. Ideal probability density distributions

2.1.6. Distributions of sums, averages and ratios

2.1.7. Non-ideal distributions: the correction-factor approach 2.1.8. Non-ideal distributions: the Fourier method

2.2.2. Introduction

2.2.3. Origin specification

2.2.6. Direct methods in real and reciprocal space: Sayre’s equation

2.2.7. Scheme of procedure for phase determination: the small-molecule case

2.2.10. Direct methods in macromolecular crystallography

2.3. Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing (L. Tong, .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

244

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2.3.2. Interpretation of Patterson maps .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

247

M. G. Rossmann and E. Arnold) 2.3.1. Introduction

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2.3.5. Noncrystallographic symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

258

2.3.3. Isomorphous replacement difference Pattersons 2.3.4. Anomalous dispersion

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2.3.7. Translation functions

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2.3.8. Molecular replacement

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2.3.6. Rotation functions

2.3.9. Conclusions

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2.4.3. Anomalous-scattering method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

284

2.4.4. Isomorphous replacement and anomalous scattering in protein crystallography .. .. .. .. .. .. .. .. .. .. .. .. .. ..

287

2.4.5. Anomalous scattering of neutrons and synchrotron radiation. The multiwavelength method .. .. .. .. .. .. .. .. .. ..

293

2.4. Isomorphous replacement and anomalous scattering (M. Vijayan and S. Ramaseshan) 2.4.1. Introduction

2.4.2. Isomorphous replacement method

2.5. Electron diffraction and electron microscopy in structure determination (J. M. Cowley, J. C. H. Spence, M. Tanaka, .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

297

2.5.1. Foreword (J. M. Cowley and J. C. H. Spence) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

297

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2.5.3. Point-group and space-group determination by convergent-beam electron diffraction (M. Tanaka) .. .. .. .. .. .. .. ..

307

B. K. Vainshtein, B. B. Zvyagin, P. A. Penczek and D. L. Dorset)

2.5.2. Electron diffraction and electron microscopy (J. M. Cowley)

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CONTENTS .. .. .. .. .. .. .. .. .. .. ..

356

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2.5.7. Single-particle reconstruction (P. A. Penczek) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.5.4. Electron-diffraction structure analysis (EDSA) (B. K. Vainshtein and B. B. Zvyagin) 2.5.5. Image reconstruction (B. K. Vainshtein)

2.5.6. Three-dimensional reconstruction (B. K. Vainshtein and P. A. Penczek) 2.5.8. Direct phase determination in electron crystallography (D. L. Dorset)

PART 3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.4. Angle between two vectors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

404

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3.1.6. Permutation tensors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

405

3.1. Distances, angles, and their standard uncertainties (D. E. Sands) 3.1.1. Introduction 3.1.2. Scalar product 3.1.3. Length of a vector 3.1.5. Vector product

3.1.7. Components of vector product .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.11. Mean values .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.12. Computation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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410

3.1.8. Some vector relationships 3.1.9. Planes

3.1.10. Variance–covariance matrices

3.2. The least-squares plane (R. E. Marsh and V. Schomaker) 3.2.1. Introduction

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Appendix A3.2.1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

416

3.2.2. Least-squares plane based on uncorrelated, isotropic weights 3.2.3. The proper least-squares plane, with Gaussian weights

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418

3.3.1. Graphics (R. Diamond) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

418

3.3.2. Molecular modelling, problems and approaches (R. Diamond) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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449

3.3. Molecular modelling and graphics (R. Diamond and L. M. D. Cranswick)

3.3.3. Implementations (R. Diamond)

3.3.4. Graphics software for the display of small and medium-sized molecules (L. M. D. Cranswick) 3.4. Accelerated convergence treatment of R 3.4.1. Introduction

n

lattice sums (D. E. Williams)

3.4.2. Definition and behaviour of the direct-space sum .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

449

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3.4.4. Preliminary derivation to obtain a formula which accelerates the convergence of an Rn sum over lattice points X(d) .. ..

450

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452

3.4.3. Preliminary description of the method

3.4.5. Extension of the method to a composite lattice 3.4.6. The case of n ¼ 1 (Coulombic lattice energy) 3.4.7. The cases of n ¼ 2 and n ¼ 3

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454

3.4.8. Derivation of the accelerated convergence formula via the Patterson function

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454

3.4.9. Evaluation of the incomplete gamma function .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

454

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455

3.4.11. Reference formulae for particular values of n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

455

3.4.12. Numerical illustrations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

456

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474

3.4.10. Summation over the asymmetric unit and elimination of intramolecular energy terms

3.5. Extensions of the Ewald method for Coulomb interactions in crystals (T. A. Darden) 3.5.1. Introduction

3.5.2. Lattice sums of point charges

3.5.3. Generalization to Gaussian- and Hermite-based continuous charge distributions 3.5.4. Computational efficiency

ix

CONTENTS

PART 4. DIFFUSE SCATTERING AND RELATED TOPICS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

484

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484

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

487

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

488

4.1. Thermal diffuse scattering of X-rays and neutrons (B. T. M. Willis) 4.1.1. Introduction

4.1.2. Dynamics of three-dimensional crystals

4.1.3. Scattering of X-rays by thermal vibrations 4.1.4. Scattering of neutrons by thermal vibrations

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

489

4.1.6. Measurement of elastic constants .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

490

.. .. .. .. .. .. .. .. .. .. ..

492

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

492

4.1.5. Phonon dispersion relations

4.2. Disorder diffuse scattering of X-rays and neutrons (F. Frey, H. Boysen and H. Jagodzinski) 4.2.1. Introduction

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

493

4.2.3. Qualitative treatment of structural disorder .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

495

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

507

4.2.5. Quantitative interpretation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

509

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

526

4.2.7. Computer simulations and modelling .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

528

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

530

4.2.2. Basic scattering theory

4.2.4. General guidelines for analysing a disorder problem 4.2.6. Disorder diffuse scattering from aperiodic crystals 4.2.8. Experimental techniques and data evaluation

4.3. Diffuse scattering in electron diffraction (J. M. Cowley and J. K. Gjønnes) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

540

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

540

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

541

4.3.3. Kinematical and pseudo-kinematical scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

542

4.3.1. Introduction

4.3.2. Inelastic scattering

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

542

4.3.5. Multislice calculations for diffraction and imaging .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

544

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

544

4.4. Scattering from mesomorphic structures (P. S. Pershan) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

547

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

547

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

549

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

551

4.4.4. Phases with in-plane order .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

554

4.3.4. Dynamical scattering: Bragg scattering effects

4.3.6. Qualitative interpretation of diffuse scattering of electrons

4.4.1. Introduction

4.4.2. The nematic phase

4.4.3. Smectic-A and smectic-C phases

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

561

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

561

4.5. Polymer crystallography (R. P. Millane and D. L. Dorset) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

567

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

567

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

568

4.5.3. Electron crystallography of polymers (D. L. Dorset) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

583

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

590

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

590

4.4.5. Discotic phases 4.4.6. Other phases

4.5.1. Overview (R. P. Millane and D. L. Dorset) 4.5.2. X-ray fibre diffraction analysis (R. P. Millane)

4.6. Reciprocal-space images of aperiodic crystals (W. Steurer and T. Haibach) 4.6.1. Introduction

4.6.2. The n-dimensional description of aperiodic crystals 4.6.3. Reciprocal-space images

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

591

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

598

.. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

621

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

626

4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals

PART 5. DYNAMICAL THEORY AND ITS APPLICATIONS 5.1. Dynamical theory of X-ray diffraction (A. Authier)

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

626

5.1.2. Fundamentals of plane-wave dynamical theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

626

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

630

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

633

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

633

5.1.1. Introduction

5.1.3. Solutions of plane-wave dynamical theory 5.1.4. Standing waves

5.1.5. Anomalous absorption

x

CONTENTS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

634

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

638

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

640

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

642

5.2. Dynamical theory of electron diffraction (A. F. Moodie, J. M. Cowley and P. Goodman) .. .. .. .. .. .. .. .. .. .. .. .. ..

647

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

647

5.1.6. Intensities of plane waves in transmission geometry 5.1.7. Intensity of plane waves in reflection geometry 5.1.8. Real waves

Appendix A5.1.1. Basic equations

5.2.1. Introduction

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

647

5.2.3. Forward scattering

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

647

5.2.4. Evolution operator

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

648

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

648

5.2.2. The defining equations

5.2.5. Projection approximation – real-space solution

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

648

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

649

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

649

5.2.9. Translational invariance .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

650

5.2.6. Semi-reciprocal space 5.2.7. Two-beam approximation 5.2.8. Eigenvalue approach

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

650

5.2.11. Dispersion surfaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

650

5.2.12. Multislice .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

651

5.2.10. Bloch-wave formulations

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

651

5.2.14. Approximations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

652

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

654

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

654

5.3.2. Comparison between X-rays and neutrons with spin neglected .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

654

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

655

5.3.4. Extinction in neutron diffraction (nonmagnetic case) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

658

5.3.5. Effect of external fields on neutron scattering by perfect crystals

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

659

5.3.6. Experimental tests of the dynamical theory of neutron scattering

5.2.13. Born series

5.3. Dynamical theory of neutron diffraction (M. Schlenker and J.-P. Guigay) 5.3.1. Introduction

5.3.3. Neutron spin, and diffraction by perfect magnetic crystals

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

659

5.3.7. Applications of the dynamical theory of neutron scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

660

Author index

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

665

Subject index

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

675

xi

Preface By Uri Shmueli The purpose of Volume B of International Tables for Crystallography is to provide the user or reader with accounts of some well established topics, of importance to the science of crystallography, which are related in one way or another to the concepts of reciprocal lattice and, more generally, reciprocal space. Efforts have been made to extend the treatment of the various topics to include X-ray, electron, and neutron diffraction techniques, and thereby do some justice to the inclusion of the present Volume in the new series of International Tables for Crystallography. An important crystallographic aspect of symmetry in reciprocal space, space-group-dependent expressions of trigonometric structure factors, already appears in Volume I of International Tables for X-ray Crystallography, and preliminary plans for incorporating this and other crystallographic aspects of reciprocal space in the new edition of International Tables date back to 1972. However, work on a volume of International Tables for Crystallography, largely dedicated to the subject of reciprocal space, began over ten years later. The present structure of Volume B, as determined in the years preceding the 1984 Hamburg congress of the International Union of Crystallography (IUCr), is due to (i) computer-controlled production of concise structure-factor tables, (ii) the ability to introduce many more aspects of reciprocal space – as a result of reducing the effort of producing the above tables, as well as their volume, and (iii) suggestions by the National Committees and individual crystallographers of some additional interesting topics. It should be pointed out that the initial plans for the present Volume and Volume C (Mathematical, Physical and Chemical Tables, edited by Professor A. J. C. Wilson), were formulated and approved during the same period.

The obviously delayed publication of Volume B is due to several reasons. Some minor delays were caused by a requirement that potential contributors should be approved by the Executive Committee prior to issuing relevant invitations. Much more serious delays were caused by authors who failed to deliver their contributions. In fact, some invited contributions had to be excluded from this first edition of Volume B. Some of the topics here treated are greatly extended, considerably updated or modern versions of similar topics previously treated in the old Volumes I, II, and IV. Most of the subjects treated in Volume B are new to International Tables. I gratefully thank Professor A. J. C. Wilson, for suggesting that I edit this Volume and for sharing with me his rich editorial experience. I am indebted to those authors of Volume B who took my requests and deadlines seriously, and to the Computing Center of Tel Aviv University for computing facilities and time. Special thanks are due to Mrs Z. Stein (Tel Aviv University) for skilful assistance in numeric and symbolic programming, involved in my contributions to this Volume. I am most grateful to many colleagues–crystallographers for encouragement, advice, and suggestions. In particular, thanks are due to Professors J. M. Cowley, P. Goodman and C. J. Humphreys, who served as Chairmen of the Commission on Electron Diffraction during the preparation of this Volume, for prompt and expert help at all stages of the editing. The kind assistance of Dr J. N. King, the Executive Secretary of the IUCr, is also gratefully acknowledged. Last, but certainly not least, I wish to thank Mr M. H. Dacombe, the Technical Editor of the IUCr, and his staff for the skilful and competent treatment of the variety of drafts and proofs out of which this Volume arose.

Preface to the second edition By Uri Shmueli The first edition of Volume B appeared in 1993, and was followed by a corrected reprint in 1996. Although practically all the material for the second edition was available in early 1997, its publication was delayed by the decision to translate all of Volume B, and indeed all the other volumes of International Tables for Crystallography, to Standard Generalized Markup Language (SGML) and thus make them available also in an electronic form suitable for modern publishing procedures. During the preparation of the second edition, most chapters that appeared in the first edition have been corrected and/or revised, some were rather extensively updated, and five new chapters were added. The overall structure of the second edition is outlined below. After an introductory chapter, Part 1 presents the reader with an account of structure-factor formalisms, an extensive treatment of the theory, algorithms and crystallographic applications of Fourier methods, and treatments of symmetry in reciprocal space. These are here enriched with more advanced aspects of representations of space groups in reciprocal space. In Part 2, these general accounts are followed by detailed expositions of crystallographic statistics, the theory of direct methods, Patterson techniques, isomorphous replacement and anomalous scattering, and treatments of the role of electron

microscopy and diffraction in crystal structure determination. The latter topic is here enhanced by applications of direct methods to electron crystallography. Part 3, Dual Bases in Crystallographic Computing, deals with applications of reciprocal space to molecular geometry and ‘best’-plane calculations, and contains a treatment of the principles of molecular graphics and modelling and their applications; it concludes with the presentation of a convergence-acceleration method, of importance in the computation of approximate lattice sums. Part 4 contains treatments of various diffuse-scattering phenomena arising from crystal dynamics, disorder and low dimensionality (liquid crystals), and an exposition of the underlying theories and/or experimental evidence. The new additions to this part are treatments of polymer crystallography and of reciprocal-space images of aperiodic crystals. Part 5 contains introductory treatments of the theory of the interaction of radiation with matter, the so-called dynamical theory, as applied to X-ray, electron and neutron diffraction techniques. The chapter on the dynamical theory of neutron diffraction is new. I am deeply grateful to the authors of the new contributions for making their expertise available to Volume B and for their excellent collaboration. I also take special pleasure in thanking

xiii

PREFACE those authors of the first edition who revised and updated their contributions in view of recent developments. Last but not least, I wish to thank all the authors for their contributions and their patience, and am grateful to those authors who took my requests seriously. I hope that the updating and revision of future editions will be much easier and more expedient, mainly because of the new format of International Tables. Four friends and greatly respected colleagues who contributed to the second edition of Volume B are no longer with us. These are Professors Arthur J. C. Wilson, Peter Goodman, Verner Schomaker and Boris K. Vainshtein. I asked Professors Michiyoshi Tanaka, John Cowley and Douglas Dorset if they were prepared to answer queries related to the contributions of the late Peter Goodman and Boris K. Vainshtein to Chapter 2.5. I am most grateful for their prompt agreement.

This editorial work was carried out at the School of Chemistry and the Computing Center of Tel Aviv University. The facilities they put at my disposal are gratefully acknowledged on my behalf and on behalf of the IUCr. I wish to thank many colleagues for interesting conversations and advice, and in particular Professor Theo Hahn with whom I discussed at length problems regarding Volume B and International Tables in general. Given all these expert contributions, the publication of this volume would not have been possible without the expertise and devotion of the Technical Editors of the IUCr. My thanks go to Mrs Sue King, for her cooperation during the early stages of the work on the second edition of Volume B, while the material was being collected, and to Dr Nicola Ashcroft, for her collaboration during the final stages of the production of the volume, for her most careful and competent treatment of the proofs, and last but not least for her tactful and friendly attitude.

Preface to the third edition By Uri Shmueli The second edition of Volume B appeared in 2001. Plans for the third edition included the addition of new chapters and sections, the substantial revision of several chapters that existed in the second edition and minor revisions and updating of existing chapters. The overall structure of Volume B remained unchanged. In Part 1, Chapter 1.5 on classifications of space-group representations in reciprocal space has been extensively revised. In Part 2, Chapter 2.2 on direct methods has been considerably extended to include applications of these methods to macromolecular crystallography. Chapter 2.3 on Patterson and molecular replacement techniques has been updated and extended. Section 2.5.3 on convergent-beam electron diffraction has been completely rewritten by a newly invited author, and Section 2.5.6 on three-dimensional reconstruction has been updated and extended by a newly invited author, who has also added Section 2.5.7 on single-particle reconstruction. The Foreword to Chapter 2.5 on electron diffraction and microscopy has also been revised. In Part 3, Chapter 3.3 on computer graphics and molecular modelling has been enriched by Section 3.3.4 on the implementation of molecular graphics to small and medium-sized molecules, and a comprehensive Chapter 3.5 on modern extensions of Ewald methods has been added, dealing with (i) inclusion of fast Fourier transforms in the computation of sums and (ii) departure from the point-charge model in Ewald summations. In Part 4, Chapter 4.1 on thermal diffuse scattering of X-rays and neutrons has been updated, and Chapter 4.2 on disorder diffuse scattering of X-rays and neutrons has been extensively revised and updated. Minor updates and corrections have also been made to several existing chapters and sections in all the parts of the volume.

My gratitude is extended to the authors of new contributions and to the authors of the first and second editions of the volume for significant revisions of their chapters and sections in view of new developments. I wish to thank all the authors for their excellent collaboration and for sharing with the International Tables for Crystallography their expertise. I hope that the tradition of keeping the contributions up to date will also persist in future editions of Volume B. This will be aided by significant improvements in various aspects of technical editing which were already apparent in the preparation of this edition. Three greatly respected friends and colleagues who contributed to this and previous editions of Volume B passed away after the second edition of Volume B was published. These are Professors John Cowley, Boris Zvyagin and Donald Williams. I asked Professors John Spence, Douglas Dorset and Pawel Penczek to take care of any questions about the articles of the late John Cowley, Boris Zvyagin and Boris Vainshtein in Chapter 2.5, and Dr Bill Smith to answer any questions about Chapter 3.4 by the late Donald Williams. They all agreed promptly and I am most grateful for this. My editorial work was carried out at the School of Chemistry of Tel Aviv University and I wish to acknowledge gratefully the facilities that were put at my disposal. I am grateful to many friends and colleagues for interesting conversations and exchanges related to this volume. Thanks are also due to my friends from the IUCr office in Chester for their helpful interest. Finally, I think that the publication of the third edition of Volume B would not have been possible without the competent, tactful and friendly collaboration of Dr Nicola Ashcroft, the Technical Editor of this project during all the stages of the preparation of this edition.

xiv

International Tables for Crystallography (2010). Vol. B, Chapter 1.1, pp. 2–9.

1.1. Reciprocal space in crystallography By U. Shmueli

be shown (e.g. Buerger, 1941; also Shmueli, 2007) that this equation is given by

1.1.1. Introduction The purpose of this chapter is to provide an introduction to several aspects of reciprocal space, which are of general importance in crystallography and which appear in the various chapters and sections to follow. We first summarize the basic definitions and briefly inspect some fundamental aspects of crystallography, while recalling that they can be usefully and simply discussed in terms of the concept of the reciprocal lattice. This introductory section is followed by a summary of the basic relationships between the direct and associated reciprocal lattices. We then introduce the elements of tensor-algebraic formulation of such dual relationships, with emphasis on those that are important in many applications of reciprocal space to crystallographic algorithms. We proceed with a section that demonstrates the role of mutually reciprocal bases in transformations of coordinates and conclude with a brief outline of some important analytical aspects of reciprocal space, most of which are further developed in other parts of this volume.

hx þ ky þ lz ¼ n;

where h, k and l, known as Miller indices of the (hkl) lattice plane, are (under the above assumption) relatively prime integers (i.e. do not have a common factor other than þ1 or 1). In this equation, x, y and z are the coordinates of any point lying in the plane and are expressed as fractions of the magnitudes of the basis vectors a, b and c of the direct lattice, and n is an integer denoting the serial number of the lattice plane within the family of parallel and equidistant ðhklÞ planes. The interplanar spacing is denoted by dhkl, the value n ¼ 0 corresponding to the ðhklÞ plane passing through the origin. Let r ¼ xa þ yb þ zc and rL ¼ ua þ vb þ wc, where u, v, w are any integers, denote the position vectors of the point xyz and a lattice point uvw lying in the plane (1.1.2.3), respectively, and assume that r and rL are different vectors. If the plane normal is denoted by N, where N is proportional to the vector product of two in-plane lattice vectors, the vector form of the equation of the lattice plane becomes

1.1.2. Reciprocal lattice in crystallography The notion of mutually reciprocal triads of vectors dates back to the introduction of vector calculus by J. Willard Gibbs in the 1880s (e.g. Wilson, 1901). This concept appeared to be useful in the early interpretations of diffraction from single crystals (Ewald, 1913; Laue, 1914) and its first detailed exposition and the recognition of its importance in crystallography can be found in Ewald’s (1921) article. The following free translation of Ewald’s (1921) introduction, presented in a somewhat different notation, may serve the purpose of this section:

N
ðr  rL Þ ¼ 0

ðfor i 6¼ kÞ

ð1:1:2:1Þ

and

ai
bi ¼ 1;

ð1:1:2:2Þ

where i and k may each equal 1, 2 or 3. The first equation, (1.1.2.1), says that each vector bk is perpendicular to two vectors ai, as follows from the vanishing scalar products. Equation (1.1.2.2) provides the norm of the vector bi : the length of this vector must be chosen such that the projection of bi on the direction of ai has the length 1=ai , where ai is the magnitude of the vector ai . . . .

ðs  s0 Þ
rL ¼ n;

ð1:1:2:4Þ

ð1:1:2:5Þ

where s0 and s are the wavevectors of the incident and scattered beams, respectively, and n is an arbitrary integer. Since rL ¼ ua þ vb þ wc, where u, v and w are unrestricted integers, equation (1.1.2.5) is equivalent to the equations of Laue:

The consequences of equations (1.1.2.1) and (1.1.2.2) were elaborated by Ewald (1921) and are very well documented in the subsequent literature, crystallographic as well as other. As is well known, the reciprocal lattice occupies a rather prominent position in crystallography and there are nearly as many accounts of its importance as there are crystallographic texts. It is not intended to review its applications, in any detail, in the present section; this is done in the remaining chapters and sections of the present volume. It seems desirable, however, to mention by way of an introduction some fundamental geometrical, physical and mathematical aspects of crystallography, and try to give a unified demonstration of the usefulness of mutually reciprocal bases as an interpretive tool. Let us assume that the coordinates of all the (direct) lattice points are integers. This can only be true for P-type lattices. Consider the equation of a lattice plane in the direct lattice. It can Copyright © 2010 International Union of Crystallography

or N
r ¼ N
rL :

For equations (1.1.2.3) and (1.1.2.4) to be identical, the plane normal N must satisfy the requirement that N
rL ¼ n, where n is an (unrestricted) integer. While the Miller indices of lattice planes in P-type lattices must be relatively prime, if the direct lattice is based on a non-primitive unit cell (any centring type) the Miller indices of some lattice planes are no longer relatively prime (e.g. Nespolo, 2015). Let us now consider the basic diffraction relations (e.g. Lipson & Cochran, 1966). Suppose a parallel beam of monochromatic radiation, of wavelength
, falls on a lattice of identical point scatterers. If it is assumed that the scattering is elastic, i.e. there is no change of the wavelength during this process, the wavevectors of the incident and scattered radiation have the same magnitude, which can conveniently be taken as 1=
. A consideration of path and phase differences between the waves outgoing from two point scatterers separated by the lattice vector rL (defined as above) shows that the condition for their maximum constructive interference is given by

To the set of ai, there corresponds in the vector calculus a set of ‘reciprocal vectors’ bi , which are defined (by Gibbs) by the following properties:

ai
bk ¼ 0

ð1:1:2:3Þ

h
a ¼ h;

h
b ¼ k;

h
c ¼ l;

ð1:1:2:6Þ

where h ¼ s  s0 is the diffraction vector, and h, k and l are integers corresponding to orders of diffraction from the threedimensional lattice (Lipson & Cochran, 1966). The diffraction vector thus has to satisfy a condition that is analogous to that imposed on the normal to a lattice plane. The next relevant aspect to be commented on is the Fourier expansion of a function having the periodicity of the crystal lattice. Such functions are e.g. the electron density, the density of nuclear matter and the electrostatic potential in the crystal, which are the operative definitions of crystal structure in X-ray, neutron and electron-diffraction methods of crystal structure determination. A Fourier expansion of such a periodic function may be

2

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY thought of as a superposition of waves (e.g. Buerger, 1959), with wavevectors related to the interplanar spacings dhkl, in the crystal lattice. Denoting the wavevector of a Fourier wave by g (a function of hkl), the phase of the Fourier wave at the point r in the crystal is given by 2g
r, and the triple Fourier series corresponding to the expansion of the periodic function, say G(r), can be written as P GðrÞ ¼ CðgÞ expð2ig
rÞ; ð1:1:2:7Þ

It follows that, at least in the present case, algebraic simplicity goes together with ease of interpretation, which certainly accounts for much of the importance of the reciprocal lattice in crystallography. The second alternative of reducing the matrix in (1.1.2.11) to a unit matrix, a transformation of (1.1.2.8) to a Cartesian system, leads to non-integral components of the vectors, which makes any interpretation of v or rL much less transparent. However, transformations to Cartesian systems are often very useful in crystallographic computing and will be discussed below (see also Chapters 2.3 and 3.3 in this volume). We shall, in what follows, abandon all the temporary notation used above and write the reciprocal-lattice vector as

g

where C(g) are the amplitudes of the Fourier waves, or Fourier coefficients, which are related to the experimental data. Numerous examples of such expansions appear throughout this volume. The permissible wavevectors in the above expansion are restricted by the periodicity of the function G(r). Since, by definition, GðrÞ ¼ Gðr þ rL Þ, where rL is a direct-lattice vector, the right-hand side of (1.1.2.7) must remain unchanged when r is replaced by r þ rL. This, however, can be true only if the scalar product g
rL is an integer. Each of the above three aspects of crystallography may lead, independently, to a useful introduction of the reciprocal vectors, and there are many examples of this in the literature. It is interesting, however, to consider the representation of the equation v
rL ¼ n;

h ¼ ha þ kb þ lc or h ¼ h1 a1 þ h2 a2 þ h3 a3 ¼

or

hi ai ;

ð1:1:2:13Þ

and denote the direct-lattice vectors by rL ¼ ua þ vb þ wc, as above, or by rL ¼ u1 a1 þ u2 a2 þ u3 a3 ¼

3 P

ui ai :

ð1:1:2:14Þ

i¼1

The representations (1.1.2.13) and (1.1.2.14) are used in the tensor-algebraic formulation of the relationships between mutually reciprocal bases (see Section 1.1.4 below).

which is common to all three, in its most convenient form. Obviously, the vector v which stands for the plane normal, the diffraction vector, and the wavevector in a Fourier expansion, may still be referred to any permissible basis and so may rL, by an appropriate transformation. Let v ¼ UA þ VB þ WC, where A, B and C are linearly independent vectors. Equation (1.1.2.8) can then be written as

or, in matrix notation, 0 1 0 1 A u ðUVWÞ@ B A
ðabcÞ@ v A ¼ n; w C

3 P i¼1

ð1:1:2:8Þ

ðUA þ VB þ WCÞ
ðua þ vb þ wcÞ ¼ n;

ð1:1:2:12Þ

1.1.3. Fundamental relationships We now present a brief derivation and a summary of the most important relationships between the direct and the reciprocal bases. The usual conventions of vector algebra are observed and the results are presented in the conventional crystallographic notation. Equations (1.1.2.1) and (1.1.2.2) now become

ð1:1:2:9Þ

a
b ¼ a
c ¼ b
a ¼ b
c ¼ c
a ¼ c
b ¼ 0

ð1:1:3:1Þ

and

ð1:1:2:10Þ

a
a ¼ b
b ¼ c
c ¼ 1;

ð1:1:3:2Þ

respectively, and the relationships are obtained as follows. 0

A
a ðUVWÞ@ B
a C
a

A
b B
b C
b

10

1

A
c u B
c A@ v A ¼ n: C
c w

1.1.3.1. Basis vectors ð1:1:2:11Þ

It is seen from (1.1.3.1) that a must be proportional to the vector product of b and c, a ¼ Kðb  cÞ;

The simplest representation of equation (1.1.2.8) results when the matrix of scalar products in (1.1.2.11) reduces to a unit matrix. This can be achieved (i) by choosing the basis vectors ABC to be orthonormal to the basis vectors abc, while requiring that the components of rL be integers, or (ii) by requiring that the bases ABC and abc coincide with the same orthonormal basis, i.e. expressing both v and rL , in (1.1.2.8), in the same Cartesian system. If we choose the first alternative, it is seen that: (1) The components of the vector v, and hence those of N, h and g, are of necessity integers, since u, v and w are already integral. The components of v include Miller indices in the case of the lattice plane, they coincide with the orders of diffraction from a three-dimensional lattice of scatterers, and correspond to the summation indices in the triple Fourier series (1.1.2.7). (2) The basis vectors A, B and C are reciprocal to a, b and c, as can be seen by comparing the scalar products in (1.1.2.11) with those in (1.1.2.1) and (1.1.2.2). In fact, the bases ABC and abc are mutually reciprocal. Since there are no restrictions on the integers U, V and W, the vector v belongs to a lattice which, on account of its basis, is called the reciprocal lattice.

and, since a
a ¼ 1, the proportionality constant K equals 1=½a
ðb  cÞ. The mixed product a
ðb  cÞ can be interpreted as the positive volume of the unit cell in the direct lattice only if a, b and c form a right-handed set. If the above condition is fulfilled, we obtain a ¼

bc ; V

b ¼

ca ; V

ab V

ð1:1:3:3Þ

a  b ; V

ð1:1:3:4Þ

c ¼

and analogously a¼

b  c ; V

b¼

c  a ; V

c¼

where V and V  are the volumes of the unit cells in the associated direct and reciprocal lattices, respectively. Use has been made of the fact that the mixed product, say a
ðb  cÞ, remains unchanged under cyclic rearrangement of the vectors that appear in it.

3

1. GENERAL RELATIONSHIPS AND TECHNIQUES 0 1 x 1.1.3.2. Volumes @ y A;  x ¼ The reciprocal relationship of V and V follows readily. We z have from equations (1.1.3.2), (1.1.3.3) and (1.1.3.4) ða  bÞ
ða  b Þ ¼ 1: VV 

c
c ¼

and

0

ðA  BÞ
ðC  DÞ ¼ ðA
CÞðB
DÞ  ðA
DÞðB
CÞ; ð1:1:3:5Þ

0

and equations (1.1.3.1) and (1.1.3.2), it is seen that V  ¼ 1=V.

The relationships of the angles ; ;  between the pairs of vectors (b, c), (c, a) and (a, b), respectively, and the angles  ;  ;   between the corresponding pairs of reciprocal basis vectors, can be obtained by simple vector algebra. For example, we have from (1.1.3.3): (i) b
c ¼ b c cos  , with and

ab sin  c ¼ V

0



cos  cos   cos  : sin  sin 

ð1:1:3:6Þ

Similarly, cos  ¼

cos  cos   cos  sin  sin 

cos  cos   cos  : sin  sin 

ð1:1:3:8Þ

1

C bc cos  A: c2

c
a

c
b

c
c

a2 B   ¼ @ b a cos  

a b cos   b2

c a cos 

c b cos 

1 a c cos  C b c cos  A:

ð1:1:3:12Þ

ð1:1:3:14Þ

c2

det ðGÞ ¼ ½a
ðb  cÞ2 ¼ V 2

ð1:1:3:15Þ

det ðG Þ ¼ ½a
ðb  c Þ2 ¼ V 2 ;

ð1:1:3:16Þ

þ 2 cos  cos  cos Þ1=2

ð1:1:3:17Þ

and V  ¼ a b c ð1  cos2   cos2   cos2   þ 2 cos  cos  cos   Þ1=2 :

ð1:1:3:18Þ

The following algorithm has been found useful in computational applications of the above relationships to calculations in reciprocal space (e.g. data reduction) and in direct space (e.g. crystal geometry). (1) Input the direct unit-cell parameters and construct the matrix of the metric tensor [cf. equation (1.1.3.12)]. (2) Compute the determinant of the matrix G and find the inverse matrix, G1 ; this inverse matrix is just G , the matrix of the metric tensor of the reciprocal basis (see also Section 1.1.4 below). (3) Use the elements of G, and equation (1.1.3.14), to obtain the parameters of the reciprocal unit cell. The direct and reciprocal sets of unit-cell parameters, as well as the corresponding metric tensors, are now available for further calculations. Explicit relations between direct- and reciprocal-lattice parameters, valid for the various crystal systems, are given in most

1.1.3.4. Matrices of metric tensors Various computational and algebraic aspects of mutually reciprocal bases are most conveniently expressed in terms of the metric tensors of these bases. The tensors will be treated in some detail in the next section, and only the definitions of their matrices are given and interpreted below. Consider the length of the vector r ¼ xa þ yb þ zc. This is given by ð1:1:3:9Þ

and can be written in matrix form as jrj ¼ ½xT Gx1=2 ;

ca cos 

b2 cb cos 

ac cos 

V ¼ abcð1  cos2   cos2   cos2 

cos  cos    cos  : sin  sin  

jrj ¼ ½ðxa þ yb þ zcÞ
ðxa þ yb þ zcÞ1=2

c
c

ab cos 

and a direct expansion of the determinants, from (1.1.3.12) and (1.1.3.14), leads to

The expressions for the cosines of the direct angles in terms of those of the reciprocal ones are analogous to (1.1.3.6)–(1.1.3.8). For example, cos  ¼

c
b

a2

and

ð1:1:3:7Þ

and cos   ¼

c
a

ð1:1:3:11Þ

The matrices G and G are of fundamental importance in crystallographic computations and transformations of basis vectors and coordinates from direct to reciprocal space and vice versa. Examples of applications are presented in Part 3 of this volume and in the remaining sections of this chapter. It can be shown (e.g. Buerger, 1941) that the determinants of G and G equal the squared volumes of the direct and reciprocal unit cells, respectively. Thus,

ðc  aÞ
ða  bÞ : V2

If we make use of the identity (1.1.3.5), and compare the two expressions for b
c , we readily obtain cos  ¼

1

This is the matrix of the metric tensor of the direct basis, or briefly the direct metric. The corresponding reciprocal metric is given by 0   1 a
a a
b a
c B C G ¼ @ b
a b
b b
c A ð1:1:3:13Þ

and (ii) b
c  ¼

a
c

C b
b b
cA

B ¼ @ ba cos 

1.1.3.3. Angular relationships

ca sin  b ¼ V

a
b

B G ¼ @b
a

If we make use of the vector identity



a
a

xT ¼ ðxyzÞ

ð1:1:3:10Þ

where

4

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY textbooks on crystallography [see also Chapters 1.1 and 1.2 of Volume C (Koch, 2004)].

Let us multiply both sides of (1.1.4.1) and (1.1.4.2), on the right, by the vectors am, m = 1, 2, or 3, i.e. by the reciprocal vectors to the basis a1 a2 a3. We obtain from (1.1.4.1) m xk ak
am ¼ xk m k ¼ x ;

1.1.4. Tensor-algebraic formulation The present section summarizes the tensor-algebraic properties of mutually reciprocal sets of basis vectors, which are of importance in the various aspects of crystallography. This is not intended to be a systematic treatment of tensor algebra; for more thorough expositions of the subject the reader is referred to relevant crystallographic texts (e.g. Patterson, 1967; Sands, 1982), and other texts in the physical and mathematical literature that deal with tensor algebra and analysis. Let us first recall that symbolic vector and matrix notations, in which basis vectors and coordinates do not appear explicitly, are often helpful in qualitative considerations. If, however, an expression has to be evaluated, the various quantities appearing in it must be presented in component form. One of the best ways to achieve a concise presentation of geometrical expressions in component form, while retaining much of their ‘transparent’ symbolic character, is their tensor-algebraic formulation.

where m k is the Kronecker symbol which equals 1 when k ¼ m and equals zero if k 6¼ m, and by comparison with (1.1.4.2) we have xm ¼ x0k Tkm ;

where Tkm ¼ a0k
am is an element of the required transformation matrix. Of course, the same transformation could have been written as xm ¼ Tkm x0k ;

We shall adhere to the following conventions: (i) Notation for direct and reciprocal basis vectors:

xm xn ¼ Tpm Tqn x0p x0q ;

Subscripted quantities are associated in tensor algebra with covariant, and superscripted with contravariant transformation properties. Thus the basis vectors of the direct lattice are represented as covariant quantities and those of the reciprocal lattice as contravariant ones. (ii) Summation convention: if an index appears twice in an expression, once as subscript and once as superscript, a summation over this index is thereby implied and the summation sign is omitted. For example, PP i x Tij x j will be written xi Tij x j

Qmn ¼ Tpm Tqn Q0pq :

ð1:1:4:6Þ

1.1.4.3. Scalar products The expression for the scalar product of two vectors, say u and v, depends on the bases to which the vectors are referred. If we admit only the covariant and contravariant bases defined above, we have four possible types of expression: ðIÞ u ¼ ui ai ; v ¼ vi ai

j

u
v ¼ ui v j ðai
aj Þ  ui v j gij ; i

ð1:1:4:7Þ

i

ðIIÞ u ¼ ui a ; v ¼ vi a u
v ¼ ui vj ðai
a j Þ  ui vj gij ;

since both i and j conform to the convention. Such repeating indices are often called dummy indices. The implied summation over repeating indices is also often used even when the indices are at the same level and the coordinate system is Cartesian; there is no distinction between contravariant and covariant quantities in Cartesian frames of reference (see Chapter 3.3). (iii) Components (coordinates) of vectors referred to the covariant basis are written as contravariant quantities, and vice versa. For example,

i

ð1:1:4:8Þ

i

ðIIIÞ u ¼ u ai ; v ¼ vi a

u
v ¼ ui vj ðai
a j Þ  ui vj ij ¼ ui vi ; i

ð1:1:4:9Þ

i

ðIVÞ u ¼ ui a ; v ¼ v ai u
v ¼ ui v j ðai
aj Þ  ui v j ij ¼ ui vi :

ð1:1:4:10Þ

(i) The sets of scalar products gij ¼ ai
aj (1.1.4.7) and gij ¼ ai
a j (1.1.4.8) are known as the metric tensors of the covariant (direct) and contravariant (reciprocal) bases, respectively; the corresponding matrices are presented in conventional notation in equations (1.1.3.11) and (1.1.3.13). Numerous applications of these tensors to the computation of distances and angles in crystals are given in Chapter 3.1. (ii) Equations (1.1.4.7) to (1.1.4.10) furnish the relationships between the covariant and contravariant components of the same vector. Thus, comparing (1.1.4.7) and (1.1.4.9), we have

r ¼ xa þ yb þ zc ¼ x1 a1 þ x2 a2 þ x3 a3 ¼ xi ai h ¼ ha þ kb þ lc ¼ h1 a1 þ h2 a2 þ h3 a3 ¼ hi ai :

1.1.4.2. Transformations A familiar concept but a fundamental one in tensor algebra is the transformation of coordinates. For example, suppose that an atomic position vector is referred to two unit-cell settings as follows: r ¼ x ak

ð1:1:4:5Þ

the same transformation law applies to the components of a contravariant tensor of rank two, the components of which are referred to the primed basis and are to be transformed to the unprimed one:

a ¼ a1 ; b ¼ a2 ; c ¼ a3  a ¼ a1 ; b ¼ a2 ; c  ¼ a3 :

k

ð1:1:4:4Þ

where Tkm ¼ am
a0k. A tensor is a quantity that transforms as the product of coordinates, and the rank of a tensor is the number of transformations involved (Patterson, 1967; Sands, 1982). E.g. the product of two coordinates, as in the above example, transforms from the a0 basis to the a basis as

1.1.4.1. Conventions

i

ð1:1:4:3Þ

vi ¼ v j gij :

ð1:1:4:1Þ

ð1:1:4:11Þ

Similarly, using (1.1.4.8) and (1.1.4.10) we obtain the inverse relationship

and r ¼ x0k a0k :

vi ¼ vj gij :

ð1:1:4:2Þ

5

ð1:1:4:12Þ

1. GENERAL RELATIONSHIPS AND TECHNIQUES The corresponding relationships between covariant and contravariant bases can now be obtained if we refer a vector, say v, to each of the bases

equations (1.1.4.9) or (1.1.4.10) is almost as efficient as it would be if the coordinates were referred to a Cartesian system. For example, the right-hand side of the vector identity (1.1.3.5), which is employed in the computation of dihedral angles, can be written as

v ¼ vi ai ¼ vk ak ;

ðAi Ci ÞðBj Dj Þ  ðAk Dk ÞðBl Cl Þ:

and make use of (1.1.4.11) and (1.1.4.12). Thus, e.g., vi ai ¼ ðvk gik Þai ¼ vk ak :

This is a typical application of reciprocal space to ordinary directspace computations. (iv) We wish to derive a tensor formulation of the vector product, along similar lines to those of Chapter 3.1. As with the scalar product, there are several such formulations and we choose that which has both vectors, say u and v, and the resulting product, u  v, referred to a covariant basis. We have

Hence ak ¼ gik ai

ð1:1:4:13Þ

ak ¼ gik ai :

ð1:1:4:14Þ

and, similarly,

u  v ¼ ui ai  v j aj

ij

(iii) The tensors gij and g are symmetric, by definition. (iv) It follows from (1.1.4.11) and (1.1.4.12) or (1.1.4.13) and (1.1.4.14) that the matrices of the direct and reciprocal metric tensors are mutually inverse, i.e. 0 11 0 11 1 g11 g12 g13 g12 g13 g @ g21 g22 g23 A ¼ @ g21 g22 g23 A; ð1:1:4:15Þ g31 g32 g33 g31 g32 g33

¼ ui v j ðai  aj Þ:

If we make use of the relationships (1.1.3.3) between the direct and reciprocal basis vectors, it can be verified that ai  aj ¼ Vekij ak ;

1.1.4.4. Examples There are numerous applications of tensor notation in crystallographic calculations, and many of them appear in the various chapters of this volume. We shall therefore present only a few examples. (i) The (squared) magnitude of the diffraction vector h ¼ hi ai is given by 4 sin2  ¼ hi hj gij :
2

u  v ¼ Vekij ui v j ak ¼ Vglk ekij ui v j al ;

N P

fðjÞ expðhT b ð jÞ hÞ expð2ihT r ð jÞ Þ;

ð1:1:4:16Þ

ð1:1:4:17Þ

j¼1

where b ð jÞ is the matrix of the anisotropic displacement tensor of the jth atom. In tensor notation, with the quantities referred to their natural bases, the structure factor can be written as Fðh1 h2 h3 Þ ¼

N P

i fð jÞ expðhi hk ik ð jÞ Þ expð2ihi xð jÞ Þ;

ð1:1:4:21Þ

since by (1.1.4.13), ak ¼ glk al . (v) The rotation operator. The general formulation of an expression for the rotation operator is of interest in crystal structure determination by Patterson techniques (see Chapter 2.3) and in molecular modelling (see Chapter 3.3), and another well known crystallographic application of this device is the derivation of the translation, libration and screw-motion tensors by the method of Schomaker & Trueblood (1968), discussed in Part 8 of Volume C (IT C, 2004) and in Chapter 1.2 of this volume. A digression on an elementary derivation of the above seems to be worthwhile. Suppose we wish to rotate the vector r, about an axis coinciding with the unit vector k, through the angle  and in the positive sense, i.e. an observer looking in the direction of þk will see r rotating in the clockwise sense. The vectors r, k and the rotated (target) vector r0 are referred to an origin on the axis of rotation (see Fig. 1.1.4.1). Our purpose is to express r0 in terms of r, k and  by a general vector formula, and represent the components of the rotated vectors in coordinate systems that might be of interest. Let us decompose the vector r and the (target) vector r0 into their components which are parallel ðkÞ and perpendicular ð?Þ to the axis of rotation:

This concise relationship is a starting point in a derivation of unitcell parameters from experimental data. (ii) The structure factor, including explicitly anisotropic displacement tensors, can be written in symbolic matrix notation as FðhÞ ¼

ð1:1:4:20Þ

where V is the volume of the unit cell and the antisymmetric tensor ekij equals þ1; 1, or 0 according as kij is an even permutation of 123, an odd permutation of 123 or any two of the indices kij have the same value, respectively. We thus have

and their determinants are mutually reciprocal.

jhj2 ¼

ð1:1:4:19Þ

ð1:1:4:18Þ

j¼1

and similarly concise expressions can be written for the derivatives of the structure factor with respect to the positional and displacement parameters. The summation convention applies only to indices denoting components of vectors and tensors; the atom subscript j in (1.1.4.18) clearly does not qualify, and to indicate this it has been surrounded by parentheses. (iii) Geometrical calculations, such as those described in the chapters of Part 3, may be carried out in any convenient basis but there are often some definite advantages to computations that are referred to the natural, non-Cartesian bases (see Chapter 3.1). Usually, the output positional parameters from structure refinement are available as contravariant components of the atomic position vectors. If we transform them by (1.1.4.11) to their covariant form, and store these covariant components of the atomic position vectors, the computation of scalar products using

r ¼ rk þ r?

ð1:1:4:22Þ

r0 ¼ r0k þ r0? :

ð1:1:4:23Þ

and

It can be seen from Fig. 1.1.4.1 that the parallel components of r and r0 are rk ¼ r0k ¼ kðk
rÞ

ð1:1:4:24Þ

r? ¼ r  kðk
rÞ:

ð1:1:4:25Þ

and thus

6

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY x0i ¼ Rij x j ;

ð1:1:4:30Þ

where Rij ¼ ki kj ð1  cos Þ þ ij cos  þ Vgim empj kp sin 

ð1:1:4:31Þ

is a matrix element of the rotation operator R which carries the vector r into the vector r0 . Of course, the representation (1.1.4.31) of R depends on our choice of reference bases. If all the vectors are referred to a Cartesian basis, that is three orthogonal unit vectors, the direct and reciprocal metric tensors reduce to a unit tensor, there is no difference between covariant and contravariant quantities, and equation (1.1.4.31) reduces to Rij ¼ ki kj ð1  cos Þ þ ij cos  þ eipj kp sin ;

ð1:1:4:32Þ

where all the indices have been taken as subscripts, but the summation convention is still observed. The relative simplicity of (1.1.4.32), as compared to (1.1.4.31), often justifies the transformation of all the vector quantities to a Cartesian basis. This is certainly the case for any extensive calculation in which covariances of the structural parameters are not considered. Fig. 1.1.4.1. Derivation of the general expression for the rotation operator. The figure illustrates schematically the decompositions and other simple geometrical considerations required for the derivation outlined in equations (1.1.4.22)–(1.1.4.28).

1.1.5. Transformations 1.1.5.1. Transformations of coordinates It happens rather frequently that a vector referred to a given basis has to be re-expressed in terms of another basis, and it is then required to find the relationship between the components (coordinates) of the vector in the two bases. Such situations have already been indicated in the previous section. The purpose of the present section is to give a general method of finding such relationships (transformations), and discuss some simplifications brought about by the use of mutually reciprocal and Cartesian bases. We do not assume anything about the bases, in the general treatment, and hence the tensor formulation of Section 1.1.4 is not appropriate at this stage. Let

Only a suitable expression for r0? is missing. We can find this by decomposing r0? into its components (i) parallel to r? and (ii) parallel to k  r?. We have, as in (1.1.4.24),

 r r? 0 k  r ? k  r? 0 ð1:1:4:26Þ
r? þ
r? : r0? ¼ ? jr? j jr? j jk  r? j jk  r? j We observe, using Fig. 1.1.4.1, that jr0? j ¼ jr? j ¼ jk  r? j and k  r? ¼ k  r;

r¼

3 P

uj ð1Þcj ð1Þ

ð1:1:5:1Þ

uj ð2Þcj ð2Þ

ð1:1:5:2Þ

j¼1

and, further, and

r0?
r? ¼ jr? j2 cos 

r¼

and

3 P j¼1

r0?
ðk  r? Þ ¼ k
ðr0?  r? Þ ¼ jr? j2 sin ;

be the given and required representations of the vector r, respectively. Upon the formation of scalar products of equations (1.1.5.1) and (1.1.5.2) with the vectors of the second basis, and employing again the summation convention, we obtain

since the unit vector k is perpendicular to the plane containing the vectors r? and r0? . Equation (1.1.4.26) now reduces to r0? ¼ r? cos  þ ðk  rÞ sin 

ð1:1:4:27Þ

uk ð1Þ½ck ð1Þ
cl ð2Þ ¼ uk ð2Þ½ck ð2Þ
cl ð2Þ;

and equations (1.1.4.23), (1.1.4.25) and (1.1.4.27) lead to the required result r0 ¼ kðk
rÞð1  cos Þ þ r cos  þ ðk  rÞ sin :

l ¼ 1; 2; 3

ð1:1:5:3Þ

or uk ð1ÞGkl ð12Þ ¼ uk ð2ÞGkl ð22Þ;

l ¼ 1; 2; 3;

ð1:1:5:4Þ

ð1:1:4:28Þ where Gkl ð12Þ ¼ ck ð1Þ
cl ð2Þ and Gkl ð22Þ ¼ ck ð2Þ
cl ð2Þ. Similarly, if we choose the basis vectors cl ð1Þ, l = 1, 2, 3, as the multipliers of (1.1.5.1) and (1.1.5.2), we obtain

The above general expression can be written as a linear transformation by referring the vectors to an appropriate basis or bases. We choose here r ¼ x j aj, r0 ¼ x0i ai and assume that the components of k are available in the direct and reciprocal bases. If we make use of equations (1.1.4.9) and (1.1.4.21), (1.1.4.28) can be written as

uk ð1ÞGkl ð11Þ ¼ uk ð2ÞGkl ð21Þ;

l ¼ 1; 2; 3;

ð1:1:5:5Þ

and Gkl ð21Þ ¼ ck ð2Þ
cl ð1Þ. where Gkl ð11Þ ¼ ck ð1Þ
cl ð1Þ Rewriting (1.1.5.4) and (1.1.5.5) in symbolic matrix notation, we have

x0i ¼ ki ðk j x j Þð1  cos Þ þ ij x j cos  þ Vgim empj kp x j sin ;

uT ð1ÞGð12Þ ¼ uT ð2ÞGð22Þ;

ð1:1:4:29Þ or briefly

leading to

7

ð1:1:5:6Þ

1. GENERAL RELATIONSHIPS AND TECHNIQUES u ð1Þ ¼ u ð2ÞfGð22Þ½Gð12Þ1 g

r ¼ X k ek ;

uT ð2Þ ¼ uT ð1ÞfGð12Þ½Gð22Þ1 g;

ð1:1:5:7Þ

where the Cartesian basis vectors are: e1 ¼ rL =jrL j, e2 ¼ r =jr j and e3 ¼ e1  e2 , and the vectors rL and r are given by

uT ð1ÞGð11Þ ¼ uT ð2ÞGð21Þ;

ð1:1:5:8Þ

T

T

and

ð1:1:5:12Þ

rL ¼ ui ai and r ¼ hk ak ;

and where ui and hk , i, k = 1, 2, 3, are arbitrary integers. The vectors rL and r must of course be chosen to be mutually perpendicular, rL
r ¼ ui hi ¼ 0. The X 1 ðXÞ axis of the Cartesian system thus coincides with a direct-lattice vector, and the X 2 ðYÞ axis is parallel to a vector in the reciprocal lattice. Since the basis in (1.1.5.12) is a Cartesian one, the required transformations are given by equations (1.1.5.10) as

leading to uT ð1Þ ¼ uT ð2ÞfGð21Þ½Gð11Þ1 g and uT ð2Þ ¼ uT ð1ÞfGð11Þ½Gð21Þ1 g:

xi ¼ X k ðT 1 Þik and X i ¼ xk Tki ;

ð1:1:5:9Þ

ð1:1:5:13Þ

where Tki ¼ ak
ei, k, i = 1, 2, 3, form the matrix of the scalar products. If we make use of the relationships between covariant and contravariant basis vectors, and the tensor formulation of the vector product, given in Section 1.1.4 above (see also Chapter 3.1), we obtain

Equations (1.1.5.7) and (1.1.5.9) are symbolic general expressions for the transformation of the coordinates of r from one representation to the other. In the general case, therefore, we require the matrices of scalar products of the basis vectors, G(12) and G(22) or G(11) and G(21) – depending on whether the basis ck ð2Þ or ck ð1Þ, k = 1, 2, 3, was chosen to multiply scalarly equations (1.1.5.1) and (1.1.5.2). Note, however, the following simplifications. (i) If the bases ck ð1Þ and ck ð2Þ are mutually reciprocal, each of the matrices of mixed scalar products, G(12) and G(21), reduces to a unit matrix. In this important special case, the transformation is effected by the matrices of the metric tensors of the bases in question. This can be readily seen from equations (1.1.5.7) and (1.1.5.9), which then reduce to the relationships between the covariant and contravariant components of the same vector [see equations (1.1.4.11) and (1.1.4.12) above]. (ii) If one of the bases, say ck ð2Þ, is Cartesian, its metric tensor is by definition a unit tensor, and the transformations in (1.1.5.7) reduce to

1 g ui jrL j ki 1 Tk2 ¼  hk jr j V Tk3 ¼ e ui gpl hl : jrL jjr j kip Tk1 ¼

ð1:1:5:14Þ

Note that the other convenient choice, e1 / r and e2 / rL , interchanges the first two columns of the matrix T in (1.1.5.14) and leads to a change of the signs of the elements in the third column. This can be done by writing ekpi instead of ekip, while leaving the rest of Tk3 unchanged.

uT ð1Þ ¼ uT ð2Þ½Gð12Þ1

1.1.6. Some analytical aspects of the reciprocal space 1.1.6.1. Continuous Fourier transform

and T

T

u ð2Þ ¼ u ð1ÞGð12Þ:

Of great interest in crystallographic analyses are Fourier transforms and these are closely associated with the dual bases examined in this chapter. Thus, e.g., the inverse Fourier transform of the electron-density function of the crystal R FðhÞ ¼ ðrÞ expð2ih
rÞ d3 r; ð1:1:6:1Þ

ð1:1:5:10Þ

The transformation matrix is now the mixed matrix of the scalar products, whether or not the basis ck ð1Þ, k = 1, 2, 3, is also Cartesian. If, however, both bases are Cartesian, the transformation can also be interpreted as a rigid rotation of the coordinate axes (see Chapter 3.3). It should be noted that the above transformations do not involve any shift of the origin. Transformations involving such shifts, notably the symmetry transformations of the space group, are treated rather extensively in Volume A of International Tables for Crystallography (2005) [see e.g. Part 5 there (Arnold, 2005)].

cell

where ðrÞ is the electron-density function at the point r and the integration extends over the volume of a unit cell, is the fundamental model of the contribution of the distribution of crystalline matter to the intensity of the scattered radiation. For the conventional Bragg scattering, the function given by (1.1.6.1), and known as the structure factor, may assume nonzero values only if h can be represented as a reciprocal-lattice vector. Chapter 1.2 is devoted to a discussion of the structure factor of the Bragg reflection, while Chapters 4.1, 4.2 and 4.3 discuss circumstances under which the scattering need not be confined to the points of the reciprocal lattice only, and may be represented by reciprocal-space vectors with non-integral components.

1.1.5.2. Example This example deals with the construction of a Cartesian system in a crystal with given basis vectors of its direct lattice. We shall also require that the Cartesian system bear a clear relationship to at least one direction in each of the direct and reciprocal lattices of the crystal; this may be useful in interpreting a physical property which has been measured along a given lattice vector or which is associated with a given lattice plane. For a better consistency of notation, the Cartesian components will be denoted as contravariant. The appropriate version of equations (1.1.5.1) and (1.1.5.2) is now r ¼ xi ai

1.1.6.2. Discrete Fourier transform The electron density ðrÞ in (1.1.6.1) is one of the most common examples of a function which has the periodicity of the crystal. Thus, for an ideal (infinite) crystal the electron density ðrÞ can be written as ðrÞ ¼ ðr þ ua þ vb þ wcÞ;

ð1:1:6:2Þ

ð1:1:5:11Þ and, as such, it can be represented by a three-dimensional Fourier series of the form

and

8

ðrÞ ¼

P

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY CðgÞ expð2ig
rÞ; ð1:1:6:3Þ ðr þ rL Þ ¼ ðrÞ;

g

we must have expðik
rL Þ ¼ 1. Of course, this can be so only if the wavevector k equals 2 times a vector in the reciprocal lattice. It is also seen from equation (1.1.6.7) that the wavevector appearing in the phase factor can be reduced to a unit cell in the reciprocal lattice (the basis vectors of which contain the 2 factor), or to the equivalent polyhedron known as the Brillouin zone (e.g. Ziman, 1969). This periodicity in reciprocal space is of prime importance in the theory of solids. Some Brillouin zones are discussed in detail in Chapter 1.5.

where the periodicity requirement (1.1.6.2) enables one to represent all the g vectors in (1.1.6.3) as vectors in the reciprocal lattice (see also Section 1.1.2 above). If we insert the series (1.1.6.3) in the integrand of (1.1.6.1), interchange the order of summation and integration and make use of the fact that an integral of a periodic function taken over the entire period must vanish unless the integrand is a constant, equation (1.1.6.3) reduces to the conventional form 1X FðhÞ expð2ih
rÞ; ð1:1:6:4Þ ðrÞ ¼ V h

I wish to thank Professor D. W. J. Cruickshank for bringing to my attention the contribution of M. von Laue (Laue, 1914), who was the first to introduce general reciprocal bases to crystallography.

where V is the volume of the unit cell in the direct lattice and the summation ranges over all the reciprocal lattice. Fourier transforms, discrete as well as continuous, are among the most important mathematical tools of crystallography. The discussion of their mathematical principles, the modern algorithms for their computation and their numerous applications in crystallography form the subject matter of Chapter 1.3. Many more examples of applications of Fourier methods in crystallography are scattered throughout this volume and the crystallographic literature in general.

References Arnold, H. (2005). Transformations in crystallography. In International Tables for Crystallography, Vol. A, Space-Group Symmetry, edited by Th. Hahn, Part 5. Heidelberg: Springer. Ashcroft, N. W. & Mermin, N. D. (1975). Solid State Physics. Philadelphia: Saunders College. ¨ ber die Quantenmechanik der Elektronen in Bloch, F. (1928). U Kristallgittern. Z. Phys. 52, 555–600. Buerger, M. J. (1941). X-ray Crystallography. New York: John Wiley. Buerger, M. J. (1959). Crystal Structure Analysis. New York: John Wiley. Ewald, P. P. (1913). Zur Theorie der Interferenzen der Ro¨ntgenstrahlen in Kristallen. Phys. Z. 14, 465–472. Ewald, P. P. (1921). Das reziproke Gitter in der Strukturtheorie. Z. Kristallogr. 56, 129–156. International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Heidelberg: Springer. International Tables for Crystallography (2004). Vol. C, Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers. Koch, E. (2004). In International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, edited by E. Prince, Chapters 1.1 and 1.2. Dordrecht: Kluwer Academic Publishers. Laue, M. (1914). Die Interferenzerscheinungen an Ro¨ntgenstrahlen, hervorgerufen durch das Raumgitter der Kristalle. Jahrb. Radioakt. Elektron. 11, 308–345. Lipson, H. & Cochran, W. (1966). The Determination of Crystal Structures. London: Bell. Nespolo, M. (2015). The ash heap of crystallography: restoring forgotten basic knowledge. J. Appl. Cryst. 48, 1290–1298. Patterson, A. L. (1967). In International Tables for X-ray Crystallography, Vol. II, Mathematical Tables, edited by J. S. Kasper & K. Lonsdale, pp. 5–83. Birmingham: Kynoch Press. Sands, D. E. (1982). Vectors and Tensors in Crystallography. New York: Addison-Wesley. Schomaker, V. & Trueblood, K. N. (1968). On the rigid-body motion of molecules in crystals. Acta Cryst. B24, 63–76. Shmueli, U. (2007). Theories and Techniques of Crystal Structure Determination, Section 1.2. Oxford University Press. Wilson, E. B. (1901). Vector Analysis. New Haven: Yale University Press. Ziman, J. M. (1969). Principles of the Theory of Solids. Cambridge University Press.

1.1.6.3. Bloch’s theorem It is in order to mention briefly the important role of reciprocal space and the reciprocal lattice in the field of the theory of solids. At the basis of these applications is the periodicity of the crystal structure and the effect it has on the dynamics (cf. Chapter 4.1) and electronic structure of the crystal. One of the earliest, and still most important, theorems of solid-state physics is due to Bloch (1928) and deals with the representation of the wavefunction of an electron which moves in a periodic potential. Bloch’s theorem states that: The eigenstates of the one-electron Hamiltonian - 2 =2mÞr2 þ UðrÞ, where U(r) is the crystal potential and h ¼ ðh Uðr þ rL Þ ¼ UðrÞ for all rL in the Bravais lattice, can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice.

Thus ðrÞ ¼ expðik
rÞuðrÞ;

ð1:1:6:5Þ

where uðr þ rL Þ ¼ uðrÞ

ð1:1:6:6Þ

and k is the wavevector. The proof of Bloch’s theorem can be found in most modern texts on solid-state physics (e.g. Ashcroft & Mermin, 1975). If we combine (1.1.6.5) with (1.1.6.6), an alternative form of the Bloch theorem results: ðr þ rL Þ ¼ expðik
rL Þ ðrÞ: In the important case where the wavefunction i.e.

ð1:1:6:7Þ is itself periodic,

9

references

International Tables for Crystallography (2010). Vol. B, Chapter 1.2, pp. 10–23.

1.2. The structure factor By P. Coppens

1.2.1. Introduction The structure factor is the central concept in structure analysis by diffraction methods. Its modulus is called the structure amplitude. The structure amplitude is a function of the indices of the set of scattering planes h, k and l, and is defined as the amplitude of scattering by the contents of the crystallographic unit cell, expressed in units of scattering. For X-ray scattering, that unit is the scattering by a single electron (2.82
1015 m), while for neutron scattering by atomic nuclei, the unit of scattering length of 1014 m is commonly used. The complex form of the structure factor means that the phase of the scattered wave is not simply related to that of the incident wave. However, the observable, which is the scattered intensity, must be real. It is proportional to the square of the scattering amplitude (see, e.g., Lipson & Cochran, 1966). The structure factor is directly related to the distribution of scattering matter in the unit cell which, in the X-ray case, is the electron distribution, time-averaged over the vibrational modes of the solid. In this chapter we will discuss structure-factor expressions for X-ray and neutron scattering, and, in particular, the modelling that is required to obtain an analytical description in terms of the features of the electron distribution and the vibrational displacement parameters of individual atoms. We concentrate on the most basic developments; for further details the reader is referred to the cited literature.

AðSÞ ¼ F^ fðrÞg;

ð1:2:2:4bÞ

where F^ is the Fourier transform operator. 1.2.3. Scattering by a crystal: definition of a structure factor In a crystal of infinite size, ðrÞ is a three-dimensional periodic function, as expressed by the convolution PPP unit cell ðrÞ  ðr  na  mb  pcÞ; ð1:2:3:1Þ crystal ðrÞ ¼ n m p

where n, m and p are integers, and  is the Dirac delta function. Thus, according to the Fourier convolution theorem, P PP AðSÞ ¼ F^ fðrÞg ¼ F^ funit cell ðrÞgF^ fðr  na  mb  pcÞg; n m p

ð1:2:3:2Þ which gives AðSÞ ¼ F^ funit cell ðrÞg

P PP h

k

ðS  ha  kb  lc Þ:

ð1:2:3:3Þ

l

Expression (1.2.3.3) is valid for a crystal with a very large number of unit cells, in which particle-size broadening is negligible. Furthermore, it does not account for multiple scattering of the beam within the crystal. Because of the appearance of the delta function, (1.2.3.3) implies that S = H with H ¼ ha þ kb þ lc . The first factor in (1.2.3.3), the scattering amplitude of one unit cell, is defined as the structure factor F: R FðHÞ ¼ F^ funit cell ðrÞg ¼ unit cell ðrÞ expð2
iH  rÞ dr: ð1:2:3:4Þ

1.2.2. General scattering expression for X-rays The total scattering of X-rays contains both elastic and inelastic components. Within the first-order Born approximation (Born, 1926) it has been treated by several authors (e.g. Waller & Hartree, 1929; Feil, 1977) and is given by the expression
2 P
R Itotal ðSÞ ¼ Iclassical
n expð2
iS  rj Þ 0 dr
; ð1:2:2:1Þ n

1.2.4. The isolated-atom approximation in X-ray diffraction where Iclassical is the classical Thomson scattering of an X-ray beam by a free electron, which is equal to ðe2 =mc2 Þ2 ð1 þ cos2 2Þ=2 for an unpolarized beam of unit intensity, is the n-electron space-wavefunction expressed in the 3n coordinates of the electrons located at rj and the integration is over the coordinates of all electrons. S is the scattering vector of length 2 sin =. The coherent elastic component of the scattering, in units of the scattering of a free electron, is given by
R
P Icoherent; elastic ðSÞ ¼
0
expð2
iS  rj Þj 0 drj2 : ð1:2:2:2Þ

To a reasonable approximation, the unit-cell density can be described as a superposition of isolated, spherical atoms located at rj . P unit cell ðrÞ ¼ atom; j ðrÞ  ðr  rj Þ: ð1:2:4:1Þ j

Substitution in (1.2.3.4) gives P P FðHÞ ¼ F^ fatom; j gF^ fðr  rj Þg ¼ fj expð2
iH  rj Þ j

j

ð1:2:4:2aÞ

j

or

If integration is performed over all coordinates but those of the jth electron, one obtains after summation over all electrons R Icoherent; elastic ðSÞ ¼ j ðrÞ expð2
iS  rÞ drj2 ; ð1:2:2:3Þ

Fðh; k; lÞ ¼

P

fj exp 2
iðhxj þ kyj þ lzj Þ

j

¼

P

fj fcos 2
ðhxj þ kyj þ lzj Þ

j

þ i sin 2
ðhxj þ kyj þ lzj Þg:

where ðrÞ is the electron distribution. The scattering amplitude AðSÞ is then given by R AðSÞ ¼ ðrÞ expð2
iS  rÞ dr ð1:2:2:4aÞ

fj ðSÞ, the spherical atomic scattering factor, or form factor, is the Fourier transform of the spherically averaged atomic density j ðrÞ, in which the polar coordinate r is relative to the nuclear position. fj ðSÞ can be written as (James, 1982)

or Copyright © 2010 International Union of Crystallography

ð1:2:4:2bÞ

10

1.2. THE STRUCTURE FACTOR

Z

excited state and the scattering length is essentially real and independent of the energy of the incoming neutron. In either case, b is independent of the Bragg angle , unlike the X-ray form factor, since the nuclear dimensions are very small relative to the wavelength of thermal neutrons. The scattering length is not the same for different isotopes of an element. A random distribution of isotopes over the sites occupied by that element leads to an incoherent contribution, such that effectively total ¼ coherent þ incoherent. Similarly for nuclei with nonzero spin, a spin incoherent scattering occurs as the spin states are, in general, randomly distributed over the sites of the nuclei. For free or loosely bound nuclei, the scattering length is modified by bfree ¼ ½M=ðm þ MÞb, where M is the mass of the nucleus and m is the mass of the neutron. This effect is of consequence only for the lightest elements. It can, in particular, be of significance for hydrogen atoms. With this in mind, the structure-factor expression for elastic scattering can be written as P FðHÞ ¼ bj; coherent exp 2
iðhxj þ kyj þ lzj Þ ð1:2:4:2dÞ

j ðrÞ expð2
iS  rÞ dr

fj ðSÞ ¼ atom

Z
Z2
Z1 ¼

j ðrÞ expð2
iSr cos #Þr2 sin # dr d# d’

¼0 ’¼0 r¼0

Z1 Z1 sin 2
Sr 2 dr  4
r2 j ðrÞj0 ð2
SrÞ dr ¼ 4
r j ðrÞ 2
Sr 0

0

 h j0 i;

ð1:2:4:3Þ

where j0 ð2
SrÞ is the zero-order spherical Bessel function. j ðrÞ represents either the static or the dynamic density of atom j. In the former case, the effect of thermal motion, treated in Section 1.2.9 and following, is not included in the expression. When scattering is treated in the second-order Born approximation, additional terms occur which are in particular of importance for X-ray wavelengths with energies close to absorption edges of atoms, where the participation of free and bound excited states in the scattering process becomes very important, leading to resonance scattering. [Resonance scattering is referred to as anomalous scattering in the older literature, but this misnomer is avoided in the current chapter.] Inclusion of such contributions leads to two extra terms, which are both wavelength- and scattering-angle-dependent: fj ðS; Þ ¼ fj 0 ðSÞ þ fj0 ðS; Þ þ ifj00 ðS; Þ:

j

by analogy to (1.2.4.2b). 1.2.5.2. Magnetic scattering The interaction between the magnetic moments of the neutron and the unpaired electrons in solids leads to magnetic scattering. The total elastic scattering including both the nuclear and magnetic contributions is given by

ð1:2:4:4Þ

jFðHÞj2total ¼ jFN ðHÞ þ QðHÞ  k^ j2 ;

The treatment of resonance effects is beyond the scope of this chapter. We note however (a) that to a reasonable approximation the S-dependence of fj0 and fj00 can be neglected, (b) that fj0 and fj00 are not independent, but related through the Kramers–Kronig transformation, and (c) that in an anisotropic environment the atomic scattering factor becomes anisotropic, and accordingly is described as a tensor property. Detailed descriptions and appropriate references can be found in Materlick et al. (1994) and in Section 4.2.6 of IT C (2004). The structure-factor expressions (1.2.4.2) can be simplified when the crystal class contains nontrivial symmetry elements. For example, when the origin of the unit cell coincides with a centre of symmetry ðx; y; z ! x; y; zÞ the sine term in (1.2.4.2b) cancels when the contributions from the symmetry-related atoms are added, leading to the expression N=2 P

FðHÞ ¼ 2

fj cos 2
ðhxj þ kyj þ lzj Þ;

ð1:2:5:1aÞ

where the unit vector k^ describes the polarization vector for the neutron spin, FN ðHÞ is given by (1.2.4.2b) and Q is defined by Z mc b b expð2
iH  rÞ dr: H
½MðrÞ
H ð1:2:5:2aÞ Q¼ eh MðrÞ is the vector field describing the electron-magnetization b is a unit vector parallel to H. distribution and H Q is thus proportional to the projection of M onto a direction orthogonal to H in the plane containing M and H. The magnitude of this projection depends on sin , where  is the angle between Q and H, which prevents magnetic scattering from being a truly three-dimensional probe. If all moments MðrÞ are collinear, as may be achieved in paramagnetic materials by applying an external field, and for the maximum signal (H orthogonal to M), (1.2.5.2a) becomes Z mc Q ¼ MðHÞ ¼ MðrÞ expð2
iH  rÞ dr ð1:2:5:2bÞ eh

ð1:2:4:2cÞ

j¼1

where the summation is over the unique half of the unit cell only. Further simplifications occur when other symmetry elements are present. They are treated in Chapter 1.4, which also contains a complete list of symmetry-specific structure-factor expresions valid in the spherical-atom isotropic-temperaturefactor approximation.

and (1.2.5.1a) gives jFj2total ¼ jFN ðHÞ  MðHÞj2

ð1:2:5:1bÞ

and jFj2total ¼ jFN ðHÞ þ MðHÞj2

1.2.5. Scattering of thermal neutrons 1.2.5.1. Nuclear scattering

for neutrons parallel and antiparallel to MðHÞ, respectively.

The scattering of neutrons by atomic nuclei is described by the atomic scattering length b, related to the total cross section total by the expression total ¼ 4
b2 . At present, there is no theory of nuclear forces which allows calculation of the scattering length, so that experimental values are to be used. Two types of nuclei can be distinguished (Squires, 1978). In the first type, the scattering is a resonance phenomenon and is associated with the formation of a compound nucleus (consisting of the original nucleus plus a neutron) with an energy close to that of an excited state. In the second type, the compound nucleus is not near an

1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism A first improvement beyond the isolated-atom formalism is to allow for changes in the radial dependence of the atomic electron distribution. Such changes may be due to electronegativity differences which lead to the transfer of electrons between the valence shells of different atoms. The electron transfer introduces a change in

11

1. GENERAL RELATIONSHIPS AND TECHNIQUES ylm ð; Þ ¼ Nlm Plm ðcos Þ sin m’

the screening of the nuclear charge by the electrons and therefore affects the radial dependence of the atomic electron distribution (Coulson, 1961). A change in radial dependence of the density may also occur in a purely covalent bond, as, for example, in the H2 molecule (Ruedenberg, 1962). It can be expressed as 0valence ðrÞ ¼ 3 valence ð rÞ

¼ ð1Þm ðYlm  Yl; m Þ=ð2iÞ:

The normalization constants Nlm are defined by the conditions R 2 ylmp d ¼ 1; ð1:2:7:3aÞ

ð1:2:6:1Þ

(Coppens et al., 1979), where 0 is the modified density and is an expansion/contraction parameter, which is > 1 for valence-shell contraction and < 1 for expansion. The 3 factor results from the normalization requirement. The valence density is usually defined as the outer electron shell from which charge transfer occurs. The inner or core electrons are much less affected by the change in occupancy of the outer shell and, in a reasonable approximation, retain their radial dependence. The corresponding structure-factor expression is P FðHÞ ¼ ½fPj; core fj; core ðHÞ þ Pj; valence fj; valence ðH= Þg

which are appropriate for normalization of wavefunctions. An alternative definition is used for charge-density basis functions: R R jdlmp j d ¼ 2 for l > 0 and jdlmp j d ¼ 1 for l ¼ 0: ð1:2:7:3bÞ The functions ylmp and dlmp differ only in the normalization constants. For the spherically symmetric function d00 , a population parameter equal to one corresponds to the function being populated by one electron. For the nonspherical functions with l > 0, a population parameter equal to one implies that one electron has shifted from the negative to the positive lobes of the function. The functions ylmp and dlmp can be expressed in Cartesian coordinates, such that

j


expð2
iH  rj Þ;

ð1:2:7:2cÞ

ð1:2:6:2Þ

where Pj; core and Pj; valence are the number of electrons (not necessarily integral) in the core and valence shell, respectively, and the atomic scattering factors fj; core and fj; valence are normalized to one electron. Here and in the following sections, the resonant-scattering contributions are incorporated in the core scattering.

ylmp ¼ Mlm clmp

ð1:2:7:4aÞ

dlmp ¼ Llm clmp ;

ð1:2:7:4bÞ

and

where the clmp are Cartesian functions. The relations between the various definitions of the real spherical harmonic functions are summarized by

1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion 1.2.7.1. Direct-space description of aspherical atoms Even though the spherical-atom approximation is often adequate, atoms in a crystal are in a nonspherical environment; therefore, an accurate description of the atomic electron density requires nonspherical density functions. In general, such density functions can be written in terms of the three polar coordinates r,  and ’. Under the assumption that the radial and angular parts can be separated, one obtains for the density function:
ðr; ; ’Þ ¼ RðrÞð; ’Þ:

in which the direction of the arrows and the corresponding conversion factors Xlm define expressions of the type (1.2.7.4). The expressions for clmp with l  4 are listed in Table 1.2.7.1, together with the normalization factors Mlm and Llm . The spherical harmonic functions are mutually orthogonal and form a complete set, which, if taken to sufficiently high order, can be used to describe any arbitrary angular function. The spherical harmonic functions are often referred to as multipoles since each represents the components of the charge distribution R ðrÞ, which gives nonzero contribution to the integral lmp ¼ ðrÞclmp r l dr, where lmp is an electrostatic multipole moment. Terms with increasing l are referred to as monopolar ðl ¼ 0Þ, dipolar ðl ¼ 1Þ, quadrupolar ðl ¼ 2Þ, octapolar ðl ¼ 3Þ, hexadecapolar ðl ¼ 4Þ, triacontadipolar ðl ¼ 5Þ and hexacontatetrapolar ðl ¼ 6Þ. Site-symmetry restrictions for the real spherical harmonics as given by Kara & Kurki-Suonio (1981) are summarized in Table 1.2.7.2. In cubic space groups, the spherical harmonic functions as defined by equations (1.2.7.2) are no longer linearly independent. The appropriate basis set for this symmetry consists of the ‘Kubic harmonics’ of Von der Lage & Bethe (1947). Some low-order terms are listed in Table 1.2.7.3. Both wavefunction and densityfunction normalization factors are specified in Table 1.2.7.3. A related basis set of angular functions has been proposed by Hirshfeld (1977). They are of the form cosn k , where k is the angle with a specified set of ðn þ 1Þðn þ 2Þ=2 polar axes. The Hirshfeld functions are identical to a sum of spherical harmonics

ð1:2:7:1Þ

The angular functions  are based on the spherical harmonic functions Ylm defined by   1=2 2l þ 1 ðl  jmjÞ! Ylm ð; ’Þ ¼ ð1Þm Plm ðcos Þ expðim’Þ; 4
ðl þ jmjÞ! ð1:2:7:2aÞ with l  m  l, where Plm ðcos Þ are the associated Legendre polynomials (see Arfken, 1970). djmj Pl ðxÞ ; dxjmj  1 dl  Pl ðxÞ ¼ l l ðx2  1Þl : l!2 dx

Plm ðxÞ ¼ ð1  x2 Þjmj=2

The real spherical harmonic functions ylmp, 0  m  l, p ¼ þ or  are obtained as a linear combination of Ylm :  1=2 ð2l þ 1Þðl  jmjÞ! ylmþ ð; Þ ¼ Plm ðcos Þ cos m’ 2
ð1 þ m0 Þðl þ jmjÞ! ¼ Nlm Plm ðcos Þ cos m’ ¼ ð1Þm ðYlm þ Yl; m Þ

ð1:2:7:2bÞ

and

12

1.2. THE STRUCTURE FACTOR Table 1.2.7.1. Real spherical harmonic functions (x, y, z are direction cosines) Normalization for wavefunctions, Mlmp §

Normalization for density functions, Llmp }

l

Symbol

C†

Angular function, clmp ‡

Expression

Numerical value

Expression

Numerical value

0

00

1

ð1=4
Þ1=2

0.28209

1=4


0.07958

1

11þ 11 10

1 1 1

1 9 x= y ; z

ð3=4
Þ1=2

0.48860

1=


0.31831

20

1=2

3z2  1

ð5=16
Þ1=2

0.31539

pffiffiffi 3 3 8


0.20675

21þ 21 22þ 22

3 3 6 6

9 xz > > = yz 2 2 ðx  y Þ=2 > > ; xy

ð15=4
Þ1=2

1.09255

3=4

0.75

30

1=2

5z3  3z

ð7=16
Þ1=2

0.37318

10 13


0.24485

31þ 31

3=2 3=2

x½5z2  1 y½5z2  1

ð21=32
Þ1=2

0.45705

½ar þ ð14=5Þ  ð
=4Þ1 ††

0.32033

32þ 32

15 15

ðx2  y2 Þz 2xyz

ð105=16
Þ1=2

1.44531

1

1

33þ 33

15 15

x3  3xy2 y3 þ 3x2 y

ð35=32
Þ1=2

0.59004

4=3


0.42441

40 41þ 41

1=8 5=2 5=2

35z4  30z2 þ 3 x½7z3  3z 3 y½7z  3z

ð9=256
Þ1=2

0.10579 0.66905

‡‡ 735 pffiffiffi 512 7 þ 196

0.06942

ð45=32
Þ1=2

42þ 42

15=2 15=2

ðx2  y2 Þ½7z2  1 2xy½7z2  1

ð45=64
Þ1=2

0.47309

pffiffiffi 105 7 pffiffiffi 4ð136 þ 28 7Þ

0.33059

43þ 43

105 105

ðx3  3xy2 Þz ðy3 þ 3x2 yÞz

ð315=32
Þ1=2

1.77013

5=4

1.25

44þ 44

105 105

x4  6x2 y2 þ y4 4x3 y  4xy3

ð315=256
Þ1=2

0.62584

15=32

0.46875

50 51þ 51

1=8

ð11=256
Þ1=2

2

3

4

5

15=8













63z5  70z3  15z ð21z4  14z2 þ 1Þx 4 2 ð21z  14z þ 1Þy

52þ 52

105=2

ð3z3  zÞðx2  y2 Þ 2xyð3z3  zÞ

53þ 53

105=2

ð9z2  1Þðx3  3xy2 Þ ð9z2  1Þð3x2 y  y3 Þ

54þ 54

945

zðx4  6x2 y2 þ y4 Þ zð4x3 y  4xy3 Þ

55þ 55

945

x5  10x3 y2 þ 5xy4 5x4 y  10x2 y3 þ y5







0.11695

—

0.07674

1=2

ð165=256
Þ

0.45295

—

0.32298

ð1155=64
Þ1=2

2.39677

—

1.68750

ð385=512
Þ1=2

0.48924

—

0.34515

ð3465=256
Þ1=2

2.07566

—

1.50000

ð693=512
Þ1=2

0.65638

—

0.50930

with l ¼ n, n  2, n  4; . . . ð0; 1Þ for n > 1, as shown elsewhere (Hirshfeld, 1977). The radial functions RðrÞ can be selected in different manners. Several choices may be made, such as Rl ðrÞ ¼

nl þ3 nðlÞ r expð l rÞ ðnl þ 2Þ!

0.47400

multipole (Hansen & Coppens, 1978). Values for the exponential coefficient l may be taken from energy-optimized coefficients for isolated atoms available in the literature (Clementi & Raimondi, 1963). A standard set has been proposed by Hehre et al. (1969). In the bonded atom, such values are affected by changes in nuclear screening due to migrations of charge, as described in part by equation (1.2.6.1). Other alternatives are:

(Slater type function); ð1:2:7:5aÞ

Rl ðrÞ ¼

where the coefficient nl may be selected by examination of products of hydrogenic orbitals which give rise to a particular

13

nþ1 n r expðr2 Þ n!

ðGaussian functionÞ ð1:2:7:5bÞ

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.7.1 (cont.) Normalization for wavefunctions, Mlmp §

Normalization for density functions, Llmp }

l

Symbol

C†

Angular function, clmp ‡

Expression

Numerical value

Expression

Numerical value

6

60 61þ 61

1=16

231z6  315z4 þ 105z2  5 ð33z5  30z3 þ 5zÞx 5 3 ð33z  30z þ 5zÞy

ð13=1024
Þ1=2

0.06357

—

0.04171

1=2

0.58262

—

0.41721

ð1365=2048
Þ1=2

0.46060

—

0.32611

ð1365=512
Þ1=2

0.92121

—

0.65132

ð819=1024
Þ1=2

0.50457

—

0.36104

ð9009=512
Þ1=2

2.36662

—

1.75000

ð3003=2048
Þ1=2

0.68318

—

0.54687

ð15=1024
Þ1=2

0.06828

—

0.04480

ð105=4096
Þ1=2

0.09033

—

0.06488

ð315=2048
Þ1=2

0.22127

—

0.15732

ð315=4096
Þ1=2

0.15646

—

0.11092

ð3465=1024
Þ1=2

1.03783

—

0.74044

ð3465=4096
Þ1=2

0.51892

—

0.37723

ð45045=2048
Þ1=2

2.6460

—

2.00000

ð6435=4096
Þ1=2

0.70716

—

0.58205

7

21=8

ð273=256
Þ



62þ 62

105=8

ð33z4  18z2 þ 1Þðx2  y2 Þ 2xyð33z4  18z2 þ 1Þ

63þ 63

315=2

ð11z3  3zÞðx3  3xy2 Þ ð11z3  3zÞð3x2 y  3yÞ

64þ 64

945=2

ð11z2  1Þðx4  6x2 y2 þ y4 Þ ð11z2  1Þð4x3 y  4xy3 Þ

65þ 65

10395

zðx5  10x3 y2 þ 5xy4 Þ zð5x4 y  10x2 y3 þ y5 Þ

66þ 66

10395

x6  15x4 y2 þ 15x2 y4  y6 6x5 y  20x3 y3 þ 6xy5

70 71þ 71

1=16









429z7  693z5 þ 315z3  35z ð429z6  495z4 þ 135z2  5Þx 6 4 2 ð429z  495z þ 135z  5Þy

7=16

72þ 72

63=8

ð143z5  110z3 þ 15zÞðx2  y2 Þ 2xyð143z5  110z3 þ 15zÞ

73þ 73

315=8

ð143z4  66z2 þ 3Þðx3  3xy2 Þ ð143z4  66z2 þ 3Þð3x2 y  y3 Þ

74þ 74

3465=2

ð13z3  3zÞðx4  6x2 y2 þ y4 Þ ð13z3  3zÞð4x3 y  4xy3 Þ

75þ 75

10395=2

ð13z3  1Þðx5  10x3 y2 þ 5xy4 Þ ð13z3  1Þð5x4 y  10x2 y3 þ y5 Þ

76þ 76

135135

zðx6  15x4 y2 þ 15x2 y4  y6 Þ zð6x5 y þ 20x3 y3  6xy5 Þ

77þ 77

135135

x7  21x5 y2 þ 35x3 y4  7xy6 7x6 y  35x4 y3 þ 21x2 y5  y7













m’ † Common factor such that Clm clmp ¼ Plm ðcos Þcos ‡ x ¼ sin  cos ’, y ¼ sin  sin ’, z ¼ cos : § As defined by ylmp ¼ Mlmp clmp where clmp are Cartesian functions. } Paturle & sin m’ . Coppens (1988), by dlmp ¼ Llmp clmp where clmp are Cartesian functions. †† ar = arctan (2). ‡‡ Nang ¼ fð14A5  14A5þ þ 20A3þ  20A3 þ 6A  6Aþ Þ2
g1 where pffiffiffiffiffiffiffi as defined 1=2 A ¼ ½ð30  480Þ=70 .

or

r  Rl ðrÞ ¼ r l L2lþ2 ðrÞ exp  n 2

For trigonally bonded atoms in organic molecules the l = 3 terms are often found to be the most significantly populated deformation functions.

ðLaguerre functionÞ; ð1:2:7:5cÞ

1.2.7.2. Reciprocal-space description of aspherical atoms The aspherical-atom form factor is obtained by substitution of (1.2.7.6) in expression (1.2.4.3a): R fj ðSÞ ¼ j ðrÞ expð2
iS  rÞ dr: ð1:2:4:3aÞ

where L is a Laguerre polynomial of order n and degree (2l + 2). In summary, in the multipole formalism the atomic density is described by atomic ðrÞ ¼ Pc core þ P 3 valence ð rÞ þ

lP max l¼0

03

0

Rl ð rÞ

l P P

Plmp dlmp ðr=rÞ;

In order to evaluate the integral, the scattering operator expð2
iS  rÞ must be written as an expansion of products of spherical harmonic functions. In terms of the complex spherical harmonic functions, the appropriate expression is (Weiss & Freeman, 1959; Cohen-Tannoudji et al., 1977)

ð1:2:7:6Þ

m¼0 p

in which the leading terms are those of the kappa formalism [expressions (1.2.6.1), (1.2.6.2)]; the subscript p is either + or . The expansion in (1.2.7.6) is frequently truncated at the hexadecapolar (l = 4) level. For atoms at positions of high site symmetry the first allowed functions may occur at higher l values.

1 P l P  expð2
iS  rÞ ¼ 4
i l jl ð2
SrÞYlm ð; ’ÞYlm ð ; Þ: l¼0 m¼l

ð1:2:7:7aÞ

14

1.2. THE STRUCTURE FACTOR In (1.2.7.8b) and (1.2.7.8c), hjl i, the Fourier–Bessel transform, is the radial integral defined as R hjl i ¼ jl ð2
SrÞRl ðrÞr2 dr ð1:2:7:9Þ

Table 1.2.7.2. Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981) ,  and j are integers. Symmetry

Choice of coordinate axes

Indices of allowed ylmp , dlmp

1 1 2 m 2=m 222 mm2 mmm 4 4 4=m 422 4mm 4 2m

Any Any 2kz m?z 2kz; m ? z 2kz; 2ky 2kz; m ? y m ? z; m ? y; m ? x 4kz 4kz 4kz; m ? z 4kz; 2ky 4kz; m ? y 4kz; 2kx m?y 4kz; m ? z; m ? x 3kz 3kz 3kz; 2ky 2kx

All ðl; m; Þ ð2; m; Þ ðl; 2; Þ ðl; l  2j; Þ ð2; 2; Þ ð2; 2; þÞ, ð2 þ 1; 2; Þ ðl; 2; þÞ ð2; 2; þÞ ðl; 4; Þ ð2; 4; Þ, ð2 þ 1; 4 þ 2; Þ ð2; 4; Þ ð2; 4; þÞ, ð2 þ 1; 4; Þ ðl; 4; þÞ ð2; 4; þÞ, ð2 þ 1; 4 þ 2; Þ ð2; 4; þÞ, ð2 þ 1; 4 þ 2; þÞ ð2; 4; þÞ ðl; 3; Þ ð2; 3; Þ ð2; 3; þÞ; ð2 þ 1; 3; Þ ð3 þ 2j; 3; þÞ, ð3 þ 2j þ 1; 3; Þ ðl; 3; þÞ ðl; 6; þÞ; ðl; 6 þ 3; Þ ð2; 3; þÞ ð2; 6; þÞ; ð2; 6 þ 3; Þ ðl; 6; Þ ð2; 6; Þ; ð2 þ 1; 6 þ 3; Þ ð2; 6; Þ ð2; 6; þÞ; ð2 þ 1; 6; Þ ðl; 6; þÞ ð2; 6; þÞ; ð2 þ 1; 6 þ 3; þÞ ð2; 6; þÞ; ð2 þ 1; 6 þ 3; Þ ð2; 6; þÞ

4=mmm 3 3 32

3m 3 m 6 6 6=m 622 6mm 6 m2 6=mmm

3kz; m ? y m?x 3kz; m ? y m?x 6kz 6kz 6kz; m ? z 6kz; 2ky 6kz; mky 6kz; m ? y m?x 6kz; m ? z; m ? y

of which hj0 i in expression (1.2.4.3) is a special case. The functions hjl i for Hartree–Fock valence shells of the atoms are tabulated in scattering-factor tables (IT IV, 1974). Expressions for the evaluation of hjl i using the radial function (1.2.7.5a–c) have been given by Stewart (1980) and in closed form for (1.2.7.5a) by Avery & Watson (1977) and Su & Coppens (1990). The closed-form expressions are listed in Table 1.2.7.4. Expressions (1.2.7.8) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fourier-transform invariant. The scattering factors flmp ðSÞ of the aspherical density functions Rl ðrÞdlmp ð; Þ in the multipole expansion (1.2.7.6) are thus given by flmp ðSÞ ¼ 4
i l hjl idlmp ð ; Þ:

The reciprocal-space spherical harmonic functions in this expression are identical to the functions given in Table 1.2.7.1, except for the replacement of the direction cosines x, y and z by the direction cosines of the scattering vector S. 1.2.8. Fourier transform of orbital products If the wavefunction is written as a sum over normalized Slater determinants, each representing an antisymmetrized combination of occupied molecular orbitals i P expressed as linear combinations of atomic orbitals ’ , i.e. i ¼ ci ’ , the electron density is given by (Stewart, 1969a) P 2 PP P ’ ðrÞ’ ðrÞ; ð1:2:8:1Þ ðrÞ ¼ ni i ¼ i



i

For a multi-Slater determinant wavefunction the electron density is expressed in terms of the occupied natural spin orbitals, leading again to (1.2.8.2) but with noninteger values for the coefficients ni . The summation (1.2.8.1) consists of one- and two-centre terms for which ’ and ’ are centred on the same or on different nuclei, respectively. The latter represent the overlap density, which is only significant if ’ ðrÞ and ’ ðrÞ have an appreciable value in the same region of space.

ð1:2:7:8aÞ where jl is the lth-order spherical Bessel function (Arfken, 1970), and  and ’, and  are the angular coordinates of r and S, respectively. For the Fourier transform of the real spherical harmonic functions, the scattering operator is expressed in terms of the real spherical harmonics:

1.2.8.1. One-centre orbital products If the atomic basis consists of hydrogenic type s, p, d, f, . . . orbitals, the basis functions may be written as

1 l X X ðl  mÞ! i l jl ð2
SrÞð2  m0 Þð2l þ 1Þ ðl þ mÞ! l¼0 m¼0


Plm ðcos ÞPlm ðcos Þ cos½mð  Þ;



with ni = 1 or 2. The coefficients P are the populations of the orbital product density functions  ðrÞ’ ðrÞ and are given by P P ¼ ni ci ci : ð1:2:8:2Þ

The Fourier transform of the productRof a complex spherical harmonic function with normalization jYlm j2 d ¼ 1 and an arbitrary radial function Rl ðrÞ follows from the orthonormality properties of the spherical harmonic functions, and is given by R R Ylm Rl ðrÞ expð2
iS  rÞ d ¼ 4
i l jl ð2
SrÞRl ðrÞr2 drYlm ð ; Þ;

expð2
iS  rÞ ¼

ð1:2:7:8dÞ

’ðr; ; ’Þ ¼ Rl ðrÞYlm ð; ’Þ

ð1:2:7:7bÞ

ð1:2:8:3aÞ

or which leads to R ylmp ð; ’ÞRl ðrÞ expð2
iS  rÞ d ¼ 4
i l hjl iylmp ð ; Þ: ð1:2:7:8bÞ

’ðr; ; ’Þ ¼ Rl ðrÞylmp ð; ’Þ;

ð1:2:8:3bÞ

which gives for corresponding values of the orbital products

Since ylmp occurs on both sides, the expression is independent of the normalization selected. Therefore, for the Fourier transform of the density functions dlmp R dlmp ð; ’ÞRl ðrÞ expð2
iS  rÞ d ¼ 4
i l hjl idlmp ð ; Þ: ð1:2:7:8cÞ

’ ðrÞ’ ðrÞ ¼ Rl ðrÞRl 0 ðrÞYlm ð; ’ÞYl 0 m0 ð; ’Þ

ð1:2:8:4aÞ

’ ðrÞ’ ðrÞ ¼ Rl ðrÞRl 0 ðrÞylmp ð; ’Þyl 0 m0 p0 ð; ’Þ;

ð1:2:8:4bÞ

and

15

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.7.3. ‘Kubic harmonic’ functions R
R 2
P l (a) Coefficients in the expression Klj ¼ kmpj ylmp with normalization 0 0 jKlj j2 sin  d d’ ¼ 1 (Kara & Kurki-Suonio, 1981). mp

Even l

mp

l

j

0+

0

1

1

2+

4+

4

1

1 7 1=2 2 3

1 5 1=2 2 3

6

1

0.76376 1 1 1=2

0.64550 1=2  12 72

6

2

2 2

6+

1 1=2 4 11

 14 51=2 0.55902

0.82916

10

1

1 1=2 8 33

1 7 1=2 4 3

1 65 1=2 8 3

1

0.71807 1 65 1=2

0.38188 1=2  14 112

0.58184 1=2  18 187 6

8

6

0.58630

0.41143

0.69784

1 247 1=2 8 6

1 19 1=2 16 3

1 1=2 16 85

10

2

l

j

2

3

1

1

7

1

1 13 1=2 2 6

1 11 1=2 2 16

0.73598

0.41458

9

1

1 1=2 43

2

0.43301 1 17 1=2

 14 131=2 0.90139 1=2  12 76

0.80202

9

10+

0.93541

0.35355

8

8+

2

0.15729 4

0.57622

6

6

8

0.54006

0.84163

l (b) Coefficients kmpj and density normalization factors Nlj in the expression Klj ¼ Nlj

P l m’ kmpj ulmp where ulm ¼ Plm ðcos Þcos sin m’ (Su & Coppens, 1994). mp

Even l l

j

0

1

Nlj

mp

1=4
¼ 0:079577

1

0+

2+

4+

6+

4

1

0.43454

1

þ1=168

6

1

0.25220

1

1=360

6

2

0.020833

8

1

0.56292

10

1

10

2

l

j

3

1

0.066667

7

1

0.014612

1

1=1560

9

1

0.0059569

1

1=2520

9

2

0.00014800

8+

10+

1=792

1 1

1/5940

0.36490

1

1/5460

0.0095165

1

1 1 672
5940 1 1 4320
5460 1 1  456
43680

1=43680 2

4

6

8

1

1=4080

1

(c) Density-normalized Kubic harmonicsRasR linear combinations of density-normalized spherical harmonic functions. Coefficients in the expression Klj ¼
2
Density-type normalization is defined as 0 0 jKlj j sin  d d’ ¼ 2  l0. Even l

mp

l

j

0+

0

1

1

4

1

0.78245

6

1

0.37790

6

2

l

j

3

1

1

7

1

0.73145

2+

4+ 0.57939 0.91682

0.50000

0.83848 2

6+

4

6 0.63290

16

8

8+

10+

P 00 l kmpj dlmp . mp

1.2. THE STRUCTURE FACTOR Table 1.2.7.3 (cont.) (d) Index rules for cubic symmetries (Kurki-Suonio, 1977; Kara & Kurki-Suonio, 1981). l

j

23 T

m3 Th

432 O

4 3m Td

m3 m Oh

0 3 4 6 6 7 8 9 9 10 10

1 1 1 1 2 1 1 1 2 1 2


















































respectively, where it has been assumed that the radial function depends only on l. Because the spherical harmonic functions form a complete set, their products can be expressed as a linear combination of spherical harmonics. The coefficients in this expansion are the Clebsch–Gordan coefficients (Condon & Shortley, 1957), defined by PP Mmm0 Ylm ð; ’ÞYl 0 m0 ð; ’Þ ¼ CLll 0 YLM ð; ’Þ ð1:2:8:5aÞ

R
0

0 0

0

0

yLMP ð; ’Þ;

ð1:2:8:5cÞ

Mmm The CLll vanish, unless L þ l þ l is even, jl  l j < L < l þ l 0 0 and M ¼ m þ m . The corresponding expression for ylmp is

ylmp ð; ’Þyl 0 m0 p0 ð; ’Þ ¼

PP

C

0

L M 0

Mmm0 Lll 0 P

0

0

0

with M ¼ jm þ m j and jm  m j for p ¼ p , and M ¼ jm þ m j 0 0 0 and jm  m j for p ¼ p and P ¼ p
p . 0 Values of C and C for l  2 are given in Tables 1.2.8.1 and 1.2.8.2. They Rare valid for the functions Ylm and ylmp with R normalization jYlm j2 d ¼ 1 and y2lmp d ¼ 1. By using (1.2.8.5a) or (1.2.8.5c), the one-centre orbital products are expressed as a sum of spherical harmonic functions. It follows that the one-centre orbital product density basis set is formally equivalent to the multipole description, both in real and in reciprocal space. To obtain the relation between orbital products and the charge-density functions, the right-hand side of (1.2.8.5c) has to be multiplied by the ratio of the normalization constants, as the wavefunctions ylmp and charge-density functions dlmp are normalized in a different way as described by (1.2.7.3a) and (1.2.7.3b). Thus ylmp ð; ’Þyl 0 m0 p0 ð; ’Þ ¼

PP

RLMP C

L M

0

Mmm0 Lll 0 P







Since the crystal is subject to vibrational oscillations, the observed elastic scattering intensity is an average over all normal modes of the crystal. Within the Born–Oppenheimer approximation, the theoretical electron density should be calculated for each set of nuclear coordinates. An average can be obtained by taking into account the statistical weight of each nuclear configuration, which may be expressed by the probability distribution function Pðu1 ; . . . ; uN Þ for a set of displacement coordinates u1 ; . . . ; uN . In general, if ðr; u1 ; . . . ; uN Þ is the electron density corresponding to the geometry defined by u1 ; . . . ; uN, the timeaveraged electron density is given by R hðrÞi ¼ ðr; u1 ; . . . ; uN ÞPðu1 ; . . . ; uN Þ du1 . . . duN : ð1:2:9:1Þ

R2
 sin  d d’YLM ð; ’ÞYlm ð; ’ÞYl 0 m0 ð; ’Þ: ð1:2:8:5bÞ

0




1.2.9. The atomic temperature factor

or the equivalent definition 0






& Stewart, 1973; Avery & Ørmen, 1979) atomic orbitals. The expressions can also be used for Hartree–Fock atomic functions, as expansions in terms of Gaussian- (Stewart, 1969b, 1970; Stewart & Hehre, 1970; Hehre et al., 1970) and Slater-type (Clementi & Roetti, 1974) functions are available for many atoms.

L M

Mmm CLll ¼ 0





When the crystal can be considered as consisting of perfectly following rigid entities, which may be molecules or atoms, expression (1.2.9.1) simplifies: R hrigid group ðrÞi ¼ r:g:; static ðr  uÞPðuÞ du ¼ r:g:; static  PðuÞ: ð1:2:9:2Þ In the approximation that the atomic electrons perfectly follow the nuclear motion, one obtains hatom ðrÞi ¼ atom; static ðrÞ  PðuÞ:

ð1:2:9:3Þ

The Fourier transform of this convolution is the product of the Fourier transforms of the individual functions:

dLMP ð; ’Þ; ð1:2:8:6Þ

hf ðHÞi ¼ f ðHÞTðHÞ:

where RLMP ¼ MLMP (wavefunction)=LLMP (density function). The normalization constants Mlmp and Llmp are given in Table 1.2.7.1, while the coefficients in the expressions (1.2.8.6) are listed in Table 1.2.8.3.

ð1:2:9:4Þ

Thus TðHÞ, the atomic temperature factor, is the Fourier transform of the probability distribution PðuÞ. 1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation

1.2.8.2. Two-centre orbital products Fourier transform of the electron density as described by (1.2.8.1) requires explicit expressions for the two-centre orbital product scattering. Such expressions are described in the literature for both Gaussian (Stewart, 1969b) and Slater-type (Bentley

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian, centred at the equilibrium position. For the three-dimensional isotropic harmonic oscillator, the distribution is

17

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.7.4. Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Su & Coppens, 1990) hjk i 

R1 0

N

r expðZrÞjk ðKrÞ dr; K ¼ 4
sin =:

N k

1

2

3

4

5

6

7

0

1 K2 þ Z2

2Z ðK2 þ Z2 Þ2

2ð3Z2  K2 Þ ðK2 þ Z2 Þ3

24ZðZ2  K2 Þ ðK2 þ Z2 Þ4

24ð5Z2  10K2 Z2 þ K4 Þ ðK2 þ Z2 Þ5

240ZðK2  3Z2 Þð3K2  Z2 Þ ðK2 þ Z2 Þ6

720ð7Z6  35K2 Z4 þ 21K4 Z2  K6 Þ ðK2 þ Z2 Þ7

40320ðZ7  7K2 Z5 þ 7K4 Z3  K6 ZÞ ðK2 þ Z2 Þ8

2K ðK2 þ Z2 Þ2

8KZ ðK2 þ Z2 Þ3

8Kð5Z2  K2 Þ ðK2 þ Z2 Þ4

48KZð5Z2  3K2 Þ ðK2 þ Z2 Þ5

48Kð35Z4  42K2 Z2 þ 3K4 Þ ðK2 þ Z2 Þ6

1920KZð7Z4  14K2 Z2 þ 3K4 Þ ðK2 þ Z2 Þ7

5760Kð21Z6  63K2 Z4 þ 27K4 Z2  K6 Þ ðK2 þ Z2 Þ8

8K2 ðK2 þ Z2 Þ3

48K2 Z ðK2 þ Z2 Þ4

48K2 ð7Z2  K2 Þ ðK2 þ Z2 Þ5

384K2 Zð7Z2  3K2 Þ ðK2 þ Z2 Þ6

1152K2 ð21Z4  18K2 Z2 þ K4 Þ ðK2 þ Z2 Þ7

11520K2 Zð21Z4  30K2 Z2 þ 5K4 Þ ðK2 þ Z2 Þ8

48K3 ðK2 þ Z2 Þ4

384K3 Z ðK2 þ Z2 Þ5

384K3 ð9Z2  K2 Þ ðK2 þ Z2 Þ6

11520K3 Zð3Z2  K2 Þ ðK2 þ Z2 Þ7

11520K3 ð33Z4  22K2 Z2 þ K4 Þ ðK2 þ Z2 Þ8

384K4 ðK2 þ Z2 Þ5

3840K4 Z ðK2 þ Z2 Þ6

3840K4 ð11Z2  K2 Þ ðK2 þ Z2 Þ7

46080K4 Zð11Z2  3K2 Þ ðK2 þ Z2 Þ8

3840K5 ðK2 þ Z2 Þ6

46080K5 Z ðK2 þ Z2 Þ7

40680K5 ð13Z2  K2 Þ ðK2 þ Z2 Þ8

46080K6 ðK2 þ Z2 Þ7

645120K6 Z ðK2 þ Z2 Þ8

1

2

3

4

5

8

6

7

645120K7 ðK2 þ Z2 Þ8

PðuÞ ¼ ð2
hu2 iÞ3=2 expfjuj2 =2hu2 ig;

around a vector k ð1 ; 2 ; 3 Þ, with length corresponding to the magnitude of the rotation, results in a displacement r, such that

ð1:2:10:1Þ

where hu2 i is the mean-square displacement in any direction. The corresponding trivariate normal distribution to be used for anisotropic harmonic motion is, in tensor notation, jr1 j1=2 j k PðuÞ ¼ expf 12 r1 jk ðu u Þg: ð2
Þ3=2

jr1 j1=2 expf 12 ðuÞT r1 ðuÞg; ð2
Þ3=2

2

ð1:2:10:2aÞ

3 2 1 5; 0

ð1:2:11:2Þ

0 D ¼ 4 3 2

3 0 1

or in Cartesian tensor notation, assuming summation over repeated indices, ri ¼ Dij rj ¼ "ijk k rj

ð1:2:10:2bÞ

ð1:2:10:3aÞ

ri ¼ Dij rj þ ti :

or TðHÞ ¼ expf2
2 HT rHg:

ð1:2:11:3Þ

where the permutation operator "ijk equals +1 for i, j, k a cyclic permutation of the indices 1, 2, 3, or 1 for a noncyclic permutation, and zero if two or more indices are equal. For i = 1, for example, only the "123 and "132 terms occur. Addition of a translational displacement gives

where the superscript T indicates the transpose. The characteristic function, or Fourier transform, of PðuÞ is TðHÞ ¼ expf2
2  jk hj hk g

ð1:2:11:1Þ

with

Here r is the variance–covariance matrix, with covariant components, and jr1 j is the determinant of the inverse of r. Summation over repeated indices has been assumed. The corresponding equation in matrix notation is PðuÞ ¼

r ¼ ðk
rÞ ¼ Dr

ð1:2:10:3bÞ

ð1:2:11:4Þ

When a rigid body undergoes vibrations the displacements vary with time, so suitable averages must be taken to derive the mean-square displacements. If the librational and translational motions are independent, the cross products between the two terms in (1.2.11.4) average to zero and the elements of the meansquare displacement tensor of atom n, Uijn , are given by

With the change of variable b jk ¼ 2
2  jk, (1.2.10.3a) becomes TðHÞ ¼ expfb jk hj hk g:

n U11 ¼ þL22 r23 þ L33 r22  2L23 r2 r3 þ T11 n U22 ¼ þL33 r21 þ L11 r23  2L13 r1 r3 þ T22 n U33 ¼ þL11 r22 þ L22 r21  2L12 r1 r2 þ T33

1.2.11. Rigid-body analysis The treatment of rigid-body motion of molecules or molecular fragments was developed by Cruickshank (1956) and expanded into a general theory by Schomaker & Trueblood (1968). The theory has been described by Johnson (1970b) and by Dunitz (1979). The latter reference forms the basis for the following treatment. The most general motions of a rigid body consist of rotations about three axes, coupled with translations parallel to each of the axes. Such motions correspond to screw rotations. A libration

n U12 ¼ L33 r1 r2  L12 r23 þ L13 r2 r3 þ L23 r1 r3 þ T12 n U13 ¼ L22 r1 r3 þ L12 r2 r3  L13 r22 þ L23 r1 r2 þ T13

ð1:2:11:5Þ

n U23 ¼ L11 r2 r3 þ L12 r1 r3  L13 r1 r2  L23 r21 þ T23 ;

where the coefficients Lij ¼ hi j i and Tij ¼ hti tj i are the elements of the 3
3 libration tensor L and the 3
3 translation tensor T, respectively. Since pairs of terms such as hti tj i and htj ti i

18

1.2. THE STRUCTURE FACTOR Table 1.2.8.1. Products of complex spherical harmonics as defined by equation (1.2.7.2a)

Table 1.2.8.2. Products of real spherical harmonics as defined by equations (1.2.7.2b) and (1.2.7.2c)

Y00 Y00 = 0.28209479Y00 Y10 Y00 = 0.28209479Y10 Y10 Y10 = 0.25231325Y20 + 0.28209479Y00 Y11 Y00 = 0.28209479Y11 Y11 Y10 = 0.21850969Y21 Y11 Y11 = 0.30901936Y22 Y11 Y11 = 0.12615663Y20 + 0.28209479Y00 Y20 Y00 = 0.28209479Y20 Y20 Y10 = 0.24776669Y30 + 0.25231325Y10 Y20 Y11 = 0.20230066Y31  0.12615663Y11 Y20 Y20 = 0.24179554Y40 + 0.18022375Y20 + 0.28209479Y00 Y21 Y00 = 0.28209479Y21 Y21 Y10 = 0.23359668Y31 + 0.21850969Y11 Y21 Y11 = 0.26116903Y32 Y21 Y11 = 0.14304817Y30 + 0.21850969Y10 Y21 Y20 = 0.22072812Y41 + 0.09011188Y21 Y21 Y21 = 0.25489487Y42 + 0.22072812Y22 Y21 Y21 = 0.16119702Y40 + 0.09011188Y20 + 0.28209479Y00 Y22 Y00 = 0.28209479Y22 Y22 Y10 = 0.18467439Y32 Y22 Y11 = 0.31986543Y33 Y22 Y11 = 0.08258890Y31 + 0.30901936Y11 Y22 Y20 = 0.15607835Y42  0.18022375Y22 Y22 Y21 = 0.23841361Y43 Y22 Y21 = 0.09011188Y41 + 0.22072812Y21 Y22 Y22 = 0.33716777Y44 Y22 Y22 = 0.04029926Y40  0.18022375Y20 + 0.28209479Y00

y00 y00 = 0.28209479y00 y10 y00 = 0.28209479y10 y10 y10 = 0.25231325y20 + 0.28209479y00 y11 y00 = 0.28209479y11 y11 y10 = 0.21850969y21 y11 y11 = 0.21850969y22+  0.12615663y20 + 0.28209479y00 y11+ y11 = 0.21850969y22 y20 y00 = 0.28209479y20 y20 y10 = 0.24776669y30 + 0.25231325y10 y20 y11 = 0.20230066y31  0.12615663y11 y20 y20 = 0.24179554y40 + 0.18022375y20 + 0.28209479y00 y21 y00 = 0.28209479y21 y21 y10 = 0.23359668y31 + 0.21850969y11 y21 y11 =  0.18467439y32+  0.14304817y30 + 0.21850969y10 y21 y11 = 0.18467469y32 y21 y20 = 0.22072812y41 + 0.09011188y21 y21 y21 =  0.18022375y42+  0.15607835y22+  0.16119702y40 + 0.09011188y20 + 0.28209479y00 y21+ y21 = 0.18022375y42 + 0.15607835y22 y22 y00 = 0.28209479y22 y22 y10 = 0.18467439y32 y22 y11 =  0.22617901y33+  0.05839917y31+ + 0.21850969y11+ y22 y11 = 0.22617901y33  0.05839917y31 0.21850969y11 y22 y20 = 0.15607835y42  0.18022375y22 y22 y21 =  0.16858388y43+  0.06371872y41+ + 0.15607835y21+ y22 y21 = 0.16858388y43  0.06371872y41 0.15607835y21 y22 y22 =  0.23841361y44+ + 0.04029926y40  0.18022375y20 + 0.28209479y00 y22+ y22 = 0.23841361y44

correspond to averages over the same two scalar quantities, the T and L tensors are symmetrical. If a rotation axis is correctly oriented, but incorrectly positioned, an additional translation component perpendicular to the rotation axes is introduced. The rotation angle and the parallel component of the translation are invariant to the position of the axis, but the perpendicular component is not. This implies that the L tensor is unaffected by any assumptions about the position of the libration axes, whereas the T tensor depends on the assumptions made concerning the location of the axes. The quadratic correlation between librational and translational motions can be allowed for by including in (1.2.11.5) cross terms of the type hDik tj i, or, with (1.2.11.3),

S31 r1  S32 r2 þ ðS22  S11 Þr3 : As the diagonal elements occur as differences in this expression, a constant may be added to each of the diagonal terms without changing the observational equations. In other words, the trace of S is indeterminate. In terms of the L; T and S tensors, the observational equations are Uij ¼ Gijkl Lkl þ Hijkl Skl þ Tij :

The arrays Gijkl and Hijkl involve the atomic coordinates ðx; y; zÞ ¼ ðr1 ; r2 ; r3 Þ, and are listed in Table 1.2.11.1. Equations (1.2.11.9) for each of the atoms in the rigid body form the observational equations, from which the elements of T; L and S can be derived by a linear least-squares procedure. One of the diagonal elements of S must be fixed in advance or some other suitable constraint applied because of the indeterminacy of TrðSÞ. It is common practice to set TrðSÞ equal to zero. There are thus eight elements of S to be determined, as well as the six each of L and T, for a total of 20 variables. A shift of origin leaves L invariant, but it intermixes T and S. If the origin is located at a centre of symmetry, for each atom at r with vibration tensor Un there will be an equivalent atom at r with the same vibration tensor. When the observational equations for these two atoms are added, the terms involving elements of S disappear since they are linear in the components of r. The other terms, involving elements of the T and L tensors, are simply doubled, like the Un components. The physical meaning of the T and L tensor elements is as follows. Tij li lj is the mean-square amplitude of translational vibration in the direction of the unit vector l with components l1 ; l2 ; l3 along the Cartesian axes and Lij li lj is the mean-square amplitude of libration about an axis in this direction. The quantity Sij li lj represents the mean correlation between libration

Uij ¼ hDik Djl irk rl þ hDik tj þ Djk ti irk þ hti tj i ¼ Aijkl rk rl þ Bijk rk þ hti tj i;

ð1:2:11:6Þ

which leads to the explicit expressions such as U11 ¼ hr1 i2 ¼ h23 ir22 þ h22 ir23  2h2 3 ir2 r3  2h3 t1 ir2  2h2 t1 ir3 þ ht12 i; U12 ¼ hr1 r2 i ¼ h23 ir1 r2 þ h1 3 ir2 r3 þ h2 3 ir1 r3  h1 2 ir23 þ h3 t1 ir1  h1 t1 ir3  h3 t2 ir2 þ h2 t2 ir3 þ ht1 t2 i: ð1:2:11:7Þ The products of the type hi tj i are the components of an additional tensor, S, which unlike the tensors T and L is unsymmetrical, since hi tj i is different from hj ti i. The terms involving elements of S may be grouped as h3 t1 ir1  h3 t2 ir2 þ ðh2 t2 i  h1 t1 iÞr3

ð1:2:11:9Þ

ð1:2:11:8Þ

or

19

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.8.3. Products of two real spherical harmonic functions ylmp in terms of the density functions dlmp defined by equation (1.2.7.3b)

Table 1.2.11.1. The arrays Gijkl and Hijkl to be used in the observational equations Uij ¼ Gijkl Lkl þ Hijkl Skl þ Tij [equation (1.2.11.9)] Gijkl

y00 y00 = 1.0000d00 y10 y00 = 0.43301d10 y10 y10 = 0.38490d20 + 1.0d00 y11 y00 = 0.43302d11 y11 y10 = 0.31831d21 y11 y11 = 0.31831d22+  0.19425d20 + 1.0d00 y11+ y11 = 0.31831d22 y20 y00 = 0.43033d20 y20 y10 = 0.37762d30 + 0.38730d10 y20 y11 = 0.28864d31  0.19365d11 y20 y20 = 0.36848d40 + 0.27493d20 + 1.0d00 y21 y00 = 0.41094d21 y21 y10 = 0.33329d31 + 0.33541d11 y21 y11 = 0.26691d32+  0.21802d30 + 0.33541d10 y21 y11 = 0.26691d32 y21 y20 = 0.31155d41 + 0.13127d21 y21 y21 = 0.25791d42+  0.22736d22+  0.24565d40 + 0.13747d20 + 1.0d00 y21+ y21 = 0.25790d42 + 0.22736d22 y22 y00 = 0.41094d22 y22 y10 = 0.26691d32 y22 y11 =  0.31445d33+  0.083323d31+ + 0.33541d11+ y22 y11 = 0.31445d33  0.083323d31 0.33541d11 y22 y20 = 0.22335d42  0.26254d22 y22 y21 =  0.23873d43+  0.089938d41+ + 0.22736d21+ y22 y21 = 0.23873d43  0.089938d41 0.22736d21 y22 y22 =  0.31831d44+ + 0.061413d40  0.27493d20 + 1.0d00 y22+ y22 = 0.31831d44

kl

b S23  b S32 b L22 þ b L33

b 2 ¼

b S31  b S13 b L11 þ b L33

b 3 ¼

33

23

31

12

z2 0 x2 0 xz 0

y2 x2 0 0 0 xy

2yz 0 0 x2 xy xz

0 2xz 0 xy y2 yz

0 0 2xy xz yz z2

ij

11

22

33

23

31

12

32

13

21

11 22 33 23 31 12

0 0 0 0 y z

0 0 0 x 0 z

0 0 0 x y 0

0 0 2x 0 z 0

2y 0 0 0 0 x

0 2z 0 y 0 0

0 2x 0 0 0 y

0 0 2y z 0 0

2z 0 0 0 x 0

displacements of the L2 and L3 axes are obtained by permutation of the indices. If all three axes are appropriately displaced, only the diagonal terms of S remain. Referred to the principal axes of L, they represent screw correlations along these axes and are independent of origin shifts. The elements of the reduced T are r

TII ¼ b TII 

P

ðb SKI Þ2 =b LKK

K6¼I r

TIJ ¼ b TIJ 

P b SKIb SKJ =b LKK ;

J 6¼ I:

ð1:2:11:12Þ

K

The resulting description of the average rigid-body motion is in terms of six independently distributed instantaneous motions – three screw librations about nonintersecting axes (with screw pitches given by b S11 =b L11 etc.) and three translations. The parameter set consists of three libration and three translation amplitudes, six angles of orientation for the principal axes of L and T, six coordinates of axis displacement, and three screw pitches, one of which has to be chosen arbitrarily, again for a total of 20 variables. Since diagonal elements of S enter into the expression for r TIJ , the indeterminacy of TrðSÞ introduces a corresponding indeterminacy in r T. The constraint TrðSÞ ¼ 0 is unaffected by the various rotations and translations of the coordinate systems used in the course of the analysis.

b S12  b S21 ; b L11 þ b L22

1.2.12. Treatment of anharmonicity The probability distribution (1.2.10.2) is valid in the case of rectilinear harmonic motion. If the deviations from Gaussian shape are not too large, distributions may be used which are expansions with the Gaussian distribution as the leading term. Three such distributions are discussed in the following sections.

in which the carets indicate quantities referred to the principal axis system. The description of the averaged motion can be simplified further by shifting to three generally nonintersecting libration axes, one each for each principal axis of L . Shifts of the L1 axis in the L2 and L3 directions by b S13 =b L11 and 1b S12 =b L11 ; 2 ¼ b 3 ¼ b

22

0 z2 y2 yz 0 0

kl

ð1:2:11:10Þ

1

11

11 22 33 23 31 12 Hijkl

about the axis l and translation parallel to this axis. This quantity, like Tij li lj, depends on the choice of origin, although the sum of the two quantities is independent of the origin. The nonsymmetrical tensor S can be written as the sum of a symmetric tensor with elements SSij ¼ ðSij þ Sji Þ=2 and a skewsymmetric tensor with elements SAij ¼ ðSij  Sji Þ=2. Expressed in terms of principal axes, SS consists of three principal screw correlations hI tI i. Positive and negative screw correlations correspond to opposite senses of helicity. Since an arbitrary constant may be added to all three correlation terms, only the differences between them can be determined from the data. The skew-symmetric part SA is equivalent to a vector ðk
tÞ=2 with components ðk
tÞi =2 ¼ ðj tk  k tj Þ=2, involving correlations between a libration and a perpendicular translation. The components of SA can be reduced to zero, and S made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes S also minimizes the trace of T. In terms of the coordinate system based on the principal axes of L , the required origin shifts b i are b 1 ¼

ij

1.2.12.1. The Gram–Charlier expansion The three-dimensional Gram–Charlier expansion, introduced into thermal-motion treatment by Johnson & Levy (1974), is an expansion of a function in terms of the zero and higher derivatives of a normal distribution (Kendall & Stuart, 1958). If Dj is the operator d/du j ,

ð1:2:11:11Þ

respectively, annihilate the S12 and S13 terms of the symmetrized S tensor and simultaneously effect a further reduction in TrðTÞ (the presuperscript denotes the axis that is shifted, the subscript the direction of the shift component). Analogous equations for

20

1.2. THE STRUCTURE FACTOR 1.2.12.2. The cumulant expansion

Table 1.2.12.1. Some Hermite polynomials (Johnson & Levy, 1974; Zucker & Schulz, 1982)

A second statistical expansion which has been used to describe the atomic probability distribution is that of Edgeworth (Kendall & Stuart, 1958; Johnson, 1969). It expresses the function PðuÞ as  1 1 PðuÞ ¼ exp j Dj þ jk Dj Dk  jkl Dj Dk Dl 2! 3!  1 jklm þ Dj Dk Dl Dm  . . . P0 ðuÞ: ð1:2:12:5aÞ 4!

H(u) = 1 Hj(u) = wj Hjk(u) = wjwk  pjk Hjkl(u) = wjwkwl  (wj pkl + wk plj + wl pjk) = wjwkwl  3w( j pkl) Hjklm(u) = wjwkwlwm  6w( jwk plm) + 3pj( k plm) Hjklmn(u) = wjwkwlwmwn  10w( lwmwn pjk) + 15w( npjk plm) Hjklmnp(u) = wjwkwlwmwnwp  15w( jwkwlwm pjk) + 45w( jwk plm pnp)  15pj( k plm pnp) where wj  pjk uk and pjk are the elements of  1, defined in expression (1.2.10.2). Indices between brackets indicate that the term is to be averaged over all permutations which produce distinct terms, keeping in mind that pjk ¼ pkj and wj wk ¼ wk wj as illustrated for Hjkl .

1 jk 1 c Dj Dk  c jkl Dj Dk Dl þ . . . 2! 3! c1 . . . cr D1 Dr P0 ðuÞ; þ ð1Þr r!

Like the moments  of a distribution, the cumulants are descriptive constants. They are related to each other (in the onedimensional case) by the identity  t2 tr  t2  tr exp 1 t þ 2 þ . . . r þ . . . ¼ 1 þ 1 t þ 2 þ . . . þ r : 2! r! 2! r! ð1:2:12:5bÞ

PðuÞ ¼ ½1  c j Dj þ

When it is substituted for t, (1.2.12.5b) is the characteristic function, or Fourier transform of PðtÞ (Kendall & Stuart, 1958). The first two terms in the exponent of (1.2.12.5a) can be omitted if the expansion is around the equilibrium position and the harmonic term is properly described by P0 ðuÞ. The Fourier transform of (1.2.12.5a) is, by analogy with the left-hand part of (1.2.12.5b) (with t replaced by 2
ih),   ð2
iÞ3 jkl ð2
iÞ4 jklm hj hk hl þ hj hk hl hm þ . . . T0 ðHÞ TðHÞ ¼ exp 3! 4!   4 2 ¼ exp 
3 i jkl hj hk hl þ
4 jklm hj hk hl hm þ . . . T0 ðHÞ; 3 3

ð1:2:12:1Þ

where P0 ðuÞ is the harmonic distribution, i ¼ 1; 2 or 3, and the operator D1 . . . Dr is the rth partial derivative @ r =ð@u1 . . . @ur Þ. Summation is again implied over repeated indices. The differential operators D may be eliminated by the use of three-dimensional Hermite polynomials H1 ...r defined, by analogy with the one-dimensional Hermite polynomials, by the expression D1 . . . Dr expð12jk1 u j uk Þ ¼ ð1Þr H1 ...r ðuÞ expð12jk1 u j uk Þ; ð1:2:12:2Þ

ð1:2:12:6Þ

which gives  1 1 1 PðuÞ ¼ 1 þ c jkl Hjkl ðuÞ þ c jklm Hjklm ðuÞ þ c jklmn Hjklmn ðuÞ 3! 4! 5!  1 þ c jklmnp Hjklmnp ðuÞ þ . . . P0 ðuÞ; ð1:2:12:3Þ 6!

where the first two terms have been omitted. Expression (1.2.12.6) is similar to (1.2.12.4) except that the entire series is in the exponent. Following Schwarzenbach (1986), (1.2.12.6) can be developed in a Taylor series, which gives  ð2
iÞ3 jkl ð2
iÞ4 jklm hj hk hl þ hj hk hl hm þ . . . TðHÞ ¼ 1 þ 3! 4!   ð2
iÞ6 jklmp 6! jkl mnp þ hj hk hl hm hn hp þ 6! 2!ð3!Þ2 þ higher-order terms T0 ðHÞ: ð1:2:12:7Þ

where the first and second terms have been omitted since they are equivalent to a shift of the mean and a modification of the harmonic term only. The permutations of j; k; l . . . here, and in the following sections, include all combinations which produce different terms. The coefficients c, defined by (1.2.12.1) and (1.2.12.2), are known as the quasimoments of the frequency function PðuÞ (Kutznetsov et al., 1960). They are related in a simple manner to the moments of the function (Kendall & Stuart, 1958) and are invariant to permutation of indices. There are 10, 15, 21 and 28 components of c for orders 3, 4, 5 and 6, respectively. The multivariate Hermite polynomials are functions of the elements of jk1 and of uk, and are given in Table 1.2.12.1 for orders  6 (IT IV, 1974; Zucker & Schulz, 1982). The Fourier transform of (1.2.12.3) is given by  4 2 TðHÞ ¼ 1 
3 ic jkl hj hk hl þ
4 c jklm hj hk hl hm 3 3 4 þ
5 ic jklmn hj hk hl hm hn 15  4 6 jklmnp 
c hj hk hl hm hn hp þ . . . T0 ðHÞ; ð1:2:12:4Þ 45

This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982), clearly shows the relation to the Gram–Charlier expansion (1.2.12.4), and corresponds to the probability distribution [analogous to (1.2.12.3)]  1 1 PðuÞ ¼ P0 ðuÞ 1 þ jkl Hjkl ðuÞ þ jklm Hjklm ðuÞ þ . . . 3! 4!   1 jklmnp þ 10 jkl mnp Hjklmnp þ 6! þ higher-order terms : ð1:2:12:8Þ The relation between the cumulants jkl and the quasimoments c jkl are apparent from comparison of (1.2.12.8) and (1.2.12.4): c jkl ¼ jkl c jklm ¼ jklm c jklmn ¼ jklmn c jklmnp ¼ jklmnp þ 10 jkl mnp :

where T0 ðHÞ is the harmonic temperature factor. TðHÞ is a power-series expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary.

21

ð1:2:12:9Þ

1. GENERAL RELATIONSHIPS AND TECHNIQUES The sixth- and higher-order cumulants and quasimoments differ. Thus the third-order cumulant jkl contributes not only to the coefficient of Hjkl, but also to higher-order terms of the probability distribution function. This is also the case for cumulants of higher orders. It implies that for a finite truncation of (1.2.12.6), the probability distribution cannot be represented by a finite number of terms. This is a serious difficulty when a probability distribution is to be derived from an experimental temperature factor of the cumulant type.

the cases tested, and because its truncation is equivalent in real and reciprocal space. The latter is also true for the one-particle potential model, which is mathematically related to the Gram– Charlier expansion by the interchange of the real- and reciprocalspace expressions. The terms of the OPP model have a specific physical meaning. The model allows prediction of the temperature dependence of the temperature factor (Willis, 1969; Coppens, 1980), provided the potential function itself can be assumed to be temperature independent. It has recently been shown that the Edgeworth expansion (1.2.12.5a) always has negative regions (Scheringer, 1985b). This implies that it is not a realistic description of a vibrating atom.

1.2.12.3. The one-particle potential (OPP) model When an atom is considered as an independent oscillator vibrating in a potential well VðuÞ, its distribution may be described by Boltzmann statistics. PðuÞ ¼ N expfVðuÞ=kTg;

1.2.13. The generalized structure factor

ð1:2:12:10Þ

In the generalized structure-factor formalism developed by Dawson (1975), the complex nature of both the atomic scattering factor and the generalized temperature factor are taken into account. We write for the atomic scattering factor:

R

with N, the normalization constant, defined by PðuÞ du ¼ 1. The classical expression (1.2.12.10) is valid in the high-temperature limit for which kT VðuÞ. Following Dawson (1967) and Willis (1969), the potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates:

0

Tj ðHÞ ¼ Tj; c ðHÞ þ iTj; a ðHÞ

V ¼ V0 þ j u j þ jk u j uk þ jkl u j uk ul þ jklm u j uk ul um þ . . . :

FðHÞ ¼ AðHÞ þ iBðHÞ;

The equilibrium condition gives j ¼ 0. Substitution into (1.2.12.10) leads to an expression which may be simplified by the assumption that the leading term is the harmonic component represented by jk : 0

0

ð1:2:13:1bÞ

ð1:2:13:2Þ

where the subscripts c and a refer to the centrosymmetric and noncentrosymmetric components of the underlying electron distribution, respectively. Substitution in (1.2.4.2) gives for the real and imaginary components A and B of FðHÞ P 0 AðHÞ ¼ ðfj; c þ fj Þ½cosð2
H  rj ÞTc  sinð2
H  rj ÞTa 

PðuÞ ¼ N expf jk u j uk g 0

ð1:2:13:1aÞ

and

ð1:2:12:11Þ


f1  jkl u j uk ul  jklm u j uk ul um  . . .g;

00

fj ðHÞ ¼ fj; c ðHÞ þ ifj; a ðHÞ þ fj þ ifj

j

ð1:2:12:12Þ

00

 ðfj; a þ fj Þ½cosð2
H  rj ÞTa þ sinð2
H  rj ÞTc 

0

in which ¼ =kT etc. and the normalization factor N depends on the level of truncation. The probability distribution is related to the spherical harmonic expansion. The ten products of the displacement parameters u j uk ul, for example, are linear combinations of the seven octapoles ðl ¼ 3Þ and three dipoles ðl ¼ 1Þ (Coppens, 1980). The thermal probability distribution and the aspherical atom description can be separated only because the latter is essentially confined to the valence shell, while the former applies to all electrons which follow the nuclear motion in the atomic scattering model. The Fourier transform of the OPP distribution, in a general coordinate system, is (Johnson, 1970a; Scheringer, 1985a)  4 2 0 0 TðHÞ ¼ T0 ðHÞ 1 
3 ijkl G jkl ðHÞ þ
4 jklm G jklm ðHÞ 3 3  4 4 0 0 þ
5 i"jklmn G jklmn ðHÞ 
6 i’jklmnp G jklmnp ðHÞ . . . ; 15 45

ð1:2:13:3aÞ and BðHÞ ¼

P

0

ðfj; c þ fj Þ½cosð2
H  rj ÞTa þ sinð2
H  rj ÞTc 

j 00

þ ðfj; a þ fj Þ½cosð2
H  rj ÞTc  sinð2
H  rj ÞTa  ð1:2:13:3bÞ (McIntyre et al., 1980; Dawson, 1967). Expressions (1.2.13.3) illustrate the relation between valencedensity anisotropy and anisotropy of thermal motion. 1.2.14. Conclusion This chapter summarizes mathematical developments of the structure-factor formalism. The introduction of atomic asphericity into the formalism and the treatment of thermal motion are interlinked. It is important that the complexities of the thermal probability distribution function can often be reduced by very low temperature experimentation. Results obtained with the multipole formalism for atomic asphericity can be used to derive physical properties and d-orbital populations of transition-metal atoms (IT C, 2004). In such applications, the deconvolution of the charge density and the thermal vibrations is essential. This deconvolution is dependent on the adequacy of the models summarized here.

ð1:2:12:13Þ where T0 is the harmonic temperature factor and G represents the Hermite polynomials in reciprocal space. If the OPP temperature factor is expanded in the coordinate system which diagonalizes jk, simpler expressions are obtained in which the Hermite polynomials are replaced by products of the displacement coordinates u j (Dawson et al., 1967; Coppens, 1980; Tanaka & Marumo, 1983).

The author would like to thank several of his colleagues who gave invaluable criticism of earlier versions of this manuscript. Corrections and additions were made following comments by P. J. Becker, D. Feil, N. K. Hansen, G. McIntyre, E. N. Maslen, S. Ohba, C. Scheringer and D. Schwarzenbach. Z. Su contributed to the revised version of the manuscript. Support of this work by the US

1.2.12.4. Relative merits of the three expansions The relative merits of the Gram–Charlier and Edgeworth expansions have been discussed by Zucker & Schulz (1982), Kuhs (1983), and by Scheringer (1985b). In general, the Gram–Charlier expression is found to be preferable because it gives a better fit in

22

1.2. THE STRUCTURE FACTOR National Science Foundation (CHE8711736 and CHE9317770) is gratefully acknowledged.

Johnson, C. K. (1970b). An introduction to thermal-motion analysis. In Crystallographic Computing, edited by F. R. Ahmed, S. R. Hall & C. P. Huber, pp. 207–219. Copenhagen: Munksgaard. Johnson, C. K. & Levy, H. A. (1974). Thermal motion analysis using Bragg diffraction data. In International Tables for X-ray Crystallography (1974), Vol. IV, pp. 311–336. Birmingham: Kynoch Press. Kara, M. & Kurki-Suonio, K. (1981). Symmetrized multipole analysis of orientational distributions. Acta Cryst. A37, 201–210. Kendall, M. G. & Stuart, A. (1958). The Advanced Theory of Statistics. London: Griffin. Kuhs, W. F. (1983). Statistical description of multimodal atomic probability structures. Acta Cryst. A39, 148–158. Kurki-Suonio, K. (1977). Symmetry and its implications. Isr. J. Chem. 16, 115–123. Kutznetsov, P. I., Stratonovich, R. L. & Tikhonov, V. I. (1960). Theory Probab. Its Appl. (USSR), 5, 80–97. Lipson, H. & Cochran, W. (1966). The Determination of Crystal Structures. London: Bell. McIntyre, G. J., Moss, G. & Barnea, Z. (1980). Anharmonic temperature factors of zinc selenide determined by X-ray diffraction from an extended-face crystal. Acta Cryst. A36, 482–490. Materlik, G., Sparks, C. J. & Fischer, K. (1994). Resonant Anomalous X-ray Scattering. Theory and Applications. Amsterdam: NorthHolland. Paturle, A. & Coppens, P. (1988). Normalization factors for spherical harmonic density functions. Acta Cryst. A44, 6–7. Ruedenberg, K. (1962). The nature of the chemical bond. Phys. Rev. 34, 326–376. Scheringer, C. (1985a). A general expression for the anharmonic temperature factor in the isolated-atom-potential approach. Acta Cryst. A41, 73–79. Scheringer, C. (1985b). A deficiency of the cumulant expansion of the anharmonic temperature factor. Acta Cryst. A41, 79–81. Schomaker, V. & Trueblood, K. N. (1968). On the rigid-body motion of molecules in crystals. Acta Cryst. B24, 63–76. Schwarzenbach, D. (1986). Private communication. Squires, G. L. (1978). Introduction to the Theory of Thermal Neutron Scattering. Cambridge University Press. Stewart, R. F. (1969a). Generalized X-ray scattering factors. J. Chem. Phys. 51, 4569–4577. Stewart, R. F. (1969b). Small Gaussian expansions of atomic orbitals. J. Chem. Phys. 50, 2485–2495. Stewart, R. F. (1970). Small Gaussian expansions of Slater-type orbitals. J. Chem. Phys. 52, 431–438. Stewart, R. F. (1980). Electron and Magnetization Densities in Molecules and Solids, edited by P. J. Becker, pp. 439–442. New York: Plenum. Stewart, R. F. & Hehre, W. J. (1970). Small Gaussian expansions of atomic orbitals: second-row atoms. J. Chem. Phys. 52, 5243–5247. Su, Z. & Coppens, P. (1990). Closed-form expressions for Fourier–Bessel transforms of Slater-type functions. J. Appl. Cryst. 23, 71–73. Su, Z. & Coppens, P. (1994). Normalization factors for Kubic harmonic density functions. Acta Cryst. A50, 408–409. Tanaka, K. & Marumo, F. (1983). Willis formalism of anharmonic temperature factors for a general potential and its application in the least-squares method. Acta Cryst. A39, 631–641. Von der Lage, F. C. & Bethe, H. A. (1947). A method for obtaining electronic functions and eigenvalues in solids with an application to sodium. Phys. Rev. 71, 612–622. Waller, I. & Hartree, D. R. (1929). Intensity of total scattering X-rays. Proc. R. Soc. London Ser. A, 124, 119–142. Weiss, R. J. & Freeman, A. J. (1959). X-ray and neutron scattering for electrons in a crystalline field and the determination of outer electron configurations in iron and nickel. J. Phys. Chem. Solids, 10, 147–161. Willis, B. T. M. (1969). Lattice vibrations and the accurate determination of structure factors for the elastic scattering of X-rays and neutrons. Acta Cryst. A25, 277–300. Zucker, U. H. & Schulz, H. (1982). Statistical approaches for the treatment of anharmonic motion in crystals. I. A comparison of the most frequently used formalisms of anharmonic thermal vibrations. Acta Cryst. A38, 563–568.

References Arfken, G. (1970). Mathematical Models for Physicists, 2nd ed. New York, London: Academic Press. Avery, J. & Ørmen, P.-J. (1979). Generalized scattering factors and generalized Fourier transforms. Acta Cryst. A35, 849–851. Avery, J. & Watson, K. J. (1977). Generalized X-ray scattering factors. Simple closed-form expressions for the one-centre case with Slater-type orbitals. Acta Cryst. A33, 679–680. Bentley, J. & Stewart, R. F. (1973). Two-centre calculations for X-ray scattering. J. Comput. Phys. 11, 127–145. Born, M. (1926). Quantenmechanik der Stoszvorga¨nge. Z. Phys. 38, 803. Clementi, E. & Raimondi, D. L. (1963). Atomic screening constants from SCF functions. J. Chem. Phys. 38, 2686–2689. Clementi, E. & Roetti, C. (1974). Roothaan–Hartree–Fock atomic wavefunctions. At. Data Nucl. Data Tables, 14, 177–478. Cohen-Tannoudji, C., Diu, B. & Laloe, F. (1977). Quantum Mechanics. New York: John Wiley and Paris: Hermann. Condon, E. V. & Shortley, G. H. (1957). The theory of atomic spectra. London, New York: Cambridge University Press. Coppens, P. (1980). Thermal smearing and chemical bonding. In Electron and Magnetization Densities in Molecules and Solids, edited by P. J. Becker, pp. 521–544. New York: Plenum. Coppens, P., Guru Row, T. N., Leung, P., Stevens, E. D., Becker, P. J. & Yang, Y. W. (1979). Net atomic charges and molecular dipole moments from spherical-atom X-ray refinements, and the relation between atomic charges and shape. Acta Cryst. A35, 63–72. Coulson, C. A. (1961). Valence. Oxford University Press. Cruickshank, D. W. J. (1956). The analysis of the anisotropic thermal motion of molecules in crystals. Acta Cryst. 9, 754–756. Dawson, B. (1967). A general structure factor formalism for interpreting accurate X-ray and neutron diffraction data. Proc. R. Soc. London Ser. A, 248, 235–288. Dawson, B. (1975). Studies of atomic charge density by X-ray and neutron diffraction – a perspective. In Advances in Structure Research by Diffraction Methods. Vol. 6, edited by W. Hoppe & R. Mason. Oxford: Pergamon Press. Dawson, B., Hurley, A. C. & Maslen, V. W. (1967). Anharmonic vibration in fluorite-structures. Proc. R. Soc. London Ser. A, 298, 289–306. Dunitz, J. D. (1979). X-ray analysis and the structure of organic molecules. Ithaca, London: Cornell University Press. Feil, D. (1977). Diffraction physics. Isr. J. Chem. 16, 103–110. Hansen, N. K. & Coppens, P. (1978). Testing aspherical atom refinements on small-molecule data sets. Acta Cryst. A34, 909–921. Hehre, W. J., Ditchfield, R., Stewart, R. F. & Pople, J. A. (1970). Selfconsistent molecular orbital methods. IV. Use of Gaussian expansions of Slater-type orbitals. Extension to second-row molecules. J. Chem. Phys. 52, 2769–2773. Hehre, W. J., Stewart, R. F. & Pople, J. A. (1969). Self-consistent molecular orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J. Chem. Phys. 51, 2657–2664. Hirshfeld, F. L. (1977). A deformation density refinement program. Isr. J. Chem. 16, 226–229. International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers. International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. James, R. W. (1982). The optical principles of the diffraction of X-rays. Woodbridge: Oxbow Press. Johnson, C. K. (1969). Addition of higher cumulants to the crystallographic structure-factor equation: a generalized treatment for thermalmotion effects. Acta Cryst. A25, 187–194. Johnson, C. K. (1970a). Series expansion models for thermal motion. ACA Program and Abstracts, 1970 Winter Meeting, Tulane University, p. 60.

23

references

International Tables for Crystallography (2010). Vol. B, Chapter 1.3, pp. 24–113.

1.3. Fourier transforms in crystallography: theory, algorithms and applications By G. Bricogne

of new ideas and new applications, and as such can have any hope at all of remaining useful in the long run. These conditions have led to the following choices: (i) the mathematical theory of the Fourier transformation has been cast in the language of Schwartz’s theory of distributions which has long been adopted in several applied fields, in particular electrical engineering, with considerable success; the extra work involved handsomely pays for itself by allowing the three different types of Fourier transformations to be treated together, and by making all properties of the Fourier transform consequences of a single property (the convolution theorem). This is particularly useful in all questions related to the sampling theorem; (ii) the various numerical algorithms have been presented as the consequences of basic algebraic phenomena involving Abelian groups, rings and finite fields; this degree of formalization greatly helps the subsequent incorporation of symmetry; (iii) the algebraic nature of space groups has been reemphasized so as to build up a framework which can accommodate both the phenomena used to factor the discrete Fourier transform and those which underlie the existence (and lead to the classification) of space groups; this common ground is found in the notion of module over a group ring (i.e. integral representation theory), which is then applied to the formulation of a large number of algorithms, many of which are new; (iv) the survey of the main types of crystallographic computations has tried to highlight the roles played by various properties of the Fourier transformation, and the ways in which a better exploitation of these properties has been the driving force behind the discovery of more powerful methods. In keeping with this philosophy, the theory is presented first, followed by the crystallographic applications. There are ‘forward references’ from mathematical results to the applications which later invoke them (thus giving ‘real-life’ examples rather than artificial ones), and ‘backward references’ as usual. In this way, the internal logic of the mathematical developments – the surest guide to future innovations – can be preserved, whereas the alternative solution of relegating these to appendices tends on the contrary to obscure that logic by subordinating it to that of the applications. It is hoped that this attempt at an overall presentation of the main features of Fourier transforms and of their ubiquitous role in crystallography will be found useful by scientists both within and outside the field.

1.3.1. General introduction Since the publication of Volume II of International Tables, most aspects of the theory, computation and applications of Fourier transforms have undergone considerable development, often to the point of being hardly recognizable. The mathematical analysis of the Fourier transformation has been extensively reformulated within the framework of distribution theory, following Schwartz’s work in the early 1950s. The computation of Fourier transforms has been revolutionized by the advent of digital computers and of the Cooley– Tukey algorithm, and progress has been made at an everaccelerating pace in the design of new types of algorithms and in optimizing their interplay with machine architecture. These advances have transformed both theory and practice in several fields which rely heavily on Fourier methods; much of electrical engineering, for instance, has become digital signal processing. By contrast, crystallography has remained relatively unaffected by these developments. From the conceptual point of view, old-fashioned Fourier series are still adequate for the quantitative description of X-ray diffraction, as this rarely entails consideration of molecular transforms between reciprocal-lattice points. From the practical point of view, three-dimensional Fourier transforms have mostly been used as a tool for visualizing electron-density maps, so that only moderate urgency was given to trying to achieve ultimate efficiency in these relatively infrequent calculations. Recent advances in phasing and refinement methods, however, have placed renewed emphasis on concepts and techniques long used in digital signal processing, e.g. flexible sampling, Shannon interpolation, linear filtering, and interchange between convolution and multiplication. These methods are iterative in nature, and thus generate a strong incentive to design new crystallographic Fourier transform algorithms making the fullest possible use of all available symmetry to save both storage and computation. As a result, need has arisen for a modern and coherent account of Fourier transform methods in crystallography which would provide: (i) a simple and foolproof means of switching between the three different guises in which the Fourier transformation is encountered (Fourier transforms, Fourier series and discrete Fourier transforms), both formally and computationally; (ii) an up-to-date presentation of the most important algorithms for the efficient numerical calculation of discrete Fourier transforms; (iii) a systematic study of the incorporation of symmetry into the calculation of crystallographic discrete Fourier transforms; (iv) a survey of the main types of crystallographic computations based on the Fourier transformation. The rapid pace of progress in these fields implies that such an account would be struck by quasi-immediate obsolescence if it were written solely for the purpose of compiling a catalogue of results and formulae ‘customized’ for crystallographic use. Instead, the emphasis has been placed on a mode of presentation in which most results and formulae are derived rather than listed. This does entail a substantial mathematical overhead, but has the advantage of preserving in its ‘native’ form the context within which these results are obtained. It is this context, rather than any particular set of results, which constitutes the most fertile source Copyright © 2010 International Union of Crystallography

1.3.2. The mathematical theory of the Fourier transformation 1.3.2.1. Introduction The Fourier transformation and the practical applications to which it gives rise occur in three different forms which, although they display a similar range of phenomena, normally require distinct formulations and different proof techniques: (i) Fourier transforms, in which both function and transform depend on continuous variables; (ii) Fourier series, which relate a periodic function to a discrete set of coefficients indexed by n-tuples of integers; (iii) discrete Fourier transforms, which relate finite-dimensional vectors by linear operations representable by matrices.

24

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY At the same time, the most useful property of the Fourier transformation – the exchange between multiplication and convolution – is mathematically the most elusive and the one which requires the greatest caution in order to avoid writing down meaningless expressions. It is the unique merit of Schwartz’s theory of distributions (Schwartz, 1966) that it affords complete control over all the troublesome phenomena which had previously forced mathematicians to settle for a piecemeal, fragmented theory of the Fourier transformation. By its ability to handle rigorously highly ‘singular’ objects (especially
-functions, their derivatives, their tensor products, their products with smooth functions, their translates and lattices of these translates), distribution theory can deal with all the major properties of the Fourier transformation as particular instances of a single basic result (the exchange between multiplication and convolution), and can at the same time accommodate the three previously distinct types of Fourier theories within a unique framework. This brings great simplification to matters of central importance in crystallography, such as the relations between (a) periodization, and sampling or decimation; (b) Shannon interpolation, and masking by an indicator function; (c) section, and projection; (d) differentiation, and multiplication by a monomial; (e) translation, and phase shift. All these properties become subsumed under the same theorem. This striking synthesis comes at a slight price, which is the relative complexity of the notion of distribution. It is first necessary to establish the notion of topological vector space and to gain sufficient control (or, at least, understanding) over convergence behaviour in certain of these spaces. The key notion of metrizability cannot be circumvented, as it underlies most of the constructs and many of the proof techniques used in distribution theory. Most of Section 1.3.2.2 builds up to the fundamental result at the end of Section 1.3.2.2.6.2, which is basic to the definition of a distribution in Section 1.3.2.3.4 and to all subsequent developments. The reader mostly interested in applications will probably want to reach this section by starting with his or her favourite topic in Section 1.3.4, and following the backward references to the relevant properties of the Fourier transformation, then to the proof of these properties, and finally to the definitions of the objects involved. Hopefully, he or she will then feel inclined to follow the forward references and thus explore the subject from the abstract to the practical. The books by Dieudonne´ (1969) and Lang (1965) are particularly recommended as general references for all aspects of analysis and algebra.

R e ðzÞ ¼ 12ðz þ z
Þ;

f : x 7 ! f ðxÞ will be used; the plain arrow ! will be reserved for denoting limits, as in
p x lim 1 þ ¼ ex : !1 p If X is any set and S is a subset of X, the indicator function s of S is the real-valued function on X defined by S ðxÞ ¼ 1 ¼0

if x 2 S if x 2 = S:

1.3.2.2.1. Metric and topological notions in Rn The set Rn can be endowed with the structure of a vector space of dimension n over R, and can be made into a Euclidean space by treating its standard basis as an orthonormal basis and defining the Euclidean norm:
kxk ¼

n P

x2i

1=2 :

i¼1

By misuse of notation, x will sometimes also designate the column vector of coordinates of x 2 Rn ; if these coordinates are referred to an orthonormal basis of Rn, then kxk ¼ ðxT xÞ1=2 ; where xT denotes the transpose of x. The distance between two points x and y defined by dðx; yÞ ¼ kx  yk allows the topological structure of R to be transferred to Rn, making it a metric space. The basic notions in a metric space are those of neighbourhoods, of open and closed sets, of limit, of continuity, and of convergence (see Section 1.3.2.2.6.1). A subset S of Rn is bounded if sup kx  yk < 1 as x and y run through S; it is closed if it contains the limits of all convergent sequences with elements in S. A subset K of Rn which is both bounded and closed has the property of being compact, i.e. that whenever K has been covered by a family of open sets, a finite subfamily can be found which suffices to cover K. Compactness is a very useful topological property for the purpose of proof, since it allows one to reduce the task of examining infinitely many local situations to that of examining only finitely many of them.

ðn times; n  1Þ;

so that an element x of Rn is an n-tuple of real numbers: x ¼ ðx1 ; . . . ; xn Þ:

1.3.2.2.2. Functions over Rn Let ’ be a complex-valued function over Rn . The support of ’, denoted Supp ’, is the smallest closed subset of Rn outside which ’ vanishes identically. If Supp ’ is compact, ’ is said to have compact support.

Similar meanings will be attached to Z and N . The symbol C will denote the set of complex numbers. If z 2 C, its modulus will be denoted by jzj, its conjugate by z
(not z ), and its real and imaginary parts by R e ðzÞ and I m ðzÞ: n

1 ðz  z
Þ: 2i

If X is a finite set, then jXj will denote the number of its elements. If mapping f sends an element x of set X to the element f ðxÞ of set Y, the notation

1.3.2.2. Preliminary notions and notation Throughout this text, R will denote the set of real numbers, Z the set of rational (signed) integers and N the set of natural (unsigned) integers. The symbol Rn will denote the Cartesian product of n copies of R: Rn ¼ R
. . .
R

I m ðzÞ ¼

n

25

1. GENERAL RELATIONSHIPS AND TECHNIQUES If t 2 R , the translate of ’ by t, denoted t ’, is defined by n

ðiÞ ðiiÞ

ðt ’ÞðxÞ ¼ ’ðx  tÞ:

ðiiiÞ Its support is the geometric translate of that of ’:

ðivÞ ðvÞ ðviÞ

Supp t ’ ¼ fx þ tjx 2 Supp ’g:

ðviiÞ If A is a nonsingular linear transformation in Rn , the image of ’ by A, denoted A# ’, is defined by #

xp ¼ xp1 1 . . . xpnn @f Di f ¼ ¼ @i f @xi @jpj f p . . . @xnn q  p if and only if qi  pi for all i ¼ 1; . . . ; n Dp f ¼ Dp1 1 . . . Dpnn f ¼

p @x1 1

p  q ¼ ðp1  q1 ; . . . ; pn  qn Þ p! ¼ p !
. . .
p !
 1
 n
 p1 pn p ¼
...
: q q1 qn

Leibniz’s formula for the repeated differentiation of products then assumes the concise form

1

ðA ’ÞðxÞ ¼ ’½A ðxÞ:

Dp ðfgÞ ¼

Its support is the geometric image of Supp ’ under A: Supp A# ’ ¼ fAðxÞjx 2 Supp ’g:

X
p  Dpq fDq g; q qp

while the Taylor expansion of f to order m about x ¼ a reads

If S is a nonsingular affine transformation in Rn of the form

f ðxÞ ¼

X 1 ½Dp f ðaÞðx  aÞp þ oðkx  akm Þ: p! jpjm

SðxÞ ¼ AðxÞ þ b

#

In certain sections the notation rf will be used for the gradient vector of f, and the notation ðrrT Þf for the Hessian matrix of its mixed second-order partial derivatives:

#

with A linear, the image of ’ by S is S ’ ¼ b ðA ’Þ, i.e.

0 @ 1 0 @f 1 B @x1 C B @x1 C C C B B C B . C B .. C; rf ¼ B ... C; r¼B C C B B C C B B @ @ A @ @f A @xn @xn 1 0 2 @f @2 f ... B @x2 @x1 @xn C C B 1 C B .. C .. B .. T ðrr Þf ¼ B . C: . . C B C B 2 @ @f @2 f A ... @xn @x1 @x2n

ðS# ’ÞðxÞ ¼ ’½A1 ðx  bÞ:

Its support is the geometric image of Supp ’ under S: Supp S# ’ ¼ fSðxÞjx 2 Supp ’g:

It may be helpful to visualize the process of forming the image of a function by a geometric operation as consisting of applying that operation to the graph of that function, which is equivalent to applying the inverse transformation to the coordinates x. This use of the inverse later affords the ‘left-representation property’ [see Section 1.3.4.2.2.2(e)] when the geometric operations form a group, which is of fundamental importance in the treatment of crystallographic symmetry (Sections 1.3.4.2.2.4, 1.3.4.2.2.5).

1.3.2.2.4. Integration, Lp spaces The Riemann integral used in elementary calculus suffers from the drawback that vector spaces of Riemann-integrable functions over Rn are not complete for the topology of convergence in the mean: a Cauchy sequence of integrable functions may converge to a nonintegrable function. To obtain the property of completeness, which is fundamental in functional analysis, it was necessary to extend the notion of integral. This was accomplished by Lebesgue [see Berberian (1962), Dieudonne´ (1970), or Chapter 1 of Dym & McKean (1972) and the references therein, or Chapter 9 of Sprecher (1970)], and entailed identifying functions which differed only on a subset of zero measure in Rn (such functions are said to be equal ‘almost everywhere’). The vector spaces Lp ðRn Þ consisting of function classes f modulo this identification for which

1.3.2.2.3. Multi-index notation When dealing with functions in n variables and their derivatives, considerable abbreviation of notation can be obtained through the use of multi-indices. A multi-index p 2 Nn is an n-tuple of natural integers: p ¼ ðp1 ; . . . ; pn Þ. The length of p is defined as

jpj ¼

n P

pi ;

i¼1

and the following abbreviations will be used:

26

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY
kfkp ¼

R

p

n

R F1 : x 7 ! Fðx; yÞ dn y Rn R F2 : y 7 ! Fðx; yÞ dm x

1=p

j f ðxÞj d x

0 form a fundamental system of neighbourhoods of 0. The corresponding topology makes E a normed space. This topology is metrizable, since it is equivalent to that derived from the translation-invariant distance dðx; yÞ ¼ ðx  yÞ. Normed spaces which are complete, i.e. in which all Cauchy sequences converge, are called Banach spaces; they constitute the natural setting for the study of differential calculus. A semi-norm  on a vector space E is a positive real-valued function on E
E which satisfies (i0 ) and (iii0 ) but not (ii0 ). Given a set  of semi-norms on E such that any pair (x, y) in E
E is separated by at least one  2 , let B be the set of those subsets ; r of E defined by a condition of the form ðxÞ  r with  2  and r > 0; and let S be the set of finite intersections of elements of B. Then there exists a unique topology on E for which S is a fundamental system of neighbourhoods of 0. This topology is uniformizable since it is equivalent to that derived from the family of translation-invariant pseudo-distances ðx; yÞ 7 ! ðx  yÞ. It is metrizable if and only if it can be constructed by the above procedure with  a countable set of semi-norms. If furthermore E is complete, E is called a Fre´chet space. If E is a topological vector space over C, its dual E is the set of all linear mappings from E to C (which are also called linear forms, or linear functionals, over E). The subspace of E consisting of all linear forms which are continuous for the topology of E is called the topological dual of E and is denoted E0 . If the topology on E is metrizable, then the continuity of a linear form T 2 E0 at f 2 E can be ascertained by means of sequences, i.e. by checking that the sequence ½Tð fj Þ of complex numbers converges to Tð f Þ in C whenever the sequence ð fj Þ converges to f in E.

ðiiiÞ

R

Rn


ðxÞ’ðxÞ dn x ¼ ’ð0Þ

for any function ’ sufficiently well behaved near x ¼ 0. This is related to the problem of finding a unit for convolution (Section 1.3.2.2.4). As will now be seen, this definition is still unsatisfactory. Let the sequence ð f Þ in L1 ðRn Þ be an approximate convolution unit, e.g.

f ðxÞ ¼

  1=2 2

expð122 kxk2 Þ:

Then for any well behaved function ’ the integrals R Rn

f ðxÞ’ðxÞ dn x

exist, and the sequence of their numerical values tends to ’ð0Þ. It is tempting to combine this with (iii) to conclude that
is the limit of the sequence ð f Þ as  ! 1. However, lim f ðxÞ ¼ 0

as  ! 1

almost everywhere in Rn and the crux of the problem is that

28

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY R ’ð0Þ ¼ lim f ðxÞ’ðxÞ dn x !1 Rn i Rh lim fv ðxÞ ’ðxÞ dn x ¼ 0 6¼ Rn

(c) DK ðÞ is the subspace of DðÞ consisting of functions whose (compact) support is contained within a fixed compact subset K of . When  is unambiguously defined by the context, we will simply write E ; D; DK . It sometimes suffices to require the existence of continuous derivatives only up to finite order m inclusive. The corresponding spaces are then denoted E ðmÞ ; DðmÞ ; DðmÞ K with the convention that if m ¼ 0, only continuity is required. The topologies on these spaces constitute the most important ingredients of distribution theory, and will be outlined in some detail.

!1

because the sequence ð f Þ does not satisfy the hypotheses of Lebesgue’s dominated convergence theorem. Schwartz’s solution to this problem is deceptively simple: the regular behaviour one is trying to capture is an attribute not of the sequence of functions ð f Þ, but of the sequence of continuous linear functionals

1.3.2.3.3.1. Topology on E ðÞ It is defined by the family of semi-norms

R T : ’ 7 ! f ðxÞ’ðxÞ dn x Rn

’ 2 E ðÞ 7 ! p; K ð’Þ ¼ sup jDp ’ðxÞj; x2K

which has as a limit the continuous functional

where p is a multi-index and K a compact subset of . A fundamental system S of neighbourhoods of the origin in E ðÞ is given by subsets of E ðÞ of the form

T : ’ 7 ! ’ð0Þ: It is the latter functional which constitutes the proper definition of
. The previous paradoxes arose because one insisted on writing down the simple linear operation T in terms of an integral. The essence of Schwartz’s theory of distributions is thus that, rather than try to define and handle ‘generalized functions’ via sequences such as ð f Þ [an approach adopted e.g. by Lighthill (1958) and Erde´lyi (1962)], one should instead look at them as continuous linear functionals over spaces of well behaved functions. There are many books on distribution theory and its applications. The reader may consult in particular Schwartz (1965, 1966), Gel’fand & Shilov (1964), Bremermann (1965), Tre`ves (1967), Challifour (1972), Friedlander (1982), and the relevant chapters of Ho¨rmander (1963) and Yosida (1965). Schwartz (1965) is especially recommended as an introduction.

Vðm; "; KÞ ¼ f’ 2 E ðÞjjpj  m ) p;K ð’Þ < "g for all natural integers m, positive real ", and compact subset K of . Since a countable family of compact subsets K suffices to cover , and since restricted values of " of the form " ¼ 1=N lead to the same topology, S is equivalent to a countable system of neighbourhoods and hence E ðÞ is metrizable. Convergence in E may thus be defined by means of sequences. A sequence ð’ Þ in E will be said to converge to 0 if for any given Vðm; "; KÞ there exists 0 such that ’ 2 Vðm; "; KÞ whenever  > 0 ; in other words, if the ’ and all their derivatives Dp ’ converge to 0 uniformly on any given compact K in . 1.3.2.3.3.2. Topology on Dk ðÞ It is defined by the family of semi-norms

1.3.2.3.2. Rationale The guiding principle which leads to requiring that the functions ’ above (traditionally called ‘test functions’) should be well behaved is that correspondingly ‘wilder’ behaviour can then be accommodated in theR limiting behaviour of the f while still keeping the integrals Rn f ’ dn x under control. Thus (i) to minimize restrictions on the limiting behaviour of the f at infinity, the ’’s will be chosen to have compact support; (ii) to minimize restrictions on the local behaviour of the f, the ’’s will be chosen infinitely differentiable. To ensure further the continuity of functionals such as T with respect to the test function ’ as the f go increasingly wild, very strong control will have to be exercised in the way in which a sequence ð’j Þ of test functions will be said to converge towards a limiting ’: conditions will have to be imposed not only on the values of the functions ’j, but also on those of all their derivatives. Hence, defining a strong enough topology on the space of test functions ’ is an essential prerequisite to the development of a satisfactory theory of distributions.

’ 2 DK ðÞ 7 ! p ð’Þ ¼ sup jDp ’ðxÞj; x2K

where K is now fixed. The fundamental system S of neighbourhoods of the origin in DK is given by sets of the form Vðm; "Þ ¼ f’ 2 DK ðÞjjpj  m ) p ð’Þ < "g: It is equivalent to the countable subsystem of the Vðm; 1=NÞ, hence DK ðÞ is metrizable. Convergence in DK may thus be defined by means of sequences. A sequence ð’ Þ in DK will be said to converge to 0 if for any given Vðm; "Þ there exists 0 such that ’ 2 Vðm; "Þ whenever  > 0 ; in other words, if the ’ and all their derivatives Dp ’ converge to 0 uniformly in K. 1.3.2.3.3.3. Topology on DðÞ It is defined by the fundamental system of neighbourhoods of the origin consisting of sets of the form

1.3.2.3.3. Test-function spaces With this rationale in mind, the following function spaces will be defined for any open subset  of Rn (which may be the whole of Rn ): (a) E ðÞ is the space of complex-valued functions over  which are indefinitely differentiable; (b) DðÞ is the subspace of E ðÞ consisting of functions with (unspecified) compact support contained in Rn ;

VððmÞ; ð"ÞÞ   ¼ ’ 2 DðÞjjpj  m ) sup jDp ’ðxÞj < " for all  ; kxk

29

1. GENERAL RELATIONSHIPS AND TECHNIQUES (iii) The linear map ’ 7 ! ð1Þp Dp ’ðaÞ is a distribution of order m ¼ jpj>0, and hence is P not a measure. (iv) The linear map ’ 7 ! >0 ’ðÞ ðÞ is a distribution of infinite order on R: the order of differentiation is bounded for each ’ (because ’ has compact support) but is not as ’ varies. (v) If ðp Þ is a sequence of multi-indices p ¼ ðp1 ; . . . ; pn Þ such P that jp j ! 1 as  ! 1, then the linear map ’ 7 ! >0 ðDp ’Þðp Þ is a distribution of infinite order on Rn .

where (m) is an increasing sequence ðm Þ of integers tending to þ1 and (") is a decreasing sequence ð" Þ of positive reals tending to 0, as  ! 1. This topology is not metrizable, because the sets of sequences (m) and (") are essentially uncountable. It can, however, be shown to be the inductive limit of the topology of the subspaces DK , in the following sense: V is a neighbourhood of the origin in D if and only if its intersection with DK is a neighbourhood of the origin in DK for any given compact K in . A sequence ð’ Þ in D will thus be said to converge to 0 in D if all the ’ belong to some DK (with K a compact subset of  independent of ) and if ð’ Þ converges to 0 in DK . As a result, a complex-valued functional T on D will be said to be continuous for the topology of D if and only if, for any given compact K in , its restriction to DK is continuous for the topology of DK , i.e. maps convergent sequences in DK to convergent sequences in C. This property of D, i.e. having a nonmetrizable topology which is the inductive limit of metrizable topologies in its subspaces DK, conditions the whole structure of distribution theory and dictates that of many of its proofs.

1.3.2.3.6. Distributions associated to locally integrable functions R Let f nbe a complex-valued function over  such that K j f ðxÞj d x exists for any given compact K in ; f is then called locally integrable. The linear mapping from DðÞ to C defined by R ’ 7 ! f ðxÞ’ðxÞ dn x 

may then be shown to be continuous over DðÞ. It thus defines a distribution Tf 2 D 0 ðÞ: R hTf ; ’i ¼ f ðxÞ’ðxÞ dn x:

ðmÞ 1.3.2.3.3.4. Topologies on E ðmÞ ; DðmÞ k ;D These are defined similarly, but only involve conditions on derivatives up to order m.



As the continuity of Tf only requires that ’ 2 Dð0Þ ðÞ, Tf is actually a Radon measure. It can be shown that two locally integrable functions f and g define the same distribution, i.e.

1.3.2.3.4. Definition of distributions A distribution T on  is a linear form over DðÞ, i.e. a map T : ’ 7 ! hT; ’i

hTf ; ’i ¼ hTK ; ’i which associates linearly a complex number hT; ’i to any ’ 2 DðÞ, and which is continuous for the topology of that space. In the terminology of Section 1.3.2.2.6.2, T is an element of D 0 ðÞ, the topological dual of DðÞ. Continuity over D is equivalent to continuity over DK for all compact K contained in , and hence to the condition that for any sequence ð’ Þ in D such that (i) Supp ’ is contained in some compact K independent of , (ii) the sequences ðjDp ’ jÞ converge uniformly to 0 on K for all multi-indices p; then the sequence of complex numbers hT; ’ i converges to 0 in C. If the continuity of a distribution T requires (ii) for jpj  m only, T may be defined over DðmÞ and thus T 2 D 0 ðmÞ ; T is said to be a distribution of finite order m. In particular, for m ¼ 0; Dð0Þ is the space of continuous functions with compact support, and a distribution T 2 D 0 ð0Þ is a (Radon) measure as used in the theory of integration. Thus measures are particular cases of distributions. Generally speaking, the larger a space of test functions, the smaller its topological dual:

for all ’ 2 D;

if and only if they are equal almost everywhere. The classes of locally integrable functions modulo this equivalence form a vector space denoted L1loc ðÞ; each element of L1loc ðÞ may therefore be identified with the distribution Tf defined by any one of its representatives f. 1.3.2.3.7. Support of a distribution A distribution T 2 D 0 ðÞ is said to vanish on an open subset ! of  if it vanishes on all functions in Dð!Þ, i.e. if hT; ’i ¼ 0 whenever ’ 2 Dð!Þ. The support of a distribution T, denoted Supp T, is then defined as the complement of the set-theoretic union of those open subsets ! on which T vanishes; or equivalently as the smallest closed subset of  outside which T vanishes. When T ¼ Tf for f 2 L1loc ðÞ, then Supp T ¼ Supp f, so that the two notions coincide. Clearly, if Supp T and Supp ’ are disjoint subsets of , then hT; ’i ¼ 0. It can be shown that any distribution T 2 D 0 with compact support may be extended from D to E while remaining continuous, so that T 2 E 0 ; and that conversely, if S 2 E 0, then its restriction T to D is a distribution with compact support. Thus, the topological dual E 0 of E consists of those distributions in D 0 which have compact support. This is intuitively clear since, if the condition of having compact support is fulfilled by T, it needs no longer be required of ’, which may then roam through E rather than D.

m < n ) DðmÞ  DðnÞ ) D 0 ðnÞ  D 0 ðmÞ : This clearly results from the observation that if the ’’s are allowed to be less regular, then less wildness can be accommodated in T if the continuity of the map ’ 7 ! hT; ’i with respect to ’ is to be preserved.

1.3.2.3.8. Convergence of distributions A sequence ðTj Þ of distributions will be said to converge in D 0 to a distribution T as j ! 1 if, for any given ’ 2 D, the sequence of complex numbers ðhTj ; ’iÞ converges in C to the complex number hT;P ’i. 1 A series j¼0 Tj of distributions will be said to converge in D 0 and to havePdistribution S as its sum if the sequence of partial k sums Sk ¼ j¼0 converges to S.

1.3.2.3.5. First examples of distributions (i) The linear map ’ 7 ! h
; ’i ¼ ’ð0Þ is a measure (i.e. a zeroth-order distribution) called Dirac’s measure or (improperly) Dirac’s ‘
-function’. (ii) The linear map ’ 7 ! h
ðaÞ ; ’i ¼ ’ðaÞ is called Dirac’s measure at point a 2 Rn .

30

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY These definitions of convergence in D 0 assume that the limits T and S are known in advance, and are distributions. This raises the question of the completeness of D 0 : if a sequence ðTj Þ in D 0 is such that the sequence ðhTj ; ’iÞ has a limit in C for all ’ 2 D, does the map

h@2ij T; ’i ¼ h@j T; @i ’i ¼ hT; @2ij ’i; hDp T; ’i ¼ ð1Þjpj hT; Dp ’i; hT; ’i ¼ hT; ’i; where  is the Laplacian operator. The derivatives of Dirac’s
distribution are

’ 7 ! lim hTj ; ’i j!1

hDp
; ’i ¼ ð1Þjpj h
; Dp ’i ¼ ð1Þjpj Dp ’ð0Þ: 0

define a distribution T 2 D ? In other words, does the limiting process preserve continuity with respect to ’? It is a remarkable theorem that, because of the strong topology on D, this is actually the case. An analogous statement holds for series. This notion of convergence does not coincide with any of the classical notions used for ordinary functions: for example, the sequence ð’ Þ with ’ ðxÞ ¼ cos x converges to 0 in D 0 ðRÞ, but fails to do so by any of the standard criteria. An example of convergent sequences of distributions is provided by sequences which converge to
. If ð f Þ is a sequence n of locally R summable functions on R such that (i) kxk< b f ðxÞ dn x ! 1 as  ! 1 for all b > 0; R (ii) akxk1=a j f ðxÞj dn x ! 0 as  ! 1 for all 0 < a < 1; R (iii) there exists d > 0 and M > 0 such that kxk< d j f ðxÞj dn x < M for all ; then the sequence ðTf Þ of distributions converges to
in D 0 ðRn Þ.

It is remarkable that differentiation is a continuous operation for the topology on D 0 : if a sequence ðTj Þ of distributions converges to distribution T, then the sequence ðDp Tj Þ of derivatives converges to Dp T for any multi-index p, since as j ! 1 hDp Tj ; ’i ¼ ð1Þjpj hTj ; Dp ’i ! ð1Þjpj hT; Dp ’i ¼ hDp T; ’i: An analogous statement holds for series: any convergent series of distributions may be differentiated termwise to all orders. This illustrates how ‘robust’ the constructs of distribution theory are in comparison with those of ordinary function theory, where similar statements are notoriously untrue. (b) Differentiation under the duality bracket Limiting processes and differentiation may also be carried out under the duality bracket h; i as under the integral sign with ordinary functions. Let the function ’ ¼ ’ðx; Þ depend on a parameter  2  and a vector x 2 Rn in such a way that all functions

1.3.2.3.9. Operations on distributions As a general rule, the definitions are chosen so that the operations coincide with those on functions whenever a distribution is associated to a function. Most definitions consist in transferring to a distribution T an operation which is well defined on ’ 2 D by ‘transposing’ it in the duality product hT; ’i; this procedure will map T to a new distribution provided the original operation maps D continuously into itself.

’ : x 7 ! ’ðx; Þ be in DðRn Þ for all  2 . Let T 2 D0 ðRn Þ be a distribution, let IðÞ ¼ hT; ’ i

1.3.2.3.9.1. Differentiation (a) Definition and elementary properties If T is a distribution on Rn , its partial derivative @i T with respect to xi is defined by

and let 0 2  be given parameter value. Suppose that, as  runs through a small enough neighbourhood of 0, (i) all the ’ have their supports in a fixed compact subset K of Rn ; (ii) all the derivatives Dp ’ have a partial derivative with respect to  which is continuous with respect to x and . Under these hypotheses, IðÞ is differentiable (in the usual sense) with respect to  near 0 , and its derivative may be obtained by ‘differentiation under the h; i sign’:

h@i T; ’i ¼ hT; @i ’i

for all ’ 2 D. This does define a distribution, because the partial differentiations ’ 7 ! @i ’ are continuous for the topology of D. Suppose that T ¼ Tf with f a locally integrable function such that @i f exists and is almost everywhere continuous. Then integration by parts along the xi axis gives R Rn

dI ¼ hT; @ ’ i: d (c) Effect of discontinuities When a function f or its derivatives are no longer continuous, the derivatives Dp Tf of the associated distribution Tf may no longer coincide with the distributions associated to the functions Dp f . In dimension 1, the simplest example is Heaviside’s unit step function Y ½YðxÞ ¼ 0 for x < 0; YðxÞ ¼ 1 for x  0:

@i f ðxl ; . . . ; xi ; . . . ; xn Þ’ðxl ; . . . ; xi ; . . . ; xn Þ dxi ¼ ð f ’Þðxl ; . . . ; þ1; . . . ; xn Þ  ð f ’Þðxl ; . . . ; 1; . . . ; xn Þ R  f ðxl ; . . . ; xi ; . . . ; xn Þ@i ’ðxl ; . . . ; xi ; . . . ; xn Þ dxi ; Rn

hðTY Þ0 ; ’i ¼ hðTY Þ; ’0 i ¼ 

the integrated term vanishes, since ’ has compact support, showing that @i Tf ¼ T@i f . The test functions ’ 2 D are infinitely differentiable. Therefore, transpositions like that used to define @i T may be repeated, so that any distribution is infinitely differentiable. For instance,

þ1 R

’0 ðxÞ dx ¼ ’ð0Þ ¼ h
; ’i:

0

Hence ðTY Þ0 ¼
, a result long used ‘heuristically’ by electrical engineers [see also Dirac (1958)].

31

1. GENERAL RELATIONSHIPS AND TECHNIQUES Let f be infinitely differentiable for x < 0 and x > 0 but have discontinuous derivatives f ðmÞ at x ¼ 0 [ f ð0Þ being f itself] with jumps m ¼ f ðmÞ ð0þÞ  f ðmÞ ð0Þ. Consider the functions:

0

h1; i h1;

i¼0

reflects the fact that has compact support. To specify T in the whole of D, it suffices to specify the value of hT; ’0 i where ’0 2 D is such that h1; ’0 i ¼ 1: then any ’ 2 D may be written uniquely as

g0 ¼ f  0 Y g1 ¼ g00  1 Y          gk ¼ g0k1  k Y:

’ ¼ ’0 þ

0

with

The gk are continuous, their derivatives g0k are continuous almost everywhere [which implies that ðTgk Þ0 ¼ Tg0k and g0k ¼ f ðkþ1Þ almost everywhere]. This yields immediately:

 ¼ ’  ’0 ;

 ¼ h1; ’i;

Rx ðxÞ ¼ ðtÞ dt; 0

ðTf Þ0 ¼ Tf 0 þ 0
ðTf Þ00 ¼ Tf 00 þ 0
0 þ 1


and T is defined by

                

hT; ’i ¼ hT; ’0 i  hS; i:

ðTf ÞðmÞ ¼ Tf ðmÞ þ 0
ðm1Þ þ . . . þ m1
:                 

The freedom in the choice of ’0 means that T is defined up to an additive constant.

Thus the ‘distributional derivatives’ ðTf ÞðmÞ differ from the usual functional derivatives Tf ðmÞ by singular terms associated with discontinuities. In dimension n, let f be infinitely differentiable everywhere except on a smooth hypersurface S, across which its partial derivatives show discontinuities. Let 0 and  denote the discontinuities of f and its normal derivative @ ’ across S (both 0 and  are functions of position on S), and let
ðSÞ and @
ðSÞ be defined by

1.3.2.3.9.3. Multiplication of distributions by functions The product T of a distribution T on Rn by a function  over n R will be defined by transposition: hT; ’i ¼ hT; ’i

for all ’ 2 D:

In order that T be a distribution, the mapping ’ 7 ! ’ must send DðRn Þ continuously into itself; hence the multipliers  must be infinitely differentiable. The product of two general distributions cannot be defined. The need for a careful treatment of multipliers of distributions will become clear when it is later shown (Section 1.3.2.5.8) that the Fourier transformation turns convolutions into multiplications and vice versa. If T is a distribution of order m, then  needs only have continuous derivatives up to order m. For instance,
is a distribution of order zero, and 
¼ ð0Þ
is a distribution provided  is continuous; this relation is of fundamental importance in the theory of sampling and of the properties of the Fourier transformation related to sampling (Sections 1.3.2.6.4, 1.3.2.6.6). More generally, Dp
is a distribution of order jpj, and the following formula holds for all  2 DðmÞ with m ¼ jpj:

R h
ðSÞ ; ’i ¼ ’ dn1 S S R h@
ðSÞ ; ’i ¼  @ ’ dn1 S: S

Integration by parts shows that @i Tf ¼ T@i f þ 0 cos i
ðSÞ ; where i is the angle between the xi axis and the normal to S along which the jump 0 occurs, and that the Laplacian of Tf is given by

ðDp
Þ ¼ ðTf Þ ¼ Tf þ 
ðSÞ þ @ ½0
ðSÞ :


 X p ðDpq Þð0ÞDq
: ð1Þjpqj q qp

The derivative of a product is easily shown to be

The latter result is a statement of Green’s theorem in terms of distributions. It will be used in Section 1.3.4.4.3.5 to calculate the Fourier transform of the indicator function of a molecular envelope.

@i ðTÞ ¼ ð@i ÞT þ ð@i TÞ and generally for any multi-index p

1.3.2.3.9.2. Integration of distributions in dimension 1 The reverse operation from differentiation, namely calculating the ‘indefinite integral’ of a distribution S, consists in finding a distribution T such that T 0 ¼ S. For all  2 D such that  ¼ 0 with 2 D, we must have

X
p  D ðTÞ ¼ ðDpq Þð0ÞDq T: q qp p

hT; i ¼ hS; i:

1.3.2.3.9.4. Division of distributions by functions Given a distribution S on Rn and an infinitely differentiable multiplier function , the division problem consists in finding a distribution T such that T ¼ S.

This condition defines T in a ‘hyperplane’ H of D, whose equation

32

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY If  never vanishes, T ¼ S= is the unique answer. If n ¼ 1, and if  has only isolated zeros of finite order, it can be reduced to a collection of cases where the multiplier is xm, for which the general solution can be shown to be of the form T¼Uþ

hA# T; ’i ¼ jdet AjhT; ðA1 Þ# ’i: This formula will be shown later (Sections 1.3.2.6.5, 1.3.4.2.1.1) to be the basis for the definition of the reciprocal lattice. In particular, if A ¼ I, where I is the identity matrix, A is an inversion through a centre of symmetry at the origin, and denoting A# ’ by ’ we have:

m1 P

ci
ðiÞ ;

i¼0

where U is a particular solution of the division problem xm U ¼ S and the ci are arbitrary constants. In dimension n > 1, the problem is much more difficult, but is of fundamental importance in the theory of linear partial differential equations, since the Fourier transformation turns the problem of solving these into a division problem for distributions [see Ho¨rmander (1963)].

hT ; ’i ¼ hT; ’ i: T is called an even distribution if T ¼ T, an odd distribution if T ¼ T. If A ¼ I with  > 0, A is called a dilation and hA# T; ’i ¼ n hT; ðA1 Þ# ’i:

1.3.2.3.9.5. Transformation of coordinates Let  be a smooth nonsingular change of variables in Rn , i.e. an infinitely differentiable mapping from an open subset  of Rn to 0 in Rn , whose Jacobian

Writing symbolically
as
ðxÞ and A#
as
ðx=Þ, we have:
ðx=Þ ¼ n
ðxÞ:

  @ðxÞ JðÞ ¼ det @x

If n ¼ 1 and f is a function with isolated simple zeros xj, then in the same symbolic notation

vanishes nowhere in . By the implicit function theorem, the inverse mapping  1 from 0 to  is well defined. If f is a locally summable function on , then the function  # f defined by


½ f ðxÞ ¼

X

1

j

j f 0 ðxj Þj


ðxj Þ;

ð # f ÞðxÞ ¼ f ½ 1 ðxÞ

where each j ¼ 1=j f 0 ðxj Þj is analogous to a ‘Lorentz factor’ at zero xj.

is a locally summable function on 0 , and for any ’ 2 Dð0 Þ we may write:

1.3.2.3.9.6. Tensor product of distributions The purpose of this construction is to extend Fubini’s theorem to distributions. Following Section 1.3.2.2.5, we may define the tensor product L1loc ðRm Þ  L1loc ðRn Þ as the vector space of finite linear combinations of functions of the form

R

ð # f ÞðxÞ’ðxÞ dn x ¼

0

R

f ½ 1 ðxÞ’ðxÞ dn x

0

¼

R

f ðyÞ’½ðyÞjJðÞj dn y

by x ¼ ðyÞ:

f  g : ðx; yÞ 7 ! f ðxÞgðyÞ;

0

where x 2 Rm ; y 2 Rn ; f 2 L1loc ðRm Þ and g 2 L1loc ðRn Þ. Let Sx and Ty denote the distributions associated to f and g, respectively, the subscripts x and y acting as mnemonics for Rm and Rn . It follows from Fubini’s theorem (Section 1.3.2.2.5) that f  g 2 L1loc ðRm
Rn Þ, and hence defines a distribution over Rm
Rn ; the rearrangement of integral signs gives

In terms of the associated distributions hT# f ; ’i ¼ hTf ; jJðÞjð 1 Þ# ’i: This operation can be extended to an arbitrary distribution T by defining its image  # T under coordinate transformation  through

hSx  Ty ; ’x; y i ¼ hSx ; hTy ; ’x; y ii ¼ hTy ; hSx ; ’x; y ii

h # T; ’i ¼ hT; jJðÞjð 1 Þ# ’i;

for all ’x; y 2 DðRm
Rn Þ. In particular, if ’ðx; yÞ ¼ uðxÞvðyÞ with u 2 DðRm Þ; v 2 DðRn Þ, then

which is well defined provided that  is proper, i.e. that  1 ðKÞ is compact whenever K is compact. For instance, if  : x 7 ! x þ a is a translation by a vector a in Rn , then jJðÞj ¼ 1;  # is denoted by a, and the translate a T of a distribution T is defined by

hS  T; u  vi ¼ hS; uihT; vi: This construction can be extended to general distributions S 2 D 0 ðRm Þ and T 2 D 0 ðRn Þ. Given any test function ’ 2 DðRm
Rn Þ, let ’x denote the map y 7 ! ’ðx; yÞ; let ’y denote the map x 7 ! ’ðx; yÞ; and define the two functions

ðxÞ ¼ hT; ’x i and !ðyÞ ¼ hS; ’y i. Then, by the lemma on differentiation under the h; i sign of Section 1.3.2.3.9.1,

2 DðRm Þ; ! 2 DðRn Þ, and there exists a unique distribution S  T such that

ha T; ’i ¼ hT; a ’i: Let A : x 7 ! Ax be a linear transformation defined by a nonsingular matrix A. Then JðAÞ ¼ det A, and

33

1. GENERAL RELATIONSHIPS AND TECHNIQUES The latter condition is met, in particular, if S or T has compact support. The support of S  T is easily seen to be contained in the closure of the vector sum

hS  T; ’i ¼ hS; i ¼ hT; !i: S  T is called the tensor product of S and T. With the mnemonic introduced above, this definition reads identically to that given above for distributions associated to locally integrable functions:

A þ B ¼ fx þ yjx 2 A; y 2 Bg: Convolution by a fixed distribution S is a continuous operation for the topology on D 0 : it maps convergent sequences ðTj Þ to convergent sequences ðS  Tj Þ. Convolution is commutative: S  T ¼ T  S. The convolution of p distributions T1 ; . . . ; Tp with supports A1 ; . . . ; Ap can be defined by

hSx  Ty ; ’x; y i ¼ hSx ; hTy ; ’x; y ii ¼ hTy ; hSx ; ’x; y ii: The tensor product of distributions is associative: ðR  SÞ  T ¼ R  ðS  TÞ:

hT1  . . .  Tp ; ’i ¼ hðT1 Þx1  . . .  ðTp Þxp ; ’ðx1 þ . . . þ xp Þi

Derivatives may be calculated by

whenever the following generalized support condition: ‘the set fðx1 ; . . . ; xp Þjx1 2 A1 ; . . . ; xp 2 Ap ; x1 þ . . . þ xp 2 Kg is compact in ðRn Þ p for all K compact in Rn ’

Dpx Dqy ðSx  Ty Þ ¼ ðDpx Sx Þ  ðDqy Ty Þ:

is satisfied. It is then associative. Interesting examples of associativity failure, which can be traced back to violations of the support condition, may be found in Bracewell (1986, pp. 436–437). It follows from previous definitions that, for all distributions T 2 D 0 , the following identities hold: (i)
 T ¼ T:
is the unit convolution; (ii)
ðaÞ  T ¼ a T: translation is a convolution with the corresponding translate of
; (iii) ðDp
Þ  T ¼ Dp T: differentiation is a convolution with the corresponding derivative of
; (iv) translates or derivatives of a convolution may be obtained by translating or differentiating any one of the factors: convolution ‘commutes’ with translation and differentiation, a property used in Section 1.3.4.4.7.7 to speed up least-squares model refinement for macromolecules. The latter property is frequently used for the purpose of regularization: if T is a distribution,  an infinitely differentiable function, and at least one of the two has compact support, then T   is an infinitely differentiable ordinary function. Since sequences ð Þ of such functions  can be constructed which have compact support and converge to
, it follows that any distribution T can be obtained as the limit of infinitely differentiable functions T  . In topological jargon: DðRn Þ is ‘everywhere dense’ in D 0 ðRn Þ. A standard function in D which is often used for such proofs is defined as follows: put

The support of a tensor product is the Cartesian product of the supports of the two factors. 1.3.2.3.9.7. Convolution of distributions The convolution f  g of two functions f and g on Rn is defined by ð f  gÞðxÞ ¼

R

f ðyÞgðx  yÞ dn y ¼

Rn

R

f ðx  yÞgðyÞ dn y

Rn

whenever the integral exists. This is the case when f and g are both in L1 ðRn Þ; then f  g is also in L1 ðRn Þ. Let S, T and W denote the distributions associated to f, g and f  g; respectively: a change of variable immediately shows that for any ’ 2 DðRn Þ, hW; ’i ¼

R

f ðxÞgðyÞ’ðx þ yÞ dn x dn y:

Rn
Rn

Introducing the map  from Rn
Rn to Rn defined by ðx; yÞ ¼ x þ y, the latter expression may be written: hSx  Ty ; ’ i


 1 1

ðxÞ ¼ exp  A 1  x2

(where denotes the composition of mappings) or by a slight abuse of notation:

¼0

hW; ’i ¼ hSx  Ty ; ’ðx þ yÞi: A difficulty arises in extending this definition to general distributions S and T because the mapping  is not proper: if K is compact in Rn , then  1 ðKÞ is a cylinder with base K and generator the ‘second bisector’ x þ y ¼ 0 in Rn
Rn . However, hS  T; ’ i is defined whenever the intersection between Supp ðS  TÞ ¼ ðSupp SÞ
ðSupp TÞ and  1 ðSupp ’Þ is compact. We may therefore define the convolution S  T of two distributions S and T on Rn by

for jxj  1; for jxj  1;

with Zþ1 A¼


exp 

 1 dx 1  x2

1

(so that is in D and is normalized), and put 1 x

" ðxÞ ¼

in dimension 1; " " n Y

" ðxÞ ¼

" ðxj Þ in dimension n:

hS  T; ’i ¼ hS  T; ’ i ¼ hSx  Ty ; ’ðx þ yÞi whenever the following support condition is fulfilled: ‘the set fðx; yÞjx 2 A; y 2 B; x þ y 2 Kg is compact in Rn
Rn for all K compact in Rn ’.

j¼1

34

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY Another related result, also proved by convolution, is the structure theorem: the restriction of a distribution T 2 D 0 ðRn Þ to a bounded open set  in Rn is a derivative of finite order of a continuous function. Properties (i) to (iv) are the basis of the symbolic or operational calculus (see Carslaw & Jaeger, 1948; Van der Pol & Bremmer, 1955; Churchill, 1958; Erde´lyi, 1962; Moore, 1971) for solving integro-differential equations with constant coefficients by turning them into convolution equations, then using factorization methods for convolution algebras (Schwartz, 1965).

F ½A# f  ¼ jdet Aj½ðA1 Þ # F ½ f  T

and similarly for F
. The matrix ðA1 ÞT is called the contragredient of matrix A. Under an affine change of coordinates x 7 ! SðxÞ ¼ Ax þ b with nonsingular matrix A, the transform of S# f is given by F ½S# f ðnÞ ¼ F ½b ðA# f ÞðnÞ

¼ expð2 in  bÞF ½A# f ðnÞ

1.3.2.4. Fourier transforms of functions 1.3.2.4.1. Introduction Given a complex-valued function f on Rn subject to suitable regularity conditions, its Fourier transform F ½ f  and Fourier cotransform F
½ f  are defined as follows: F ½ f ðÞ ¼

R

¼ expð2 in  bÞjdet AjF ½ f ðAT nÞ with a similar result for F
, replacing i by +i. 1.3.2.4.2.3. Conjugate symmetry The kernels of the Fourier transformations F and F
satisfy the following identities:

n

f ðxÞ expð2 in  xÞ d x

Rn

F
½ f ðÞ ¼

R

f ðxÞ expðþ2 in  xÞ dn x; expð 2 in  xÞ ¼ exp ½ 2 in  ðxÞ ¼ exp ½ 2 iðnÞ  x:

Rn

Pn where n  x ¼ i¼1 i xi is the ordinary scalar product. The terminology and sign conventions given above are the standard ones in mathematics; those used in crystallography are slightly different (see Section 1.3.4.2.1.1). These transforms enjoy a number of remarkable properties, whose natural settings entail different regularity assumptions on f : for instance, properties relating to convolution are best treated in L1 ðRn Þ, while Parseval’s theorem requires the Hilbert space structure of L2 ðRn Þ. After a brief review of these classical properties, the Fourier transformation will be examined in a space S ðRn Þ particularly well suited to accommodating the full range of its properties, which will later serve as a space of test functions to extend the Fourier transformation to distributions. There exists an abundant literature on the ‘Fourier integral’. The books by Carslaw (1930), Wiener (1933), Titchmarsh (1948), Katznelson (1968), Sneddon (1951, 1972), and Dym & McKean (1972) are particularly recommended.

As a result the transformations F and F
themselves have the following ‘conjugate symmetry’ properties [where the notation f ðxÞ ¼ f ðxÞ of Section 1.3.2.2.2 will be used]: 

F ½ f ðnÞ ¼ F ½
f ðnÞ ¼ F ½
f ðnÞ F ½ f ðnÞ ¼ F ½
f ðnÞ:

Therefore, (i) f real , f ¼ f
, F ½ f  ¼ F ½ f  , F ½ f ðnÞ ¼ F ½ f ðnÞ : F ½ f  is said to possess Hermitian symmetry; (ii) f centrosymmetric , f ¼ f , F ½ f  ¼ F ½
f ; (iii) f real centrosymmetric , f ¼ f
¼ f , F ½ f  ¼ F ½ f  ¼  F ½ f  , F ½ f  real centrosymmetric. Conjugate symmetry is the basis of Friedel’s law (Section 1.3.4.2.1.4) in crystallography.

1.3.2.4.2. Fourier transforms in L1 1.3.2.4.2.1. Linearity Both transformations F and F
are obviously linear maps from L1 to L1 when these spaces are viewed as vector spaces over the field C of complex numbers.

1.3.2.4.2.4. Tensor product property Another elementary property of F is its naturality with respect to tensor products. Let u 2 L1 ðRm Þ and v 2 L1 ðRn Þ, and let F x ; F y ; F x; y denote the Fourier transformations in L1 ðRm Þ; L1 ðRn Þ and L1 ðRm
Rn Þ, respectively. Then

1.3.2.4.2.2. Effect of affine coordinate transformations F and F
turn translations into phase shifts:

F x; y ½u  v ¼ F x ½u  F y ½v:

Furthermore, if f 2 L1 ðRm
Rn Þ, then F y ½ f  2 L1 ðRm Þ as a function of x and F x ½ f  2 L1 ðRn Þ as a function of y, and

F ½a f ðnÞ ¼ expð2 in  aÞF ½ f ðnÞ F
½a f ðnÞ ¼ expðþ2 in  aÞF
½ f ðnÞ:

F x; y ½ f  ¼ F x ½F y ½ f  ¼ F y ½F x ½ f :

Under a general linear change of variable x 7 ! Ax with nonsingular matrix A, the transform of A# f is F ½A# f ðnÞ ¼

R

This is easily proved by using Fubini’s theorem and the fact that ðn; gÞ  ðx; yÞ ¼ n  x þ g  y, where x; n 2 Rm, y; g 2 Rn . This property may be written:

f ðA1 xÞ expð2 in  xÞ dn x

Rn

¼

R

f ðyÞ expð2 iðAT nÞ  yÞjdet Aj dn y

F x; y ¼ F x  F y :

Rn

by x ¼ Ay T

¼ jdet AjF ½ f ðA nÞ

1.3.2.4.2.5. Convolution property If f and g are summable, their convolution f  g exists and is summable, and

i.e.

35

1. GENERAL RELATIONSHIPS AND TECHNIQUES F ½ f  gðnÞ ¼

 R R Rn

 f ðyÞgðx  yÞ dn y expð2 in  xÞ dn x:

k2 F ½ f k1  k f 0 k1

Rn

so that jF ½ f ðÞj decreases faster than 1=jj ! 1. This result can be easily extended to several dimensions and to any multi-index m: if f is summable and has continuous summable partial derivatives up to order jmj, then

With x ¼ y þ z, so that expð2 in  xÞ ¼ expð2 in  yÞ expð2 in  zÞ;

m

F ½Dm f ðnÞ ¼ ð2 inÞ F ½ f ðnÞ

and with Fubini’s theorem, rearrangement of the double integral gives:

and

F ½ f  g ¼ F ½ f 
F ½g

kð2 nÞm F ½ f k1  kDm f k1 :

and similarly

Similar results hold for F
, with 2 in replaced by 2 in. Thus, the more differentiable f is, with summable derivatives, the faster F ½ f  and F
½ f  decrease at infinity. The property of turning differentiation into multiplication by a monomial has many important applications in crystallography, for instance differential syntheses (Sections 1.3.4.2.1.9, 1.3.4.4.7.2, 1.3.4.4.7.5) and moment-generating functions [Section 1.3.4.5.2.1(c)].

F
½ f  g ¼ F
½ f 
F
½g:

Thus the Fourier transform and cotransform turn convolution into multiplication. 1.3.2.4.2.6. Reciprocity property In general, F ½ f  and F
½ f  are not summable, and hence cannot be further transformed; however, as they are essentially bounded, their products with the Gaussians Gt ðÞ ¼ expð2 2 kk2 tÞ are summable for all t > 0, and it can be shown that

1.3.2.4.2.9. Decrease at infinity Conversely, assume that f is summable on Rn and that f decreases fast enough at infinity for xm f also to be summable, for some multi-index m. Then the integral defining F ½ f  may be subjected to the differential operator Dm , still yielding a convergent integral: therefore Dm F ½ f  exists, and

f ¼ lim F
½Gt F ½ f  ¼ lim F ½Gt F
½ f ; t!0

t!0

Dm ðF ½ f ÞðnÞ ¼ F ½ð2 ixÞm f ðnÞ

where the limit is taken in the topology of the L1 norm k:k1 . Thus F and F
are (in a sense) mutually inverse, which justifies the common practice of calling F
the ‘inverse Fourier transformation’.

with the bound kDm ðF ½ f Þk1 ¼ kð2 xÞm f k1 :

1.3.2.4.2.7. Riemann–Lebesgue lemma If f 2 L1 ðRn Þ, i.e. is summable, then F ½ f  and F
½ f  exist and are continuous and essentially bounded:

Similar results hold for F
, with 2 ix replaced by 2 ix. Thus, the faster f decreases at infinity, the more F ½ f  and F
½ f  are differentiable, with bounded derivatives. This property is the converse of that described in Section 1.3.2.4.2.8, and their combination is fundamental in the definition of the function space S in Section 1.3.2.4.4.1, of tempered distributions in Section 1.3.2.5, and in the extension of the Fourier transformation to them.

kF ½ f k1 ¼ kF
½ f k1  k f k1 : In fact one has the much stronger property, whose statement constitutes the Riemann–Lebesgue lemma, that F ½ f ðnÞ and F
½ f ðnÞ both tend to zero as knk ! 1. 1.3.2.4.2.8. Differentiation Let us now suppose that n ¼ 1 and that f 2 L1 ðRÞ is differentiable with f 0 2 L1 ðRÞ. Integration by parts yields F ½ f 0 ðÞ ¼

1.3.2.4.2.10. The Paley–Wiener theorem An extreme case of the last instance occurs when f has compact support: then F ½ f  and F
½ f  are so regular that they may be analytically continued from Rn to Cn where they are entire functions, i.e. have no singularities at finite distance (Paley & Wiener, 1934). This is easily seen for F ½ f : giving vector n 2 Rn a vector g 2 Rn of imaginary parts leads to

þ1 R

f 0 ðxÞ expð2 i  xÞ dx

1

¼ ½ f ðxÞ expð2 i  xÞþ1 1 þ1 R þ 2 i f ðxÞ expð2 i  xÞ dx:

F ½ f ðn þ igÞ ¼

R

f ðxÞ exp½2 iðn þ igÞ  x dn x

Rn

1

¼ F ½expð2 g  xÞf ðnÞ;

Since f 0 is summable, f has a limit when x ! 1, and this limit must be 0 since f is summable. Therefore

where the latter transform always exists since expð2 g  xÞf is summable with respect to x for all values of g. This analytic continuation forms the basis of the saddlepoint method in probability theory [Section 1.3.4.5.2.1( f )] and leads to the use of maximum-entropy distributions in the statistical theory of direct phase determination [Section 1.3.4.5.2.2(e)].

F ½ f 0 ðÞ ¼ ð2 iÞF ½ f ðÞ

with the bound

36

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY By Liouville’s theorem, an entire function in Cn cannot vanish identically on the complement of a compact subset of Rn without vanishing everywhere: therefore F ½ f  cannot have compact support if f has, and hence DðRn Þ is not stable by Fourier transformation.

ð f  gÞðxÞ ¼

R

f ðx  yÞgðyÞ dn y ¼

Rn

R f
ðy  xÞgðyÞ dn y; Rn

i.e.

2

1.3.2.4.3. Fourier transforms in L Let f belong to L2 ðRn Þ, i.e. be such that
k f k2 ¼

R

ð f  gÞðxÞ ¼ ðx f
 ; gÞ:

1=2 j f ðxÞj2 dn x < 1:

Invoking the isometry property, we may rewrite the right-hand side as

Rn

ðF ½x f
; F ½gÞ ¼ ðexpð2 ix  nÞF ½ f n ; F ½gn Þ R ¼ ðF ½ f 
F ½gÞðxÞ

2

1.3.2.4.3.1. Invariance of L F ½ f  and F
½ f  exist and are functions in L2 , i.e. F L2 ¼ L2 ,
F L2 ¼ L2 .

Rn


expðþ2 ix  nÞ dn n ¼ F
½F ½ f 
F ½g;

1.3.2.4.3.2. Reciprocity F ½F
½ f  ¼ f and F
½F ½ f  ¼ f , equality being taken as ‘almost everywhere’ equality. This again leads to calling F
the ‘inverse

so that the initial identity yields the convolution theorem. To obtain the converse implication, note that

Fourier transformation’ rather than the Fourier cotransformation.

ð f ; gÞ ¼

1.3.2.4.3.3. Isometry F and F
preserve the L2 norm:

R

f ðyÞgðyÞ dn y ¼ ð f
 gÞð0Þ

Rn

¼ F
½F ½ f
 
F ½gð0Þ R ¼ F ½ f ðnÞF ½gðnÞ dn n ¼ ðF ½ f ; F ½gÞ;

kF ½ f k2 ¼ kF
½ f k2 ¼ k f k2 (Parseval’s/Plancherel’s theorem):

Rn

This property, which may be written in terms of the inner product (,) in L2 ðRn Þ as

where conjugate symmetry (Section 1.3.2.4.2.2) has been used. These relations have an important application in the calculation by Fourier transform methods of the derivatives used in the refinement of macromolecular structures (Section 1.3.4.4.7).

ðF ½ f ; F ½gÞ ¼ ðF
½ f ; F
½gÞ ¼ ð f ; gÞ; implies that F and F
are unitary transformations of L2 ðRn Þ into itself, i.e. infinite-dimensional ‘rotations’.

1.3.2.4.4. Fourier transforms in S 1.3.2.4.4.1. Definition and properties of S The duality established in Sections 1.3.2.4.2.8 and 1.3.2.4.2.9 between the local differentiability of a function and the rate of decrease at infinity of its Fourier transform prompts one to consider the space S ðRn Þ of functions f on Rn which are infinitely differentiable and all of whose derivatives are rapidly decreasing, so that for all multi-indices k and p

1.3.2.4.3.4. Eigenspace decomposition of L2 Some light can be shed on the geometric structure of these rotations by the following simple considerations. Note that F 2 ½ f ðxÞ ¼

R

F ½ f ðnÞ expð2 ix  nÞ dn n

Rn

¼ F
½F ½ f ðxÞ ¼ f ðxÞ

ðxk Dp f ÞðxÞ ! 0 as kxk ! 1:

so that F 4 (and similarly F
4 ) is the identity map. Any eigenvalue of F or F
is therefore a fourth root of unity, i.e. 1 or i, and L2 ðRn Þ splits into an orthogonal direct sum

The product of f 2 S by any polynomial over Rn is still in S (S is an algebra over the ring of polynomials). Furthermore, S is invariant under translations and differentiation. If f 2 S , then its transforms F ½ f  and F
½ f  are (i) infinitely differentiable because f is rapidly decreasing; (ii) rapidly decreasing because f is infinitely differentiable; hence F ½ f  and F
½ f  are in S : S is invariant under F and F
. Since L1  S and L2  S , all properties of F and F
already encountered above are enjoyed by functions of S , with all restrictions on differentiability and/or integrability lifted. For instance, given two functions f and g in S , then both fg and f  g are in S (which was not the case with L1 nor with L2 ) so that the reciprocity theorem inherited from L2

H0  H1  H2  H3 ; where F (respectively F
) acts in each subspace Hk ðk ¼ 0; 1; 2; 3Þ by multiplication by ðiÞk. Orthonormal bases for these subspaces can be constructed from Hermite functions (cf. Section 1.3.2.4.4.2) This method was used by Wiener (1933, pp. 51–71). 1.3.2.4.3.5. The convolution theorem and the isometry property In L2 , the convolution theorem (when applicable) and the Parseval/Plancherel theorem are not independent. Suppose that f, g, f
g and f  g are all in L2 (without questioning whether these properties are independent). Then f  g may be written in terms of the inner product in L2 as follows:

F ½F
½ f  ¼ f

and

F
½F ½ f  ¼ f

allows one to state the reverse of the convolution theorem first established in L1 :

37

1. GENERAL RELATIONSHIPS AND TECHNIQUES F ½ fg ¼ F ½ f   F ½g

F ½DHm ðÞ ¼ ð2 inÞF ½Hm ðÞ

F
½ fg ¼ F
½ f   F
½g:

F ½2 xHm ðÞ ¼ iDðF ½Hm ÞðÞ:

Combination of this with the induction hypothesis yields 1.3.2.4.4.2. Gaussian functions and Hermite functions Gaussian functions are particularly important elements of S . In dimension 1, a well known contour integration (Schwartz, 1965, p. 184) yields F ½expð x2 ÞðÞ ¼ F
½expð x2 ÞðÞ ¼ expð 2 Þ;

F ½Hmþ1 ðÞ ¼ ðiÞ

mþ1

½ðDHm ÞðÞ  2 Hm ðÞ

¼ ðiÞ

mþ1

Hmþ1 ðÞ;

thus proving that Hm is an eigenfunction of F for eigenvalue ðiÞm for all m  0. The same proof holds for F
, with eigenvalue im . If these eigenfunctions are normalized as

which shows that the ‘standard Gaussian’ expð x2 Þ is invariant under F and F
. By a tensor product construction, it follows that the same is true of the standard Gaussian

ð1Þm 21=4 Hm ðxÞ; m!2m m=2

H m ðxÞ ¼ pffiffiffiffiffiffi

GðxÞ ¼ expð kxk2 Þ then it can be shown that the collection of Hermite functions fH m ðxÞgm0 constitutes an orthonormal basis of L2 ðRÞ such that H m is an eigenfunction of F (respectively F
) for eigenvalue ðiÞm (respectively im ). In dimension n, the same construction can be extended by tensor product to yield the multivariate Hermite functions

in dimension n: F ½GðnÞ ¼ F
½GðnÞ ¼ GðnÞ:

In other words, G is an eigenfunction of F and F
for eigenvalue 1 (Section 1.3.2.4.3.4). A complete system of eigenfunctions may be constructed as follows. In dimension 1, consider the family of functions Hm ¼

Dm G2 G

H m ðxÞ ¼ H m1 ðx1 Þ
H m2 ðx2 Þ
. . .
H mn ðxn Þ

(where m  0 is a multi-index). These constitute an orthonormal basis of L2 ðRn Þ, with H m an eigenfunction of F (respectively F
) for eigenvalue ðiÞjmj (respectively ijmj ). Thus the subspaces Hk of Section 1.3.2.4.3.4 are spanned by those H m with jmj k mod 4 ðk ¼ 0; 1; 2; 3Þ. General multivariate Gaussians are usually encountered in the nonstandard form

ðm  0Þ;

where D denotes the differentiation operator. The first two members of the family H0 ¼ G;

GA ðxÞ ¼ expð12xT  AxÞ;

H1 ¼ 2DG;

where A is a symmetric positive-definite matrix. Diagonalizing A as EKET with EET the identity matrix, and putting A1=2 ¼ EK1=2 ET , we may write

are such that F ½H0  ¼ H0 , as shown above, and DGðxÞ ¼ 2 xGðxÞ ¼ ið2 ixÞGðxÞ ¼ iF ½DGðxÞ;

"
hence

GA ðxÞ ¼ G

A 2

1=2 # x

F ½H1  ¼ ðiÞH1 :

i.e. We may thus take as an induction hypothesis that GA ¼ ½ð2 A1 Þ1=2 # G;

m

F ½Hm  ¼ ðiÞ Hm :

hence (by Section 1.3.2.4.2.3) The identity
D

m

2

D G G

 ¼

D

mþ1

G

G

2

m



DG D G G G

2

1

F ½GA  ¼ jdetð2 A Þj

1=2

"


1=2 ## A G; 2

i.e.

may be written

F ½GA ðnÞ ¼ jdetð2 A1 Þj1=2 G½ð2 A1 Þ

Hmþ1 ðxÞ ¼ ðDHm ÞðxÞ  2 xHm ðxÞ; and the two differentiation theorems give:

i.e. finally

38

1=2

n;

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY Z 1 F ½GA  ¼ j detð2 A1 Þj1=2 G4 2 A1 : f ðxÞ ¼ n F h; ! ½ f ðnÞ expðþi!n  xÞ dn x k Rn

This result is widely used in crystallography, e.g. to calculate form factors for anisotropic atoms (Section 1.3.4.2.2.6) and to obtain transforms of derivatives of Gaussian atomic densities (Section 1.3.4.4.7.10).

with k real positive. The consistency condition between h, k and ! is

1.3.2.4.4.3. Heisenberg’s inequality, Hardy’s theorem The result just obtained, which also holds for F
, shows that the ‘peakier’ GA , the ‘broader’ F ½GA . This is a general property of the Fourier transformation, expressed in dimension 1 by the Heisenberg inequality (Weyl, 1931):
Z

hk ¼

The usual choices are: ðiÞ ! ¼ 2 ; h ¼ k ¼ 1 ðas hereÞ; ðiiÞ ! ¼ 1; h ¼ 1; k ¼ 2 ðin probability theory


Z  2 2 x j f ðxÞj dx  jF ½ f ðÞj d 2

2

1  16 2


Z

2

j f ðxÞj dx

2 : j!j

2

pffiffiffiffiffiffi ðiiiÞ ! ¼ 1; h ¼ k ¼ 2

;

and in solid-state physicsÞ; ðin much of classical analysisÞ:

It should be noted that conventions (ii) and (iii) introduce numerical factors of 2 in convolution and Parseval formulae, while (ii) breaks the symmetry between F and F
.

where, by a beautiful theorem of Hardy (1933), equality can only be attained for f Gaussian. Hardy’s theorem is even stronger: if both f and F ½ f  behave at infinity as constant multiples of G, then each of them is everywhere a constant multiple of G; if both f and F ½ f  behave at infinity as constant multiples of G
monomial, then each of them is a finite linear combination of Hermite functions. Hardy’s theorem is invoked in Section 1.3.4.4.5 to derive the optimal procedure for spreading atoms on a sampling grid in order to obtain the most accurate structure factors. The search for optimal compromises between the confinement of f to a compact domain in x-space and of F ½ f  to a compact domain in -space leads to consideration of prolate spheroidal wavefunctions (Pollack & Slepian, 1961; Landau & Pollack, 1961, 1962).

1.3.2.4.6. Tables of Fourier transforms The books by Campbell & Foster (1948), Erde´lyi (1954) and Magnus et al. (1966) contain extensive tables listing pairs of functions and their Fourier transforms. Bracewell (1986) lists those pairs particularly relevant to electrical engineering applications. 1.3.2.5. Fourier transforms of tempered distributions 1.3.2.5.1. Introduction It was found in Section 1.3.2.4.2 that the usual space of test functions D is not invariant under F and F
. By contrast, the space S of infinitely differentiable rapidly decreasing functions is invariant under F and F
, and furthermore transposition formulae such as

1.3.2.4.4.4. Symmetry property A final formal property of the Fourier transform, best established in S , is its symmetry: if f and g are in S , then by Fubini’s theorem

hF ½ f ; gi ¼ h f ; F ½gi


 R R hF ½ f ; gi ¼ f ðxÞ expð2 in  xÞ dn x gðnÞ dn n Rn Rn
 R R ¼ f ðxÞ gðnÞ expð2 in  xÞ dn n dn x Rn

hold for all f ; g 2 S . It is precisely this type of transposition which was used successfully in Sections 1.3.2.3.9.1 and 1.3.2.3.9.3 to define the derivatives of distributions and their products with smooth functions. This suggests using S instead of D as a space of test functions ’, and defining the Fourier transform F ½T of a distribution T by

Rn

¼ hf ; F ½gi: This possibility of ‘transposing’ F (and F
) from the left to the right of the duality bracket will be used in Section 1.3.2.5.4 to extend the Fourier transformation to distributions.

hF ½T; ’i ¼ hT; F ½’i

1.3.2.4.5. Various writings of Fourier transforms Other ways of writing Fourier transforms in Rn exist besides the one used here. All have the form

F h; ! ½ f ðnÞ ¼

1 hn

Z

whenever T is capable of being extended from D to S while remaining continuous. It is this latter proviso which will be subsumed under the adjective ‘tempered’. As was the case with the construction of D 0, it is the definition of a sufficiently strong topology (i.e. notion of convergence) in S which will play a key role in transferring to the elements of its topological dual S 0 (called tempered distributions) all the properties of the Fourier transformation. Besides the general references to distribution theory mentioned in Section 1.3.2.3.1 the reader may consult the books by Zemanian (1965, 1968). Lavoine (1963) contains tables of Fourier transforms of distributions.

f ðxÞ expði!n  xÞ dn x;

Rn

where h is real positive and ! real nonzero, with the reciprocity formula written:

39

1. GENERAL RELATIONSHIPS AND TECHNIQUES arguments for ’ and F ½’, respectively, the notation Tx and F ½Tn will be used to indicate which variables are involved. When T is a distribution with compact support, its Fourier transform may be written

1.3.2.5.2. S as a test-function space A notion of convergence has to be introduced in S ðRn Þ in order to be able to define and test the continuity of linear functionals on it. A sequence ð’j Þ of functions in S will be said to converge to 0 if, for any given multi-indices k and p, the sequence ðxk Dp ’j Þ tends to 0 uniformly on Rn . It can be shown that DðRn Þ is dense in S ðRn Þ. Translation is continuous P for this topology. For any linear differential operator PðDÞ ¼ p ap Dp and any polynomial QðxÞ over Rn , ð’j Þ ! 0 implies ½QðxÞ
PðDÞ’j  ! 0 in the topology of S . Therefore, differentiation and multiplication by polynomials are continuous for the topology on S . The Fourier transformations F and F
are also continuous for the topology of S . Indeed, let ð’j Þ converge to 0 for the topology on S . Then, by Section 1.3.2.4.2,

F ½Tx n ¼ hTx ; expð2 in  xÞi

since the function x 7 ! expð2 in  xÞ is in E while Tx 2 E 0. It can be shown, as in Section 1.3.2.4.2, to be analytically continuable into an entire function over Cn . 1.3.2.5.5. Transposition of basic properties The duality between differentiation and multiplication by a monomial extends from S to S 0 by transposition: p

m

p

m

F ½Dpx Tx n ¼ ð2 inÞ F ½Tx n

p

kð2 nÞ D ðF ½’j Þk1  kD ½ð2 xÞ ’j k1 :

Dpn ðF ½Tx n Þ ¼ F ½ð2 ixÞp Tx n :

The right-hand side tends to 0 as j ! 1 by definition of convergence in S , hence knkm Dp ðF ½’j Þ ! 0 uniformly, so that ðF ½’j Þ ! 0 in S as j ! 1. The same proof applies to F
.

Analogous formulae hold for F
, with i replaced by i. The formulae expressing the duality between translation and phase shift, e.g.

1.3.2.5.3. Definition and examples of tempered distributions A distribution T 2 D 0 ðRn Þ is said to be tempered if it can be extended into a continuous linear functional on S . If S 0 ðRn Þ is the topological dual of S ðRn Þ, and if S 2 S 0 ðRn Þ, then its restriction to D is a tempered distribution; conversely, if T 2 D 0 is tempered, then its extension to S is unique (because D is dense in S ), hence it defines an element S of S 0. We may therefore identify S 0 and the space of tempered distributions. A distribution with compact support is tempered, i.e. S 0  E 0 . By transposition of the corresponding properties of S , it is readily established that the derivative, translate or product by a polynomial of a tempered distribution is still a tempered distribution. These inclusion relations may be summarized as follows: since S contains D but is contained in E , the reverse inclusions hold for the topological duals, and hence S 0 contains E 0 but is contained in D 0 . A locally summable function f on Rn will be said to be of polynomial growth if j f ðxÞj can be majorized by a polynomial in kxk as kxk ! 1. It is easily shown that such a function f defines a tempered distribution Tf via hTf ; ’i ¼

R

F ½a Tx n ¼ expð2 ia  nÞF ½Tx n

a ðF ½Tx n Þ ¼ F ½expð2 ia  xÞTx n ; between a linear change of variable and its contragredient, e.g. F ½A# T ¼ jdet Aj½ðA1 Þ # F ½T; T

are obtained similarly by transposition from the corresponding identities in S . They give a transposition formula for an affine change of variables x 7 ! SðxÞ ¼ Ax þ b with nonsingular matrix A: F ½S# T ¼ expð2 in  bÞF ½A# T

¼ expð2 in  bÞjdet Aj½ðA1 ÞT # F ½T; with a similar result for F
, replacing i by +i. Conjugate symmetry is obtained similarly: 

F ½T
 ¼ F ½T; ½T F ½T
 ¼ F ½T;

f ðxÞ’ðxÞ dn x:

Rn

with the same identities for F
. The tensor product property also transposes to tempered distributions: if U 2 S 0 ðRm Þ; V 2 S 0 ðRn Þ,

In particular, polynomials over Rn define tempered distributions, and so do functions in S . The latter remark, together with the transposition identity (Section 1.3.2.4.4), invites the extension of F and F
from S to S 0.

F ½Ux  Vy  ¼ F ½Un  F ½Vg F
½Ux  Vy  ¼ F
½Un  F
½Vg :

1.3.2.5.4. Fourier transforms of tempered distributions The Fourier transform F ½T and cotransform F
½T of a tempered distribution T are defined by

1.3.2.5.6. Transforms of
-functions Since
has compact support,

hF ½T; ’i ¼ hT; F ½’i hF
½T; ’i ¼ hT; F
½’i

F ½
x n ¼ h
x ; expð2 in  xÞi ¼ 1n ;

for all test functions ’ 2 S . Both F ½T and F
½T are themselves tempered distributions, since the maps ’ 7 ! F ½’ and ’ 7 ! F
½’ are both linear and continuous for the topology of S . In the same way that x and n have been used consistently as

i:e: F ½
 ¼ 1:

It is instructive to show that conversely F ½1 ¼
without invoking the reciprocity theorem. Since @j 1 ¼ 0 for all j ¼ 1; . . . ; n, it follows from Section 1.3.2.3.9.4 that F ½1 ¼ c
; the

40

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY S , then the associated distribution Tf is in O C0 ; and that conversely if T is in O C0 ;  T is in S for all 2 D. The two spaces O M and O C0 are mapped into each other by the

constant c can be determined by using the invariance of the standard Gaussian G established in Section 1.3.2.4.3:

Fourier transformation

hF ½1x ; Gx i ¼ h1n ; Gn i ¼ 1;

F ðO M Þ ¼ F
ðO M Þ ¼ O C0

hence c ¼ 1. Thus, F ½1 ¼
. The basic properties above then read (using multi-indices to denote differentiation): m

F ½
ðmÞ x n ¼ ð2 inÞ ; F ½
a n ¼ expð2 ia  nÞ;

F ½xm n ¼ ð2 iÞ

F ðO C0 Þ ¼ F
ðO C0 Þ ¼ O M

and the convolution theorem takes the form

jmj ðmÞ
n ;

F ½expð2 ia  xÞn ¼
a ;

F ½S ¼ F ½  F ½S F ½S  T ¼ F ½S
F ½T

with analogous relations for F
, i becoming i. Thus derivatives of
are mapped to monomials (and vice versa), while translates of
are mapped to ‘phase factors’ (and vice versa).

R

F ½’ðnÞ expð2 in  xÞ dn n

Rn

¼

R

Rn

S 2 S 0 ; T 2 O C0 ; F ½T 2 O M :

The same identities hold for F
. Taken together with the reciprocity theorem, these show that F and F
establish mutually inverse isomorphisms between O M and O C0 , and exchange multiplication for convolution in S 0 . It may be noticed that most of the basic properties of F and F
may be deduced from this theorem and from the properties of
. Differentiation operators Dm and translation operators a are convolutions with Dm
and a
; they are turned, respectively, into multiplication by monomials ð 2 inÞm (the transforms of Dm
) or by phase factors expð 2 in  aÞ (the transforms of a
). Another consequence of the convolution theorem is the duality established by the Fourier transformation between sections and projections of a function and its transform. For instance, in R3 , the projection of f ðx; y; zÞ on the x, y plane along the z axis may be written

1.3.2.5.7. Reciprocity theorem The previous results now allow a self-contained and rigorous proof of the reciprocity theorem between F and F
to be given, whereas in traditional settings (i.e. in L1 and L2 ) the implicit handling of
through a limiting process is always the sticking point. Reciprocity is first established in S as follows: F
½F ½’ðxÞ ¼

S 2 S 0 ;  2 O M ; F ½ 2 O C0 ;

F ½x ’ðnÞ dn n

¼ h1; F ½x ’i ¼ hF ½1; x ’i

ð
x 
y  1z Þ  f ;

¼ hx
; ’i ¼ ’ðxÞ

its Fourier transform is then ð1  1 
 Þ
F ½ f ;

and similarly F ½F
½’ðxÞ ¼ ’ðxÞ:

which is the section of F ½ f  by the plane  ¼ 0, orthogonal to the z axis used for projection. There are numerous applications of this property in crystallography (Section 1.3.4.2.1.8) and in fibre diffraction (Section 1.3.4.5.1.3).

The reciprocity theorem is then proved in S 0 by transposition: F
½F ½T ¼ F ½F
½T ¼ T

for all T 2 S 0 :

1.3.2.5.9. L2 aspects, Sobolev spaces The special properties of F in the space of square-integrable functions L2 ðRn Þ, such as Parseval’s identity, can be accommodated within distribution theory: if u 2 L2 ðRn Þ, then Tu is a tempered distribution in S 0 (the map u 7 ! Tu being continuous) and it can be shown that S ¼ F ½Tu  is of the form Sv , where u ¼ F ½u is the Fourier transform of u in L2 ðRn Þ. By Plancherel’s theorem, kuk2 ¼ kvk2 . This embedding of L2 into S 0 can be used to derive the convolution theorem for L2 . If u and v are in L2 ðRn Þ, then u  v can be shown to be a bounded continuous function; thus u  v is not in L2 , but it is in S 0 , so that its Fourier transform is a distribution, and

Thus the Fourier cotransformation F
in S 0 may legitimately be called the ‘inverse Fourier transformation’. The method of Section 1.3.2.4.3 may then be used to show that F and F
both have period 4 in S 0 . 1.3.2.5.8. Multiplication and convolution Multiplier functions ðxÞ for tempered distributions must be infinitely differentiable, as for ordinary distributions; furthermore, they must grow sufficiently slowly as kxk ! 1 to ensure that ’ 2 S for all ’ 2 S and that the map ’ 7 ! ’ is continuous for the topology of S . This leads to choosing for multipliers the subspace O M consisting of functions  2 E of polynomial growth. It can be shown that if f is in O M , then the associated distribution Tf is in S 0 (i.e. is a tempered distribution); and that conversely if T is in S 0 ;  T is in O M for all 2 D. Corresponding restrictions must be imposed to define the space O C0 of those distributions T whose convolution S  T with a tempered distribution S is still a tempered distribution: T must be such that, for all ’ 2 S ; ðxÞ ¼ hTy ; ’ðx þ yÞi is in S ; and such that the map ’ 7 ! be continuous for the topology of S . This implies that S is ‘rapidly decreasing’. It can be shown that if f is in

F ½u  v ¼ F ½u
F ½v:

Spaces of tempered distributions related to L2 ðRn Þ can be defined as follows. For any real s, define the Sobolev space Hs ðRn Þ to consist of all tempered distributions S 2 S 0 ðRn Þ such that ð1 þ jnj2 Þs=2 F ½Sn 2 L2 ðRn Þ:

41

1. GENERAL RELATIONSHIPS AND TECHNIQUES periodic. Periodic functions over Rn may thus be identified with functions over Rn =Zn , and this identification preserves the notions of convergence, local summability and differentiability. Given ’0 2 DðRn Þ, we may define

These spaces play a fundamental role in the theory of partial differential equations, and in the mathematical theory of tomographic reconstruction – a subject not unrelated to the crystallographic phase problem (Natterer, 1986). 1.3.2.6. Periodic distributions and Fourier series 1.3.2.6.1. Terminology Let Zn be the subset of Rn consisting of those points with (signed) integer coordinates; it is an n-dimensional lattice, i.e. a free Abelian group on n generators. A particularly simple set of n generators is given by the standard basis of Rn, and hence Zn will be called the standard lattice in Rn. Any other ‘nonstandard’ n-dimensional lattice  in Rn is the image of this standard lattice by a general linear transformation. If we identify any two points in Rn whose coordinates are congruent modulo Zn, i.e. differ by a vector in Zn , we obtain the standard n-torus Rn =Zn . The latter may be viewed as ðR=ZÞn, i.e. as the Cartesian product of n circles. The same identification may be carried out modulo a nonstandard lattice , yielding a nonstandard n-torus Rn =. The correspondence to crystallographic terminology is that ‘standard’ coordinates over the standard 3-torus R3 =Z3 are called ‘fractional’ coordinates over the unit cell; while Cartesian coordinates, e.g. in a˚ngstro¨ms, constitute a set of nonstandard coordinates. Finally, we will denote by I the unit cube ½0; 1n and by C" the subset

’ðxÞ ¼

m2Zn

1.3.2.6.4. Fourier transforms of periodic distributions The content of this section is perhaps the central result in the relation between Fourier theory and crystallography (Section 1.3.4.2.1.1). Let T ¼ r  T 0 with r defined as in Section 1.3.2.6.2. Then r 2 S 0 , T 0 2 E 0 hence T 0 2 O C0 , so that T 2 S 0 : Zn -periodic distributions are tempered, hence have a Fourier transform. The convolution theorem (Section 1.3.2.5.8) is applicable, giving: F ½T ¼ F ½r
F ½T 0 

1.3.2.6.2. Zn -periodic distributions in Rn A distribution T 2 D 0 ðRn Þ is called periodic with period lattice n Z (or Zn-periodic) if m T ¼ T for all m 2 Zn (in crystallography the period lattice is the direct lattice). Given with compact support T 0 2 E 0 ðRn Þ, then P a distribution n 0 T ¼ m2Zn m T is a Z -periodic P distribution. Note that we may write T ¼ r  T 0, where r ¼ m2Zn
ðmÞ consists of Dirac
’s at all nodes of the period lattice Zn. Conversely, any Zn -periodic distribution T may be written as r  T 0 for some T 0 2 E 0. To retrieve such a ‘motif’ T 0 from T, a function will be constructed in such a way that 2D (hence has compact support) and r  ¼ 1; then T 0 ¼ T. Indicator functions (Section 1.3.2.2) such as 1 or C1=2 cannot be used directly, since they are discontinuous; but regularized versions of them may be constructed by convolution (see Section 1.3.2.3.9.7) as 0 ¼ C" 

, with " and such that 0 ðxÞ ¼ 1 on C1=2 and 0 ðxÞ ¼ 0 outside C3=4. Then the function

and similarly for F
. Since F ½
ðmÞ ðÞ ¼ expð2 in  mÞ, formally F ½rn ¼

m

P

expð2 in  mÞ ¼ Q;

m2Zn

say. It is Preadily shown that Q is tempered and periodic, so that Q ¼ l2Zn l ð QÞ, while the periodicity of r implies that ½expð2 ij Þ  1 Q ¼ 0;

j ¼ 1; . . . ; n:

Since the first factors have single isolated zeros at j ¼ 0 in C3=4 , Q ¼ c
(see Section 1.3.2.3.9.4) and hence by periodicity Q ¼ cr; convoluting with C1 shows that c ¼ 1. Thus we have the fundamental result:

0 m2Zn

ðm ’0 ÞðxÞ

since the sum only contains finitely many nonzero terms; ’ is periodic, and ’~ 2 DðRn =Zn Þ. Conversely, if ’~ 2 DðRn =Zn Þ we may define ’ 2 E ðRn Þ periodic by ’ðxÞ ¼ ’~ ð~xÞ, and ’0 2 DðRn Þ by putting ’0 ¼ ’ with constructed as above. By transposition, a distribution T~ 2 D 0 ðRn =Zn Þ defines a unique periodic distribution T 2 D 0 ðRn Þ by hT; ’0 i ¼ hT~ ; ’~ i; conversely, T 2 D 0 ðRn Þ periodic defines uniquely T~ 2 D 0 ðRn =Zn Þ by hT~ ; ’~ i ¼ hT; ’0 i. We may therefore identify Zn -periodic distributions over Rn with distributions over Rn =Zn . We will, however, use mostly the former presentation, as it is more closely related to the crystallographer’s perception of periodicity (see Section 1.3.4.1).

C" ¼ fx 2 Rn kxj j < " for all j ¼ 1; . . . ; ng:

¼P

P

0

has the desired property. The sum in the denominator contains at most 2n nonzero terms at any given point x and acts as a smoothly varying ‘multiplicity correction’.

so that F ½T ¼ r
F ½T 0 ;

1.3.2.6.3. Identification with distributions over R =Z Throughout this section, ‘periodic’ will mean ‘Zn -periodic’. Let s 2 R, and let [s] denote the largest integer  s. For x ¼ ðx1 ; . . . ; xn Þ 2 Rn , let x~ be the unique vector ð~x1 ; . . . ; x~ n Þ with x~ j ¼ xj  ½xj . If x; y 2 Rn, then x~ ¼ y~ if and only if x  y 2 Zn. The image of the map x 7 ! x~ is thus Rn modulo Zn, or Rn =Zn . If f is a periodic function over Rn , then x~ ¼ y~ implies f ðxÞ ¼ f ðyÞ; we may thus define a function f~ over Rn =Zn by putting f~ ð~xÞ ¼ f ðxÞ for any x 2 Rn such that x  x~ 2 Zn . Conversely, if f~ is a function over Rn =Zn , then we may define a function f over Rn by putting f ðxÞ ¼ f~ ð~xÞ, and f will be n

n

i.e., according to Section 1.3.2.3.9.3, F ½Tn ¼

P l2Zn

F ½T 0 ðlÞ

ðlÞ :

The right-hand side is a weighted lattice distribution, whose nodes l 2 Zn are weighted by the sample values F ½T 0 ðlÞ of the transform of the motif T 0 at those nodes. Since T 0 2 E 0, the latter values may be written

42

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY F ½T 0 ðlÞ ¼ hTx0 ; expð2 il  xÞi:

R ¼ j det Aj½ðA1 ÞT # r: R is a lattice distribution:

By the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), T 0 is a derivative of finite order of a continuous function; therefore, from Section 1.3.2.4.2.8 and Section 1.3.2.5.8, F ½T 0 ðlÞ grows at most polynomially as klk ! 1 (see also P Section 1.3.2.6.10.3 about this property). Conversely, let W ¼ l2Zn wl
ðlÞ be a weighted lattice distribution such that the weights wl grow at most polynomially as klk ! 1. Then W P is a tempered distribution, whose Fourier cotransform Tx ¼ l2Zn wl expðþ2 il  xÞ is periodic. If T is now written as r  T 0 for some T 0 2 E 0, then by the reciprocity theorem

R ¼

l2Zn

Although the choice of T 0 is not unique, and need not yield back the same motif as may have been used to build T initially, different choices of T 0 will lead to the same coefficients wl because of the periodicity of expð2 il  xÞ. The Fourier transformation thus establishes a duality between periodic distributions and weighted lattice distributions. The pair of relations

n2

P

¼ jdet Aj1

l2Zn

n2


ðnÞ

F ½T 0 ðnÞ
ðnÞ T

F ½T 0 ½ðA1 Þ l
½ðA1 ÞT l

so that F ½T is a weighted reciprocal-lattice distribution, the weight attached to node n 2  being jdet Aj1 times the value F ½T 0 ðnÞ of the Fourier transform of the motif T 0. This result may be further simplified if T and its motif T 0 are referred to the standard period lattice Zn by defining t and t0 so that T ¼ A# t, T 0 ¼ A# t0 , t ¼ r  t0 . Then

are referred to as the Fourier analysis and the Fourier synthesis of T, respectively (there is a discrepancy between this terminology and the crystallographic one, see Section 1.3.4.2.1.1). In other words, any periodic distribution T 2 S 0 may be represented by a Fourier series (ii), whose coefficients are calculated by (i). The convergence of (ii) towards T in S 0 will be investigated later (Section 1.3.2.6.10).

F ½T 0 ðnÞ ¼ jdet AjF ½t0 ðAT nÞ;

hence T

F ½T 0 ½ðA1 Þ l ¼ jdet AjF ½t0 ðlÞ;

1.3.2.6.5. The case of nonstandard period lattices Let  denote P the nonstandard lattice consisting of all vectors of the form j¼1 mj aj , where the mj are rational integers and n a1 ; . . . ; an are n linearly independent vectors P in R . Let R be the corresponding lattice distribution: R ¼ x2
ðxÞ . Let A be the nonsingular n
n matrix whose successive columns are the coordinates of vectors a1 ; . . . ; an in the standard basis of Rn ; A will be called the period matrix of , and the mapping x 7 ! Ax will be denoted by A. According to Section 1.3.2.3.9.5 we have P

P

¼ jdet Aj1

l2Zn

hR; ’i ¼

P

F ½T ¼ jdet Aj1 R  F ½T 0 

wl ¼ hTx0 ; expð2 il  xÞi P Tx ¼ wl expðþ2 il  xÞ

ðiiÞ


½ðA1 ÞT l ¼

associated with the reciprocal lattice  whose basis vectors a1 ; . . . ; an are the columns of ðA1 ÞT . Since the latter matrix is equal to the adjoint matrix (i.e. the matrix of co-factors) of A divided by det A, the components of the reciprocal basis vectors can be written down explicitly (see Section 1.3.4.2.1.1 for the crystallographic case n ¼ 3). A distribution T will be called -periodic if n T ¼ T for all n 2 ; as previously, T may be written R  T 0 for some motif distribution T 0 with compact support. By Fourier transformation,

wl ¼ F ½T 0 ðlÞ ¼ hTx0 ; expð2 il  xÞi:

ðiÞ

P

so that F ½T ¼

P l2Zn

F ½t0 ðlÞ
½ðA1 ÞT l

in nonstandard coordinates, while F ½t ¼

P l2Zn

F ½t0 ðlÞ
ðlÞ

’ðAmÞ ¼ hr; ðA1 Þ# ’i ¼ jdet Aj1 hA# r; ’i

m2Zn

in standard coordinates. The reciprocity theorem may then be written:

for any ’ 2 S , and hence R ¼ jdet Aj1 A# r. By Fourier transformation, according to Section 1.3.2.5.5,

ðiiiÞ ðivÞ

F ½R ¼ jdet Aj1 F ½A# r ¼ ½ðA1 Þ # F ½r ¼ ½ðA1 Þ # r; T

T

Wn ¼ jdet Aj1 hTx0 ; expð2 in  xÞi; P Tx ¼ Wn expðþ2 in  xÞ

n 2 K

n2

in nonstandard coordinates, or equivalently:

which we write:

ðvÞ ðviÞ

F ½R ¼ jdet Aj1 R

wl ¼ htx0 ; expð2 il  xÞi; l 2 Zn P tx ¼ wl expðþ2 il  xÞ l2Zn

with

43

1. GENERAL RELATIONSHIPS AND TECHNIQUES in standard coordinates. It gives an n-dimensional Fourier series representation for any periodic distribution over Rn . The convergence of such series in S 0 ðRn Þ will be examined in Section 1.3.2.6.10.

and Poisson’s summation formula for a lattice with period matrix A reads: P m2Zn

1.3.2.6.6. Duality between periodization and sampling Let T 0 be a distribution with compact support (the ‘motif’). Its Fourier transform F
½T 0  is analytic (Section 1.3.2.5.4) and may thus be used as a multiplier. We may rephrase the preceding results as follows: (i) if T 0 is ‘periodized by R’ to give R  T 0, then F
½T 0  is ‘sampled by R ’ to give jdet Aj1 R  F
½T 0 ; (ii) if F
½T 0  is ‘sampled by R ’ to give R  F
½T 0 , then T 0 is ‘periodized by R’ to give jdet AjR  T 0. Thus the Fourier transformation establishes a duality between the periodization of a distribution by a period lattice  and the sampling of its transform at the nodes of lattice  reciprocal to . This is a particular instance of the convolution theorem of Section 1.3.2.5.8. At this point it is traditional to break the symmetry between F and F
which distribution theory has enabled us to preserve even in the presence of periodicity, and to perform two distinct identifications: (i) a -periodic distribution T will be handled as a distribution T~ on Rn =, was done in Section 1.3.2.6.3; P (ii) a weighted lattice distribution W ¼ l2Zn Wl
½ðA1 ÞT l will be identified with the collection fWl jl 2 Zn g of its n-tuply indexed coefficients.

GB ðAmÞ ¼ jdet Aj1 jdetð2 B1 Þj1=2


P l2Zn

G4 2 B1 ½ðA1 ÞT l

or equivalently P m2Zn

GC ðmÞ ¼ jdetð2 C1 Þj1=2

P l2Zn

G4 2 C1 ðlÞ

with C ¼ AT BA: 1.3.2.6.8. Convolution of Fourier series Let S ¼ R  S0 and T ¼ R  T 0 be two -periodic distributions, the motifs S0 and T 0 having compact support. The convolution S  T does not exist, because S and T do not satisfy the support condition (Section 1.3.2.3.9.7). However, the three distributions R, S0 and T 0 do satisfy the generalized support condition, so that their convolution is defined; then, by associativity and commutativity: R  S0  T 0 ¼ S  T 0 ¼ S0  T:

1.3.2.6.7. The Poisson summation formula Let ’ 2 S , so that F ½’ 2 S . Let R be the lattice distribution associated to lattice , with period matrix A, and let R be associated to the reciprocal lattice . Then we may write:

By Fourier transformation and by the convolution theorem: R
F ½S0  T 0  ¼ ðR
F ½S0 Þ
F ½T 0  ¼ F ½T 0 
ðR
F ½S0 Þ:

hR; ’i ¼ hR; F
½F ½’i ¼ hF
½R; F ½’i

Let fUn gn2 , fVn gn2 and fWn gn2 be the sets of Fourier coefficients associated to S, T and S  T 0 ð¼ S0  TÞ, respectively. Identifying the coefficients of
n for n 2  yields the forward version of the convolution theorem for Fourier series:

¼ jdet Aj1 hR ; F ½’i i.e. P x2

’ðxÞ ¼ jdet Aj1

P

Wn ¼ jdet AjUn Vn : F ½’ðnÞ:

n2

The backward version of the theorem requires that T be infinitely differentiable. The distribution S
T is then well defined and its Fourier coefficients fQn gn2 are given by

This identity, which also holds for F
, is called the Poisson summation formula. Its usefulness follows from the fact that the speed of decrease at infinity of ’ and F ½’ are inversely related (Section 1.3.2.4.4.3), so that if one of the series (say, the left-hand side) is slowly convergent, the other (say, the right-hand side) will be rapidly convergent. This procedure has been used by Ewald (1921) [see also Bertaut (1952), Born & Huang (1954)] to evaluate lattice sums (Madelung constants) involved in the calculation of the internal electrostatic energy of crystals (see Chapter 3.4 in this volume on convergence acceleration techniques for crystallographic lattice sums and Chapter 3.5 on modern extensions of the Ewald summation method). When ’ is a multivariate Gaussian

Qn ¼

P g2

Ug Vng :

1.3.2.6.9. Toeplitz forms, Szego¨’s theorem Toeplitz forms were first investigated by Toeplitz (1907, 1910, 1911a). They occur in connection with the ‘trigonometric moment problem’ (Shohat & Tamarkin, 1943; Akhiezer, 1965) and probability theory (Grenander, 1952) and play an important role in several direct approaches to the crystallographic phase problem [see Sections 1.3.4.2.1.10, 1.3.4.5.2.2(e)]. Many aspects of their theory and applications are presented in the book by Grenander & Szego¨ (1958).

’ðxÞ ¼ GB ðxÞ ¼ expð12xT BxÞ; then

1.3.2.6.9.1. Toeplitz forms Let f 2 L1 ðR=ZÞ be real-valued, so that its Fourier coefficients satisfy the relations cm ð f Þ ¼ cm ð f Þ. The Hermitian form in n þ 1 complex variables

F ½’ðnÞ ¼ jdetð2 B1 Þj1=2 GB1 ðnÞ;

44

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY Tn ½ f ðuÞ ¼

n P n P

finite, then for any continuous function FðÞ defined in the interval [m, M] we have

u c  u

¼0 ¼0 nþ1 1 X FððnÞ lim  Þ ¼ n!1 n þ 1 ¼1

is called the nth Toeplitz form associated to f. It is a straightforward consequence of the convolution theorem and of Parseval’s identity that Tn ½ f  may be written:

0 ¼0

1.3.2.6.9.2. The Toeplitz–Carathe´odory–Herglotz theorem It was shown independently by Toeplitz (1911b), Carathe´odory (1911) and Herglotz (1911) that a function f 2 L1 is almost everywhere non-negative if and only if the Toeplitz forms Tn ½ f  associated to f are positive semidefinite for all values of n. This is equivalent to the infinite system of determinantal inequalities c0 B c1 B Dn ¼ detB B  @  cn

c1 c0 c1  

 c1   

    c1

1 cn  C C  C C0 c1 A c0

lim ðnÞ 1 ¼ m ¼ ess inf f ;

ðnÞ lim nþ1 ¼ M ¼ ess sup f :

n!1

n!1

ðnÞ Thus, when f  0, the condition number ðnÞ nþ1 =1 of Tn ½ f  tends towards the ‘essential dynamic range’ M=m of f. (ii) Let FðÞ ¼ s where s is a positive integer. Then

for all n:

nþ1 1 X s lim ½ðnÞ   ¼ n!1 n þ 1 ¼1

Z1

½ f ðxÞs dx:

0

(iii) Let m > 0, so that ðnÞ  > 0, and let Dn ð f Þ ¼ det Tn ð f Þ. Then

1.3.2.6.9.3. Asymptotic distribution of eigenvalues of Toeplitz forms The eigenvalues of the Hermitian form Tn ½ f  are defined as the n þ 1 real roots of the characteristic equation detfTn ½ f  g ¼ 0. They will be denoted by

Dn ð f Þ ¼

nþ1 Q

ðnÞ  ;

¼1

hence

ðnÞ ðnÞ ðnÞ 1 ; 2 ; . . . ; nþ1 :

log Dn ð f Þ ¼

It is easily shown that if m  f ðxÞ  M for all x, then m  ðnÞ   M for all n and all  ¼ 1; . . . ; n þ 1. As n ! 1 these bounds, and the distribution of the ðnÞ within these bounds, can be made more precise by introducing two new notions. (i) Essential bounds: define ess inf f as the largest m such that f ðxÞ  m except for values of x forming a set of measure 0; and define ess sup f similarly. (ii) Equal distribution. For each n, consider two sets of n þ 1 real numbers: and

0

1.3.2.6.9.4. Consequences of Szego¨’s theorem (i) If the ’s are ordered in ascending order, then

The Dn are called Toeplitz determinants. Their application to the crystallographic phase problem is described in Section 1.3.4.2.1.10.

ðnÞ ðnÞ aðnÞ 1 ; a2 ; . . . ; anþ1 ;

F½ f ðxÞ dx:

In other words, the eigenvalues ðnÞ of the Tn and the values f ½=ðn þ 2Þ of f on a regular subdivision of ]0, 1[ are equally distributed. Further investigations into the spectra of Toeplitz matrices may be found in papers by Hartman & Wintner (1950, 1954), Kac et al. (1953), Widom (1965), and in the notes by Hirschman & Hughes (1977).

n

2

R1 P

Tn ½ f ðuÞ ¼ u expð2 ixÞ

f ðxÞ dx:

0

Z1

nþ1 P

log ðnÞ  :

¼1

Putting FðÞ ¼ log , it follows that lim ½Dn ð f Þ

n!1

1=ðnþ1Þ

¼ exp

1 R

 log f ðxÞ dx :

0

Further terms in this limit were obtained by Szego¨ (1952) and interpreted in probabilistic terms by Kac (1954).

ðnÞ ðnÞ bðnÞ 1 ; b2 ; . . . ; bnþ1 :

1.3.2.6.10. Convergence of Fourier series The investigation of the convergence of Fourier series and of more general trigonometric series has been the subject of intense study for over 150 years [see e.g. Zygmund (1976)]. It has been a constant source of new mathematical ideas and theories, being directly responsible for the birth of such fields as set theory, topology and functional analysis. This section will briefly survey those aspects of the classical results in dimension 1 which are relevant to the practical use of Fourier series in crystallography. The books by Zygmund (1959), Tolstov (1962) and Katznelson (1968) are standard references in the field, and Dym & McKean (1972) is recommended as a stimulant.

ðnÞ Assume that for each  and each n, jaðnÞ  j < K and jb j < K with ðnÞ ðnÞ K independent of  and n. The sets fa g and fb g are said to be equally distributed in ½K; þK if, for any function F over ½K; þK, nþ1 1 X ðnÞ ½FðaðnÞ  Þ  Fðb Þ ¼ 0: n!1 n þ 1 ¼1

lim

We may now state an important theorem of Szego¨ (1915, 1920). Let f 2 L1 , and put m ¼ ess inf f , M ¼ ess sup f . If m and M are

45

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1

is the Feje´r kernel. Fp has over Dp the advantage of being everywhere positive, so that the Cesa`ro sums Cp ð f Þ of a positive function f are always positive. The de la Valle´e Poussin kernel

1.3.2.6.10.1. Classical L theory The space L1 ðR=ZÞ consists of (equivalence classes of) complex-valued functions f on the circle which are summable, i.e. for which k f k1

R1

Vp ðxÞ ¼ 2F2pþ1 ðxÞ  Fp ðxÞ

j f ðxÞj dx < þ 1:

0

1

has a trapezoidal distribution of coefficients and is such that cm ðVp Þ ¼ 1 if jmj  p þ 1; therefore Vp  f is a trigonometric polynomial with the same Fourier coefficients as f over that range of values of m. The Poisson kernel

1

It is a convolution algebra: If f and g are in L , then f  g is in L . The mth Fourier coefficient cm ð f Þ of f, cm ð f Þ ¼

R1

f ðxÞ expð2 imxÞ dx

1 X Pr ðxÞ ¼ 1 þ 2 rm cos 2 mx

0

m¼1

is bounded: jcm ð f Þj  k f k1 , and by the Riemann–Lebesgue lemma cm ð f Þ ! 0 as m ! 1. By the convolution theorem, cm ð f  gÞ ¼ cm ð f Þcm ðgÞ. The pth partial sum Sp ð f Þ of the Fourier series of f, Sp ð f ÞðxÞ ¼

P

¼

1  r2 1  2r cos 2 mx þ r2

with 0  r < 1 gives rise to an Abel summation procedure [Tolstov (1962, p. 162); Whittaker & Watson (1927, p. 57)] since

cm ð f Þ expð2 imxÞ;

jmjp

ðPr  f ÞðxÞ ¼ may be written, by virtue of the convolution theorem, as Sp ð f Þ ¼ Dp  f , where Dp ðxÞ ¼

X

expð2 imxÞ ¼

jmjp

P m2Z

cm ð f Þrjmj expð2 imxÞ:

Compared with the other kernels, Pr has the disadvantage of not being a trigonometric polynomial; however, Pr is the real part of the Cauchy kernel (Cartan, 1961; Ahlfors, 1966):

sin½ð2p þ 1Þ x sin x

 Pr ðxÞ ¼ R e

is the Dirichlet kernel. Because Dp comprises numerous slowly decaying oscillations, both positive and negative, Sp ð f Þ may not converge towards f in a strong sense as p ! 1. Indeed, spectacular pathologies are known to exist where the partial sums, examined pointwise, diverge everywhere (Zygmund, 1959, Chapter VIII). When f is piecewise continuous, but presents isolated jumps, convergence near these jumps is marred by the Gibbs phenomenon: Sp ð f Þ always ‘overshoots the mark’ by about 9%, the area under the spurious peak tending to 0 as p ! 1 but not its height [see Larmor (1934) for the history of this phenomenon]. By contrast, the arithmetic mean of the partial sums, also called the pth Cesa`ro sum, Cp ð f Þ ¼

and hence provides a link between trigonometric series and analytic functions of a complex variable. Other methods of summation involve forming a moving average of f by convolution with other sequences of functions p ðxÞ besides Dp of Fp which ‘tend towards
’ as p ! 1. The convolution is performed by multiplying the Fourier coefficients of f by those of p, so that one forms the quantities S0p ð f ÞðxÞ ¼

1 ½S ð f Þ þ . . . þ Sp ð f Þ; pþ1 0

cm ðp Þcm ð f Þ expð2 imxÞ:

For instance the ‘sigma factors’ of Lanczos (Lanczos, 1966, p. 65), defined by m ¼

sin½m =p ; m =p

lead to a summation procedure whose behaviour is intermediate between those using the Dirichlet and the Feje´r kernels; it corresponds to forming a moving average of f by convolution with

Cp ð f Þ ¼ Fp  f ; where X


P jmjp

converges to f in the sense of the L1 norm: kCp ð f Þ  f k1 ! 0 as p ! 1. If furthermore f is continuous, then the convergence is uniform, i.e. the error is bounded everywhere by a quantity which goes to 0 as p ! 1. It may be shown that

 jmj expð2 imxÞ pþ1 jmjp  2 1 sinðp þ 1Þ x ¼ pþ1 sin x

Fp ðxÞ ¼

 1 þ r expð2 ixÞ 1  r expð2 ixÞ

p ¼ p½1=ð2pÞ; 1=ð2pÞ Dp ;

1

which is itself the convolution of a ‘rectangular pulse’ of width 1=p and of the Dirichlet kernel of order p. A review of the summation problem in crystallography is given in Section 1.3.4.2.1.3.

46

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 2

Let fwm gm2Z be a sequence of complex numbers with jwm j growing at most polynomially as jmj ! 1, say jwm j  CjmjK . Then the sequence fwm =ð2 imÞKþ2 gm2Z is in ‘2 and even defines a continuous function f 2 L2 ðR=ZÞ and an associated tempered distribution Tf 2 D 0 ðR=ZÞ. Differentiation of Tf ðK þ 2Þ times then yields a tempered distribution whose Fourier transform leads to the original sequence of coefficients. Conversely, by the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), the motif T 0 of a Z-periodic distribution is a derivative of finite order of a continuous function; hence its Fourier coefficients will grow at most polynomially with jmj as jmj ! 1. Thus distribution theory allows the manipulation of Fourier series whose coefficients exhibit polynomial growth as their order goes to infinity, while those derived from functions had to tend to 0 by virtue of the Riemann–Lebesgue lemma. The distributiontheoretic approach to Fourier series holds even in the case of general dimension n, where classical theories meet with even more difficulties (see Ash, 1976) than in dimension 1.

1.3.2.6.10.2. Classical L theory The space L2 ðR=ZÞ of (equivalence classes of) squareintegrable complex-valued functions f on the circle is contained in L1 ðR=ZÞ, since by the Cauchy–Schwarz inequality k f k21

¼


1 R




1 R

2 j f ðxÞj
1 dx

0

2

j f ðxÞj dx


1 R

0



1 dx ¼ k f k22  1: 2

0

Thus all the results derived for L1 hold for L2 , a great simplification over the situation in R or Rn where neither L1 nor L2 was contained in the other. However, more can be proved in L2 , because L2 is a Hilbert space (Section 1.3.2.2.4) for the inner product ð f ; gÞ ¼

R1

f ðxÞgðxÞ dx;

1.3.2.7. The discrete Fourier transformation

0

1.3.2.7.1. Shannon’s sampling theorem and interpolation formula Let ’ 2 E ðRn Þ be such that  ¼ F ½’ has compact support K. Let ’ be sampled at the nodes of a lattice , yielding the lattice distribution R
’. The Fourier transform of this sampled version of ’ is

and because the family of functions fexpð2 imxÞgm2Z constitutes an orthonormal Hilbert basis for L2 . The sequence of Fourier coefficients cm ð f Þ of f 2 L2 belongs to the space ‘2 ðZÞ of square-summable sequences: P m2Z

jcm ð f Þj2 < 1:

F ½R
’ ¼ jdet AjðR  Þ;

Conversely, every element c ¼ ðcm Þ of ‘2 is the sequence of Fourier coefficients of a unique function in L2 . The inner product ðc; dÞ ¼

P m2Z

which is essentially  periodized by period lattice  ¼ ð Þ, with period matrix A. Let us assume that  is such that the translates of K by different period vectors of  are disjoint. Then we may recover  from R   by masking the contents of a ‘unit cell’ V of  (i.e. a fundamental domain for the action of  in Rn ) whose boundary does not meet K. If V is the indicator function of V , then

cm dm

makes ‘2 into a Hilbert space, and the map from L2 to ‘2 established by the Fourier transformation is an isometry (Parseval/Plancherel):

 ¼ V
ðR  Þ:

k f kL2 ¼ kcð f Þk‘2

Transforming both sides by F
yields 
’ ¼ F V


or equivalently:

 1  F ½R
’ ; jdet Aj

ð f ; gÞ ¼ ðcð f Þ; cðgÞÞ: i.e. This is a useful property in applications, since (f, g) may be calculated either from f and g themselves, or from their Fourier coefficients cð f Þ and cðgÞ (see Section 1.3.4.4.6) for crystallographic applications). By virtue of the orthogonality of the basis fexpð2 imxÞgm2Z, the partial sum Sp ð f Þ is the best mean-square fit to f in the linear subspace of L2 spanned by fexpð2 imxÞgjmjp, and hence (Bessel’s inequality) P

jcm ð f Þj2 ¼ k f k22 

jmjp

P




 1
’¼ F ½V   ðR
’Þ V since jdet Aj is the volume V of V . This interpolation formula is traditionally credited to Shannon (1949), although it was discovered much earlier by Whittaker (1915). It shows that ’ may be recovered from its sample values on  (i.e. from R
’) provided  is sufficiently fine that no overlap (or ‘aliasing’) occurs in the periodization of  by the dual lattice . The interpolation kernel is the transform of the normalized indicator function of a unit cell of  containing the support K of . If K is contained in a sphere of radius 1= and if  and  are rectangular, the length of each basis vector of  must be greater than 2=, and thus the sampling interval must be smaller than =2. This requirement constitutes the Shannon sampling criterion.

jcM ð f Þj2  k f k22 :

jMjp

1.3.2.6.10.3. The viewpoint of distribution theory The use of distributions enlarges considerably the range of behaviour which can be accommodated in a Fourier series, even in the case of general dimension n where classical theories meet with even more difficulties than in dimension 1.

47

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1.3.2.7.2. Duality between subdivision and decimation of period lattices 1.3.2.7.2.1. Geometric description of sublattices Let A be a period lattice in Rn with matrix A, and let A be the lattice reciprocal to A, with period matrix ðA1 ÞT . Let B ; B; B be defined similarly, and let us suppose that A is a sublattice of B , i.e. that B  A as a set. The relation between A and B may be described in two different fashions: (i) multiplicatively, and (ii) additively. (i) We may write A ¼ BN for some nonsingular matrix N with integer entries. N may be viewed as the period matrix of the coarser lattice A with respect to the period basis of the finer lattice B. It will be more convenient to write A ¼ DB, where D ¼ BNB1 is a rational matrix (with integer determinant since det D ¼ det N) in terms of which the two lattices are related by

ðiiiÞ

l

RA ¼

ðiiÞ ðiÞ ðiiÞ where

TB=A ¼

 TA=B ¼

SB=A ¼

ðl þ DB Þ

which may be viewed as the n-dimensional equivalent of the Euclidean algorithm for integer division: l is the ‘remainder’ of the division by A of a vector in B , the quotient being the matrix D. 1.3.2.7.2.2. Sublattice relations for reciprocal lattices Let us now consider the two reciprocal lattices A and B . Their period matrices ðA1 ÞT and ðB1 ÞT are related by: ðB1 ÞT ¼ ðA1 ÞT NT , where NT is an integer matrix; or equivalently by ðB1 ÞT ¼ DT ðA1 ÞT. This shows that the roles are reversed in that B is a sublattice of A , which we may write:



ðiiÞ

¼

[



ðl þ


ðl  Þ

1 T ; jdet Dj B=A

SA=B ¼

1 T : jdet Dj A=B

ði0 Þ

RA ¼ D# ðSB=A  RA Þ

ðii0 Þ

RB ¼ SB=A  ðD# RB Þ

ði0 Þ

RB ¼ ðDT Þ# ðSA=B  RB Þ

ðii0 Þ

RA ¼ SA=B  ½ðDT Þ# RA :

These identities show that period subdivision by convolution with SB=A (respectively SA=B ) on the one hand, and period decimation by ‘dilation’ by D# on the other hand, are mutually inverse operations on RA and RB (respectively RA and RB ). 1.3.2.7.2.4. Relation between Fourier transforms Finally, let us consider the relations between the Fourier transforms of these lattice distributions. Recalling the basic relation of Section 1.3.2.6.5,

B ¼ DT A

A

l  2A =B

Since jdet Dj ¼ ½B : A  ¼ ½A : B , convolution with SB=A and SA=B has the effect of averaging the translates of a distribution under the elements (or ‘cosets’) of the residual lattices B =A and A =B , respectively. This process will be called ‘coset averaging’. Eliminating RA and RB between (i) and (ii), and RA and RB between ðiÞ and ðiiÞ , we may write:

l 2B =A

ðiÞ

P

are (finite) residual-lattice distributions. We may incorporate the factor 1=jdet Dj in (i) and ðiÞ into these distributions and define

represents B as the disjoint union of ½B : A  translates of A : B =A is a finite lattice with ½B : A  elements, called the residual lattice of B modulo A. The two descriptions are connected by the relation ½B : A  ¼ det D ¼ det N, which follows from a volume calculation. We may also combine (i) and (ii) into B ¼


ðl Þ

and

l 2B =A

ðiiiÞ

P l 2B =A

ðl þ  A Þ

[

ðl  þ DT A Þ:

2A =B

1 D# RB jdet Dj RB ¼ TB=A  RA 1 ðDT Þ# RA RB ¼ jdet Dj  RA ¼ TA=B  RB

ðiÞ

(ii) Call two vectors in B congruent modulo A if their difference lies in A . Denote the set of congruence classes (or ‘cosets’) by B =A, and the number of these classes by ½B : A . The ‘coset decomposition’ [



1.3.2.7.2.3. Relation between lattice distributions The above relations between lattices may be rewritten in terms of the corresponding lattice distributions as follows:

A ¼ DB :

B ¼

[

A ¼

1 R jdet Aj A 1 T   RB ¼ jdet DBj A=B

 1 1  TA=B RB ¼  jdet Dj jdet Bj

F ½RA  ¼

B Þ:

l  2A =B

The residual lattice A =B is finite, with ½A : B  ¼ det D ¼ det N ¼ ½B : A , and we may again combine ðiÞ and ðiiÞ into

48

by (ii)

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY i.e.

restricting a function to a discrete additive subgroup of the domain over which it is initially given. F ½RA  ¼ SA=B  F ½RB 

ðivÞ

1.3.2.7.2.5. Sublattice relations in terms of periodic distributions The usual presentation of this duality is not in terms of lattice distributions, but of periodic distributions obtained by convolving them with a motif. Given T 0 2 E 0 ðRn Þ, let us form RA  T 0 , then decimate its transform ð1=jdet AjÞRA
F
½T 0  by keeping only its values at the points of the coarser lattice B ¼ DT A ; as a result, RA is replaced by ð1=jdet DjÞRB, and the reverse transform then yields

and similarly: F ½RB  ¼ SB=A  F ½RA :

ðvÞ

Thus RA (respectively RB ), a decimated version of RB (respectively RA ), is transformed by F into a subdivided version of F ½RB  (respectively F ½RA ). The converse is also true: 1 R jdet Bj B 1 1 ðDT Þ# RA ¼ jdet Bj jdet Dj
 1 T #  R ¼ ðD Þ jdet Aj A

1 R  T 0 ¼ SB=A  ðRA  T 0 Þ jdet Dj B

F ½RB  ¼

which is the coset-averaged version of the original RA  T 0 . The converse situation is analogous to that of Shannon’s sampling theorem. Let a function ’ 2 E ðRn Þ whose transform  ¼ F ½’ has compact support be sampled as RB
’ at the nodes of B. Then

by (i)

F ½RB
’ ¼

i.e. #

ðiv0 Þ

F ½RB  ¼ ðDT Þ F ½RA 

ðv Þ

F ½RA 

¼D

#

1 ðR  Þ jdet Dj A ¼ SA=B  ðRB  Þ

F ½RA
’ ¼

F ½RB :

Thus RB (respectively RA ), a subdivided version of RA (respectively RB ) is transformed by F into a decimated version of F ½RA  (respectively F ½RB ). Therefore, the Fourier transform exchanges subdivision and decimation of period lattices for lattice distributions. Further insight into this phenomenon is provided by applying F
to both sides of (iv) and (v) and invoking the convolution theorem: ðiv00 Þ ðv00 Þ

1.3.2.7.3. Discretization of the Fourier transformation Let ’0 2 E ðRn Þ be such that 0 ¼ F ½’0  has compact support (’0 is said to be band-limited). Then ’ ¼ RA  ’0 is A -periodic, and  ¼ F ½’ ¼ ð1=jdet AjÞRA
0 is such that only a finite number of points A of A have a nonzero Fourier coefficient 0 ðA Þ attached to them. We may therefore find a decimation B ¼ DT A of A such that the distinct translates of Supp 0 by vectors of B do not intersect. The distribution  can be uniquely recovered from RB   by the procedure of Section 1.3.2.7.1, and we may write:

These identities show that multiplication by the transform of the period-subdividing distribution SA=B (respectively SB=A ) has the effect of decimating RB to RA (respectively RA to RB ). They clearly imply that, if l 2 B =A and l  2 A =B , then

l ¼0

1 R  ðRA
0 Þ jdet Aj B 1 R
ðRB  0 Þ ¼ jdet Aj A 1  R  ½TA=B ¼
ðRB  0 Þ; jdet Aj B

ði:e: if l belongs

RB   ¼

to the class of A Þ; ¼ 0 if l 6¼ 0; 
F ½SB=A ðl Þ ¼ 1 if l  ¼ 0

by (ii) ;

hence becomes periodized more finely by averaging over the cosets of A =B. With this finer periodization, the various copies of Supp  may start to overlap (a phenomenon called ‘aliasing’), indicating that decimation has produced too coarse a sampling of ’.

RA ¼ F
½SA=B 
RB RB ¼ F
½SB=A 
RA :

F
½SA=B ðl Þ ¼ 1 if

1 ðR  Þ jdet Bj B

is periodic with period lattice B. If the sampling lattice B is decimated to A ¼ DB, the inverse transform becomes

and similarly 0

by (ii);

ði:e: if l  belongs to the class of B Þ;



¼ 0 if l 6¼ 0:  these rearrangements being legitimate because 0 and TA=B have compact supports which are intersection-free under the action of B . By virtue of its B -periodicity, this distribution is entirely ~ with respect to B : characterized by its ‘motif’ 

Therefore, the duality between subdivision and decimation may be viewed as another aspect of that between convolution and multiplication. There is clearly a strong analogy between the sampling/ periodization duality of Section 1.3.2.6.6 and the decimation/ subdivision duality, which is viewed most naturally in terms of subgroup relationships: both sampling and decimation involve

~ ¼ 

49

1 T 
ðRB  0 Þ: jdet Aj A=B

1. GENERAL RELATIONSHIPS AND TECHNIQUES Similarly, ’ may be uniquely recovered by Shannon interpolation from the distribution sampling its values at the nodes of B ¼ D1 A ðB is a subdivision of B ). By virtue of its A periodicity, this distribution is completely characterized by its motif:

P

~ ðl  Þ ¼ 

’~ ðl Þ expðþ2 il  l  Þ:

l 2B =A

Now the decimation/subdivision relations between A and B may be written:

’~ ¼ TB=A
’ ¼ TB=A
ðRA  ’0 Þ: A ¼ DB ¼ BN; Let l 2 B =A and l  2 A =B , and define the two sets of coefficients ð1Þ ’~ ðl Þ ~ ðl  Þ ð2Þ 

so that

for any kA 2 A ðall choices of kA give the same ’~ Þ; ¼ 0 ðl  þ kB Þ for the unique kB (if it exists) such that l  þ kB 2 Supp 0 ; ¼0 if no such kB exists:

l  ¼ ðA1 ÞT k 

l   l ¼ l  l  ¼ k   ðN1 k Þ: P

!¼

l 2B =A

~ ½ðA1 ÞT k   by ðk  Þ, the relation Denoting ’~ ðBk Þ by ðk Þ and  between ! and  may be written in the equivalent form

’~ ðl Þ
ðl Þ

ðiÞ

and P l  2A =B

~ ðl  Þ
ðl  Þ : 

ðiiÞ

where the summations are now over finite residual lattices in standard form. Equations (i) and (ii) describe two mutually inverse linear transformations F ðNÞ and F
ðNÞ between two vector spaces WN and WN of dimension jdet Nj. F ðNÞ [respectively F
ðNÞ] is the discrete Fourier (respectively inverse Fourier) transform associated to matrix N. The vector spaces WN and WN may be viewed from two different standpoints: (1) as vector spaces of weighted residual-lattice distributions, of  ; the canonical basis of WN the form ðxÞTB=A and ðxÞTA=B  (respectively WN ) then consists of the
ðkÞ for k 2 Zn =NZn [respectively
ðk Þ for k  2 Zn =NT Zn ]; (2) as vector spaces of weight vectors for the jdet Nj
-functions  involved in the expression for TB=A (respectively TA=B ); the  canonical basis of WN (respectively WN ) consists of weight vectors uk (respectively vk ) giving weight 1 to element k (respectively k  ) and 0 to the others. These two spaces are said to be ‘isomorphic’ (a relation denoted ffi), the isomorphism being given by the one-to-one correspondence:

RA  ! ¼ F ½RB   F
½RA  ! ¼ RB  :

ðiiÞ

By (i), RA  ! ¼ jdet BjRB
F ½. Both sides are weighted lattice distributions concentrated at the nodes of B, and equating the weights at kB ¼ l þ kA gives ’~ ðl Þ ¼

X 1 ~ ðl  Þ exp½2 il   ðl þ kA Þ:  jdet Dj l  2 = A

B

Since l  2 A, l   kA is an integer, hence ’~ ðl Þ ¼

X 1 ~ ðl  Þ expð2 il   l Þ:  jdet Dj l  2 = A

X 1 ðk  Þ exp½2 ik   ðN1 k Þ jdet Nj k 2Zn =NT Zn X ðk Þ exp½þ2 ik   ðN1 k Þ; ðk  Þ ¼ ðk Þ ¼

k 2Zn =NZn

The relation between ! and  has two equivalent forms: ðiÞ

for k  2 Zn

with ðA1 ÞT ¼ ðB1 ÞT ðN1 ÞT , hence finally

Define the two distributions

¼

for k 2 Zn

l ¼ Bk

¼ ’ðl þ kA Þ

B

By (ii), we have

!¼

P

¼

P

ðk Þ
ðkÞ

$

¼

P

¼

P

k

1 1  R  ½TA=B
ðRB  0 Þ ¼ F
½RA  !: jdet Aj B jdet Aj

k

P

ðk  Þ
ðk Þ

$

k

ðk  Þvk :

The second viewpoint will be adopted, as it involves only linear algebra. However, it is most helpful to keep the first one in mind and to think of the data or results of a discrete Fourier transform as representing (through their sets of unique weights) two periodic lattice distributions related by the full, distribution-theoretic Fourier transform. We therefore view WN (respectively WN ) as the vector space of complex-valued functions over the finite residual lattice B =A (respectively A =B ) and write:

Both sides are weighted lattice distributions concentrated at the nodes of B, and equating the weights at kA ¼ l  þ kB gives ~ ðl  Þ ¼ 

ðk Þuk

k

’~ ðl Þ exp½þ2 il  ðl  þ kB Þ:

l 2B =A

Since l 2 B, l  kB is an integer, hence

50

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY F
ðNÞ ¼ F
ð1 Þ  F
ð2 Þ  . . .  F
ðn Þ;

WN ffi LðB =A Þ ffi LðZn =NZn Þ WN ffi LðA =B Þ ffi LðZn =NT Zn Þ where since a vector such as is in fact the function k 7 ! ðk Þ. The two spaces WN and WN may be equipped with the following Hermitian inner products: ð’; ÞW ¼

P


 k j k j ½F
j kj ; kj ¼ exp þ2 i : j

’ðk Þ ðk Þ

k

ð; ÞW  ¼

P

1.3.2.7.5. Properties of the discrete Fourier transform The DFT inherits most of the properties of the Fourier transforms, but with certain numerical factors (‘Jacobians’) due to the transition from continuous to discrete measure. (1) Linearity is obvious. (2) Shift property. If ða Þðk Þ ¼ ðk  aÞ and ða Þðk  Þ ¼ ðk   a Þ, where subtraction takes place by modular vector arithmetic in Zn =NZn and Zn =NT Zn , respectively, then the following identities hold:

ðk  Þ ðk  Þ;

k

which makes each of them into a Hilbert space. The canonical bases fuk jk 2 Zn =NZn g and fvk jk  2 Zn =NT Zn g and WN and WN are orthonormal for their respective product. 1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT) By virtue of definitions (i) and (ii),

F
ðNÞ½k ðk  Þ ¼ exp½þ2 ik   ðN1 k ÞF
ðNÞ½ ðk  Þ

1 X exp½2 ik   ðN1 k Þuk jdet Nj k X F
ðNÞuk ¼ exp½þ2 ik   ðN1 k Þvk

F ðNÞ½k  ðk Þ ¼ exp½2 ik   ðN1 k ÞF ðNÞ½ ðk Þ:

F ðNÞvk  ¼

(3) Differentiation identities. Let vectors w and W be constructed from ’0 2 E ðRn Þ as in Section 1.3.2.7.3, hence be related by the DFT. If Dp w designates the vector of sample values of Dpx ’0 at the points of B =A, and Dp W the vector of values of Dpn 0 at points of A =B, then for all multi-indices p ¼ ðp1 ; p2 ; . . . ; pn Þ

k

so that F ðNÞ and F
ðNÞ may be represented, in the canonical bases of WN and WN , by the following matrices: 1 exp½2 ik   ðN1 k Þ jdet Nj ¼ exp½þ2 ik   ðN1 k Þ:

ðDp wÞðk Þ ¼ F
ðNÞ½ðþ2 ik  Þp Wðk Þ ðDp WÞðk  Þ ¼ F ðNÞ½ð2 ik Þp wðk  Þ

½F ðNÞkk ¼ ½F
ðNÞk k

or equivalently

When N is symmetric, Zn =NZn and Zn =NT Zn may be identified in a natural manner, and the above matrices are symmetric. When N is diagonal, say N ¼ diagð1 ; 2 ; . . . ; n Þ, then the tensor product structure of the full multidimensional Fourier transform (Section 1.3.2.4.2.4)

p

F ðNÞ½Dp wðk  Þ ¼ ðþ2 ik  Þ Wðk  Þ F
ðNÞ½Dp Wðk Þ ¼ ð2 ik Þ wðk Þ: p

F x ¼ F x1  F x2  . . .  F xn

(4) Convolution property. Let u 2 WN and U 2 WN (respectively w and W) be related by the DFT, and define

gives rise to a tensor product structure for the DFT matrices. The tensor product of matrices is defined as follows: 0

a11 B B .. AB¼@ .

...

an1 B

...

ðu  wÞðk Þ ¼

1 a1n B .. C: . A

ðU  WÞðk  Þ ¼

P k 0 2Zn =NZn

uðk 0 Þwðk  k 0 Þ

P

0

0

Uðk  ÞWðk   k  Þ:

0 k  2Zn =NT Zn

ann B

Then

Let the index vectors k and k  be ordered in the same way as the elements in a Fortran array, e.g. for k with k 1 increasing fastest, k 2 next fastest, . . . ; k n slowest; then

F
ðNÞ½U  Wðk Þ ¼ jdet Njuðk Þwðk Þ F ðNÞ½u  wðk  Þ ¼ Uðk  ÞWðk  Þ

F ðNÞ ¼ F ð1 Þ  F ð2 Þ  . . .  F ðn Þ;

and 1 ðU  WÞðk  Þ jdet Nj F ðNÞ½U
Wðk Þ ¼ ðu  wÞðk Þ:

where

F
ðNÞ½u
wðk  Þ ¼

½F ðj Þkj ; kj ¼


 k j k j 1 exp 2 i ; j j

Since addition on Zn =NZn and Zn =NT Zn is modular, this type of convolution is called cyclic convolution.

and

51

1. GENERAL RELATIONSHIPS AND TECHNIQUES (5) Parseval/Plancherel property. If u, w, U, W are as above, then

theory of certain Lie groups and coding theory – to list only a few. The interested reader may consult Auslander & Tolimieri (1979); Auslander, Feig & Winograd (1982, 1984); Auslander & Tolimieri (1985); Tolimieri (1985). One-dimensional algorithms are examined first. The Sande mixed-radix version of the Cooley–Tukey algorithm only calls upon the additive structure of congruence classes of integers. The prime factor algorithm of Good begins to exploit some of their multiplicative structure, and the use of relatively prime factors leads to a stronger factorization than that of Sande. Fuller use of the multiplicative structure, via the group of units, leads to the Rader algorithm; and the factorization of short convolutions then yields the Winograd algorithms. Multidimensional algorithms are at first built as tensor products of one-dimensional elements. The problem of factoring the DFT in several dimensions simultaneously is then examined. The section ends with a survey of attempts at formalizing the interplay between algorithm structure and computer architecture for the purpose of automating the design of optimal DFT code. It was originally intended to incorporate into this section a survey of all the basic notions and results of abstract algebra which are called upon in the course of these developments, but time limitations have made this impossible. This material, however, is adequately covered by the first chapter of Tolimieri et al. (1989) in a form tailored for the same purposes. Similarly, the inclusion of numerous detailed examples of the algorithms described here has had to be postponed to a later edition, but an abundant supply of such examples may be found in the signal processing literature, for instance in the books by McClellan & Rader (1979), Blahut (1985), and Tolimieri et al. (1989).

1 ðU; WÞW  jdet Nj 1 ðu; wÞW : ðF
ðNÞ½u; F
ðNÞ½wÞW ¼ jdet Nj

ðF ðNÞ½U; F ðNÞ½WÞW ¼

(6) Period 4. When N is symmetric, so that the ranges of indices k and k  can be identified, it makes sense to speak of powers of F ðNÞ and F
ðNÞ. Then the ‘standardized’ matrices ð1=jdet Nj1=2 ÞF ðNÞ and ð1=jdet Nj1=2 ÞF
ðNÞ are unitary matrices whose fourth power is the identity matrix (Section 1.3.2.4.3.4); their eigenvalues are therefore 1 and i. 1.3.3. Numerical computation of the discrete Fourier transform 1.3.3.1. Introduction The Fourier transformation’s most remarkable property is undoubtedly that of turning convolution into multiplication. As distribution theory has shown, other valuable properties – such as the shift property, the conversion of differentiation into multiplication by monomials, and the duality between periodicity and sampling – are special instances of the convolution theorem. This property is exploited in many areas of applied mathematics and engineering (Campbell & Foster, 1948; Sneddon, 1951; Champeney, 1973; Bracewell, 1986). For example, the passing of a signal through a linear filter, which results in its being convolved with the response of the filter to a
-function ‘impulse’, may be modelled as a multiplication of the signal’s transform by the transform of the impulse response (also called transfer function). Similarly, the solution of systems of partial differential equations may be turned by Fourier transformation into a division problem for distributions. In both cases, the formulations obtained after Fourier transformation are considerably simpler than the initial ones, and lend themselves to constructive solution techniques. Whenever the functions to which the Fourier transform is applied are band-limited, or can be well approximated by bandlimited functions, the discrete Fourier transform (DFT) provides a means of constructing explicit numerical solutions to the problems at hand. A great variety of investigations in physics, engineering and applied mathematics thus lead to DFT calculations, to such a degree that, at the time of writing, about 50% of all supercomputer CPU time is alleged to be spent calculating DFTs. The straightforward use of the defining formulae for the DFT leads to calculations of size N 2 for N sample points, which become unfeasible for any but the smallest problems. Much ingenuity has therefore been exerted on the design and implementation of faster algorithms for calculating the DFT (McClellan & Rader, 1979; Nussbaumer, 1981; Blahut, 1985; Brigham, 1988). The most famous is that of Cooley & Tukey (1965) which heralded the age of digital signal processing. However, it had been preceded by the prime factor algorithm of Good (1958, 1960), which has lately been the basis of many new developments. Recent historical research (Goldstine, 1977, pp. 249–253; Heideman et al., 1984) has shown that Gauss essentially knew the Cooley–Tukey algorithm as early as 1805 (before Fourier’s 1807 work on harmonic analysis!); while it has long been clear that Dirichlet knew of the basis of the prime factor algorithm and used it extensively in his theory of multiplicative characters [see e.g. Chapter I of Ayoub (1963), and Chapters 6 and 8 of Apostol (1976)]. Thus the computation of the DFT, far from being a purely technical and rather narrow piece of specialized numerical analysis, turns out to have very rich connections with such central areas of pure mathematics as number theory (algebraic and analytic), the representation

1.3.3.2. One-dimensional algorithms Throughout this section we will denote by eðtÞ the expression expð2 itÞ, t 2 R. The mapping t 7 ! eðtÞ has the following properties: eðt1 þ t2 Þ ¼ eðt1 Þeðt2 Þ eðtÞ ¼ eðtÞ ¼ ½eðtÞ1 eðtÞ ¼ 1 , t 2 Z: Thus e defines an isomorphism between the additive group R=Z (the reals modulo the integers) and the multiplicative group of complex numbers of modulus 1. It follows that the mapping ‘ 7 ! eð‘=NÞ, where ‘ 2 Z and N is a positive integer, defines an isomorphism between the one-dimensional residual lattice Z=N Z and the multiplicative group of Nth roots of unity. The DFT on N points then relates vectors X and X in W and  W through the linear transformations:

FðNÞ : F
ðNÞ :

1 X   X ðk Þeðk k=NÞ N k 2Z=N Z X X  ðk Þ ¼ XðkÞeðk k=NÞ:

XðkÞ ¼

k2Z=N Z

1.3.3.2.1. The Cooley–Tukey algorithm The presentation of Gentleman & Sande (1966) will be followed first [see also Cochran et al. (1967)]. It will then be reinterpreted in geometric terms which will prepare the way for the treatment of multidimensional transforms in Section 1.3.3.3. Suppose that the number of sample points N is composite, say N ¼ N1 N2 . We may write k to the base N1 and k to the base N2 as follows:

52

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY k ¼ k1 þ N1 k2 k ¼ k2 þ k1 N2

k1 2 Z=N1 Z; k1 2 Z=N1 Z;

The Cooley–Tukey factorization may also be derived from a geometric rather than arithmetic argument. The decomposition k ¼ k1 þ N1 k2 is associated to a geometric partition of the residual lattice Z=N Z into N1 copies of Z=N2 Z , each translated by k1 2 Z=N1 Z and ‘blown up’ by a factor N1 . This partition in turn induces a (direct sum) decomposition of X as

k2 2 Z=N2 Z k2 2 Z=N2 Z:

The defining relation for F
ðNÞ may then be written: X  ðk2 þ k1 N2 Þ ¼

X

X

Xðk1 þ N1 k2 Þ

X¼

P

k1 2Z=N1 Z k2 2Z=N2 Z

k1

   ðk2 þ k1 N2 Þðk1 þ N1 k2 Þ :
e N1 N2

where Xk1 ðkÞ ¼ XðkÞ if k k1 mod N1 ;

The argument of e½: may be expanded as

¼0

k2 k1 k1 k1 k2 k2 þ þ þ k1 k2 ; N N1 N2

X  ðk2 þ k1 N2 Þ ( "
 #) X
k2 k1  X kk e Xðk1 þ N1 k2 Þe 2 2 ¼ N N2 k1 k2
  kk
e 1 1 : N1

X  ðk Þ ¼

k2 2 Z=N2 Z;

k1 2 Z=N1 Z;

(iii) form the N2 vectors Zk2 of length N1 by the prescription
  kk Zk2 ðk1 Þ ¼ e 2 1 Yk1 ðk2 Þ; N

k1 2 Z=N1 Z;

X
k k  1 Yk1 ðk2 Þ; e N k

the periodization by N2 being reflected by the fact that Yk1 does not depend on k1 . Writing k ¼ k2 þ k1 N2 and expanding k k1 shows that the phase shift contains both the twiddle factor eðk2 k1 =NÞ and the kernel eðk1 k1 =N1 Þ of F
ðN1 Þ. The Cooley– Tukey algorithm is thus naturally associated to the coset decomposition of a lattice modulo a sublattice (Section 1.3.2.7.2). It is readily seen that essentially the same factorization can be obtained for FðNÞ, up to the complex conjugation of the twiddle factors. The normalizing constant 1=N arises from the normalizing constants 1=N1 and 1=N2 in FðN1 Þ and FðN2 Þ, respectively. Factors of 2 are particularly simple to deal with and give rise to a characteristic computational structure called a ‘butterfly loop’. If N ¼ 2M, then two options exist: (a) using N1 ¼ 2 and N2 ¼ M leads to collecting the evennumbered coordinates of X into Y0 and the odd-numbered coordinates into Y1

(ii) calculate the N1 transforms Yk1 on N2 points: Yk1 ¼ F
ðN2 Þ½Yk1 ;

by decimation by N1 F
ðNÞ½Xk1  is related and phase shift by

1

This computation may be decomposed into five stages, as follows: (i) form the N1 vectors Yk1 of length N2 by the prescription k1 2 Z=N1 Z;

otherwise:

According to (i), Xk1 is related to Yk1 and offset by k1 . By Section 1.3.2.7.2, to F
ðN2 Þ½Yk1  by periodization by N2 eðk k1 =NÞ, so that

and the last summand, being an integer, may be dropped:

Yk1 ðk2 Þ ¼ Xðk1 þ N1 k2 Þ;

Xk1 ;

k2 2 Z=N2 Z;

Y0 ðk2 Þ ¼ Xð2k2 Þ; Y1 ðk2 Þ ¼ Xð2k2 þ 1Þ;

k2 ¼ 0; . . . ; M  1; k2 ¼ 0; . . . ; M  1;

(iv) calculate the N2 transforms Zk2 on N1 points: Zk2 ¼ F
ðN1 Þ½Zk2 ;

and writing:

k2 2 Z=N2 Z;

X  ðk2 Þ ¼ Y0 ðk2 Þ þ eðk2 =NÞY1 ðk2 Þ; k2 ¼ 0; . . . ; M  1; X  ðk2 þ MÞ ¼ Y0 ðk2 Þ  eðk2 =NÞY1 ðk2 Þ;

(v) collect X  ðk2 þ k1 N2 Þ as Zk ðk1 Þ. 2 If the intermediate transforms in stages (ii) and (iv) are performed in place, i.e. with the results overwriting the data, then at stage (v) the result X  ðk2 þ k1 N2 Þ will be found at address k1 þ N1 k2 . This phenomenon is called scrambling by ‘digit reversal’, and stage (v) is accordingly known as unscrambling. The initial N-point transform F
ðNÞ has thus been performed as N1 transforms F
ðN2 Þ on N2 points, followed by N2 transforms F
ðN1 Þ on N1 points, thereby reducing the arithmetic cost from ðN1 N2 Þ2 to N1 N2 ðN1 þ N2 Þ. The phase shifts applied at stage (iii) are traditionally called ‘twiddle factors’, and the transposition between k1 and k2 can be performed by the fast recursive technique of Eklundh (1972). Clearly, this procedure can be applied recursively if N1 and N2 are themselves composite, leading to an overall arithmetic cost of order N log N if N has no large prime factors.

k2 ¼ 0; . . . ; M  1: This is the original version of Cooley & Tukey, and the process of formation of Y0 and Y1 is referred to as ‘decimation in time’ (i.e. decimation along the data index k). (b) using N1 ¼ M and N2 ¼ 2 leads to forming Z0 ðk1 Þ ¼ Xðk1 Þ þ Xðk1 þ MÞ;


 k Z1 ðk1 Þ ¼ ½Xðk1 Þ  Xðk1 þ MÞe 1 ; N

53

k1 ¼ 0; . . . ; M  1; k1 ¼ 0; . . . ; M  1;

1. GENERAL RELATIONSHIPS AND TECHNIQUES then obtaining separately the even-numbered and odd-numbered components of X by transforming Z0 and Z1 :

‘¼

d P

‘i qi Qi mod N

i¼1

X  ð2k1 Þ ¼ Z0 ðk1 Þ;

k1 ¼ 0; . . . ; M  1;

X  ð2k1 þ 1Þ ¼ Z1 ðk1 Þ;

k1 ¼ 0; . . . ; M  1:

is the solution. Indeed, ‘ ‘j qj Qj mod Nj

This version is due to Sande (Gentleman & Sande, 1966), and the process of separately obtaining even-numbered and oddnumbered results has led to its being referred to as ‘decimation in frequency’ (i.e. decimation along the result index k ). By repeated factoring of the number N of sample points, the calculation of FðNÞ and F
ðNÞ can be reduced to a succession of stages, the smallest of which operate on single prime factors of N. The reader is referred to Gentleman & Sande (1966) for a particularly lucid analysis of the programming considerations which help implement this factorization efficiently; see also Singleton (1969). Powers of two are often grouped together into factors of 4 or 8, which are advantageous in that they require fewer complex multiplications than the repeated use of factors of 2. In this approach, large prime factors P are detrimental, since they require a full P2 -size computation according to the defining formula.

because all terms with i 6¼ j contain Nj as a factor; and qj Qj 1 mod Nj by the defining relation for qj . It may be noted that ðqi Qi Þðqj Qj Þ 0 ðqj Qj Þ qj Qj

Z=N1 Z
Z=N2 Z
. . .
Z=Nd Z

via the two mutually inverse mappings: (i) ‘ 7 ! ð‘1 ; ‘2 ; . . . ; ‘d Þ by ‘ ‘j mod Nj for each j; Pd (ii) ð‘1 ; ‘2 ; . . . ; ‘d Þ 7 ! ‘ by ‘ ¼ i¼1 ‘i qi Qi mod N. The mapping defined by (ii) is sometimes called the ‘CRT reconstruction’ of ‘ from the ‘j. These two mappings have the property of sending sums to sums and products to products, i.e: ðiÞ

‘ þ ‘0 7 ! ð‘1 þ ‘01 ; ‘2 þ ‘02 ; . . . ; ‘d þ ‘0d Þ

ðiiÞ

‘‘0 7 ! ð‘1 ‘01 ; ‘2 ‘02 ; . . . ; ‘d ‘0d Þ ð‘1 þ ‘01 ; ‘2 þ ‘02 ; . . . ; ‘d þ ‘0d Þ 7 ! ‘ þ ‘0 ð‘1 ‘01 ; ‘2 ‘02 ; . . . ; ‘d ‘0d Þ 7 ! ‘‘0

1.3.3.2.2.2. The Chinese remainder theorem Let N ¼ N1 N2 . . . Nd be factored into a product of pairwise coprime integers, so that g.c.d. ðNi ; Nj Þ ¼ 1 for i 6¼ j. Then the system of congruence equations

(the last proof requires using the properties of the idempotents qj Qj ). This may be described formally by stating that the CRT establishes a ring isomorphism: Z=N Z ffi ðZ=N1 ZÞ
. . .
ðZ=Nd ZÞ:

j ¼ 1; . . . ; d;

has a unique solution ‘ mod N. In other words, each ‘ 2 Z=N Z is associated in a one-to-one fashion to the d-tuple ð‘1 ; ‘2 ; . . . ; ‘d Þ of its residue classes in Z=N1 Z; Z=N2 Z; . . . ; Z=Nd Z. The proof of the CRT goes as follows. Let Qj ¼

mod N; j ¼ 1; . . . ; d;

so that the qj Qj are mutually orthogonal idempotents in the ring Z=N Z, with properties formally similar to those of mutually orthogonal projectors onto subspaces in linear algebra. The analogy is exact, since by virtue of the CRT the ring Z=N Z may be considered as the direct product

1.3.3.2.2. The Good (or prime factor) algorithm 1.3.3.2.2.1. Ring structure on Z=N Z The set Z=N Z of congruence classes of integers modulo an integer N [see e.g. Apostol (1976), Chapter 5] inherits from Z not only the additive structure used in deriving the Cooley–Tukey factorization, but also a multiplicative structure in which the product of two congruence classes mod N is uniquely defined as the class of the ordinary product (in Z) of representatives of each class. The multiplication can be distributed over addition in the usual way, endowing Z=N Z with the structure of a commutative ring. If N is composite, the ring Z=N Z has zero divisors. For example, let N ¼ N1 N2 , let n1 N1 mod N, and let n2 N2 mod N: then n1 n2 0 mod N. In the general case, a product of nonzero elements will be zero whenever these elements collect together all the factors of N. These circumstances give rise to a fundamental theorem in the theory of commutative rings, the Chinese Remainder Theorem (CRT), which will now be stated and proved [see Apostol (1976), Chapter 5; Schroeder (1986), Chapter 16].

‘ ‘j mod Nj ;

mod N for i 6¼ j;

2

1.3.3.2.2.3. The prime factor algorithm The CRT will now be used to factor the N-point DFT into a tensor product of d transforms, the jth of length Nj . Let the indices k and k be subjected to the following mappings: (i) k 7 ! ðk1 ; k2 ; . . . ; kd Þ; kj 2 Z=Nj Z, by kj k mod Nj for each j, with reconstruction formula

N Y ¼ Ni : Nj i6¼j

Since g.c.d. ðNj ; Qj Þ ¼ 1 there exist integers nj and qj such that

k¼

d P

ki qi Qi mod N;

i¼1

nj Nj þ qj Qj ¼ 1;

j ¼ 1; . . . ; d; (ii) k 7 ! ðk1 ; k2 ; . . . ; kd Þ; kj 2 Z=Nj Z, by kj qj k mod Nj for each j, with reconstruction formula

then the integer

54

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 

k ¼

d P

ki Qi

require twiddle factors). Thus, the DFT on a prime number of points remains undecomposable.

mod N:

i¼1

1.3.3.2.3. The Rader algorithm The previous two algorithms essentially reduce the calculation of the DFT on N points for N composite to the calculation of smaller DFTs on prime numbers of points, the latter remaining irreducible. However, Rader (1968) showed that the p-point DFT for p an odd prime can itself be factored by invoking some extra arithmetic structure present in Z=pZ.

Then 

k k¼


d P

ki Qi



i¼1

¼

d P

d P

! kj qj Qj

mod N

j¼1

ki kj Qi qj Qj mod N:

i; j¼1

1.3.3.2.3.1. N an odd prime The ring Z=pZ ¼ f0; 1; 2; . . . ; p  1g has the property that its p  1 nonzero elements, called units, form a multiplicative group UðpÞ. In particular, all units r 2 UðpÞ have a unique multiplicative inverse in Z=pZ, i.e. a unit s 2 UðpÞ such that rs 1 mod p. This endows Z=pZ with the structure of a finite field. Furthermore, UðpÞ is a cyclic group, i.e. consists of the successive powers gm mod p of a generator g called a primitive root mod p (such a g may not be unique, but it always exists). For instance, for p ¼ 7, Uð7Þ ¼ f1; 2; 3; 4; 5; 6g is generated by g ¼ 3, whose successive powers mod 7 are:

Cross terms with i 6¼ j vanish since they contain all the factors of N, hence k k ¼

d P

qj Q2j kj kj mod N

j¼1

¼

d P

ð1  nj Nj ÞQj kj kj mod N:

j¼1

Dividing by N, which may be written as Nj Qj for each j, yields g0 ¼ 1;

d Qj  kk X ¼ ð1  nj Nj Þ k k mod 1 N Nj Qj j j j¼1  d
X 1 ¼  nj kj kj mod 1; Nj j¼1

g1 ¼ 3;

g2 ¼ 2;

g3 ¼ 6;

g4 ¼ 4;

g5 ¼ 5



[see Apostol (1976), Chapter 10]. The basis of Rader’s algorithm is to bring to light a hidden regularity in the matrix FðpÞ by permuting the basis vectors uk and vk of LðZ=pZÞ as follows:

and hence 

kk N

d X kj kj j¼1

Nj

u00 ¼ u0 u0m ¼ uk

mod 1:

v00

with k ¼ gm ;

m ¼ 1; . . . ; p  1;

¼ v0

v0m ¼ vk



with k ¼ gm ;

m ¼ 1; . . . ; p  1;

Therefore, by the multiplicative property of eð:Þ,
 
  O d kj kj kk : e e N Nj j¼1

where g is a primitive root mod p. With respect to these new bases, the matrix representing F
ðpÞ will have the following elements:

Let X 2 LðZ=N ZÞ be described by a one-dimensional array XðkÞ indexed by k. The index mapping (i) turns X into an element of LðZ=N1 Z
. . .
Z=Nd ZÞ described by a d-dimensional array N Xðk1 ;N . . . ; kd Þ; the latter may be transformed by F
ðN1 Þ . . . F
ðNd Þ into a new array X  ðk1 ; k2 ; . . . ; kd Þ. Finally, the one-dimensional array of results X  ðk Þ will be obtained by reconstructing k according to (ii). The prime factor algorithm, like the Cooley–Tukey algorithm, reindexes a 1D transform to turn it into d separate transforms, but the use of coprime factors and CRT index mapping leads to the further gain that no twiddle factors need to be applied between the successive transforms (see Good, 1971). This makes up for the cost of the added complexity of the CRT index mapping. The natural factorization of N for the prime factor algorithm is thus its factorization into prime powers: F
ðNÞ is then the tensor product of separate transforms (one for each prime power factor  Nj ¼ pj j ) whose results can be reassembled without twiddle factors. The separate factors pj within each Nj must then be dealt with by another algorithm (e.g. Cooley–Tukey, which does

element ð0; 0Þ ¼ 1 element ð0; m þ 1Þ ¼ 1

for all m ¼ 0; . . . p  2;



element ðm þ 1; 0Þ ¼ 1 for all m ¼ 0; . . . ; p  2;
  kk element ðm þ 1; m þ 1Þ ¼ e p 

¼ eðgðm þmÞ=p Þ for all m ¼ 0; . . . ; p  2:

Thus the ‘core’ C
ðpÞ of matrix F
ðpÞ, of size ðp  1Þ
ðp  1Þ, formed by the elements with two nonzero indices, has a so-called skew-circulant structure because element ðm ; mÞ depends only on m þ m. Simplification may now occur because multiplication by C
ðpÞ is closely related to a cyclic convolution. Introducing the notation CðmÞ ¼ eðgm=p Þ we may write the relation Y ¼ F
ðpÞY in the permuted bases as

55

1. GENERAL RELATIONSHIPS AND TECHNIQUES Y  ð0Þ ¼

P

Zðk2 Þ ¼ Xðpk2 Þ;

YðkÞ

k2 2 Z=p1 Z

k

Y  ðm þ 1Þ ¼ Yð0Þ þ

p2 P

Cðm þ mÞYðm þ 1Þ

(the p1 -periodicity follows implicity from the fact that the transform on the right-hand side is independent of k1 2 Z=pZ). Finally, the contribution X1 from all k 2 Uðp Þ may be calculated by reindexing by the powers of a primitive root g modulo p, i.e. by writing

m¼0

¼ Yð0Þ þ

p2 P

Cðm  mÞZðmÞ

m¼0

¼ Yð0Þ þ ðC  ZÞðm Þ;

m ¼ 0; . . . ; p  2;



X1 ðgm Þ ¼

where Z is defined by ZðmÞ ¼ Yðp  m  2Þ, m ¼ 0; . . . ; p  2. Thus Y may be obtained by cyclic convolution of C and Z, which may for instance be calculated by

qP  1

Xðgm Þeðgðm

 þmÞ=p

Þ

m¼0

then carrying out the multiplication by the skew-circulant matrix core as a convolution. Thus the DFT of size p may be reduced to two DFTs of size 1 p (dealing, respectively, with p-decimated results and p-decimated data) and a convolution of size q ¼ p1 ðp  1Þ. The latter may be ‘diagonalized’ into a multiplication by purely real or purely imaginary numbers (because gðq =2Þ ¼ 1) by two DFTs, whose factoring in turn leads to DFTs of size p1 and p  1. This method, applied recursively, allows the complete decomposition of the DFT on p points into arbitrarily small DFTs.

C  Z ¼ Fðp  1Þ½F
ðp  1Þ½C
F
ðp  1Þ½Z; where
denotes the component-wise multiplication of vectors. Since p is odd, p  1 is always divisible by 2 and may even be highly composite. In that case, factoring F
ðp  1Þ by means of the Cooley–Tukey or Good methods leads to an algorithm of complexity p log p rather than p2 for F
ðpÞ. An added bonus is that, because gðp1Þ=2 ¼ 1, the elements of F
ðp  1Þ½C can be shown to be either purely real or purely imaginary, which halves the number of real multiplications involved.

1.3.3.2.3.3. N a power of 2 When N ¼ 2 , the same method can be applied, except for a slight modification in the calculation of X1 . There is no primitive root modulo 2 for  > 2: the group Uð2 Þ is the direct product of two cyclic groups, the first (of order 2) generated by 1, the second (of order N=4) generated by 3 or 5. One then uses a representation

1.3.3.2.3.2. N a power of an odd prime This idea was extended by Winograd (1976, 1978) to the treatment of prime powers N ¼ p, using the cyclic structure of the multiplicative group of units Uðp Þ. The latter consists of all those elements of Z=p Z which are not divisible by p, and thus has q ¼ p1 ðp  1Þ elements. It is cyclic, and there exist primitive roots g modulo p such that

k ¼ ð1Þm1 5m2 Uðp Þ ¼ f1; g; g2 ; g3 ; . . . ; gq 1 g:

k ¼ ð1Þm1 5m2

The p1 elements divisible by p, which are divisors of zero, have to be treated separately just as 0 had to be treated separately for N ¼ p. When k 62 Uðp Þ, then k ¼ pk1 with k1 2 Z=p1 Z. The results X  ðpk1 Þ are p-decimated, hence can be obtained via the p1 -point DFT of the p1 -periodized data Y:

and the reindexed core matrix gives rise to a two-dimensional convolution. The latter may be carried out by means of two 2D DFTs on 2
ðN=4Þ points.



1.3.3.2.4. The Winograd algorithms The cyclic convolutions generated by Rader’s multiplicative reindexing may be evaluated more economically than through DFTs if they are re-examined within a new algebraic setting, namely the theory of congruence classes of polynomials [see, for instance, Blahut (1985), Chapter 2; Schroeder (1986), Chapter 24]. The set, denoted K½X, of polynomials in one variable with coefficients in a given field K has many of the formal properties of the set Z of rational integers: it is a ring with no zero divisors and has a Euclidean algorithm on which a theory of divisibility can be built. Given a polynomial PðzÞ, then for every WðzÞ there exist unique polynomials QðzÞ and RðzÞ such that

X  ðpk1 Þ ¼ F
ðp1 Þ½Yðk1 Þ with Yðk1 Þ ¼

P k2 2Z=pZ



Xðk1 þ p1 k2 Þ:

When k 2 Uðp Þ, then we may write X  ðk Þ ¼ X0 ðk Þ þ X1 ðk Þ;

WðzÞ ¼ PðzÞQðzÞ þ RðzÞ X0



X1

where contains the contributions from k 2 = Uðp Þ and those from k 2 Uðp Þ. By a converse of the previous calculation, X0 arises from p-decimated data Z, hence is the p1 -periodization of the p1 -point DFT of these data:

and degree ðRÞ < degree ðPÞ:

X0 ðp1 k1 þ k2 Þ ¼ F
ðp1 Þ½Zðk2 Þ

RðzÞ is called the residue of HðzÞ modulo PðzÞ. Two polynomials H1 ðzÞ and H2 ðzÞ having the same residue modulo PðzÞ are said to be congruent modulo PðzÞ, which is denoted by

with

56

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY H1 ðzÞ H2 ðzÞ mod PðzÞ:

WðzÞ UðzÞVðzÞ mod ðzN  1Þ:

If HðzÞ 0 mod PðzÞ; HðzÞ is said to be divisible by PðzÞ. If HðzÞ only has divisors of degree zero in K½X, it is said to be irreducible over K (this notion depends on K). Irreducible polynomials play in K½X a role analogous to that of prime numbers in Z, and any polynomial over K has an essentially unique factorization as a product of irreducible polynomials. There exists a Chinese remainder theorem (CRT) for polynomials. Let PðzÞ ¼ P1 ðzÞ . . . Pd ðzÞ be factored into a product of pairwise coprime polynomials [i.e. Pi ðzÞ and Pj ðzÞ have no common factor for i 6¼ j]. Then the system of congruence equations

Now the polynomial zN  1 can be factored over the field of rational numbers into irreducible factors called cyclotomic polynomials: if d is the number of divisors of N, including 1 and N, then zN  1 ¼

where the cyclotomics Pi ðzÞ are well known (Nussbaumer, 1981; Schroeder, 1986, Chapter 22). We may now invoke the CRT, and exploit the ring isomorphism it establishes to simplify the calculation of WðzÞ from UðzÞ and VðzÞ as follows: (i) compute the d residual polynomials

has a unique solution HðzÞ modulo PðzÞ. This solution may be constructed by a procedure similar to that used for integers. Let Qj ðzÞ ¼ PðzÞ=Pj ðzÞ ¼

Q

Pi ðzÞ;

i¼1

j ¼ 1; . . . ; d;

HðzÞ Hj ðzÞ mod Pj ðzÞ;

d Q

Pi ðzÞ:

Ui ðzÞ UðzÞ mod Pi ðzÞ;

i ¼ 1; . . . ; d;

Vi ðzÞ VðzÞ mod Pi ðzÞ;

i ¼ 1; . . . ; d;

i6¼j

(ii) compute the d polynomial products

Then Pj and Qj are coprime, and the Euclidean algorithm may be used to obtain polynomials pj ðzÞ and qj ðzÞ such that

Wi ðzÞ Ui ðzÞVi ðzÞ mod Pi ðzÞ;

i ¼ 1; . . . ; d;

pj ðzÞPj ðzÞ þ qj ðzÞQj ðzÞ ¼ 1: (iii) use the CRT reconstruction formula just proved to recover WðzÞ from the Wi ðzÞ:

With Si ðzÞ ¼ qi ðzÞQi ðzÞ, the polynomial HðzÞ ¼

d P

Si ðzÞHi ðzÞ mod PðzÞ

WðzÞ

i¼1

When N is not too large, i.e. for ‘short cyclic convolutions’, the Pi ðzÞ are very simple, with coefficients 0 or 1, so that (i) only involves a small number of additions. Furthermore, special techniques have been developed to multiply general polynomials modulo cyclotomic polynomials, thus helping keep the number of multiplications in (ii) and (iii) to a minimum. As a result, cyclic convolutions can be calculated rapidly when N is sufficiently composite. It will be recalled that Rader’s multiplicative indexing often gives rise to cyclic convolutions of length p  1 for p an odd prime. Since p  1 is highly composite for all p  50 other than 23 and 47, these cyclic convolutions can be performed more efficiently by the above procedure than by DFT. These combined algorithms are due to Winograd (1977, 1978, 1980), and are known collectively as ‘Winograd small FFT algorithms’. Winograd also showed that they can be thought of as bringing the DFT matrix F to the following ‘normal form’:

K½X mod P ffi ðK½X mod P1 Þ
. . .
ðK½X mod Pd Þ:

These results will now be applied to the efficient calculation of cyclic convolutions. Let U ¼ ðu0 ; u1 ; . . . ; uN1 Þ and V ¼ ðv0 ; v1 ; . . . ; vN1 Þ be two vectors of length N, and let W ¼ ðw0 ; w1 ; . . . ; wN1 Þ be obtained by cyclic convolution of U and V: N1 P

um vnm ;

n ¼ 0; . . . ; N  1:

m¼0

The very simple but crucial result is that this cyclic convolution may be carried out by polynomial multiplication modulo ðzN  1Þ: if UðzÞ ¼

N1 P

F ¼ CBA; where A is an integer matrix with entries 0, 1, defining the ‘preadditions’, B is a diagonal matrix of multiplications, C is a matrix with entries 0, 1, i, defining the ‘postadditions’. The elements on the diagonal of B can be shown to be either real or pure imaginary, by the same argument as in Section 1.3.3.2.3.1. Matrices A and C may be rectangular rather than square, so that intermediate results may require extra storage space.

ul zl

l¼0

VðzÞ ¼

N1 P

vm zm

m¼0

WðzÞ ¼

Si ðzÞWi ðzÞ mod ðzN  1Þ:

i¼1

is easily shown to be the desired solution. As with integers, it can be shown that the 1:1 correspondence between HðzÞ and Hj ðzÞ sends sums to sums and products to products, i.e. establishes a ring isomorphism:

wn ¼

d P

N1 P

wn zn

n¼0

then the above relation is equivalent to

57

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1.3.3.3. Multidimensional algorithms

N ¼ N1 N2 . . . Nd1 Nd

From an algorithmic point of view, the distinction between one-dimensional (1D) and multidimensional DFTs is somewhat blurred by the fact that some factoring techniques turn a 1D transform into a multidimensional one. The distinction made here, however, is a practical one and is based on the dimensionality of the indexing sets for data and results. This section will therefore be concerned with the problem of factoring the DFT when the indexing sets for the input data and output results are multidimensional.

and hence NT ¼ NTd NTd1 . . . NT2 NT1 : Then the coset decomposition formulae corresponding to these successive decimations (Section 1.3.2.7.1) can be combined as follows:

1.3.3.3.1. The method of successive one-dimensional transforms The DFT was defined in Section 1.3.2.7.4 in an n-dimensional setting and it was shown that when the decimation matrix N is diagonal, say N ¼ diagðN ð1Þ ; N ð2Þ ; . . . ; N ðnÞ Þ, then F
ðNÞ has a tensor product structure:

Zn ¼

[ k1

¼

[

ðk1 þ N1 Zn Þ (

" k1 þ N1

k1

F
ðNÞ ¼ F
ðN ð1Þ Þ  F
ðN ð2Þ Þ  . . .  F
ðN ðnÞ Þ:

[

#) ðk2 þ N2 Z Þ n

k2

¼ ... [ [ ¼ ... ðk1 þ N1 k2 þ . . . þ N1 N2
. . .
Nd1 kd þ NZn Þ

This may be rewritten as follows:

k1

F
ðNÞ ¼ ½F
ðN ð1Þ Þ  IN ð2Þ  . . .  IN ðnÞ 
½INð1Þ  F
ðN ð2Þ Þ  . . .  IN ðnÞ 

kd

with kj 2 Zn =Nj Zn . Therefore, any k 2 Z=NZn may be written uniquely as


... k ¼ k1 þ N1 k2 þ . . . þ N1 N2
. . .
Nd1 kd :


½INð1Þ  IN ð2Þ  . . .  F
ðN ðnÞ ; Similarly:

where the I’s are identity matrices and
denotes ordinary matrix multiplication. The matrix within each bracket represents a onedimensional DFT along one of the n dimensions, the other dimensions being left untransformed. As these matrices commute, the order in which the successive 1D DFTs are performed is immaterial. This is the most straightforward method for building an n-dimensional algorithm from existing 1D algorithms. It is known in crystallography under the name of ‘Beevers–Lipson factorization’ (Section 1.3.4.3.1), and in signal processing as the ‘row– column method’.

Zn ¼

[

ðkd þ NTd Zn Þ

kd

¼ ... [ [ ¼ ... ðkd þ NTd kd1 þ . . . þ NTd
. . .
NT2 k1 kd

k1

þN Z Þ T

n

so that any k 2 Zn =NT Zn may be written uniquely as

1.3.3.3.2. Multidimensional factorization Substantial reductions in the arithmetic cost, as well as gains in flexibility, can be obtained if the factoring of the DFT is carried out in several dimensions simultaneously. The presentation given here is a generalization of that of Mersereau & Speake (1981), using the abstract setting established independently by Auslander, Tolimieri & Winograd (1982). Let us return to the general n-dimensional setting of Section 1.3.2.7.4, where the DFT was defined for an arbitrary decimation matrix N by the formulae (where jNj denotes jdet Nj):

k ¼ kd þ NTd kd1 þ . . . þ NTd
. . .
NT2 k1 with kj 2 Zn =NTj Zn . These decompositions are the vector analogues of the multi-radix number representation systems used in the Cooley–Tukey factorization. We may then write the definition of F
ðNÞ with d ¼ 2 factors as X  ðk2 þ NT2 k1 Þ ¼

PP

Xðk1 þ N1 k2 Þ

k1 k2

FðNÞ : F
ðNÞ :

1 X   X ðk Þe½k  ðN1 kÞ jNj k X XðkÞe½k  ðN1 kÞ X  ðk Þ ¼

T 1 1
e½ðkT 2 þ k1 N2 ÞN2 N1 ðk1 þ N1 k2 Þ:

XðkÞ ¼

The argument of e(–) may be expanded as

k  1  k2  ðN1 k1 Þ þ k1  ðN1 1 k1 Þ þ k2  ðN2 k2 Þ þ k1  k2 :

with k 2 Zn =NZn ;

The first summand may be recognized as a twiddle factor, the second and third as the kernels of F
ðN1 Þ and F
ðN2 Þ, respectively, while the fourth is an integer which may be dropped. We are thus led to a ‘vector-radix’ version of the Cooley–Tukey algorithm, in which the successive decimations may be introduced in all n dimensions simultaneously by general integer matrices. The computation may be decomposed into five stages analogous to those of the one-dimensional algorithm of Section 1.3.3.2.1:

k 2 Zn =NT Zn :

1.3.3.3.2.1. Multidimensional Cooley–Tukey factorization Let us now assume that this decimation can be factored into d successive decimations, i.e. that

58

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY "

(i) form the jN1 j vectors Yk1 of shape N2 by Yk1 ðk2 Þ ¼ Xðk1 þ N1 k2 Þ;

k1 2 Zn =N1 Zn ;

Zk2 ðk1 Þ ¼

k2 2 Zn =N2 Zn ;

¼

P k2

e½k2



ðN1 2 k2 ÞYk1 ðk2 Þ;

n

k1 2 Zn =N1 Zn ;

k2 2 Zn =NT2 Zn ; (iv) calculate the jN2 j transforms Zk2 on jN1 j points: Zk2 ðk1 Þ ¼

P k1

e½k1  ðN1 1 k1 ÞZk2 ðk1 Þ;

Xðk1 þ Mk2 Þ

i.e. the 2n parity classes of results, corresponding to the different k2 2 Zn =2Zn , are obtained separately. When the dimension n is 2 and the decimating matrix is diagonal, this analysis reduces to the ‘vector radix FFT’ algorithms proposed by Rivard (1977) and Harris et al. (1977). These lead to substantial reductions in the number M of multiplications compared to the row–column method: M is reduced to 3M=4 by simultaneous 2
2 factoring, and to 15M=32 by simultaneous 4
4 factoring. The use of a nondiagonal decimating matrix may bring savings in computing time if the spectrum of the band-limited function under study is of such a shape as to pack more compactly in a nonrectangular than in a rectangular lattice (Mersereau, 1979). If, for instance, the support K of the spectrum  is contained in a sphere, then a decimation matrix producing a close packing of these spheres will yield an aliasing-free DFT algorithm with fewer sample points than the standard algorithm using a rectangular lattice.

(iii) form the jN2 j vectors Zk2 of shape N1 by Zk2 ðk1 Þ ¼ e½k2  ðN1 k1 ÞYk1 ðk2 Þ;

k2 2Zn =2Zn

X  ðk2 þ 2k1 Þ ¼ Zk2 ðk1 Þ;

k1 2 Z =N1 Z ; n

ð1Þ

k2 k2


e½k2  ðN1 k1 Þ; Zk2 ¼ F
ðMÞ½Zk2 ;

(ii) calculate the jN1 j transforms Yk1 on jN2 j points: Yk1 ðk2 Þ

#

P

k2 2 Zn =NT2 Zn ;

(v) collect X  ðk2 þ NT2 k1 Þ as Zk ðk1 Þ. 2 The initial jNj-point transform F
ðNÞ can thus be performed as jN1 j transforms F
ðN2 Þ on jN2 j points, followed by jN2 j transforms F
ðN1 Þ on jN1 j points. This process can be applied successively to all d factors. The same decomposition applies to FðNÞ, up to the complex conjugation of twiddle factors, the normalization factor 1=jNj being obtained as the product of the factors 1=jNj j in the successive partial transforms FðNj Þ. The geometric interpretation of this factorization in terms of partial transforms on translates of sublattices applies in full to this n-dimensional setting; in particular, the twiddle factors are seen to be related to the residual translations which place the sublattices in register within the big lattice. If the intermediate transforms are performed in place, then the quantity

1.3.3.3.2.2. Multidimensional prime factor algorithm Suppose that the decimation matrix N is diagonal N ¼ diag ðN ð1Þ ; N ð2Þ ; . . . ; N ðnÞ Þ and let each diagonal element be written in terms of its prime factors: N ðiÞ ¼

m Q

jÞ pði; ; j

j¼1

X  ðkd þ NTd kd1 þ . . . þ NTd NTd1
. . .
NT2 k1 Þ

where m is the total number of distinct prime factors present in the N ðiÞ. The CRT may be used to turn each 1D transform along dimension i ði ¼ 1; . . . ; nÞ into a multidimensional transform with a separate ‘pseudo-dimension’ for each distinct prime factor of N ðiÞ ; the number i , of these pseudo-dimensions is equal to the cardinality of the set:

will eventually be found at location k1 þ N1 k2 þ . . . þ N1 N2
. . .
Nd1 kd ; so that the final results will have to be unscrambled by a process which may be called ‘coset reversal’, the vector equivalent of digit reversal. Factoring by 2 in all n dimensions simultaneously, i.e. taking N ¼ 2M, leads to ‘n-dimensional butterflies’. Decimation in time corresponds to the choice N1 ¼ 2I; N2 ¼ M, so that k1 2 Zn =2Zn is an n-dimensional parity class; the calculation then proceeds by

f j 2 f1; . . . ; mgjði; jÞ > 0 for some ig: The full n-dimensional transform thus becomes -dimensional, Pn with ¼ i¼1 i . We may now permute the pseudo-dimensions so as to bring into contiguous position those corresponding to the same prime factor pj ; the m resulting groups of pseudo-dimensions are said to define ‘p-primary’ blocks. The initial transform is now written as a tensor product of m p-primary transforms, where transform j is on

Yk1 ðk2 Þ ¼ Xðk1 þ 2k2 Þ; k1 2 Zn =2Zn ; k2 2 Zn =MZn ; Yk1 ¼ F
ðMÞ½Yk1 ; k1 2 Zn =2Zn ; P  X  ðk2 þ MT k1 Þ ¼ ð1Þk1 k1

jÞ pjð1; jÞ
pjð2; jÞ
. . .
pðn; j

k1 2Zn =2Zn


e½k2  ðN1 k1 ÞYk1 ðk2 Þ:

points [by convention, dimension i is not transformed if ði; jÞ ¼ 0]. These p-primary transforms may be computed, for instance, by multidimensional Cooley–Tukey factorization (Section 1.3.3.3.1), which is faster than the straightforward row– column method. The final results may then be obtained by reversing all the permutations used.

Decimation in frequency corresponds to the choice N1 ¼ M, N2 ¼ 2I, so that k2 2 Zn =2Zn labels ‘octant’ blocks of shape M; the calculation then proceeds through the following steps:

59

1. GENERAL RELATIONSHIPS AND TECHNIQUES The extra gain with respect to the multidimensional Cooley– Tukey method is that there are no twiddle factors between p-primary pieces corresponding to different primes p. The case where N is not diagonal has been examined by Guessoum & Mersereau (1986).

Qk2 ðzÞ ¼ Rk2 ðzÞ ¼

1.3.3.3.2.3. Nesting of Winograd small FFTs Suppose that the CRT has been used as above to map an n-dimensional DFT to a -dimensional DFT. For each  ¼ 1; . . . ; [ runs over those pairs (i, j) such that ði; jÞ > 0], the Rader/Winograd procedure may be applied to put the matrix of the th 1D DFT in the CBA normal form of a Winograd small FFT. The full DFT matrix may then be written, up to permutation of data and results, as O

P P k2



X  ðk1 ; k2 Þ ¼ Rk2 ð!k1 Þ ¼

ðC B A Þ:

N1 P

A well known property of the tensor product of matrices allows this to be rewritten as

C




¼1

O

!

O

B


f ðk2 Þ ¼

!

X  ðk1 ; k2 Þ ¼

P k2

A

N1 P

f ðk1 k2 Þ

 



!k1 k2 k2 Qk1 k2 ð!k1 Þ 

¼ Sk1 k2 ð!k1 Þ where Sk ðzÞ ¼

P k2



zk k2 Qk2 ðzÞ:



Since only the value of polynomial Sk ðzÞ at z ¼ !k1 is involved in the result, the computation of Sk may be carried out modulo the  unique cyclotomic polynomial PðzÞ such that Pð!k1 Þ ¼ 0. Thus, if we define:

1.3.3.3.2.4. The Nussbaumer–Quandalle algorithm Nussbaumer’s approach views the DFT as the evaluation of certain polynomials constructed from the data (as in Section 1.3.3.2.4). For instance, putting ! ¼ eð1=NÞ, the 1D N-point DFT

Tk ðzÞ ¼

P k2

X  ðk Þ ¼



for any function f over Z=N Z. We may thus write:

and thus to form a matrix in which the combined pre-addition, multiplication and post-addition matrices have been precomputed. This procedure, called nesting, can be shown to afford a reduction of the arithmetic operation count compared to the row–column method (Morris, 1978). Clearly, the nesting rearrangement need not be applied to all dimensions, but can be restricted to any desired subset of them.

N1 P

k2



!k2 k2 Qk2 ð!k1 Þ:

k2 ¼0

¼1

¼1

P

Let us now suppose that k1 is coprime to N. Then k1 has a unique inverse modulo N (denoted by 1=k1 ), so that multiplication by k1 simply permutes the elements of Z=N Z and hence

k2 ¼0

!



!k2 k2 Qk2 ðzÞ;

this may be rewritten:

¼1

O

Xðk1 ; k2 Þzk1

k1



zk k2 Qk2 ðzÞ mod PðzÞ

k

XðkÞ!k

we may write:

k¼0



X  ðk1 ; k2 Þ ¼ Tk1 k2 ð!k1 Þ

may be written 

X  ðk Þ ¼ Qð!k Þ;

or equivalently
 k  X  k1 ; 2 ¼ Tk2 ð!k1 Þ: k1

where the polynomial Q is defined by

QðzÞ ¼

N1 P

XðkÞzk :

For N an odd prime p, all nonzero values of k1 are coprime with p so that the p
p-point DFT may be calculated as follows: (1) form the polynomials

k¼0

Let us consider (Nussbaumer & Quandalle, 1979) a 2D transform of size N
N: X  ðk1 ; k2 Þ ¼

N1 P P N1



Tk2 ðzÞ ¼

PP



Xðk1 ; k2 Þzk1 þk2 k2 mod PðzÞ

k1 k2



Xðk1 ; k2 Þ!k1 k1 þk2 k2 :

for k2 ¼ 0; . . . ; p  1;  (2) evaluate Tk2 ð!k1 Þ for k1 ¼ 0; . . . ; p  1;     (3) put X ðk1 ; k2 =k1 Þ ¼ Tk2 ð!k1 Þ; (4) calculate the terms for k1 ¼ 0 separately by

k1 ¼0 k2 ¼0

By introduction of the polynomials

60

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY X



ð0; k2 Þ

¼

" P P k2

# 

Xðk1 ; k2 Þ !k2 k2 :

k1

Step (1) is a set of p ‘polynomial transforms’ involving no multiplications; step (2) consists of p DFTs on p points each since if Tk2 ðzÞ ¼

P k1

Yk2 ðk1 Þzk1

then 

Tk2 ð!k1 Þ ¼

P k1



Yk2 ðk1 Þ!k1 k1 ¼ Yk2 ðk1 Þ;

step (3) is a permutation; and step (4) is a p-point DFT. Thus the 2D DFT on p
p points, which takes 2p p-point DFTs by the row–column method, involves only ðp þ 1Þ p-point DFTs; the other DFTs have been replaced by polynomial transforms involving only additions. This procedure can be extended to n dimensions, and reduces the number of 1D p-point DFTs from npn1 for the row–column method to ðpn  1Þ=ðp  1Þ, at the cost of introducing extra additions in the polynomial transforms. A similar algorithm has been formulated by Auslander et al. (1983) in terms of Galois theory.

Fig. 1.3.3.1. A few global algorithms for computing a 400-point DFT. CT: Cooley–Tukey factorization. PF: prime factor (or Good) factorization. W: Winograd algorithm.

arithmetic on addresses, although some shortcuts may be found (Uhrich, 1969; Burrus & Eschenbacher, 1981); (ii) reduction in the f.p. multiplication count usually leads to a large increase in the f.p. addition count (Morris, 1978); (iii) nesting can increase execution speed, but causes a loss of modularity and hence complicates program development (Silverman, 1977; Kolba & Parks, 1977). Many of the mathematical developments above took place in the context of single-processor serial computers, where f.p. addition is substantially cheaper than f.p. multiplication but where integer address arithmetic has to compete with f.p. arithmetic for processor cycles. As a result, the alternatives to the Cooley–Tukey algorithm hardly ever led to particularly favourable trade-offs, thus creating the impression that there was little to gain by switching to more exotic algorithms. The advent of new machine architectures with vector and/or parallel processing features has greatly altered this picture (Pease, 1968; Korn & Lambiotte, 1979; Fornberg, 1981; Swartzrauber, 1984): (i) pipelining equalizes the cost of f.p. addition and f.p. multiplication, and the ideal ‘blend’ of the two types of operations depends solely on the number of adder and multiplier units available in each machine; (ii) integer address arithmetic is delegated to specialized arithmetic and logical units (ALUs) operating concurrently with the f.p. units, so that complex reindexing schemes may be used without loss of overall efficiency. Another major consideration is that of data flow [see e.g. Nawab & McClellan (1979)]. Serial machines only have few registers and few paths connecting them, and allow little or no overlap between computation and data movement. New architectures, on the other hand, comprise banks of vector registers (or ‘cache memory’) besides the usual internal registers, and dedicated ALUs can service data transfers between several of them simultaneously and concurrently with computation. In this new context, the devices described in Sections 1.3.3.2 and 1.3.3.3 for altering the balance between the various types of arithmetic operations, and reshaping the data flow during the computation, are invaluable. The field of machine-dependent DFT algorithm design is thriving on them [see e.g. Temperton (1983a,b,c, 1985); Agarwal & Cooley (1986, 1987)].

1.3.3.3.3. Global algorithm design 1.3.3.3.3.1. From local pieces to global algorithms The mathematical analysis of the structure of DFT computations has brought to light a broad variety of possibilities for reducing or reshaping their arithmetic complexity. All of them are ‘analytic’ in that they break down large transforms into a succession of smaller ones. These results may now be considered from the converse ‘synthetic’ viewpoint as providing a list of procedures for assembling them: (i) the building blocks are one-dimensional p-point algorithms for p a small prime; (ii) the low-level connectors are the multiplicative reindexing methods of Rader and Winograd, or the polynomial transform reindexing method of Nussbaumer and Quandalle, which allow the construction of efficient algorithms for larger primes p, for prime powers p, and for p-primary pieces of shape p
. . .
p ; (iii) the high-level connectors are the additive reindexing scheme of Cooley–Tukey, the Chinese remainder theorem reindexing, and the tensor product construction; (iv) nesting may be viewed as the ‘glue’ which seals all elements. The simplest DFT may then be carried out into a global algorithm in many different ways. The diagrams in Fig. 1.3.3.1 illustrate a few of the options available to compute a 400-point DFT. They may differ greatly in their arithmetic operation counts. 1.3.3.3.3.2. Computer architecture considerations To obtain a truly useful measure of the computational complexity of a DFT algorithm, its arithmetic operation count must be tempered by computer architecture considerations. Three main types of trade-offs must be borne in mind: (i) reductions in floating-point (f.p.) arithmetic count are obtained by reindexing, hence at the cost of an increase in integer

61

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1.3.3.3.3.3. The Johnson–Burrus family of algorithms In order to explore systematically all possible algorithms for carrying out a given DFT computation, and to pick the one best suited to a given machine, attempts have been made to develop: (i) a high-level notation of describing all the ingredients of a DFT computation, including data permutation and data flow; (ii) a formal calculus capable of operating on these descriptions so as to represent all possible reorganizations of the computation; (iii) an automatic procedure for evaluating the performance of a given algorithm on a specific architecture. Task (i) can be accomplished by systematic use of a tensor product notation to represent the various stages into which the DFT can be factored (reindexing, small transforms on subsets of indices, twiddle factors, digit-reversal permutations). Task (ii) may for instance use the Winograd CBA normal form for each small transform, then apply N the rules governing the rearrangement of tensor product and ordinary product
operations on matrices. The matching of these rearrangements to the architecture of a vector and/or parallel computer can be formalized algebraically [see e.g. Chapter 2 of Tolimieri et al. (1989)]. Task (iii) is a complex search which requires techniques such as dynamic programming (Bellman, 1958). Johnson & Burrus (1983) have proposed and tested such a scheme to identify the optimal trade-offs between prime factor nesting and Winograd nesting of small Winograd transforms. In step (ii), they further decomposed the pre-addition matrix A and post-addition matrix C into several factors, so that the number of design options available becomes very large: the N-point DFT when N has four factors can be calculated in over 1012 distinct ways. This large family of nested algorithms contains the prime factor algorithm and the Winograd algorithms as particular cases, but usually achieves greater efficiency than either by reducing the f.p. multiplication count while keeping the number of f.p. additions small. There is little doubt that this systematic approach will be extended so as to incorporate all available methods of restructuring the DFT.

FðHÞ ¼ F
½ðHÞ ¼ hx ; expð2 iH  XÞi: F is still a well behaved function (analytic, by Section 1.3.2.4.2.10) because  has been assumed to have compact support. If the sample is assumed to be an infinite crystal, so that  is now a periodic distribution, the customary limiting process by which it is shown that F becomes a discrete series of peaks at reciprocal-lattice points (see e.g. von Laue, 1936; Ewald, 1940; James, 1948a p. 9; Lipson & Taylor, 1958, pp. 14–27; Ewald, 1962, pp. 82–101; Warren, 1969, pp. 27–30) is already subsumed under the treatment of Section 1.3.2.6. 1.3.4.2. Crystallographic Fourier transform theory 1.3.4.2.1. Crystal periodicity 1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors Let  be the distribution of electrons in a crystal. Then, by definition of a crystal,  is -periodic for some period lattice  (Section 1.3.2.6.5) so that there exists a motif distribution 0 with compact support such that  ¼ R  0 ; P where R ¼ x2
ðXÞ. The lattice  is usually taken to be the finest for which the above representation holds. Let  have a basis ða1 ; a2 ; a3 Þ over the integers, these basis vectors being expressed in terms of a standard orthonormal basis ðe1 ; e2 ; e3 Þ as ak ¼

3 P

ajk ej :

j¼1

Then the matrix 0

a11 A ¼ @ a21 a31

1.3.4. Crystallographic applications of Fourier transforms 1.3.4.1. Introduction

a12 a22 a32

1 a13 a23 A a33

is the period matrix of  (Section 1.3.2.6.5) with respect to the unit lattice with basis ðe1 ; e2 ; e3 Þ, and the volume V of the unit cell is given by V ¼ jdet Aj. By Fourier transformation

The central role of the Fourier transformation in X-ray crystallography is a consequence of the kinematic approximation used in the description of the scattering of X-rays by a distribution of electrons (Bragg, 1915; Duane, 1925; Havighurst, 1925a,b; Zachariasen, 1945; James, 1948a, Chapters 1 and 2; Lipson & Cochran, 1953, Chapter 1; Bragg, 1975). Let ðXÞ be the density of electrons in a sample of matter contained in a finite region V which is being illuminated by a parallel monochromatic X-ray beam with wavevector K0 . Then the far-field amplitude scattered in a direction corresponding to wavevector K ¼ K0 þ H is proportional to

F
½ ¼ R
F
½0 ;

P where R ¼ H2
ðHÞ is the lattice distribution associated to the reciprocal lattice . The basis vectors ða1 ; a2 ; a3 Þ have coordinates in ðe1 ; e2 ; e3 Þ given by the columns of ðA1 ÞT , whose expression in terms of the cofactors of A (see Section 1.3.2.6.5) gives the familiar formulae involving the cross product of vectors for n ¼ 3. The H-distribution F of scattered amplitudes may be written

R FðHÞ ¼ ðXÞ expð2 iH  XÞ d3 X V

F ¼ F
½H ¼

¼ F
½ðHÞ

P H2

F
½0 ðHÞ
ðHÞ ¼

P H2

FH
ðHÞ

¼ hx ; expð2 iH  XÞi: and is thus a weighted reciprocal-lattice distribution, the weight FH attached to each node H 2  being the value at H of the transform F
½0  of the motif 0. Taken in conjunction with the assumption that the scattering is elastic, i.e. that H only changes the direction but not the magnitude of the incident wavevector K0 , this result yields the usual forms (Laue or Bragg) of the

In certain model calculations, the ‘sample’ may contain not only volume charges, but also point, line and surface charges. These singularities may be accommodated by letting  be a distribution, and writing

62

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 

diffraction conditions: H 2  , and simultaneously H lies on the Ewald sphere. By the reciprocity theorem, 0 can be recovered if F is known for all H 2  as follows [Section 1.3.2.6.5, e.g. (iv)]: x ¼

1.3.4.2.1.2. Structure factors in terms of form factors In many cases,  0 is a sum of translates of atomic electrondensity distributions. Assume there are n distinct chemical types of atoms, with Nj identical isotropic atoms of type j described by an electron distribution  j about their centre of mass. According to quantum mechanics each  j is a smooth rapidly decreasing function of x, i.e.  j 2 S , hence  0 2 S and (ignoring the effect of thermal agitation)

1 X F expð2 iH  XÞ: V H2 H

These relations may be rewritten in terms of standard, or ‘fractional crystallographic’, coordinates by putting

 0 ðxÞ ¼

" Nj n P P

#  j ðx  xkj Þ ;

j¼1 kj ¼1

X ¼ Ax;

H ¼ ðA1 ÞT h; which may be written (Section 1.3.2.5.8)

so that a unit cell of the crystal corresponds to x 2 R3 =Z3, and that h 2 Z3 . Defining   and  0 by

0

  ¼

n P

"  j 

kj ¼1

j¼1

¼

1 # A ; V

0 ¼

1 # 0 A  V

FðhÞ ¼ 3

ðxÞ d x; ðXÞ d X ¼ 

3

0


ðxk Þ j

:

By Fourier transformation:

so that 3

!#

Nj P

0

n P

(

"

kj ¼1

j¼1

3

 ðXÞ d X ¼   ðxÞ d x;

Nj P

F
½ j ðhÞ


#) expð2 ih  xkj Þ

:

Defining the form factor fj of atom j as a function of h to be

we have F
½ h ¼

FðhÞ ¼ ¼

P

fj ðhÞ ¼ F
½ j ðhÞ

FðhÞ
ðhÞ ;

h2Z3 h 0x ; expð2 ih R 0

 xÞi

we have 3

  ðxÞ expð2 ih  xÞ d x

0

if   2

L1loc ðR3 =Z3 Þ;

R3 =Z3

 x ¼

P

FðhÞ ¼

FðhÞ expð2 ih  xÞ:

n P

" fj ðhÞ


kj ¼1

j¼1

h2Z3

Nj P

# expð2 ih  xkj Þ :

If X ¼ Ax and H ¼ ðA1 ÞT h are the real- and reciprocal-space ˚ and A ˚ 1, and if j ðkXkÞ is the spherically coordinates in A symmetric electron-density function for atom type j, then

These formulae are valid for an arbitrary motif distribution  0 , provided the convergence of the Fourier series for   is considered from the viewpoint of distribution theory (Section 1.3.2.6.10.3). The experienced crystallographer may notice the absence of the familiar factor 1=V from the expression for   just given. This is because we use the (mathematically) natural unit for  , the electron per unit cell, which matches the dimensionless nature of the crystallographic coordinates x and of the associated volume element d3 x. The traditional factor 1=V was the result of the somewhat inconsistent use of x as an argument but of d3 X as a ˚ 3). volume element to obtain  in electrons per unit volume (e.g. A A fortunate consequence of the present convention is that nuisance factors of V or 1=V, which used to abound in convolution or scalar product formulae, are now absent. It should be noted at this point that the crystallographic terminology regarding F and F
differs from the standard mathematical terminology introduced in Section 1.3.2.4.1 and applied to periodic distributions in Section 1.3.2.6.4: F is the inverse Fourier transform of  rather than its Fourier transform, and the calculation of  is called a Fourier synthesis in crystallography even though it is mathematically a Fourier analysis. The origin of this discrepancy may be traced to the fact that the mathematical theory of the Fourier transformation originated with the study of temporal periodicity, while crystallography deals with spatial periodicity; since the expression for the phase factor of a plane wave is exp½2 iðt  K  XÞ, the difference in sign between the contributions from time versus spatial displacements makes this conflict unavoidable.

Z1 fj ðHÞ ¼

4 kXk2 j ðkXkÞ

sinð2 kHkkXkÞ dkXk: 2 kHkkXk

0

More complex expansions are used for electron-density studies (see Chapter 1.2 in this volume). Anisotropic Gaussian atoms may be dealt with through the formulae given in Section 1.3.2.4.4.2. 1.3.4.2.1.3. Fourier series for the electron density and its summation The convergence of the Fourier series for    ðxÞ ¼

P

FðhÞ expð2 ih  xÞ

h2Z3

is usually examined from the classical point of view (Section 1.3.2.6.10). The summation of multiple Fourier series meets with considerable difficulties, because there is no natural order in Zn to play the role of the natural order in Z (Ash, 1976). In crystallography, however, the structure factors FðhÞ are often obtained within spheres kHk  1 for increasing resolution (decreasing ). Therefore, successive estimates of   are most

63

1. GENERAL RELATIONSHIPS AND TECHNIQUES I ðxÞ are both real, F R ðhÞ and F I ðhÞ are both Since  R ðxÞ and  Hermitian symmetric, hence

naturally calculated as the corresponding partial sums (Section 1.3.2.6.10.1): P

ÞðxÞ ¼ S ð

FðhÞ expð2 ih  xÞ:

FðhÞ ¼ F R ðhÞ þ iF I ðhÞ;

kðA1 ÞT hk1

while

This may be written

FðhÞ ¼ F R ðhÞ  iF I ðhÞ:

S ð ÞðxÞ ¼ ðD   ÞðxÞ;

Thus FðhÞ 6¼ FðhÞ, so that Friedel’s law is violated. The components F R ðhÞ and F I ðhÞ, which do obey Friedel’s law, may be expressed as:

where D is the ‘spherical Dirichlet kernel’ P

D ðxÞ ¼

expð2 ih  xÞ:

kðA1 ÞT hk1

F R ðhÞ ¼ 12½FðhÞ þ FðhÞ; 1 F I ðhÞ ¼ ½FðhÞ  FðhÞ: 2i

D exhibits numerous negative ripples around its central peak. Thus the ‘series termination errors’ incurred by using S ð Þ instead of   consist of negative ripples around each atom, and may lead to a Gibbs-like phenomenon (Section 1.3.2.6.10.1) near a molecular boundary. As in one dimension, Cesa`ro sums (arithmetic means of partial sums) have better convergence properties, as they lead to a convolution by a ‘spherical Feje´r kernel’ which is everywhere positive. Thus Cesa`ro summation will always produce positive approximations to a positive electron density. Other positive summation kernels were investigated by Pepinsky (1952) and by Waser & Schomaker (1953).

1.3.4.2.1.5. Parseval’s identity and other L2 theorems By Section 1.3.2.4.3.3 and Section 1.3.2.6.10.2, P h2Z3

FðhÞ ¼

R3 =Z3

¼

jðXÞj2 d3 X:

R3 =

P

FðhÞGðhÞ ¼

h2Z3

3

 ðxÞ expð2 ih  xÞ d x

R

 ðxÞ ðxÞ d3 x

R 3 = Z3

R3 =Z3

R

R

j ðxÞj2 d3 x ¼ V

Usually  ðxÞ is real and positive, hence j ðxÞj ¼  ðxÞ, but the identity remains valid even when  ðxÞ is made complex-valued by the presence of anomalous scatterers. If fGh g is the collection of structure factors belonging to another electron density  ¼ A#  with the same period lattice as , then

1.3.4.2.1.4. Friedel’s law, anomalous scatterers If the wavelength  of the incident X-rays is far from any absorption edge of the atoms in the crystal, there is a constant phase shift in the scattering, and the electron density may be considered to be real-valued. Then R

R

jFðhÞj2 ¼

¼V

R

ðXÞðXÞ d3 X:

R3 =

 ðxÞ exp½2 iðhÞ  x d3 x

R3 =Z3

ðxÞ ¼  ðxÞ: ¼ FðhÞ since 

Thus, norms and inner products may be evaluated either from structure factors or from ‘maps’.

FðhÞ ¼ jFðhÞj expði’ðhÞÞ;

1.3.4.2.1.6. Convolution, correlation and Patterson function Let   ¼ r  0 and   ¼ r   0 be two electron densities referred to crystallographic coordinates, with structure factors fFh gh2Z3 and fGh gh2Z3 , so that

Thus if

then

 x ¼ jFðhÞj ¼ jFðhÞj

and

P

FðhÞ expð2 ih  xÞ;

h2Z3

’ðhÞ ¼ ’ðhÞ:

 x ¼

P

GðhÞ expð2 ih  xÞ:

h2Z3

This is Friedel’s law (Friedel, 1913). The set fFh g of Fourier coefficients is said to have Hermitian symmetry. If  is close to some absorption edge(s), the proximity to resonance induces an extra phase shift, whose effect may be represented by letting  ðxÞ take on complex values. Let R

The distribution ! ¼ r  ð 0   0 Þ is well defined, since the generalized support condition (Section 1.3.2.3.9.7) is satisfied. The forward version of the convolution theorem implies that if !x ¼

I

 ðxÞ ¼   ðxÞ þ i  ðxÞ

P

WðhÞ expð2 ih  xÞ;

h2Z3

and correspondingly, by termwise Fourier transformation

then

FðhÞ ¼ F R ðhÞ þ iF I ðhÞ:

WðhÞ ¼ FðhÞGðhÞ:

64

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 0

0

If either   or   is infinitely differentiable, then the distribution ¼  
  exists, and if we analyse it as x

P

¼

to crystals of the radially averaged correlation function used by Warren & Gingrich (1934) in the study of powders.

YðhÞ expð2 ih  xÞ;

h2Z3

1.3.4.2.1.7. Sampling theorems, continuous transforms, interpolation Shannon’s sampling and interpolation theorem (Section 1.3.2.7.1) takes two different forms, according to whether the property of finite bandwidth is assumed in real space or in reciprocal space. (1) The most usual setting is in reciprocal space (see Sayre, 1952c). Only a finite number of diffraction intensities can be recorded and phased, and for physical reasons the cutoff criterion is the resolution  ¼ 1=kHkmax . Electron-density maps are thus calculated as partial sums (Section 1.3.4.2.1.3), which may be written in Cartesian coordinates as

then the backward version of the convolution theorem reads: P

YðhÞ ¼

FðhÞGðh  kÞ:

k2Z3

The cross correlation ½ ;  between   and  is the Z3 -periodic distribution defined by: ¼  0 :   0 are locally integrable, If  0 and   0 ðxÞ ðx þ tÞ d3 x

R3

R

¼

FðHÞ expð2 iH  XÞ:

H2 ; kHk1

R

½ ;ðtÞ  ¼

P

S ðÞðXÞ ¼

S ðÞ is band-limited, the support of its spectrum being contained in the solid sphere  defined by kHk  1. Let  be the indicator function of . The transform of the normalized version of  is (see below, Section 1.3.4.4.3.5)

 ðxÞ ðx þ tÞ d3 x:

R3 =Z3

Let P

ðtÞ ¼

33 F ½ ðXÞ 4 3 kXk : ¼ 3 ðsin u  u cos uÞ where u ¼ 2 u 

I ðXÞ ¼

KðhÞ expð2 ih  tÞ:

h2Z3

The combined use of the shift property and of the forward convolution theorem then gives immediately:

By Shannon’s theorem, it suffices to calculate S ðÞ on an integral subdivision  of the period lattice  such that the sampling criterion is satisfied (i.e. that the translates of  by vectors of  do not overlap). Values of S ðÞ may then be calculated at an arbitrary point X by the interpolation formula:

KðhÞ ¼ FðhÞGðhÞ; hence the Fourier series representation of ½ ;:  ½ ;ðtÞ  ¼

P

S ðÞðXÞ ¼

FðhÞGðhÞ expð2 ih  tÞ:

P

I ðX  YÞS ðÞðYÞ:

Y2

h2Z3

(2) The reverse situation occurs whenever the support of the motif  0 does not fill the whole unit cell, i.e. whenever there exists a region M (the ‘molecular envelope’), strictly smaller than the unit cell, such that the translates of M by vectors of r do not overlap and that

Clearly, ½ ;  ¼ ð½ ;  Þ, as shown by the fact that permuting F and G changes KðhÞ into its complex conjugate. The auto-correlation of   is defined as ½ ;   and is called the Patterson function of  . If   consists of point atoms, i.e.  0 ¼

N P j¼1

0 ¼  0 : M


Zj
ðxj Þ ;

Þ: Defining the ‘interference It then follows that   ¼ r  ðM
 function’ G as the normalized indicator function of M according to

then " ½ ;   ¼ r 

N P N P j¼1 k¼1

# Zj Zk
ðxj xk Þ

GðgÞ ¼

contains information about interatomic vectors. It has the Fourier series representation ½ ;  ðtÞ ¼

P

1 F
½M ðgÞ volðMÞ

we may invoke Shannon’s theorem to calculate the value

F
½ 0 ðnÞ at an arbitrary point n of reciprocal space from its sample values FðhÞ ¼ F
½ 0 ðhÞ at points of the reciprocal lattice

jFðhÞj2 expð2 ih  tÞ;

as

h2Z3

F
½ 0 ðnÞ ¼

P h2Z3

and is therefore calculable from the diffraction intensities alone. It was first proposed by Patterson (1934, 1935a,b) as an extension

65

Gðn  hÞFðhÞ:

1. GENERAL RELATIONSHIPS AND TECHNIQUES This aspect of Shannon’s theorem constitutes the mathematical basis of phasing methods based on geometric redundancies created by solvent regions and/or noncrystallographic symmetries (Bricogne, 1974). The connection between Shannon’s theorem and the phase problem was first noticed by Sayre (1952b). He pointed out that the Patterson function of  , written as ½ ;   ¼ r  ð  0   0 Þ, may be viewed as consisting of a motif 0 ¼   0   0 (containing all the internal interatomic vectors) which is periodized by convolution with r. As the translates of 0 by vectors of Z3 do overlap, the sample values of the intensities jFðhÞj2 at nodes of the reciprocal lattice do not provide enough data to interpolate intensities jFðnÞj2 at arbitrary points of reciprocal space. Thus the loss of phase is intimately related to the impossibility of intensity interpolation, implying in return that any indication of intensity values attached to nonintegral points of the reciprocal lattice is a potential source of phase information.

Let 1 u1 B C u ¼ @ ... A 0

un be a primitive integral vector, i.e. g.c.d. ðu1 ; . . . ; un Þ ¼ 1. Then an n
n integral matrix P with det P ¼ 1 having u as its first column can be constructed by induction as follows. For n ¼ 1 the result is trivial. For n ¼ 2 it can be solved by means of the Euclidean algorithm, which yields z1 ; z2 such that u1 z2  u2 z1 ¼ 1, so that we may take
P¼

Note that, if

1.3.4.2.1.8. Sections and projections It was shown at the end of Section 1.3.2.5.8 that the convolution theorem establishes, under appropriate assumptions, a duality between sectioning a smooth function (viewed as a multiplication by a
-function in the sectioning coordinate) and projecting its transform (viewed as a convolution with the function 1 everywhere equal to 1 as a function of the projection coordinate). This duality follows from the fact that F and F
map 1xi to
xi and
xi to 1xi (Section 1.3.2.5.6), and from the tensor product property (Section 1.3.2.5.5). In the case of periodic distributions, projection and section must be performed with respect to directions or subspaces which are integral with respect to the period lattice if the result is to be periodic; furthermore, projections must be performed only on the contents of one repeating unit along the direction of projection, or else the result would diverge. The same relations then hold between principal central sections and projections of the electron density and the dual principal central projections and sections of the weighted reciprocal lattice, e.g. P

 ðx1 ; 0; 0Þ $


z¼

 1; 2 ðx3 Þ ¼

P


u¼

R

R=Z

u1 dz



0

1 z2 B . C C z¼B @ .. A zn


and

u1



d

Fðh1 ; h2 ; h3 Þ; are primitive. By the inductive hypothesis there is an integral 2
2 matrix V with

Fðh1 ; h2 ; h3 Þ;




 ðx1 ; x2 ; x3 Þ dx1 dx2 $ Fð0; 0; h3 Þ;

R2 =Z2

 1 ðx2 ; x3 Þ ¼



with d ¼ g.c.d. ðu2 ; . . . ; un Þ so that both

h3

R

z1 z2

is a solution, then z þ mu is another solution for any m 2 Z. For n  3, write

h1 ; h2

 ðx1 ; x2 ; 0Þ $

 z1 : z2

u1 u2

 ðx1 ; x2 ; x3 Þ dx1

u1



d

$ Fð0; h2 ; h3 Þ

as its first column, and an integral ðn  1Þ
ðn  1Þ matrix Z with z as its first column, with det V ¼ 1 and det Z ¼ 1. Now put

etc. When the sections are principal but not central, it suffices to use the shift property of Section 1.3.2.5.5. When the sections or projections are not principal, they can be made principal by changing to new primitive bases B and B for  and  , respectively, the transition matrices P and P to these new bases being related by P ¼ ðP1 ÞT in order to preserve duality. This change of basis must be such that one of these matrices (say, P) should have a given integer vector u as its first column, u being related to the line or plane defining the section or projection of interest. The problem of constructing a matrix P given u received an erroneous solution in Volume II of International Tables (Patterson, 1959), which was subsequently corrected in 1962. Unfortunately, the solution proposed there is complicated and does not suggest a general approach to the problem. It therefore seems worthwhile to record here an effective procedure which solves this problem in any dimension n (Watson, 1970).


P¼




1

V

Z

In2

 ;

i.e. 0

1 B0 B P¼B B0 @: 0

0 z2 z3 : zn

The first column of P is

66

0   : 

10 : 0 u1 Bd : C CB B : C CB 0 : : A@ : 0 : 

  0 : 0

0 0 1 : 0

: : : : :

1 0 0C C 0C C: :A 1

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 1 u1 B dz2 C C B B : C ¼ u; C B @ : A dzn 0

where

 ¼

j¼1

ðr ÞðxÞ ¼ ½ðrrT Þ ðxÞ ¼

P

ð4 2 hhT ÞFðhÞ expð2 ih  xÞ;

h ¼ P h 0 ; and a step of Newton iteration towards the nearest stationary point of   will proceed by x 7 ! x  f½ðrrT Þ ðxÞg1 ðr ÞðxÞ: The modern use of Fourier transforms to speed up the computation of derivatives for model refinement will be described in Section 1.3.4.4.7. The converse property is also useful: it relates the derivatives of the continuous transform F
½0  to the moments of 0 :

The transform is then

@m1 þm2 þm3 F
½0  m1 þm2 þm3 m1 m2 m3 0
X1 X2 X3 x ðHÞ: m m m ðHÞ ¼ F ½ð2 iÞ @X1 1 @X2 2 @X3 3

½F  ð
h 
k  F
½½z1 ; z2  Þ
ð1h  1k 
l Þ;

For jmj ¼ 2 and H ¼ 0, this identity gives the well known relation between the Hessian matrix of the transform F
½0  at the origin of reciprocal space and the inertia tensor of the motif 0. This is a particular case of the moment-generating properties of F
, which will be further developed in Section 1.3.4.5.2.

giving for coefficient ðh; kÞ: Fðh; k; lÞ expf2 il½ðz1 þ z2 Þ=2g


l2Z

sin lðz1  z2 Þ : l

1.3.4.2.1.10. Toeplitz forms, determinantal inequalities and Szego¨’s theorem The classical results presented in Section 1.3.2.6.9 can be readily generalized to the case of triple Fourier series; no new concept is needed, only an obvious extension of the notation. Let   be real-valued, so that Friedel’s law holds and FðhÞ ¼ FðhÞ. Let H be a finite set of indices comprising the origin: H ¼ fh0 ¼ 0; h1 ; . . . ; hn g. Then the Hermitian form in n þ 1 complex variables

1.3.4.2.1.9. Differential syntheses Another particular instance of the convolution theorem is the duality between differentiation and multiplication by a monomial (Sections 1.3.2.4.2.8, 1.3.2.5.8). In the present context, this result may be written 

ð2 ihÞFðhÞ expð2 ih  xÞ;

h2Z3

½ 
ð1x  1y  ½z1 ; z2  Þ  ð
x 
y  1z Þ:

F


P h2Z3

and an appeal to the tensor product property. Booth (1945a) made use of the convolution theorem to form the Fourier coefficients of ‘bounded projections’, which provided a compromise between 2D and 3D Fourier syntheses. If it is desired to compute the projection on the (x, y) plane of the electron density lying between the planes z ¼ z1 and z ¼ z2 , which may be written as

X

@Xj2

is the Laplacian of . The second formula has been used with jmj ¼ 1 or 2 to compute ‘differential syntheses’ and refine the location of maxima (or other stationary points) in electron-density maps. Indeed, the values at x of the gradient vector r  and Hessian  are readily obtained as matrix ðrrT Þ

and its determinant is 1, QED. The incremental step from dimension n  1 to dimension n is the construction of 2
2 matrix V, for which there exist infinitely many solutions labelled by an integer mn1 . Therefore, the collection of matrices P which solve the problem is labelled by n  1 arbitrary integers ðm1 ; m2 ; . . . ; mn1 Þ. This freedom can be used to adjust the shape of the basis B. Once P has been chosen, the calculation of general sections and projections is transformed into that of principal sections and projections by the changes of coordinates: x ¼ Px0 ;

3 X @2 

 @m1 þm2 þm3  ðHÞ @X1m1 @X2m2 @X3m3

¼ ð2 iÞm1 þm2 þm3 H1m1 H2m2 H3m3 FðAT HÞ

ðuÞ ¼ TH ½

n P

Fðhj  hk Þuj uk

j; k¼0

in Cartesian coordinates, and is called the Toeplitz form of order H associated to  . By the convolution theorem and Parseval’s identity,



 @m1 þm2 þm3   m1 þm2 þm3 m1 m2 m3 F
h1 h2 h3 FðhÞ m2 m3 ðhÞ ¼ ð2 iÞ 1 @xm @x @x 2 3 1



2

P

n TH ½ ðuÞ ¼  ðxÞ uj expð2 ihj  xÞ d3 x:



j¼0 R3 =Z3 R

in crystallographic coordinates. A particular case of the first formula is 4 2

P

kHk2 FðAT HÞ expð2 iH  XÞ ¼ ðXÞ;

If   is almost everywhere non-negative, then for all H the forms  are positive semi-definite and therefore all Toeplitz deterTH ½  are non-negative, where minants DH ½

H2

67

1. GENERAL RELATIONSHIPS AND TECHNIQUES surveyed briefly in Section 1.3.4.2.2.3 for the purpose of establishing further terminology and notation, after recalling basic notions and results concerning groups and group actions in Section 1.3.4.2.2.2.

DH ½  ¼ detf½Fðhj  hk Þg: The Toeplitz–Carathe´odory–Herglotz theorem given in Section 1.3.2.6.9.2 states that the converse is true: if DH ½  0 for all H, then   is almost everywhere non-negative. This result is known in the crystallographic literature through the papers of Karle & Hauptman (1950), MacGillavry (1950), and Goedkoop (1950), following previous work by Harker & Kasper (1948) and Gillis (1948a,b). Szego¨’s study of the asymptotic distribution of the eigenvalues of Toeplitz forms as their order tends to infinity remains valid. Some precautions are needed, however, to define the notion of a sequence ðHk Þ of finite subsets of indices tending to infinity: it suffices that the Hk should consist essentially of the reciprocallattice points h contained within a domain of the form k (k-fold dilation of ) where  is a convex domain in R3 containing the origin (Widom, 1960). Under these circumstances, the eigenvalues ðnÞ  become equidistributed  of the Toeplitz forms THk ½ with the sample values  ðnÞ0 of   on a grid satisfying the Shannon sampling criterion for the data in Hk (cf. Section 1.3.2.6.9.3). A particular consequence of this equidistribution is that the ðnÞ geometric means of the ðnÞ  and of the  0 are equal, and hence as in Section 1.3.2.6.9.4 ( g1=jHk j ¼ exp lim fDHk ½

k!1

R

1.3.4.2.2.2. Groups and group actions The books by Hall (1959) and Scott (1964) are recommended as reference works on group theory. (a) Left and right actions Let G be a group with identity element e, and let X be a set. An action of G on X is a mapping from G
X to X with the property that, if g x denotes the image of ðg; xÞ, then (i) ðg1 g2 Þx ¼ g1 ðg2 xÞ (ii)

ex ¼ x

for all g1 ; g2 2 G and all x 2 X; for all x 2 X:

An element g of G thus induces a mapping Tg of X into itself defined by Tg ðxÞ ¼ gx, with the ‘representation property’: (iii) Tg1 g2 ¼ Tg1 Tg2

for all g1 ; g2 2 G:

Since G is a group, every g has an inverse g1 ; hence every mapping Tg has an inverse Tg1 , so that each Tg is a permutation of X. Strictly speaking, what has just been defined is a left action. A right action of G on X is defined similarly as a mapping ðg; xÞ 7 ! xg such that

) log  ðxÞ d3 x ;

R3 =Z3

where jHk j denotes the number of reflections in Hk . Complementary terms giving a better comparison of the two sides were obtained by Widom (1960, 1975) and Linnik (1975). This formula played an important role in the solution of the 2D Ising model by Onsager (1944) (see Montroll et al., 1963). It is also encountered in phasing methods involving the ‘Burg entropy’ (Britten & Collins, 1982; Narayan & Nityananda, 1982; Bricogne, 1982, 1984, 1988).

ði0 Þ xðg1 g2 Þ ¼ ðxg1 Þg2 ðii0 Þ xe ¼ x

for all g1 ; g2 2 G and all x 2 X; for all x 2 X:

The mapping Tg0 defined by Tg0 ðxÞ ¼ xg then has the ‘rightrepresentation’ property: ðiii0 Þ Tg0 1 g2 ¼ Tg0 2 Tg0 1

1.3.4.2.2. Crystal symmetry 1.3.4.2.2.1. Crystallographic groups The description of a crystal given so far has dealt only with its invariance under the action of the (discrete Abelian) group of translations by vectors of its period lattice . Let the crystal now be embedded in Euclidean 3-space, so that it may be acted upon by the group Mð3Þ of rigid (i.e. distancepreserving) motions of that space. The group Mð3Þ contains a normal subgroup Tð3Þ of translations, and the quotient group Mð3Þ=Tð3Þ may be identified with the 3-dimensional orthogonal group Oð3Þ. The period lattice  of a crystal is a discrete uniform subgroup of Tð3Þ. The possible invariance properties of a crystal under the action of Mð3Þ are captured by the following definition: a crystallographic group is a subgroup  of Mð3Þ if (i)  \ Tð3Þ ¼ , a period lattice and a normal subgroup of ; (ii) the factor group G ¼ = is finite. The two properties are not independent: by a theorem of Bieberbach (1911), they follow from the assumption that  is a discrete subgroup of Mð3Þ which operates without accumulation point and with a compact fundamental domain (see Auslander, 1965). These two assumptions imply that G acts on  through an integral representation, and this observation leads to a complete enumeration of all distinct ’s. The mathematical theory of these groups is still an active research topic (see, for instance, Farkas, 1981), and has applications to Riemannian geometry (Wolf, 1967). This classification of crystallographic groups is described elsewhere in these Tables (Wondratschek, 2005), but it will be

for all g1 ; g2 2 G:

The essential difference between left and right actions is of course not whether the elements of G are written on the left or right of those of X: it lies in the difference between (iii) and (iii0 ). In a left action the product g1 g2 in G operates on x 2 X by g2 operating first, then g1 operating on the result; in a right action, g1 operates first, then g2 . This distinction will be of importance in Sections 1.3.4.2.2.4 and 1.3.4.2.2.5. In the sequel, we will use left actions unless otherwise stated. (b) Orbits and isotropy subgroups Let x be a fixed element of X. Two fundamental entities are associated to x: (1) the subset of G consisting of all g such that gx ¼ x is a subgroup of G, called the isotropy subgroup of x and denoted Gx ; (2) the subset of X consisting of all elements gx with g running through G is called the orbit of x under G and is denoted Gx. Through these definitions, the action of G on X can be related to the internal structure of G, as follows. Let G=Gx denote the collection of distinct left cosets of Gx in G, i.e. of distinct subsets of G of the form gGx . Let jGj; jGx j; jGxj and jG=Gx j denote the numbers of elements in the corresponding sets. The number jG=Gx j of distinct cosets of Gx in G is also denoted ½G : Gx  and is called the index of Gx in G; by Lagrange’s theorem ½G : Gx  ¼ jG=Gx j ¼

68

jGj : jGx j

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY Now if g1 and g2 are in the same coset of Gx, then g2 ¼ g1 g0 with g0 2 Gx , and hence g1 x ¼ g2 x; the converse is obviously true. Therefore, the mapping from cosets to orbit elements

Cg ðhÞ ¼ ghg1 : Indeed, Cg ðhkÞ ¼ Cg ðhÞCg ðkÞ and ½Cg ðhÞ1 ¼ Cg1 ðhÞ. In particular, Cg operates on the set of subgroups of G, two subgroups H and K being called conjugate if H ¼ Cg ðKÞ for some g 2 G; for example, it is easily checked that Ggx ¼ Cg ðGx Þ. The orbits under this action are the conjugacy classes of subgroups of G, and the isotropy subgroup of H under this action is called the normalizer of H in G. If fHg is a one-element orbit, H is called a self-conjugate or normal subgroup of G; the cosets of H in G then form a group G=H called the factor group of G by H. Let G and H be two groups, and suppose that G acts on H by automorphisms of H, i.e. in such a way that

gGx 7 ! gx establishes a one-to-one correspondence between the distinct left cosets of Gx in G and the elements of the orbit of x under G. It follows that the number of distinct elements in the orbit of x is equal to the index of Gx in G: jGj ; jGx j

jGxj ¼ ½G : Gx  ¼

and that the elements of the orbit of x may be listed without repetition in the form

gðh1 h2 Þ ¼ gðh1 Þgðh2 Þ gðeH Þ ¼ eH 1

Gx ¼ fxj 2 G=Gx g:

gðh Þ ¼ ðgðhÞÞ

Similar definitions may be given for a right action of G on X. The set of distinct right cosets Gx g in G, denoted Gx \G, is then in one-to-one correspondence with the distinct elements in the orbit xG of x.

Then the symbols [g, h] with g 2 G, h 2 H form a group K under the product rule: ½g1 ; h1 ½g2 ; h2  ¼ ½g1 g2 ; h1 g1 ðh2 Þ

(c) Fundamental domain and orbit decomposition The group properties of G imply that two orbits under G are either disjoint or equal. The set X may thus be written as the disjoint union X¼

[

{associativity checks; [eG ; eH ] is the identity; ½g; h has inverse ½g1 ; g1 ðh1 Þ}. The group K is called the semi-direct product of H by G, denoted K ¼ H . EM , it is possible and convenient to replace the limits of integration in equation (2.1.8.3) by infinity. Thus R1 pðEÞ expð
ikEÞ dE ¼ hexpð
ikEÞi: ð2:1:8:4Þ Ck ¼ 1

Equation (2.1.8.4) shows that Ck is a Fourier transform of the p.d.f. pðEÞ and, as such, it is the value of the corresponding characteristic function at the point tk ¼
k [i.e., Ck ¼ Cð
kÞ, where the characteristic function CðtÞ is defined by equation (2.1.4.1)]. It is also seen that Ck is the expected value of the exponential expð
ikEÞ. It follows that the feasibility of the present approach depends on one’s ability to evaluate the characteristic function in closed form without the knowledge of the p.d.f.; this is analogous to the problem of evaluating absolute moments of the structure factor for the correction-factor approach, discussed in Section 2.1.7. Fortunately, in crystallographic applications these calculations are feasible, provided individual isotropic motion is assumed. The formal expression for the p.d.f. of jEj, for any centrosymmetric space group, is therefore   1 P pðjEjÞ ¼  1 þ 2 Ck cosð
kjEjÞ ; ð2:1:8:5Þ

pðjEjÞ ¼ 22 jEj

Du J0 ðu jEjÞ;

where Du ¼

Cðu Þ J12 ðu Þ

Cðu Þ ¼

N=g Q

Cju ;

ð2:1:8:13Þ

j¼1

where J1 ðxÞ is the Bessel function of the first kind, and u is the uth root of the equation J0 ðxÞ ¼ 0; the atomic contribution Cju to equation (2.1.8.13) is computed as Cju ¼ Cðnj u Þ:

ð2:1:8:14Þ

The roots u are tabulated in the literature (e.g. Abramowitz & Stegun, 1972), but can be most conveniently computed as follows. The first five roots are given by

ð2:1:8:6Þ

1 ¼ 2:4048255577

m

ð2:1:8:12Þ

and

where use is made of the assumption that pðEÞ ¼ pðEÞ, and the Fourier coefficients are evaluated from equation (2.1.8.4). The p.d.f. of jEj for a noncentrosymmetric space group is obtained by first deriving the joint p.d.f. of the real and imaginary parts of E and then integrating out its phase. The general expression for E is

where ’ is the phase of E. The required joint p.d.f. is PP pðA; BÞ ¼ ð2 =4Þ Cmn exp½
iðmA þ nBÞ
;

ð2:1:8:11Þ

u¼1

k¼1

E ¼ A þ iB ¼ jEj cos ’ þ ijEj sin ’;

1 P

2 ¼ 5:5200781103 3 ¼ 8:6537279129

ð2:1:8:7Þ

n

4 ¼ 11:7915344390 and introducing polar pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi coordinates m ¼ r sin  and n ¼ r cos , where r ¼ m2 þ n2 and  ¼ tan1 ðm=nÞ, we have

5 ¼ 14:9309177085

208

2.1. STATISTICAL PROPERTIES OF THE WEIGHTED RECIPROCAL LATTICE and the higher ones can be obtained from McMahon’s approximation (cf. Abramowitz & Stegun, 1972) 1 124 120928 401743168 u ¼ þ  þ  þ . . . ; ð2:1:8:15Þ 8 3ð8 Þ3 15ð8Þ5 105ð8Þ7

equation (2.1.8.26) is then obtained by making use of the assumptions of independence and uniformity in an analogous manner to that detailed in equations (2.1.8.12)–(2.1.8.22) above. The right-hand side of equation (2.1.8.26) is to be used as a Fourier coefficient of the double Fourier series given by (2.1.8.9). Since, however, this coefficient depends on ðm2 þ n2 Þ1=2 alone rather than on m and n separately, the p.d.f. of jEj for P1 can also be represented by a Fourier–Bessel series [cf. equation (2.1.8.11)] with coefficient

where ¼ ðu  14Þ
. For u > 5 the values given by equation (2.1.8.15) have a relative error less than 1011 so that no refinement of roots of higher orders is needed (Shmueli et al., 1984). Numerical computations of single Fourier–Bessel series are of course faster than those of the double Fourier series, but both representations converge fairly rapidly.

Du ¼

2.1.8.3. Simple examples

E¼2

nj cos #j ;

with

T

#j ¼ 2
h  rj ;

2.1.8.4. A more complicated example We now illustrate the methodology of deriving characteristic functions for space groups of higher symmetries, following the method of Rabinovich et al. (1991a,b). The derivation is performed for the space group P6 (No. 174). According to Table A1.4.3.6, the real and imaginary parts of the normalized structure factor are given by

ð2:1:8:16Þ

j¼1

and the Fourier coefficient is Ck ¼ hexpð
ikEÞi * " #+ N=2 P ¼ exp 2
ik nj cos #j j¼1

* ¼

N=2 Q

ð2:1:8:17Þ A¼2

+

¼2

hexpð2
iknj cos #j Þi

¼



N=2 Q

ð1=2
Þ

j¼1

R


ð2:1:8:20Þ

B¼2

expð2
iknj cos #Þ d#

N=2 Q

¼2




J0 ð2
knj Þ:

nj cos #j

and

B¼

j¼1

N P

nj sin #j :

N Q

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp½
inj m2 þ n2 sinð#j þ Þ


¼

nj cos j

3 P

sin jk ;

ð2:1:8:29Þ

k¼1

Note that j1 þ j2 þ j3 ¼ 0, i.e., one of these contributions depends on the other two; this is a recurring problem in calculations pertaining to trigonal and hexagonal systems. For brevity, we write directly the general form of the characteristic function from which the functional form of the Fourier coefficient can be readily obtained. The characteristic function is given by

ð2:1:8:23Þ

Cðt1 ; t2 Þ ¼ hexp½iðt1 A þ t2 BÞ
i ð2:1:8:30Þ    N=6 3 Q P ¼ exp 2inj cos j ðt1 cos jk þ t2 sin jk Þ j¼1

k¼1



¼

ð2:1:8:24Þ

N=6 Q j¼1

 exp 2inj t cos j

ð2:1:8:31Þ 3 P

ðsin  cos jk

k¼1

 þ cos  sin jk Þ    N=6 3 Q P ¼ exp 2inj t cos j sinðjk þ Þ ;

ð2:1:8:25Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J0 ð
nj m2 þ n2 Þ:

N=6 P

j ¼ 2
lzj :

j¼1

N Q

nj ½SðhkiÞcðlzÞ
j

j3 ¼ 2
ðixj þ hyj Þ;

j¼1

+

N=6 P

j2 ¼ 2
ðkxj þ iyj Þ;

j¼1

¼

ð2:1:8:28Þ

j1 ¼ 2
ðhxj þ kyj Þ;

These expressions for A and B are substituted in equation (2.1.8.10), resulting in * + N Q Cmn ¼ exp½
inj ðm cos #j þ n sin #j Þ
*

cos jk

where

ð2:1:8:22Þ

Equation (2.1.8.20) is obtained from equation (2.1.8.19) if we make use of the assumption of independence, the assumption of uniformity allows us to rewrite equation (2.1.8.20) as (2.1.8.21), and the expression in the braces in the latter equation is just a definition of the Bessel function J0 ð2
knj Þ (e.g. Abramowitz & Stegun, 1972). Let us now consider the Fourier coefficient of the p.d.f. of jEj for the noncentrosymmetric space group P1. We have N P

3 P k¼1

j¼1

j¼1

A¼

nj cos j

j¼1

ð2:1:8:21Þ ¼

N=6 P

and

N=2 Q j¼1

nj ½CðhkiÞcðlzÞ
j

j¼1

ð2:1:8:19Þ

j¼1

¼

N=6 P j¼1

ð2:1:8:18Þ

expð2
iknj cos #j Þ

ð2:1:8:27Þ

where u is the uth root of the equation J0 ðxÞ ¼ 0.

Consider the Fourier coefficient of the p.d.f. of jEj for the centrosymmetric space group P1 . The normalized structure factor is given by N=2 P

N 1 Y J ðn  Þ; J12 ðu Þ j¼1 0 j u

j¼1

ð2:1:8:26Þ

ð2:1:8:32Þ

k¼1

ð2:1:8:33Þ

j¼1

where  ¼ tan1 ðt1 =t2 Þ, t ¼ ðt12 þ t22 Þ1=2 and the assumption of independence was used. If we further employ the assumption of

Equation (2.1.8.24) leads to (2.1.8.25) by introducing polar coordinates analogous to those leading to equation (2.1.8.8), and

209

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.1.8.1. Atomic contributions to characteristic functions for pðjEjÞ The table lists symbolic expressions for the atomic contributions to exact characteristic functions (abbreviated as c.f.) for pðjEjÞ, to be computed as single Fourier series (centric), double Fourier series (acentric) and single Fourier–Bessel series (acentric), as defined in Sections 2.1.8.1 and 2.1.8.2. The symbolic expressions are defined in Section 2.1.8.5. The table is arranged by point groups, space groups and parities of the reflection indices analogously to the table of moments, Table 2.1.7.1, and covers all the space groups and statistically different parities of hkl up to and including space group Fd3 . The expressions are valid for atoms in general positions, for general reflections and presume the absence of noncrystallographic symmetry and of dispersive scatterers. The p.d.f.’s of jEj for any centrosymmetric space group are computed from a single Fourier series [cf. (2.1.8.5)]. For a noncentrosymmetric space group the p.d.f. of jEj must be computed from a double Fourier series [cf. (2.1.8.9)] – if the characteristic function in the table depends on  – and may be computed from a single Fourier–Bessel series [cf. (2.1.8.11)] if it does not depend on . However, the p.d.f. of jEj for any noncentrosymmetric space group may also be computed from a double Fourier series (cf. Section 2.1.8.1). Space group(s)

g

Atomic c.f.

Remarks

Space group(s) Point groups: 4 2m, 4 m2

Point group: 1 P1 Point group: 1 P1 Point groups: 2, m All P

1 2

J0 ðtnj Þ

4

J02 ðtnj Þ J02 ð2tnj Þ

All P

4

J02 ð2t1 nj Þ

All C

8

J02 ð4t1 nj Þ

All C

2

Point group: 222

Qð1Þ j ðt; Þ

16

Qð1Þ j ð2t; Þ

16

Qð1Þ j ð2t; Þ

2h þ l ¼ 2n

16

Qð2Þ j ð2t; Þ

2h þ l ¼ 2n þ 1

All P

16

Qð1Þ j ð2t1 ; 0Þ

I4=mmm, I4=mcm

32

Qð1Þ j ð4t1 ; 0Þ

I41 =amd, I41 =acd

32

Qð1Þ j ð4t1 ; 0Þ

l ¼ 2n

32

Qð1Þ j ð4t1 ;
=4Þ

l ¼ 2n þ 1

ðaÞ

All P

4

Lj ðt; Þ

All C and I

8

Lj ð2t; Þ

16

Lj ð4t; Þ

Point group: 3

Point group: mm2 All P

4

Lj ðt; 0Þ

All C and I

8

Lj ð2t; 0Þ

Fmm2

16

Lj ð4t; 0Þ

Fdd2

16

Lj ð4t; 0Þ

h þ k þ l ¼ 2n

16

Lj ð4t;
=4Þ

h þ k þ l ¼ 2n þ 1

All P and R Point group: 3

3

J03 ðtnj Þ

All P and R

6

J03 ð2t1 nj Þ

6

Tj ðt; ÞðdÞ

Point group: 32 All P and R Point group: 3m

Point group: mmm 8

Lj ð2t1 ; 0Þ

All C and I Fmmm

16 32

Lj ð4t1 ; 0Þ Lj ð8t1 ; 0Þ

Fddd

32

Lj ð8t1 ; 0Þ

h þ k þ l ¼ 2n

32

Lj ð8t1 ;
=4Þ

h þ k þ l ¼ 2n þ 1

All P

P3m1, P31m, R3m

6

Tj ðt;
=2Þ

P3c1, P31c, R3c

6

Tj ðt;
=2Þ

6

Tj ðt; 0Þ

Point group: 3 m P3 m1, P3 1m, R3 m P3 c1, P3 1c, R3 c

Point group: 4 P4, P42

4

Remarks

Point group: 4=mmm

Point group: 2=m

F222

Atomic c.f.

8

All P I4 2m, I4 m2, I4 c2 I4 2d

J0 ð2t1 nj Þ

g

12

Tj ð2t1 ;
=2Þ

12

Tj ð2t1 ;
=2Þ

12

Tj ð2t1 ; 0Þ

Lj ðt; 0Þ

l ¼ 2n ðPÞ, h þ k þ l ¼ 2n ðRÞ l ¼ 2n þ 1 ðPÞ, h þ k þ l ¼ 2n þ 1 ðRÞ

l ¼ 2n ðPÞ, h þ k þ l ¼ 2n ðRÞ l ¼ 2n þ 1 ðPÞ, h þ k þ l ¼ 2n þ 1 ðRÞ

4

Lj ðt; 0Þ

l ¼ 2n

Point group: 6

4

Lj ðt;
=4Þ

l ¼ 2n þ 1

P6

6

Hjð1Þ ðt;
=2ÞðeÞ

I4

8

Lj ð2t; 0Þ

P61 †

6

Hjð1Þ ðt;
=2Þ

l ¼ 6n

I41

8 8

Lj ð2t; 0Þ Lj ð2t;
=4Þ

6 6

Hjð2Þ ðt; 0Þðf Þ Hjð2Þ ðt;
=2Þ

l ¼ 6n þ 1, 6n þ 5 l ¼ 6n þ 2, 6n þ 4

Point group: 4 P4

6

Hjð1Þ ðt; 0Þ

l ¼ 6n þ 3

4

Lj ðt; Þ

6

Hjð1Þ ðt;
=2Þ

l ¼ 3n

I 4

8

Lj ð2t; Þ

6

Hjð2Þ ðt;
=2Þ

l ¼ 3n 1

6

Hjð1Þ ðt;
=2Þ

l ¼ 2n

6

Hjð1Þ ðt; 0Þ

l ¼ 2n þ 1

6

Hjð1Þ ðt; Þ

P41 †

2h þ l ¼ 2n 2h þ l ¼ 2n þ 1 P62 †

Point group: 4=m All P

P63 8

Lj ð2t1 ; 0Þ

I4=m

16

Lj ð4t1 ; 0Þ

I41 =a

16 16

Lj ð4t1 ; 0Þ Lj ð4t1 ;
=4Þ

P422, P4212, P4222, P42212

8

ðbÞ Qð1Þ j ðt; Þ

P4122,† P41212†

8

Qð1Þ j ðt; Þ

l ¼ 2n

8

ðcÞ Qð2Þ j ðt; Þ

l ¼ 2n þ 1

Point group: 6 P6

l ¼ 2n l ¼ 2n þ 1

Point group: 6=m

Point group: 422

I422

16

Qð1Þ j ð2t; Þ

I4122

16

Qð1Þ j ð2t; Þ

2k þ l ¼ 2n

16

Qð2Þ j ð2t; Þ

2k þ l ¼ 2n þ 1

All P I4mm, I4cm

8 16

Qð1Þ j ðt; 0Þ Qð1Þ j ð2t; 0Þ

I41md, I41cd

16

Qð1Þ j ð2t; 0Þ

2k þ l ¼ 2n

16

Qð1Þ j ð2t;
=4Þ

2k þ l ¼ 2n þ 1

P6=m

12

Hjð1Þ ð2t1 ;
=2Þ

P63 =m

12

Hjð1Þ ð2t1 ;
=2Þ

l ¼ 2n

12

Hjð1Þ ð2t1 ; 0Þ

l ¼ 2n þ 1

P622

12

P61 22†

12

~ jð1Þ ðt;
=2, H 
=2; ÞðgÞ ~ jð1Þ ðt;
=2, H 
=2; Þ ~ jð2Þ ðt; 0; 0; ÞðhÞ H ~ jð2Þ ðt;
=2;
=2; Þ H ~ jð1Þ ðt; 0; 0; Þ H

Point group: 622

12 12

Point group: 4mm

12 P62 22†

12 12

210

~ jð1Þ ðt;
=2, H 
=2; Þ ~ jð2Þ ðt;
=2;
=2; Þ H

l ¼ 6n l ¼ 6n þ 1, 6n þ 5 l ¼ 6n þ 2, 6n þ 4 l ¼ 6n þ 3 l ¼ 3n l ¼ 3n 1

2.1. STATISTICAL PROPERTIES OF THE WEIGHTED RECIPROCAL LATTICE If we change the variable  to 0  , sinð þ Þ becomes sin 0 and ik ¼ ik0 þ ik. Hence  N=6 1 R
Q P 3 ð1=2
Þ d expð3ikÞJk ð2nj t cos Þ : Cðt1 ; t2 Þ ¼

Table 2.1.8.1 (cont.) Space group(s)

g

P63 22

12

Atomic c.f. ~ jð1Þ ðt;
=2, H 
=2; Þ ~ jð1Þ ðt; 0; 0; Þ H

12

Remarks l ¼ 2n

j¼1




k¼1

l ¼ 2n þ 1

ð2:1:8:37Þ

Point group: 6mm P6mm

12

P6cc

12

~ jð1Þ ðt;
=2;
=2; 0Þ H ~ jð1Þ ðt;
=2;
=2; 0Þ H

l ¼ 2n

12

~ jð1Þ ðt;
=2, H

l ¼ 2n þ 1

12


=2; 0Þ ~ jð1Þ ðt;
=2;
=2; 0Þ H ~ jð1Þ ðt; 0; 0; 0Þ H

l ¼ 2n þ 1

~ jð1Þ ðt; ; ; 0Þ H ~ jð1Þ ðt; ; ; 0Þ H

l ¼ 2n

P63 cm, P63 mc

12 Point groups: 6 2m, 6 m2 P6 2m, P6 m2 P6 2c, P6 c2

12 12

~ jð1Þ ðt;  þ
=2, H  
=2; 0Þ

12

The imaginary part of the summation, involving Bessel functions of odd orders, vanishes upon integration and the latter is restricted to the positive quadrant in . Thus, upon replacing cosines by sines (this is permissible at this stage) the atomic contribution to the characteristic function becomes 
=2 R 3 Cj ðt; Þ ¼ ð2=
Þ J0 ð2nj t sin Þ 0  1 P cosð6kÞJk3 ð2nj t sin Þ d þ2

l ¼ 2n

l ¼ 2n þ 1

k¼1

ð2:1:8:38Þ

Point group: 6=mmm P6=mmm

24

P6=mcc

24

~ jð1Þ ð2t1 ;
=2;
=2; 0Þ H ~ jð1Þ ð2t1 ;
=2;
=2; 0Þ H

and a double Fourier series must be used for the p.d.f.

l ¼ 2n

24

~ jð1Þ ð2t1 ,
=2, H 
=2; 0Þ ~ jð1Þ ð2t1 ;
=2;
=2; 0Þ H ~ jð1Þ ð2t1 ; 0; 0; 0Þ H

P23, P213

12

L3j ðt; Þ

I23, I21 3

24

L3j ð2t; Þ

F23 Point group: m3 Pm3 , Pn3 , Pa3

48

L3j ð4t; Þ

24

L3j ð2t1 ; 0Þ

Im3 , Ia3 Fm3

48

L3j ð4t1 ; 0Þ

96

L3j ð8t1 ; 0Þ

Fd3

96

L3j ð8t1 ; 0Þ

h þ k þ l ¼ 2n

96

L3j ð8t1 ;
=4Þ

h þ k þ l ¼ 2n þ 1

24 P63 =mcm, P63 =mmc

24

l ¼ 2n þ 1

2.1.8.5. Atomic characteristic functions

l ¼ 2n

Expressions for the atomic contributions to the characteristic functions were obtained by Rabinovich et al. (1991a) for a wide range of space groups, by methods similar to those described above. These expressions are collected in Table 2.1.8.1 in terms of symbols which are defined below. The following abbreviations are used in the subsequent definitions of the symbols:

l ¼ 2n þ 1

Point group: 23

s ¼ 2anj sinð Þ; c ¼ 2anj cosð Þ and  ¼ 2anj sinð 2
=3 þ Þ; and the symbols appearing in Table 2.1.8.1 are given below: ðaÞ

† And the enantiomorphous space group.

uniformity, while remembering that the angular variables jk are not independent, the characteristic function can be written as 
N=6 R
Q Cðt1 ; t2 Þ ¼ ð1=2
Þ d ½1=ð2
Þ2
j¼1

k¼1

¼ J04 ðanj Þ þ 2 ðbÞ




R
R
R


k¼1 1 P cosð6kÞJk6 ðanj Þ; ¼ J06 ðanj Þ þ 2 k¼1 D h iE ðeÞ ð1Þ Hj ða; Þ ¼ R Sð1Þ ð ; a; ; 0Þ ; j D h iE ðf Þ ð2Þ Hj ða; Þ ¼ R Sð2Þ ; j ð ; a; ; 0Þ

  ðgÞ ~ ð1Þ Hj ða; 1 ; 2 ; Þ ¼ R Sjð1Þ ð ; a; 1 ; Þ  ð1Þ  Sj ð ; a; 2 ; Þ ;

  ð2Þ ð2Þ ðhÞ ~ Hj ða; 1 ; 2 ; Þ ¼ R Sj ð ; a; 1 ; Þ  ð2Þ  Sj ð ; a; 2 ; Þ ;

where ð2:1:8:35Þ

is the Fourier representation of the periodic delta function. Equation (2.1.8.34) then becomes   N=6 1 R
Q P Cðt1 ; t2 Þ ¼ d ð1=2
Þ ð1=2
Þ j¼1






R


k¼1

 exp ik þ 2inj t cos sinð þ Þ d

2 2 Qð1Þ j ða; Þ ¼ hJ0 ðsþ ÞJ0 ðs Þi ;

Qð2Þ j ða; Þ ¼ hJ0 ðsþ ÞJ0 ðs ÞJ0 ðcþ ÞJ0 ðc Þi ; 1 P ðdÞ Tj ða; Þ ¼ expð6ikÞJk6 ðanj Þ

ð2:1:8:34Þ

1 1 X expðikÞ 2
k¼1

cosð4kÞJk4 ðanj Þ;

ðcÞ

k¼1

2
ðÞ ¼

1 P k¼1

d1 d2 d3 2
ð1 þ 2 þ 3 Þ 


  3 P sinðk þ Þ ;  exp 2inj t cos 

Lj ða; Þ ¼ hJ0 ðsþ ÞJ0 ðs Þi 1 P ¼ cosð4kÞJk4

3 :




ð2:1:8:36Þ

where

211

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION 1 P Sð1Þ e3ik Jk3 ðsþ Þ j ð ; a; ; Þ ¼ k¼1

and Sð2Þ j ð ; a; ; Þ ¼

1 P

e3ik Jk ðsþ ÞJk ðþ ÞJk ð Þ:

k¼1

The averages appearing in the above summary are, in general, computed as


=2

R f ð Þ ¼ ð2=
Þ f ð Þ d ;

ð2:1:8:39Þ

0

~ jð2Þ which are computed as except Hjð2Þ and H


=3

R f ð Þ ¼ ð3=
Þ f ð Þ d ;

ð2:1:8:40Þ

0

Fig. 2.1.8.1. The same recalculated histogram as in Fig. 2.1.7.1 along with the centric correction-factor p.d.f. [equation (2.1.7.5)], truncated after two, three, four and five terms (dashed lines), and with that accurately computed for the correct space-group Fourier p.d.f. [equations (2.1.8.5) and (2.1.8.22)] (solid line).

where f ð Þ is any of the atomic characteristic functions indicated above. The superscripts preceding the symbols in the above summary are appended to the corresponding symbols in Table 2.1.8.1 on their first occurrence.

product of the k characteristic functions, each of which is related to one of these special positions; the same property of the characteristic function as that in Section 2.1.4.1 is here utilized.

2.1.8.6. Other non-ideal Fourier p.d.f.’s As pointed out above, the representation of the p.d.f.’s by Fourier series is also applicable to effects of noncrystallographic symmetry. Thus, Shmueli et al. (1985) obtained the following Fourier coefficient for the bicentric distribution in the space group P1 : " #
=2 R N=4 Q Ck ¼ ð2=
Þ J0 ð4
knj cos #Þ d#; ð2:1:8:41Þ 0

2.1.8.7. Comparison of the correction-factor and Fourier approaches The need for theoretical non-ideal distributions was exemplified by Fig. 2.1.7.1, referred to above, and the performance of the two approaches described above, for this particular example, is shown in Fig. 2.1.8.1. Briefly, the Fourier p.d.f. shows an excellent agreement with the histogram of recalculated jEj values, while the agreement attained by the Hermite correction factor is much less satisfactory, even for the (longest available to us) five-term expansion. It must be pointed out that (i) the inadequacy of ‘short’ correction factors, in the example shown, is due to the large deviation from the ideal behaviour and (ii) the number of terms used there in the Fourier summation is twenty, whereafter the summation is terminated. Obviously, the computation of twenty (or more) Fourier coefficients is easier than that of five terms in the correction factor. The convergence of the Fourier series is very satisfactory. It appears that the (analytically) exact Fourier approach is the preferred one in cases of large or intermediate deviations, while the correction-factor approach may cope well with small ones. As far as the availability of symmetrydependent centric and acentric p.d.f.’s is concerned, correction factors are available for all the space groups (see Table 2.1.7.1), while Fourier coefficients of p.d.f.’s are available for the first 206 space groups (see Table 2.1.8.1). It should be pointed out that p.d.f.’s based on the correction-factor method cope very well with cubic symmetries higher than Fd3 , even if the asymmetric unit of the space group is strongly heterogeneous (Rabinovich et al., 1991b). Both approaches described in this section are related to the characteristic function of the required p.d.f. The correction-factor p.d.f.’s (2.1.7.5) and (2.1.7.6) can be obtained by expanding the logarithm of the appropriate characteristic function in a series of cumulants [e.g. equation (2.1.4.13); see also Shmueli & Wilson (1982)], truncating the series and performing its term-by-term Fourier inversion. The Fourier p.d.f., on the other hand, is computed by forming a Fourier series whose coefficients are exact analytical forms of the characteristic function at points related to the summation indices [e.g. equations (2.1.8.5), (2.1.8.9) and (2.1.8.11), and Table 2.1.8.1] and truncating the series when the terms become small enough.

j¼1

to be used with equation (2.1.8.5). Furthermore, if we use the important property of the characteristic function as outlined in Section 2.1.4.1, it is easy to write down the Fourier coefficient for a P1 asymmetric unit containing a centrosymmetric fragment centred at a noncrystallographic centre and a number of atoms not related by symmetry. This Fourier for the above partially bicentric arrangement is a product of expressions (2.1.8.17) and (2.1.8.41), with the appropriate number of atoms in each factor (Shmueli & Weiss, 1985a). While the purely bicentric p.d.f. obtained by using (2.1.8.41) with (2.1.8.5) is significantly different from the ideal bicentric p.d.f. given by equation (2.1.5.13) only when the atomic composition is sufficiently heterogeneous, the above partially bicentric p.d.f. appears to be a useful development even for an equal-atom structure. The problem of the coexistence of several noncrystallographic centres of symmetry within the asymmetric unit of P1 , and its effect on the p.d.f. of jEj, was examined by Shmueli, Weiss & Wilson (1989) by the Fourier method. The latter study indicates that the strongest effect is produced by the presence of a single noncrystallographic centre. Another kind of noncrystallographic symmetry is that arising from the presence of centrosymmetric fragments in a noncentrosymmetric structure – the subcentric arrangement already discussed in Section 2.1.5.4. A Fourier-series representation of a non-ideal p.d.f. corresponding to this case was developed by Shmueli, Rabinovich & Weiss (1989), and was also applied to the mathematically equivalent effects of dispersion and presence of heavy scatterers in centrosymmetric special positions in a noncentrosymmetric space group. A variety of other non-ideal p.d.f.’s occur when heavy atoms are present in special positions (Shmueli & Weiss, 1988). Without going into the details of this development, it can be noted that if the atoms are distributed among k types of Wyckoff positions, the characteristic function corresponding to the p.d.f. of jEj is a

212

2.1. STATISTICAL PROPERTIES OF THE WEIGHTED RECIPROCAL LATTICE Rogers, D. & Wilson, A. J. C. (1953). The probability distribution of X-ray intensities. V. A note on some hypersymmetric distributions. Acta Cryst. 6, 439–449. Shmueli, U. (1979). Symmetry- and composition-dependent cumulative distributions of the normalized structure amplitude for use in intensity statistics. Acta Cryst. A35, 282–286. Shmueli, U. (1982). A study of generalized intensity statistics: extension of the theory and practical examples. Acta Cryst. A38, 362–371. Shmueli, U. & Kaldor, U. (1981). Calculation of even moments of the trigonometric structure factor. Methods and results. Acta Cryst. A37, 76–80. Shmueli, U. & Kaldor, U. (1983). Moments of the trigonometric structure factor. Acta Cryst. A39, 615–621. Shmueli, U., Rabinovich, S. & Weiss, G. H. (1989). Exact conditional distribution of a three-phase invariant in the space group P1. I. Derivation and simplification of the Fourier series. Acta Cryst. A45, 361–367. Shmueli, U., Rabinovich, S. & Weiss, G. H. (1990). Exact random-walk models in crystallographic statistics. V. Non-symmetrically bounded distributions of structure-factor magnitudes. Acta Cryst. A46, 241–246. Shmueli, U. & Weiss, G. H. (1985a). Centric, bicentric and partially bicentric intensity statistics. In Structure and Statistics in Crystallography, edited by A. J. C. Wilson, pp. 53–66. Guilderland: Adenine Press. Shmueli, U. & Weiss, G. H. (1985b). Exact joint probability distributions for centrosymmetric structure factors. Derivation and application to the
1 relationship in the space group P1 . Acta Cryst. A41, 401–408. Shmueli, U. & Weiss, G. H. (1986). Exact joint distribution of Eh , Ek and Eh+k, and the probability for the positive sign of the triple product in the space group P1 . Acta Cryst. A42, 240–246. Shmueli, U. & Weiss, G. H. (1987). Exact random-walk models in crystallographic statistics. III. Distributions of jEj for space groups of low symmetry. Acta Cryst. A43, 93–98. Shmueli, U. & Weiss, G. H. (1988). Exact random-walk models in crystallographic statistics. IV. P.d.f.’s of jEj allowing for atoms in special positions. Acta Cryst. A44, 413–417. Shmueli, U., Weiss, G. H. & Kiefer, J. E. (1985). Exact random-walk models in crystallographic statistics. II. The bicentric distribution in the space group P1 . Acta Cryst. A41, 55–59. Shmueli, U., Weiss, G. H., Kiefer, J. E. & Wilson, A. J. C. (1984). Exact random-walk models in crystallographic statistics. I. Space groups P1 and P1. Acta Cryst. A40, 651–660. Shmueli, U., Weiss, G. H. & Wilson, A. J. C. (1989). Explicit Fourier representations of non-ideal hypercentric p.d.f.’s of jEj. Acta Cryst. A45, 213–217. Shmueli, U. & Wilson, A. J. C. (1981). Effects of space-group symmetry and atomic heterogeneity on intensity statistics. Acta Cryst. A37, 342– 353. Shmueli, U. & Wilson, A. J. C. (1982). Intensity statistics: non-ideal distributions in theory and practice. In Crystallographic Statistics: Progress and Problems, edited by S. Ramaseshan, M. F. Richardson & A. J. C. Wilson, pp. 83–97. Bangalore: Indian Academy of Sciences. Shmueli, U. & Wilson, A. J. C. (1983). Generalized intensity statistics: the subcentric distribution and effects of dispersion. Acta Cryst. A39, 225–233. Spiegel, M. R. (1974). Theory and Problems of Fourier Analysis. Schaum’s Outline Series. New York: McGraw-Hill. Srinivasan, R. & Parthasarathy, S. (1976). Some Statistical Applications in X-ray Crystallography. Oxford: Pergamon Press. Stuart, A. & Ord, K. (1994). Kendall’s Advanced Theory of Statistics, Vol. 1, Distribution Theory, 6th ed. London: Edward Arnold. Weiss, G. H. & Kiefer, J. E. (1983). The Pearson random walk with unequal step sizes. J. Phys. A, 16, 489–495. Wilson, A. J. C. (1942). Determination of absolute from relative intensity data. Nature (London), 150, 151–152. Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–320. Wilson, A. J. C. (1950). The probability distribution of X-ray intensities. III. Effects of symmetry elements on zones and rows. Acta Cryst. 3, 258–261. Wilson, A. J. C. (1952). Hypercentric and hyperparallel distributions of X-ray intensities. Research (London), 5, 588–589. Wilson, A. J. C. (1956). The probability distribution of X-ray intensities. VII. Some sesquicentric distributions. Acta Cryst. 9, 143–144.

References Abramowitz, M. & Stegun, I. A. (1972). Handbook of Mathematical Functions. New York: Dover. Barakat, R. (1974). First-order statistics of combined random sinusoidal waves with application to laser speckle patterns. Opt. Acta, 21, 903– 921. Bernstein, S. (1922). Sur la the´ore`me limite du calcul des probabilite´s. Math. Ann. 85, 237–241. Bernstein, S. (1927). Sur l’extension du the´ore`me limite du calcul des probabilite´s aux sommes de quantite´s de´pendantes. Math. Ann. 97, 1–59. Crame´r, H. (1951). Mathematical Methods of Statistics. Princeton University Press. Faggiani, R., Lippert, B. & Lock, C. J. L. (1980). Heavy transition metal complexes of biologically important molecules. 4. Crystal and molecular structure of pentahydroxonium chloro(uracilatoN(1))(ethylenediamine)platinum(II)chloride (H5O2)[PtCl(NH2CH2CH2NH2)(C4H5N2O2)]Cl, and chloro(thyminato-N(1))(ethylenediamine)platinum(II), PtCl(NH2CH2CH2NH2)(C5H5N2O2). Inorg. Chem. 19, 295–300. French, S. & Wilson, K. (1978). On the treatment of negative intensity observations. Acta Cryst. A34, 517–525. Gerhard, O. E. (1993). Line-of-sight velocity profiles in spherical galaxies: breaking the degeneracy between anisotropy and mass. Mon. Not. R. Astron. Soc. 265, 213–230. Giacovazzo, C. (1977). On different probabilistic approaches to quartet theory. Acta Cryst. A33, 50–54. Giacovazzo, C. (1980). Direct Methods in Crystallography. London: Academic Press. Harker, D. (1953). The meaning of the average of jFj2 for large values of interplanar spacing. Acta Cryst. 6, 731–736. Hauptman, H. & Karle, J. (1953). Solution of the Phase Problem. I. The Centrosymmetric Crystal. Am. Crystallogr. Assoc. Monograph No. 3. Dayton, Ohio: Polycrystal Book Service. Howells, E. R., Phillips, D. C. & Rogers, D. (1950). The probability distribution of X-ray intensities. II. Experimental investigation and the X-ray detection of centers of symmetry. Acta Cryst. 3, 210–214. International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Heidelberg: Springer. International Tables for Crystallography (2004). Vol. C, Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers. Kendall, M. & Stuart, A. (1977). The Advanced Theory of Statistics, Vol. 1, 4th ed. London: Griffin. Lipson, H. & Woolfson, M. M. (1952). An extension of the use of intensity statistics. Acta Cryst. 5, 680–682. Lomer, T. R. & Wilson, A. J. C. (1975). Scaling of intensities. Acta Cryst. B31, 646–647. Marel, R. P. van der & Franx, M. (1993). A new method for the identification of non-Gaussian line profiles in elliptical galaxies. Astrophys. J. 407, 525–539. Myller-Lebedeff, W. (1907). Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen. Math. Ann. 64, 388– 416. Nigam, G. D. (1972). On the compensation of X-ray intensity. Indian J. Pure Appl. Phys. 10, 655–656. Nigam, G. D. & Wilson, A. J. C. (1980). Compensation of excess intensity in space group P2. Acta Cryst. A36, 832–833. Rabinovich, D. & Shakked, Z. (1984). A new approach to structure determination of large molecules by multi-dimensional search methods. Acta Cryst. A40, 195–200. Rabinovich, S., Shmueli, U., Stein, Z., Shashua, R. & Weiss, G. H. (1991a). Exact random-walk models in crystallographic statistics. VI. P.d.f.’s of jEj for all plane groups and most space groups. Acta Cryst. A47, 328–335. Rabinovich, S., Shmueli, U., Stein, Z., Shashua, R. & Weiss, G. H. (1991b). Exact random-walk models in crystallographic statistics. VII. An all-space-group study of the effects of atomic heterogeneity on the p.d.f.’s of jEj. Acta Cryst. A47, 336–340. Rayleigh, Lord (1879). Investigations in optics with special reference to the spectroscope. Philos. Mag. 8, 261–274. Rogers, D. (1950). The probability distribution of X-ray intensities. IV. New methods of determining crystal classes and space groups. Acta Cryst. 3, 455–464.

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Wilson, A. J. C. (1964). The probability distribution of X-ray intensities. VIII. A note on compensation for excess average intensity. Acta Cryst. 17, 1591–1592. Wilson, A. J. C. (1975). Effect of neglect of dispersion on apparent scale and temperature factors. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 325–332. Copenhagen: Munksgaard. Wilson, A. J. C. (1976). Statistical bias in least-squares refinement. Acta Cryst. A32, 994–996. Wilson, A. J. C. (1978a). On the probability of measuring the intensity of a reflection as negative. Acta Cryst. A34, 474–475. Wilson, A. J. C. (1978b). Variance of X-ray intensities: effect of dispersion and higher symmetries. Acta Cryst. A34, 986–994. Wilson, A. J. C. (1978c). Statistical bias in scaling factors: Erratum. Acta Cryst. B34, 1749. Wilson, A. J. C. (1979). Problems of resolution and bias in the experimental determination of the electron density and other densities in crystals. Acta Cryst. A35, 122–130.

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International Tables for Crystallography (2010). Vol. B, Chapter 2.2, pp. 215–243.

2.2. Direct methods By C. Giacovazzo

2.2.1. List of symbols and abbreviations atomic scattering factor of jth atom atomic number of jth atom number of atoms in the unit cell order of the point group

fj Zj N m

½
r
p ; ½
r
q ; ½
r
N ; . . . ¼

p P j¼1

Zjr ;

q P j¼1

Zjr ;

N P

2.2.3. Origin specification (a) Once the origin has been chosen, the symmetry operators Cs  ðRs ; Ts Þ and, through them, the algebraic form of the s.f. remain fixed. A shift of the origin through a vector with coordinates X0 transforms ’h into

Zjr ; . . .

’0h ¼ ’h  2h  X0

j¼1

½
r
N is always abbreviated to
r when N is the number of atoms in the cell p q N P P P P P P fj2 ; fj2 ; fj2 ; . . . p; q; N; . . . ¼ j¼1

s.f. n.s.f. cs. ncs. s.i. s.s. C ¼ ðR; TÞ ’h

j¼1

j¼1

and the symmetry operators Cs into C0s ¼ ðR0s ; T0s Þ, where R0s ¼ Rs ; T0s ¼ Ts þ ðRs  IÞX0

s ¼ 1; 2; . . . ; m:

ð2:2:3:2Þ

(b) Allowed or permissible origins (Hauptman & Karle, 1953, 1959) for a given algebraic form of the s.f. are all those points in direct space which, when taken as origin, maintain the same symmetry operators Cs. The allowed origins will therefore correspond to those points having the same symmetry environment in the sense that they are related to the symmetry elements in the same way. For instance, if Ts ¼ 0 for s ¼ 1; . . . ; 8, then the allowed origins in Pmmm are the eight inversion centres. To each functional form of the s.f. a set of permissible origins will correspond. (c) A translation between permissible origins will be called a permissible or allowed translation. Trivial allowed translations correspond to the lattice periods or to their multiples. A change of origin by an allowed translation does not change the algebraic form of the s.f. Thus, according to (2.2.3.2), all origins allowed by a fixed functional form of the s.f. will be connected by translational vectors Xp such that

structure factor normalized structure factor centrosymmetric noncentrosymmetric structure invariant structure seminvariant symmetry operator; R is the rotational part, T the translational part phase of the structure factor Fh ¼ jFh j expði’h Þ

2.2.2. Introduction Direct methods are today the most widely used tool for solving small crystal structures. They work well both for equal-atom molecules and when a few heavy atoms exist in the structure. In recent years the theoretical background of direct methods has been improved to take into account a large variety of prior information (the form of the molecule, its orientation, a partial structure, the presence of pseudosymmetry or of a superstructure, the availability of isomorphous data or of data affected by anomalous-dispersion effects, . . . ). Owing to this progress and to the increasing availability of powerful computers, the phase problem for small molecules has been solved in practice: a number of effective, highly automated packages are today available to the scientific community. The combination of direct methods with so-called direct-space methods have recently allowed the ab initio crystal structure solution of proteins. The present limit of complexity is about 2500 non-hydrogen atoms in the asymmetric unit, but diffraction data ˚ ) are required. Trials are under way to at atomic resolution (~1 A ˚ and have shown some success. bring this limit to 1.5 A The theoretical background and tables useful for origin specification are given in Section 2.2.3; in Section 2.2.4 the procedures for normalizing structure factors are summarized. Phase-determining formulae (inequalities, probabilistic formulae for triplet, quartet and quintet invariants, and for one- and twophase s.s.’s, determinantal formulae) are given in Section 2.2.5. In Section 2.2.6 the connection between direct methods and related techniques in real space is discussed. Practical procedures for solving small-molecule crystal structures are described in Sections 2.2.7 and 2.2.8, and references to the most extensively used packages are given in Section 2.2.9. The integration of direct methods, isomorphous replacement and anomalous-dispersion techniques is briefly discussed in Section 2.2.10. The reader interested in a more detailed description of the topic is referred to a recent textbook (Giacovazzo, 1998). Copyright © 2010 International Union of Crystallography

ð2:2:3:1Þ

s ¼ 1; 2; . . . ; m;

ðRs  IÞXp ¼ V;

ð2:2:3:3Þ

where V is a vector with zero or integer components. In centred space groups, an origin translation corresponding to a centring vector Bv does not change the functional form of the s.f. Therefore all vectors Bv represent permissible translations. Xp will then be an allowed translation (Giacovazzo, 1974) not only when, as imposed by (2.2.3.3), the difference T0s  Ts is equal to one or more lattice units, but also when, for any s, the condition s ¼ 1; 2; . . . ; m;

ðRs  IÞXp ¼ V þ Bv ;

 ¼ 0; 1 ð2:2:3:4Þ

is satisfied. We will call any set of cs. or ncs. space groups having the same allowed origin translations a Hauptman–Karle group (H–K group). The 94 ncs. primitive space groups, the 62 primitive cs. groups, the 44 ncs. centred space groups and the 30 cs. centred space groups can be collected into 13, 4, 14 and 5 H–K groups, respectively (Hauptman & Karle, 1953, 1956; Karle & Hauptman, 1961; Lessinger & Wondratschek, 1975). In Tables 2.2.3.1–2.2.3.4 the H–K groups are given together with the allowed origin translations. (d) Let us consider a product of structure factors FhA11  FhA22  . . .  FhAnn ¼

n Q j¼1

A

Fhj j

¼ exp i

n P j¼1

Aj being integer numbers.

215

! Aj ’hj

n Q j¼1

jFhj jAj ;

ð2:2:3:5Þ

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Pn

The factor j¼1 Aj ’hj is the phase of the product (2.2.3.5). A structure invariant (s.i.) is a product (2.2.3.5) such that n P

Aj hj ¼ 0:

is satisfied. The second condition means that at least one Aq exists that is not congruent to zero modulo each of the components of xs . If (2.2.3.10) is not satisfied for any n-set of integers Aj, the phases ’hj are linearly semindependent. If (2.2.3.10) is valid for n ¼ 1 and A ¼ 1, then h1 is said to be linearly semidependent and ’h1 is an s.s. It may be concluded that a seminvariant phase is linearly semidependent, and, vice versa, that a phase linearly semidependent is an s.s. In Tables 2.2.3.1–2.2.3.4 the allowed variations (which are those due to the allowed origin translations) for the semindependent phases are given for every H–K group. If ’h1 is linearly semindependent its value can be fixed arbitrarily because at least one origin compatible with the given value exists. Once ’h1 is assigned, the necessary condition to be able to fix a second phase ’h2 is that it should be linearly semindependent of ’h1 . Similarly, the necessary condition to be able arbitrarily to assign a third phase ’h3 is that it should be linearly semindependent from ’h1 and ’h2 . In general, the number of linearly semindependent phases is equal to the dimension of the seminvariant vector xs (see Tables 2.2.3.1–2.2.3.4). The reader will easily verify in (h, k, l) P (2, 2, 2) that the three phases ’oee, ’eoe , ’eoo define the origin (o indicates odd, e even). (h) From the theory summarized so far it is clear that a number of semindependent phases ’hj , equal to the dimension of the seminvariant vector xs , may be arbitrarily assigned in order to fix the origin. However, it is not always true that only one allowed origin compatible with the given phases exists. An additional condition is required such that only one permissible origin should lie at the intersection of the lattice planes corresponding to the origin-fixing reflections (or on the lattice plane h if one reflection is sufficient to define the origin). It may be shown that the condition is verified if the determinant formed with the vectors seminvariantly associated with the origin reflections, reduced modulo xs, has the value 1. In other words, such a determinant should be primitive modulo xs. For example, in P1
the three reflections

ð2:2:3:6Þ

j¼1

Since jFhj j are usually known from experiment, it is often said that s.i.’s are combinations of phases n P j¼1

Aj ’hj ;

ð2:2:3:7Þ

for which (2.2.3.6) holds. F0 , Fh Fh , Fh Fk Fhþk , Fh Fk Fl Fhþkþl , Fh Fk Fl Fp Fhþkþlþp are examples of s.i.’s for n ¼ 1; 2; 3; 4; 5. The value of any s.i. does not change with an arbitrary shift of the space-group origin and thus it will depend on the crystal structure only. (e) A structure seminvariant (s.s.) is a product of structure factors [or a combination of phases (2.2.3.7)] whose value is unchanged when the origin is moved by an allowed translation. Let Xp ’s be the permissible origin translations of the space group. Then the product (2.2.3.5) [or the sum (2.2.3.7)] is an s.s., if, in accordance with (2.2.3.1), n P

Aj ðhj  Xp Þ ¼ r;

p ¼ 1; 2; . . .

ð2:2:3:8Þ

j¼1

where r is a positive integer, null or a negative integer. Conditions (2.2.3.8) can be written in the following more useful form (Hauptman & Karle, 1953): n P j¼1

Aj hsj  0

ðmod xs Þ;

ð2:2:3:9Þ

where hsj is the vector seminvariantly associated with the vector hj and xs is the seminvariant modulus. In Tables 2.2.3.1–2.2.3.4, the reflection hs seminvariantly associated with h ¼ ðh; k; lÞ, the seminvariant modulus xs and seminvariant phases are given for every H–K group. The symbol of any group (cf. Giacovazzo, 1974) has the structure hs Lxs, where L stands for the lattice symbol. This symbol is underlined if the space group is cs. By definition, if the class of permissible origin has been chosen, that is to say, if the algebraic form of the symmetry operators has been fixed, then the value of an s.s. does not depend on the origin but on the crystal structure only. (f ) Suppose that we have chosen the symmetry operators Cs and thus fixed the functional form of the s.f.’s and the set of allowed origins. In order to describe the structure in direct space a unique reference origin must be fixed. Thus the phasedetermining process must also require a unique permissible origin congruent to the values assigned to the phases. More specifically, at the beginning of the structure-determining process by direct methods we shall assign as many phases as necessary to define a unique origin among those allowed (and, as we shall see, possibly to fix the enantiomorph). From the theory developed so far it is obvious that arbitrary phases can be assigned to one or more s.f.’s if there is at least one allowed origin which, fixed as the origin of the unit cell, will give those phase values to the chosen reflections. The concept of linear dependence will help us to fix the origin. (g) n phases ’hj are linearly semidependent (Hauptman & Karle, 1956) when the n vectors hsj seminvariantly associated with the hj are linearly dependent modulo xs, xs being the seminvariant modulus of the space group. In other words, when n P j¼1

Aj hsj  0

ðmod xs Þ;

Aq 6 0

ðmod xs Þ

h1 ¼ ð345Þ; h2 ¼ ð139Þ; h3 ¼ ð784Þ define the origin uniquely because



3 4 5


reduced mod ð2;2;2Þ
1

1 3 9

1

!


7 8 4
 1

0 1 0


1

1

¼ 1: 0


Furthermore, in P4mm ½hs ¼ ðh þ k; lÞ; xs ¼ ð2; 0Þ
h1 ¼ ð5; 2; 0Þ;

h2 ¼ ð6; 2; 1Þ

define the origin uniquely since



7 0
reduced mod ð2;0Þ
1



8 1
!
0


0

¼ 1: 1


(i) If an s.s. or an s.i. has a general value ’ for a given structure, it will have a value ’ for the enantiomorph structure. If ’ ¼ 0,  the s.s. has the same value for both enantiomorphs. Once the origin has been assigned, in ncs. space groups the sign of a given s.s. ’ 6¼ 0,  can be assigned to fix the enantiomorph. In practice it is often advisable to use an s.s. or an s.i. whose value is as near as possible to =2. 2.2.4. Normalized structure factors 2.2.4.1. Definition of normalized structure factor The normalized structure factors E (see also Chapter 2.1) are calculated according to (Hauptman & Karle, 1953)

ð2:2:3:10Þ

jEh j2 ¼ jFh j2 =hjFh j2 i;

216

ð2:2:4:1Þ

2.2. DIRECT METHODS Table 2.2.3.1. Allowed origin translations, seminvariant moduli and phases for centrosymmetric primitive space groups H–K group ðh; k; lÞPð2; 2; 2Þ P1


Space group

ðh þ k; lÞPð2; 2Þ 4 P m

Pmna

ðh þ k þ lÞPð2Þ

P3


R3


P3
1m

R3
m

P

2 m

Pcca

P

42 m

4 P cc n

P

21 m

Pbam

P

4 n

P

42 mc m

P3
1c

R3
c

P

2 c

Pccn

P

42 n

P

42 cm m

P3
m1

Pm3


P

21 c

Pbcm

P

4 mm m

P

42 bc n

P3
c1

Pn3


Pmmm

Pnnm

P

4 cc m

P

42 nm n

P

6 m

Pa3


Pnnn

Pmmn

4 P bm n

P

42 bc m

P

63 m

Pm3
m

Pccm

Pbcn

4 P nc n

P

42 nm m

P

6 mm m

Pn3
n

Pban

Pbca

P

4 bm m

P

42 mc n

P

6 cc m

Pm3
n

Pmma

Pnma

P

4 nc m

P

42 cm n

P

63 cm m

Pn3
m

(0, 0, 0);

ð0; 12 ; 12Þ

(0, 0, 0)

63 mc m (0, 0, 0)

ð12 ; 0; 0Þ;

ð12 ; 0; 12Þ

ð0; 0; 12Þ

ð0; 0; 12Þ

ð12 ; 12 ; 12Þ

ð0; 12 ; 0Þ;

ð12 ; 12 ; 0Þ

ð12 ; 12 ; 0Þ

ð0; 0; 12Þ;

ð12 ; 12 ; 12Þ

ð12 ; 12 ; 12Þ

Pnna Allowed origin translations

ðlÞPð2Þ 4 P mm n

P

(0, 0, 0)

Vector hs seminvariantly associated with h ¼ ðh; k; lÞ

ðh; k; lÞ

ðh þ k; lÞ

(l)

ðh þ k þ lÞ

Seminvariant modulus xs

(2, 2, 2)

(2, 2)

(2)

(2)

Seminvariant phases

’eee

’eee ; ’ooe

’eee ; ’eoe ’oee ; ’ooe

’eee ; ’ooe ’oeo ; ’eoo

Number of semindependent phases to be specified

3

2

1

1

where jFh j2 is the squared observed structure-factor magnitude on the absolute scale and hjFh j2 i is the expected value of jFh j2. hjFh j2 i depends on the available a priori information. Often, but not always, this may be considered as a combination of several typical situations. We mention: (a) No structural information. The atomic positions are considered random variables. Then hjFh j2 i ¼ "h

N P

fj2 ¼ "h

P

(c) P atomic groups with a known configuration, correctly oriented, but with unknown position (Main, 1976). Then a certain group of interatomic vectors rj1 j2 is fixed and ! Mi P P P P hjFh j2 i ¼ "h fj1 fj2 exp 2ih  rj1 j2 : Nþ i¼1 j1 6¼j2 ¼1

The above formula has been derived on the assumption that primitive positional random variables are uniformly distributed over the unit cell. Such an assumption may be considered unfavourable (Giacovazzo, 1988) in space groups for which the allowed shifts of origin, consistent with the chosen algebraic form for the symmetry operators Cs, are arbitrary displacements along any polar axes. Thanks to the indeterminacy in the choice of origin, the first of the shifts si (to be applied to the ith fragment in order to translate atoms in the correct positions) may be restricted to a region which is smaller than the unit cell (e.g. in P2 we are free to specify the origin along the diad axis by restricting s1 to the family of vectors fs1 g of type ½x0z
). The practical consequence is that hjFh j2 i is significantly modified in polar space groups if h satisfies

N

j¼1

so that Eh ¼

ð"h

Fh P

NÞ

1=2

:

ð2:2:4:2Þ

"h takes account of the effect of space-group symmetry (see Chapter 2.1). (b) P atomic groups having a known configuration but with unknown orientation and position (Main, 1976). Then a certain number of interatomic distances rj1 j2 are known and ! Mi P X X X sin 2qrj1 j2 2 ; þ fj 1 fj 2 hjFh j i ¼ "h N 2qrj1 j2 i¼1 j 6¼j ¼1 1

h  s1 ¼ 0;

2

where s1 belongs to the family of restricted vectors fs1 g. (d) Atomic groups correctly positioned. Then (Main, 1976; Giacovazzo, 1983a)

where Mi is the number of atoms in the ith molecular fragment and q ¼ jhj.

217

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION P hjFh j i ¼ jFp; h j2 þ "h q ; Often substructures are not ideal: e.g. atoms related by pseudotranslational symmetry are ideally located but of different type (replacive deviations from ideality); or they are equal but where Fp;h is the structure factor of the partial known structure not ideally located (displacive deviations); or a combination of and q are the atoms with unknown positions. the two situations occurs. In these cases a correlation exists (e) A pseudotranslational symmetry is present. Let between the substructure and the superstructure. It has been u1 ; u2 ; u3 ; . . . be the pseudotranslation vectors of order shown (Mackay, 1953; Cascarano et al., 1988a) that the scattering n1 ; n2 ; n3 ; . . ., respectively. Furthermore, let p be the number of power of the substructural part may be estimated via a statistical atoms (symmetry equivalents included) whose positions are analysis of diffraction data for ideal pseudotranslational related by pseudotranslational symmetry and q the number of symmetry or for displacive deviations from it, while it is not atoms (symmetry equivalents included) whose positions are not estimable in the case of replacive deviations. related by any pseudotranslation. Then (Cascarano et al., 1985a,b) P P 2.2.4.2. Definition of quasi-normalized structure factor hjFh j2 i ¼ "h ðh p þ q Þ; When probability theory is not used, the quasi-normalized structure factors E h and the unitary structure factors Uh are often where used. E h and Uh are defined according to ðn n n . . .Þh h ¼ 1 2 3 jE h j2 ¼ "h jEh j2 m ! . P N and h is the number of times for which algebraic congruences Uh ¼ Fh fj : j¼1 h  Rs ui  0 ðmod 1Þ for i ¼ 1; 2; 3; . . . 2

PN Since j¼1 fj is the largest possible value for Fh ; Uh represents the fraction of Fh with respect to its largest possible value. Therefore

are simultaneously satisfied when s varies from 1 to m. If h ¼ 0 then Fh is said to be a superstructure reflection, otherwise it is a substructure reflection.

Table 2.2.3.2. Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric primitive space groups H–K group Space group

ðh; k; lÞPð0; 0; 0Þ

ðh; k; lÞPð2; 0; 2Þ

ðh; k; lÞPð0; 2; 0Þ

ðh; k; lÞPð2; 2; 2Þ

ðh; k; lÞPð2; 2; 0Þ

ðh þ k; lÞPð2; 0Þ

P1

P2

Pm

P222

Pmm2

P4

ðh þ k; lÞPð2; P4


P21

Pc

P2221

Pmc21

P41

P422

P21 21 2

Pcc2

P42

P421 2

P21 21 21

Pma2

P43

P41 22

Pca21

P4mm

P41 21 2

Pnc2

P4bm

P42 22

Pmn21 Pba2

P42 cm P42 nm

P42 21 2 P43 22

Pna21

P4cc

Pnn2

P4nc

P43 21 2 P4
2m

P42 mc P42 bc

P4
2c P4
21 m P4
21 c P4
m2 P4
c2 P4
b2 P4
n2

Allowed origin translations

(x, y, z)

(0, y, 0)

(x, 0, z)

(0, 0, 0)

(0, 0, z)

(0, 0, z)

(0, 0, 0)

ð0; y; 12Þ

ðx; 12 ; zÞ

ð12 ; 0; 0Þ

ð0; 12 ; zÞ

ð12 ; 12 ; zÞ

ð0; 0; 12Þ

ð12 ; y; 0Þ

ð0; 12 ; 0Þ

ð12 ; 0; zÞ

ð12 ; 12 ; 0Þ

ð12 ; y; 12Þ

ð0; 0; 12Þ ð0; 12 ; 12Þ ð12 ; 0; 12Þ ð12 ; 12 ; 0Þ ð12 ; 12 ; 12Þ

ð12 ; 12 ; zÞ

ð12 ; 12 ; 12Þ

Vector hs seminvariantly associated with h ¼ ðh; k; lÞ

(h, k, l)

(h, k, l)

(h, k, l)

(h, k, l)

(h, k, l)

ðh þ k; lÞ

ðh þ k; lÞ

Seminvariant modulus xs

(0, 0, 0)

(2, 0, 2)

(0, 2, 0)

(2, 2, 2)

(2, 2, 0)

(2, 0)

(2, 2)

Seminvariant phases

’000

’e0e

’0e0

’eee

’ee0

’ee0 ’oo0

’eee ’ooe

Allowed variations for the semindependent phases

k1k

k1k, k2k if k¼0

k1k, k2k if h¼l¼0

k2k

k1k, k2k if l¼0

k1k, k2k if l¼0

k2k

Number of semindependent phases to be specified

3

3

3

3

3

2

2

218

e 2.2.3.2.

þ k; lÞPð2; 0Þ

mm

2.2. DIRECT METHODS assumed that all the atoms are at rest. hjF o j2 i depends upon the structural information that is available (see Section 2.2.4.1 for some examples). Equation (2.2.4.3) may be rewritten as

0  jUh j  1: If atoms are equal, then Uh ¼ E h =N 1=2 . 2.2.4.3. The calculation of normalized structure factors

  hIi ln ¼  ln K  2Bs2 ; hjF o j2 i

N.s.f.’s cannot be calculated by applying (2.2.4.1) to observed s.f.’s because: (a) the observed magnitudes Ih (already corrected for Lp factor, absorption, . . . ) are on a relative scale; (b) hjFh j2 i cannot be calculated without having estimated the vibrational motion of the atoms. This is usually obtained by the well known Wilson plot (Wilson, 1942), according to which observed data are divided into ranges of s2 ¼ sin2 =2 and averages of intensity hIh i are taken in each shell. Reflection multiplicities and other effects of spacegroup symmetry on intensities must be taken into account when such averages are calculated. The shells are symmetrically overlapped in order to reduce statistical fluctuations and are restricted so that the number of reflections in each shell is reasonably large. For each shell KhIi ¼ hjFj2 i ¼ hjF o j2 i expð2Bs2 Þ

which plotted at various s2 should be a straight line of which the slope (2B) and intercept (ln K) on the logarithmic axis can be obtained by applying a linear least-squares procedure. Very often molecular geometries produce perceptible departures from linearity in the logarithmic Wilson plot. However, the more extensive the available a priori information on the structure is, the closer, on the average, are the Wilson-plot curves to their least-squares straight lines. Accurate estimates of B and K require good strategies (Rogers & Wilson, 1953) for: (1) treatment of weak measured data. If weak data are set to zero, there will be bias in the statistics. Methods are, however, available (French & Wilson, 1978) that provide an a posteriori estimate of weak (even negative) intensities by means of Bayesian statistics.

ð2:2:4:3Þ

should be obtained, where K is the scale factor needed to place X-ray intensities on the absolute scale, B is the overall thermal parameter and hjF o j2 i is the expected value of jFj2 in which it is Table 2.2.3.2 (cont.)

ðh þ k; lÞPð2; 2Þ

ðh  k; lÞPð3; 0Þ

ð2h þ 4k þ 3lÞPð6Þ

ðlÞPð0Þ

ðlÞPð2Þ

ðh þ k þ lÞPð0Þ

ðh þ k þ lÞPð2Þ

P4


P3

P312

P31m

P321

R3

R32

P422 P421 2

P31 P32

P31 12 P32 12

P31c P6

P31 21 P32 21

R3m R3c

P23 P21 3

P41 22

P3m1

P622

P432

P3c1

P6 P6
m2

P61

P41 21 2

P65

P61 22

P42 32

P6
c2

m

P42 22

P64

P65 22

P43 32

cm

P42 21 2

P63

P62 22

nm

P43 22

P62

P64 22

P41 32 P4
3m

c

P43 21 2 P4
2m P4
2c

P6mm

P63 22 P6
2m

P4
21 m P4
21 c

P63 mc

c mc

bc

P6cc P63 cm

P4
b2 P4
n2 (0, 0, 0)

(0, 0, z)

(0, 0, 0)

; zÞ

ð0; 0; 12Þ ð12 ; 12 ; 0Þ

ð13 ; 23 ; zÞ ð23 ; 13 ; zÞ

ð0; 0; 12Þ ð13 ; 23 ; 0Þ

0)

k, k2k if ¼0

P6
2c

P4
m2 P4
c2

0, z)

þ k; lÞ

P4
3n

ð12 ; 12 ; 12Þ

(0, 0, z)

(0, 0, 0)

(x, x, x)

ð0; 0; 12Þ

(0, 0, 0) ð12 ; 12 ; 12Þ

ð13 ; 23 ; 12Þ ð23 ; 13 ; 0Þ ð23 ; 13 ; 12Þ

ðh þ k; lÞ

ðh  k; lÞ

ð2h þ 4k þ 3lÞ

(l)

(l)

ðh þ k þ lÞ

ðh þ k þ lÞ

(2, 2)

(3, 0)

(6)

(0)

(2)

(0)

(2)

’eee ’ooe

’hk0 if h  k ¼ 0 (mod 3)

’hkl if 2h þ 4k þ 3l ¼ 0 (mod 6)

’hk0

’hke

’h; k; h
þk


’eee ; ’ooe ’oeo ; ’ooe

k2k

k1k, k3k if l ¼ 0

k2k if h  k (mod 3) k3k if l  0 (mod 2)

k1k

k2k

k1k

k2k

2

2

1

1

1

1

1

219

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.2.3.3. Allowed origin translations, seminvariant moduli and phases for centrosymmetric non-primitive space groups H–K group ðh; lÞCð2; 2Þ 2 C m 2 C c

Space groups

Allowed origin translations

ðk; lÞIð2; 2Þ

ðh þ k þ lÞFð2Þ

ðlÞIð2Þ 4 I m 4 I 1 a

Immm

Fmmm

Ibam

Fddd

Cmcm

Ibca

Fm3


I

4 mm m

Im3
m

Cmca

Imma

Fd3


I

4 cm m

Ia3
d

Cmmm

Fm3
m

I

41 md a

Cccm

Fm3
c

I

41 cd a

Cmma

Fd3
m

Ccca

Fd3
c

I Im3
Ia3


(0, 0, 0)

(0, 0, 0)

(0, 0, 0)

(0, 0, 0)

ð0; 0; 12Þ

ð0; 0; 12Þ

ð12 ; 12 ; 12Þ

ð0; 0; 12Þ

ð12 ; 0; 0Þ

ð0; 12 ; 0Þ

ð12 ; 0; 12Þ

ð12 ; 0; 0Þ

Vector hs seminvariantly associated with h ¼ ðh; k; lÞ

ðh; lÞ

ðk; lÞ

ðh þ k þ lÞ

(l)

ðh; k; lÞ

Seminvariant modulus xs

(2, 2)

(2, 2)

(2)

(2)

(1, 1, 1)

Seminvariant phases

’eee

’eee

’eee

’eoe ; ’eee ’ooe ; ’oee

All

Number of semindependent phases to be specified

2

2

1

1

0

(2) treatment of missing weak data (Rogers et al., 1955; Vickovic´ & Viterbo, 1979). All unobserved reflections may assume

(0, 0, 0)

where the subscript ‘o min’ refers to the minimum observed intensity. Once K and B have been estimated, Eh values can be obtained from experimental data by

 ¼ jFo min j2 =3 for cs. space groups

jEh j2 ¼

 ¼ jFo min j2 =2 for ncs. space groups,

KIh ; o 2 hjFh j i expð2Bs2 Þ

Table 2.2.3.4. Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric non-primitive space groups H–K group Space group

ðk; lÞCð0; 2Þ

ðh; lÞCð0; 0Þ

ðh; lÞCð2; 0Þ

ðh; lÞCð2; 2Þ

ðh; lÞAð2; 0Þ

ðh; lÞIð2; 0Þ

ðh; lÞIð2; 2Þ

C2

Cm

Cmm2

C222

Amm2

Imm2

I222

Cc

Cmc21 Ccc2

C2221

Abm2 Ama2

Iba2 Ima2

I21 21 21

(0, 0, 0)

Aba2

Allowed origin translations

(0, y, 0)

(x, 0, z)

ð0; y; 12Þ

(0, 0, z)

(0, 0, 0)

(0, 0, z)

(0, 0, z)

ð12 ; 0; zÞ

ð0; 0; 12Þ

ð12 ; 0; zÞ

ð12 ; 0; zÞ

ð0; 0; 12Þ

ð12 ; 0; 0Þ

ð0; 12 ; 0Þ

ð12 ; 0; 12Þ

ð12 ; 0; 0Þ

Vector hs seminvariantly associated with h ¼ ðh; k; lÞ

(k, l)

(h, l)

(h, l)

(h, l)

(h, l)

(h, l)

(h, l)

Seminvariant modulus xs

(0, 2)

(0, 0)

(2, 0)

(2, 2)

(2, 0)

(2, 0)

(2, 2)

Seminvariant phases

’e0e

’0e0

’ee0

’eee

’ee0

’ee0

’eee

Allowed variations for the semindependent phases

k1k, k2k if k¼0

k1k

k1k, k2k if l¼0

k2k

k1k, k2k if l¼0

k1k, k2k if l¼0

k2k

Number of semindependent phases to be specified

2

2

2

2

2

2

2

220

2.2. DIRECT METHODS hjFho j2 i

jFho j2

where is the expected value of for the reflection h on the basis of the available a priori information. 2.2.4.4. Probability distributions of normalized structure factors Under some fairly general assumptions (see Chapter 2.1) probability distribution functions for the variable jEj for cs. and ncs. structures are (see Fig. 2.2.4.1) rffiffiffi   2 E2 exp  djEj ð2:2:4:4Þ 1
PðjEjÞ djEj ¼  2 and 1 PðjEjÞ djEj

¼ 2jEj expðjEj2 Þ djEj;

ð2:2:4:5Þ

respectively. Corresponding cumulative functions are (see Fig. 2.2.4.2) rffiffiffi ZjEj    2 2 t jEj dt ¼ erf pffiffiffi ; exp  1
NðjEjÞ ¼  2 2

Fig. 2.2.4.1. Probability density functions for cs. and ncs. crystals.

Significant developments are the derivation of inequalities and the introduction of probabilistic techniques via the use of joint probability distribution methods (Hauptman & Karle, 1953).

0

ZjEj ¼

1 NðjEjÞ

2t expðt2 Þ dt ¼ 1  expðjEj2 Þ:

0

2.2.5.1. Inequalities among structure factors Some moments of the distributions (2.2.4.4) and (2.2.4.5) are listed in Table 2.2.4.1. In the absence of other indications for a given crystal structure, a cs. or an ncs. space group will be preferred according to whether the statistical tests yield values closer to column 2 or to column 3 of Table 2.2.4.1. For further details about the distribution of intensities see Chapter 2.1.

An extensive system of inequalities exists for the coefficients of a Fourier series which represents a positive function. This can restrict the allowed values for the phases of the s.f.’s in terms of measured structure-factor magnitudes. Harker & Kasper (1948) derived two types of inequalities: Type 1. A modulus is bound by a combination of structure factors: jUh j2 

2.2.5. Phase-determining formulae From the earliest periods of X-ray structure analysis several authors (Ott, 1927; Banerjee, 1933; Avrami, 1938) have tried to determine atomic positions directly from diffraction intensities.

e 2.2.3.4.

m 1X a ðhÞUhðIRs Þ ; m s¼1 s

ð2:2:5:1Þ

where m is the order of the point group and as ðhÞ ¼ expð2ih  Ts Þ.

Table 2.2.3.4 (cont.)

lÞIð2; 0Þ

ðh; lÞIð2; 2Þ

ðh þ k þ lÞFð2Þ

ðh þ k þ lÞFð4Þ

ðlÞIð0Þ

ðlÞIð2Þ

ð2k  lÞIð4Þ

ðlÞFð0Þ

m2

I222

F432

F222

I4

I422

Fmm2

I23

a2 a2

I21 21 21

F41 32

F23 F 4
3m F 4
3c

I41 I4mm

I41 22 I 4
2m I 4
2d

I 4
I 4
m2 I 4
c2

Fdd2

I21 3 I432

I4cm

I41 32 I 4
3m I 4
3d

I41 md I41 cd 0, z)

(0, 0, 0)

(0, 0, 0)

(0, 0, 0)

0; zÞ

ð0; 0; 12Þ

ð12 ; 12 ; 12Þ

ð14 ; 14 ; 14Þ

(0, 0, z)

I

(0, 0, 0)

(0, 0, 0)

ð0; 0; 12Þ

ð0; 0; 12Þ

ð0; 12 ; 0Þ

ð12 ; 12 ; 12Þ

ð12 ; 0; 34Þ

ð12 ; 0; 0Þ

ð34 ; 34 ; 34Þ

ð12 ; 0; 14Þ

(0, 0, z)

(0, 0, 0)

(h, l)

ðh þ k þ lÞ

ðh þ k þ lÞ

(l)

(l)

ð2k  lÞ

(l)

ðh; k; lÞ

0)

(2, 2)

(2)

(4)

(0)

(2)

(4)

(0)

(1, 1, 1)

0

’eee

’eee

’hkl with hþkþl 0 (mod 4)

’hk0

’hke

’hkl with ð2k  lÞ  0 (mod 4)

’hk0

All

k2k

k2k

k2k if h þ k þ l  0 (mod 2) k4k if h þ k þ l  1 (mod 2)

k1k

k2k

k2k if h þ k þ l  0 (mod 2) k4k if 2k  l  1 (mod 2)

k1k

All

2

1

1

1

1

1

1

0

l)

1k, k2k if ¼0

221

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.2.4.1. Moments of the distributions (2.2.4.4) and (2.2.4.5) RðEs Þ is the percentage of n.s.f.’s with amplitude greater than the threshold Es . Criterion

Fig. 2.2.4.2. Cumulative distribution functions for cs. and ncs. crystals.

0.798

0.886

hjEj2 i hjEj3 i

1.000 1.596

1.000 1.329

hjEj4 i

3.000

2.000

hjEj5 i

6.383

3.323

hjEj6 i

15.000

6.000

hjE2  1ji

0.968

0.736

hðE2  1Þ2 i

2.000

1.000

hðE2  1Þ3 i

8.000

2.000

hjE2  1j3 i R(1)

8.691 0.320

2.415 0.368

R(2)

0.050

0.018

R(3)

0.003

0.0001

Uh2  0:5 þ 0:5U2h :

P21 :

jUh; k; l j2  1 2 Uh; k; l

For m ¼ 3, equation (2.2.5.3) becomes


U0 Uh Uk


U0 Uhk

 0; D3 ¼

Uh
U U U0
k kh

 0:5 þ 0:5U2h; 2k; 2l

jUh; k; l j2  0:5 þ 0:5ð1Þk U2h; 0; 2l :

The meaning of each inequality is easily understandable: in P1
, for example, U2h; 2k; 2l must be positive if jUh; k; l j is large enough. Type 2. The modulus of the sum or of the difference of two structure factors is bound by a combination of structure factors: ( m m X 1 X 2 jUh  Uh0 j  as ðhÞUhðIRs Þ þ as ðh0 ÞUh0 ðIRs Þ m s¼1 s¼1 " #) m X as ðh0 ÞUhh0 Rs 2R e ð2:2:5:2Þ

from which 1  jUh j2  jUk j2  jUhk j2 þ 2jUh Uk Uhk j cos h; k  0; ð2:2:5:4Þ where h; k ¼ ’h  ’k  ’hk : If the moduli jUh j, jUk j, jUhk j are large enough, (2.2.5.4) is not satisfied for all values of h; k. In cs. structures the eventual check that one of the two values of h; k does not satisfy (2.2.5.4) brings about the unambiguous identification of the sign of the product Uh Uk Uhk . It was observed (Gillis, 1948) that ‘there was a number of cases in which both signs satisfied the inequality, one of them by a comfortable margin and the other by only a relatively small margin. In almost all such cases it was the former sign which was the correct one. That suggests that the method may have some power in reserve in the sense that there are still fundamentally stronger inequalities to be discovered’. Today we identify this power in reserve in the use of probability theory.

s¼1

where R e stands for ‘real part of’. Equation (2.2.5.2) applied to P1 gives jUh  Uh0 j2  2  2jUhh0 j cos ’hh0 : A variant of (2.2.5.2) valid for cs. space groups is ðUh  Uh0 Þ2  ð1  Uhþh0 Þð1  Uhh0 Þ: After Harker & Kasper’s contributions, several other inequalities were discovered (Gillis, 1948; Goedkoop, 1950; Okaya & Nitta, 1952; de Wolff & Bouman, 1954; Bouman, 1956; Oda et al., 1961). The most general are the Karle–Hauptman inequalities (Karle & Hauptman, 1950):


U0 Uh1 Uh2 . . . Uhn



Uh1 U0 Uh1 h2 . . . Uh1 hn



U0 . . . Uh2 hn

 0: ð2:2:5:3Þ Dm ¼
Uh2 Uh2 h1
. ..

.. .. ..
.. .
. . .

U U U ... U
hn

hn h1

hn h2

Noncentrosymmetric distribution

hjEji

Applied to low-order space groups, (2.2.5.1) gives P1 : P1
:

Centrosymmetric distribution

2.2.5.2. Probabilistic phase relationships for structure invariants For any space group (see Section 2.2.3) there are linear combinations of phases with cosines that are, in principle, fixed by the jEj magnitudes alone (s.i.’s) or by the jEj values and the trigonometric form of the structure factor (s.s.’s). This result greatly stimulated the calculation of conditional distribution functions

0

PðjfRgÞ;

ð2:2:5:5Þ

P where Rh ¼ jEh j,  ¼ Ai ’hi is an s.i. or an s.s. and fRg is a suitable set of diffraction magnitudes. The method was first proposed by Hauptman & Karle (1953) and was developed further by several authors (Bertaut, 1955a,b, 1960; Klug, 1958; Naya et al., 1964, 1965; Giacovazzo, 1980a). From a probabilistic point of view the crystallographic problem is clear: the joint distribution PðEh1 ; . . . ; Ehn Þ, from which the conditional distributions (2.2.5.5) can be derived, involves a number of normalized structure factors each of which is a linear sum of random vari-

The determinant can be of any order but the leading column (or row) must consist of U’s with different indices, although, within the column, symmetry-related U’s may occur. For n ¼ 2 and h2 ¼ 2h1 ¼ 2h, equation (2.2.5.3) reduces to


U0 Uh U2h


D3 ¼

Uh U0 Uh

 0;
U Uh U0
2h which, for cs. structures, gives the Harker & Kasper inequality

222

2.2. DIRECT METHODS

Fig. 2.2.5.1. Curves of (2.2.5.6) for some values of G ¼ 2
3
23=2 jEh Ek Ehk j.

ables (the atomic contributions to the structure factors). So, for the probabilistic interpretation of the phase problem, the atomic positions and the reciprocal vectors may be considered as random variables. A further problem is that of identifying, for a given , a suitable set of magnitudes jEj on which  primarily depends. The formulation of the nested neighbourhood principle first (Hauptman, 1975) fixed the idea of defining a sequence of sets of reflections each contained in the succeeding one and having the property that any s.i. or s.s. may be estimated via the magnitudes constituting the various neighbourhoods. A subsequent more general theory, the representation method (Giacovazzo, 1977a, 1980b), arranges for any  the set of intensities in a sequence of subsets in order of their expected effectiveness (in the statistical sense) for the estimation of . In the following sections the main formulae estimating loworder invariants and seminvariants or relating phases to other phases and diffraction magnitudes are given.

Fig. 2.2.5.2. Variance (in square radians) as a function of .

Vh ¼

j¼1

where Pþ is the probability that Eh is positive and k ranges over the set of known values Ek Ehk. The larger the absolute value of the argument of tanh, the more reliable is the phase indication. An auxiliary formula exploiting all the jEj’s in reciprocal space in order to estimate a single  is the B3; 0 formula (Hauptman & Karle, 1958; Karle & Hauptman, 1958) given by

ð2:2:5:6Þ

PN where
n ¼ j¼1 Zjn, Zj is the atomic number of the jth atom and In is the modified Bessel function of order n. In Fig. 2.2.5.1 the distribution PðÞ is shown for different values of G. The conditional probability distribution for ’h , given a set of ð’kj þ ’hkj Þ and Gj ¼ 2
3
23=2 Rh Rkj Rhkj , is given (Karle & Hauptman, 1956; Karle & Karle, 1966) by Pð’h Þ ¼ ½2I0 ðÞ
1 exp½ cosð’h  h Þ
;

jEh1 Eh2 Eh1 h2 j cosð’h1 þ ’h2  ’h1 þh2 Þ ’ ChðjEk jp  jEjp ÞðjEh1 þk jp  jEjp ÞðjEh1 þh2 þk jp  jEjp Þik 

ð2:2:5:7Þ

"  ¼

r P j¼1

" þ

#2 Gh; kj cosð’kj þ ’hkj Þ

r P j¼1

P tan h ¼ P

#2 Gh; kj sinð’kj þ ’hkj Þ ;

2
6
81=2 þ ðjEh1 j2 þ jEh2 j2 þ jEh1 þh2 j2 Þ . . . ;
4
43=2

ð2:2:5:12Þ

where C is a constant which differs for cs. and ncs. crystals, jEjp is the average value of jEjp and p is normally chosen to be some small number. Several modifications of (2.2.5.12) have been proposed (Hauptman, 1964, 1970; Karle, 1970a; Giacovazzo, 1977b). A recent formula (Cascarano, Giacovazzo, Camalli et al., 1984) exploits information contained within the second representation of , that is to say, within the collection of special quintets (see Section 2.2.5.6):

where 2

ð2:2:5:10Þ

which is plotted in Fig. 2.2.5.2. Equation (2.2.5.9) is the so-called tangent formula. According to (2.2.5.10), the larger is  the more reliable is the relation ’h ¼ h . For an equal-atom structure
3
23=2 ¼ N 1=2. The basic conditional formula for sign determination of Eh in cs. crystals is Cochran & Woolfson’s (1955) formula ! r P 3=2 ð2:2:5:11Þ Pþ ¼ 12 þ 12 tanh
3
2 jEh j Ekj Ehkj ;

2.2.5.3. Triplet relationships The basic formula for the estimation of the triplet phase  ¼ ’h  ’k  ’hk given the parameter G ¼ 2
3
23=2  Rh Rk Rhk is Cochran’s (1955) formula PðÞ ¼ ½2I0 ðGÞ
1 expðG cos Þ;

1 X 2 I2n ðÞ þ ½I0 ðÞ
1 n2 3 n¼1 1 X I2nþ1 ðÞ  4½I0 ðÞ
1 ; ð2n þ 1Þ2 n¼0

ð2:2:5:8Þ

’h1 þ ’h2  ’h1 þh2 þ ’k  ’k ; j Gh; kj

sinð’kj þ ’hkj Þ

j Gh; kj cosð’kj þ ’hkj Þ

:

where k is a free vector. The formula retains the same algebraic form as (2.2.5.6), but

ð2:2:5:9Þ

G¼

h is the most probable value for ’h . The variance of ’h may be obtained from (2.2.5.7) and is given by

223

2Rh1 Rh2 Rh3 pffiffiffiffi ð1 þ QÞ; N

ð2:2:5:13Þ

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION where h. . .iR means rotational average. The average of the exponential term extends over all orientations of the triangle formed by the atoms j, k and l, and is given (Hauptman, 1965) by

where ½h3 ¼ ðh1 þ h2 Þ
, Q¼

X k

P0 m

i¼1 Ak; i =N   ; P0 m 1 þ "h1 "h2 "h3 þ i¼1 Bk; i 2N

Bðz; tÞ ¼ hexp½2iðh  r þ h0  r0 Þ
i  1=2 X 1  t2n ¼ 2 Jð4nþ1Þ=2 ðzÞ; 2z n¼0 ðn!Þ

Ak; i ¼ "k ½"h1 þkRi ð"h2 kRi þ "h3 kRi Þ þ "h2 þkRi ð"h1 kRi þ "h3 kRi Þ þ "h3 þkRi ð"h1 kRi þ "h2 kRi Þ
;

where

Bk; i ¼ "h1 ½"k ð"h1 þkRi þ "h1 kRi Þ

z ¼ 2½q2 r2 þ 2qrq0 r0 cos ’q cos ’r þ q02 r02
1=2

þ "h2 þkRi "h3 kRi þ "h2 kRi "h3 þkRi
þ "h2 ½"k ð"h2 þkRi þ "h2 kRi Þ

and

þ "h1 þkRi "h3 kRi þ "h1 kRi "h3 þkRi


t ¼ ½22 qrq0 r0 sin ’q sin ’r
=z;

þ "h3 ½"k ð"h3 þkRi þ "h3 kRi Þ

q, q0 , r and r0 are the magnitudes of h, h0 , r and r0 , respectively; ’q and ’r are the angles ðh; h0 Þ and ðr; r0 Þ, respectively. (c) Randomly positioned but correctly oriented atomic groups Then

þ "h1 þkRi "h2 kRi þ "h1 kRi "h2 þkRi
; P0 m " ¼ jEj2  1; ð"h1 "h2 "h3 þ i¼1 Bk; i Þ is assumed to be zero if it is experimentally negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplications in the contributions. G may be positive or negative. In particular, if G < 0 the triplet is estimated negative. The accuracy with which the value of  is estimated strongly depends on "k . Thus, in practice, only a subset of reciprocal space (the reflections k with large values of ") may be used for estimating . (2.2.5.13) proved to be quite useful in practice. Positive triplet cosines are ranked in order of reliability by (2.2.5.13) markedly better than by Cochran’s parameters. Negative estimated triplet cosines may be excluded from the phasing process and may be used as a figure of merit for finding the correct solution in a multisolution procedure.

gi ðh1 ; h2 ; h3 Þ ¼

exp½2QR1 R2 R3 cosð  qÞ
; 2I0 ð2QR1 R2 R3 Þ

 exp½2iðh1  Rs rkj þ h2  Rs rlk Þ
; where the summations over j, k, l are taken over all the atoms in the ith group. A modified expression for gi has to be used in polar space groups for special triplets (Giacovazzo, 1988). Translation functions [see Chapter 2.3; for an overview, see also Beurskens et al. (1987)] are also used to determine the position of a correctly oriented molecular fragment. Such functions can work in direct space [expressed as Patterson convolutions (Buerger, 1959; Nordman, 1985) or electron-density convolutions (Rossmann et al., 1964; Argos & Rossmann, 1980)] or in reciprocal space [expressed as correlation functions (Crowther & Blow, 1967; Karle, 1972; Langs, 1985) or residual functions (Rae, 1977)]. Both the probabilistic methods and the translation functions are quite efficient tools: the decision as to which one to use is often a personal choice. (d) Atomic groups correctly positioned Let p be the number of atoms with known position, q the number of atoms with unknown position, Fp and Fq the corresponding structure factors. Tangent recycling methods (Karle, 1970b) may be used for recovering the complete crystal structure. The phase ’p; h is accepted in the starting set as a useful approximation of ’h if jFp; h j > jFh j, where is the fraction of the total scattering power contained in the fragment and where jFh j is associated with jEh j > 1:5. Tangent recycling methods are applied (Beurskens et al., 1979) with greater effectiveness to difference s.f.’s F ¼ ðjFj  jFp jÞ expði’p Þ. The weighted tangent formula uses Fh values in order to convert them to more probable Fq; h values. From a probabilistic point of view (Giacovazzo, 1983a; Camalli et al., 1985) the distribution of ’h, given E0p; h and some products ðE0k  E0p; k ÞðE0hk  E0p; hk Þ, is the von Mises function

ð2:2:5:14Þ

where Pp

Q expðiqÞ ¼

i¼1 gi ðh1 ; h2 ; h3 Þ 2 1=2 hjFh1 j i hjFh2 j2 i1=2 hjFh3 j2 i1=2

fj fk fl

s¼1 j; k; l

2.2.5.4. Triplet relationships using structural information A strength of direct methods is that no knowledge of structure is required for their application. However, when some a priori information is available, it should certainly be a weakness of the methods not to make use of this knowledge. The conditional distribution of  given Rh Rk Rhk and the first three of the five kinds of a priori information described in Section 2.2.4.1 is (Main, 1976; Heinermann, 1977a) PðÞ ’

m P P

:

h1 ; h2 ; h3 stand for h, k, h þ k, and R1 ; R2 ; R3 for Rh ; Rk ; Rhk . The quantities hjFhi j2 i have been calculated in Section 2.2.4.1 according to different categories: gi ðh1 ; h2 ; h3 Þ is a suitable average of the product of three scattering factors for the ith atomic group, p is the number of atomic groups in the cell including those related by symmetry elements. We have the following categories. (a) No structural information (2.2.5.14) then reduces to (2.2.5.6). (b) Randomly positioned and randomly oriented atomic groups Then P gi ðh1 ; h2 ; h3 Þ ¼ fj fk fl hexp½2iðh1  rkj þ h2  rlj Þ
iR ;

Pð’h j . . .Þ ¼ ½2I0 ðÞ
1 exp½ cosð’h  h Þ
;

ð2:2:5:15Þ

where h, the most probable value of ’h, is given by tan h ’ 02 =01 ; 2

02

ð2:2:5:16Þ 02

 ¼ 1 þ 2

j; k; l

and

224

2.2. DIRECT METHODS

Fig. 2.2.5.3. Distributions (2.2.5.18) (solid curve) and (2.2.5.20) (dashed curve) for the indicated jEj values in three typical cases.



P 01 ¼ 2R0h R E0p; h þ q1=2 k ðE0k  E0p; k Þ io  ðE0hk  E0p; hk Þ

case in which deviations both of replacive and of displacive type from ideal pseudo-translational symmetry occur. 2.2.5.5. Quartet phase relationships



P 02 ¼ 2R0h I E0p; h þ q1=2 k ðE0k  E0p; k Þ io  ðE0hk  E0p; hk Þ :

In early papers (Hauptman & Karle, 1953; Simerska, 1956) the phase  ¼ ’h þ ’k þ ’l  ’hþkþl

R and I stand for ‘real P and imaginary part of’, respectively.

was always expected to be zero. Schenk (1973a,b) [see also Hauptman (1974)] suggested that  primarily depends on the seven magnitudes: Rh ; Rk ; Rl ; Rhþkþl , called basis magnitudes, and Rhþk ; Rhþl ; Rkþl , called cross magnitudes. The conditional probability of  in P1 given seven magnitudes ðR1 ¼ Rh ; . . . ; R4 ¼ Rhþkþl ; R5 ¼ Rhþk ; R6 ¼ Rhþl ; R7 ¼ Rkþl Þ according to Hauptman (1975) is

1=2

Furthermore, E0 ¼ F= q is a pseudo-normalized s.f. If no pair ð’k ; ’hk Þ is known, then 01 ¼ 2R0h R0p; h cos ’p; h 02 ¼ 2R0h R0p; h sin ’p; h

and (2.2.5.15) reduces to Sim’s (1959) equation Pð’h Þ ’ ½2I0 ðGÞ
1 exp½G cosð’h  ’p; h Þ
;

P7 ðÞ ¼

ð2:2:5:17Þ

where G ¼ 2R0h R0p; h. In this case ’p; h is the most probable value of ’h. (e) Pseudotranslational symmetry is present Substructure and superstructure reflections are then described by different forms of the structure-factor equation (Bo¨hme, 1982; Gramlich, 1984; Fan et al., 1983), so that probabilistic formulae estimating triplet cosines derived on the assumption that atoms are uniformly dispersed in the unit cell cannot hold. In particular, the reliability of each triplet also depends on, besides Rh ; Rk ; Rhk , the actual h, k, h  k indices and on the nature of the pseudotranslation. It has been shown (Cascarano et al., 1985b; Cascarano, Giacovazzo & Luic´, 1987) that (2.2.5.7), (2.2.5.8), (2.2.5.9) still hold provided Gh; kj is replaced by G0h; kj

1 expð2B cos ÞI0 ð2
3
23=2 R5 Y5 Þ L  I0 ð2
3
23=2 R6 Y6 ÞI0 ð2
3
23=2 R7 Y7 Þ;

where L is a suitable normalizing constant which can be derived numerically, B ¼
23 ð3
32 
2
4 ÞR1 R2 R3 R4 Y5 ¼ ½R21 R22 þ R23 R24 þ 2R1 R2 R3 R4 cos 
1=2 Y6 ¼ ½R23 R21 þ R22 R24 þ 2R1 R2 R3 R4 cos 
1=2 Y7 ¼ ½R22 R23 þ R21 R24 þ 2R1 R2 R3 R4 cos 
1=2 : For equal atoms
23 ð3
32 
2
4 Þ ¼ 2=N. Denoting pffiffiffiffi Z5 ¼ 2Y5 = N ;

2Rh Rkj Rhkj ; ¼ pffiffiffiffiffiffiffiffiffi Nh; k

C ¼ R1 R2 R3 R4 =N; pffiffiffiffi pffiffiffiffi Z6 ¼ 2Y6 = N ; Z7 ¼ 2Y7 = N

gives where factors E and ni are defined according to Section 2.2.4.1, Nh;k ¼

ðh ½
2
p þ ½
2
q Þðk ½
2
p þ ½
2
q Þðhk ½
2
p þ ½
2
q Þ fð =mÞ½
3
p ðn21 n22 n23 . . .Þ þ ½
3
q g2

P7 ðÞ ¼ ;

and is the number of times for which

1 expð4C cos Þ L  I0 ðR5 Z5 ÞI0 ðR6 Z6 ÞI0 ðR7 Z7 Þ:

ð2:2:5:18Þ

Fig. 2.2.5.3 shows the distribution (2.2.5.18) for three typical cases. It is clear from the figure that the cosine estimated near  or in the middle range will be in poorer agreement with the true values than the cosine near 0 because of the relatively larger values of the variance. In principle, however, the formula is able to estimate negative or enantiomorph-sensitive quartet cosines from the seven magnitudes. In the cs. case (2.2.5.18) is replaced (Hauptman & Green, 1976) by

hRs  u1  0 ðmod 1Þ hRs  u2  0 ðmod 1Þ hRs  u3  0 ðmod 1Þ . . . kRs  u1  0 ðmod 1Þ kRs  u2  0 ðmod 1Þ kRs  u3  0 ðmod 1Þ . . . ðh  kÞRs  u1  0 ðmod 1Þ ðh  kÞRs  u2  0 ðmod 1Þ ðh  kÞRs  u3  0 ðmod 1Þ . . .

are simultaneously satisfied when s varies from 1 to m. The above formulae have been generalized (Cascarano et al., 1988b) to the

225

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.2.5.1. List of quartets symmetry equivalent to  ¼ 1 in the class mmm Quartets

Basis vectors

1

(1, 2, 3) ð1
; 2; 3Þ

2

ð1; 2; 3
Þ ð1
; 2; 3
Þ

3 4 5

ð1
; 2; 3Þ

6

ð1; 2; 3Þ ð1
; 2; 3Þ

7

10

ð1
; 2; 3
Þ ð1; 2; 3
Þ ð1
; 2; 3
Þ

11

ð1
; 2; 3Þ

8 9

P ’

Cross vectors ð1
; 5; 3
Þ ð1; 5; 3
Þ ð1
; 5; 3Þ (1, 5, 3) ð1
; 5; 3
Þ ð1
; 5
; 3
Þ ð1; 5
; 3
Þ ð1
; 5; 3Þ ð1
; 5
; 3Þ ð1; 5
; 3Þ ð1
; 5
; 3
Þ

1 expð 2CÞ coshðR5 Z5 Þ L  coshðR6 Z6 Þ coshðR7 Z7 Þ;

ð1
; 5
; 8Þ ð1
; 5
; 8Þ

ð1; 2
; 8
Þ ð1; 2
; 8
Þ

(0, 7, 0)

ð1
; 5
; 8Þ ð1
; 5
; 8Þ

ð1; 2
; 8
Þ ð1; 2
; 8
Þ

(0, 7, 0)

ð1; 5
; 8Þ ð1
; 5; 8Þ

ð1; 2
; 8
Þ ð1; 2
; 8
Þ

(0, 7, 0) ð2
; 7; 0Þ ð0; 3
; 0Þ

ð1
; 5; 8Þ ð1; 5
; 8Þ

ð1; 2
; 8
Þ ð1; 2
; 8
Þ

ð0; 3
; 0Þ ð2
; 7; 0Þ

ð0; 3
; 5Þ

ð1
; 5; 8Þ ð1
; 5; 8Þ

ð1; 2
; 8
Þ ð1; 2
; 8
Þ

ð0; 3
; 0Þ ð0; 3
; 0Þ

(0, 7, 5) ð2
; 7; 5Þ

(1, 5, 8)

ð1; 2
; 8
Þ

ð2
; 3
; 0Þ

(0, 7, 11)

ð0; 3
; 11Þ ð2
; 3
; 11Þ

(0, 7, 0)

ð0; 3
; 5Þ ð2
; 3
; 5Þ ð0; 3
; 11Þ (0, 7, 11) ð2
; 7; 11Þ

ð2
; 0; 5Þ (0, 0, 5) ð2
; 0; 11) (0, 0, 11) (0, 0, 5) ð2
; 0; 5Þ (0, 0, 5) (0, 0, 11) ð2
; 0; 11Þ (0, 0, 11) (0, 0, 5)

or PðjR1 ; . . . ; R4 Þ ’

ð2:2:5:19Þ

1 expð2C cos Þ; L000

where P is the probability that the sign of E1 E2 E3 E4 is positive or negative, and 1 Z5 ¼ 1=2 ðR1 R2  R3 R4 Þ; N 1 Z6 ¼ 1=2 ðR1 R3  R2 R4 Þ; N 1 Z7 ¼ 1=2 ðR1 R4  R2 R3 Þ: N

respectively. (b) in the same situations, we have for cs. cases 1 P ’ 0 expð CÞ coshðR5 Z5 Þ coshðR6 Z6 Þ; L

The normalized probability may be derived by Pþ =ðPþ þ P Þ. More simple probabilistic formulae were derived independently by Giacovazzo (1975, 1976):

or

P7 ðÞ ¼ ½2I0 ðGÞ
1 expðG cos Þ;

ð2:2:5:20Þ

2Cð1 þ "5 þ "6 þ "7 Þ 1 þ Q=ð2NÞ

ð2:2:5:21Þ

or P ’

P ¼

Q ¼ ð"1 "2 þ "3 "4 Þ"5 þ ð"1 "3 þ "2 "4 Þ"6 þ ð"1 "4 þ "2 "3 Þ"7 ð2:2:5:22Þ

G¼

and "i ¼ ðjEi j2  1Þ. Q is never allowed to be negative. According to (2.2.5.20) cos  is expected to be positive or negative according to whether ð"5 þ "6 þ "7 þ 1Þ is positive or negative: the larger is C, the more reliable is the phase indication. For N  150, (2.2.5.18) and (2.2.5.20) are practically equivalent in all cases. If N is small, (2.2.5.20) is in good agreement with (2.2.5.18) for quartets strongly defined as positive or negative, but in poor agreement for enantiomorph-sensitive quartets (see Fig. 2.2.5.3). In cs. cases the sign probability for E1 E2 E3 E4 is Pþ ¼ 12 þ 12 tanhðG=2Þ;

2Cð1 þ "5 Þ ; 1 þ Q=ð2NÞ

Q ¼ ð"1 "2 þ "3 "4 Þ"5 :

In space groups with symmetry higher than P1
more symmetryequivalent quartets can exist of the type ¼ ’hR þ ’kR þ ’lR þ ’ðhþkþlÞR ; where R ; R ; R ; R are rotation matrices of the space group. The set f g is called the first representation of . In this case  primarily depends on more than seven magnitudes. For example, let us consider in Pmmm the quartet  ¼ ’123 þ ’1
53
þ ’1
5
8 þ ’12
8
:

ð2:2:5:23Þ

Quartets symmetry equivalent to  and respective cross terms are given in Table 2.2.5.1. Experimental tests on the application of the representation concept to quartets have been made (Busetta et al., 1980). It was shown that quartets with more than three cross magnitudes are more accurately estimated than other quartets. Also, quartets with a cross reflection which is systematically absent were shown to be of significant importance in direct methods. In this context it is noted that systematically absent reflections are not usually included in the set of diffraction data. This custom, not exceptionable when only triplet relations are used, can give rise to a loss of information when quartets are used. In fact the usual programs of direct methods discard quartets as soon as one of the cross reflections is not measured, so that systematic absences are

where G is defined by (2.2.5.21). All three cross magnitudes are not always in the set of measured reflections. From marginal distributions the following formulae arise (Giacovazzo, 1977c; Heinermann, 1977b): (a) in the ncs. case, if R7, or R6 and R7 , or R5 and R6 and R7 , are not in the measurements, then (2.2.5.18) is replaced by 1 PðjR1 ; . . . ; R6 Þ ’ 0 expð2C cos ÞI0 ðR5 Z5 ÞI0 ðR6 Z6 Þ; L or PðjR1 ; . . . ; R5 Þ ’

1 expðCÞ ’ 0:5 þ 0:5 tanhðCÞ; L000

respectively. Equations (2.2.5.20) and (2.2.5.23) are easily modifiable when some cross magnitudes are not in the measurements. If Ri is not measured then (2.2.5.20) or (2.2.5.23) are still valid provided that in G it is assumed that "i ¼ 0. For example, if R7 and R6 are not in the data then (2.2.5.21) and (2.2.5.22) become

where G¼

1 coshðR5 Z5 Þ; L00

1 I ðR Z Þ; L00 0 5 5

226

2.2. DIRECT METHODS 15 P dealt with in the same manner as those reflections which are "i ; A ¼ outside the sphere of measurements. i¼6 B ¼ "6 "13 þ "6 "15 þ "6 "14 þ "7 "11 þ "7 "15 þ "7 "12 þ "8 "10 þ "8 "14 þ "8 "12 þ "10 "15 þ "10 "9 þ "11 "14 þ "11 "9 þ "13 "9 þ "13 "12 ;

2.2.5.6. Quintet phase relationships A quintet phase

D ¼ "1 "2 "6 þ "1 "3 "7 þ "1 "4 "8 þ "1 "5 "9 þ "1 "10 "15

 ¼ ’h þ ’k þ ’l þ ’m þ ’hþkþlþm

þ "1 "11 "14 þ "1 "13 "12 þ "2 "3 "10 þ "2 "4 "11 þ "2 "5 "12 þ "2 "7 "15 þ "2 "8 "14 þ "2 "13 "9 þ "3 "4 "13

may be considered as the sum of three suitable triplets or the sum of a triplet and a quartet, i.e.

þ "3 "5 "14 þ "3 "6 "15 þ "3 "8 "12 þ "3 "11 "9 þ "4 "5 "15 þ "4 "6 "14 þ "4 "7 "12 þ "4 "10 "9 þ "5 "6 "13 þ "5 "7 "11

 ¼ ð’h þ ’k  ’hþk Þ þ ð’l þ ’m  ’lþm Þ

þ "5 "8 "10 :

þ ð’hþk þ ’lþm þ ’hþkþlþm Þ or

For cs. cases (2.2.5.24) reduces to Pþ ’ 0:5 þ 0:5 tanhðG=2Þ:

 ¼ ð’h þ ’k  ’hþk Þ þ ð’l þ ’m þ ’hþkþlþm þ ’hþk Þ:

Positive or negative quintets may be identified according to whether G is larger or smaller than zero. If Ri is not measured then (2.2.5.24) and (2.2.5.25) are still valid provided that in (2.2.5.25) "i ¼ 0. If the symmetry is higher than in P1
then more symmetryequivalent quintets can exist of the type

It depends primarily on 15 magnitudes: the five basis magnitudes Rh ;

Rk ;

Rl ;

Rm ;

Rhþkþlþm ;

and the ten cross magnitudes Rhþk ; Rkþm ;

Rhþl ;

Rhþm ;

Rkþlþm ;

Rkþl ;

Rhþlþm ;

Rlþm ;

Rhþkþm ;

Rhþkþl :

¼ ’hR þ ’kR þ ’lR þ ’mR þ ’ðhþkþlþmÞR" ; where R ; . . . ; R" are rotation matrices of the space groups. The set f g is called the first representation of . In this case  primarily depends on more than 15 magnitudes which all have to be taken into account for a careful estimation of  (Giacovazzo, 1980a). A wide use of quintet invariants in direct-methods procedures is prevented for two reasons: (a) the large correlation of positive quintet cosines with positive triplets; (b) the large computing time necessary for p their ffiffiffiffi estimation [quintets are phase relationships of order 1=ðN N Þ, so a large number of quintets have to be estimated in order to pick up a sufficient percentage of reliable ones].

In the following we will denote R1 ¼ Rh ;

R2 ¼ Rk ; . . . ;

R15 ¼ Rhþkþl :

Conditional distributions of  in P1 and P1
given the 15 magnitudes have been derived by several authors and allow in favourable circumstances in ncs. space groups the quintets having  near 0 or near  or near =2 to be identified. Among others, we remember: (a) the semi-empirical expression for P15 ðÞ suggested by Van der Putten & Schenk (1977): " ! # 15 15 X Y 1 2 Pðj . . .Þ ’ exp 6  Rj 2C cos  I0 ð2Rj Yj Þ; L j¼6 j¼6

2.2.5.7. Determinantal formulae In a crystal structure with N identical atoms the joint probability distribution of n normalized s.f.’s Eh1 þk ; Eh2 þk ; . . . ; Ehn þk under the following conditions: (a) the structure is kept fixed whereas k is the primitive random variable; (b) Ehi hj ; i; j ¼ 1; . . . ; n, have values which are known a priori; is given (Tsoucaris, 1970) [see also Castellano et al. (1973) and Heinermann et al. (1979)] by

where C ¼ N 3=2 R1 R2 R3 R4 R5 and Yj is an expression related to the jth of the ten quartets connected with the quintet ; (b) the formula by Fortier & Hauptman (1977), valid in P1
, which is able to predict the sign of a quintet by means of an expression which involves a summation over 1024 sets of signs; (c) the expression by Giacovazzo (1977d), according to which P15 ðÞ ’ ½2I0 ðGÞ


1

expðG cos Þ;

PðE1 ; E2 ; . . . ; En Þ ¼ ð2Þn=2 D1=2 expð12Qn Þ n

G¼

2C 1þAþB 1=2 1 þ D=ð2NÞ 1 þ 6ðNÞ

ð2:2:5:27Þ

for cs. structures and PðE1 ; E2 ; . . . ; En Þ ¼ ð2Þn D1=2 expðQn Þ n

ð2:2:5:24Þ

ð2:2:5:28Þ

for ncs. structures. In (2.2.5.27) and (2.2.5.28) we have denoted

where

ð2:2:5:26Þ

Dn ¼ ;

Qn ¼

n P

pq Ep E q

p; q¼1

Ej ¼ Ehj þk ;

ð2:2:5:25Þ

Upq ¼ Uhp hq ;

j; p; q ¼ 1; . . . ; n:

pq is an element of k1, and k is the covariance matrix with elements

and where

227

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION hEhp þk Ehq þk i ¼ Uhp hq determinants with a magic-integer approach. The computing

time, however, was larger than that required by standard
1 U12 . . . U1q . . . U1n


computing techniques. The use of K–H matrices has been made

U 1 . . . U2q . . . U2n

21 faster and more effective by de Gelder et al. (1990) (see also de



. . . . . . Gelder, 1992). They developed a phasing procedure (CRUNCH) ..
.. .. .. ..
..

which uses random phases as starting points for the maximization k¼

: of the K–H determinants.
Up1 Up2 . . . Upq . . . Upn



. ..

.. .. .. ..
. 2.2.5.8. Algebraic relationships for structure seminvariants .
. . .
. .



U According to the representations method (Giacovazzo, 1977a, Un2 . . . Unq . . . 1 n1 1980a,b): (i) any s.s.  may be estimated via one or more s.i.’s f g, whose  is a K–H determinant: therefore Dn  0. Let us call values differ from  by a constant arising because of symmetry;


1 ... U1n Eh1 þk
U12 (ii) two types of s.s.’s exist, first-rank and second-rank s.s.’s, with


U21 1 ... U2n Eh2 þk
different algebraic properties:

1
..
; .. .. .. (iii) conditions characterizing s.s.’s of first rank for any space nþ1 ¼
... .

. . . N
group may be expressed in terms of seminvariant moduli and
U Un2 ... 1 Ehn þk

n1
seminvariantly associated vectors. For example, for all the space
E Eh2 k . . . Ehn k N
h1 k groups with point group 422 [Hauptman–Karle group ðh þ k; lÞ P(2, 2)] the one-phase s.s.’s of first rank are characterized by the K–H determinant obtained by adding to k the last column ðh; k; lÞ  0 mod ð2; 2; 0Þ or ð2; 0; 2Þ or ð0; 2; 2Þ and line formed by E1 ; E2 ; . . . ; En, and E 1 ; E 2 ; . . . ; E n , respectively. Then (2.2.5.27) and (2.2.5.28) may be written ðh  k; lÞ  0 mod ð0; 2Þ or ð2; 0Þ: PðE1 ; E2 ; . . . ; En Þ n=2

¼ ð2Þ

Dn1=2



nþ1  Dn exp N 2Dn

The more general expressions for the s.s.’s of first rank are (a)  ¼ ’u ¼ ’hðIR Þ for one-phase s.s.’s; (b)  ¼ ’u1 þ ’u2 ¼ ’h1 h2 R þ ’h2 h1 R for two-phase s.s.’s; (c)  ¼ ’u1 þ ’u2 þ ’u3 ¼ ’h1 h2 R þ ’h2 h3 R þ ’h3 h1 R for three-phase s.s.’s;

ð2:2:5:29Þ

and PðE1 ; E2 ; . . . ; En Þ



nþ1  Dn ; ¼ ð2Þn D1=2 exp N n Dn

ðdÞ  ¼ ’u1 þ ’u2 þ ’u3 þ ’u4 ¼ ’h1 h2 R þ ’h2 h3 R þ ’h3 h4 R þ ’h4 h1 R

ð2:2:5:30Þ

for four-phase s.s.’s; etc. In other words: (a) ’u is an s.s. of first rank if at least one h and at least one rotation matrix R exist such that u ¼ hðI  R Þ. ’u may be estimated via the special triplet invariants

respectively. Because Dn is a constant, the maximum values of the conditional joint probabilities (2.2.5.29) and (2.2.5.30) are obtained when nþ1 is a maximum. Thus the maximum determinant rule may be stated (Tsoucaris, 1970; Lajze´rowicz & Lajze´rowicz, 1966): among all sets of phases which are compatible with the inequality

f g ¼ ’u  ’h þ ’hR :

nþ1 ðE1 ; E2 ; . . . ; En Þ  0

ð2:2:5:33Þ

The set f g is called the first representation of ’u. (b)  ¼ ’u1 þ ’u2 is an s.s. of first rank if at least two vectors h1 and h2 and two rotation matrices R and R exist such that  u1 ¼ h1  h2 R ð2:2:5:34Þ u2 ¼ h2  h1 R :

the most probable one is that which leads to a maximum value of nþ1 . If only one phase, i.e. ’q , is unknown whereas all other phases and moduli are known then (de Rango et al., 1974; Podjarny et al., 1976) for cs. crystals 8 9 < = n P ð2:2:5:31Þ P ðEq Þ ’ 0:5 þ 0:5 tanh jEq j pq Ep ; : ; p¼1

 may then be estimated via the special quartet invariants f g ¼ ’u1 R þ ’u2  ’h2 þ ’h2 R R

ð2:2:5:35aÞ

f g ¼ f’u1 þ ’u2 R  ’h1 þ ’h1 R R g:

ð2:2:5:35bÞ

p6¼q

and and for ncs. crystals Pð’q Þ ¼ ½2I0 ðGq Þ


1

expfGq cosð’q  q Þg;

ð2:2:5:32Þ For example,  ¼ ’123 þ ’7
2
5
in P21 may be estimated via

where Gq expðiq Þ ¼ 2jEq j

n P

f g ¼ ’123 þ ’7
2
5
 ’3
K1
þ ’3K1 pq Ep :

and

p6¼q¼1

f g ¼ ’123 þ ’72
5  ’4K4 þ ’4
K4
; Equations (2.2.5.31) and (2.2.5.32) generalize (2.2.5.11) and (2.2.5.7), respectively, and reduce to them for n ¼ 3. Fourth-order determinantal formulae estimating triplet invariants in cs. and ncs. crystals, and making use of the entire data set, have recently been secured (Karle, 1979, 1980a). Advantages, limitations and applications of determinantal formulae can be found in the literature (Heinermann et al., 1979; de Rango et al., 1975, 1985). Taylor et al. (1978) combined K–H

where K is a free index. The set of special quartets (2.2.5.35a) and (2.2.5.35b) constitutes the first representations of . Structure seminvariants of the second rank can be characterized as follows: suppose that, for a given seminvariant , it is not possible to find a vectorial index h and a rotation matrix R such that   ’h þ ’hR is a structure invariant. Then  is a structure

228

2.2. DIRECT METHODS The second representation of ’H is the set of special quintets

seminvariant of the second rank and a set of structure invariants can certainly be formed, of type

f g ¼ f’H  ’h þ ’hRn þ ’kRj  ’kRj g

f g ¼  þ ’hRp  ’hRq þ ’lRi  ’lRj ;

provided that h and Rn vary over the vectors and matrices for which (2.2.5.36) is compatible, k over the asymmetric region of the reciprocal space, and Rj over the rotation matrices in the space group. Formulae estimating ’H via the second representation in all the space groups [all the base and cross magnitudes of the quintets (2.2.5.40) now constitute the a priori information] have been secured (Giacovazzo, 1978; Cascarano & Giacovazzo, 1983; Cascarano, Giacovazzo, Calabrese et al., 1984). Such formulae contain, besides the contribution of order N 1=2 provided by the first representation, a supplementary (not negligible) contribution of order N 3=2 arising from quintets. Denoting

by means of suitable indices h and l and rotation matrices Rp ; Rq ; Ri and Rj . As an example, for symmetry class 222, ’240 or ’024 or ’204 are s.s.’s of the first rank while ’246 is an s.s. of the second rank. The procedure may easily be generalized to s.s.’s of any order of the first and of the second rank. So far only the role of onephase and two-phase s.s.’s of the first rank in direct procedures is well documented (see references quoted in Sections 2.2.5.9 and 2.2.5.10). 2.2.5.9. Formulae estimating one-phase structure seminvariants of the first rank

E1 ¼ EH ; E2 ¼ Eh ; E3 ¼ Ek ; E4; j ¼ EhþkRj ; E5; j ¼ EHþkRj ;

Let EH be our one-phase s.s. of the first rank, where H ¼ hðI  Rn Þ:

ð2:2:5:36Þ

formulae (2.2.5.37), (2.2.5.38), (2.2.5.39) still hold provided that P h; n Gh; n is replaced by X 0 jE j Ah; k; n X H Gh; n þ ; 3=2 1 þ B 2N h; k; n h; n h; k; n

In general, more than one rotation matrix Rn and more than one vector h are compatible with (2.2.5.36). The set of special triplets f g ¼ f’H  ’h þ ’hRn g is the first representation of EH. In cs. space groups the probability that EH > 0, given jEH j and the set fjEh jg, may be estimated (Hauptman & Karle, 1953; Naya et al., 1964; Cochran & Woolfson, 1955) by P Pþ ðEH Þ ’ 0:5 þ 0:5 tanh Gh; n ð1Þ2hTn ; ð2:2:5:37Þ

where 2

0

Ri ¼Rj Rj þRi Rn ¼0

m m X X X 6 "5; j þ "1 "4; i "4; j þ "2 "3 "4; j Bh; k; n ¼ 4"1 "3

þ "2

m is the number of symmetry operators and H4 ðEÞ ¼ E4  6E2 þ 3 is the Hermite polynomial of order four. Bh; k; n is assumed to be zero if it is computed negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplication in the contributions.

If ’H is a general phase then ’H is distributed according to

tan H ¼ 

1 expf cosð’H  H Þg; L

2.2.5.10. Formulae estimating two-phase structure seminvariants of the first rank



Two-phase s.s.’s of the first rank were first evaluated in some cs. space groups by the method of coincidence by Grant et al. (1957); the idea was extended to ncs. space groups by Debaerdemaeker & Woolfson (1972), and in a more general way by Giacovazzo (1977e, f). The technique was based on the combination of the two triplets

Gh; n sin 2h  Tn

h; n

P



ð2:2:5:39Þ

Gh; n cos 2h  Tn

h; n

with a reliability measured by ( 2 P ¼ Gh; n sin 2h  Tn

’h1 þ ’h2 ’ ’h1 þh2 ’h1 þ ’h2 R ’ ’h1 þh2 R ;

h; n

þ

 P

2 )1=2 Gh; n cos 2h  Tn

j¼1

3 , X 7 1 "4; i "5; j þ 4"1 H4 ðE2 Þ5 ð2NÞ:

Rj ¼Ri Rj þRi Rn ¼0

ð2:2:5:38Þ

P

Rj ¼Ri Rn Ri ¼Rj Rn

j¼1

h; n



Rj þRi Rn ¼0

2

In (2.2.5.37), the summation over n goes within the set of matrices Rn for which (2.2.5.35a,b) is compatible, and h varies within the set of vectors which satisfy (2.2.5.36) for each Rn . Equation P (2.2.5.36) is actually a generalized way of writing the so-called 1 relationships (Hauptman & Karle, 1953). If ’H is a phase restricted by symmetry to H and H þ  in an ncs. space group then (Giacovazzo, 1978) ( ) X Pð’H ¼ H Þ ’ 0:5 þ 0:5 tanh Gh; n cosðH  2h  Tn Þ :

where

Rj ¼Ri Rn Ri ¼Rj Rn

3 , m X X "3 7  " 1 " " 5 N; 2 j¼1 4; j 2 Rj ¼Ri 4; i 5; j

pffiffiffiffi Gh; n ¼ jEH j"h =ð2 N Þ; and " ¼ jEj2  1:

Pð’H Þ ’

1

X B X C 6 "4; i "5; j þ "4; i "4; j A Ah; k; n ¼ 4ð2jE2 j2  1Þ"3 @

h; n

where

ð2:2:5:40Þ

which, subtracted from one another, give

:

’h1 þh2 R  ’h1 þh2 ’ ’h2 R  ’h2 ’ 2h  T:

h; n

229

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION " #n N N If all four jEj’s are sufficiently large, an estimate of the two-phase P P n j ðr  rj Þ ’ nj ðr  rj Þ ðrÞ ¼ seminvariant ’h1 þh2 R  ’h1 þh2 is available. j¼1 j¼1 Probability distributions valid in P21 according to the neighbourhood principle have been given by Hauptman & Green and its Fourier transform gives (1978). Finally, the theory of representations was combined by R n Giacovazzo (1979a) with the joint probability distribution ðrÞ expð2ih  rÞ dV n Fh ¼ method in order to estimate two-phase s.s.’s in all the space V groups. N P ¼ n f j expð2ih  rj Þ: ð2:2:6:2Þ According to representation theory, the problem is that of j¼1 evaluating  ¼ ’u1 þ ’u2 via the special quartets (2.2.5.35a) and (2.2.5.35b). Thus, contributions of order N 1 will appear in the n n fj is the scattering factor for the jth peak of ðrÞ: probabilistic formulae, which will be functions of the basis and of R n the cross magnitudes of the quartets (2.2.5.35) . Since more pairs j ðrÞ expð2ih  rÞ dr: n fj ðhÞ ¼ of matrices R and R can be compatible with (2.2.5.34), and for V each pair ðR ; R Þ more pairs of vectors h1 and h2 may satisfy We now introduce the condition that all atoms are equal, so (2.2.5.34), several quartets can in general be exploited for estithat fj  f and n fj  n f for any j. From (2.2.6.1) and (2.2.6.2) we mating . The simplest case occurs in P1
where the two quartets may write (2.2.5.35) suggest the calculation of the six-variate distribution function ðu1 ¼ h1 þ h2 ; u2 ¼ h1  h2 Þ f F h ¼ n F h ¼ n n F h ; ð2:2:6:3Þ nf PðE ; E ; E ;E ;E ;E Þ h1

h2

h1 þh2

h1 h2

2h1

2h2

where n is a function which corrects for the difference of shape of the atoms with electron distributions ðrÞ and n ðrÞ. Since

which leads to the probability formula   jEh1 þh2 Eh1 h2 j A þ  P ’ 0:5 þ 0:5 tanh ; 1þB 2N

n ðrÞ ¼ ðrÞ . . . ðrÞ þ1 1 X ¼ n F . . . Fhn exp½2iðh1 þ . . . þ hn Þ  r
; V h1 ; ...; hn h1

where Pþ is the probability that the product Eh1 þh2 Eh1 h2 is positive, and

1

the Fourier transform of both sides gives Z þ1 1 X F . . . Fhn exp½2iðh  h1  . . .  hn Þ  r
dV n Fh ¼ V n h1 ; ...; hn h1

A ¼ "h1 þ "h2 þ 2"h1 "h2 þ "h1 "2h1 þ "h2 "2h2 B ¼ ð"h1 "h2 "u1 þ "h1 "h2 "u2

¼ It may be seen that in favourable cases Pþ < 0:5. For the sake of brevity, the probabilistic formulae for the general case are not given and the reader is referred to the original papers.

1

from which the following relation arises:

2.2.6. Direct methods in real and reciprocal space: Sayre’s equation The statistical treatment suggested by Wilson for scaling observed intensities corresponds, in direct space, to the origin peak of the Patterson function, so it is not surprising that a general correspondence exists between probabilistic formulation in reciprocal space and algebraic properties in direct space. For a structure containing atoms which are fully resolved from one another, the operation of raising ðrÞ to the nth power retains the condition of resolved atoms but changes the shape of each atom. Let N P

N P

j ðr  rj Þ;

Fh1 Fh2 . . . Fhh1 h2 ...hn1 :

ð2:2:6:4Þ

where As and Bs are adjustable parameters of ðsin Þ=. Equation (2.2.6.6) can easily be generalized to the case of structures containing resolved atoms of more than two types (von Eller, 1973). Besides the algebraic properties of the electron density, Patterson methods also can be developed so that they provide phase indications. For example, it is possible to find the reciprocal counterpart of the function R Pn ðu1 ; u2 ; . . . ; un Þ ¼ ðrÞ ðr þ u1 Þ . . . ðr þ un Þ dV: ð2:2:6:7Þ

j ðr  rj Þ expð2ih  rÞ dV

fj expð2ih  rj Þ:

V n1 h1 ; ...; hn1

If the structure contains resolved isotropic atoms of two types, P and Q, it is impossible to find a factor 2 such that the relation Fh ¼ 2 2 Fh holds, since this would imply values of 2 such that ð2 f ÞP ¼ 2 ð f ÞP and ð2 f ÞQ ¼ 2 ð f ÞQ simultaneously. However, the following relationship can be stated (Woolfson, 1958): A X B X Fk Fhk þ 2s F FF ; ð2:2:6:6Þ Fh ¼ s V k V k; l k l hkl

j¼1 V

¼

þ1 X

For n ¼ 2, equation (2.2.6.4) reduces to Sayre’s (1952) equation [but see also Hughes (1953)] 1X F h ¼ 2 FF : ð2:2:6:5Þ V k k hk

where j ðrÞ is an atomic function and rj is the coordinate of the ‘centre’ of the atom. Then the Fourier transform of the electron density can be written as N R P

1

1

j¼1

Fh ¼

þ1 X

1 F F . . . Fhh1 h2 ...hn1 ; V n1 h1 ; ...; hn1 h1 h2

F h ¼ n

ðrÞ ¼

V

1

þ "u1 "u2 "2h1 þ "u1 "u2 "2h2 Þ=ð2NÞ:

ð2:2:6:1Þ

j¼1

V

If the atoms do not overlap

230

2.2. DIRECT METHODS For n ¼ 1 the function (2.2.6.7) coincides with the usual Patterson function PðuÞ; for n ¼ 2, (2.2.6.7) reduces to the double Patterson function P2 ðu1 ; u2 Þ introduced by Sayre (1953). Expansion of P2 ðu1 ; u2 Þ as a Fourier series yields 1 X P2 ðu1 ; u2 Þ ¼ 2 E E E exp½2iðh1  u1 þ h2  u2 Þ
: V h ; h h1 h2 h3 1

Stage 4: Definition of the origin and enantiomorph. This stage is carried out according to the theory developed in Section 2.2.3. Phases chosen for defining the origin and enantiomorph, onephase seminvariants estimated at stage 2, and symbolic phases described at stage 5 are the only phases known at the beginning of the phasing procedure. This set of phases is conventionally referred to as the starting set, from which iterative application of the tangent formula will derive new phase estimates. Stage 5: Assignment of one or more (symbolic or numerical) phases. In complex structures the number of phases assigned for fixing the origin and the enantiomorph may be inadequate as a basis for further phase determination. Furthermore, only a few one-phase s.s.’s can be determined with sufficient reliability to make them qualify as members of the starting set. Symbolic phases may then be associated with some (generally from 1 to 6) high-modulus reflections (symbolic addition procedures). Iterative application of triplet relations leads to the determination of other phases which, in part, will remain expressed by symbols (Karle & Karle, 1966). In other procedures (multisolution procedures) each symbol is assigned four phase values in turn: =4; 3=4; 5=4; 7=4. If p symbols are used, in at least one of the possible 4p solutions each symbolic phase has unit probability of being within 45 of its true value, with a mean error of 22:5 . To find a good starting set a convergence method (Germain et al., 1970) is used according to which: (a) P hh i ¼ Gj I1 ðGj Þ=I0 ðGj Þ

2

ð2:2:6:8Þ Vice versa, the value of a triplet invariant may be considered as the Fourier transform of the double Patterson. Among the main results relating direct- and reciprocal-space properties it may be remembered: (a) from the properties of P2 ðu1 ; u2 Þ the following relationship may be obtained (Vaughan, 1958) Eh1 Eh2 Eh1 þh2  N 3=2 ’ A1 hðjEk j2  1ÞðjEh1 þk j2  1ÞðjEh2 þk j2  1Þik  B1 ; which is clearly related to (2.2.5.12); (b) the zero points in the Patterson function provide information about the value of a triplet invariant (Anzenhofer & Hoppe, 1962; Allegra, 1979); (c) the Hoppe sections (Hoppe, 1963) of the double Patterson provide useful information for determining the triplet signs (Krabbendam & Kroon, 1971; Simonov & Weissberg, 1970); (d) one phase s.s.’s of the first rank can be estimated via the Fourier transform of single Harker sections of the Patterson (Ardito et al., 1985), i.e. Z 1 PðuÞ expð2ih  uÞ du; ð2:2:6:9Þ FH  expð2ih  Tn Þ L

j

is calculated for all reflections ( j runs over the set of triplets containing h); (b) the reflection is found with smallest hi not already in the starting set; it is retained to define the origin if the origin cannot be defined without it; (c) the reflection is eliminated if it is not used for origin definition. Its hi is recorded and hi values for other reflections are updated; (d) the cycle is repeated from (b) until all reflections are eliminated; (e) the reflections with the smallest hi at the time of elimination go into the starting set; ( f ) the cycle from (a) is repeated until all reflections have been chosen. Stage 6: Application of tangent formula. Phases are determined in reverse order of elimination in the convergence procedure. In order to ensure that poorly determined phases ’kj and ’hkj have little effect in the determination of other phases a weighted tangent formula is normally used (Germain et al., 1971): P j wk whkj jEkj Ehkj j sinð’kj þ ’hkj Þ ; ð2:2:7:1Þ tan ’h ¼ P j j wkj whkj jEkj Ehkj j cosð’kj þ ’hkj Þ

HSðI; Cn Þ

where (see Section 2.2.5.9) H ¼ hðI  Rn Þ is the s.s., u varies over the complete Harker section corresponding to the operator Cn [in symbols HSðI; Cn Þ] and L is a constant which takes into account the dimensionality of the Harker section. If no spurious peak is on the Harker section, then (2.2.6.9) is an exact relationship. Owing to the finiteness of experimental data and to the presence of spurious peaks, (2.2.6.9) cannot be considered in practice an exact relation: it works better when heavy atoms are in the chemical formula. More recently (Cascarano, Giacovazzo, Luic´ et al., 1987), a special least-squares procedure has been proposed for discriminating spurious peaks among those lying on Harker sections and for improving positional and thermal parameters of heavy atoms. (e) translation and rotation functions (see Chapter 2.3), when defined in direct space, always have their counterpart in reciprocal space.

where wh ¼ min ð0:2; 1Þ:

2.2.7. Scheme of procedure for phase determination: the smallmolecule case A traditional procedure for phase assignment may be schematically presented as follows: Stage 1: Normalization of s.f.’s. See Section 2.2.4. Stage 2: (Possible) estimation of one-phase s.s.’s. The computing program recognizes the one-phase s.s.’s and applies the proper formulae (see Section 2.2.5.9). Each phase is associated with a reliability value, to allow the user to regard as known only those phases with reliability higher than a given threshold. Stage 3: Search of the triplets. The reflections are listed for decreasing jEj values and, related to each jEj Pvalue, all possible triplets are reported pffiffiffiffi(this is the so-called 2 list). The value G ¼ 2jEh Ek Ehk j= N is associated with every triplet for an evaluation of its efficiency. Usually reflections with jEj < Es (Es may range from 1.2 to 1.6) are omitted from this stage onward.

Once a large number of contributions are available in (2.2.7.1) for a given ’h , then the value of h quickly becomes greater than 5, and so assigns an unrealistic unitary weight to ’h. In this respect a different weighting scheme may be proposed (Hull & Irwin, 1978) according to which w¼

Rx expðx2 Þ expðt2 Þ dt;

ð2:2:7:2Þ

0

where x ¼ =hi and ¼ 1:8585 is a constant chosen so that w ¼ 1 when x ¼ 1. Except for , the right-hand side of (2.2.7.2) is the Dawson integral which assumes its maximum value at x ¼ 1 (see Fig. 2.2.7.1): when  > hi or  < hi then w < 1 and so the agreement between  and hi is promoted. Alternative weighting schemes for the tangent formula are frequently used [for example, see Debaerdemaeker et al. (1985)]. In one (Giacovazzo, 1979b), the values kj and hkj (which are

231

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION P ðdÞ NQEST ¼ Gj cos j ; j

where G is defined by (2.2.5.21) and  ¼ ’h  ’k  ’l  ’hkl are quartet invariants characterized by large basis magnitudes and small cross magnitudes (De Titta et al., 1975; Giacovazzo, 1976). Since G is expected to be negative as well as cos , the value of NQEST is expected to be positive and a maximum for the correct solution. Figures of merit are then combined as ABSFOM  ABSFOMmin CFOM ¼ w1 ABSFOMmax  ABSFOMmin PSI0max  PSI0 þ w2 PSI0max  PSI0min Rmax  R þ w3 Rmax  Rmin NQEST  NQESTmin þ w4 ; NQESTmax  NQESTmin

Fig. 2.2.7.1. The form of w as given by (2.2.7.2).

usually available in direct procedures) are considered as additional a priori information so that (2.2.7.1) may be replaced by P j j sinð’kj þ ’hkj Þ P tan ’h ’ ; ð2:2:7:3Þ j j cosð’kj þ ’hkj Þ where j is the solution of the equation D1 ð j Þ ¼ D1 ðGj ÞD1 ðkj ÞD1 ðhkj Þ:

where wi are empirical weights proportional to the confidence of the user in the various FOMs. Different FOMs are often used by some authors in combination with those described above: for example, enantiomorph triplets and quartets are supplementary FOMs (Van der Putten & Schenk, 1977; Cascarano, Giacovazzo & Viterbo, 1987). Different schemes of calculating and combining FOMs are also used: one scheme (Cascarano, Giacovazzo & Viterbo, 1987) uses

ð2:2:7:4Þ

In (2.2.7.4), pffiffiffiffi Gj ¼ 2jEh Ekj Ehkj j N or the corresponding second representation parameter, and D1 ðxÞ ¼ I1 ðxÞ=I0 ðxÞ is the ratio of two modified Bessel functions. In order to promote (in accordance with the aims of Hull and Irwin) the agreement between  and hi, the distribution of  may be used (Cascarano, Giacovazzo, Burla et al., 1984; Burla et al., 1987); in particular, the first two moments of the distribution: accordingly, 

1=3 ð  hiÞ2 w ¼ exp 2
2

P ða1Þ

ABSFOM ¼

P h

h =

P

hh i;

h

which is expected to be unity for the correct solution.

ðbÞ


P

P
h k Ek Ehk PSI0 ¼ P P :  2 1=2 h k jEk Ehk j

The summation over k includes (Cochran & Douglas, 1957) the strong jEj’s for which phases have been determined, and indices h correspond to very small jEh j. Minimal values of PSI0 ( 1.20) are expected to be associated with the correct solution. P ðcÞ

R ¼

wj Gj cosðj  j Þ þ wj Gj cos j P ; s:i:þs:s: wj Gj D1 ðGj Þ

where the first summation in the numerator extends over symmetry-restricted one-phase and two-phase s.s.’s (see Sections 2.2.5.9 and 2.2.5.10), and the second summation in the numerator extends over negative triplets estimated via the second representation formula [equation (2.2.5.13)] and over negative quartets. The value of CPHASE is expected to be close to unity for the correct solution. (a2) h for strong triplets and Ek Ehk contributions for PSI0 triplets may be considered random variables: the agreements between their actual and their expected distributions are considered as criteria for identifying the correct solution. (a3) correlation among some FOMs is taken into account. According to this scheme, each FOM (as well as the CFOM) is expected to be unity for the correct solution. Thus one or more figures are available which constitute a sort of criterion (on an absolute scale) concerning the correctness of the various solutions: FOMs (and CFOM) ’ 1 probably denote correct solutions, CFOMs 1 should indicate incorrect solutions. Stage 8: Interpretation of E maps. This is carried out in up to four stages (Koch, 1974; Main & Hull, 1978; Declercq et al., 1973): (a) peak search; (b) separation of peaks into potentially bonded clusters; (c) application of stereochemical criteria to identify possible molecular fragments; (d) comparison of the fragments with the expected molecular structure.

may be used, where
2 is the estimated variance of . Stage 7: Figures of merit. The correct solution is found among several by means of figures of merit (FOMs) which are expected to be extreme for the correct solution. Largely used are (Germain et al., 1970) ðaÞ

CPHASE ¼

h jh

P

 hh ij : h h hi 2.2.8. Other multisolution methods applied to small molecules In very complex structures a large initial set of known phases seems to be a basic requirement for a structure to be determined.

That is, the Karle & Karle (1966) residual between the actual and the estimated ’s. After scaling of h on hh i the correct solution should be characterized by the smallest R values.

232

2.2. DIRECT METHODS m1 x þ m2 y þ m3 z þ b  0 ðmod 1Þ;

Table 2.2.8.1. Magic-integer sequences for small numbers of phases (n) together with the number of sets produced and the root-mean-square error in the phases

n

No. of sets

Sequence

1

1

2

2

3

3

3

4

4

5

7

8

9

5

8

11

13

14

15

6 7

13 21

18 29

21 34

23 37

24 39

25 40

41

8

34

47

55

60

63

65

66

5

67

where b is a phase constant which arises from symmetry translation. It may be expected that the ‘best’ value of the unknown x, y, z corresponds to a maximum of the function P ðx; y; zÞ ¼ jE1 E2 E3 j cos 2ðm1 x þ m2 y þ m3 z þ bÞ;

R.m.s. error ( )

4

26

12

29

20

37

32

42

50

45

80 128

47 48

206

49

with 0  x; y; z < 1. It should be noticed that is a Fourier summation which can easily be evaluated. In fact, is essentially a figure of merit for a large number of phases evaluated in terms of a small number of magic-integer Pvariables and gives a measure of the internal consistency of map 2 relationships. The generally presents several peaks and therefore can provide several solutions for the variables. (2) The random-start method These are procedures which try to solve crystal structures by starting from random initial phases (Baggio et al., 1978; Yao, 1981). They may be so described: (a) A number of reflections (say NUM  100 or larger) at the bottom of the CONVERGE map are selected. These, and the relationships which link them, form the system for which trial phases will be found. (b) A pseudo-random number generator is used to generate M sets of NUM random phases. Each of the M sets is refined and extended by the tangent formula or similar methods. (3) Accurate calculation of s.i.’s and s.s.’s with 1, 2, 3, 4, . . . , n phases Having a large set of good phase relationships allows one to overcome difficulties in the early stages and in the refinement process of the phasing procedure. Accurate estimates of s.i.’s and s.s.’s may be achieved by the application of techniques such as the representation method or the neighbourhood principle (Hauptman, 1975; Giacovazzo, 1977a, 1980b). So far, secondrepresentation formulae are available for triplets and one-phase seminvariants; in particular, reliably estimated negative triplets can be recognized, which is of great help in the phasing process (Cascarano, Giacovazzo, Camalli et al., 1984). Estimation of higher-order s.s.’s with upper representations or upper neighbourhoods is rather difficult, both because the procedures are time consuming and because the efficiency of the present joint probability distribution techniques deteriorates with complexity. However, further progress can be expected in the field. (4) Modified tangent formulae and least-squares determination and refinement of phases The problem of deriving the individual phase angles from triplet relationships is greatly overdetermined: indeed the number of triplets, in fact, greatly exceeds the number of phases so that any ’h may be determined by a least-squares approach (Hauptman et al., 1969). The function to be minimized may be P w ½cosð’h  ’k  ’hk Þ  Ck
2 P M¼ k k ; wk

This aim can be achieved, for example, by introducing a large number of permutable phases into the initial set. However, the introduction of every new symbol implies a fourfold increase in computing time, which, even in fast computers, quickly leads to computing-time limitations. On the other hand, a relatively large starting set is not in itself enough to ensure a successful structure determination. This is the case, for example, when the triplet invariants used in the initial steps differ significantly from zero. New strategies have therefore been devised to solve more complex structures. (1) Magic-integer methods In the classical procedure described in Section 2.2.7, the unknown phases in the starting set are assigned all combinations of the values =4; 3=4. For n unknown phases in the starting set, 4n sets of phases arise by quadrant permutation; this is a number that increases very rapidly with n. According to White & Woolfson (1975), phases can be represented for a sequence of n integers by the equations ’i ¼ mi x ðmod 2Þ;

i ¼ 1; . . . ; n:

ð2:2:8:1Þ

The set of equations can be regarded as the parametric equation of a straight line in n-dimensional phase space. The nature and size of errors connected with magic-integer representations have been investigated by Main (1977) who also gave a recipe for deriving magic-integer sequences which minimize the r.m.s. errors in the represented phases (see Table 2.2.8.1). To assign a phase value, the variable x in equation (2.2.8.1) is given a series of values at equal intervals in the range 0 < x < 2. The enantiomorph is defined by exploring only the appropriate half of the n-dimensional space. A different way of using the magic-integer method (Declercq et al., 1975) is the primary–secondary P–S method which may be described schematically in the following way: (a) Origin- and enantiomorph-fixing phases are chosen and some one-phase s.s.’s are estimated. (b) Nine phases [this is only an example: very long magicinteger sequences may be used to represent primary phases (Hull et al., 1981; Debaerdemaeker & Woolfson, 1983)] are represented with the approximated relationships: 8 8 8 < ’i1 ¼ 3x < ’j1 ¼ 3y < ’p1 ¼ 3z ’i2 ¼ 4x ’j2 ¼ 4y ’p ¼ 4z : ’ ¼ 5x : ’ ¼ 5y : ’ 2 ¼ 5z: i3 j3 p3

where Ck is the estimate of the cosine obtained by probabilistic or other methods. Effective least-squares procedures based on linear equations (Debaerdemaeker & Woolfson, 1983; Woolfson, 1977) can also be used. A triplet relationship is usually represented by ð’p  ’q  ’r þ bÞ  0 ðmod 2Þ;

Phases in (a) and (b) consistitute the primary set. P(c) The phases in the secondary set are those defined through 2 relationships involving pairs of phases from the primary set: they, too, can be expressed in magic-integer form. (d) All the triplets that link together the phases in the combined primary and secondary set are now found, other than triplets used to obtain secondary reflections from the primary ones. The general algebraic form of these triplets will be

ð2:2:8:2Þ

where b is a factor arising from translational symmetry. If (2.2.8.2) is expressed in cycles and suitably weighted, then it may be written as wð’p  ’q  ’r þ bÞ ¼ wn; where n is some integer. If the integers were known then the equation would appear (in matrix notation) as

233

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION AU ¼ C; ð2:2:8:3Þ The maximum value Hmax ¼ log V is reached for a uniform prior pðrÞ ¼ 1=V. giving the least-squares solution The strength of the restrictions introduced by p(r) is not measured by HðpÞ but by HðpÞ  Hmax , given by 1 T T U ¼ ðA AÞ A C: ð2:2:8:4Þ R HðpÞ  Hmax ¼  pðrÞ log½ pðrÞ=mðrÞ
dr; V When approximate phases are available, the nearest integers may be found and equations (2.2.8.3) and (2.2.8.4) constitute the basis where mðrÞ ¼ 1=V. Accordingly, if a prior prejudice m(r) exists, for further refinement. which maximizes H, the revised relative entropy is Modified tangent procedures are also used, such as (Sint & Schenk, 1975; Busetta, 1976) R SðpÞ ¼  pðrÞ log½ pðrÞ=mðrÞ
dr: P V j Gh; kj sinð’kj þ ’hkj  j Þ tan ’h ’ P ; Gh; kj cosð’kj þ ’hkj  j Þ The maximization problem was solved by Jaynes (1957). If Gj ðpÞ are linear constraint functionals defined by given constraint where j is an estimate for the triplet phase sum ð’h  functions Cj ðrÞ and constraint values cj, i.e. ’kj  ’hkj Þ. R (5) Techniques based on the positivity of Karle–Hauptman Gj ðpÞ ¼ pðrÞCj ðrÞ dr ¼ cj ; determinants V (The main formulae have been briefly described in Section 2.2.5.7.) The maximum determinant rule has been applied to the most unbiased probability density p(r) under prior prejudice solve small structures (de Rango, 1969; Vermin & de Graaff, m(r) is obtained by maximizing the entropy of p(r) relative to 1978) via determinants of small order. It has, however, been m(r). A standard variational technique suggests that the found that their use (Taylor et al., 1978) is not of sufficient power constrained maximization is equivalent to the unconstrained to justify the larger amount of computing time required by the maximization of the functional technique as compared to that required by the tangent formula. P SðpÞ þ j Gj ðpÞ; (6) Tangent techniques using simultaneously triplets, j quartets, . . . The availability of a large number of phase relationships, in where the j ’s are Lagrange multipliers whose values can be particular during the first stages of a direct procedure, makes the determined from the constraints. phasing process easier. However, quartets are sums of two Such a technique has been applied to the problem of finding triplets with a common reflection. If the phase of this reflection good electron-density maps in different ways by various authors (and/or of the other cross terms) is known then the quartet (Wilkins et al., 1983; Bricogne, 1984; Navaza, 1985; Navaza et al., probability formulae described in Section 2.2.5.5 cannot hold. 1983). Similar considerations may be made for quintet relationships. Maximum entropy methods are strictly connected with tradiThus triplet, quartet and quintet formulae described in the tional direct methods: in particular it has been shown that: preceding paragraphs, if used without modifications, will certainly (a) the maximum determinant rule (see Section 2.2.5.7) is introduce systematic errors in the tangent refinement process. strictly connected (Britten & Collins, 1982; Piro, 1983; Narayan & A method which takes into account correlation between Nityananda, 1982; Bricogne, 1984); triplets and quartets has been described (Giacovazzo, 1980c) [see (b) the construction of conditional probability distributions of also Freer & Gilmore (1980) for a first application], according to structure factors amounts precisely to a reciprocal-space which P P evaluation of the entropy functional SðpÞ (Bricogne, 1984). G sinð’k þ ’hk Þ  G0 sinð’k þ ’l þ ’hkl Þ Maximum entropy methods are under strong development: k k; l P 0 tan ’h ’ P ; important contributions can be expected in the near future even G cosð’k þ ’hk Þ  G cosð’k þ ’l þ ’hkl Þ if a multipurpose robust program has not yet been written. k k; l where G0 takes into account both the magnitudes of the cross terms of the quartet and the fact that their phases may be known. (7) Integration of Patterson techniques and direct methods (Egert & Sheldrick, 1985) [see also Egert (1983, and references therein)] A fragment of known geometry is oriented in the unit cell by real-space Patterson rotation search (see Chapter 2.3) and its position is found by application of a translation function (see Section 2.2.5.4 and Chapter 2.3) or by maximizing the weighted sum of the cosines of a small number of strong translationsensitive triple phase invariants, starting from random positions. Suitable FOMs rank the most reliable solutions. (8) Maximum entropy methods A common starting point for all direct methods is a stochastic process according to which crystal structures are thought of as being generated by randomly placing atoms in the asymmetric unit of the unit cell according to some a priori distribution. A non-uniform prior distribution of atoms p(r) gives rise to a source of random atomic positions with entropy (Jaynes, 1957) R HðpÞ ¼  pðrÞ log pðrÞ dr:

2.2.9. Some references to direct-methods packages: the smallmolecule case Some references for direct-methods packages are given below. Other useful packages using symbolic addition or multisolution procedures do exist but are not well documented. CRUNCH: Gelder, R. de, de Graaff, R. A. G. & Schenk, H. (1993). Automatic determination of crystal structures using Karle– Hauptman matrices. Acta Cryst. A49, 287–293. DIRDIF: Beurskens, P. T., Beurskens G., de Gelder, R., Garcia-Granda, S., Gould, R. O., Israel, R. & Smits, J. M. M. (1999). The DIRDIF-99 program system. Crystallography Laboratory, University of Nijmegen, The Netherlands. MITHRIL: Gilmore, C. J. (1984). MITHRIL. An integrated direct-methods computer program. J. Appl. Cryst. 17, 42–46. MULTAN88: Main, P., Fiske, S. J., Germain, G., Hull, S. E., Declercq, J.-P., Lessinger, L. & Woolfson, M. M. (1999). Crystallographic software: teXsan for Windows. http://www. rigaku.com/downloads/journal/Vol15.1.1998/texsan.pdf. PATSEE: Egert, E. & Sheldrick, G. M. (1985). Search for a fragment of known geometry by integrated Patterson and direct methods. Acta Cryst. A41, 262–268.

V

234

2.2. DIRECT METHODS SAPI: Fan, H.-F. (1999). Crystallographic software: teXsan for Windows. http://www.rigaku.com/downloads/journal/ Vol15.1.1998/texsan.pdf. SnB: Weeks, C. M. & Miller, R. (1999). The design and implementation of SnB version 2.0. J. Appl. Cryst. 32, 120–124. SHELX97 and SHELXS: Sheldrick, G. M. (2000). The SHELX home page. http://shelx.uni-ac.gwdg.de/SHELX/. SHELXD: Sheldrick, G. M. (1998). SHELX: applications to macromolecules. In Direct methods for solving macromolecular structures, edited by S. Fortier, pp. 401–411. Dordrecht: Kluwer Academic Publishers. SIR97: Altomare, A., Burla, M. C., Camalli, M., Cascarano, G. L., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G., Polidori, G. & Spagna, R. (1999). SIR97: a new tool for crystal structure determination and refinement. J. Appl. Cryst. 32, 115– 119. SIR2004: Burla, M. C., Caliandro, R., Camalli, M., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C., Polidori, G. & Spagna, R. (2005). SIR2004: an improved tool for crystal structure determination and refinement. J. Appl. Cryst. 38, 381–388. XTAL3.6.1: Hall, S. R., du Boulay, D. J. & Olthof-Hazekamp, R. (1999). Xtal3.6 crystallographic software. http://xtal. sourceforge.net/.

(2) low-resolution data create additional problems for direct methods since the number of available phase relationships per reflection is small. Sheldrick (1990) suggested that direct methods are not expected to succeed if fewer than half of the reflections in the ˚ are observed with jFj > 4
ðjFjÞ (a condition range 1.1–1.2 A seldom satisfied by protein data). The most complete analysis of the problem has been made by Giacovazzo, Guagliardi et al. (1994). They observed that the expected value of  (see Section 2.2.7) suggested by the tangent formula for proteins is comparable with the variance of the  parameter. In other words, for proteins the signal determining the phase is comparable with the noise, and therefore the phase indication is expected to be unreliable. Quite relevant results have recently been obtained by integrating direct methods with some additional experimental information. In particular, we will describe the combination of direct methods with: (a) direct-space techniques for the ab initio crystal structure solution of proteins; (b) isomorphous-replacement (SIR–MIR) techniques; (c) anomalous-dispersion (SAD–MAD) techniques; (d) molecular replacement. Point (d) will not be treated here, as it is described extensively in IT F, Part 13.

2.2.10. Direct methods in macromolecular crystallography 2.2.10.1. Introduction

2.2.10.2. Ab initio crystal structure solution of proteins Ab initio techniques do not require prior information of any atomic positions. The recent tremendous increase in computing speed led to direct methods evolving towards the rapid development of multisolution techniques. The new algorithms of the program Shake-and-Bake (Weeks et al., 1994; Weeks & Miller, 1999; Hauptman et al., 1999) allowed an impressive extension of the structural complexity amenable to direct phasing. In particular we mention: (a) the minimal principle (De Titta et al., 1994), according to which the phase problem is considered as a constrained global optimization problem; (b) the refinement procedure, which alternately uses direct- and reciprocal-space techniques; and (c) the parameter-shift optimization technique (Bhuiya & Stanley, 1963), which aims at reducing the value of the minimal function (Hauptman, 1991; De Titta et al., 1994). An effective variant of Shake-and Bake is SHELXD (Sheldrick, 1998) which cyclically alternates tangent refinement in reciprocal space with peak-list optimisation procedures in real space (Sheldrick & Gould, 1995). Detailed information on these programs is available in IT F (2001), Part 16. A different approach is used by ACORN (Foadi et al., 2000), which first locates a small fragment of the molecule (eventually by molecular-replacement techniques) to obtain a useful nonrandom starting set of phases, and then refines them by means of solvent-flattening techniques. The program SIR2004 (Burla et al., 2005) uses the tangent formula as well as automatic Patterson techniques to obtain a first imperfect structural model; then direct-space techniques are used to refine the model. The Patterson approach is based on the use of the superposition minimum function (Buerger, 1959; Richardson & Jacobson, 1987; Sheldrick, 1992; Pavelcı´k, 1988; Pavelcı´k et al., 1992; Burla et al., 2004). It may be worth noting that even this approach is of multisolution type: up to 20 trial solutions are provided by using as pivots the highest maxima in the superposition minimum function. It is today possible to solve structures up to 2500 non-hydrogen ˚) atoms in the asymmetric unit provided data at atomic (about 1 A resolution are available. Proteins with data at quasi-atomic ˚ ) can also be solved, but with resolution (say up to 1.5–1.6 A greater difficulties (Burla et al., 2005). A simple evaluation of the potential of the ab initio techniques suggests that the structural complexity range and the resolution limits amenable to the ab

The smallest protein molecules contain about 400 nonhydrogen atoms, so they cannot be solved ab initio by the algorithms specified in Sections 2.2.7 and 2.2.8. However, traditional direct methods are applied for: (a) improvement of the accuracy of the available phases (refinement process); (b) extension of phases from lower to higher resolution (phaseextension process). The application of standard tangent techniques to (a) and (b) has not been found to be very satisfactory (Coulter & Dewar, 1971; Hendrickson et al., 1973; Weinzierl et al., 1969). Tangent methods, in fact, require atomicity and non-negativity of the electron density. Both these properties are not satisfied if data do ˚ ). Because of series not extend to atomic resolution (d > 1.2 A termination and other errors the electron-density map at d > ˚ presents large negative regions which will appear as false 1.2 A peaks in the squared structure. However, tangent methods use only a part of the information given by the Sayre equation (2.2.6.5). In fact, (2.2.6.5) express two equations relating the radial and angular parts of the two sides, so obtaining a large degree of overdetermination of the phases. To achieve this Sayre (1972) [see also Sayre & Toupin (1975)] suggested minimizing (2.2.10.1) by least squares as a function of the phases:
2

P

P

ah Fh  Fk Fhk
: h

ð2:2:10:1Þ

k

Even if tests on rubredoxin (extensions of phases from 2.5 to ˚ resolution) and insulin (Cutfield et al., 1975) (from 1.9 to 1.5 A ˚ resolution) were successful, the limitations of the method 1.5 A are its high cost and, especially, the higher efficiency of the leastsquares method. Equivalent considerations hold for the application of determinantal methods to proteins [see Podjarny et al. (1981); de Rango et al. (1985) and literature cited therein]. A question now arises: why is the tangent formula unable to solve protein structures? Fan et al. (1991) considered the question from a first-principle approach and concluded that: (1) the triplet phase probability distribution is very flat for proteins (N is very large) and close to the uniform distribution;

235

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION initio approach could be larger in the near future. The approach will profit by general technical advances like the increasing speed of computers and by the greater efficiency of informatic tools (e.g. faster Fourier-transform techniques). It could also profit from new specific crystallographic algorithms (for example, Oszla´nyi & Su¨to, 2004). It is of particular interest that extrapolating moduli and phases of nonmeasured reflections beyond the experimental resolution limit makes the ab initio phasing process more efficient, and leads to crystal structure solution even in cases in which the standard programs do not succeed (Caliandro et al., 2005a,b). Moreover, the use of the extrapolated values improves the quality of the final electron-density maps and makes it easier to recognize the correct one among several trial structures.

Three-phase structure invariants were evaluated by considering that eight invariants exist for a given triple of indices h, k, l ðh þ k þ l ¼ 0Þ:

2.2.10.3. Integration of direct methods with isomorphous replacement techniques

has to be studied, from which eight conditional probability densities can be obtained:
Pði
jEh j; jEk j; jEl j; jGh j; jGk j; jGl jÞ

fj expð2ih  rj Þ;

¼ 02

l

6 ¼

h

þ ’k þ

l

7 ¼

k

þ ’l

8 ¼

h

þ

l:

h

þ

k

þ

Q1 ¼ 2½
3 =
23=2
p jEh Ek El j þ 2½
3 =
23=2
H h k l ;

ð2:2:10:2Þ

where indices p and H warn that parameters have to be calculated over protein atoms and over heavy atoms, respectively, and

j¼1 hÞ

þ

for j ¼ 1; . . . ; 8. The analytical expressions of Qj are too intricate and are not given here (the reader is referred to the original paper). We only say that Qj may be positive or negative, so that reliable triplet phase estimates near 0 or near  are possible: the larger jQj j, the more reliable the phase estimate. A useful interpretation of the formulae in terms of experimental parameters was suggested by Fortier et al. (1984): according to them, distributions do not depend, as in the case of the traditional three-phase invariants, on the total number of atoms per unit cell but rather on the scattering difference between the native protein and the derivative (that is, on the scattering of the heavy atoms in the derivative). Hauptman’s formulae were generalized by Giacovazzo et al. (1988): the new expressions were able to take into account the resolution effects on distribution parameters. The formulae are completely general and include as special cases native protein and heavy-atom isomorphous derivatives as well as X-ray and neutron diffraction data. Their complicated algebraic forms are easily reduced to a simple expression in the case of a native protein heavy-atom derivative: in particular, the reliability parameter for 1 is

where the subscripts d and p denote the derivative and the protein, respectively. Denote also by fj and gj atomic scattering factors for the atom labelled j in a pair of isomorphous structures, and let Eh and Gh denote corresponding normalized structure factors. Then

Gh ¼ jGh j expði

k

’ ½2I0 ðQj Þ
1 exp½Qj cos j


F ¼ jFd j  jFp j

N 1=2 P

5 ¼ ’h þ

PðEh ; Ek ; El ; Gh ; Gk ; Gl Þ

2.2.10.4. SIR–MIR case: one-step procedures The theoretical basis was established by Hauptman (1982a): his primary interest was to establish the two-phase and threephase structure invariants by exploiting the experimental information provided by isomorphous data. The protein phases could be directly assigned via a tangent procedure. Let us denote the modulus of the isomorphous difference as

Eh ¼ jEh j expði’h Þ ¼ 20

2 ¼ ’h þ ’k þ l  4 ¼ h þ ’k þ ’l

So, for the estimation of any j , the joint probability distribution

SIR–MIR cases are characterized by a situation in which there is one native protein and one or more heavy-atom substructures. In this situation the phasing procedure may be a two-step process: in the first stage the heavy-atom positions are identified by Patterson techniques (Rossmann, 1961; Okaya et al., 1955) or by direct methods (Mukherjee et al., 1989). In the second step the protein phases are estimated by exploiting the substructure information. Direct methods are able to contribute to both steps (see Sections 2.2.10.5 and 2.2.10.6). In Section 2.2.10.4 we show that direct methods are also able to suggest alternative one-step procedures by estimating structure invariants from isomorphous data.

N 1=2 P

1 ¼ ’h þ ’k þ ’l 3 ¼ ’h þ k þ ’l

gj expð2ih  rj Þ;

j¼1

 ¼ ðFd  Fp Þ=ð

P

fj2 Þ1=2 H :

where mn ¼

N P

 is a pseudo-normalized difference (with respect to the heavyatom structure) between moduli of structure factors. Equation (2.2.10.2) may be compared with Karle’s (1983) algebraic rule: if the sign of h k l is plus then the value of 1 is estimated to be zero; if its sign is minus then the expected value of 1 is close to . In practice Karle’s rule agrees with (2.2.10.2) only if the Cochran-type term in (2.2.10.2) may be neglected. Furthermore, (2.2.10.2) shows that large reliability values do not depend on the triple product of structure-factor differences, but on the triple product of pseudo-normalized differences. A similar mathematical approach has been applied to estimate quartet invariants via isomorphous data. The result may be summarized as follows: a quartet is a phase relationship of order NH1 (Giacovazzo & Siliqi, 1996a,b; see also Kyriakidis et al., 1996), with reliability factor equal to

fjm gnj :

j¼1

The conditional probability of the two-phase structure invariant  ¼ ’h  h given jEh j and jGh j is (Hauptman, 1982a)
Pð
jEj; jGjÞ ’ ½2I0 ðQÞ
1 expðQ cos Þ; where Q ¼ jEGj½2=ð1  2 Þ
; 1=2  ¼ 11 =ð1=2 20 02 Þ:

236

2.2. DIRECT METHODS G¼

2h k l hþkþl Q4 NH         1 þ 2hþk  1 þ 2hþl  1 þ 2kþl  1 ;

A sounder procedure has been suggested by Giacovazzo et al. (2004): they studied, for the SIR case, the joint probability distribution function PðEH ; Ep ; Ed Þ

ð2:2:10:3Þ

under the following assumptions: (a) the atomic positions of the native protein structure and the positions of the heavy atoms in the derivative structure are the primitive random variables of the probabilistic approach; (b)

where Q4 is a suitable normalizing factor. As previously stressed, equations (2.2.10.2) and (2.2.10.3) are valid if the lack of isomorphism and the errors in the measurements are assumed to be negligible. At first sight this approach seems more appealing than the traditional two-step procedures, however it did not prove to be competitive with them. The main reason is the absence in the Hauptman and Giacovazzo approaches of a probabilistic treatment of the errors: such a treatment, on the contrary, is basic for the traditional SIR–MIR techniques [see Blow & Crick (1959) and Terwilliger & Eisenberg (1987) for two related approaches]. The problem of the errors in the probabilistic scenario defined by the joint probability distribution functions approach has recently been overcome by Giacovazzo et al. (2001). In their probabilistic calculations the following assumptions were made:

jFd j expði’d Þ ¼ jFp j expði’p Þ þ jFH j expði’H Þ þ jd j expðid Þ ð2:2:10:6Þ is the structure factor of the derivative. Then the conditional distribution PðRH jRp ; Rd Þ may be derived, from which hRH jRp ; Rd i may be obtained. In terms of structure factors " # P P H H 2 2 2  hjd j i þ P  hjFH j i ¼ P 2 2 iso : H þhjd j i H þhjd j i ð2:2:10:7Þ

jFdj j expði’j Þ ¼ jFp j expði’p Þ þ FHj expði’Hj Þ þ jj j expðij Þ; ð2:2:10:4Þ

The effect of the errors on the evaluation of the moduli |FH|2 may be easily derived: if hjd j2 i ¼ 0, equation (2.2.10.7) confirms Blow and Rossmann’s approximation hjFH j2 i ’ jFj2 . If hjd j2 i 6¼ 0 Blow and Rossmann’s estimate should be affected by a systematic error, increasing with hjd j2 i.

where j refers to the jth derivative. jj j expðij Þ is the error, which can include model as well as measurement errors. A more realistic expression for the reliability factor G of triplet invariants is obtained by including the expression (2.2.10.4) in the probabilistic approach. Then the reliability parameter of the triplet invariants is transformed into (Giacovazzo et al., 2001) G¼

2.2.10.6. SIR–MIR case: protein phasing by direct methods Let us suppose that the various heavy-atom substructures have been determined. They may be used as additional prior information for a more accurate estimate of the ’p values. To this purpose the distributions

2½
3 =
23=2
p Rp1 Rp2 Rp3 þ 2½
3 =
23=2
H

 1 2  3 ; 2 2 2 ½1 þ ð
1 ÞH
½1 þ ð
2 ÞH
½1 þ ð
3 ÞH


PðEp ; E0d jE0H Þ  PðEp ; E0d1 ; . . . ; E0dn jE0H1 ; . . . ; E0Hn Þ

ð2:2:10:8Þ

ð2:2:10:5Þ may be used under the assumption (2.2.10.6). E0dj and E0Hj , for j = 1, . . . , n, are the structure factors of the jth derivative and of the jth heavy-atom substructure, respectively, both normalized with respect to the protein. Any joint probability density (2.2.10.8) may be reliably approximated by a multidimensional Gaussian distribution (Giacovazzo & Siliqi, 2002), from which the following conditional distribution is obtained:

where ð
2 ÞH ¼ jj2 =ðfj2 ÞH. Equation (2.2.10.5) suggests how the error influences the reliability of the triplet estimate: even quite a small value of jj2 may be critical if the scattering power of the heavy-atom substructure is a very small percentage of the derivative scattering power. A one-step procedure has been implemented in a computer program (Giacovazzo et al., 2002): it has been shown that the method is able to derive automatically, from the experimental data and without any user intervention, good quality (i.e. perfectly interpretable) electron-density maps.

Pð’p jRp ; R0d ; E0H Þ ’ ½2I0 ðGÞ
1 exp½p cosð’p  p Þ
where p, the expected value of ’p, is given by Pn T j¼1 Gj sin ’Hj ¼ tan p ¼ Pn B G cos ’ j Hj j¼1

2.2.10.5. SIR–MIR case: the two-step procedure. Finding the heavy-atom substructure by direct methods

and Gj ¼ 2jFHj jF=2j . p ¼ ðT 2 þ B2 Þ1=2 is the reliability factor of the phase estimate. A robust phasing procedure has been established which, starting from the observed moduli jFp j; jFdj j; j ¼ 1; . . . ; n, is able to automatically provide, without any user intervention, a highquality electron-density map of the protein (Giacovazzo et al., 2002).

The first trials for finding the heavy-atom substructure were based on the following assumption: the modulus of the isomorphous difference, F ¼ jFd j  jFp j; is assumed at a first approximation as an estimate of the heavyatom structure factor FH. Perutz (1956) approximated |FH|2 with the difference ðjFd j2  jFp j2 Þ. Blow (1958) and Rossmann (1960) suggested a better approximation: jFH j2 ’ jFj2 . A deeper analysis was performed by Phillips (1966), Dodson & Vijayan (1971), Blessing & Smith (1999) and Grosse-Kunstleve & Brunger (1999). The use of direct methods requires the normalization of jFj and application of the tangent formula (Wilson, 1978).

2.2.10.7. Integration of anomalous-dispersion techniques with direct methods If the frequency of the radiation is close to an absorption edge of an atom, then that atom will scatter the X-rays anomalously (see Chapter 2.4) according to f ¼ f 0 þ if 00. This results in the breakdown of Friedel’s law. It was soon realized that the Bijvoet difference could also be used in the determination of phases (Peerdeman & Bijvoet, 1956; Ramachandran & Raman, 1956;

237

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

¼ Eh Ek El ¼ Rh Rk Rl expðih; k Þ;


¼ Eh Ek El ¼ Gh Gk Gl expðih
; k
Þ;
 ¼ 1ðh; k  h
; k
Þ;

Okaya & Pepinsky, 1956). Since then, a great deal of work has been done both from algebraic (see Chapter 2.4) and from probabilistic points of view. In this section we are only interested in the second. SAD (single anomalous dispersion) and MAD (multiple anomalous dispersion) techniques can be used. Both are characterized by one protein structure and one anomalous-scatterer substructure. The experimental diffraction data differ only because of the different anomalous scattering (not because of different anomalous-scatterer substructures). In the MAD case the anomalous-scatterer substructure is in some way ‘overdetermined’ by the data and, therefore, it is more convenient to use a two-step procedure: first define the positions of the anomalous scatterers, and then estimate the protein phase values. For completeness, we describe the one-step procedures in Section 2.2.10.8. These are based on the estimation of the structure invariants and on the application of the tangent formula. The two-step procedures are described in the Sections 2.2.10.9 and 2.2.10.10.

2

and 00 is the contribution of the imaginary part of , which may be approximated in favourable conditions by

00 ¼ 2f 00 ½ fh0 fk0 þ fh0 fl0 þ fk fl
 ½1 þ SðR2h þ R2k þ R2l  3Þ
; where S is a suitable scale factor.
( and Equation (2.2.10.10) gives two possible values for    ). Only if Rh Rk Rhþk is large enough may this phase ambiguity be resolved by choosing the angle nearest to zero. The evaluation of triplet phases by means of anomalous dispersion has been further pursued by Hauptman (1982b) and independently by Giacovazzo (1983b). Owing to the breakdown of Friedel’s law there are eight distinct triplet invariants which can contemporaneously be exploited:

2.2.10.8. The SAD case: the one-step procedures Probability distributions of diffraction intensities and of selected functions of diffraction intensities for dispersive structures have been given by various authors [Parthasarathy & Srinivasan (1964), see also Srinivasan & Parthasarathy (1976) and relevant literature cited therein]. We describe here some probabilistic formulae for estimating invariants of low order. (a) Estimation of two-phase structure invariants. The conditional probability distribution of  ¼ ’h þ ’h given Rh and Gh (normalized moduli of Fh and Fh , respectively) (Hauptman, 1982b; Giacovazzo, 1983b) is PðjRh ; Gh Þ ’ ½2I0 ðQÞ
1 exp½Q cosð  qÞ
;

ð2:2:10:9Þ

2Rh Gh 2 pffiffiffi ½c1 þ c22
1=2 ; c c c sin q ¼ 2 2 2 1=2 ; cos q ¼ 2 1 2 1=2 ; ½c1 þ c2
½c1 þ c2
N P P c1 ¼ ðfj0 2  fj00 2 Þ= ; P

R2 ¼ jEk j

R3 ¼ jEhþk j

G1 ¼ jEh j G2 ¼ jEk j

R3 ¼ jEhk j

’1 ¼ ’h 1 ¼ ’h

’3 ¼ ’hþk 3 ¼ ’hk ;

’2 ¼ ’k 2 ¼ ’k

The definitions of  and ! are rather extensive and so the reader is referred to the published papers. We only add that  is always positive and that !, the expected value of , may lie anywhere between 0 and 2. Understanding the role of the various parameters in equation (2.2.10.11) is not easy. Giacovazzo et al. (2003) found an equivalent simpler expression from which interpretable estimates of the parameters were obtained. In the same paper the limitations of the approach (versus the twostep procedures) were clarified.

;

c ¼ ½1  ðc21 þ c22 Þ
2 ; N P P ¼ ð fj0 2 þ fj00 2 Þ: j¼1

q is the most probable value of : a large value of the parameter Q suggests that the phase relation  ¼ q is reliable. Large values of Q are often available in practice: q, however, may be considered an estimate of jj rather than of  because the enantiomorph is not fixed in (2.2.10.9). A formula for the estimation of  in centrosymmetric structures has been provided by Giacovazzo (1987). (b) Estimation of triplet invariants. Kroon et al. (1977) first incorporated anomalous diffraction in order to estimate triplet invariants. Their work was based on an analysis of the complex double Patterson function. Subsequent probabilistic considerations (Heinermann et al., 1978) confirmed their results, which can be so expressed: j j2  j
j2 ; 4 00 ½12 ðj j2 þ j
j2 Þ  j 00 j2
1=2

6 ¼ ’h þ ’k þ ’l 8 ¼ ’h þ ’k  ’l :

ð2:2:10:11Þ

j¼1


¼ sin 

4 ¼ ’h þ ’k  ’l

5 ¼ ’h þ ’k þ ’l ; 7 ¼ ’h  ’k þ ’l ;

Hauptman and Giacovazzo found the following conditional distribution:

1 PðjRj ; Gj ; j ¼ 1; 2; 3Þ ’ 2 I0 ðÞ exp½ cos ð  !Þ
:

j¼1

fj0 fj00 =

3 ¼ ’h  ’k þ ’l ;

R1 ¼ jEh j

Q¼

N P

2 ¼ ’h þ ’k þ ’l

Given

where

c2 ¼ 2

1 ¼ ’ h þ ’ k þ ’ l ;

2.2.10.9. SAD–MAD case: the two-step procedures. Finding the anomalous-scatterer substructure by direct methods The anomalous-scatterer substructure is traditionally determined by the techniques suggested by Karle and Hendrickson (Karle, 1980b; Hendrickson, 1985; Pa¨hler et al., 1990; Terwilliger, 1994). The introduction of selenium into proteins as selenomethionine encouraged the second-generation direct methods programs [Shake and Bake by Miller et al. (1994); Half bake by Sheldrick (1998); SIR2000-N by Burla et al. (2001); ACORN by Foadi et al. (2000)] to locate Se atoms. Since the number of Se atoms may be quite large (up to 200), direct methods rather than Patterson techniques seem to be preferable. Shake and Bake, Half Bake and ACORN obtain the coordinates of the anomalous scatterers from a single-wavelength set of data. When more sets of diffraction data are available the solutions obtained by the other sets are used to confirm the correct solution.

ð2:2:10:10Þ

where ðh þ k þ l ¼ 0Þ,

238

2.2. DIRECT METHODS various ano values was also provided (see also Schneider & Sheldrick, 2002) for predicting the most informative combinations.

A different approach has been suggested in two recent papers (Burla et al., 2002; Burla, Carrozzini et al., 2003): the estimates of the amplitudes of the structure factors of the anomalously scattering substructure are derived, via the rigorous method of the joint probability distribution functions, from the experimental diffraction moduli relative to n wavelengths. To do that, first the joint distribution

2.2.10.10. SAD–MAD case: protein phasing by direct methods Once the anomalous-scatterer substructure has been found, þ   the corresponding structure factors Eþ a1 ; . . . ; Ean ; Ea1 ; . . . ; Ean are known in modulus and phase. Then the conditional joint probability distribution   þ   þ þ   P Eþ 1 ; . . . ; En ; E1 ; . . . ; En jEa1 ; . . . ; Ean ; Ea1 ; . . . ; Ean

þ þ    Pn ¼ PðAoa ; Aþ 1 ; A2 ; . . . ; An ; A1 ; A2 ; . . . ; An ; þ þ    Boa ; Bþ 1 ; B2 ; . . . ; Bn ; B1 ; B2 ; . . . ; Bn Þ

¼ ð2nþ1Þ ðdet KÞ1=2 expð 12 TT K1 TÞ þ   is calculated, where Aoa, Boa, Eoa, Aþ i , Bi , Ai , Bi are the real and þ  imaginary components of Eoa, Ei , Ei , respectively, K is a symmetric square matrix of order (4n + 2), K1 = {ij} is its inverse, and T is a suitable vector with components defined in þ  terms of the variables Aoa ; Aþ 1 ; A2 ; . . . ; Bn . Eoa is the normalized structure factor of the anomalous scatterer substructure calculated by neglecting anomalous scattering components. Then the conditional distribution

may be calculated (Giacovazzo & Siliqi, 2004), from which the conditional distribution   þ  P ’þ 1 jEai ; Eai ; Ri ; Gi ; i ¼ 1; . . . ; 2 may be derived. þ It has been shown that the most probable phase of ’þ 1 , say 1 , is the phase of the vector

PðRoa jR1 ; . . . ; Rn ; G1 ; . . . ; Gn Þ

n

 P þ  

wþ j Eaj þ wj Eaj

is derived, from which hRoa jR1 ; . . . ; Gn i ¼

1=2 1 2 ð=11 Þ ½1

j¼1 2

þ 4X =ð11 Þ


1=2

ð2:2:10:12Þ

 

 þ 

½wjp ðEþ aj  Eap Þ þ wnþj;nþp Eaj  Eap


n P

þ

j;p¼1;p>j

is obtained, where 2

Q21

þ

Q22

X ¼ þ Q1 ¼ 12 R1 þ 13 R2 þ . . . þ 1;nþ1 Rn þ 1;nþ2 G1 þ . . .

  

wj;nþp Eþ aj  Eap


ð2:2:10:14Þ

j;p¼1

and the reliability parameter of the phase estimate is nothing other than the modulus of (2.2.10.14). The first term in (2.2.10.14) is a Sim-like contribution; the other terms, through the weights w, take into account the errors and the experimental differences ðRj  Rp Þ, ðGj  Gp Þ and ðRj  Gp Þ.

þ 1;2nþ1 Gn Q2 ¼ 1;2nþ3 R1 þ 1;2nþ4 R2 þ . . . þ 1;3nþ2 Rn þ . . .  1;3nþ3 G1  . . .  1;4nþ2 Gn : The standard deviation of the estimate is also calculated:

1=2 h  i1=2 ;
Roa ¼ hR2oa j . . .i  hRoa j:::i2 ¼ 1  1 4 11

References Allegra, G. (1979). Derivation of three-phase invariants from the Patterson function. Acta Cryst. A35, 213–220. Altomare, A., Burla, M. C., Camalli, M., Cascarano, G. L., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G., Polidori, G. & Spagna, R. (1999). SIR97: a new tool for crystal structure determination and refinement. J. Appl. Cryst. 32, 115–119. Anzenhofer, K. & Hoppe, W. (1962). Phys. Verh. Mosbach. 13, 119. Ardito, G., Cascarano, G., Giacovazzo, C. & Luic´, M. (1985). 1-Phase seminvariants and Harker sections. Z. Kristallogr. 172, 25–34. Argos, P. & Rossmann, M. G. (1980). Molecular replacement method. In Theory and Practice of Direct Methods in Crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 381–389. New York: Plenum. Avrami, M. (1938). Direct determination of crystal structure from X-ray data. Phys. Rev. 54, 300–303. Baggio, R., Woolfson, M. M., Declercq, J.-P. & Germain, G. (1978). On the application of phase relationships to complex structures. XVI. A random approach to structure determination. Acta Cryst. A34, 883–892. Banerjee, K. (1933). Determination of the signs of the Fourier terms in complete crystal structure analysis. Proc. R. Soc. London Ser. A, 141, 188–193. Bertaut, E. F. (1955a). La me´thode statistique en cristallographie. I. Acta Cryst. 8, 537–543. Bertaut, E. F. (1955b). La me´thode statistique en cristallographie. II. Quelques applications. Acta Cryst. 8, 544–548. Bertaut, E. F. (1960). Ordre logarithmique des densite´s de re´partition. I. Acta Cryst. 13, 546–552. Beurskens, P. T., Beurskens, G., de Gelder, R., Garcia-Granda, S., Gould, R. O., Israel, R. & Smits, J. M. M. (1999). The DIRDIF-99 program system. Crystallography Laboratory, University of Nijmegen, The Netherlands. Beurskens, P. T., Gould, R. O., Bruins Slot, H. J. & Bosman, W. P. (1987). Translation functions for the positioning of a well oriented molecular fragment. Z. Kristallogr. 179, 127–159.

from which

1=2 hRoa j . . .i ð=4Þ þ ðX 2 Þ=11 ¼ :
Roa 1  ð=4Þ

n P

ð2:2:10:13Þ

The advantage of the above approach is that the estimates can simultaneously exploit both the anomalous and the dispersive differences. The computing procedure proposed by Burla, Carrozzini et al. (2003) is the following: (i) The sets Sj , j = 1, . . . , n, of the observed magnitudes (say |F+|, |F|) are stored for all the n wavelengths. (ii) The Wilson method is applied to put the sets Sj on their absolute scales. (iii) Equations (2.2.10.12) and (2.2.10.13) are applied to obtain the values hRoa j . . .i and hRoa j . . .i=
Roa . (iv) The triplet invariants involving the reflections with the highest hRoa j . . .i=
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239

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when anomalous scatterers are present. Acta Cryst. A39, 585–592. Giacovazzo, C. (1987). One wavelength technique: estimation of centrosymmetrical two-phase invariants in dispersive structures. Acta Cryst. A43, 73–75. Giacovazzo, C. (1988). New probabilistic formulas for finding the positions of correctly oriented atomic groups. Acta Cryst. A44, 294–300. Giacovazzo, C. (1998). Direct Phasing in Crystallography. New York: IUCr, Oxford University Press. Giacovazzo, C., Cascarano, G. & Zheng, C.-D. (1988). On integrating the techniques of direct methods and isomorphous replacement. A new probabilistic formula for triplet invariants. Acta Cryst. A44, 45–51.

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2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Koch, M. H. J. (1974). On the application of phase relationships to complex structures. IV. Automatic interpretation of electron-density maps for organic structures. Acta Cryst. A30, 67–70. Krabbendam, H. & Kroon, J. (1971). A relation between structure factor, triple products and a single Patterson vector, and its application to sign determination. Acta Cryst. A27, 362–367. Kroon, J., Spek, A. L. & Krabbendam, H. (1977). Direct phase determination of triple products from Bijvoet inequalities. Acta Cryst. A33, 382–385. Kyriakidis, C. E., Peschar, R. & Schenk, H. (1996). The estimation of four-phase structure invariants using the single difference of isomorphous structure factors. Acta Cryst. A52, 77–87. Lajze´rowicz, J. & Lajze´rowicz, J. (1966). Loi de distribution des facteurs de structure pour un re´partition non uniforme des atomes. Acta Cryst. 21, 8–12. Langs, D. A. (1985). Translation functions: the elimination of structuredependent spurious maxima. Acta Cryst. A41, 305–308. Lessinger, L. & Wondratschek, H. (1975). Seminvariants for space groups I 4
2m and I 4
d. Acta Cryst. A31, 521. Mackay, A. L. (1953). A statistical treatment of superlattice reflexions. Acta Cryst. 6, 214–215. Main, P. (1976). Recent developments in the MULTAN system. The use of molecular structure. In Crystallographic Computing Techniques, edited by F. R. Ahmed, pp. 97–105. Copenhagen: Munksgaard. Main, P. (1977). On the application of phase relationships to complex structures. XI. A theory of magic integers. Acta Cryst. A33, 750–757. Main, P., Fiske, S. J., Germain, G., Hull, S. E., Declercq, J.-P., Lessinger, L. & Woolfson, M. M. (1999). Crystallographic software: teXsan for Windows. http://www.rigaku.com/downloads/journal/Vol15.1.1998/ texsan.pdf. Main, P. & Hull, S. E. (1978). The recognition of molecular fragments in E maps and electron density maps. Acta Cryst. A34, 353–361. Miller, R., Gallo, S. M., Khalak, H. G. & Weeks, C. M. (1994). SnB: crystal structure determination via shake-and-bake. J. Appl. Cryst. 27, 613–621. Mukherjee, A. K., Helliwell, J. R. & Main, P. (1989). The use of MULTAN to locate the positions of anomalous scatterers. Acta Cryst. A45, 715–718. Narayan, R. & Nityananda, R. (1982). The maximum determinant method and the maximum entropy method. Acta Cryst. A38, 122–128. Navaza, J. (1985). On the maximum-entropy estimate of the electron density function. Acta Cryst. A41, 232–244. Navaza, J., Castellano, E. E. & Tsoucaris, G. (1983). Constrained density modifications by variational techniques. Acta Cryst. A39, 622–631. Naya, S., Nitta, I. & Oda, T. (1964). A study on the statistical method for determination of signs of structure factors. Acta Cryst. 17, 421–433. Naya, S., Nitta, I. & Oda, T. (1965). Affinement tridimensional du sulfanilamide . Acta Cryst. 19, 734–747. Nordman, C. E. (1985). Introduction to Patterson search methods. In Crystallographic Computing 3. Data Collection, Structure Determination, Proteins and Databases, edited by G. M. Sheldrick, G. Kruger & R. Goddard, pp. 232–244. Oxford: Clarendon Press. Oda, T., Naya, S. & Taguchi, I. (1961). Matrix theoretical derivation of inequalities. II. Acta Cryst. 14, 456–458. Okaya, J. & Nitta, I. (1952). Linear structure factor inequalities and the application to the structure determination of tetragonal ethylenediamine sulphate. Acta Cryst. 5, 564–570. Okaya, Y. & Pepinsky, R. (1956). New formulation and solution of the phase problem in X-ray analysis of non-centric crystals containing anomalous scatterers. Phys. Rev. 103, 1645–1647. Okaya, Y., Saito, Y. & Pepinsky, R. (1955). New method in X-ray crystal structure determination involving the use of anomalous dispersion. Phys. Rev. 98, 1857–1858. Oszla´nyi, G. & Su¨to, A. (2004). Ab initio structure solution by charge flipping. Acta Cryst. A60, 134–141. Ott, H. (1927). Zur Methodik der Struckturanalyse. Z. Kristallogr. 66, 136–153. Pa¨hler, A., Smith, J. L. & Hendrickson, W. A. (1990). A probability representation for phase information from multiwavelength anomalous dispersion. Acta Cryst. A46, 537–540. Parthasarathy, S. & Srinivasan, R. (1964). The probability distribution of Bijvoet differences. Acta Cryst. 17, 1400–1407. Pavelcı´k, F. (1988). Patterson-oriented automatic structure determination: getting a good start. Acta Cryst. A44, 724–729.

Hauptman, H. & Karle, J. (1959). Table 2. Equivalence classes, seminvariant vectors and seminvariant moduli for the centered centrosymmetric space groups, referred to a primitive unit cell. Acta Cryst. 12, 93–97. Hauptman, H. A. (1991). In Crystallographic Computing 5: from Chemistry to Biology, edited by D. Moras, A. D. Podjarny & J. C. Thierry. IUCr/Oxford University Press. Hauptman, H. A., Xu, H., Weeks, C. M. & Miller, R. (1999). Exponential Shake-and-Bake: theoretical basis and applications. Acta Cryst. A55, 891–900. Heinermann, J. J. L. (1977a). The use of structural information in the phase probability of a triple product. Acta Cryst. A33, 100–106. Heinermann, J. J. L. (1977b). Thesis. University of Utrecht. Heinermann, J. J. L., Krabbendam, H. & Kroon, J. (1979). The joint probability distribution of the structure factors in a Karle–Hauptman matrix. Acta Cryst. A35, 101–105. Heinermann, J. J. L., Krabbendam, H., Kroon, J. & Spek, A. L. (1978). Direct phase determination of triple products from Bijvoet inequalities. II. A probabilistic approach. Acta Cryst. A34, 447–450. Hendrickson, W. A. (1985). Analysis of protein structure from diffraction measurement at multiple wavelengths. Trans. Am. Crystallogr. Assoc. 21, 11–21. Hendrickson, W. A., Love, W. E. & Karle, J. (1973). Crystal structure ˚ resolution. J. Mol. Biol. 74, analysis of sea lamprey hemoglobin at 2 A 331–361. Hoppe, W. (1963). Phase determination and zero points in the Patterson function. Acta Cryst. 16, 1056–1057. Hughes, E. W. (1953). The signs of products of structure factors. Acta Cryst. 6, 871. Hull, S. E. & Irwin, M. J. (1978). On the application of phase relationships to complex structures. XIV. The additional use of statistical information in tangent-formula refinement. Acta Cryst. A34, 863–870. Hull, S. E., Viterbo, D., Woolfson, M. M. & Shao-Hui, Z. (1981). On the application of phase relationships to complex structures. XIX. Magicinteger representation of a large set of phases: the MAGEX procedure. Acta Cryst. A37, 566–572. International Tables for Crystallography (2001). Vol. F, Macromolecular Crystallography, edited by M. G. Rossmann & E. Arnold. Dordrecht: Kluwer Academic Publishers. Jaynes, E. T. (1957). Information theory and statistical mechanics. Phys. Rev. 106, 620–630. Karle, J. (1970a). An alternative form for B3.0 , a phase determining formula. Acta Cryst. B26, 1614–1617. Karle, J. (1970b). Partial structures and use of the tangent formula and translation functions. In Crystallographic Computing, pp. 155–164. Copenhagen: Munksgaard. Karle, J. (1972). Translation functions and direct methods. Acta Cryst. B28, 820–824. Karle, J. (1979). Triple phase invariants: formula for centric case from fourth-order determinantal joint probability distributions. Proc. Natl Acad. Sci. USA, 76, 2089–2093. Karle, J. (1980a). Triplet phase invariants: formula for acentric case from fourth-order determinantal joint probability distributions. Proc. Natl Acad. Sci. USA, 77, 5–9. Karle, J. (1980b). Some developments in anomalous dispersion for the structural investigation of macromolecular systems in biology. Int. J. Quantum Chem. Quantum Biol. Symp. 7, 357–367. Karle, J. (1983). A simple rule for finding and distinguishing triplet phase invariants with values near 0 or  with isomorphous replacement data. Acta Cryst. A39, 800–805. Karle, J. & Hauptman, H. (1950). The phases and magnitudes of the structure factors. Acta Cryst. 3, 181–187. Karle, J. & Hauptman, H. (1956). A theory of phase determination for the four types of non-centrosymmetric space groups 1P222, 2P22, 3P1 2, 3P2 2. Acta Cryst. 9, 635–651. Karle, J. & Hauptman, H. (1958). Phase determination from new joint probability distributions: space group P1. Acta Cryst. 11, 264–269. Karle, J. & Hauptman, H. (1961). Seminvariants for non-centrosymmetric space groups with conventional centered cells. Acta Cryst. 14, 217–223. Karle, J. & Karle, I. L. (1966). The symbolic addition procedure for phase determination for centrosymmetric and non-centrosymmetric crystals. Acta Cryst. 21, 849–859. Klug, A. (1958). Joint probability distributions of structure factors and the phase problem. Acta Cryst. 11, 515–543.

242

2.2. DIRECT METHODS Pavelcı´k, F., Kuchta, L. & Sivy´, J. (1992). Patterson-oriented automatic structure determination. Utilizing Patterson peaks. Acta Cryst. A48, 791–796. Peerdeman, A. F. & Bijvoet, J. M. (1956). The indexing of reflexions in investigations involving the use of the anomalous scattering effect. Acta Cryst. 9, 1012–1015. Perutz, M. F. (1956). Isomorphous replacement and phase determination in non-centrosymmetric space groups. Acta Cryst. 9, 867–873. Phillips, D. C. (1966). Advances in protein crystallography. In Advances in Structure Research by Diffraction Methods, Vol. 2, edited by R. Brill & R. Mason, pp. 75–140. New York: John Wiley. Piro, O. E. (1983). Information theory and the phase problem in crystallography. Acta Cryst. A39, 61–68. Podjarny, A. D., Schevitz, R. W. & Sigler, P. B. (1981). Phasing lowresolution macromolecular structure factors by matricial direct methods. Acta Cryst. A37, 662–668. Podjarny, A. D., Yonath, A. & Traub, W. (1976). Application of multivariate distribution theory to phase extension for a crystalline protein. Acta Cryst. A32, 281–292. Rae, A. D. (1977). The use of structure factors to find the origin of an oriented molecular fragment. Acta Cryst. A33, 423–425. Ramachandran, G. N. & Raman, S. (1956). A new method for the structure analysis of non-centrosymmetric crystals. Curr. Sci. (India), 25, 348. Rango, C. de (1969). Thesis. Paris. Rango, C. de, Mauguen, Y. & Tsoucaris, G. (1975). Use of high-order probability laws in phase refinement and extension of protein structures. Acta Cryst. A31, 227–233. Rango, C. de, Mauguen, Y., Tsoucaris, G., Dodson, E. J., Dodson, G. G. & ˚ spacing Taylor, D. J. (1985). The extension and refinement of the 1.9 A ˚ spacing in 2Zn insulin by determinantal isomorphous phases to 1.5 A methods. Acta Cryst. A41, 3–17. Rango, C. de, Tsoucaris, G. & Zelwer, C. (1974). Phase determination from the Karle–Hauptman determinant. II. Connexion between inequalities and probabilities. Acta Cryst. A30, 342–353. Richardson, J. W. & Jacobson, R. A. (1987). Patterson and Pattersons, edited by J. P. Glusker, B. K. Patterson & M. Rossi, pp. 310–317. Oxford University Press. Rogers, D., Stanley, E. & Wilson, A. J. C. (1955). The probability distribution of intensities. VI. The influence of intensity errors on the statistical tests. Acta Cryst. 8, 383–393. Rogers, D. & Wilson, A. J. C. (1953). The probability distribution of X-ray intensities. V. A note on some hypersymmetric distributions. Acta Cryst. 6, 439–449. Rossmann, M. G. (1960). The accurate determination of the position and shape of heavy-atom replacement groups in proteins. Acta Cryst. 13, 221–226. Rossmann, M. G. (1961). The position of anomalous scatterers in protein crystals. Acta Cryst. 14, 383–388. Rossmann, M. G., Blow, D. M., Harding, M. M. & Coller, E. (1964). The relative positions of independent molecules within the same asymmetric unit. Acta Cryst. 17, 338–342. Sayre, D. (1952). The squaring method: a new method for phase determination. Acta Cryst. 5, 60–65. Sayre, D. (1953). Double Patterson function. Acta Cryst. 6, 430–431. Sayre, D. (1972). On least-squares refinement of the phases of crystallographic structure factors. Acta Cryst. A28, 210–212. Sayre, D. & Toupin, R. (1975). Major increase in speed of least-squares phase refinement. Acta Cryst. A31, S20. Schenk, H. (1973a). Direct structure determination in P1 and other non-centrosymmetric symmorphic space groups. Acta Cryst. A29, 480–481. Schenk, H. (1973b). The use of phase relationships between quartets of reflexions. Acta Cryst. A29, 77–82. Schneider, T. R. & Sheldrick, G. M. (2002). Substructure solution with SHELXD. Acta Cryst. D58, 1772–1779. Sheldrick, G. M. (1990). Phase annealing in SHELX-90: direct methods for larger structures. Acta Cryst. A46, 467–473. Sheldrick, G. M. (1992). Crystallographic Computing 5, edited by D. Moras, A. D. Podjarny & J. C. Thierry, pp. 145–157. Oxford University Press.

Sheldrick, G. M. (1998). SHELX: applications to macromolecules. In Direct Methods for Solving Macromolecular Structures, edited by S. Fortier, pp. 401–411. Dordrecht: Kluwer Academic Publishers. Sheldrick, G. M. (2000). The SHELX home page. http://shelx.uni-ac. gwdg.de/SHELX/. Sheldrick, G. M. & Gould, R. O. (1995). Structure solution by iterative peaklist optimization and tangent expansion in space group P1. Acta Cryst. B51, 423–431. Sim, G. A. (1959). The distribution of phase angles for structures containing heavy atoms. II. A modification of the normal heavy-atoms method for non-centrosymmetrical structures. Acta Cryst. 12, 813–815. Simerska, M. (1956). Czech. J. Phys. 6, 1. Simonov, V. I. & Weissberg, A. M. (1970). Calculation of the signs of structure amplitudes by a binary function section of interatomic vectors. Sov. Phys. Dokl. 15, 321–323. Sint, L. & Schenk, H. (1975). Phase extension and refinement in noncentrosymmetric structures containing large molecules. Acta Cryst. A31, S22. Srinivasan, R. & Parthasarathy, S. (1976). Some Statistical Applications in X-ray Crystallography. Oxford: Pergamon Press. Taylor, D. J., Woolfson, M. M. & Main, P. (1978). On the application of phase relationships to complex structures. XV. Magic determinants. Acta Cryst. A34, 870–883. Terwilliger, T. C. (1994). MAD phasing: Bayesian estimates of FA. Acta Cryst. D50, 11–16. Terwilliger, T. C. & Eisenberg, D. (1987). Isomorphous replecement: effect of errors on the phase probability distribution. Acta Cryst. A43, 6–13. Tsoucaris, G. (1970). A new method for phase determination. The maximum determinant rule. Acta Cryst. A26, 492–499. Van der Putten, N. & Schenk, H. (1977). On the conditional probability of quintets. Acta Cryst. A33, 856–858. Vaughan, P. A. (1958). A phase-determining procedure related to the vector-coincidence method. Acta Cryst. 11, 111–115. Vermin, W. J. & de Graaff, R. A. G. (1978). The use of Karle–Hauptman determinants in small-structure determinations. Acta Cryst. A34, 892– 894. Vickovic´, I. & Viterbo, D. (1979). A simple statistical treatment of unobserved reflexions. Application to two organic substances. Acta Cryst. A35, 500–501. Weeks, C. M., DeTitta, G. T., Hauptman, H. A., Thuman, P. & Miller, R. (1994). Structure solution by minimal-function phase refinement and Fourier filtering. II. Implementation and applications. Acta Cryst. A50, 210–220. Weeks, C. M. & Miller, R. (1999). The design and implementation of SnB version 2.0. J. Appl. Cryst. 32, 120–124. Weinzierl, J. E., Eisenberg, D. & Dickerson, R. E. (1969). Refinement of protein phases with the Karle–Hauptman tangent fomula. Acta Cryst. B25, 380–387. White, P. & Woolfson, M. M. (1975). The application of phase relationships to complex structures. VII. Magic integers. Acta Cryst. A31, 53–56. Wilkins, S. W., Varghese, J. N. & Lehmann, M. S. (1983). Statistical geometry. I. A self-consistent approach to the crystallographic inversion problem based on information theory. Acta Cryst. A39, 47–60. Wilson, A. J. C. (1942). Determination of absolute from relative X-ray intensity data. Nature (London), 150, 151–152. Wilson, K. S. (1978). The application of MULTAN to the analysis of isomorphous derivatives in protein crystallography. Acta Cryst. B34, 1599–1608. Wolff, P. M. de & Bouman, J. (1954). A fundamental set of structure factor inequalities. Acta Cryst. 7, 328–333. Woolfson, M. M. (1958). Crystal and molecular structure of p,p0 dimethoxybenzophenone by the direct probability method. Acta Cryst. 11, 277–283. Woolfson, M. M. (1977). On the application of phase relationships to complex structures. X. MAGLIN – a successor to MULTAN. Acta Cryst. A33, 219–225. Yao, J.-X. (1981). On the application of phase relationships to complex structures. XVIII. RANTAN – random MULTAN. Acta Cryst. A37, 642–664.

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International Tables for Crystallography (2010). Vol. B, Chapter 2.3, pp. 244–281.

2.3. Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing By L. Tong, M. G. Rossmann and E. Arnold

space group to the Laue symmetry is produced by the translation of all vectors to the Patterson origin and the introduction of a centre of symmetry. The latter is a consequence of the relationship between the vectors AB and BA. The Patterson symmetries for all 230 space groups are tabulated in IT A (2005). An analysis of Patterson peaks can be obtained by considering N atoms with form factors fi in the unit cell. Then

2.3.1. Introduction 2.3.1.1. Background Historically, the Patterson has been used in a variety of ways to effect the solutions of crystal structures. While some simple structures (Ketelaar & de Vries, 1939; Hughes, 1940; Speakman, 1949; Shoemaker et al., 1950) were solved by direct analysis of Patterson syntheses, alternative methods have largely superseded this procedure. An early innovation was the heavy-atom method which depends on the location of a small number of relatively strong scatterers (Harker, 1936). Image-seeking methods and Patterson superposition techniques were first contemplated in the late 1930s (Wrinch, 1939) and applied sometime later (Beevers & Robertson, 1950; Clastre & Gay, 1950; Garrido, 1950a; Buerger, 1959). This experience provided the encouragement for computerized vector-search methods to locate individual atoms automatically (Mighell & Jacobson, 1963; Kraut, 1961; Hamilton, 1965; Simpson et al., 1965) or to position known molecular fragments in unknown crystal structures (Nordman & Nakatsu, 1963; Huber, 1965). The Patterson function has been used extensively in conjunction with the isomorphous replacement method (Rossmann, 1960; Blow, 1958) or anomalous dispersion (Rossmann, 1961a) to determine the position of heavy-atom substitution. Pattersons have been used to detect the presence and relative orientation of multiple copies of a given chemical motif in the crystallographic asymmetric unit in the same or different crystals (Rossmann & Blow, 1962). Finally, the orientation and placement of known molecular structures (‘molecular replacement’) into unknown crystal structures can be accomplished via Patterson techniques. The function, introduced by Patterson in 1934 (Patterson, 1934a,b), is a convolution of electron density with itself and may be defined as R PðuÞ ¼
ðxÞ

ðu þ xÞ dx;

Fh ¼

Using Friedel’s law, jFh j2 ¼ Fh
Fh #
N " N P P ¼ fi expð2ih
xi Þ fj expð2ih
xj Þ ; i¼1

j¼1

which can be decomposed to jFh j2 ¼

N P

fi2 þ

i¼1

N P N P

fi fj exp½2ih
ðxi  xj Þ:

ð2:3:1:3Þ

i6¼j

On substituting (2.3.1.3) in (2.3.1.2), we see that the Patterson consists of the sum of N 2 total interactions of which N are of weight fi2 at the origin and NðN  1Þ are of weight fi fj at xi  xj . The weight of a peak in a real cell is given by R wi ¼
i ðxÞ dx ¼ Zi ðthe atomic numberÞ; U

ð2:3:1:1Þ where U is the volume of the atom i. By analogy, the weight of a peak in a Patterson (form factor fi fj ) will be given by

where PðuÞ is the ‘Patterson’ function at u,
ðxÞ is the crystal’s periodic electron density and V is the volume of the unit cell. The Patterson function, or F 2 series, can be calculated directly from the experimentally derived X-ray intensities as hemisphere X

fi expð2ih
xi Þ:

i¼1

V

2 PðuÞ ¼ 2 V

N P

2

jFh j cos 2h
u:

R wij ¼ Pij ðuÞ du ¼ Zi Zj : U

Although the maximum height of a peak will depend on the spread of the peak, it is reasonable to assume that heights of peaks in a Patterson are proportional to the products of the atomic numbers of the interacting atoms. There are a total of N 2 interactions in a Patterson due to N atoms in the crystal cell. These can be represented as an N  N square matrix whose elements uij, wij indicate the position and weight of the peak produced between atoms i and j (Table 2.3.1.1). The N vectors corresponding to the diagonal of this matrix are located at the Patterson origin and arise from the convolution of each atom with itself. This leaves NðN  1Þ vectors whose locations depend on the relative positions of all of the atoms in the crystal cell and whose weights depend on the atom types related by the vector. Complete specification of the unique non-origin Patterson vectors requires description of only the NðN  1Þ=2 elements in either the upper or the lower triangle of this matrix, since the two sets of vectors represented by the two triangles are related by a centre of symmetry

ð2:3:1:2Þ

h

The derivation of (2.3.1.2) from (2.3.1.1) can be found in this volume (see Section 1.3.4.2.1.6) along with a discussion of the physical significance and symmetry of the Patterson function, although the principal properties will be restated here. The Patterson can be considered to be a vector map of all the pairwise interactions between the atoms in a unit cell. The vectors in a Patterson correspond to vectors in the real (direct) crystal cell but translated to the Patterson origin. Their weights are proportional to the product of densities at the tips of the vectors in the real cell. The Patterson unit cell has the same size as the real crystal cell. The symmetry of the Patterson comprises the Laue point group of the crystal cell plus any additional lattice symmetry due to Bravais centring. The reduction of the real Copyright © 2010 International Union of Crystallography

244

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES for example, Jacobson et al. (1961), Braun et al. (1969) and Nordman (1980a)]. Since Patterson’s original work, other workers have suggested that the Patterson function itself might be modified; Fourier inversion of the modified Patterson then provides a new and perhaps more tractable set of structure factors (McLachlan & Harker, 1951; Simonov, 1965; Raman, 1966; Corfield & Rosenstein, 1966). Karle & Hauptman (1964) suggested that an improved set of structure factors could be obtained from an origin-removed Patterson modified such that it was everywhere non-negative and that Patterson density values less than a bonding distance from the origin were set to zero. Nixon (1978) was successful in solving a structure which had previously resisted solution by using a set of structure factors which had been obtained from a Patterson in which the largest peaks had been attenuated. The N origin peaks [see expression (2.3.1.3)] may be removed from the Patterson by using coefficients

Table 2.3.1.1. Matrix representation of Patterson peaks The N  N matrix represents the position uij and weights wij of atomic interactions in a Patterson arising from N atoms at xi and weight wi in the real cell. x1 ; w1

x2 ; w2

...

xN ; wN

u11 ¼ x1  x1 ,

u12 ¼ x1  x2 ,

...

u1N ¼ x1  xN ,

w11 ¼ w21

w12 ¼ w1 w2

x2 ; w2 .. .

x2  x1 ; w2 w1 .. .

0, w22 .. .

... .. .

x2  xN ; w2 wN .. .

xN ; wN

xN  x1 ; wN w1

xN  x2 ; wN w2

...

0, w2N

x1 ; w1

w1N ¼ w1 wN

½uij  xi  xj ¼ uij  ðxj  xi Þ. Patterson vector positions are usually represented as huvwi, where u, v and w are expressed as fractions of the Patterson cell axes. 2.3.1.2. Limits to the number of resolved vectors If we assume a constant number of atoms per unit volume, the number of atoms N in a unit cell increases in direct proportion with the volume of the unit cell. Since the number of non-origin peaks in the Patterson function is NðN  1Þ and the Patterson cell is the same size as the real cell, the problem of overlapping peaks in the Patterson function becomes severe as N increases. To make matters worse, the breadth of a Patterson peak is roughly equal to the sum of the breadth of the original atoms. The effective width of a Patterson peak will also increase with increasing thermal motion, although this effect can be artificially reduced by sharpening techniques. Naturally, a loss of attainable resolution at high scattering angles will affect the resolution of atomic peaks in the real cell as well as peaks in the Patterson cell. If U is the van der Waals volume per average atom, then the fraction of the cell occupied by atoms will be f ¼ NU=V. Similarly, the fraction of the cell occupied by Patterson peaks will be 2UNðN  1Þ=V or 2f ðN  1Þ. With the reasonable assumption that f ’ 0:1 for a typical organic crystal, then the cell can contain at most five atoms ðN  5Þ for there to be no overlap, other than by coincidence, of the peaks in the Patterson. As N increases there will occur a background of peaks on which are superimposed features related to systematic properties of the structure. The contrast of selected Patterson peaks relative to the general background level can be enhanced by a variety of techniques. For instance, the presence of heavy atoms not only enhances the size of a relatively small number of peaks but ordinarily ensures a larger separation of the peaks due to the light-atom skeleton on which the heavy atoms are hung. That is, the factor f (above) is substantially reduced. Another example is the effect of systematic atomic arrangements (e.g. -helices or aromatic rings) resulting in multiple peaks which stand out above the background. In the isomorphous replacement method, isomorphous difference Pattersons are utilized in which the contrast of the Patterson interactions between the heavy atoms is enhanced by removal of the predominant interactions which involve the rest of the structure.

jFh; mod j2 ¼ jFh j2 

N P

fi2 :

i¼1

A Patterson function using these modified coefficients will retain all interatomic vectors. However, the observed structure factors Fh must first be placed on an absolute scale (Wilson, 1942) in order to match the scattering-factor term. In practice, Patterson origins can also be removed by using coefficients       Fh;mod 2 ¼ Fh 2 hFh 2 i;  2 where hFh  i is the average reflection intensity, usually calculated in several resolution shells. This formula has the advantage that the observed structure factors do not need to be on absolute scale. Analogous to origin removal, the vector interactions due to atoms in known positions can also be removed from the Patterson function. Patterson showed that non-origin Patterson peaks arising from known atoms 1 and 2 may be removed by using the expression jFh; mod j2 ¼ jFh j2 

N P

fi2 ti2  2f1 f2 t1 t2 cos 2h
ðx1  x2 Þ;

i¼1

where x1 and x2 are the positions of atoms 1 and 2 and t1 and t2 are their respective thermal correction factors. Using onedimensional Fourier series, Patterson illustrated how interactions due to known atoms can obscure other information. Patterson also introduced a means by which the peaks in a Patterson function may be artificially sharpened. He considered the effect of thermal motion on the broadening of electrondensity peaks and consequently their Patterson peaks. He suggested that the F 2 coefficients could be corrected for thermal effects by simulating the atoms as point scatterers and proposed using a modified set of coefficients

2.3.1.3. Modifications: origin removal, sharpening etc. A. L. Patterson, in his first in-depth exposition of his newly discovered F 2 series (Patterson, 1935), introduced the major modifications to the Patterson which are still in use today. He illustrated, with one-dimensional Fourier series, the techniques of removing the Patterson origin peak, sharpening the overall function and also removing peaks due to atoms in special positions. Each one of these modifications can improve the interpretability of Pattersons, especially those of simple structures. Whereas the recommended extent of such modifications is controversial (Buerger, 1966), most studies which utilize Patterson functions do incorporate some of these techniques [see,

jFh; sharp j2 ¼ jFh j2 =f
2 ; where f
, the average scattering factor per electron, is given by f
¼

N P i¼1

245

fi

N P i¼1

Zi :

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.3.1.2. (c) The point Patterson of the two homometric structures in (a) and (b). The latter are constructed by taking points at a and 12 M0 , where M0 is a cell diagonal, and adding a third point which is (a) at 34 M0 þ a or (b) at 1 0 4 M þ a. [Reprinted with permission from Patterson (1944).]

Fig. 2.3.1.1. Effect of ‘sharpening’ Patterson coefficients. (1) shows a mean distribution of jFj2 values with resolution, ðsin Þ=. The normal decline of this curve is due to increasing destructive interference between different portions within diffuse atoms at larger Bragg angles. (2) shows the distribution of ‘sharpened’ coefficients. (3) shows the theoretical distribution of jFj2 produced by a point-atom structure. To represent such a structure with a Fourier series would require an infinite series in order to avoid large errors caused by truncation.

of k was empirically chosen as 23). This approach was subsequently further developed and generalized by Wunderlich (1965). 2.3.1.4. Homometric structures and the uniqueness of structure solutions; enantiomorphic solutions

A common formulation for this type of sharpening expresses the atomic scattering factors at a given angle in terms of an overall isotropic thermal parameter B as

Interpretation of any Patterson requires some assumption, such as the existence of discrete atoms. A complete interpretation might also require an assumption of the number of atoms and may require other external information (e.g. bond lengths, bond angles, van der Waals separations, hydrogen bonding, positive density etc.). To what extent is the solution of a Patterson function unique? Clearly the greater is the supply of external information, the fewer will be the number of possible solutions. Other constraints on the significance of a Patterson include the error involved in measuring the observed coefficients and the resolution limit to which they have been observed. The larger the error, the larger the number of solutions. When the error on the amplitudes is infinite, it is only the other physical constraints, such as packing, which limit the structural solutions. Alternative solutions of the same Patterson are known as homometric structures. During their investigation of the mineral bixbyite, Pauling & Shappell (1930) made the disturbing observation that there were two solutions to the structure, with different arrangements of atoms, which yielded the same set of calculated intensities. Specifically, atoms occupying equipoint set 24d in space group Ið21 =aÞ3
can be placed at either x; 0; 14 or x; 0; 14 without changing the calculated intensities. Yet the two structures were not chemically equivalent. These authors resolved the ambiguity by placing the oxygen atoms in question at positions which gave the most acceptable bonding distances with the rest of the structure. Patterson interpreted the above ambiguity in terms of the F 2 series: the distance vector sets or Patterson functions of the two structures were the same since each yielded the same calculated intensities (Patterson, 1939). He defined such a pair of structures a homometric pair and called the degenerate vector set which they produced a homometric set. Patterson went on to investigate the likelihood of occurrence of homometric structures and, indeed, devoted a great deal of his time to this matter. He also developed algebraic formalisms for examining the occurrence of homometric pairs and multiplets in selected one-dimensional sets of points, such as cyclotomic sets, and also sets of points along a line (Patterson, 1944). Some simple homometric pairs are illustrated in Fig. 2.3.1.2. Drawing heavily from Patterson’s inquiries into the structural uniqueness allowed by the diffraction data, Hosemann, Bagchi and others have given formal definitions of the different types of homometric structures (Hosemann & Bagchi, 1954). They suggested a classification divided into pseudohomometric structures and homomorphs, and used an integral equation representing a convolution operation to express their examples of finite homometric structures. Other workers have chosen various

f ðsÞ ¼ f0 expðBs2 Þ: The Patterson coefficients then become Z Fh; sharp ¼ PNtotal Fh : i¼l f ðsÞ The normalized structure factors, E (see Chapter 2.2), which are used in crystallographic direct methods, are also a common source of sharpened Patterson coefficients ðE2  1Þ. Although the centre positions and total contents of Patterson peaks are unaltered by sharpening, the resolution of individual peaks may be enhanced. The degree of sharpening can be controlled by adjusting the size of the assumed B factor; Lipson & Cochran (1966, pp. 165–170) analysed the effect of such a choice on Patterson peak shape. All methods of sharpening Patterson coefficients aim at producing a point atomic representation of the unit cell. In this quest, the high-resolution terms are enhanced (Fig. 2.3.1.1). Unfortunately, this procedure must also produce a serious Fourier truncation error which will be seen as large ripples about each Patterson peak (Gibbs, 1898). Consequently, various techniques have been used (mostly unsuccessfully) in an attempt to balance the advantages of sharpening with the disadvantages of truncation errors. Schomaker and Shoemaker [unpublished; see Lipson & Cochran (1966, p. 168)] used a function jFh; sharp j2 ¼


 jFh j2 2 2 2 s ; s exp  p f
2

in which s is the length of the scattering vector, to produce a Patterson synthesis which is less sensitive to errors in the loworder terms. Jacobson et al. (1961) used a similar function, jFh; sharp j2 ¼


 jFh j2  2 2 s ; ðk þ s Þ exp  p f
2

which they rationalize as the sum of a scaled exponentially sharpened Patterson and a gradient Patterson function (the value

246

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES

Fig. 2.3.2.1. Origin selection in the interpretation of a Patterson of a onedimensional centrosymmetric structure.

means for describing homometric structures [Buerger (1959, pp. 41–50), Menzer (1949), Bullough (1961, 1964), Hoppe (1962)]. Since a Patterson function is centrosymmetric, the Pattersons of a crystal structure and of its mirror image are identical. Thus the enantiomeric ambiguity present in noncentrosymmetric crystal structures cannot be overcome by using the Patterson alone and represents a special case of homometric structures. Assignment of the correct enantiomorph in a crystal structure analysis is generally not possible unless a recognizable fragment of known chirality emerges (e.g. l-amino acids in proteins, d-riboses in nucleic acids, the known framework of steroids and other natural products, the right-handed twist of -helices, the left-handed twist of successive strands in a -sheet, the fold of a known protein subunit etc.) or anomalous-scattering information is available and can be used to resolve the ambiguity (Bijvoet et al., 1951). It is frequently necessary to select arbitrarily one enantiomorph over another in the early stages of a structure solution. Structure-factor phases calculated from a single heavy atom in space group P1, P2 or P21 (for instance) will be centrosymmetric and both enantiomorphs will be present in Fourier calculations based on these phases. In other space groups (e.g. P21 21 21 ), the selected heavy atom is likely to be near one of the planes containing the 21 axes and thus produce a weaker ‘ghost’ image of the opposite enantiomorph. The mixture of the two overlapped enantiomorphic solutions can cause interpretive difficulties. As the structure solution progresses, the ‘ghosts’ are exorcized owing to the dominance of the chosen enantiomorph in the phases.

Fig. 2.3.2.2. The c-axis projection of cuprous chloride azomethane complex (C2H6Cl2Cu2N2). The space group is P1
with one molecule per unit cell. [Adapted from and reprinted with permission from Woolfson (1970, p. 321).]

height to the product of the electrons in each atom. Although this function has not been found very useful in practice, it is useful for demonstrating the presence of weak enantiomorphic images in a given tentative structure determination. 2.3.2. Interpretation of Patterson maps 2.3.2.1. Simple solutions in the triclinic cell. Selection of the origin A hypothetical one-dimensional centrosymmetric crystal structure containing an atom at x and at x and the corresponding Patterson is illustrated in Fig. 2.3.2.1. There are two different centres of symmetry which may be chosen as convenient origins. If the atoms are of equal weight, we expect Patterson vectors at positions u ¼ 2x with weights equal to half the origin peak. There are two symmetry-related peaks, u1 and u2 (Fig. 2.3.2.1) in the Patterson. It is an arbitrary choice whether u1 ¼ 2x or u2 ¼ 2x. This choice is equivalent to selecting the origin at the centre of symmetry I or II in the real structure (Fig. 2.3.2.1). Similarly in a three-dimensional P1
cell, the Patterson will contain peaks at huvwi which can be used to solve for the atom coordinates h2x; 2y; 2zi. Solving for the same coordinates by starting from symmetric representations of the same vector will lead to alternate origin choices. For example, use of h1 þ u; 1 þ v; wi will lead to translating the origin by ðþ 12 ; þ 12 ; 0Þ relative to the solution based on huvwi. There are eight distinct inversion centres in P1
, each one of which represents a valid origin choice. Although any choice of origin would be allowable, an inversion centre is convenient because then the structure factors are all real. Typically, one of the vector peaks closest to the Patterson origin is selected to start the solution, usually in the calculated asymmetric unit of the Patterson. Care must be exercised in selecting the same origin for all atomic positions by considering cross-vectors between atoms. Examine, for example, the c-axis Patterson projection of a cuprous chloride azomethane complex (C2H6Cl2Cu2N2) in P1
as shown in Fig. 2.3.2.2. The largest Patterson peaks should correspond to vectors arising from Cu ðZ ¼ 29Þ and Cl ðZ ¼ 17Þ atoms. There will be copper atoms at xCu ðxCu ; yCu Þ and xCu ðxCu ; yCu Þ as well as chlorine atoms at analogous positions. The interaction matrix is

2.3.1.5. The Patterson synthesis of the second kind Patterson also defined a second, less well known, function (Patterson, 1949) as Z P ðuÞ ¼ ¼


ðu þ xÞ

ðu  xÞ dx 2 V2

hemisphere X

Fh2 cosð22h
u  2h Þ:

h

This function can be computed directly only for centrosymmetric structures. It can be calculated for noncentrosymmetric structures when the phase angles are known or assumed. It will represent the degree to which the known or assumed structure has a centre of symmetry at u. That is, the product of the density at u þ x and u  x is large when integrated over all values x within the unit cell. Since atoms themselves have a centre of symmetry, the function will contain peaks at each atomic site roughly proportional in height to the square of the number of electrons in each atom plus peaks at the midpoint between atoms proportional in

xCu ; 29 xCl ; 17 xCu ; 29 xCl ; 17

247

xCu ; 29

xCl ; 17

xCu ; 29

xCl ; 17

0; 841

xCu  xCl ; 493 0; 289

2xCu ; 841 xCl þ xCu ; 493

xCu þ xCl ; 493 2xCl ; 289

0; 841

xCu  xCl ; 493 0; 289

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.3.2.1. Coordinates of Patterson peaks for C2H6Cl2Cu2N2 projection Height

u

v

Number in diagram (Fig. 2.3.2.2)

7

0.33

0.34

I

7

0.18

0.97

II

6

0.16

0.40

III

3

0.49

0.29

IV

3 2

0.02 0.30

0.59 0.75

V VI

2

0.12

0.79

VII

xmn ¼ ½T m x1n þ tm ; where ½T m  and tm are the rotation matrix and translation vector, respectively, for the mth crystallographic symmetry operator. The Patterson of this crystal will contain vector peaks which arise from atoms interacting with other atoms both in the same and in different crystallographic asymmetric units. The set of ðMNÞ2 Patterson vector interactions for this crystal is represented in a matrix in Table 2.3.2.2. Upon dissection of this diagram we see that there are MN origin vectors, M½ðN  1ÞN vectors from atom interactions with other atoms in the same crystallographic asymmetric unit and ½MðM  1ÞN 2 vectors involving atoms in separate asymmetric units. Often a number of vectors of special significance relating symmetry-equivalent atoms emerge from this milieu of Patterson vectors and such ‘Harker vectors’ constitute the subject of the next section.

which shows that the Patterson should contain the following types of vectors:

Position 2xCu 2xCl xCu  xCl xCu þ xCl

Weight 841 289 493 493

Multiplicity 1 1 2 2

Total weight 841 289 986 986

2.3.2.2. Harker sections Soon after Patterson introduced the F 2 series, Harker (1936) recognized that many types of crystallographic symmetry result in a concentration of vectors at characteristic locations in the Patterson. Specifically, he showed that atoms related by rotation axes produce vectors in characteristic planes of the Patterson, and that atoms related by mirror planes or reflection glide planes produce vectors on characteristic lines. Similarly, noncrystallographic symmetry operators produce analogous concentrations of vectors. Harker showed how special sections through a threedimensional function could be computed using one- or twodimensional summations. With the advent of powerful computers, it is usual to calculate a full three-dimensional Patterson synthesis. Nevertheless, ‘Harker’ planes or lines are often the starting point for a structure determination. It should, however, be noted that non-Harker vectors (those not due to interactions between symmetry-related atoms) can appear by coincidence in a Harker section. Table 2.3.2.3 shows the position in a Patterson of Harker planes and lines produced by all types of crystallographic symmetry operators. Buerger (1946) noted that Harker sections can be helpful in space-group determination. Concentrations of vectors in appropriate regions of the Patterson should be diagnostic for the presence of some symmetry elements. This is particularly useful where these elements (such as mirror planes) are not directly detected by systematic absences. Buerger also developed a systematic method of interpreting Harker peaks which he called implication theory [Buerger (1959, Chapter 7)].

The coordinates of the largest Patterson peaks are given in Table 2.3.2.1 for an asymmetric half of the cell chosen to span 0 ! 12 in u and 0 ! 1 in v. Since the three largest peaks are in the same ratio (7:7:6) as the three largest expected vector types (986:986:841), it is reasonable to assume that peak III corresponds to the copper–copper interaction at 2xCu . Hence, xCu ¼ 0:08 and yCu ¼ 0:20. Peaks I and II should be due to the double-weight Cu–Cl vectors at xCu  xCl and xCu þ xCl . Now suppose that peak I is at position xCu þ xCl , then xCl ¼ 0:25 and yCl ¼ 0:14. Peak II should now check out as the remaining double-weight Cu–Cl interaction at xCu  xCl . Indeed, xCu  xCl ¼ h0:17; 0:06i ¼ h0:17; 0:06i which agrees tolerably well with the position of peak II. The chlorine position also predicts the position of a peak at 2xCl with weight 289; peak IV confirms the chlorine assignment. In fact, this Patterson can be solved also for the lighter nitrogen- and carbon-atom positions which account for the remainder of the vectors listed in Table 2.3.2.1. However, the simplest way to complete the structure determination is probably to compute a Fourier synthesis using phases calculated from the heavier copper and chlorine positions. Consider now a real cell with M crystallographic asymmetric units, each of which contains N atoms. Let us define xmn, the position of the nth atom in the mth crystallographic unit, by

Table 2.3.2.2. Square matrix representation of vector interactions in a Patterson of a crystal with M crystallographic asymmetric units each containing N atoms Peak positions um1n1; m2n2 correspond to vectors between the atoms xm1n1 and xm2n2 where xmn is the nth atom in the mth crystallographic asymmetric unit. The corresponding weights are wn1 wn2 . The outlined blocks I1 and IM represent vector interactions between atoms in the same crystallographic asymmetric units (there are M such blocks). The off-diagonal blocks IIM1 and II1M represent vector interactions between atoms in crystal asymmetric units 1 and M; there are MðM  1Þ blocks of this type. The significance of diagonal elements of block IIM1 is that they represent Harker-type interactions between symmetry-equivalent atoms (see Section 2.3.2.2).

x11 ; w1

x11 ; w1

x12 ; w2

...

x1N ; wN

0, w21

u11; 12 ; w1 w2

...

u11; 1N ; w1 wN

0, w22 .. .

...

u12; 1N ; w2 wN .. .

x12 ; w2 .. .

xM1 ; w1

xM2 ; w2

...

0, w2N

x1N ; wN .. .

...

Block I1

.. .

xM1 ; w1

uM1; 11 ; w21

uM1; 12 ; w1 w2

xM2 ; w2 .. .

uM2; 11 ; w2 w1

uM2; 12 ; w22 .. .

..

Block II1M .

... uMN; 1N ; w2N

xMN ; wN Block IIM1

Block IM

248

.. .

...

xMN ; wN

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES A general strategy for determining heavy atoms from the Patterson usually involves the following steps. (1) List the number and type of atoms in the cell. (2) Construct the interaction matrix for the heaviest atoms to predict the positions and weights of the largest Patterson vectors. Group recurrent vectors and notice vectors with special properties, such as Harker vectors. (3) Compute the Patterson using any desiredPmodifications. Placing the map on an absolute scale ½Pð000Þ ¼ Z2  is convenient but not necessary. (4) Examine Harker sections and derive trial atom coordinates from vector positions. (5) Check the trial coordinates using other vectors in the predicted set. Correlate enantiomorphic choice and origin choice for independent sites. (6) Include the next-heaviest atoms in the interpretation if possible. In particular, use the cross-vectors with the heaviest atoms. (7) Use the best heavy-atom model to initiate phasing. Detailed and instructive examples of using Pattersons to find heavy-atom positions are found in almost every textbook on crystal structure analysis [see, for example, Buerger (1959), Lipson & Cochran (1966) and Stout & Jensen (1968)]. The determination of the crystal structure of cholesteryl iodide by Carlisle & Crowfoot (1945) provides an example of using the Patterson function to locate heavy atoms. There were two molecules, each of formula C27H45I, in the P21 unit cell. The ratio r ¼ 2:8 is clearly well over the optimal value of unity. The P(x, z) Patterson projection showed one dominant peak at h0:434; 0:084i in the asymmetric unit. The equivalent positions for P21 require that an iodine atom at xI , yI , zI generates another at xI ; 12 þ yI ; zI and thus produces a Patterson peak at h2xI ; 12 ; 2zI i. The iodine position was therefore determined as 0.217, 0.042. The y coordinate of the iodine is arbitrary for P21 yet the value of yI ¼ 0:25 is convenient, since an inversion centre in the two-atom iodine structure is then exactly at the origin, making all calculated phases 0 or . Although the presence of this extra symmetry caused some initial difficulties in the interpretation of the steroid backbone, Carlisle and Crowfoot successfully separated the enantiomorphic images. Owing to the presence of the perhaps too heavy iodine atom, however, the structure of the carbon skeleton could not be defined very precisely. Nevertheless, all critical stereochemical details were adequately illuminated by this determination. In the cholesteryl iodide example, a number of different yet equivalent origins could have been selected. Alternative origin choices include all combinations of x  12 and z  12. A further example of using the Patterson to find heavy atoms will be provided in Section 2.3.5.2 on solving for heavy atoms in the presence of noncrystallographic symmetry.

Table 2.3.2.3. Position of Harker sections within a Patterson Symmetry element

Form of Pðx; y; zÞ

(a) Harker planes Axes parallel to the b axis: (i) 2

Pðx; 0; zÞ

(ii) 21

Pðx; 12 ; zÞ

Axes parallel to the c axis: (i) 2, 3, 3
, 4, 4
, 6, 6
(ii) 21 , 42 , 63

Pðx; y; 0Þ Pðx; y; 12Þ

(iii) 31 , 32 , 62 , 64

Pðx; y; 13Þ

(iv) 41 , 43

Pðx; y; 14Þ

(v) 61 , 65

Pðx; y; 16Þ

(b) Harker lines Planes perpendicular to the b axis: (i) Reflection planes

Pð0; y; 0Þ

(ii) Glide plane, glide ¼ 12 a

Pð12 ; y; 0Þ

(iii) Glide plane, glide ¼ 12 c

Pð0; y; 12Þ

(iv) Glide plane, glide ¼ 12 ða þ cÞ (v) Glide plane, glide ¼ 14 ða þ cÞ (vi) Glide plane, glide ¼ 14 ð3a þ cÞ

Pð12 ; y; 12Þ Pð14 ; y; 14Þ Pð34 ; y; 14Þ

(c) Special Harker planes Axes parallel to or containing body diagonal (111), valid for cubic space groups only:

(i)

3

(ii) 31

Equation of plane lx þ my þ nz  p ¼ 0 l ¼ m ¼ n ¼ cos 54:73561 ¼ 0:57735 p¼0 l¼m p¼ ffiffiffi n ¼ cos 54:73561 ¼ 0:57735 p ¼ 3=3

Rhombohedral threefold axes produce analogous Harker planes whose description will depend on the interaxial angle.

2.3.2.3. Finding heavy atoms The previous two sections have developed some of the useful mechanics for interpreting Pattersons. In this section, we will consider finding heavy-atom positions, in the presence of numerous light atoms, from Patterson maps. The feasibility of structure solution by the heavy-atom method depends on a number of factors which include the relative size of the heavy atom and the extent and quality of the data. A useful rule of thumb is that the ratio P 2 heavy Z r¼ P 2 light Z

2.3.2.4. Superposition methods. Image detection As early as 1939, Wrinch (1939) showed that it was possible, in principle, to recover a fundamental set of points from the vector map of that set. Unlike the Harker–Buerger implication theory (Buerger, 1946), the method that Wrinch suggested was capable of using all the vectors in a three-dimensional set, not those restricted to special lines or sections. Wrinch’s ideas were developed for vector sets of points, however, and could not be directly applied to real, heavily overlapped Pattersons of a complex structure. These ideas seem to have lain dormant until the early 1950s when a number of independent investigators developed superposition methods (Beevers & Robertson, 1950; Clastre & Gay, 1950; Garrido, 1950a; Buerger, 1950a). A Patterson can be considered as a sum of images of a molecule as seen, in turn, for each atom placed on the origin (Fig. 2.3.2.3). Thus, the deconvolution of a Patterson could proceed by superimposing each image of the molecule obtained onto the others by translating the Patterson origin to each imaging atom.

should be near unity if the heavy atom is to provide useful starting phase information (Z is the atomic number of an atom). The condition that r > 1 normally guarantees interpretability of the Patterson function in terms of the heavy-atom positions. This ‘rule’, arising from the work of Luzzati (1953), Woolfson (1956), Sim (1961) and others, is not inviolable; many ambitious determinations have been accomplished via the heavy-atom method for which r was well below 1.0. An outstanding example is vitamin B12 with formula C62H88CoO14P, which gave an r ¼ 0:14 for the cobalt atom alone (Hodgkin et al., 1957). One factor contributing to the success of such a determination is that the relative scattering power of Co is enhanced for higher scattering angles. Thus, the ratio, r, provides a conservative estimate. If the value of r is well above 1.0, the initial easier interpretation of the Patterson will come at the expense of poorly defined parameters of the lighter atoms.

249

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION m expði’h Þ ¼

N P

expð2ih
ui Þ

i¼1

(m and ’h can be calculated from the translational vectors used for the superposition), SðxÞ ¼

P

Fh2 m expð2ih
x þ ’h Þ:

h

Thus, the sum function is equivalent to a weighted ‘heavy atom’ method based on the known atoms assumed by the superposition translation vectors. The product function is somewhat more vigorous in that the images are enhanced by the product. If an image is superimposed on no image, then the product should be zero. The product function can be expressed as

Fig. 2.3.2.3. Atoms ABCD, arranged as a quadrilateral, generate a Patterson which is the sum of the images of the quadrilateral when each atom is placed on the origin in turn.

For instance, let us take a molecule consisting of four atoms ABCD arranged in the form of a quadrilateral (Fig. 2.3.2.3). Then the Patterson consists of the images of four identical quadrilaterals with atoms A, B, C and D placed on the origin in turn. The Pattersons can then be deconvoluted by superimposing two of these Pattersons after translating these (without rotation) by, for instance, the vector AB. A further improvement could be obtained by superimposing a third Patterson translated by AC. This would have the additional advantage in that ABC is a noncentrosymmetric arrangement and, therefore, selects the enantiomorph corresponding to the hand of the atomic arrangement ABC [cf. Buerger (1951, 1959)]. A basic problem is that knowledge of the vectors AB and AC also implies some knowledge of the structure at a time when the structure is not yet known. In practice ‘good-looking’ peaks, estimated to be single peaks by assessing the absolute scale of the structure amplitudes with Wilson statistics, can be assumed to be the result of single interatomic vectors within a molecule. Superposition can then proceed and the result can be inspected for reasonable chemical sense. As many apparently single peaks can be tried systematically using a computer, this technique is useful for selecting and testing a series of reasonable Patterson interpretations (Jacobson et al., 1961). Three major methods have been used for the detection of molecular images of superimposed Pattersons. These are the sum, product and minimum ‘image seeking’ functions (Raman & Lipscomb, 1961). The concept of the sum function is to add the images where they superimpose, whereas elsewhere the summed Pattersons will have a lower value owing to lack of image superposition. Therefore, the sum function determines the average Patterson density for superimposed images, and is represented analytically as

SðxÞ ¼

N P

PrðxÞ ¼

When N ¼ 2 (h and p are sets of Miller indices), PrðxÞ ¼

h




N P

Successive superpositions using the product functions will quickly be dominated by a few terms with very large coefficients. Finally, the minimum function is a clever invention of Buerger (Buerger, 1950b, 1951, 1953a,b,c; Taylor, 1953; Rogers, 1951). If a superposition is correct then each Patterson must represent an image of the structure. Whenever there are two or more images that intersect in the Patterson, the Patterson density will be greater than a single image. When two different images are superimposed, it is a reasonable hope that at least one of these is a single image. Thus by taking the value of that Patterson which is the minimum, it should be possible to select a single image and eliminate noise from interfering images as far as possible. Although the minimum function is perhaps the most powerful algorithm for image selection of well sharpened Pattersons, it is not readily amenable to Fourier representation. The minimum function was conceived on the basis of selecting positive images on a near-zero background. If it were desired to select negative images [e.g. the ðF1  F2 Þ2 correlation function discussed in Section 2.3.3.4], then it would be necessary to use a maximum function. In fact, normally, an image has finite volume with varying density. Thus, some modification of the minimum function is necessary in those cases where the image is large compared to the volume of the unit cell, as in low-resolution protein structures (Rossmann, 1961b). Nordman (1966) used the average of the Patterson values of the lowest 10 to 20 per cent of the vectors in comparing Pattersons with hypothetical point Pattersons. A similar criterion was used by High & Kraut (1966). Image-seeking methods using Patterson superposition have been used extensively (Beevers & Robertson, 1950; Garrido, 1950b; Robertson, 1951). For a review the reader is referred to Vector Space (Buerger, 1959) and a paper by Fridrichsons & Mathieson (1962). However, with the advent of computerized direct methods (see Chapter 2.2), such techniques are no longer popular. Nevertheless, they provide the conceptual framework for the rotation and translation functions (see Sections 2.3.6 and 2.3.7).

 expð2ih
ui Þ

Fh2 Fp2 exp½2iðh þ pÞ
x

p

 exp½2iðh
ui þ p
ui Þ:

Pðx þ ui Þ;

Fh2 expð2ih
xÞ

PP h

where SðxÞ is the sum function at x given by the superposition of the ith Patterson translated by ui , or P

Pðx þ ui Þ:

i¼1

i¼1

SðxÞ ¼

N Q

:

i¼1

Setting

250

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES or more bits for single density values. Treatment of vector overlap is handled differently by different investigators and the choice will depend on the degree of overlapping (Nordman & Schilling, 1970; Nordman, 1972). General Gaussian multiplicity corrections can be employed to treat coincidental overlap of independent vectors in general positions and overlap which occurs for symmetric peaks in the vicinity of a special position or mirror plane in the Patterson (G. Kamer, S. Ramakumar & P. Argos, unpublished results; Rossmann et al., 1972). 2.3.3. Isomorphous replacement difference Pattersons 2.3.3.1. Introduction Fig. 2.3.3.1. Three different cases which can occur in the relation of the native, FN , and heavy-atom derivative, FNH , structure factors for centrosymmetric reflections. FN is assumed to have a phase of 0, although analogous diagrams could be drawn when FN has a phase of . The crossover situation in (c) is clearly rare if the heavy-atom substitution is small compared to the native molecule, and can in general be neglected.

One of the initial stages in the application of the isomorphous replacement method is the determination of heavy-atom positions. Indeed, this step of a structure determination can often be the most challenging. Not only may the number of heavy-atom sites be unknown, and have incomplete substitution, but the various isomorphous compounds may also lack isomorphism. To compound these problems, the error in the measurement of the isomorphous difference in structure amplitudes is often comparable to the differences themselves. Clearly, therefore, the ease with which a particular problem can be solved is closely correlated with the quality of the data-measuring procedure. The isomorphous replacement method was used incidentally by Bragg in the solution of NaCl and KCl. It was later formalized by J. M. Robertson in the analysis of phthalocyanine where the coordination centre could be Pt, Ni and other metals (Robertson, 1935, 1936; Robertson & Woodward, 1937). In this and similar cases, there was no difficulty in finding the heavy-atom positions. Not only were the heavy atoms frequently in special positions, but they dominated the total scattering effect. It was not until Perutz and his colleagues (Green et al., 1954; Bragg & Perutz, 1954) applied the technique to the solution of haemoglobin, a protein of 68 000 Da, that it was necessary to consider methods for detecting heavy atoms. The effect of a single heavy atom, even uranium, can only have a very marginal effect on the structure amplitudes of a crystalline macromolecule. Hence, techniques had to be developed which were dependent on the difference of the isomorphous structure amplitudes rather than on the solution of the Patterson of the heavy-atom-derivative compound on its own.

2.3.2.5. Systematic computerized Patterson vector-search procedures. Looking for rigid bodies The power of the modern digital computer has enabled rapid access to the large number of Patterson density values which can serve as a lookup table for systematic vector-search procedures. In the late 1950s, investigators began to use systematic searches for the placement of single atoms, of known chemical groups or fragments and of entire known structures. Methods for locating single atoms were developed and called variously: vector verification (Mighell & Jacobson, 1963), symmetry minimum function (Kraut, 1961; Simpson et al., 1965; Corfield & Rosenstein, 1966) and consistency functions (Hamilton, 1965). Patterson superposition techniques using stored function values were often used to image the structure from the known portion. In such single-site search procedures, single atoms are placed at all possible positions in a crystal, using a search grid of the same fineness as for the stored Patterson function, preferably about one-third of the resolution of the Patterson map. Solutions are gauged to be acceptable if all expected vectors due to symmetry-related atoms are observed above a specified threshold in the Patterson. Systematic computerized Patterson search procedures for orienting and positioning known molecular fragments were also developed in the early 1960s. These hierarchical procedures rely on first using the ‘self’-vectors which depend only on the orientation of a molecular fragment. A search for the position of the fragment relative to the crystal symmetry elements then uses the cross-vectors between molecules (see Sections 2.3.6 and 2.3.7). Nordman constructed a weighted point representation of the predicted vector set for a fragment (Nordman & Nakatsu, 1963; Nordman, 1966) and successfully solved the structure of a number of complex alkaloids. Huber (1965) used the convolution molecule method of Hoppe (1957a) in three dimensions to solve a number of natural-product structures, including steroids. Various program systems have used Patterson search methods operating in real space to solve complex structures (Braun et al., 1969; Egert, 1983). Others have used reciprocal-space procedures for locating known fragments. Tollin & Cochran (1964) developed a procedure for determining the orientation of planar groups by searching for origin-containing planes of high density in the Patterson. General procedures using reciprocal-space representations for determining rotation and translation parameters have been developed and will be described in Sections 2.3.6 and 2.3.7, respectively. Although as many functions have been used to detect solutions in these Patterson search procedures as there are programs, most rely on some variation of the sum, product and minimum functions (Section 2.3.2.4). The quality of the stored Patterson density representation also varies widely, but it is now common to use 16

2.3.3.2. Finding heavy atoms with centrosymmetric projections Phases in a centrosymmetric projection will be 0 or  if the origin is chosen at the centre of symmetry. Hence, the native structure factor, FN , and the heavy-atom-derivative structure factor, FNH , will be collinear. It follows that the structure amplitude, jFH j, of the heavy atoms alone in the cell will be given by jFH j ¼ jðjFNH j  jFN jÞj þ "; where " is the error on the parenthetic sum or difference. Three different cases may arise (Fig. 2.3.3.1). Since the situation shown in Fig. 2.3.3.1(c) is rare, in general jFH j2 ’ ðjFNH j  jFN jÞ2 :

ð2:3:3:1Þ

Thus, a Patterson computed with the square of the differences between the native and derivative structure amplitudes of a centrosymmetric projection will approximate to a Patterson of the heavy atoms alone. The approximation (2.3.3.1) is valid if the heavy-atom substitution is small enough to make jFH j jFNH j for most reflections, but sufficiently large to make " ðjFNH j  jFN jÞ2. It is also

251

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION containing only HH vectors. If the phase angle between FN and FNH is ’ (Fig. 2.3.3.2), then jFH j2 ¼ jFN j2 þ jFNH j2  2jFN kFNH j cos ’: In general, however, jFH j jFN j. Hence, ’ is small and jFH j2 ’ ðjFNH j  jFN jÞ2 ; which is the same relation as (2.3.3.1) for centrosymmetric approximations. Since the direction of FH is random compared to FN , p theffiffiffi root-mean-square projected length of FH onto FN will be FH = 2. Thus it follows that a better approximation is

Fig. 2.3.3.2. Vector triangle showing the relationship between FN , FNH and FH , where FNH ¼ FN þ FH.

jFH j2 ’

assumed that the native and heavy-atom-derivative data have been placed on the same relative scale. Hence, the relation (2.3.3.1) should be rewritten as

pffiffiffi 2ðjFNH j  jFN jÞ2 ;

ð2:3:3:2Þ

whichpffiffiaccounts for the assumption (Section 2.3.3.2) that ffi "3 ¼ 2"2 . The almost universal method for the initial determination of major heavy-atom sites in an isomorphous derivative utilizes a Patterson with ðjFNH j  jFN jÞ2 coefficients. Approximation (2.3.3.2) is also the basis for the refinement of heavyatom parameters in a single isomorphous replacement pair (Rossmann, 1960; Cullis et al., 1962; Terwilliger & Eisenberg, 1983).

jFH j2 ’ ðjFNH j  kjFN jÞ2 ; where k is an experimentally determined scale factor (see Section 2.3.3.7). Uncertainty in the determination of k will contribute further to ", albeit in a systematic manner. Centrosymmetric projections were used extensively for the determination of heavy-atom sites in early work on proteins such as haemoglobin (Green et al., 1954), myoglobin (Bluhm et al., 1958) and lysozyme (Poljak, 1963). However, with the advent of ˚ limit) faster data-collecting techniques, low-resolution (e.g. a 5 A three-dimensional data are to be preferred for calculating difference Pattersons. For noncentrosymmetric reflections, the approximation (2.3.3.1) is still valid but less exact (Section 2.3.3.3). However, the larger number of three-dimensional differences compared to projection differences will enhance the signal of the real Patterson peaks relative to the noise. If there are N terms in the Patterson pffiffiffiffisynthesis, then the peak-to-noise ratio will be proportionally N and 1/". With the subscripts 2 and 3 representing two- and three-dimensional syntheses, respectively, the latter will be more powerful than the former whenever

2.3.3.4. Correlation functions In the most general case of a triclinic space group, it will be necessary to select an origin arbitrarily, usually coincident with a heavy atom. All other heavy atoms (and subsequently also the macromolecular atoms) will be referred to this reference atom. However, the choice of an origin will be independent in the interpretation of each derivative’s difference Patterson. It will then be necessary to correlate the various, arbitrarily chosen, origins. The same problem occurs in space groups lacking symmetry axes perpendicular to the primary rotation axis (e.g. P21 ; P6 etc.), although only one coordinate, namely parallel to the unique rotation axis, will require correlation. This problem gave rise to some concern in the 1950s. Bragg (1958), Blow (1958), Perutz (1956), Hoppe (1959) and Bodo et al. (1959) developed a variety of techniques, none of which were entirely satisfactory. Rossmann (1960) proposed the ðFNH1  FNH2 Þ2 synthesis and applied it successfully to the heavy-atom determination of horse haemoglobin. This function gives positive peaks ðH1
H1Þ at the end of vectors between the heavy-atom sites in the first compound, positive peaks ðH2
H2Þ between the sites in the second compound, and negative peaks between sites in the first and second compound (Fig. 2.3.3.3). It is thus the negative peaks which provide the necessary correlation. The function is unique in that it is a Patterson containing significant information in both positive and negative peaks. Steinrauf (1963) suggested using the coefficients ðjFNH1 j  jFN jÞ
ðjFNH2 j  jFN jÞ in order to eliminate the positive H1
H1 and H2
H2 vectors. Although the problem of correlation was a serious concern in the early structural determination of proteins during the late 1950s and early 1960s, the problem has now been bypassed. Blow & Rossmann (1961) and Kartha (1961) independently showed that it was possible to compute usable phases from a single isomorphous replacement (SIR) derivative. This contradicted the previously accepted notion that it was necessary to have at least two isomorphous derivatives to be able to determine a noncentrosymmetric reflection’s phase (Harker, 1956). Hence, currently, the procedure used to correlate origins in different derivatives is to compute SIR phases from the first compound and apply them to a difference electron-density map of the second heavy-atom derivative. Thus, the origin of the second

pffiffiffiffiffiffi pffiffiffiffiffiffi N3 N2 > : "3 "2 pffiffiffi Now, as "3 ’ 2"2, it follows that N3 must be greater than 2N2 if the three-dimensional noncentrosymmetric computation is to be more powerful. This condition must almost invariably be true. 2.3.3.3. Finding heavy atoms with three-dimensional methods A Patterson of a native biomacromolecular structure (coefficients FN2 ) can be considered as being, at least approximately, a vector map of all the light atoms (carbons, nitrogens, oxygens, some sulfurs, and also phosphorus for nucleic acids) other than hydrogen atoms. These interactions will be designated as LL. Similarly, a Patterson of the heavy-atom derivative will contain HH þ HL þ LL interactions, where H represents the heavy atoms. Thus, a true difference Patterson, with coefficients 2 FNH  FN2 , will contain only the interactions HH þ HL. In general, the carpet of HL vectors completely dominates the HH vectors except for very small proteins such as insulin (Adams et al., 1969). Therefore, it would be preferable to compute a Patterson containing only HH interactions in order to interpret the map in terms of specific heavy-atom sites. Blow (1958) and Rossmann (1960) showed that a Patterson with ðjFNH j  jFN jÞ2 coefficients approximated to a Patterson

252

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES E2h

 P

2  2 NP N P P ahi þ bhi ¼ L þ ðahi ahj þ bhi bhj Þ:

L

L

i6¼j

Therefore, CP ¼

P

" 2h

Lþ2

PP

# ðahi ahj þ bhi bhj Þ :

i6¼j

h

P But h 2h must be independent of the number, L, of heavy-atom sites per cell. Thus the criterion can be rewritten as CP0 ¼

P

" 2h

PP

# ðahi ahj þ bhi bhj Þ :

ð2:3:3:3Þ

i6¼j

h 2

Fig. 2.3.3.3. A Patterson with coefficients ðFNH1  FNH2 Þ will be equivalent to a Patterson whose coefficients are ðABÞ2 . However, AB ¼ FH1 þ FH2 . Thus, a Patterson with ðABÞ2 coefficients is equivalent to having negative atomic substitutions in compound 1 and positive substitutions in compound 2, or vice versa. Therefore, the Patterson will contain positive peaks for vectors of the type H1
H1 and H2
H2, but negative vector peaks for vectors of type H1
H2.

More generally, if some sites have already been tentatively determined, and if these sites give rise to the structure-factor components Ah and Bh , then  2  2 P P E2h ¼ Ah þ ahi þ Bh þ bhi :

derivative will be referred to the arbitrarily chosen origin of the first compound. More important, however, the interpretation of such a ‘feedback’ difference Fourier is easier than that of a difference Patterson. Hence, once one heavy-atom derivative has been solved for its heavy-atom sites, the solution of other derivatives is almost assured. This concept is examined more closely in the following section.

N

Following the same procedure as above, it follows that CP0 ¼

P

" 2h ðAh ah þ Bh bh Þ þ

PP

# ðahi ahj þ bhi bhj Þ ; ð2:3:3:5Þ

i6¼j

h

PL PL where ah ¼ i¼1 ahi and bh ¼ i¼1 bhi . Expression (2.3.3.5) will now be compared with the ‘feedback’ method (Dickerson et al., 1967, 1968) of verifying heavy-atom sites using SIR phasing. Inspection of Fig. 2.3.3.4 shows that the native phase, , will be determined as  ¼ ’ þ  (’ is the structure-factor phase corresponding to the presumed heavyatom positions) when jFN j > jFH j and  ¼ ’ when jFN j jFH j. Thus, an SIR difference electron density, 
ðxÞ, can be synthesized by the Fourier summation

2.3.3.5. Interpretation of isomorphous difference Pattersons Difference Pattersons have usually been manually interpreted in terms of point atoms. In more complex situations, such as crystalline viruses, a systematic approach may be necessary to analyse the Patterson. That is especially true when the structure contains noncrystallographic symmetry (Argos & Rossmann, 1976). Such methods are in principle dependent on the comparison of the observed Patterson, P1 ðxÞ, with a calculated Patterson, P2 ðxÞ. A criterion, CP , based on the sum of the Patterson densities at all test vectors within the unit-cell volume V, would be

1X mðjFNH j  jFN jÞ cosð2h
x  ’h Þ V from terms with h ¼ jFNH j  jFN j > 0 1X þ mðjFNH j  jFN jÞ cosð2h
x  ’h  Þ V from terms with h < 0 1X mjh j cosð2h
x  ’h Þ; ¼ V


ðxÞ ¼

R

CP ¼ P1 ðxÞ
P2 ðxÞ dx: V

CP can be evaluated for all reasonable heavy-atom distributions. Each different set of trial sites corresponds to a different P2 Patterson. It is then easily shown that CP ¼

ð2:3:3:4Þ

N

P

2h E2h ;

h

where m is a figure of merit of the phase reliability (Blow & Crick, 1959; Dickerson et al., 1961). Now,

where the sum is taken over all h reflections in reciprocal space, 2h are the observed differences and Eh are the structure factors of the trial point Patterson. (The symbol E is used here because of its close relation to normalized structure factors.) Let there be n noncrystallographic asymmetric units within the crystallographic asymmetric unit and m crystallographic asymmetric units within the crystal unit cell. Then there are L symmetry-related heavy-atom sites where L ¼ nm. Let the scattering contribution of the ith site have ai and bi real and imaginary structure-factor components with respect to an arbitrary origin. Hence, for reflection h

Fh ¼ Ah þ iBh ¼ FH cos ’h þ iFH sin ’h ; where Ah and Bh are the real and imaginary components of the presumed heavy-atom sites. Therefore, 
ðxÞ ¼

253

1 X mjh j ðAh cos 2h
x þ Bh sin 2h
xÞ: V jFH j

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.3.3.5. Let (a) be the original structure which contains three heavy atoms ABC in a noncentrosymmetric configuration. Then a Fourier 2  FN2 Þ coefficients, will give the Patterson shown in summation, with ðFNH (b). Displacement of the Patterson by the vector BC and selecting the common patterns yields (c). Similarly, displacement by AC gives (d). Finally, superposition of (c) on (d) gives the original figure or its enantiomorph. This series of steps demonstrates that, in principle, complete structural information is contained in an SIR derivative.

(2.3.3.5) and (2.3.3.6) are indeed rather similar. The second term in (2.3.3.5) relates to the use of the search atoms in phasing and could be included in (2.3.3.6), provided the actual feedback sites in each of the n electron-density functions tested by CSIR are omitted in turn. Thus, a systematic Patterson search and an SIR difference Fourier search are very similar in character and power.

Fig. 2.3.3.4. The phase  of the native compound (structure factor FN ) is determined either as being equal to, or 180 out of phase with, the presumed heavy-atom contribution when only a single isomorphous compound is available. In (a) is shown the case when jFN j > jFNH j and  ’ ’ þ . In (b) is shown the case when jFN j < jFNH j and  ¼ ’, where ’ is the phase of the heavy-atom structure factor FH .

2.3.3.6. Direct structure determination from difference Pattersons

If this SIR difference electron-density map shows significant peaks at sites related by noncrystallographic symmetry, then those sites will be at the position of a further set of heavy atoms. Hence, a suitable criterion for finding heavy-atom sites is CSIR ¼

n P

2 The difference Patterson computed with coefficients FHN  FN2 contains information on the heavy atoms (HH vectors) and the macromolecular structure (HL vectors) (Section 2.3.3.3). If the scaling between the jFHN j and jFN j data sets is not perfect there will also be noise. Rossmann (1961b) was partially successful in determining the low-resolution horse haemoglobin structure by using a series of superpositions based on the known heavy-atom sites. Nevertheless, Patterson superposition methods have not been used for the structure determination of proteins owing to the successful error treatment of the isomorphous replacement method in reciprocal space. However, it is of some interest here for it gives an alternative insight into SIR phasing. The deconvolution of an arbitrary molecule, represented as ‘?’, 2  FN2 Þ Patterson, is demonstrated in Fig. 2.3.3.5. from an ðFHN The original structure is shown in Fig. 2.3.3.5(a) and the corresponding Patterson in Fig. 2.3.3.5(b). Superposition with respect to one of the heavy-atom sites is shown in Fig. 2.3.3.5(c) and the other in Fig. 2.3.3.5(d). Both Figs. 2.3.3.5(c) and (d) contain a centre of symmetry because the use of only a single HH vector implies a centre of symmetry half way between the two sites. The centre is broken on combining information from all three sites (which together lack a centre of symmetry) by superimposing Figs. 2.3.3.5(c) and (d) to obtain either the original structure (Fig. 2.3.3.5a) or its enantiomorph. Thus it is clear, in principle, that there is sufficient information in a single isomorphous derivative data set, when used in conjunction with a native data set, to solve a structure completely. However, the procedure shown in Fig. 2.3.3.5 does not consider the accumulation of error in the selection of individual images when these intersect with another image. In this sense the reciprocal-space isomorphous replacement technique has greater elegance and provides more insight, whereas the alternative view given by the Patterson method was the original stimulus for the discovery of the SIR phasing technique (Blow & Rossmann, 1961).


ðxj Þ;

j¼1

or by substitution CSIR ¼

n X 1 X mjh j ðAh cos 2h
xj þ Bh sin 2h
xj Þ: V h jFH j j¼1

But ah ¼

n P

cos 2h
xj

j¼1

and

bh ¼

n P

sin 2h
xj :

j¼1

Therefore, CSIR ¼

1 X mjh j ðAh ah þ Bh bh Þ: V h jFH j

ð2:3:3:6Þ

This expression is similar to (2.3.3.5) derived by consideration of a Patterson search. It differs from (2.3.3.5) in two respects: the Fourier coefficients are different and expression (2.3.3.6) is lacking a second term. Now the figure of merit m will be small whenever jFH j is small as the SIR phase cannot be determined well under those conditions. Hence, effectively, the coefficients are a function of jh j, and the coefficients of the functions

254

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES reduced. Blundell & Johnson (1976, pp. 333–336) give a careful discussion of this subject. Suffice it to say here only that a linear scale factor is seldom acceptable as the heavy-atom-derivative crystals frequently suffer from greater disorder than the native crystals. The heavy-atom derivative should, in general, have a slightly larger mean value for the structure factors on account of the additional heavy P (Green et al., 1954). The usual P atoms effect is to make jFNH j2 = jFN j2 ’ 1:05 (Phillips, 1966). As the amount of heavy atom is usually unknown in a yet unsolved heavy-atom derivative, it is usual practice either to apply a scale factor of the form k exp½Bðsin =Þ2  or, more generally, to use local scaling (Matthews & Czerwinski, 1975). The latter has the advantage of not making any assumption about the physical nature of the relative intensity decay with resolution.

Fig. 2.3.3.6. A plot of mean isomorphous differences as a function of resolution. (a) The theoretical size of mean differences following roughly a Gaussian distribution. (b) The observed size of differences for a good isomorphous derivative where the smaller higher-order differences have been largely masked by the error of measurement. (c) Observed differences ˚ resoluwhere ‘lack of isomorphism’ dominates beyond approximately 5 A tion.

2.3.4. Anomalous dispersion 2.3.4.1. Introduction The physical basis for anomalous dispersion has been well reviewed by Ramaseshan & Abrahams (1975), James (1965), Cromer (1974) and Bijvoet (1954). As the wavelength of radiation approaches the absorption edge of a particular element, then an atom will disperse X-rays in a manner that can be defined by the complex scattering factor

Other Patterson functions for the deconvolution of SIR data have been proposed by Ramachandran & Raman (1959), as well as others. The principles are similar but the coefficients of the functions are optimized to emphasize various aspects of the signal representing the molecular structure.

f0 þ f 0 þ if 00 ; 2.3.3.7. Isomorphism and size of the heavy-atom substitution where f0 is the scattering factor of the atom without the anomalous absorption and rescattering effect, f 0 is the real correction term (usually negative), and f 00 is the imaginary component. The real term f0 þ f 0 is often written as f 0, so that the total scattering factor will be f 0 þ if 00. Values of f 0 and f 00 are tabulated in IT IV (Cromer, 1974), although their precise values are dependent on the environment of the anomalous scatterer. Unlike f0, f 0 and f 00 are almost independent of scattering angle as they are caused by absorption of energy in the innermost electron shells. Thus, the anomalous effect resembles scattering from a point atom. The structure factor of index h can now be written as

It is insufficient to discuss Patterson techniques for locating heavy-atom substitutions without also considering errors of all kinds. First, it must be recognized that most heavy-atom labels are not a single atom but a small compound containing one or more heavy atoms. The compound itself will displace water or ions and locally alter the conformation of the protein or nucleic acid. Hence, a simple Gaussian approximation will suffice to represent individual heavy-atom scatterers responsible for the difference between native and heavy-atom derivatives. Furthermore, the heavy-atom compound often introduces small global structural changes which can be detected only at higher resolution. These problems were considered with some rigour by Crick & Magdoff (1956). In general, lack of isomorphism is exhibited by an increase in the size of the isomorphous differences with increasing resolution (Fig. 2.3.3.6). Crick & Magdoff (1956) also derived the approximate expression

Fh ¼

N P

fj0 expð2ih
xj Þ þ i

j¼1

N P

fj00 expð2ih
xj Þ:

ð2:3:4:1Þ

j¼1

(Note that the only significant contributions to the second term are from those atoms that have a measurable anomalous effect at the chosen wavelength.) Let us now write the first term as A þ iB and the second as a þ ib. Then, from (2.3.4.1),

sffiffiffiffiffiffiffiffiffi 2NH fH
N P fP

to estimate the r.m.s. fractional change in intensity as a function of heavy-atom substitution. Here, NH represents the number of heavy atoms attached to a protein (or other large molecule) which contains NP light atoms. fH and fP are the scattering powers of the average heavy and protein atom, respectively. This function was tabulated by Eisenberg (1970) as a function of molecular weight (proportional to NP ). For instance, for a single, fully substituted, Hg atom the formula predicts an r.m.s. intensity change of around 25% in a molecule of 100 000 Da. However, the error of measurement of a reflection intensity is likely to be arround 10% of I, implying perhaps an error of around 14% of I on a difference measurement. Thus, the isomorphous replacement difference measurement for almost half the reflections will be buried in error for this case. Scaling of the different heavy-atom-derivative data sets onto a common relative scale is clearly important if error is to be

F ¼ ðA þ iBÞ þ iða þ ibÞ ¼ ðA  bÞ þ iðB þ aÞ:

ð2:3:4:2Þ

Therefore, jFh j2 ¼ ðA  bÞ2 þ ðB þ aÞ2 and similarly jFh
j2 ¼ ðA þ bÞ2 þ ðB þ aÞ2 ; demonstrating that Friedel’s law breaks down in the presence of anomalous dispersion. However, it is only for noncentrosymmetric reflections that jFh j 6¼ jFh
j.

255

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION jFh j2  jFh
j2 ¼ 2

P

ðfi0 fj00  fi00 fj0 Þ sin 2h
ðxi  xj Þ:

i; j

Let us now consider the significance of a Patterson in the presence of anomalous dispersion. The normal Patterson definition is given by R PðuÞ ¼
 ðxÞ
ðx þ uÞ dx V

¼

sphere 1 X jFh j2 expð2ih
uÞ V2 h

 Pc ðuÞ  iPs ðuÞ; where Fig. 2.3.4.1. (a) A model structure with an anomalous scatterer at A. (b) The corresponding Ps ðuÞ function showing positive peaks (full lines) and negative peaks (dashed lines). [Reprinted with permission from Woolfson (1970, p. 293).]

Pc ðuÞ ¼

2 V

hemisphere X

ðjFh j2 þ jFh
j2 Þ cos 2h
u

h

and Now, Ps ðuÞ ¼

sphere 1 X
ðxÞ ¼ F expð2ih
xÞ: V h h

2 V

hemisphere X

½ðA cos 2h
x  B sin 2h
xÞ

h

þ iða cos 2h
x  b sin 2h
xÞ:

ð2:3:4:3Þ

The first term in (2.3.4.3) is the usual real Fourier expression for electron density, while the second term is an imaginary component due to the anomalous scattering of a few atoms in the cell. 2.3.4.2. The Ps ðuÞ function Expression (2.3.4.3) gives the complex electron density expression in the presence of anomalous scatterers. A variety of Patterson-type functions can be derived from (2.3.4.3) for the determination of a structure. For instance, the Ps ðuÞ function gives vectors between the anomalous atoms and the ‘normal’ atoms. From (2.3.4.1) it is easy to show that

i; j

P

ðjFh j2  jFh
j2 Þ sin 2h
u:

h

2.3.4.3. The position of anomalous scatterers Anomalous scatterers can be used as an aid to phasing, when their positions are known. But the anomalous-dispersion differences (Bijvoet differences) can also be used to determine or confirm the heavy atoms which scatter anomalously (Rossmann, 1961a). Furthermore, the use of anomalous-dispersion information obviates the lack of isomorphism but, on the other hand, the differences are normally far smaller than those produced by a heavy-atom isomorphous replacement. Consider a structure of many light atoms giving rise to the structure factor Fh ðNÞ. In addition, it contains a few heavy atoms which have a significant anomalous-scattering effect. The nonanomalous component will be Fh ðHÞ and the anomalous component is F00h ðHÞ ¼ iðf 00 =f 0 ÞFh ðHÞ (Fig. 2.3.4.2a). The total structure factor will be Fh. The Friedel opposite is constructed appropriately (Fig. 2.3.4.2a). Now reflect the Friedel opposite construction across the real axis of the Argand diagram (Fig.

Fh Fh ¼ jFh j2 P ¼ ðfi0 fj0 þ fi00 fj00 Þ cos 2h
ðxi  xj Þ þ

hemisphere X

The Pc ðuÞ component is essentially the normal Patterson, in which the peak heights have been very slightly modified by the anomalous-scattering effect. That is, the peaks of Pc ðuÞ are proportional in height to ðfi0 fj0 þ fi00 fj00 Þ. The Ps ðuÞ component is more interesting. It represents vectors between the normal atoms in the unit cell and the anomalous scatterers, proportional in height to ðfi0 fj00  fi00 fj0 Þ (Okaya et al., 1955). This function is antisymmetric with respect to the change of the direction of the diffraction vector. An illustration of the function is given in Fig. 2.3.4.1. In a unit cell containing N atoms, n of which are anomalous scatterers, the Ps ðuÞ function contains only nðN  nÞ positive peaks and an equal number of negative peaks related to the former by anticentrosymmetry. The analysis of a Ps ðuÞ synthesis presents problems somewhat similar to those posed by a normal Patterson. The procedure has been used only rarely [cf. Moncrief & Lipscomb (1966) and Pepinsky et al. (1957)], probably because alternative procedures are available for small compounds and the solution of Ps ðuÞ is too complex for large biological molecules.

Hence, by using (2.3.4.2) and simplifying,


ðxÞ ¼

2 V

ðfi0 fj00  fi00 fj0 Þ sin 2h
ðxi  xj Þ:

i; j

Therefore, P jFh j2 þ jFh
j2 ¼ 2 ðfi0 fj0 þ fi00 fj00 Þ cos 2h
ðxi  xj Þ i; j

and

256

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES (Rossmann, 1961a), as well as a Patterson with coefficients 2 FISO ¼ ðjFNH j  jFH jÞ2

(Rossmann, 1960; Blow, 1958), represent Pattersons of the heavy 2 atoms. The FANO Patterson suffers from errors which may be 2 larger than the size of the Bijvoet differences, while the FISO Patterson may suffer from partial lack of isomorphism. Hence, Kartha & Parthasarathy (1965) have suggested the use of the sum of these two Pattersons, which would then have coefficients 2 2 ðFANO þ FISO Þ. However, given both SIR and anomalous-dispersion data, it is possible to make an accurate estimate of the jFH j2 magnitudes for use in a Patterson calculation [Blundell & Johnson (1976, p. 340), Matthews (1966), Singh & Ramaseshan (1966)]. In essence, the Harker phase diagram can be constructed out of three circles: the native amplitude and each of the two isomorphous Bijvoet differences, giving three circles in all (Blow & Rossmann, 1961) which should intersect at a single point thus resolving the ambiguity in the SIR data and the anomalous-dispersion data. Furthermore, the phase ambiguities are orthogonal; thus the two data sets are cooperative. It can be shown (Matthews, 1966; North, 1965) that 2 2 FN2 ¼ FNH þ FN2  ð16k2 FP2 FH2  I 2 Þ1=2 ; k Fig. 2.3.4.2. Anomalous-dispersion effect for a molecule whose light atoms contribute Fh ðNÞ and heavy atom Fh ðHÞ with a small anomalous component of F00h ðHÞ, 90 ahead of the non-anomalous Fh ðHÞ component. In (a) is seen the construction for Fh and Fh
. In (b) Fh
has been reflected across the real axis.

2

þ  2 where I ¼ FNH  FNH and k ¼ f 00 =f 0 . The sign in the thirdterm expression is  when jðNH  H Þj < =2 or + otherwise. Since, in general, jFH j is small compared to jFN j, it is reasonable to assume that the sign above is usually negative. Hence, the heavy-atom lower estimate (HLE) is usually written as

2.3.4.2b). Let the difference in phase between Fh and Fh
be ’. Thus

2 2 2 ¼ FNH þ FH2  ð16k2 FP2 FH2  I 2 Þ1=2 ; FHLE k

4jF00h ðHÞj2 ¼ jFh j2 þ jFh
j2  2jFh jjFh
j cos ’; but since ’ is very small

which is an expression frequently used to derive Patterson coefficients useful in the determination of heavy-atom positions when both SIR and anomalous-dispersion data are available.

jF00h ðHÞj2 ’ 14ðjFh j  jFh
jÞ2 :

2.3.4.4. Computer programs for automated location of atomic positions from Patterson maps

Hence, a Patterson with coefficients ðjFh j  jFh
jÞ2 will be equivalent to a Patterson with coefficients jF00h ðHÞj2 which is proportional to jFh ðHÞj2. Such a Patterson (Rossmann, 1961a) will have vectors between all anomalous scatterers with heights proportional to the number of anomalous electrons f 00. This ‘anomalous dispersion’ Patterson function has been used to find anomalous scatterers such as iron (Smith et al., 1983; Strahs & Kraut, 1968) and sulfur atoms (Hendrickson & Teeter, 1981). The anomalous signal from Se atoms in selenomethionine-substituted proteins has been found to be extremely powerful and is now routinely used for protein structure determinations (Hendrickson, 1991). Anomalous signals from halide ions or xenon atoms have also been used to solve protein structures (Dauter et al., 2000; Nagem et al., 2003; Schiltz et al., 2003). The anomalous signal from sulfur atoms, though very small (Hendrickson & Teeter, 1981), has recently been applied successfully to solve several protein structures (Debreczeni et al., 2003; Ramagopal et al., 2003; Yang et al., 2003). It is then apparent that a Patterson with coefficients

Several programs are currently used for automated systematic interpretation of (difference) Patterson maps to locate the positions of heavy atoms and/or anomalous scatterers from isomorphous replacement and anomalous-dispersion data (Weeks et al., 2003). These include Solve (Terwilliger & Berendzen, 1999), CNS (Bru¨nger et al., 1998), CCP4 (Collaborative Computational Project, Number 4, 1994) and Patsol (Tong & Rossmann, 1993). In these programs, sets of trial atomic positions (seeds) are produced based on one- and two-atom solutions to the Patterson map (see Section 2.3.2.5) (Grosse-Kunstleve & Brunger, 1999; Terwilliger et al., 1987; Tong & Rossmann, 1993). Information from a translation search with a single atom can also be used in this process (Grosse-Kunstleve & Brunger, 1999). Scoring functions have been devised to identify the likely correct solutions, based on agreements with the Patterson map or the observed isomorphous or anomalous differences, as well as the quality of the resulting electron-density map (Terwilliger, 2003b; Terwilliger & Berendzen, 1999). The power of modern computers allows the rapid screening of a large collection of trial structures, and the correct solution is found automatically in many cases, even when there is a large number of atomic positions (Weeks et al., 2003).

2 ¼ ðjFh j  jFh
jÞ2 FANO

257

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.3.5.2. The objects A1 B1 and A2 B2 are related by an improper rotation , since it is necessary to consider the sense of rotation to achieve superposition of the two objects. [Reprinted with permission from Rossmann (1972, p. 9).]

graphic symmetry can also be recognized by the existence of a closed point group within a defined volume of the lattice. Improper rotation axes are found when two molecules are arbitrarily oriented relative to each other in the same asymmetric unit or when they occur in two entirely different crystal lattices. For instance, in Fig. 2.3.5.2, the object A1 B1 can be rotated by + about the axis at P to orient it identically with A2 B2 . However, the two objects will not be coincident after a rotation of A1 B1 by  or of A2 B2 by +. The envelope around each noncrystallographic object must be known in order to define an improper rotation. In contrast, only the volume about the closed point group need be defined for proper noncrystallographic operations. Hence, the boundaries of the repeating unit need not correspond to chemically covalently linked units in the presence of proper rotations. Translational components of noncrystallographic rotation elements are said to be ‘precise’ in a direction parallel to the axis and ‘imprecise’ perpendicular to the axis (Rossmann et al., 1964). The position, but not direction, of a rotation axis is arbitrary. However, a convenient choice is one that leaves the translation perpendicular to the axis at zero after rotation (Fig. 2.3.5.3). Noncrystallographic symmetry has been used as a tool in structural analysis primarily in the study of biological molecules. This is due to the propensity of proteins to form aggregates with closed point groups, as, for instance, viruses with 532 symmetry. At best, only part of such a point group can be incorporated into the crystal lattice. Since biological materials cannot contain inversion elements, all studies of noncrystallographic symmetries have been limited to rotational axes. Reflection planes and inversion centres could also be considered in the application of molecular replacement to nonbiological materials. In this chapter, the relationship

Fig. 2.3.5.1. The two-dimensional periodic design shows crystallographic twofold axes perpendicular to the page and local noncrystallographic rotation axes in the plane of the paper (design by Audrey Rossmann). [Reprinted with permission from Rossmann (1972, p. 8).]

2.3.5. Noncrystallographic symmetry 2.3.5.1. Definitions The interpretation of Pattersons can be helped by using various types of chemical or physical information. An obvious example is the knowledge that one or two heavy atoms per crystallographic asymmetric unit are present. Another example is the exploitation of a rigid chemical framework in a portion of a molecule (Nordman & Nakatsu, 1963; Burnett & Rossmann, 1971). One extremely useful constraint on the interpretation of Pattersons is noncrystallographic symmetry. Indeed, the structural solution of large biological assemblies such as viruses is only possible because of the natural occurrence of this phenomenon. The term ‘molecular replacement’ is used for methods that utilize noncrystallographic symmetry in the solution of structures [for earlier reviews see Rossmann (1972); Argos & Rossmann (1980); and Rossmann (1990, 2001)]. These methods, which are only partially dependent on Patterson concepts, are discussed in Sections 2.3.6–2.3.8. Crystallographic symmetry applies to the whole of the threedimensional crystal lattice. Hence, the symmetry must be expressed both in the lattice and in the repeating pattern within the lattice. In contrast, noncrystallographic symmetry is valid only within a limited volume about the noncrystallographic symmetry element. For instance, the noncrystallographic twofold axes in the plane of the paper of Fig. 2.3.5.1 are true only in the immediate vicinity of each local dyad. In contrast, the crystallographic twofold axes perpendicular to the plane of the paper (Fig. 2.3.5.1) apply to every point within the lattice. Two types of noncrystallographic symmetry can be recognized: proper and improper rotations. A proper symmetry element is independent of the sense of rotation, as, for example, a fivefold axis in an icosahedral virus; a rotation either left or right by one-fifth of a revolution will leave all parts of a given icosahedral shell (but not the whole crystal) in equivalent positions. Proper noncrystallo-

x0 ¼ ½Cx þ d will be used to describe noncrystallographic symmetry, where x and x0 are position vectors, expressed as fractional coordinates, with respect to the crystallographic origin, [C] is a rotation matrix, and d is a translation vector. Crystallographic symmetry will be described as x0 ¼ ½Tx þ t;

258

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES Table 2.3.5.1. Possible types of vector searches Self-vectors

Dimension of search, n

Cross-vectors

n¼3

(1) Locate single site relative to particle centre (2)

Use information from (1) to locate particle centre

n3

(3) Simultaneous search for both (1) and (2). In general this is a six-dimensional search but may be simplified when particle is on a crystallographic symmetry axis

3n6

(4) Given (1) for more than one site, find all vectors within particle

n¼3

(5) Given information from (3), locate additional site using complete vector set

n¼3

al., 1972; Buehner et al., 1974). The GAPDH enzyme crystallized ˚ ) containing one in a P21 21 21 cell (a = 149.0, b = 139.1, c = 80.7 A tetramer per asymmetric unit. A rotation-function analysis had indicated the presence of three mutually perpendicular molecular twofold axes which suggested that the tetramer had 222 symmetry, and a locked rotation function determined the precise orientation of the tetramer relative to the crystal axes (see Table 2.3.5.2). Packing considerations led to assignment of a tentative particle centre near 12 ; 14 ; Z. An isomorphous difference Patterson was calculated for the ˚. K2HgI4 derivative of GAPDH using data to a resolution of 6.8 A From an analysis of the three Harker sections, a tentative first heavy-atom position was assigned (atom A2 at x, y, z). At this juncture, the known noncrystallographic symmetry was used to obtain a full interpretation. From Table 2.3.5.2 we see that molecular axis 2 will generate a second heavy atom with coordinates roughly 14 þ y;  14 þ x; 2Z  z (if the molecular centre was assumed to be at 12 ; 14 ; Z). Starting from the tentative coordinates of site A2, the site A1 related by molecular axis 1 was detected at about the predicted position and the second site A1 generated acceptable cross-vectors with the earlier determined site A2. Further examination enabled the completion of the set of four noncrystallographically related heavy-atom sites, such that all predicted Patterson vectors were acceptable and all four sites placed the molecular centre in the same position. Following refinement of these four sites, the corresponding SIR phases were used to find an additional set of four sites in this compound as well as in a number of other derivatives. The multiple isomorphous replacement phases, in conjunction with real-space electron-density averaging of the noncrystallographically related units, were then sufficient to solve the GAPDH structure. When investigators studied larger macromolecular aggregates such as the icosahedral viruses, which have 532 point symmetry, systematic methods were developed for utilizing the noncrystallographic symmetry to aid in locating heavy-atom sites in isomorphous heavy-atom derivatives. Argos & Rossmann (1974, 1976) introduced an exhaustive Patterson search procedure for a single heavy-atom site within the noncrystallographic asymmetric unit which has been successfully applied to the interpretation of both virus [satellite tobacco necrosis virus (STNV) (Lentz et al., 1976), southern bean mosaic virus (Rayment et al., 1978), alfalfa mosaic virus (Fukuyama et al., 1983), cowpea mosaic virus

Fig. 2.3.5.3. The position of the twofold rotation axis which relates the two piglets is completely arbitrary. The diagram on the left shows the situation when the translation is parallel to the rotation axis. The diagram on the right has an additional component of translation perpendicular to the rotation axis, but the component parallel to the axis remains unchanged. [Reprinted from Rossmann et al. (1964).]

where [T] and t are the crystallographic rotation matrix and translation vector, respectively. The noncrystallographic asymmetric unit will be defined as having n copies within the crystallographic asymmetric unit, and the unit cell will be defined as having m crystallographic asymmetric units. Hence, there are L ¼ nm noncrystallographic asymmetric units within the unit cell. Clearly, the n noncrystallographic asymmetric units cannot completely fill the volume of one crystallographic asymmetric unit. The remaining space must be assumed to be empty or to be occupied by solvent molecules which disobey the noncrystallographic symmetry. 2.3.5.2. Interpretation of Pattersons in the presence of noncrystallographic symmetry If noncrystallographic symmetry is present, an atom at a general position within the relevant volume will imply the presence of others within the same crystallographic asymmetric unit. If the noncrystallographic symmetry is known, then the positions of equivalent atoms may be generated from a single atomic position. The additional vector interactions which arise from crystallographically and noncrystallographically equivalent atoms in a crystal may be predicted and exploited in an interpretation of the Patterson function. An object in real space which has a closed point group may incorporate some of its symmetry in the crystallographic symmetry. If there are l such objects in the cell, then there will be mn=l equivalent positions within each object. The ‘self-vectors’ formed between these positions within the object will be independent of the position of the objects. This distinction is important in that the self-vectors arising from atoms interacting with other atoms within a single particle may be correctly predicted without the knowledge of the particle centre position. In fact, this distinction may be exploited in a two-stage procedure in which an atom may be first located relative to the particle centre by use of the self-vectors and subsequently the particle may be positioned relative to crystallographic symmetry elements by use of the ‘cross-vectors’ (Table 2.3.5.1). The interpretation of a heavy-atom difference Patterson for the holo-enzyme of lobster glyceraldehyde-3-phosphate dehydrogenase (GAPDH) provides an illustration of how the known noncrystallographic symmetry can aid the solution (Rossmann et

Table 2.3.5.2. Orientation of the glyceraldehyde-3-phosphate dehydrogenase molecular twofold axis in the orthorhombic cell Rotation axes

259

Polar coordinates ( ) ’

Cartesian coordinates (direction cosines) u

v

w

1

45.0

7.0

0.7018

0.7071

2

180.0–55.0

38.6

0.6402

0.5736

0.0862 0.5111

3

180.0–66.0

70.6

0.3035

0.4067

0.8616

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.3.6.1. Different types of uses for the rotation function Type of rotation function

Pattersons to be compared P1

P2

Purpose

Self

Unknown structure

Unknown structure, same cell

Finds orientation of noncrystallographic axes

Cross

Unknown structure

Unknown structure in different cell

Finds relative orientation of unknown molecules

Cross

Unknown structure

Known structure in large cell to avoid overlap of self-Patterson vectors

Determines orientation of unknown structure as preliminary to positioning and subsequent phasing with known molecule

(Stauffacher et al., 1987)] and enzyme [catalase (Murthy et al., 1981)] heavy-atom difference Pattersons. This procedure has also been implemented in the program Patsol (Tong & Rossmann, 1993). A heavy atom is placed in turn at all plausible positions within the volume of the noncrystallographic asymmetric unit and the corresponding vector set is constructed from the resulting constellation of heavy atoms. Argos & Rossmann (1976) found a spherical polar coordinate search grid to be convenient for spherical viruses. After all vectors for the current search position are predicted, the vectors are allocated to the nearest grid point and the list is sorted to eliminate recurring ones. The criterion used by Argos & Rossmann for selecting a solution is that the sum

S¼

N P

Conversely, the knowledge that the heavy-atom positions, especially the Se atoms in a selenomethionyl protein, should obey the noncrystallographic symmetry can be used to deduce the nature, orientation and position of the noncrystallographic symmetry in the crystal unit cell, with either manual or automated procedures (Buehner et al., 1974; Lu, 1999; Terwilliger, 2002a). The noncrystallographic symmetry can also serve as a powerful tool for refining the phase information derived from the heavy-atom positions (Buehner et al., 1974). 2.3.6. Rotation functions 2.3.6.1. Introduction The rotation function is designed to detect noncrystallographic rotational symmetry (see Table 2.3.6.1). The normal rotation function definition is given as (Rossmann & Blow, 1962)

Pi  NPav

i¼1

R R ¼ P1 ðuÞ
P2 ðu0 Þ du;

of the lookup Patterson density values Pi achieves a high value for a correct heavy-atom position. The sum is corrected for the carpet of cross-vectors by the second term in the sum. An additional criterion, which has been found useful for discriminating correct solutions, is a unit vector density criterion (Arnold et al., 1987)

U¼

N P

ð2:3:6:1Þ

U

where P1 and P2 are two Pattersons and U is an envelope centred at the superimposed origins. This convolution therefore measures the degree of similarity, or ‘overlap’, between the two Pattersons when P2 has been rotated relative to P1 by an amount defined by

. ðPi =ni Þ N;

u0 ¼ ½Cu:

ð2:3:6:2Þ

i¼1

The elements of [C] will depend on three rotation angles ð1 ; 2 ; 3 Þ. Thus, R is a function of these three angles. Alternatively, the matrix [C] could be used to express mirror symmetry, permitting searches for noncrystallographic mirror or glide planes. The basic concepts were first clearly stated by Rossmann & Blow (1962), although intuitive uses of the rotation function had been considered earlier. Hoppe (1957b) had also hinted at a convolution of the type given by (2.3.6.1) to find the orientation of known molecular fragments and these ideas were implemented by Huber (1965). Consider a structure of two identical units which are in different orientations. The Patterson function of such a structure consists of three parts. There will be the self-Patterson vectors of one unit, being the set of interatomic vectors which can be formed within that unit, with appropriate weights. The set of selfPatterson vectors of the other unit will be identical, but they will be rotated away from the first due to the different orientation. Finally, there will be the cross-Patterson vectors, or set of interatomic vectors which can be formed from one unit to another. The self-Patterson vectors of the two units will all lie in a volume centred at the origin and limited by the overall dimensions of the units. Some or all of the cross-Patterson vectors will lie outside this volume. Suppose the Patterson function is now superposed on a rotated version of itself. There will be no particular agreement except when one set of self-Patterson vectors of one unit has the same orientation as the self-Patterson vectors from the other unit. In this position, we would expect a maximum of agreement or ‘overlap’ between the two. Similarly,

where ni is the number of vectors expected to contribute to the Patterson density value Pi (Arnold et al., 1987). This criterion can be especially valuable for detecting correct solutions at special search positions, such as an icosahedral fivefold axis, where the number of vector lookup positions may be drastically reduced owing to the higher symmetry. An alternative, but equivalent, method for locating heavy-atom positions from isomorphous difference data is discussed in Section 2.3.3.5. Even for a single heavy-atom site at a general position in the simplest icosahedral or ðT ¼ 1Þ virus, there are 60 equivalent heavy atoms in one virus particle. The number of unique vectors corresponding to this self-particle vector set will depend on the crystal symmetry but may be as many as ð60Þð59Þ=2 ¼ 1770 for a virus particle at a general crystallographic position. Such was the case for the STNV crystals which were in space group C2 containing four virus particles at general positions. The method of Argos & Rossmann was applied successfully to a solution of ˚ resolution difference the K2HgI4 derivative of STNV using a 10 A Patterson. Application of the noncrystallographic symmetry vector search procedure to a K2Au(CN)2 derivative of human rhinovirus 14 (HRV14) crystals (space group P21 3; Z ¼ 4) has succeeded in establishing both the relative positions of heavy atoms within one particle and the positions of the virus particles relative to the crystal symmetry elements (Arnold et al., 1987). The particle position was established by incorporating interparticle vectors in the search and varying the particle position along the crystallographic threefold axis until the best fit for the predicted vector set was achieved.

260

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES the superposition of the molecular self-Patterson derived from different crystal forms can provide the relative orientation of the two crystals when the molecules are aligned. While it would be possible to evaluate R by interpolating in P2 and forming the point-by-point product with P1 within the volume U for every combination of 1 ; 2 and 3 , such a process is tedious and requires large computer storage for the Pattersons. Instead, the process is usually performed in reciprocal space where the number of independent structure amplitudes which form the Pattersons is about one-thirtieth of the number of Patterson grid points. Thus, the computation of a rotation function is carried out directly on the structure amplitudes, while the overlap definition (2.3.6.1) simply serves as a physical basis for the technique. The derivation of the reciprocal-space expression depends on the expansion of each Patterson either as a Fourier summation, the conventional approach of Rossmann & Blow (1962), or as a sum of spherical harmonics in Crowther’s (1972) analysis. The conventional and mathematically easier treatment is discussed presently, but the reader is referred also to Section 2.3.6.5 for Crowther’s elegant approach. The latter leads to a rapid technique for performing the computations, about one hundred times faster than conventional methods. Let, omitting constant coefficients, P

P1 ðuÞ ¼

Fig. 2.3.6.1. Shape of the interference function G for a spherical envelope of radius R at a distance H from the reciprocal-space origin. [Reprinted from Rossmann & Blow (1962).]

[C]. Only for those integral reciprocal-lattice points which are close to h0 will Ghp be of an appreciable size (Fig. 2.3.6.1). Thus, the number of significant terms is greatly reduced in the summation over p for every value of h, making the computation of the rotation function manageable. The radius of integration R should be approximately equal to or a little smaller than the molecular diameter. If R were roughly equal to the length of a lattice translation, then the separation of reciprocal-lattice points would be about 1=R. Hence, when H is equal to one reciprocal-lattice separation, HR ’ 1, and G is thus quite small. Indeed, all terms with HR > 1 might well be neglected. Thus, in general, the only terms that need be considered are those where h0 is within one lattice point of h. However, in dealing with a small molecular fragment for which R is small compared to the unit-cell dimensions, more reciprocallattice points must be included for the summation over p in the rotation-function expression (2.3.6.3). In practice, the equation

jFh j2 expð2ih
uÞ

h

and P

P2 ðu0 Þ ¼

jFp j2 expð2ip
u0 Þ:

p

From (2.3.6.2) it follows that P

P2 ðu0 Þ ¼

jFp j2 expð2ip½C
uÞ;

p

h þ h0 ¼ 0; and, hence, by substitution in (2.3.6.1) Rð1 ; 2 ; 3 Þ ¼

Z
P

that is

 2 jFh j expð2ih
uÞ

½C T p ¼ h

h U

"



P

# 2

¼U

P

or

jFp j expð2ip½C
uÞ du

p

jFh j

h

2

P

! 2

jFp j Ghp ;

p ¼ ½CT 1 ðhÞ;

ð2:3:6:3Þ

p

determines p, given a set of Miller indices h. This will give a nonintegral set of Miller indices. The terms included in the inner summation of (2.3.6.3) will be integral values of p around the non-integral lattice point found by solving (2.3.6.5). Details of the conventional program were given by Tollin & Rossmann (1966) and follow the principles outlined above. They discussed various strategies as to which crystal should be used to calculate the first (h) and second (p)PPatterson. Rossmann & 2 Blow (1962) noted that the factor p jFp j Ghp in expression (2.3.6.3) represents an interpolation of the squared transform of the self-Patterson of the second (p) crystal. Thus, the rotation function is a sum of the products of the two molecular transforms taken over all the h reciprocal-lattice points. Lattman & Love (1970) therefore computed the molecular transform explicitly and stored it in the computer, sampling it as required by the rotation operation. A discussion on the suitable choice of variables in the computation of rotation functions has been given by Lifchitz (1983).

where UGhp ¼

R

ð2:3:6:5Þ

expf2iðh þ p½CÞ
ug du:

U

When the volume U is a sphere, Ghp has the analytical form Ghp ¼

3ðsin    cos Þ ; 3

ð2:3:6:4Þ

where  ¼ 2HR and H ¼ h þ p½C. G is a spherical interference function whose form is shown in Fig. 2.3.6.1. The expression (2.3.6.3) represents the rotation function in reciprocal space. If h0 ¼ ½CT p in the argument of Ghp, then h0 can be seen as the point in reciprocal space to which p is rotated by

261

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.3.6.2. Relationships of the orthogonal axes X1 ; X2 ; X3 to the crystallographic axes a1 ; a2 ; a3. [Reprinted from Rossmann & Blow (1962).]

Fig. 2.3.6.4. Variables and ’ are polar coordinates which specify a direction about which the axes may be rotated through an angle . [Reprinted from Rossmann & Blow (1962).]

2.3.6.2. Matrix algebra The initial step in the rotation-function procedure involves the orthogonalization of both crystal systems. Thus, if fractional coordinates in the first crystal system are represented by x, these can be orthogonalized by a matrix [b] to give the coordinates X in units of length (Fig. 2.3.6.2); that is,

½C ¼ ½a½q½b:

Fig. 2.3.6.2 shows the mode of orthogonalization used by Rossmann & Blow (1962). With their definition it can be shown that

X ¼ ½bx: 0

1=ða1 sin 3 sin !Þ B 1=ða2 tan 1 tan !Þ B ½a ¼ @ 1=ða2 tan 3 sin !Þ 1=ða3 sin 1 tan !Þ

0

If the point X is rotated to the point X , then X0 ¼ ½qX;

ð2:3:6:6Þ

where q represents the rotation matrix relating the two vectors in the orthogonal system. Finally, X0 is converted back to fractional coordinates measured along the oblique cell dimension in the second crystal by

0 1=a2 0

1 0 1=ða2 tan 1 Þ C C A 1=ða3 sin 1 Þ

and 0

a1 sin 3 sin ! 0 ½b ¼ @ a1 cos 3 a2 a1 sin 3 cos ! 0

x0 ¼ ½aX0 :

1 0 a3 cos 1 A; a3 sin 1

Thus, by substitution, x0 ¼ ½a½qX ¼ ½a½q½bx;

where cos ! ¼ ðcos 2  cos 1 cos 3 Þ=ðsin 1 sin 3 Þ with 0  ! < . For a Patterson compared with itself, ½a ¼ ½b1 . An alternative mode of orthogonalization, used by the Protein Data Bank and most programs, is to align the a1 axis of the unit cell with the Cartesian X1 axis, and to align the a3 axis with the Cartesian X3 axis. With this definition, the orthogonalization matrix is

ð2:3:6:7Þ

and by comparison with (2.3.6.2) it follows that

0

a1 ½b ¼ @ 0 0

a2 cos 3 a2 sin 3 0

1 a3 cos 2 a3 sin 2 cos 1 A: a3 sin 2 sin 1

Other modes of orthogonalization are also possible, some of which are supported in the program GLRF (Tong & Rossmann, 1990, 1997). Both spherical ð; ; ’Þ and Eulerian ð1 ; 2 ; 3 Þ angles are used in evaluating the rotation function. The usual definitions employed are given diagrammatically in Figs. 2.3.6.3 and 2.3.6.4. They give rise to the following rotation matrices. (a) Matrix [q] in terms of Eulerian angles 1 ; 2 ; 3 :

Fig. 2.3.6.3. Eulerian angles 1 ; 2 ; 3 relating the rotated axes X10 ; X20 ; X30 to the original unrotated orthogonal axes X1 ; X2 ; X3. [Reprinted from Rossmann & Blow (1962).]

262

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES 0

 sin 1 cos 2 sin 3 B þ cos 1 cos 3 B B B  sin 1 cos 2 cos 3 B @  cos 1 sin 3 sin 1 sin 2

cos 1 cos 2 sin 3 þ sin 1 cos 3 cos 1 cos 2 cos 3  sin 1 sin 3  cos 1 sin 2

sin 2 sin 3

1

(X3) axis, instead of the Y (X2) axis. As most space groups have the unique axis along a3, the angle will define the inclination relative to the unique axis of the space group with this definition.

C C C sin 2 cos 3 C C A

2.3.6.3. Symmetry In analogy with crystal lattices, the rotation function is periodic and contains symmetry. The rotation function has a cell whose periodicity is 2 in each of its three angles. This may be written as

cos 2

and (b) matrix [q] in terms of rotation angle  and the spherical polar coordinates , ’: 0

Rð1 ; 2 ; 3 Þ  Rð1 þ 2n1 ; 2 þ 2n2 ; 3 þ 2n3 Þ 2

2

cos  þ sin cos ’ð1  cos Þ sin cos cos ’ð1  cos Þ B þ sin sin ’ sin  B B B B sin cos cos ’ð1  cos Þ cos  þ cos2 ð1  cos Þ B B  sin sin ’ sin  B B B @  sin2 sin ’ cos ’ð1  cos Þ  sin cos sin ’ð1  cos Þ  cos

sin 

or Rð; ; ’Þ  Rð þ 2n1 ;

where n1, n2 and n3 are integers. A redundancy in the definition of either set of angles leads to the equivalence of the following points:

þ sin cos ’ sin  1  sin cos ’ sin ’ð1  cos Þ C þ cos sin  C C C  sin cos sin ’ð1  cos Þ C C C  sin cos ’ sin  A 2

cos  þ sin2

Rð1 ; 2 ; 3 Þ  Rð1 þ ; 2 ; 3 þ Þ in Eulerian space or

sin2 ’ð1  cos Þ

Rð; ; ’Þ  Rð; 2  ; ’ þ Þ in polar space: These relationships imply an n glide plane perpendicular to 2 for Eulerian space or a ’ glide plane perpendicular to in polar space. In addition, the Laue symmetry of the two Pattersons themselves must be considered. This problem was first discussed by Rossmann & Blow (1962) and later systematized by Tollin et al. (1966), Burdina (1970, 1971, 1973) and Rao et al. (1980). A closely related problem was considered by Hirshfeld (1968). The rotation function will have the same value whether the Patterson density at X or ½T i X in the first crystal is multiplied by the Patterson density at X0 or ½T j X0 in the second crystal. ½T i  and ½T j  refer to the ith and jth crystallographic rotations in the orthogonalized coordinate systems of the first and second crystal, respectively. Hence, from (2.3.6.6)

Alternatively, (b) can be expressed as 0

cos  þ u2 ð1  cos Þ uvð1  cos Þ  w sin  B @ vuð1  cos Þ þ w sin  cos  þ v2 ð1  cos Þ wuð1  cos Þ  v sin  wvð1  cos Þ þ u sin  uwð1  cos Þ þ v sin 

1

C uwð1  cos Þ  u sin  A; cos  þ w2 ð1  cos Þ where u, v and w are the direction cosines of the rotation axis given by u ¼ sin

þ 2n2 ; ’ þ 2n3 Þ;

ð½T j X0 Þ ¼ ½qð½T i XÞ

cos ’;

v ¼ cos ; w ¼  sin sin ’:

or X0 ¼ ½T Tj ½q½T i X:

This latter form also demonstrates that the trace of a rotation matrix is 2 cos  þ 1. The relationship between the two sets of variables established by comparison of the elements of the two matrices yields

Thus, it is necessary to find angular relationships which satisfy the relation

 1 þ 3 ; 2    þ 3   3 tan ’ ¼  cotð2 =2Þ sin 1 sec 1 ; 2 2    3 cos ’ tan ¼ cot 1 : 2

½q ¼ ½T Tj ½q½T i 

cosð=2Þ ¼ cosð2 =2Þ cos

for given Patterson symmetries. Tollin et al. (1966) show that the Eulerian angular equivalences can be expressed in terms of the Laue symmetries of each Patterson (Table 2.3.6.2). The example given by Tollin et al. (1966) is instructive in the use of Table 2.3.6.2. They consider the determination of the Eulerian space group when P1 has symmetry Pmmm and P2 has symmetry P2=m. These Pattersons contain the proper rotation groups 222 and 2 (parallel to b), respectively. Inspection of Table 2.3.6.2 shows that these symmetries produce the following Eulerian relationships:

Since ’ and can always be chosen in the range 0 to , these equations suffice to find ð; ; ’Þ from any set ð1 ; 2 ; 3 Þ. Another definition for the polar angles is also commonly used. In this definition, the angle is measured from the Cartesian Z

263

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.3.6.2. Eulerian symmetry elements for all possible types of space-group rotations Axis

Direction

1

First crystal

Second crystal

ð þ 1 ; 2 ;  þ 3 Þ

ð þ 1 ; 2 ;  þ 3 Þ

2

[010]

ð  1 ;  þ 2 ; 3 Þ

ð1 ;  þ 2 ;   3 Þ

2

[001]

ð þ 1 ; 2 ; 3 Þ

ð1 ; 2 ;  þ 3 Þ

4

[001]

ð=2 þ 1 ; 2 ; 3 Þ

ð1 ; 2 ; =2 þ 3 Þ

3

[001]

ð2=3 þ 1 ; 2 ; 3 Þ

ð1 ; 2 ; 2=3 þ 3 Þ

6

[001]

ð=3 þ 1 ; 2 ; 3 Þ

ð1 ; 2 ; =3 þ 3 Þ

2†

[110]

ð3=2  1 ;   2 ;  þ 3 Þ

ð þ 1 ;   2 ; 3=2  3 Þ

† This axis is not unique (that is, it can always be generated by two other unique axes), but is included for completeness.

Table 2.3.6.3. Numbering of the rotation-function space groups The Laue group of the rotated Patterson map P1 is chosen from the left column and the Laue group of P2 is chosen from the upper row. 2/m, b axis unique

2/m, c axis unique

mmm

4/m

4/mmm

3

3m

6/m

1

1

11

21

31

41

51

61

71

81

91

2=m, b axis unique

2

12

22

32

42

52

62

72

82

92

2=m, c axis unique mmm

3 4

13 14

23 24

33 34

43 44

53 54

63 64

73 74

83 84

93 94

4=m

5

15

25

35

45

55

65

75

85

95

4=mmm

6

16

26

36

46

56

66

76

86

96

3

7

17

27

37

47

57

67

77

87

97

3m

8

18

28

38

48

58

68

78

88

98

6=m

9

19

29

39

49

59

69

79

89

99

10

20

30

40

50

60

70

80

90

100

1

6=mmm

(a) In the first crystal (Pmmm):

6/mmm

When these symmetry operators are combined two cells result, each of which has the space group Pbcb (Fig. 2.3.6.5). The asymmetric unit within which the rotation function need be evaluated is found from a knowledge of the Eulerian space group. In the above example, the limits of the asymmetric unit are 0  1  =2, 0  2   and 0  3  =2. Nonlinear transformations occur when using Eulerian symmetries for threefold axes along [111] (as in the cubic system) or when using polar coordinates. Hence, Eulerian angles are far more suitable for a derivation of the limits of the rotationfunction asymmetric unit. However, when searching for given molecular axes, where some plane of  need be explored, polar angles are more useful. Rao et al. (1980) have determined all possible rotationfunction Eulerian space groups, except for combinations with Pattersons of cubic space groups. They numbered these rotation groups 1 through 100 (Table 2.3.6.3) according to the combination of the Patterson Laue groups. The characteristics of each of the 100 groups are given in Table 2.3.6.4, including the limits of the asymmetric unit. In the 100 unique combinations of noncubic Laue groups, there are only 16 basic rotation-function space groups.

1 2 3 !  þ 1 ; 2 ;  þ 3 ðonefold axisÞ 1 2 3 !   1 ;  þ 2 ; 3 ðtwofold axis parallel to bÞ 1 2 3 !  þ 1 ; 2 ; 3 ðtwofold axis parallel to cÞ: (b) In the second crystal ðP2=mÞ: 1 2 3 !  þ 1 ; 2 ;  þ 3 ðonefold axisÞ 1 2 3 ! 1 ;  þ 2 ;   3 ðtwofold axis parallel to bÞ:

2.3.6.4. Sampling, background and interpretation If the origins are retained in the Pattersons, their product will form a high but constant plateau on which the rotation-function peaks are superimposed; this leads to a small apparent peak-tonoise ratio. The effect can be eliminated by removal of the origins through a modification of the Patterson coefficients. Irrespective of origin removal, a significant peak is one which is more than three r.m.s. deviations from the mean background. As in all continuous functions sampled at discrete points, a convenient grid size must be chosen. Small intervals result in an excessive computing burden, while large intervals might miss peaks. Furthermore, equal increments of angles do not represent equal changes in rotation, which can result in distorted peaks (Lattman, 1972). In general, a crude idea of a useful sampling

Fig. 2.3.6.5. Rotation space group diagram for the rotation function of a Pmmm Patterson function ðP1 Þ against a P2=m Patterson function ðP2 Þ. The Eulerian angles 1 ; 2 ; 3 repeat themselves after an interval of 2. Heights above the plane are given in fractions of a revolution. [Reprinted from Tollin et al. (1966).]

264

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES Table 2.3.6.4. Rotation-function Eulerian space groups The rotation space groups are given in Table 2.3.6.3. No. of the rotation space group

No. of equivalent positions(a)

Symbol(b)

Translation along the 1 axis(c)

Translation along the 3 axis(c)

Range of the asymmetric unit(d)

1

2

Pn

2

2

0  1 < 2,

0  2  ,

0  3 < 2

2

4

Pbn21

2

2

0  1 < 2,

0  2  =2,

0  3 < 2

3 4

4 8

Pc Pbc21

 

2 2

0  1 < , 0  1 < ,

0  2  , 0  2  =2,

0  3 < 2 0  3 < 2

5

8

Pc

=2

2

0  1 < =2,

0  2  ,

0  3 < 2

6

16

Pbc21

=2

2

0  1 < =2,

0  2  =2,

0  3 < 2

7

6

Pn

2=3

2

0  1 < 2=3,

0  2  ,

0  3 < 2

8

12

Pbn21

2=3

2

0  1 < 2=3,

0  2  =2,

0  3 < 2

9

12

Pc

=3

2

0  1 < =3,

0  2  ,

0  3 < 2

10

24

Pbc21

=3

2

0  1 < =3,

0  2  =2,

0  3 < 2

11 12

4 8

P21 nb Pbnb

2 2

2 2

0  1 < 2, 0  1  =2,

0  2  =2, 0  2 < ,

0  3 < 2 0  3 < 2

13

8

P2cb



2

0  1 < ,

0  2  =2,

0  3 < 2

14

16

Pbcb



2

0  1  =2,

0  2  =2,

0  3 < 2

15

16

P2cb

=2

2

0  1 < =2,

0  2  =2,

0  3 < 2

16

32

Pbcb

=2

2

0  1 < =2,

0  2 < ,

0  3  =2

17

12

P21 nb

2=3

2

0  1 < 2=3,

0  2  =2,

0  3 < 2

18

24

Pbnb

2=3

2

0  1 < 2=3,

0  2 < ,

0  3  =2

19 20

24 48

P2cb Pbcb

=3 =3

2 2

0  1 < =3, 0  1 < =3,

0  2  =2, 0  2 < ,

0  3 < 2 0  3  =2

21

4

Pa

2



0  1 < 2,

0  2  ,

0  3 < 

22

8

Pba2

2



0  1 < 2,

0  2  =2,

0  3 < 

23

8

Pm





0  1 < ,

0  2  ,

0  3 < 

24

16

Pbm2





0  1 < ,

0  2  =2,

0  3 < 

25

16

Pm

=2



0  1 < =2,

0  2  ,

0  3 < 

26

32

Pbm2

=2



0  1 < =2,

0  2  =2,

0  3 < 

27 28

12 24

Pa Pba2

2=3 2=3

 

0  1 < 2=3, 0  1 < 2=3,

0  2  , 0  2  =2,

0  3 <  0  3 < 

29

24

Pm

=3



0  1 < =3,

0  2  ,

0  3 < 

30

48

Pbm2

=3



0  1 < =3,

0  2  =2,

0  3 < 

31

8

P21 ab

2



0  1 < 2,

0  2  =2,

0  3 < 

32

16

Pbab

2



0  1  =2,

0  2 < ,

0  3 < 

33

16

P2mb





0  1 < ,

0  2  =2,

0  3 < 

34

32

Pbmb





0  1  =2,

0  2  =2,

0  3 < 

35 36

32 64

P2mb Pbmb

=2 =2

 

0  1 < =2, 0  1 < =2,

0  2  =2, 0  2  =2,

0  3 <  0  3  =2

37

24

P21 ab

2=3



0  1 < 2=3,

0  2  =2,

0  3 < 

38

48

Pbab

2=3



0  1 < 2=3,

0  2  =2,

0  3  =2

39

48

P2mb

=3



0  1 < =3,

0  2  =2,

0  3 < 

40

96

Pbmb

=3



0  1 < =3,

0  2  =2,

0  3  =2

41

8

Pa

2

=2

0  1 < 2,

0  2  ,

0  3 < =2

42

16

Pba2

2

=2

0  1 < 2,

0  2  =2,

0  3 < =2

43 44

16 32

Pm Pbm2

 

=2 =2

0  1 < , 0  1 < ,

0  2  , 0  2  =2,

0  3 < =2 0  3 < =2

45

32

Pm

=2

=2

0  1 < =2,

0  2  ,

0  3 < =2

46

64

Pbm2

=2

=2

0  1 < =2,

0  2  =2,

0  3 < =2

47

24

Pa

2=3

=2

0  1 < 2=3,

0  2  ,

0  3 < =2

48

48

Pba2

2=3

=2

0  1 < 2=3,

0  2  =2,

0  3 < =2

49

48

Pm

=3

=2

0  1 < =3,

0  2  ,

0  3 < =2

50

96

Pbm2

=3

=2

0  1 < =3,

0  2  =2,

0  3 < =2

51 52

16 32

P21 ab Pbab

2 2

=2 =2

0  1 < 2, 0  1 < 2,

0  2  =2, 0  2  =2,

0  3 < =2 0  3  =4

53

32

P2mb



=2

0  1 < ,

0  2  =2,

0  3 < =2

54

64

Pbmb



=2

0  1  =2,

0  2  =2,

0  3 < =2

55

64

P2mb

=2

=2

0  1 < =2,

0  2  =2,

0  3 < =2

56

128

Pbmb

=2

=2

0  1  =4,

0  2  =2,

0  3 < =2

57

48

P21 ab

2=3

=2

0  1 < 2=3,

0  2  =2,

0  3 < =2

58

96

Pbab

2=3

=2

0  1 < 2=3,

0  2  =2,

0  3  =4

265

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.3.6.4 (cont.) No. of the rotation space group

No. of equivalent positions(a)

Symbol(b)

Translation along the 1 axis(c)

Translation along the 3 axis(c)

Range of the asymmetric unit(d)

59

96

P2mb

=3

=2

0  1 < =3,

0  2  =2,

0  3 < =2

60

192

Pbmb

=3

=2

0  1  =6,

0  2  =2,

0  3 < =2

61

6

Pn

2

2=3

0  1 < 2,

0  2  ,

0  3 < 2=3

62

12

Pbn21

2

2=3

0  1 < 2,

0  2  =2,

0  3 < 2=3

63

12

Pc



2=3

0  1 < ,

0  2  ,

0  3 < 2=3

64 65

24 24

Pbc21 Pc

 =2

2=3 2=3

0  1 < , 0  1 < =2,

0  2  =2, 0  2  ,

0  3 < 2=3 0  3 < 2=3

66

48

Pbc21

=2

2=3

0  1 < =2,

0  2  =2,

0  3 < 2=3

67

18

Pn

2=3

2=3

0  1 < 2=3,

0  2  ,

0  3 < 2=3

68

36

Pbn21

2=3

2=3

0  1 < 2=3,

0  2  =2,

0  3 < 2=3

69

36

Pc

=3

2=3

0  1 < =3,

0  2  ,

0  3 < 2=3

70

72

Pbc21

=3

2=3

0  1 < =3,

0  2  =2,

0  3 < 2=3

71

12

P21 nb

2

2=3

0  1 < 2,

0  2  =2,

0  3 < 2=3

72 73

24 24

Pbnb P2cb

2 

2=3 2=3

0  1  =2, 0  1 < ,

0  2 < , 0  2  =2,

0  3 < 2=3 0  3 < 2=3

74

48

Pbcb



2=3

0  1  =2,

0  2  =2,

0  3 < 2=3

75

48

P2cb

=2

2=3

0  1 < =2,

0  2  =2,

0  3 < 2=3

76

96

Pbcb

=2

2=3

0  1  =4,

0  2  =2,

0  3 < 2=3

77

36

P21 nb

2=3

2=3

0  1 < 2=3,

0  2  =2,

0  3 < 2=3

78

72

Pbnb

2=3

2=3

0  1  =6,

0  2  ,

0  3 < 2=3

79

72

P2 cb

=3

2=3

0  1 < =3,

0  2  =2,

0  3 < 2=3

80 81

144 12

Pbcb Pa

=3 2

2=3 =3

0  1  =6, 0  1 < 2,

0  2  =2, 0  2  ,

0  3 < 2=3 0  3 < =3

82

24

Pba2

2

=3

0  1 < 2,

0  2  =2,

0  3 < =3

83

24

Pm



=3

0  1 < ,

0  2  ,

0  3 < =3

84

48

Pbm2



=3

0  1 < ,

0  2  =2,

0  3 < =3

85

48

Pm

=2

=3

0  1 < =2,

0  2  ,

0  3 < =3

86

96

Pbm2

=2

=3

0  1 < =2,

0  2  =2,

0  3 < =3

87

36

Pa

2=3

=3

0  1 < 2=3,

0  2  ,

0  3 < =3

88 89

72 72

Pba2 Pm

2=3 =3

=3 =3

0  1 < 2=3, 0  1 < =3,

0  2  =2, 0  2  ,

0  3 < =3 0  3 < =3

90

144

Pbm2

=3

=3

0  1 < =3,

0  2  =2,

0  3 < =3

91

24

P21 ab

2

=3

0  1 < 2,

0  2  =2,

0  3 < =3

92

48

Pbab

2

=3

0  1  =2,

0  2 < ,

0  3 < =3

93

48

P2mb



=3

0  1 < ,

0  2  =2,

0  3 < =3

94

96

Pbmb



=3

0  1  =2,

0  2  =2,

0  3  =2

95

96

P2mb

=2

=3

0  1 < =2,

0  2  =2,

0  3 < =3

96 97

192 72

Pbmb P21 ab

=2 2=3

=3 =3

0  1  =4, 0  1 < 2=3,

0  2  =2, 0  2  =2,

0  3 < =3 0  3 < =3

98

144

Pbab

2=3

=3

0  1 < 2=3,

0  2  =2,

0  3  =6

99

144

P2mb

=3

=3

0  1 < =3,

0  2  =2,

0  3 < =3

100

288

Pbmb

=3

=3

0  1  =6,

0  2  =2,

0  3 < =3

Notes: (a) This is the number of equivalent positions in the rotation unit cell. (b) Each symbol retains the order 1 ; 2 ; 3 . The monoclinic space groups have the b axis unique setting. (c) This is a translation symmetry: e.g. for the case of =2 translation along the 1 axis, 1 ; 2 ; 3 goes to =2 þ 1 ; 2 ; 3 and  þ 1 ; 2 ; 3 , and 3=2 þ 1 ; 2 ; 3 . All other equivalent positions in the basic rotation space group are similarly translated. (d) Several consistent sets of ranges exist but the one with the minimum range of 2 is listed.

˚ for a virus structure determination. In addiprotein or 6 to 5 A ˚ or 3.5 to tion, use of restricted resolution ranges, such as 6 to 5 A ˚ 3.0 A, has been found in numerous cases to give especially well defined results (Arnold et al., 1984). When exploring the rotation function in polar coordinates, there is no significance to the latitude ’ (Fig. 2.3.6.4) when ¼ 0. For small values of , the rotation function will be quite insensitive to ’, which therefore needs to be explored only at coarse intervals (say 45 ). As approaches the equator at 90 , optimal increments of and ’ will be about equal. A similar situation exists with Eulerian angles. When 2 ¼ 0, the rotation function ¼ 0 and will be determined by 1 þ 3 , corresponding to

interval can be obtained by considering the angle necessary to move one reciprocal-lattice point onto its neighbour (separated by a ) at the extremity of the resolution limit, R. This interval is given by  ¼ a =2ð1=RÞ ¼ 12Ra : Simple sharpening of the rotation function can be useful. This can be achieved by restricting the computations to a shell in reciprocal space or by using normalized structure factors. Useful ˚ for an average limits are frequently 10 to 6, 10 to 4 or 10 to 3.5 A

266

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES function (Rossmann et al., 1972) for use when a molecule has more than one noncrystallographic symmetry axis. It is then possible to determine the rotation-function values for each molecular axis for a chosen molecular orientation (Fig. 2.3.6.6) (see Section 2.3.6.6). Another problem in the interpretation of rotation functions is the appearance of apparent noncrystallographic symmetry that relates the self-Patterson of one molecule to the self-Patterson of a crystallographically related molecule. For example, take the case of -chymotrypsin (Blow et al., 1964). The space group is P21 with a molecular dimer in each of the two crystallographic asymmetric units. The noncrystallographic dimer axis was found to be perpendicular to the crystallographic 21 axis. The product of the crystallographic twofold in the Patterson with the orthogonal twofold in the self-Patterson vectors around the origin creates a third twofold, orthogonal to both of the other twofolds. In real space this represents a twofold screw direction relating the two dimers in the cell. In other cases, the product of the crystallographic and noncrystallographic symmetry results in symmetry which only has meaning in terms of all the vectors in the vicinity of the Patterson origin, but not in real space. Rotation-function peaks arising from such products are called Klug peaks (Johnson et al., 1975). Such peaks normally refer to the total symmetry of all the vectors around the Patterson origin and may, therefore, be much larger than the peaks due to noncrystallographic symmetry within one molecule alone. Hence the Klug peaks, if not correctly ˚ kervall et al., recognized, can lead to erroneous conclusions (A 1972). Litvin (1975) has shown how Klug peaks can be predicted. These usually occur only for special orientations of a particle with a given symmetry relative to the crystallographic symmetry axes. Prediction of Klug peaks requires the simultaneous consideration of the noncrystallographic point group, the crystallographic point group and their relative orientations. A special, but frequently occurring, situation arises when an evenfold noncrystallographic symmetry operator (e.g. 2-, 4-, 6-, 8etc. fold axes) is parallel, or nearly parallel, to a crystallographic evenfold axis or screw axis. If the crystallographic evenfold axis is, say, parallel to Z, then if the centre of molecule I is at ðX0 ; Y0 ; Z0 Þ, the centre of molecule II will be at ðX0 ; Y0 ; Z0 Þ. If molecule I has an evenfold axis parallel to Z, then for every atom (a) at ðx þ X0 ; y þ Y0 ; z þ Z0 Þ, there will be an atom (b) at ðx þ X0 ; y þ Y0 ; z þ Z0 Þ. The crystallographic symmetryequivalent positions of these two atoms in molecule II will be at (c) ðx  X0 ; y  Y0 ; z þ Z0 Þ and (d) ðx  X0 ; y  Y0 ; z þ Z0 Þ. The vectors between atoms (a) and (d) and also between atoms (b) and (c) will both have component lengths of ð2X0 ; 2Y0 ; 0Þ. The position of this vector in a Patterson map is independent of the actual atoms in the molecule and depends only on the position of the molecular noncrystallographic symmetry axis. Every atom will produce two vectors of this type, all of which will accumulate in a Patterson map to produce a large peak, which establishes the exact position of the noncrystallographic symmetry evenfold axis relative to the crystallographic axis. The position of the special peak is on the Harker section, namely at w = 0 for a crystallographic twofold axis and at w = 12 for a crystallographic 21 screw axis. If there are N atoms in the structure of the two crystallographically related dimers, then the height of the origin is proportional to N (the number of zerolength vectors). The number of vectors with length ð2X0 ; 2Y0 ; 0Þ will be twice the number of atoms in each monomer, or 2  (N/4), which is N/2. Thus the special peak should be about half the height of the Patterson map’s origin peak. In practice, the peak is often somewhat lower because the noncrystallographic symmetry and crystallographic axes might not be exactly parallel. This situation can be mitigated by computing the Patterson map with lower-resolution reflections only, as the difference in orientation between the axes is less significant when viewed at lower resolution (McKenna et al., 1992).

Fig. 2.3.6.6. The locked rotation function, L, applied to the determination of the orientation of the common cold virus (Arnold et al., 1984). There are four virus particles per cubic cell with each particle sitting on a threefold axis. The locked rotation function explores all positions of rotation about this axis and, hence, repeats itself after 120 . The locked rotation function is determined from the individual rotation-function values of the noncrystallographic symmetry directions of a 532 icosahedron. [Reprinted with permission from Arnold et al. (1984).]

varying  in polar coordinates. There will be no dependence on ð1  3 Þ. Thus Eulerian searches can often be performed more economically in terms of the variables  ¼ 1 þ 3 and  ¼ 1  3 , where 0 B B B B B B B ½q ¼ B B B B B B B @




 2 2  2 2 þ cos  sin 2
   sin  cos2 2  2   þ sin  sin2 2 2 cos  cos2

sin 2 sinð þ Þ




 2 2  2 2 þ sin  sin 2
  cos  cos2 2 2    cos  sin2 2 2 sin  cos2

 sin 2 cosð þ Þ

1 sin 2 sinð  Þ C C C C C C C C; sin 2 cosð  Þ C C C C C C A cos 2

which reduces to the simple rotation matrix 0

1 cos  sin  0 ½q ¼ @  sin  cos  0 A 0 0 1 when 2 ¼ 0. The computational effort to explore carefully a complete asymmetric unit of the rotation-function Eulerian group can be considerable. However, unless improper rotations are under investigation (as, for example, cross-rotation functions between different crystal forms of the same molecule), it is not generally necessary to perform such a global search. The number of molecules per crystallographic asymmetric unit, or the number of subunits per molecule, are often good indicators as to the possible types of noncrystallographic symmetry element. For instance, in the early investigation of insulin, the rotation function was used to explore only the  ¼ 180 plane in polar coordinates as there were only two molecules per crystallographic asymmetric unit (Dodson et al., 1966). Rotation functions of viruses, containing 532 icosahedral symmetry, are usually limited to exploration of the  = 180, 120, 72 and 144 planes [e.g. Rayment et al. (1978) and Arnold et al. (1984)]. In general, the interpretation of the rotation function is straightforward. However, noise often builds up relative to the signal in high-symmetry space groups or if the data are limited or poor. One aid to a systematic interpretation is the locked rotation

267

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION
P P

2.3.6.5. The fast rotation function Rð1 ; 2 ; 3 Þ ¼

Unfortunately, the rotation-function computations can be extremely time-consuming by conventional methods. Sasada (1964) developed a technique for rapidly finding the maximum of a given peak by looking at the slope of the rotation function. A major breakthrough came when Crowther (1972) recast the rotation function in a manner suitable for rapid computation. Only a brief outline of Crowther’s fast rotation function is given here. Details are found in the original text (Crowther, 1972) and his computer program description. Since the rotation function correlates spherical volumes of a given Patterson density with rotated versions of either itself or another Patterson density, it is likely that a more natural form for the rotation function will involve spherical harmonics rather than the Fourier components jFh j2 of the crystal representation. Thus, if the two Patterson densities P1 ðr; ; ’Þ and P2 ðr; ; ’Þ are expanded within the spherical volume of radius less than a limiting value of a, then P1 ðr; ; ’Þ ¼

P

mm0

 clmm0 dlm0 m ð2 Þ exp½iðm0 3 þ m1 Þ:

l

The coefficients clmm0 refer to a particular pair of Patterson densities and are independent of the rotation. The coefficients Dlm0 m , containing the whole rotational part, refer to rotations of spherical harmonics and are independent of the particular Patterson densities. Since the summations over m and m0 represent a Fourier synthesis, rapid calculation is possible. As polar coordinates rather than Eulerian angles provide a more graphic interpretation of the rotation function, Tanaka (1977) has recast the initial definition as R

Rð1 ; 2 ; 3 Þ ¼

½R ð1 ; 2 ; 3 ¼ 0ÞP1 ðr; ; ’Þ

sphere

 ½R ð1 ; 2 ; 3 ÞP2 ðr; ; ’Þ dV R ¼ ½P1 ðr; ; ’Þ½R 1 ð1 ; 2 ; 3 ¼ 0Þ



almn ^jl ðkln rÞY^ lm ð ; ’Þ

sphere

lmn

 R ð1 ; 2 ; 3 ÞP2 ðr; ; ’Þ dV:

and P2 ðr; ; ’Þ ¼

P l0 m0 n0

He showed that the polar coordinates are now equivalent to  ¼ 3 , ¼ 2 and ’ ¼ 1  =2. The rotation function can then be expressed as

0 bl0 m0 n0 ^jl0 ðkl0 n0 rÞY^ lm0 ð ; ’Þ;

and the rotation function would then be defined as R¼

R

Rð; ; ’Þ ¼

 P P lmm0

P1 ðr; ; ’ÞR P2 ðr; ; ’Þr2 sin

dr d d’:

Here Y^ lm ð ; ’Þ is the normalized spherical harmonic of order l; ^jl ðkln rÞ is the normalized spherical Bessel function of order l; almn , blmn are complex coefficients; and R P2 ðr; ; ’Þ represents the rotated second Patterson. The rotated spherical harmonic can then be expressed in terms of the Eulerian angles 1 ; 2 ; 3 as l P

Dlqm ð1 ; 2 ; 3 ÞY^ lq ð ; ’Þ;

where Dlqm ð1 ; 2 ; 3 Þ ¼ expðiq3 Þdlqm ð2 Þ expðim1 Þ

lmm0 n

Many oligomers of macromolecules obey simple point-group symmetry, which is maintained as noncrystallographic symmetry when they are crystallized. For example, a homo-tetramer often obeys 222 point-group symmetry, and icosahedral viruses obey 532 point-group symmetry. The locked rotation function takes advantage of this information and can greatly simplify the calculation and the interpretation of rotation functions (Fig. 2.3.6.6) (Arnold et al., 1984; Rossmann et al., 1972; Tong, 2001a; Tong & Rossmann, 1990, 1997). During the rotation-function calculation, the noncrystallographic symmetry of the crystal is locked to the presumed point group, hence the name locked rotation function. Given the noncrystallographic symmetry point group, a standard orientation can be defined which serves as a reference orientation for this point group. For example, for 222 point-group symmetry, the standard orientation can be defined such that the three twofold axes are parallel to the three Cartesian coordinate axes that are defined with respect to the crystal unit cell. Once the standard orientation is defined, any orientation of the noncrystallographic symmetry point group can be related to the standard

almn blm0 n Dlm0 m ð1 ; 2 ; 3 Þ:

Since the radial summation over n is independent of the rotation, clmm0 ¼

P

almn blmn ;

n

and hence Rð1 ; 2 ; 3 Þ ¼

P lmm0

q

2.3.6.6. Locked rotation functions

and dlqm ð2 Þ are the matrix elements of the three-dimensional rotation group. It can then be shown that P

n

0

fdlqm ð Þdlqm0 ð Þð1Þðm mÞ

permitting rapid calculation of the fast rotation function in polar coordinates. Crowther (1972) uses the Eulerian angles , , which are related to those defined by Rossmann & Blow (1962) according to 1 ¼  þ =2, 2 ¼  and 3 ¼  =2. An alternative formulation of the fast rotation function, which reduces the errors in the calculation, is implemented in AMoRe (Navaza, 1987, 1993, 1994, 2001a). New target functions derived from the principle of maximum likelihood have been implemented in conjunction with fast rotation functions in the program Phaser, which can also take advantage of partial model information in orienting unknown fragments (Storoni et al., 2004).

q¼l

Rð1 ; 2 ; 3 Þ ¼

P

 exp½iðqÞ exp½iðm0  mÞ’g;

sphere

R ð1 ; 2 ; 3 ÞY^ lm ð ; ’Þ ¼

almn blm0 n

clmm0 Dlm0 m ð1 ; 2 ; 3 Þ

or

268

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES orientation by a single set of three rotation angles that determine the rotation matrix [E]. Assume [In] (n = 1, . . . , N) is the collection of noncrystallographic symmetry point-group rotation matrices in the standard orientation. Then the operation [E] will bring the noncrystallographic symmetry point group to a new orientation and the noncrystallographic symmetry rotation matrices in this new orientation, ½qn , are given by (Tong & Rossmann, 1990)

Therefore, ½qn  represents the rotational relationship between the monomer search model and the monomers of the assembly in the crystal. An ordinary cross-rotation-function value Rn can be calculated for each of the rotations ½qn , and the locked crossrotation-function value is defined as the average

½qn  ¼ ½E½In ½E1 :

Like the locked self rotation function, the locked cross rotation function can determine the orientation of all the monomers of the noncrystallographic symmetry assembly with a single rotation.

RL ¼ ð1=NÞ

Rn :

n

ð2:3:6:8Þ

For each rotation [E], the ordinary self-rotation-function value (Rn) for each of the noncrystallographic symmetry rotation matrices in the new orientation ð½qn Þ is calculated. The locked self-rotation-function value (RL) for this rotation is defined as the average of the ordinary rotation-function values over the noncrystallographic symmetry elements

RL ð½EÞ ¼

P

2.3.7. Translation functions 2.3.7.1. Introduction The problem of determining the position of a noncrystallographic symmetry element in space, or the position of a molecule of known orientation in a unit cell, has been reviewed by Rossmann (1972), Colman et al. (1976), Karle (1976), Argos & Rossmann (1980), Harada et al. (1981) and Beurskens (1981). All methods depend on the prior knowledge of the object’s orientation implied by the rotation matrix [C]. The various translation functions, T, derived below, can only be computed given this information. The general translation function can be defined as

N 1 X R; N  1 n¼2 n

where the summation starts from 2 as it is assumed that [I1] is the identity matrix. The locked self rotation function simplifies the task of interpreting the self rotation function for the orientation of an noncrystallographic symmetry assembly. Instead of searching for N  1 peaks in the ordinary self rotation function, a single peak is sought in the locked self rotation function. It must be emphasized that this rotation ([E]) in the locked self rotation function is most often a general rotation. The locked self rotation function also reduces the noise in the rotation-function calculation by a factor of ðN  1Þ1=2 due to the averaging of the ordinary rotationfunction values (Tong & Rossmann, 1990). The symmetry of the locked self rotation function is generally rather complex and an analytical solution is often impossible (Tong & Rossmann, 1990). It depends not only on the crystallographic symmetry and the noncrystallographic symmetry, but also on the definition of the standard orientation of the noncrystallographic symmetry. For example, if the standard orientation is defined such that the twofold axes are parallel to the Cartesian coordinate axes for the 222 point group, a 90 rotation around the X, Y or Z axis, or a 120 rotation around the 111 direction, does not cause a net change to the standard orientation. Such rotations will appear as symmetry in the locked self rotation function (Tong & Rossmann, 1997). In practice, the locked self rotation function can be calculated rather quickly, especially if the fast rotation function is used. A large region of rotation space can be explored in the calculation of the locked rotation function and the solutions can then be clustered based on the resulting orientation of the noncrystallographic symmetry. For example, two rotations [E1] and [E2] that produce the same set of noncrystallographic symmetry matrices based on (2.3.6.8) are likely to be related by the symmetry of the locked self rotation function. A locked cross rotation function can also be defined to determine the orientation, [F], of the known monomer structure relative to the noncrystallographic symmetry of the molecular assembly (Navaza et al., 1998; Tong, 2001a; Tong & Rossmann, 1990, 1997). With the knowledge of [F] and the orientation of the noncrystallographic symmetry in the crystal [E], which can be determined from the locked self rotation function, the orientation of all the monomers in the crystal cell is given by

R TðSx ; Sx0 Þ ¼
1 ðxÞ

2 ðx0 Þ dx; U

where T is a six-variable function given by each of the three components that define Sx and Sx0 . Here Sx and Sx0 are equivalent reference positions of the objects, whose densities are
1 ðxÞ and
2 ðx0 Þ. The translation function searches for the optimal overlap of the two objects after they have been similarly oriented. Following the same procedure used for the rotation-function derivation, Fourier summations are substituted for
1 ðxÞ and
2 ðx0 Þ. It can then be shown that Z(

) 1 X TðSx ; Sx0 Þ ¼ jF j exp½iðh  2h
xÞ Vh h h U ( ) 1 X 0  jF j exp½iðp  2p
x Þ dx: Vp p p Using the substitution x0 ¼ ½Cx þ d and simplifying leads to TðSx ; S0x Þ ¼

1 XX jF jjF j Vh Vp h p h p  exp½iðh þ p  2p
dÞ Z  expf2iðh þ ½CT pÞ
xg dx: U

The integral is the diffraction function Ghp (2.3.6.4). If the integration is taken over the volume U, centred at Sx and Sx0 , it follows that TðSx ; Sx0 Þ ¼

2 XX jF jjF jG Vh Vp h p h p hp  cos½h þ p  2ðh
Sx þ p
Sx0 Þ:

½qn  ¼ ½E½In ½F:

269

ð2:3:7:1Þ

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.3.7.1. Crosses represent atoms in a two-dimensional model structure. The triangles are the points chosen as approximate centres of molecules A and B. AB has components t and s parallel and perpendicular, respectively, to the screw rotation axis. [Reprinted from Rossmann et al. (1964).]

Fig. 2.3.7.2. Vectors arising from the structure in Fig. 2.3.7.1. The self-vectors of molecules A and B are represented by + and
; the cross-vectors from molecules A to B and B to A by  and *. Triangles mark the position of þAB and AB . [Reprinted from Rossmann et al. (1964).]

2.3.7.2. Position of a noncrystallographic element relating two unknown structures The function (2.3.7.1) is quite general. For instance, the rotation function corresponds to a comparison of Patterson functions P1 and P2 at their origins. That is, the coefficients are F 2, phases are zero and Sx ¼ Sx0 ¼ 0. However, the determination of the translation between two objects requires the comparison of crossvectors away from the origin. Consider, for instance, the determination of the precise translation vector parallel to a rotation axis between two identical molecules of unknown structure. For simplicity, let the noncrystallographic axis be a dyad (Fig. 2.3.7.1). Fig. 2.3.7.2 shows the corresponding Patterson of the hypothetical point-atom structure. Opposite sets of cross-Patterson vectors in Fig. 2.3.7.2 are related by a twofold rotation and a translation equal to twice the precise vector in the original structure. A suitable translation function would then compare a Patterson at S with the rotated Patterson at S. Hence, substituting Sx ¼ S and Sx0 ¼ S in (2.3.7.1), TðSÞ ¼

2 XX jF j2 jFp j2 Ghp cos½2ðh  pÞ
S: V2 h p h

unknown crystal. For instance, if the structure of an enzyme has previously been determined by the isomorphous replacement method, then the structure of the same enzyme from another species can often be solved by molecular replacement [e.g. Grau et al. (1981)]. However, there are some severe pitfalls when, for instance, there are gross conformational changes [e.g. Moras et al. (1980)]. This type of translation function could also be useful in the interpolation of E maps produced by direct methods. Here there may often be confusion as a consequence of a number of molecular images related by translations (Karle, 1976; Beurskens, 1981; Egert & Sheldrick, 1985). Tollin’s (1966) Q function and Crowther & Blow’s (1967) translation function are essentially identical (Tollin, 1969) and depend on a prior knowledge of the search molecule as well as its orientation in the unknown cell. The derivation given here, however, is somewhat more general and follows the derivation of Argos & Rossmann (1980), and should be compared with the method of Harada et al. (1981). If the known molecular structure is correctly oriented into a cell (p) of an unknown structure and placed at S with respect to a defined origin, then a suitable translation function is

ð2:3:7:2Þ

TðSÞ ¼

The opposite cross-vectors can be superimposed only if an evenfold rotation between the unknown molecules exists. The translation function (2.3.7.2) is thus applicable only in this special situation. There is no published translation method to determine the interrelation of two unknown structures in a crystallographic asymmetric unit or in two different crystal forms. However, another special situation exists if a molecular evenfold axis is parallel to a crystallographic evenfold axis. In this case, the position of the noncrystallographic symmetry element can be easily determined from the large peak in the corresponding Harker section of the Patterson. In general, it is difficult or impossible to determine the positions of noncrystallographic axes (or their intersection at a molecular centre). However, the position of heavy atoms in isomorphous derivatives, which usually obey the noncrystallographic symmetry, can often determine this information.

P

jFp; obs j2 jFp ðSÞj2 :

ð2:3:7:3Þ

p

This definition is preferable to one based on an R-factor calculation as it is more amenable to computation and is independent of a relative scale factor. The structure factor Fp ðSÞ can be calculated by modifying expression (2.3.8.9) (see below). That is, " # N X UX Fp ðSÞ ¼ expð2ip
Sn Þ Fh Ghpn expð2ih
SÞ ; Vh n¼1 h where Vh is the volume of cell (h) and Sn is the position, in the nth crystallographic asymmetric unit, of cell (p) corresponding to S in known cell (h). Let

2.3.7.3. Position of a known molecular structure in an unknown unit cell The most common type of translation function occurs when looking for the position of a known molecular structure in an

Ap; n expði n Þ ¼

P h

270

Fh Ghpn expð2ih
SÞ;

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES which are the coefficients of the molecular transform for the known molecule placed into the nth asymmetric unit of the p cell. Thus Fp ðSÞ ¼

functions have been derived that estimate the amount of overlap among the models (Harada et al., 1981; Hendrickson & Ward, 1976; Rabinovich & Shakked, 1984; Simpson et al., 2001), and such considerations can frequently limit the search volume very considerably. Alternatively, a simple enumeration of the actual close contacts among different molecules in the crystal (for ˚ ) has also been found to example, C–C distances less than 3 A be an effective way of eliminating those solutions that produce unreasonable crystal packing (Jogl et al., 2001; Tong, 1993). If conformational differences are expected between the search atomic model and the actual structure, care must be taken when applying this packing check.

N UX A exp½ið þ 2p
Sn Þ Vh n¼1 p; n

or Fp ðSÞ ¼

N UX A exp½ið n þ 2pn
SÞ; Vh n¼1 p; n

2.3.7.4. Position of a noncrystallographic symmetry element in a poorly defined electron-density map

where pn ¼ ½CTn p and S ¼ S1 . Hence jFp ðSÞj2 ¼



U Vh

2 XX

n

If an initial set of poor phases, for example from an SIR derivative, are available and the rotation function has given the orientation of a noncrystallographic rotation axis, it is possible to search the electron-density map systematically to determine the translation axis position. The translation function must, therefore, measure the quality of superposition of the poor electrondensity map on itself. Hence Sx ¼ Sx0 ¼ S and the function (2.3.7.1) now becomes

Ap; n Ap; m

m

  expfi½2ðpn  pm Þ
S þ ð n  m Þg ; and then from (2.3.7.3) 

U TðSÞ ¼ Vh

2 XXX p

n

TðSÞ ¼

2

jFp; obs j Ap; n Ap; m

m

 expfi½2ðpn  pm Þ
S þ ð n  m Þg ;

ð2:3:7:4Þ

This real-space translation function has been used successfully to determine the intermolecular dyad axis for -chymotrypsin (Blow et al., 1964) and to verify the position of immunoglobulin domains (Colman & Fehlhammer, 1976).

which is a Fourier summation with known coefficients fjFp; obs j2 Ap; n Ap; m  exp½ið n  m Þg such that T(S) will be a maximum at the correct molecular position. Terms with n ¼ m in expression (2.3.7.4) can be omitted as they are independent of S and only contribute a constant to the value of T(S). For terms with n 6¼ m, the indices take on special values. For instance, if the p cell is monoclinic with its unique axis parallel to b such that p1 ¼ ðp; q; rÞ and p2 ¼ ð
p; q; r
Þ, then p1  p2 would be (2p, 0, 2r). Hence, T(S) would be a twodimensional function consistent with the physical requirement that the translation component, parallel to the twofold monoclinic axis, is arbitrary. Crowther & Blow (1967) show that if FM are the structure factors of a known molecule correctly oriented within the cell of the unknown structure at an arbitrary molecular origin, then (altering the notation very slightly from above) TðSÞ ¼

P

2 XX jF jjF jG cos½h þ p  2ðh þ pÞ
S: Vh2 h p h p hp

2.3.7.5. Locked translation function In a translation search, an atomic model with a given orientation is moved systematically through the unit cell. In such a situation, the structure-factor equation takes on the special form (Harada et al., 1981; Rae, 1977; Tong, 1993) P

Fch ¼

Fh;n expð2ihT ½Tn SÞ;

n

where S is the translation vector and the summation goes over the crystallographic symmetry operators. Fh;n is the structure factor calculated based only on the nth symmetry-related molecule, Fh;n ¼

jFobs ðpÞj2 FM ðpÞFM ðp½CÞ expð2ip
SÞ;

P

fj expf2ihT ð½Tn x0j þ tn Þg;

j

p

where x0j represents the atomic position of the model at the reference position and the summation goes over all the atoms. Noting equation (2.3.7.3), the translation function is given by

where [C] is a crystallographic symmetry operator relative to which the molecular origin is to be determined. This is of the same form as (2.3.7.4) but concerns the special case where the h cell, into which the known molecule was placed, has the same dimensions as the p cell. The translation function as defined by (2.3.7.4) is on an arbitrary scale, which makes it difficult to compare results from different calculations. Translation functions can also be defined based on the crystallographic R factor or a correlation coefficient (CC). In particular, CCs based on reflection intensities can be evaluated by Fourier methods (Navaza & Vernoslova, 1995), although it is still computationally more expensive than the evaluation of (2.3.7.4). Alternatively, the translation function can be calculated first with (2.3.7.4), and then the R factor and CC can be calculated for the resulting top solutions. A correct solution should also produce satisfactory packing arrangements of the molecular models in the crystal. Packing

TðSÞ ¼

PP

jFoh j2 jFh;n j2 PP P o 2 þ jFh j jFh;n Fh;m expf2ihT ð½Tm   ½Tn ÞSg; h

n

h

n m6¼n

ð2:3:7:5Þ where the second term is the ordinary translation function, analogous to (2.3.7.4). The first term of (2.3.7.5) depends on the orientation of the model. Maximization of this term, or its correlation coefficient equivalent, is the basis behind the Patterson-correlation refinement (Bru¨nger, 1990; Tong, 1996b) and the direct rotation function (DeLano & Bru¨nger, 1995). It is

271

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION also related to the intensity-based domain refinement (Yeates & Rini, 1990). In the presence of noncrystallographic symmetry, the locked self rotation function can be used to define the orientation of the noncrystallographic symmetry point group in the crystal. If an atomic model is available for the monomer but not for the entire oligomer, the locked cross rotation function can be used to determine the orientation of this monomer in the oligomer. The locked translation function can then be used to determine the position of this monomer relative to the centre of the noncrystallographic symmetry point group (Tong, 1996b, 2001a), which will produce a model for the entire oligomer. The centre of this oligomer in the crystal can be defined by a simple translation search. With the knowledge of the orientation of one monomer of the oligomer, the first term of (2.3.7.5) is dependent on the position of this monomer relative to the centre of the noncrystallographic symmetry oligomer (Tong, 1996b). The atomic positions of the entire noncrystallographic symmetry oligomer in the standard orientation are given by

(Kissinger et al., 1999), GLRF (part of the Replace package) (Tong, 1993, 2001a; Tong & Rossmann, 1990, 1997), Molrep (Vagin & Teplyakov, 2000) and Phaser (Storoni et al., 2004). The correct placement of an atomic model in a crystal unit cell is generally a six-dimensional problem, with three degrees of rotational freedom and three degrees of translational freedom. Systematic examination of all six degrees of freedom at the same time is computationally expensive and cannot be used routinely (Fujinaga & Read, 1987; Rabinovich & Shakked, 1984; Sheriff et al., 1999). On the other hand, directed sampling of the six degrees of freedom, driven by a stochastic or genetic algorithm (Chang & Lewis, 1997; Glykos & Kokkinidis, 2000; Kissinger et al., 1999), has been successful in solving structures. Traditionally, the calculations are divided into a rotational component (the rotation function) and a translational component (the translation function). Only a few rotation angles (for example the top few peaks of the rotation function) are manually passed to the translation function for examination (Fitzgerald, 1988). With the power of modern computers, it is now possible to perform limited six-dimensional searches, with the sampling of the rotational degrees of freedom guided by the rotation function. For example, the top peaks of the rotation function (Navaza, 1994) and their neighbours (Urzhumtsev & Podjarny, 1995) can be automatically examined by the translation function. A more general approach is to examine all rotation-function grid points with values greater than a certain threshold (Tong, 1996a). Such combined molecular replacement protocols have been found to be very powerful in solving new structures.

 Xn;j ¼ ½In  ½FX0j þ V0 ; where X0j are the atomic positions of the monomer model, centred at (0, 0, 0); [F] is the orientation of this model in the oligomer in the standard orientation; V0 is the position of this monomer relative to the centre of the oligomer; and [In] is the nth noncrystallographic symmetry rotation matrix in the standard orientation. The atomic positions of the noncrystallographic symmetry oligomer in the crystal unit cell, centred at the origin, are given by

2.3.8. Molecular replacement 2.3.8.1. Using a known molecular fragment The most straightforward application of the molecular replacement method occurs when the orientation and position of a known molecular fragment in an unknown cell have been previously determined. The simple procedure is to apply the rotation and translation operations to the known fragment. This will place it into one ‘standard’ asymmetric unit of the unknown cell. Then the crystal operators (assuming no further noncrystallographic operators are present in the unknown cell) are applied to generate the complete unit cell of the unknown structure. Structure factors can then be calculated from the rotated and translated known molecule into the unknown cell. The resultant model can be refined in numerous ways. More generally, consider a molecule placed in any crystal cell (h), within which coordinate positions shall be designated by x. Let the corresponding structure factors be Fh. It is then possible to compute the structure factors Fp for another cell (p) into which the same molecule has been placed N times related by the crystallographic symmetry operators ½C 1 ; d1 ; ½C 2 ; d2 ; . . . ; ½CN ; dN. Let the electron density at a point y1 in the first crystallographic asymmetric unit be spatially related to the point yn in the nth asymmetric unit of the p crystal such that

 xn;j ¼ ½a½EXn;j ¼ ½a½E½In  ½FX0j þ V0 ; where [E] is the orientation of the noncrystallographic symmetry in the crystal unit cell and ½a is the deorthogonalization matrix. By incorporating the calculated structure factors based on this noncrystallographic symmetry oligomer into the first term of (2.3.7.5), the locked translation function is given by P

jFoh j2 jFh j2 PP P o 2 ¼ jFh j Fh;n Fh;m expf2ihð½hm   ½hn ÞV0 g;

TL ðV0 Þ ¼

h h

n m6¼n

ð2:3:7:6Þ P where ½hn  ¼ ½a½E½I Fh;n ¼ j fj expð2ih½hn ½FX0j Þ. A P Pn  and 2 o 2 constant term h n jFh j jFh;n j has been omitted from this equation. Conceptually, the locked translation function is based on the overlap of intermolecular vectors within the noncrystallographic symmetry oligomer and the observed Patterson map (Tong, 1996b). The equation for the locked translation function, (2.3.7.6), bears remarkable resemblance to that for the ordinary Patterson-correlation translation function, (2.3.7.5), with the interchange of the crystallographic ([Tn]) and noncrystallographic symmetry ð½hn Þ parameters.


ðyn Þ ¼
ðy1 Þ;

ð2:3:8:1Þ

yn ¼ ½C n y1 þ dn :

ð2:3:8:2Þ

where

2.3.7.6. Computer programs for rotation and translation function calculations Several programs are currently in popular use for the calculation of rotation and translation functions. These include AMoRe (Navaza, 1994, 2001a), BEAST (Read, 2001b), CCP4 (Collaborative Computational Project, Number 4, 1994), CNS (Bru¨nger et al., 1998), COMO (Jogl et al., 2001), EPMR

From the definition of a structure factor, Fp ¼

N R P n¼1 U

272


ðyn Þ expð2ip
yn Þ dyn ;

ð2:3:8:3Þ

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES where the integral is taken over the volume U of one molecule. But since each molecule is identical as expressed in equation (2.3.8.1) and since (2.3.8.2) can be substituted in equation (2.3.8.3), we have Fp ¼

N R P


ðy1 Þ exp½2ip
ð½Cn y1 þ dn Þ dy1 :

Table 2.3.8.1. Molecular replacement: phase refinement as an iterative process (A)

Fobs ; 0n ; m0n !
n

(B)


n !
n (modified) (i) Use of noncrystallographic symmetry operators (ii) Definition of envelope limiting volume within which noncrystallographic symmetry is valid

ð2:3:8:4Þ

(iii) Adjustment of solvent density†

n¼1 U

(iv) Use of crystallographic operators to reconstruct modified density into a complete cell

Now let the molecule in the h crystal be related to the molecule in the first asymmetric unit of the p crystal by the noncrystallographic symmetry operation x ¼ ½Cy þ d;

(C)


n (modified) ! Fcalc; nþ1 ; calc; nþ1

(D)

ðFcalc; nþ1 ; calc; nþ1 Þ þ ðFobs ; 0 Þ ! Fobs ; 0nþ1 ; m0nþ1 (i) Assessment of reliability of new phasing set nþ1 in relation to original phasing set 0 ðwÞ

ð2:3:8:5Þ

(ii) Use of figures of merit m0 ; mnþ1 and reliability w to determine modified phasing set 0nþ1 ; m0nþ1 ‡ (iii) Consideration of nþ1 and mnþ1 where there was no prior knowledge of (a) Fobs (e.g. very low order reflections or uncollected data)

which implies
ðxÞ ¼
ðy1 Þ ¼
ðy2 Þ ¼ : . . .

ð2:3:8:6Þ

(b) 0 (e.g. no isomorphous information or phase extension) (E)

Furthermore, in the h cell 1 X
ðxÞ ¼ F expð2ih
xÞ; Vh h h

† Wang (1985); Bhat & Blow (1982); Collins (1975); Schevitz et al. (1981); Hoppe & Gassmann (1968). ‡ Rossmann & Blow (1961); Hendrickson & Lattman (1970).

ð2:3:8:7Þ interpretable. The uniqueness and validity of the solution lay in the obvious chemical correctness of the polypeptide fold and its agreement with known amino-acid-sequence data. In contrast to the earlier reciprocal-space methods, noncrystallographic symmetry was used as a method to improve poor phases rather than to determine phases ab initio. Many other applications followed rapidly, aided greatly by the versatile techniques developed by Bricogne (1976). Of particular interest is the application to the structure determination of hexokinase (Fletterick & Steitz, 1976), where the averaging occurred both between different crystal forms and within the same crystal. The most widely used procedure for real-space averaging is the ‘double sorting’ technique developed by Bricogne (1976) and also by Johnson (1978). An alternative method is to maintain the complete map stored in the computer (Nordman, 1980b). This avoids the sorting operation, but is only possible given a very large computer or a low-resolution map containing relatively few grid points. Bricogne’s double sorting technique involves generating realspace non-integral points ðDi Þ which are related to integral grid points ðIi Þ in the cell asymmetric unit by the noncrystallographic symmetry operators. The elements of the set Di are then brought back to their equivalent points in the cell asymmetric unit ðD0i Þ and sorted by their proximity to two adjacent real-space sections. The set Ii0 , calculated on a finer grid than Ii and stored in the computer memory two sections at a time, is then used for linear interpolation to determine the density values at D0i which are successively stored and summed in the related array Ii . A count is kept of the number of densities received at each Ii , resulting in a final averaged aggregate, when all real-space sections have been utilized. The density to be assigned outside the molecular envelope (defined with respect to the set Ii ) is determined by averaging the density of all unused points in Ii . The grid interval for the set Ii0 should be about one-sixth of the resolution to avoid serious errors from interpolation (Bricogne, 1976). The grid point separation in the set Ii need only be sufficient for representation of electron density, or about one-third of the resolution. Molecular replacement in real space consists of the following steps (Table 2.3.8.1): (a) calculation of electron density based on a starting phase set and observed amplitudes; (b) averaging of this density among the noncrystallographic asymmetric units or molecular copies in several crystal forms, a process which defines

and thus, by combining with (2.3.8.5), (2.3.8.6) and (2.3.8.7),
ðy1 Þ ¼

1 X F exp½2iðh½C
y1 þ h
dÞ: Vh h h

ð2:3:8:8Þ

Now using (2.3.8.4) and (2.3.8.8) it can be shown that Fp ¼

N UX X Fh Ghpn exp½2iðp
Sn  h
SÞ; Vh h n¼1

ð2:3:8:9Þ

where UGhpn ¼

R

exp½2iðp½C n   h½CÞ
u du:

Return to step (A) with 0nþ1 ; m0nþ1 and a possibly augmented set of Fobs .

ð2:3:8:10Þ

U

S is a chosen molecular origin in the h crystal and Sn is the corresponding molecular position in the nth asymmetric unit of the p crystal. 2.3.8.2. Using noncrystallographic symmetry for phase improvement The use of noncrystallographic symmetry for phase determination was proposed by Rossmann & Blow (1962, 1963) and subsequently explored by Crowther (1967, 1969) and Main & Rossmann (1966). These methods were developed in reciprocal space and were primarily concerned with ab initio phase determination. Real-space averaging of electron density between noncrystallographically related molecules was used in the structure determination of deoxyhaemoglobin (Muirhead et al., 1967) and of -chymotrypsin (Matthews et al., 1967). The improvement derived from the averaging between the two noncrystallographic units was, however, not clear in either case. The first obviously successful application was in the structure determination of lobster glyceraldehyde-3-phosphate dehydrogenase (Buehner et al., 1974; Argos et al., 1975), where the tetrameric molecule of symmetry 222 occupied one crystallographic asymmetric unit. The improvement in the essentially SIR electron-density map was considerable and the results changed from uninterpretable to

273

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION ˚ resolution. Particularly a molecular envelope as the averaging is only valid within the southern bean mosaic virus to 22.5 A range of the noncrystallographic symmetry; (c) reconstructing the impressive was the work on polyoma virus (Rayment et al., 1982; unit cell based on averaged density in every noncrystallographic Rayment, 1983; Rayment et al., 1983) where crude initial models asymmetric unit; (d) calculating structure factors from the led to an entirely unexpected breakdown of the Caspar & Klug reconstructed cell; (e) combining the new phases with others to (1962) concept of quasi-symmetry. Ab initio phasing has also obtain a weighted best-phase set; and (f ) returning to step (a) at been used by combining the electron-diffraction projection data the previous or an extended resolution. Decisions made in steps of two different crystal forms of bacterial rhodopsin (Rossmann (b) and (e) determine the rate of convergence (see Table 2.3.8.1) & Henderson, 1982). to a solution (Arnold et al., 1987). The power of the molecular replacement procedure for either 2.3.8.3. Update on noncrystallographic averaging and densityphase improvement or phase extension depends on the number modification methods of noncrystallographic asymmetric units, the size of the excluded volume expressed in terms of the ratio ðV  UNÞ=V and the Since this article was originally written, molecular replacement magnitude of the measurement error on the structure amplitudes. has been subject of a number of reviews (Rossmann, 1990), Crowther (1967, 1969) and Bricogne (1974) have investigated the including a historical background of the subject (Rossmann, dependence on the number of noncrystallographic asymmetric 2001). A series of chapters pertaining to molecular replacement units and conclude that three or more copies are sufficient to have been published in IT Volume F (Rossmann & Arnold, ensure convergence of an iterative phase improvement proce2001a), reviewing noncrystallographic symmetry (Chapter 13.1; dure in the absence of errors on the structure amplitudes. As with Blow, 2001), rotation (Chapter 13.2; Navaza, 2001b) and transthe analogous case of isomorphous replacement in which three lation (Chapter 13.3; Tong, 2001b) functions, and noncrystallodata sets ensure reasonable phase determination, additional graphic symmetry averaging for phase improvement and copies will enhance the power of the method, although their extension (Chapter 13.4; Rossmann & Arnold, 2001b). Chapters usefulness is subject to the law of diminishing returns. Another on phase improvement by density modification (Chapter 15.1; example of this principle is the sign determination of the h0l Zhang et al., 2001), optimal weighting of Fourier terms in map reflections of horse haemoglobin (Perutz, 1954) in which seven calculations (Chapter 15.2; Read, 2001a) and refinement calcushrinkage stages constituted the sampling of the transform of a lations incorporating bulk solvent correction (Chapter 18.4; single copy. Dauter et al., 2001) are also recommended reading. In an analysis of how phasing errors propagate into errors in There has been remarkable progress in the general area of calculations of electron density, Arnold & Rossmann (1986) density modification, involving improvement of real-space concluded that the ‘power’ of phase determination could be methods for averaging and reconstruction, and treatment of related to the noncrystallographic redundancy, N, the ratio of the solvent for iterative phase improvement and refinement calcumolecular envelope volume, U, to the unit cell volume, V, the lations. The use of real-space averaging between noncrystallofractional error of the structure-factor amplitudes, R and the graphically related electron density within the crystallographic fractional completeness of the data, f, by (Arnold & Rossmann, asymmetric unit has become an accepted mode of extending 1986) phase information to higher resolution, particularly for complex structures such as viruses [Acharya et al., 1989; Arnold & Rossmann, 1988; Gaykema et al., 1986; Hogle et al., 1985; Luo et al., ðNf Þ1=2 1989; Rossmann & Arnold, 2001b (IT F Chapter 13.4); Rossmann P¼ : ð2:3:8:11Þ RU=V et al., 1985, 1992]. Ab initio phase determination based on noncrystallographic redundancy has become fairly common (Chapman et al., 1992; Lunin et al., 2000; Miller et al., 2001; This semiquantitative result makes intuitive sense in that the Rossmann, 1990; Tsao et al., 1992). General programs in common noncrystallographic redundancy and solvent content terms can use for noncrystallographic symmetry averaging include be directly related to over-sampling of the molecular transform in BUSTER-TNT [Blanc et al., 2004; Roversi et al., 2000; Tronrud & reciprocal space, and, thus, are analogous in providing phasing Ten Eyck, 2001 (IT F Section 25.2.4)], CNS [Bru¨nger et al., 1998; information. The phasing power of solvent flattening/density Brunger, Adams, DeLano et al., 2001 (IT F Section 25.2.3)], modification was further analysed and shown to lead to Sayre’s DM/DMMULTI [Cowtan & Main, 1993; Cowtan et al., 2001 (IT F equations (Sayre, 1952) at a limit where the molecular envelope is Section 25.2.2); Schuller, 1996; Zhang, 1993], PHASES [Furey, sufficiently detailed and shrunken to cover sharpened and 2001 (IT F Section 25.2.1); Furey & Swaminathan, 1997], RAVE/ separated atoms (Arnold & Rossmann, 1986). This result MAVE (Jones, 1992; Kleywegt, 1996) and SOLVE/RESOLVE suggests that more detailed definitions of molecular envelopes [Terwilliger, 2002b, 2003c; Terwilliger & Berendzen, 2001 (IT F than are traditionally used could be advantageous for phase Section 14.2.2)]. improvement and extension procedures. Solvent flattening has been formulated in reciprocal space for Procedures for real-space averaging have been used extengreater computational efficiency (Leslie, 1987; Terwilliger, 1999) sively with great success. The interesting work of Wilson et al. and solvent ‘flipping’ is a powerful extension of solvent density (1981) is noteworthy for the continuous adjustment of molecular modification (Abrahams, 1997; Abrahams & Leslie, 1996). Bulkenvelope with increased map definition. Furthermore, the solvent corrections are now commonly used in crystallographic analysis of complete virus structures has only been possible as a refinement, allowing for better modelling and phase determinaconsequence of this technique (Bloomer et al., 1978; Harrison et tion of low-resolution data [Bru¨nger et al., 1998; Dauter et al., al., 1978; Abad-Zapatero et al., 1980; Liljas et al., 1982). Although 2001 (IT F Chapter 18.4)]. The problem of phase error estimation the procedure has been used primarily for phase improvement, and analysis and bias removal has been treated extensively apparently successful attempts have been made at phase exten(Cowtan, 1999; Cowtan & Main, 1996), including extension of sion (Nordman, 1980b; Gaykema et al., 1984; Rossmann et al., methods to include maximum-likelihood functions and iterative 1985). Ab initio phasing of glyceraldehyde-3-phosphate bias removal procedures [Brunger, Adams & Rice, 2001 (IT F dehydrogenase (Argos et al., 1975) was successfully attempted by Chapter 18.2); Hunt & Deisenhofer, 2003; Lamzin et al., 2001 (IT initially filling the known envelope with uniform density to F Section 25.2.5); Perrakis et al., 1997; Terwilliger, 2004]. Histodetermine the phases of the innermost reflections and then gram matching [Cowtan & Main, 1993; Lunin, 1993; Nieh & ˚ resolution. Johnson et al. gradually extending phases to 6.3 A Zhang, 1999; Refaat et al., 1996; Zhang, 1993; Zhang et al., 2001 (1976) used the same procedure to determine the structure of (IT F Chapter 15.1)] and skeletonization [Baker et al., 1993;

274

2.3. PATTERSON AND MOLECULAR REPLACEMENT TECHNIQUES X Zhang et al., 2001 (IT F Chapter 15.1)], and structural fragment Bhp Fh ð2:3:8:12Þ Fp ¼ matching procedures (Terwilliger, 2003a) have been added to the h arsenal of density-modification methods. Automated mask and molecular-envelope definition has helped to remove the tedium and increase the efficiency and quality of density-modification (Main & Rossmann, 1966), or in matrix form and symmetry-averaging procedures. Noncrystallographic symmetry averaging among different crystal forms (Perutz, 1954) F ¼ ½BF; has become increasingly common, and exploitation of the unitcell variation among flash-cooled and noncooled forms of the same crystal is a broadly applicable method for phase determiwhich is the form of the equations used by Main (1967) and by nation (Das et al., 1996; Ding et al., 1995); soaking crystals in a Crowther (1967). Colman (1974) arrived at the same conclusions series of different solvents and buffers can produce an analogous by an application of Shannon’s sampling theorem. It should be effect (Ren et al., 1995; Tong et al., 1997). Phases from noncrysnoted that the elements of [B] are dependent only on knowledge tallographic symmetry averaging and other ‘experimental’ of the noncrystallographic symmetry and the volume within sources have been incorporated into crystallographic refinement which it is valid. Substitution of approximate phases into the procedures using a number of formalisms (Arnold & Rossmann, right-hand side of (2.3.8.12) produces a set of calculated structure 1988; Rees & Lewis, 1983) including maximum likelihood (Pannu factors exactly analogous to those produced by backet al., 1998). transforming the averaged electron density in real space. The new phases can then be used in a renewed cycle of molecular repla2.3.8.4. Equivalence of real- and reciprocal-space molecular cement. The reciprocal-space molecular replacement procedure replacement has been implemented and tested in a computer program (Tong Let us proceed in reciprocal space doing exactly the same as is & Rossmann, 1995). done in real-space averaging. Thus Computationally, it has been found more convenient and faster to work in real space. This may, however, change with the advent N of vector processing in ‘supercomputers’. Obtaining improved 1X
AV ðxÞ ¼
ðx Þ; phases by substitution of current phases on the right-hand side of N n¼1 n the molecular replacement equations (2.3.8.1) seems less cumbersome than the repeated forward and backward Fourier transformation, intermediate sorting, and averaging required in where the real-space procedure. xn ¼ ½C n x þ dn : 2.3.9. Conclusions Therefore, Complete interpretation of Patterson maps is no longer used frequently in structure analysis, although most determinations of " # heavy-atom positions of isomorphous pairs are based on X X 1 1 Patterson analyses. Incorporation of the Patterson concept is
AV ðxÞ ¼ Fh expð2ih
xn Þ : N N V h crucial in many sophisticated techniques essential for the solution of complex problems, particularly in the application to biological macromolecular structures. Patterson techniques provide The next step is to perform the back-transform of the averaged important physical insights in a link between real- and reciprocalelectron density. Hence, space formulation of crystal structures and diffraction data. R Fp ¼
AV ðxÞ expð2ip
xÞ dx; This article, first written in December 1984 (by MGR and EA) U and completed in January 1986, was published in the first edition of this volume 1993, and in a mildly revised form in the second where U is the volume within the averaged part of the cell. edition in 2001. We are grateful for generous support of our Hence, substituting for
AV , laboratories from the National Science Foundation (to LT and MGR) and from the National Institutes of Health (LT, MGR and # Z" EA). We acknowledge the many authors whose insights, innoX X 1 vation and writings make up the subject matter of this review. We Fp ¼ Fh expð2ih
xn Þ expð2ip
xÞ dx; NV N h also acknowledge Sharon Wilder for her painstaking attention to U detail in preparation of the original manuscript and an article by Argos & Rossmann (1980) as the source of some material in this which is readily simplified to article. Fp ¼

U X X F G expð2ih
dn Þ: NV h h N hpn

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Setting Bhp ¼

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275

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Terwilliger, T. C. (2004). Using prime-and-switch phasing to reduce model bias in molecular replacement. Acta Cryst. D60, 2144–2149. Terwilliger, T. C. & Berendzen, J. (1999). Automated MAD and MIR structure solution. Acta Cryst. D55, 849–861. Terwilliger, T. C. & Berendzen, J. (2001). Automated MAD and MIR structure solution. In International Tables for Crystallography, Vol. F, Crystallography of Biological Macromolecules, edited by M. G. Rossmann & E. Arnold, Section 14.2.2. Dordrecht: Kluwer Academic Publishers. Terwilliger, T. C. & Eisenberg, D. (1983). Unbiased three-dimensional refinement of heavy-atom parameters by correlation of origin-removed Patterson functions. Acta Cryst. A39, 813–817. Terwilliger, T. C., Kim, S.-H. & Eisenberg, D. (1987). Generalized method of determining heavy-atom positions using the difference Patterson function. Acta Cryst. A43, 1–5. Tollin, P. (1966). On the determination of molecular location. Acta Cryst. 21, 613–614. Tollin, P. (1969). A comparison of the Q-functions and the translation function of Crowther and Blow. Acta Cryst. A25, 376–377. Tollin, P. & Cochran, W. (1964). Patterson function interpretation for molecules containing planar groups. Acta Cryst. 17, 1322–1324. Tollin, P., Main, P. & Rossmann, M. G. (1966). The symmetry of the rotation function. Acta Cryst. 20, 404–407. Tollin, P. & Rossmann, M. G. (1966). A description of various rotation function programs. Acta Cryst. 21, 872–876. Tong, L. (1993). REPLACE, a suite of computer programs for molecularreplacement calculations. J. Appl. Cryst. 26, 748–751. Tong, L. (1996a). Combined molecular replacement. Acta Cryst. A52, 782–784. Tong, L. (1996b). The locked translation function and other applications of a Patterson correlation function. Acta Cryst. A52, 476–479. Tong, L. (2001a). How to take advantage of non-crystallographic symmetry in molecular replacement: ‘locked’ rotation and translation functions. Acta Cryst. D57, 1383–1389. Tong, L. (2001b). Translation functions. In International Tables for Crystallography, Vol. F, Crystallography of Biological Macromolecules, edited by M. G. Rossmann & E. Arnold, ch. 13.3. Dordrecht: Kluwer Academic Publishers. Tong, L., Qian, C., Davidson, W., Massariol, M.-J., Bonneau, P. R., Cordingley, M. G. & Lagace´, L. (1997). Experiences from the structure determination of human cytomegalovirus protease. Acta Cryst. D53, 682–690. Tong, L. & Rossmann, M. G. (1990). The locked rotation function. Acta Cryst. A46, 783–792. Tong, L. & Rossmann, M. G. (1993). Patterson-map interpretation with noncrystallographic symmetry. J. Appl. Cryst. 26, 15–21. Tong, L. & Rossmann, M. G. (1995). Reciprocal-space molecularreplacement averaging. Acta Cryst. D51, 347–353.

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references

International Tables for Crystallography (2010). Vol. B, Chapter 2.4, pp. 282–296.

2.4. Isomorphous replacement and anomalous scattering By M. Vijayan and S. Ramaseshan†

interesting applications (Koetzle & Hamilton, 1975; Sikka & Rajagopal, 1975). More recently there has been a further revival in the development of anomalous-scattering methods with the advent of synchrotron radiation, particularly in view of the possibility of choosing any desired wavelength from a synchrotron-radiation source (Helliwell, 1984). It is clear from the foregoing that the isomorphous replacement and the anomalous-scattering methods have a long and distinguished history. It is therefore impossible to do full justice to them in a comparatively short presentation like the present one. Several procedures for the application of these methods have been developed at different times. Many, although of considerable historical importance, are not extensively used at present for a variety of reasons. No attempt has been made to discuss them in detail here; the emphasis is primarily on the state of the art as it exists now. The available literature on isomorphous replacement and anomalous scattering is extensive. The reference list given at the end of this part is representative rather than exhaustive. During the past few years, rapid developments have taken place in the isomorphous replacement and anomalous-scattering methods, particularly in the latter, as applied to macromolecular crystallography. These developments are described in detail in International Tables for Crystallography, Volume F (2001). Therefore, they have not been dealt with in this chapter. Significant developments in applications of direct methods to macromolecular crystallography have also occurred in recent years. A summary of these developments as well as the traditional direct methods on which the recent progress is based are presented in Chapter 2.2.

2.4.1. Introduction Isomorphous replacement is among the earliest methods to be employed for crystal structure determination (Cork, 1927). The power of this method was amply demonstrated in the classical X-ray work of J. M. Robertson on phthalocyanine in the 1930s using centric data (Robertson, 1936; Robertson & Woodward, 1937). The structure determination of strychnine sulfate pentahydrate by Bijvoet and others provides an early example of the application of this method to acentric reflections (Bokhoven et al., 1951). The usefulness of isomorphous replacement in the analysis of complex protein structures was demonstrated by Perutz and colleagues (Green et al., 1954). This was closely followed by developments in the methodology for the application of isomorphous replacement to protein work (Harker, 1956; Blow & Crick, 1959) and rapidly led to the first ever structure solution of two related protein crystals, namely, those of myoglobin and haemoglobin (Kendrew et al., 1960; Cullis et al., 1961b). Since then isomorphous replacement has been the method of choice in macromolecular crystallography and most of the subsequent developments in and applications of this method have been concerned with biological macromolecules, mainly proteins (Blundell & Johnson, 1976; McPherson, 1982). The application of anomalous-scattering effects has often developed in parallel with that of isomorphous replacement. Indeed, the two methods are complementary to a substantial extent and they are often treated together, as in this article. Although the most important effect of anomalous scattering, namely, the violation of Friedel’s law, was experimentally observed as early as 1930 (Coster et al., 1930), two decades elapsed before this effect was made use of for the first time by Bijvoet and his associates for the determination of the absolute configuration of asymmetric molecules as well as for phase evaluation (Bijvoet, 1949, 1954; Bijvoet et al., 1951). Since then there has been a phenomenal spurt in the application of anomalous-scattering effects (Srinivasan, 1972; Ramaseshan & Abrahams, 1975; Vijayan, 1987). A quantitative formulation for the determination of phase angles using intensity differences between Friedel equivalents was derived by Ramachandran & Raman (1956), while Okaya & Pepinsky (1956) successfully developed a Patterson approach involving anomalous effects. The anomalous-scattering method of phase determination has since been used in the structure analysis of several structures, including those of a complex derivative of vitamin B12 (Dale et al., 1963) and a small protein (Hendrickson & Teeter, 1981). In the meantime, the effect of changes in the real component of the dispersion correction as a function of the wavelength of the radiation used, first demonstrated by Mark & Szillard (1925), also received considerable attention. This effect, which is formally equivalent to that of isomorphous replacement, was demonstrated to be useful in structure determination (Ramaseshan et al., 1957; Ramaseshan, 1963). Protein crystallographers have been quick to exploit anomalous-scattering effects (Rossmann, 1961; Kartha & Parthasarathy, 1965; North, 1965; Matthews, 1966; Hendrickson, 1979) and, as in the case of the isomorphous replacement method, the most useful applications of anomalous scattering during the last two decades have been perhaps in the field of macromolecular crystallography (Kartha, 1975; Watenpaugh et al., 1975; Vijayan, 1981). In addition to anomalous scattering of X-rays, that of neutrons was also found to have

2.4.2. Isomorphous replacement method 2.4.2.1. Isomorphous replacement and isomorphous addition Two crystals are said to be isomorphous if (a) both have the same space group and unit-cell dimensions and (b) the types and the positions of atoms in both are the same except for a replacement of one or more atoms in one structure with different types of atoms in the other (isomorphous replacement) or the presence of one or more additional atoms in one of them (isomorphous addition). Consider two crystal structures with identical space groups and unit-cell dimensions, one containing N atoms and the other M atoms. The N atoms in the first structure contain subsets P and Q whereas the M atoms in the second structure contain subsets P, Q0 and R. The subset P is common to both structures in terms of atomic positions and atom types. The atomic positions are identical in subsets Q and Q0 , but at any given atomic position the atom type is different in Q and Q0 . The subset R exists only in the second structure. If FN and FM denote the structure factors of the two structures for a given reflection, ð2:4:2:1Þ

FM ¼ FP þ FQ 0 þ FR ;

ð2:4:2:2Þ

and

where the quantities on the right-hand side represent contributions from different subsets. From (2.4.2.1) and (2.4.2.2) we have

† Deceased.

Copyright © 2010 International Union of Crystallography

FN ¼ FP þ FQ

282

2.4. ISOMORPHOUS REPLACEMENT AND ANOMALOUS SCATTERING

Fig. 2.4.2.2. Relationship between
N ,
H and ’.

The sign of FH is already known and the signs of FNH and FN can be readily determined from (2.4.2.6) (Robertson & Woodward, 1937). When the data are acentric, the best one can do is to use both the possible phase angles simultaneously in a Fourier synthesis (Bokhoven et al., 1951). This double-phased synthesis, which is equivalent to the isomorphous synthesis of Ramachandran & Raman (1959), contains the structure and its inverse when the replaceable atoms have a centrosymmetric distribution (Ramachandran & Srinivasan, 1970). When the distribution is noncentrosymmetric, however, the synthesis contains peaks corresponding to the structure and general background. Fourier syntheses computed using the single isomorphous replacement method of Blow & Rossmann (1961) and Kartha (1961) have the same properties. In this method, the phase angle is taken to be the average of the two possible solutions of
N, which is always
H or
H þ 180 . Also, the Fourier coefficients are multiplied by cos ’, following arguments based on the Blow & Crick (1959) formulation of phase evaluation (see Section 2.4.4.4). Although Blow & Rossmann (1961) have shown that this method could yield interpretable protein Fourier maps, it is rarely used as such in protein crystallography as the Fourier maps computed using it usually have unacceptable background levels (Blundell & Johnson, 1976).

Fig. 2.4.2.1. Vector relationship between FN and FM ð FNH Þ.

FM
FN ¼ FH ¼ FQ0
FQ þ FR :

ð2:4:2:3Þ

The above equations are illustrated in the Argand diagram shown in Fig. 2.4.2.1. FQ and FQ0 would be collinear if all the atoms in Q were of the same type and those in Q0 of another single type, as in the replacement of chlorine atoms in a structure by bromine atoms. We have a case of ‘isomorphous replacement’ if FR ¼ 0 ðFH ¼ FQ0
FQ Þ and a case of ‘isomorphous addition’ if FQ ¼ FQ0 ¼ 0 ðFH ¼ FR Þ. Once FH is known, in addition to the magnitudes of FN and FM , which can be obtained experimentally, the two cases can be treated in an equivalent manner in reciprocal space. In deference to common practice, the term ‘isomorphous replacement’ will be used to cover both cases. Also, in as much as FM is the vector sum of FN and FH , FM and FNH will be used synonymously. Thus FM  FNH ¼ FN þ FH :

ð2:4:2:4Þ

2.4.2.2. Single isomorphous replacement method The number of replaceable (or ‘added’) atoms is usually small and they generally have high atomic numbers. Their positions are often determined by a Patterson synthesis of one type or another (see Chapter 2.3). It will therefore be assumed in the following discussion that FH is known. Then it can be readily seen by referring to Fig. 2.4.2.2 that
N ¼
H
cos
1

2 FNH
FN2
FH2 ¼
H  ’; 2FN FH

2.4.2.3. Multiple isomorphous replacement method The ambiguity in
N in a noncentrosymmetric crystal can be resolved only if at least two crystals isomorphous to it are available (Bokhoven et al., 1951). We then have two equations of the type (2.4.2.5), namely,

ð2:4:2:5Þ


N ¼
H1  ’1

when ’ is derived from its cosine function, it could obviously be positive or negative. Hence, there are two possible solutions for
N . These two solutions are distributed symmetrically about FH . One of these would correspond to the correct value of
N. Therefore, in general, the phase angle cannot be unambiguously determined using a pair of isomorphous crystals. The twofold ambiguity in phase angle vanishes when the structures are centrosymmetric. FNH ; FN and FH are all real in centric data and the corresponding phase angles are 0 or 180 . From (2.4.2.4) FNH  FN ¼ FH :

and


N ¼
H2  ’2 ;

ð2:4:2:7Þ

where subscripts 1 and 2 refer to isomorphous crystals 1 and 2, respectively. This is demonstrated graphically in Fig. 2.4.2.3 with the aid of the Harker (1956) construction. A circle is drawn with FN as radius and the origin of the vector diagram as the centre. Two more circles are drawn with FNH1 and FNH2 as radii and the ends of vectors
FH1 and
FH2 , respectively as centres. Each of these circles intersects the FN circle at two points corresponding to the two possible solutions. One of the points of intersection is common and this point defines the correct value of
N. With the assumption of perfect isomorphism and if errors are neglected, the phase circles corresponding to all the crystals would intersect at a common point if a number of isomorphous crystals were used for phase determination.

ð2:4:2:6Þ

283

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.4.2.3. Harker construction when two heavy-atom derivatives are available.

2.4.3. Anomalous-scattering method 2.4.3.1. Dispersion correction Atomic scattering factors are normally calculated on the assumption that the binding energy of the electrons in an atom is negligible compared to the energy of the incident X-rays and the distribution of electrons is spherically symmetric. The transition frequencies within the atom are then negligibly small compared to the frequency of the radiation used and the scattering power of each electron in the atom is close to that of a free electron. When this assumption is valid, the atomic scattering factor is a real positive number and its value decreases as the scattering angle increases because of the finite size of the atom. When the binding energy of the electrons is appreciable, the atomic scattering factor at any given angle is given by f0 þ f 0 þ if 00 ;

ð2:4:3:1Þ

Fig. 2.4.3.1. Variation of (a) f 0 and (b) f 00 as a function of atomic number for Cu K
and Mo K
radiations. Adapted from Fig. 3 of Srinivasan (1972).

where f0 is a real positive number and corresponds to the atomic scattering factor for a spherically symmetric collection of free electrons in the atom. The second and third terms are, respectively, referred to as the real and the imaginary components of the ‘dispersion correction’ (IT IV, 1974). f 0 is usually negative whereas f 00 is positive. For any given atom, f 00 is obviously 90 ahead of the real part of the scattering factor given by

2.4.3.2. Violation of Friedel’s law Consider a structure containing N atoms of which P are normal atoms and the remaining Q anomalous scatterers. Let FP denote the contribution of the P atoms to the structure, and FQ and F00Q the real and imaginary components of the contribution of the Q atoms. The relation between the different contributions to a reflection h and its Friedel equivalent
h is illustrated in Fig. 2.4.3.2. For simplicity we assume here that all Q atoms are of the same type. The phase angle of F00Q is then exactly 90 ahead of that of FQ. The structure factors of h and
h are denoted in the figure by FN ðþÞ and FN ð
Þ, respectively. In the absence of anomalous scattering, or when the imaginary component of the dispersion correction is zero, the magnitudes of the two structure factors are equal and Friedel’s law is obeyed; the phase angles have equal magnitudes, but opposite signs. As can be seen from Fig. 2.4.3.2, this is no longer true when F00Q has a nonzero value. Friedel’s law is then violated. A composite view of the vector relationship for h and
h can be obtained, as in Fig. 2.4.3.3, by reflecting the vectors corresponding to
h about the real axis of the vector diagram. FP and FQ corresponding to the two reflections superpose exactly, but F00Q do not. FN ðþÞ and FN ð
Þ then have different magnitudes and phases. It is easily seen that Friedel’s law is obeyed in centric data even when anomalous scatterers are present. FP and FQ are then parallel to the real axis and F00Q perpendicular to it. The vector sum of the three components is the same for h and
h. It may, however, be noted that the phase angle of the structure factor is

f ¼ f0 þ f 0 :

ð2:4:3:2Þ

The variation of f 0 and f 00 as a function of atomic number for two typical radiations is given in Fig. 2.4.3.1 (Srinivasan, 1972; Cromer, 1965). The dispersion effects are pronounced when an absorption edge of the atom concerned is in the neighbourhood of the wavelength of the incident radiation. Atoms with high atomic numbers have several absorption edges and the dispersion-correction terms in their scattering factors always have appreciable values. The values of f 0 and f 00 do not vary appreciably with the angle of scattering as they are caused by core electrons confined to a very small volume around the nucleus. An atom is usually referred to as an anomalous scatterer if the dispersion-correction terms in its scattering factor have appreciable values. The effects on the structure factors or intensities of Bragg reflections resulting from dispersion corrections are referred to as anomalous-dispersion effects or anomalous-scattering effects.

284

2.4. ISOMORPHOUS REPLACEMENT AND ANOMALOUS SCATTERING

Fig. 2.4.3.3. A composite view of the vector relationship between FN ðþÞ and FN ð
Þ.

pattern displays the same symmetry as that of the crystal in the presence of anomalous scattering. The same is true with highersymmetry space groups also. For example, consider a crystal with space group P222, containing anomalous scatterers. The magnitudes of FP are the same for all equivalent reflections; so are those of FQ and F00Q . Their phase angles, however, differ from one equivalent to another, as can be seen from Table 2.4.3.1. When F00Q ¼ 0, the magnitudes of the vector sum of FP and FQ are the same for all the equivalent reflections. The intensity pattern thus has point-group symmetry 2=m 2=m 2=m. When F00Q 6¼ 0, the equivalent reflections can be grouped into two sets in terms of their intensities: hkl, hk

l, h
k
l and h
k
l; and h
k

l, h
kl, hk
l and hk
l. The equivalents belonging to the first group have the same intensity; so have the equivalents in the second group. But the two intensities are different. Thus the symmetry of the pattern is 222, the same as that of the crystal. In general, under conditions of anomalous scattering, equivalent reflections generated by the symmetry elements in the crystal have intensities different from those of equivalent reflections generated by the introduction of an additional inversion centre in normal scattering. There have been suggestions that a reflection from the first group and another from the second group should be referred to as a ‘Bijvoet pair’ instead of a ‘Friedel pair’, when the two reflections are not inversely related. Most often, however, the terms are used synonymously. The same practice will be followed in this article.

Fig. 2.4.3.2. Vector diagram illustrating the violation of Friedel’s law when F00Q 6¼ 0.

then no longer 0 or 180 . Even when the structure is noncentrosymmetric, the effect of anomalous scattering in terms of intensity differences between Friedel equivalents varies from reflection to reflection. The difference between FN ðþÞ and FN ð
Þ is zero when
P ¼
Q or
Q þ 180 . The difference tends to the maximum possible value ð2FQ00 Þ when
P ¼
Q  90 . Intensity differences between Friedel equivalents depend also on the ratio (in terms of number and scattering power) between anomalous and normal scatterers. Differences obviously do not occur when all the atoms are normal scatterers. On the other hand, a structure containing only anomalous scatterers of the same type also does not give rise to intensity differences. Expressions for intensity differences between Friedel equivalents have been derived by Zachariasen (1965) for the most general case of a structure containing normal as well as different types of anomalous scatterers. Statistical distributions of such differences under various conditions have also been derived (Parthasarathy & Srinivasan, 1964; Parthasarathy, 1967). It turns out that, with a single type of anomalous scatterer in the structure, the ratio

2.4.3.4. Determination of absolute configuration The determination of the absolute configuration of chiral molecules has been among the most important applications of anomalous scattering. Indeed, anomalous scattering is the only effective method for this purpose and the method, first used in the early 1950s (Peerdeman et al., 1951), has been extensively employed in structural crystallography (Ramaseshan, 1963; Vos, 1975). Many molecules, particularly biologically important ones, are chiral in that the molecular structure is not superimposable on its mirror image. Chirality (handedness) arises primarily on account of the presence of asymmetric carbon atoms in the molecule. A tetravalent carbon is asymmetric when the four atoms (or groups) bonded to it are all different from one another. The substituents can then have two distinct arrangements which are mirror images

jFN2 ðþÞ
FN2 ð
Þj FN2 ðþÞ þ FN2 ð
Þ has a maximum mean value when the scattering powers of the anomalous scatterers and the normal scatterers are nearly the same (Srinivasan, 1972). Also, for a given ratio between the scattering powers, the smaller the number of anomalous scatterers, the higher is the mean ratio. 2.4.3.3. Friedel and Bijvoet pairs The discussion so far has been concerned essentially with crystals belonging to space groups P1 and P1
. In the centrosymmetric space group, the crystal and the diffraction pattern have the same symmetry, namely, an inversion centre. In P1, however, the crystals are noncentrosymmetric while the diffraction pattern has an inversion centre, in the absence of anomalous scattering. When anomalous scatterers are present in the structure ðF00Q 6¼ 0Þ, Friedel’s law breaks down and the diffraction pattern no longer has an inversion centre. Thus the diffraction

Table 2.4.3.1. Phase angles of different components of the structure factor in space group P222 Phase angle ( ) of Reflection hkl; hk

l; h
k
l; h
k
l h
k

l; h
kl; hk
l; hk
l

285

FP

FQ

F00Q


P


Q

90 þ
Q



P



Q

90

Q

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION of (or related by inversion to) each other. These optical isomers or enantiomers have the same chemical and physical properties except that they rotate the plane of polarization in opposite directions when polarized light passes through them. It is not, however, possible to calculate the sign of optical rotation, given the exact spatial arrangement or the ‘absolute configuration’ of the molecule. Therefore, one cannot distinguish between the possible enantiomorphic configurations of a given asymmetric molecule from measurements of optical rotation. This is also true of molecules with chiralities generated by overall asymmetric geometry instead of the presence of asymmetric carbon atoms in them. Normal X-ray scattering does not distinguish between enantiomers. Two structures A ðxj ; yj ; zj Þ and B ð
xj ;
yj ;
zj Þ ð j ¼ 1; . . . ; NÞ obviously produce the same diffraction pattern on account of Friedel’s law. The situation is, however, different when anomalous scatterers are present in the structure. The intensity difference between reflections h and
h, or that between members of any Bijvoet pair, has the same magnitude, but opposite sign for structures A and B. If the atomic coordinates are known, the intensities of Bijvoet pairs can be readily calculated. The absolute configuration can then be determined, i.e. one can distinguish between A and B by comparing the calculated intensities with the observed ones for a few Bijvoet pairs with pronounced anomalous effects.

in several structure determinations including that of a derivative of vitamin B12 (Dale et al., 1963). The same method was also employed in a probabilistic fashion in the structure solution of a small protein (Hendrickson & Teeter, 1981). A method for obtaining a unique, but approximate, solution for phase angles from (2.4.3.6) has also been suggested (Srinivasan & Chacko, 1970). An accurate unique solution for phase angles can be obtained if one collects two sets of intensity data using two different wavelengths which have different dispersion-correction terms for the anomalous scatterers in the structure. Two equations of the type (2.4.3.6) are then available for each reflection and the solution common to both is obviously the correct phase angle. Different types of Patterson and Fourier syntheses can also be employed for structure solution using intensity differences between Bijvoet equivalents (Srinivasan, 1972; Okaya & Pepinsky, 1956; Pepinsky et al., 1957). An interesting situation occurs when the replaceable atoms in a pair of isomorphous structures are anomalous scatterers. The phase angles can then be uniquely determined by combining isomorphous replacement and anomalous-scattering methods. Such situations normally occur in protein crystallography and are discussed in Section 2.4.4.5. 2.4.3.6. Anomalous scattering without phase change The phase determination, and hence the structure solution, outlined above relies on the imaginary component of the dispersion correction. Variation in the real component can also be used in structure analysis. In early applications of anomalous scattering, the real component of the dispersion correction was made use of to distinguish between atoms of nearly the same atomic numbers (Mark & Szillard, 1925; Bradley & Rodgers, 1934). For example, copper and manganese, with atomic numbers 29 and 25, respectively, are not easily distinguishable under normal X-ray scattering. However, the real components of the dispersion correction for the two elements are
1.129 and
3.367, respectively, when Fe K
radiation is used (IT IV, 1974). Therefore, the difference between the scattering factors of the two elements is accentuated when this radiation is used. The difference is more pronounced at high angles as the normal scattering factor falls off comparatively rapidly with increasing scattering angle whereas the dispersion-correction term does not. The structure determination of KMnO4 provides a typical example for the use of anomalous scattering without phase change in the determination of a centrosymmetric structure (Ramaseshan et al., 1957; Ramaseshan & Venkatesan, 1957). f 0 and f 00 for manganese for Cu K
radiation are
0:568 and 2.808, respectively. The corresponding values for Fe K
radiation are
3:367 and 0.481, respectively (IT IV, 1974). The data sets collected using the two radiations can now be treated as those arising from two perfectly isomorphous crystals. The intensity differences between a reflection in one set and the corresponding reflection in the other are obviously caused by the differences in the dispersion-correction terms. They can, however, be considered formally as intensity differences involving data from two perfectly isomorphous crystals. They can be used, as indeed they were, to determine the position of the manganese ion through an appropriate Patterson synthesis (see Section 2.4.4.2) and then to evaluate the signs of structure factors using (2.4.2.6) when the structure is centrosymmetric. When the structure is noncentrosymmetric, a twofold ambiguity exists in the phase angles in a manner analogous to that in the isomorphous replacement method. This ambiguity can be removed if the structure contains two different subsets of atoms Q1 and Q2 which, respectively, scatter radiations Q1 and Q2 anomalously. Data sets can then be collected with , which is scattered normally by all atoms, Q1 and Q2 . The three sets can be formally treated as those from three perfectly isomorphous structures and the phase determination effected using (2.4.2.7) (Ramaseshan, 1963).

2.4.3.5. Determination of phase angles An important application of anomalous scattering is in the determination of phase angles using Bijvoet differences (Ramachandran & Raman, 1956; Peerdeman & Bijvoet, 1956). From Figs. 2.4.3.2 and 2.4.3.3, we have FN2 ðþÞ ¼ FN2 þ FQ002 þ 2FN FQ00 cos 

ð2:4:3:3Þ

FN2 ð
Þ ¼ FN2 þ FQ002
2FN FQ00 cos :

ð2:4:3:4Þ

FN2 ðþÞ
FN2 ð
Þ : 4FN FQ00

ð2:4:3:5Þ

and

Then cos  ¼

In the above equations FN may be approximated to ½FN ðþÞ þ FN ð
Þ=2. Then  can be evaluated from (2.4.3.5) except for the ambiguity in its sign. Therefore, from Fig. 2.4.3.2, we have
N ¼
Q þ 90  :

ð2:4:3:6Þ

The phase angle thus has two possible values symmetrically distributed about F00Q . Anomalous scatterers are usually heavy atoms and their positions can most often be determined by Patterson methods.
Q can then be calculated and the two possible values of
N for each reflection evaluated using (2.4.3.6). In practice, the twofold ambiguity in phase angles can often be resolved in a relatively straightforward manner. As indicated earlier, anomalous scatterers usually have relatively high atomic numbers. The ‘heavy-atom’ phases calculated from their positions therefore contain useful information. For any given reflection, that phase angle which is closer to the heavy-atom phase, from the two phases calculated using (2.4.3.6), may be taken as the correct phase angle. This method has been successfully used

286

2.4. ISOMORPHOUS REPLACEMENT AND ANOMALOUS SCATTERING 2.4.3.7. Treatment of anomalous scattering in structure refinement

factors, because, at a given position, the heavy atom may not often be present in all the unit cells. For example, if the heavy atom is present at a given position in only half the unit cells in the crystal, then the occupancy factor of the site is said to be 0.5. For the successful determination of the heavy-atom parameters, as also for the subsequent phase determination, the data sets from the native and the derivative crystals should have the same relative scale. The different data sets should also have the same overall temperature factor. Different scaling procedures have been suggested (Blundell & Johnson, 1976) and, among them, the following procedure, based on Wilson’s (1942) statistics, appears to be the most feasible in the early stages of structure analysis. Assuming that the data from the native and the derivative crystals obey Wilson’s statistics, we have, for any range of sin2 =2 ,

The effect of anomalous scattering needs to be taken into account in the refinement of structures containing anomalous scatterers, if accurate atomic parameters are required. The effect of the real part of the dispersion correction is largely confined to the thermal parameters of anomalous scatterers. This effect can be eliminated by simply adding f 0 to the normal scattering factor of the anomalous scatterers. The effects of the imaginary component of the dispersion correction are, however, more complex. These effects could lead to serious errors in positional parameters when the space group is polar, if data in the entire diffraction sphere are not used (Ueki et al., 1966; Cruickshank & McDonald, 1967). For example, accessible data in a hemisphere are normally used for X-ray analysis when the space group is P1. If the hemisphere has say h positive, the x coordinates of all the atoms would be in error when the structure contains anomalous scatterers. The situation in other polar space groups has been discussed by Cruickshank & McDonald (1967). In general, in the presence of anomalous scattering, it is desirable to collect data for the complete sphere, if accurate structural parameters are required (Srinivasan, 1972). Methods have been derived to correct for dispersion effects in observed data from centrosymmetric and noncentrosymmetric crystals (Patterson, 1963). The methods are empirical and depend upon the refined parameters at the stage at which corrections are applied. This is obviously an unsatisfactory situation and it has been suggested that the measured structure factors of Bijvoet equivalents should instead be treated as independent observations in structure refinement (Ibers & Hamilton, 1964). The effect of dispersion corrections needs to be taken into account to arrive at the correct scale and temperature factors also (Wilson, 1975; Gilli & Cruickshank, 1973).


P ln

ð2:4:4:2Þ

and
P 2 P 2  fNj þ fHj sin2  ln ¼ ln KNH þ 2BNH 2 ; 2  hFNH i

ð2:4:4:3Þ

where fNj and fHj refer to the atomic scattering factors of protein atoms and heavy atoms, respectively. KN and KNH are the scale factors to be applied to the intensities from the native and the derivative crystals, respectively, and BN and BNH the temperature factors of the respective structure factors. Normally one would be able to derive the absolute scale factor and the temperature factor for both the data sets from (2.4.4.2) and (2.4.4.3) using the well known Wilson plot. The data from protein crystals, however, do not follow Wilson’s statistics as protein molecules contain highly non-random features. Therefore, in practice, it is difficult to fit a straight line through the points in a Wilson plot, thus rendering the parameters derived from it unreliable. (2.4.4.2) and (2.4.4.3) can, however, be used in a different way. From the two equations we obtain

2.4.4. Isomorphous replacement and anomalous scattering in protein crystallography 2.4.4.1. Protein heavy-atom derivatives Perhaps the most spectacular applications of isomorphous replacement and anomalous-scattering methods have been in the structure solution of large biological macromolecules, primarily proteins. Since its first successful application on myoglobin and haemoglobin, the isomorphous replacement method, which is often used in conjunction with the anomalous-scattering method, has been employed in the solution of scores of proteins. The application of this method involves the preparation of protein heavy-atom derivatives, i.e. the attachment of heavy atoms like mercury, uranium and lead, or chemical groups containing them, to protein crystals in a coherent manner without changing the conformation of the molecules and their crystal packing. This is only rarely possible in ordinary crystals as the molecules in them are closely packed. Protein crystals, however, contain large solvent regions and isomorphous derivatives can be prepared by replacing the disordered solvent molecules by heavy-atomcontaining groups without disturbing the original arrangement of protein molecules.

(P

) P 2 2 þ fHj fNj hFN2 i P 2 ln  2 hFNH i fNj   K sin2  ¼ ln NH þ 2ðBNH
BN Þ 2 :  KN

ð2:4:4:4Þ

The effects of structural non-randomness in the crystals obviously cancel out in (2.4.4.4). When the left-hand side of (2.4.4.4) is plotted against ðsin2 Þ=2 , it is called a comparison or difference Wilson plot. Such plots yield the ratio between the scales of the derivative and the native data, and the additional temperature factor of the derivative data. Initially, the number and the occupancy factors of heavy-atom sites are unknown, and P are 2 . roughly estimated from intensity differences to evaluate fHj These estimates usually undergo considerable revision in the course of the determination and the refinement of heavy-atom parameters. At first, heavy-atom positions are most often determined by Patterson syntheses of one type or another. Such syntheses are discussed in some detail elsewhere in Chapter 2.3. They are therefore discussed here only briefly. Equation (2.4.2.6) holds when the data are centric. FH is usually small compared to FN and FNH , and the minus sign is then relevant on the left-hand side of (2.4.2.6). Thus the difference between the magnitudes of FNH and FN , which can be obtained

2.4.4.2. Determination of heavy-atom parameters For any given reflection, the structure factor of the native protein crystal ðFN Þ, that of a heavy-atom derivative ðFNH Þ, and the contribution of the heavy atoms in that derivative ðFH Þ are related by the equation FNH ¼ FN þ FH :

2  fNj sin2  ¼ ln KN þ 2BN 2 2  hFN i

ð2:4:4:1Þ

The value of FH depends not only on the positional and thermal parameters of the heavy atoms, but also on their occupancy

287

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION experimentally, normally gives a correct estimate of the magnitude of FH for most reflections. Then a Patterson synthesis with ðFNH
FN Þ2 as coefficients corresponds to the distribution of vectors between heavy atoms, when the data are centric. But proteins are made up of l-amino acids and hence cannot crystallize in centrosymmetric space groups. However, many proteins crystallize in space groups with centrosymmetric projections. The centric data corresponding to these projections can then be used for determining heavy-atom positions through a Patterson synthesis of the type outlined above. The situation is more complex for three-dimensional acentric data. It has been shown (Rossmann, 1961) that ðFNH
FN Þ2 ’ FH2 cos2 ð
NH

H Þ

Matthews (1966). According to a still more accurate expression derived by Singh & Ramaseshan (1966), 2 þ FN2
2FNH FN cosð
N

NH Þ FH2 ¼ FNH 2 ¼ FNH þ FN2  2FNH FN

 ð1
fk½FNH ðþÞ
FNH ð
Þ=2FN g2 Þ1=2 :

The lower estimate in (2.4.4.9) is relevant when j
N

NH j < 90 and the upper estimate is relevant when j
N

NH j > 90 . The lower and the upper estimates may be referred to as FHLE and FHUE , respectively. It can be readily shown (Dodson & Vijayan, 1971) that the lower estimate would represent the correct value of FH for a vast majority of reflections. Thus, a Patterson synthesis 2 with FHLE as coefficients would yield the vector distribution of heavy atoms in the derivative. Such a synthesis would normally be superior to those with the left-hand sides of (2.4.4.5) and (2.4.4.6) as coefficients. However, when the level of heavy-atom substitution is low, the anomalous differences are also low and susceptible to large percentage errors. In such a situation, a synthesis with ðFNH
FN Þ2 as coefficients is likely to yield better 2 results than that with FHLE as coefficients (Vijayan, 1981). Direct methods employing different methodologies have also been used successfully for the determination of heavy-atom positions (Navia & Sigler, 1974). These methods, developed primarily for the analysis of smaller structures, have not yet been successful in a priori analysis of protein structures. The very size of protein structures makes the probability relations used in these methods weak. In addition, data from protein crystals do not normally extend to high enough angles to permit resolution of individual atoms in the structure and the feasibility of using many of the currently popular direct-method procedures in such a situation has been a topic of much discussion. The heavy atoms in protein derivative crystals, however, are small in number and are normally situated far apart from one another. They are thus expected to be resolved even when low-resolution X-ray data are used. In most applications, the magnitudes of the differences between FNH and FN are formally considered as the ‘observed structure factors’ of the heavy-atom distribution and conventional direct-method procedures are then applied to them. Once the heavy-atom parameters in one or more derivatives have been determined, approximate protein phase angles,
N’s, can be derived using methods described later. These phase angles can then be readily used to determine the heavy-atom parameters in a new derivative employing a difference Fourier synthesis with coefficients

ð2:4:4:5Þ

when FH is small compared to FNH and FN . Patterson synthesis with ðFNH
FN Þ2 as coefficients would, therefore, give an approximation to the heavy-atom vector distribution. An isomorphous difference Patterson synthesis of this type has been used extensively in protein crystallography to determine heavyatom positions. The properties of this synthesis have been extensively studied (Ramachandran & Srinivasan, 1970; Rossmann, 1960; Phillips, 1966; Dodson & Vijayan, 1971) and it has been shown that this Patterson synthesis would provide a good approximation to the heavy-atom vector distribution even when FH is large compared to FN (Dodson & Vijayan, 1971). As indicated earlier (see Section 2.4.3.1), heavy atoms are always anomalous scatterers, and the structure factors of any given reflection and its Friedel equivalent from a heavy-atom derivative have unequal magnitudes. If these structure factors are denoted by FNH ðþÞ and FNH ð
Þ and the real component of the heavy-atom contributions (including the real component of the dispersion correction) by FH, then it can be shown (Kartha & Parthasarathy, 1965) that  2 k ½FNH ðþÞ
FNH ð
Þ2 ¼ FH2 sin2 ð
NH

H Þ; 2

ð2:4:4:6Þ

where k ¼ ðfH þ fH0 Þ=fH00 . Here it has been assumed that all the anomalous scatterers are of the same type with atomic scattering factor fH and dispersion-correction terms fH0 and fH00 . A Patterson synthesis with the left-hand side of (2.4.4.6) as coefficients would also yield the vector distribution corresponding to the heavyatom positions (Rossmann, 1961; Kartha & Parthasarathy, 1965). However, FNH ðþÞ
FNH ð
Þ is a small difference between two large quantities and is liable to be in considerable error. Patterson syntheses of this type are therefore rarely used to determine heavy-atom positions. It is interesting to note (Kartha & Parthasarathy, 1965) that addition of (2.4.4.5) and (2.4.4.6) readily leads to  2 k ðFNH
FN Þ2 þ ½FNH ðþÞ
FNH ð
Þ2 ’ FH2 : 2

ðFNH
FN Þ expði
N Þ:

ð2:4:4:10Þ

Such syntheses are also used to confirm and to improve upon the information on heavy-atom parameters obtained through Patterson or direct methods. They are obviously very powerful when centric data corresponding to centrosymmetric projections are used. The synthesis yields satisfactory results even when the data are acentric although the difference Fourier technique becomes progressively less powerful as the level of heavy-atom substitution increases (Dodson & Vijayan, 1971). While the positional parameters of heavy atoms can be determined with a reasonable degree of confidence using the above-mentioned methods, the corresponding temperature and occupancy factors cannot. Rough estimates of the latter are usually made from the strength and the size of appropriate peaks in difference syntheses. The estimated values are then refined, along with the positional parameters, using the techniques outlined below.

ð2:4:4:7Þ

Thus, the magnitude of the heavy-atom contribution can be estimated if intensities of Friedel equivalents have been measured from the derivative crystal. FNH is then not readily available, but to a good approximation FNH ¼ ½FNH ðþÞ þ FNH ð
Þ=2:

ð2:4:4:9Þ

ð2:4:4:8Þ

A different and more accurate expression for estimating FH2 from isomorphous and anomalous differences was derived by

288

2.4. ISOMORPHOUS REPLACEMENT AND ANOMALOUS SCATTERING 2.4.4.3. Refinement of heavy-atom parameters

’¼

The least-squares method with different types of minimization functions is used for refining the heavy-atom parameters, including the occupancy factors. The most widely used method (Dickerson et al., 1961; Muirhead et al., 1967; Dickerson et al., 1968) involves the minimization of the function ’¼

P

2

wðFNH
jFN þ FH jÞ ;

P R¼

ð2:4:4:11Þ

ð2:4:4:12Þ

FHC ¼ FHC ðHCi Þ: A set of approximate protein phase angles is first calculated, employing methods described later, making use of the unrefined heavy-atom parameters. These phase angles are used to construct FN þ FH for each derivative. (2.4.4.11) is then minimized, separately for each derivative, by varying HAi for derivative A, HBi for derivative B, and HCi for derivative C. The refined values of HAi , HBi and HCi are subsequently used to calculate a new set of protein phase angles. Alternate cycles of parameter refinement and phase-angle calculation are carried out until convergence is reached. The progress of refinement may be monitored by computing an R factor defined as (Kraut et al., 1962) P

jFNH
jFN þ FH jj : FNH

ð2:4:4:14Þ

jFHLE
FH j P : FHLE

ð2:4:4:15Þ

The major advantage of using FHLE ’s in refinement is that the heavy-atom parameters in each derivative can now be refined independently of all other derivatives. Care should, however, be taken to omit from calculations all reflections for which FHUE is likely to be the correct estimate of FH. This can be achieved in practice by excluding from least-squares calculations all reflections for which FHUE has a value less than the maximum expected value of FH for the given derivative (Vijayan, 1981; Dodson & Vijayan, 1971). A major problem associated with this refinement method is concerned with the effect of experimental errors on refined parameters. The values of FNH ðþÞ
FNH ð
Þ are often comparable to the experimental errors associated with FNH ðþÞ and FNH ð
Þ. In such a situation, even random errors in FNH ðþÞ and FNH ð
Þ tend to increase systematically the observed difference between them (Dodson & Vijayan, 1971). In (2.4.4.7) and (2.4.4.9), this difference is multiplied by k or k=2, a quantity much greater than unity, and then squared. This could lead to the systematic overestimation of FHLE ’s and the consequent overestimation of occupancy factors. The situation can be improved by employing empirical values of k, evaluated using the relation (Kartha & Parthasarathy, 1965; Matthews, 1966)

FHA ¼ FHA ðHAi Þ

RK ¼

wðFHLE
FH Þ2 :

The progress of refinement may be monitored using a reliability index defined as

where the summation is over all the reflections and w is the weight factor associated with each reflection. Here FNH is the observed magnitude of the structure factor for the particular derivative and FN þ FH is the calculated structure factor. The latter obviously depends upon the protein phase angle
N, and the magnitude and the phase angle of FH which are in turn dependent on the heavy-atom parameters. Let us assume that we have three derivatives A, B and C, and that we have already determined the heavy-atom parameters HAi , HBi and HCi . Then,

FHB ¼ FHB ðHBi Þ

P

P 2 jFNH
FN j ; k¼P jFNH ðþÞ
FNH ð
Þj

ð2:4:4:16Þ

for estimating FHLE or by judiciously choosing the weighting factors in (2.4.4.14) (Dodson & Vijayan, 1971). The use of a modified form of FHLE, arrived at through statistical considerations, along with appropriate weighting factors, has also been advocated (Dodson et al., 1975). When the data are centric, (2.4.4.9) reduces to

ð2:4:4:13Þ

The above method has been successfully used for the refinement of heavy-atom parameters in the X-ray analysis of many proteins. However, it has one major drawback in that the refined parameters in one derivative are dependent on those in other derivatives through the calculation of protein phase angles. Therefore, it is important to ensure that the derivative, the heavyatom parameters of which are being refined, is omitted from the phase-angle calculation (Blow & Matthews, 1973). Even when this is done, serious problems might arise when different derivatives are related by common sites. In practice, the occupancy factors of the common sites tend to be overestimated compared to those of the others (Vijayan, 1981; Dodson & Vijayan, 1971). Yet another factor which affects the occupancy factors is the accuracy of the phase angles. The inclusion of poorly phased reflections tends to result in the underestimation of occupancy factors. It is therefore advisable to omit from refinement cycles reflections with figures of merit less than a minimum threshold value or to assign a weight proportional to the figure of merit (as defined later) to each term in the minimization function (Dodson & Vijayan, 1971; Blow & Matthews, 1973). If anomalous-scattering data from derivative crystals are available, the values of FH can be estimated using (2.4.4.7) or (2.4.4.9) and these can be used as the ‘observed’ magnitudes of the heavy-atom contributions for the refinement of heavy-atom parameters, as has been done by many workers (Watenpaugh et al., 1975; Vijayan, 1981; Kartha, 1965). If (2.4.4.9) is used for estimating FH , the minimization function has the form

FH ¼ FNH  FN :

ð2:4:4:17Þ

Here, again, the lower estimate most often corresponds to the correct value of FH. (2.4.4.17) does not involve FNH ðþÞ
FNH ð
Þ which, as indicated earlier, is prone to substantial error. Therefore, FH’s estimated using centric data are more reliable than those estimated using acentric data. Consequently, centric reflections, when available, are extensively used for the refinement of heavy-atom parameters. It may also be noted that in conditions under which FHLE corresponds to the correct estimate of FH, minimization functions (2.4.4.11) and (2.4.4.14) are identical for centric data. A Patterson function correlation method with a minimization function of the type ’¼

P

w½ðFNH
FN Þ2
FH2 2

ð2:4:4:18Þ

was among the earliest procedures suggested for heavy-atomparameter refinement (Rossmann, 1960). This procedure would obviously work well when centric reflections are used. A modified version of this procedure, in which the origins of the Patterson functions are removed from the correlation, and

289

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.4.4.2. Vector diagram indicating the calculated structure factor, DHi ð
Þ, of the ith heavy-atom derivative for an arbitrary value
for the phase angle of the structure factor of the native protein. 2 ð
Þ=2E2i ; Pi ð
Þ ¼ Ni exp½
Hi

Fig. 2.4.4.1. Distribution of intersections in the Harker construction under non-ideal conditions.

where Ni is the normalization constant and Ei is the estimated r.m.s. error. The methods for estimating Ei will be outlined later. When several derivatives are used for phase determination, the total probability of the phase angle
being the protein phase angle would be

centric and acentric data are treated separately, has been proposed (Terwilliger & Eisenberg, 1983). 2.4.4.4. Treatment of errors in phase evaluation: Blow and Crick formulation As shown in Section 2.4.2.3, ideally, protein phase angles can be evaluated if two isomorphous heavy-atom derivatives are available. However, in practice, conditions are far from ideal on account of several factors such as imperfect isomorphism, errors in the estimation of heavy-atom parameters, and the experimental errors in the measurement of intensity from the native and the derivative crystals. It is therefore desirable to use as many derivatives as are available for phase determination. When isomorphism is imperfect and errors exist in data and heavy-atom parameters, all the circles in a Harker diagram would not intersect at a single point; instead, there would be a distribution of intersections, such as that illustrated in Fig. 2.4.4.1. Consequently, a unique solution for the phase angle cannot be deduced. The statistical procedure for computing protein phase angles using multiple isomorphous replacement (MIR) was derived by Blow & Crick (1959). In their treatment, Blow and Crick assume, for mathematical convenience, that all errors, including those arising from imperfect isomorphism, could be considered as residing in the magnitudes of the derivative structure factors only. They further assume that these errors could be described by a Gaussian distribution. With these simplifying assumptions, the statistical procedure for phase determination could be derived in the following manner. Consider the vector diagram, shown in Fig. 2.4.4.2, for a reflection from the ith derivative for an arbitrary value
for the protein phase angle. Then, 2 DHi ð
Þ ¼ ½FN2 þ FHi þ 2FN FHi cosð
Hi

Þ1=2 :

ð2:4:4:21Þ

Pð
Þ ¼

Q


 P 2 Pi ð
Þ ¼ N exp
½Hi ð
Þ=2E2i  ;

ð2:4:4:22Þ

i

where the summation is over all the derivatives. A typical distribution of Pð
Þ plotted around a circle of unit radius is shown in Fig. 2.4.4.3. The phase angle corresponding to the highest value of Pð
Þ would obviously be the most probable protein phase,
M , of the given reflection. The most probable electron-density distribution is obtained if each FN is associated with the corresponding
M in a Fourier synthesis. Blow and Crick suggested a different way of using the probability distribution. In Fig. 2.4.4.3, the centroid of the probability distribution is denoted by P. The polar coordinates of P are m and
B , where m, a fractional positive number with a maximum value of unity, and
B are referred to as the ‘figure of merit’ and the ‘best phase’, respectively. One can then compute a ‘best Fourier’ with coefficients

ð2:4:4:19Þ

If
corresponds to the true protein phase angle
N, then DHi coincides with FNHi . The amount by which DHi ð
Þ differs from FNHi , namely, Hi ð
Þ ¼ FNHi
DHi ð
Þ;

ð2:4:4:20Þ

is a measure of the departure of
from
N .  is called the lack of closure. The probability for
being the correct protein phase angle could now be defined as

Fig. 2.4.4.3. The probability distribution of the protein phase angle. The point P is the centroid of the distribution.

290

2.4. ISOMORPHOUS REPLACEMENT AND ANOMALOUS SCATTERING the anomalous difference would normally be much smaller than that in the corresponding isomorphous difference. Firstly, the former is obviously free from the effects of imperfect isomorphism. Secondly, FNH ðþÞ and FNH ð
Þ are expected to have the same systematic errors as they are measured from the same crystal. These errors are eliminated in the difference between the two quantities. Therefore, as pointed out by North (1965), the r.m.s. error used for anomalous differences should be much smaller than that used for isomorphous differences. Denoting the r.m.s. error in anomalous differences by E0, the new expression for the probability distribution of protein phase angle may be written as 2 Pi ð
Þ ¼ Ni exp½
Hi ð
Þ=2E2i  0

 expf
½Hi
Hical ð
Þ2 =2Ei2 g; Fig. 2.4.4.4. Harker construction using anomalous-scattering data from a single derivative.

where Hi ¼ FNHi ðþÞ
FNHi ð
Þ

mFN expði
B Þ: and

The best Fourier is expected to provide an electron-density distribution with the lowest r.m.s. error. The figure of merit and the best phase are usually calculated using the equations P Pð
i Þ cosð
i Þ= Pð
i Þ i i P P m sin
B ¼ Pð
i Þ sinð
i Þ= Pð
i Þ;

m cos
B ¼

00 Hical ð
Þ ¼ 2FHi sinð
Di

Hi Þ:

P

i

ð2:4:4:24Þ

Here
Di is the phase angle of DHi ð
Þ [see (2.4.4.19) and Fig. 2.4.4.2]. Hical ð
Þ is the anomalous difference calculated for the assumed protein phase angle
. FNHi may be taken as the average 2 of FNHi ðþÞ and FNHi ð
Þ for calculating Hi ð
Þ using (2.4.4.20).

ð2:4:4:23Þ

i

where Pð
i Þ are calculated, say, at 5 intervals (Dickerson et al., 1961). The figure of merit is statistically interpreted as the cosine of the expected error in the calculated phase angle and it is obviously a measure of the precision of phase determination. In general, m is high when
M and
B are close to each other and low when they are far apart.

2.4.4.6. Estimation of r.m.s. error Perhaps the most important parameters that control the reliability of phase evaluation using the Blow and Crick formulation are the isomorphous r.m.s. error Ei and the anomalous r.m.s. error E0i . For a given derivative, the sharpness of the peak in the phase probability distribution obviously depends upon the value of E and that of E0 when anomalous-scattering data have also been used. When several derivatives are used, an overall underestimation of r.m.s. errors leads to artifically sharper peaks, the movement of
B towards
M, and deceptively high figures of merit. Opposite effects result when E’s are overestimated. Underestimation or overestimation of the r.m.s. error in the data from a particular derivative leads to distortions in the relative contribution of that derivative to the overall phase probability distributions. It is therefore important that the r.m.s. error in each derivative is correctly estimated. Centric reflections, when present, obviously provide the best means for evaluating E using the expression

2.4.4.5. Use of anomalous scattering in phase evaluation When anomalous-scattering data have been collected from derivative crystals, FNH ðþÞ and FNH ð
Þ can be formally treated as arising from two independent derivatives. The corresponding Harker diagram is shown in Fig. 2.4.4.4. Thus, in principle, protein phase angles can be determined using a single derivative when anomalous-scattering effects are also made use of. It is interesting to note that the information obtained from isomorphous differences, FNH
FN , and that obtained from anomalous differences, FNH ðþÞ
FNH ð
Þ, are complementary. The isomorphous difference for any given reflection is a maximum when FN and FH are parallel or antiparallel. The anomalous difference is then zero, if all the anomalous scatterers are of the same type, and
N is determined uniquely on the basis of the isomorphous difference. The isomorphous difference decreases and the anomalous difference increases as the inclination between FN and FH increases. The isomorphous difference tends to be small and the anomalous difference tends to have the maximum possible value when FN and FH are perpendicular to each other. The anomalous difference then has the predominant influence in determining the phase angle. Although isomorphous and anomalous differences have a complementary role in phase determination, their magnitudes are obviously unequal. Therefore, when FNH ðþÞ and FNH ð
Þ are treated as arising from two derivatives, the effect of anomalous differences on phase determination would be only marginal as, for any given reflection, FNH ðþÞ
FNH ð
Þ is usually much smaller than FNH
FN . However, the magnitude of the error in

E2 ¼

P

ðjFNH  FN j
FN Þ2 =n:

ð2:4:4:25Þ

n

As suggested by Blow & Crick (1959), values of E thus estimated can be used for acentric reflections as well. Once a set of approximate protein phase angles is available, Ei can be calculated as the r.m.s. lack of closure corresponding to
B [i.e.
¼
B in (2.4.4.20)] (Kartha, 1976). E0i can be similarly evaluated as the r.m.s. difference between the observed anomalous difference and the anomalous difference calculated for
B [see (2.4.4.24)]. Normally, the value of E0i is about a third of that of Ei (North, 1965). A different method, outlined below, can also be used to evaluate E and E0 when anomalous scattering is present (Vijayan, 1981; Adams, 1968). From Fig. 2.4.2.2, we have

291

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION 2 ¼ ðFNH þ FH2
FN2 Þ=2FNH FH

cos

arising from imperfect isomorphism are treated in a comprehensive manner. Although the isomorphous replacement method still remains the method of choice for the ab initio determination of protein structures, additional items of phase information from other sources are increasingly being used to replace, supplement, or extend the information obtained through the application of the isomorphous replacement. Methods have been developed for the routine refinement of protein structures (Watenpaugh et al., 1973; Huber et al., 1974; Sussman et al., 1977; Jack & Levitt, 1978; Isaacs & Agarwal, 1978; Hendrickson & Konnert, 1980) and they provide a rich source of phase information. However, the nature of the problem and the inherent limitations of the Fourier technique are such that the possibility of refinement yielding misleading results exists (Vijayan, 1980a,b). It is therefore sometimes desirable to combine the phases obtained during refinement with the original isomorphous replacement phases. The other sources of phase information include molecular replacement (see Chapter 2.3), direct methods (Hendrickson & Karle, 1973; Sayre, 1974; de Rango et al., 1975; see also Chapter 2.2) and different types of electron-density modifications (Hoppe & Gassmann, 1968; Collins, 1975; Schevitz et al., 1981; Bhat & Blow, 1982; Agard & Stroud, 1982; Cannillo et al., 1983; Raghavan & Tulinsky, 1979; Wang, 1985). The problem of combining isomorphous replacement phases with those obtained by other methods was first addressed by Rossmann & Blow (1961). The problem was subsequently examined by Hendrickson & Lattman (1970) and their method, which involves a modification of the Blow and Crick formulation, is perhaps the most widely used for combining phase information from different sources. The Blow and Crick procedure is based on an assumed Gaussian ‘lumped’ error in FNHi which leads to a lack of closure, Hi ð
Þ, in FNHi defined by (2.4.4.20). Hendrickson and Lattman make an equally legitimate assumption that the lumped error, 2 again assumed to be Gaussian, is associated with FNHi . Then, as in (2.4.4.20), we have

ð2:4:4:26Þ

and 2 FN2 ¼ FNH þ FH2
2FNH FH cos ;

ð2:4:4:27Þ

where ¼
NH

H. Using arguments similar to those used in deriving (2.4.3.5), we obtain 2 2 ¼ ½FNH ðþÞ
FNH ð
Þ=4FNH FH00 :

sin

ð2:4:4:28Þ

If FNH is considered to be equal to ½FNH ðþÞ þ FNH ð
Þ=2, we obtain from (2.4.4.28) FNH ðþÞ
FNH ð
Þ ¼ 2FH00 sin :

ð2:4:4:29Þ

We obtain what may be called iso if the magnitude of is determined from (2.4.4.26) and the quadrant from (2.4.4.28). Similarly, we obtain ano if the magnitude of is determined from (2.4.4.28) and the quadrant from (2.4.4.26). Ideally, iso and ano should have the same value and the difference between them is a measure of the errors in the data. FN obtained from (2.4.4.27) using ano may be considered as its calculated value ðFNcal Þ. Then, assuming all errors to lie in FN , we may write E2 ¼

P

ðFN
FNcal Þ2 =n:

ð2:4:4:30Þ

n

Similarly, the calculated anomalous difference ðHcal Þ may be evaluated from (2.4.4.29) using iso . Then E02 ¼

P

½jFNH ðþÞ
FNH ð
Þj
Hcal 2 =n:

ð2:4:4:31Þ

n 00 2 Hi ð
Þ ¼ FNHi
D2Hi ð
Þ;

If all errors are assumed to reside in FH , E can be evaluated in yet another way using the expression E2 ¼

P

ðFHLE
FH Þ2 =n:

ð2:4:4:33Þ

00 2 where Hi ð
Þ is the lack of closure associated with FNHi for an assumed protein phase angle
. Then the probability for
being the correct phase angle can be expressed as

ð2:4:4:32Þ

n

002 Pi ð
Þ ¼ Ni exp½
Hi ð
Þ=2E002 i ;

2.4.4.7. Suggested modifications to Blow and Crick formulation and the inclusion of phase information from other sources Modifications to the Blow and Crick procedure of phase evaluation have been suggested by several workers, although none represent a fundamental departure from the essential features of their formulation. In one of the modifications (Cullis et al., 1961a; Ashida, 1976), all Ei ’s are assumed to be the same, but the lack-of-closure error Hi for the ith derivative is measured as the distance from the mean of all intersections between phase circles to the point of intersection of the phase circle of that derivative with the phase circle of the native protein. Alternatively, individual values of Ei are retained, but the lack of closure is measured from the weighted mean of all intersections (Ashida, 1976). This is obviously designed to undo the effects of the unduly high weight given to FN in the Blow and Crick formulation. In another modification (Raiz & Andreeva, 1970; Einstein, 1977), suggested for the same purpose, the FN and FNHi circles are treated as circular bands, the width of each band being related to the error in the appropriate structure factor. A comprehensive set of modifications suggested by Green (1979) treats different types of errors separately. In particular, errors

ð2:4:4:34Þ

2 where E00i is the r.m.s. error in FNHi , which can be evaluated using methods similar to those employed for evaluating Ei . Hendrickson and Lattman have shown that the exponent in the probability expression (2.4.4.34) can be readily expressed as a linear combination of five terms in the following manner.

002
Hi ð
Þ=2E002 i ¼ Ki þ Ai cos
þ Bi sin
þ Ci cos 2


þ Di sin 2
;

ð2:4:4:35Þ

where Ki, Ai , Bi , Ci and Di are constants dependent on FN ; FHi ; FNHi and E00i . Thus, five constants are enough to store the complete probability distribution of any reflection. Expressions for the five constants have been derived for phase information from anomalous scattering, tangent formula, partial structure and molecular replacement. The combination of the phase information from all sources can then be achieved by simply taking the total value of each constant. Thus, the total probability of the protein phase angle being
is given by

292

2.4. ISOMORPHOUS REPLACEMENT AND ANOMALOUS SCATTERING Pð
Þ ¼

Q

 P P P Ps ð
Þ ¼ N exp Ks þ As cos
þ Bs sin
s s s  P P þ Cs cos 2
þ Ds sin 2
; s

the direct methods based on these criteria are not strictly applicable to structure analysis using neutron data, although it has been demonstrated that these methods could be successfully used in favourable situations in neutron crystallography (Sikka, 1969). The anomalous-scattering method is, however, in principle more powerful in the neutron case than in the X-ray case for ab initio structure determination. Thermal neutrons are scattered anomalously at appropriate wavelengths by several nuclei. In a manner analogous to (2.4.3.1), the neutron scattering length of these nuclei can be written as

s

ð2:4:4:36Þ

where Ks ; As etc. are the constants appropriate for the sth source and N is the normalization constant.

b0 þ b0 þ ib00 ¼ b þ ib00 :

2.4.4.8. Fourier representation of anomalous scatterers

ð2:4:5:1Þ

It is often useful to have a Fourier representation of only the anomalous scatterers in a protein. The imaginary component of the electron-density distribution obviously provides such a representation. When the structure is known and F N ðþÞ and F N ð
Þ have been experimentally determined, Chacko & Srinivasan (1970) have shown that this representation is obtained in a Fourier synthesis with i½FN ðþÞ þ FN ð
Þ=2 as coefficients, where FN ð
Þ, whose magnitude is F N ð
Þ, is the complex conjugate of FN ðþÞ. They also indicated a method for calculating the phase angles of FN ðþÞ and FN ð
Þ. It has been shown (Hendrickson & Sheriff, 1987) that the Bijvoet-difference Fourier synthesis proposed earlier by Kraut (1968) is an approximation of the true imaginary component of the electron density. The imaginary synthesis can be useful in identifying minor anomalous-scattering centres when the major centres are known and also in providing an independent check on the locations of anomalous scatterers and in distinguishing between anomalous scatterers with nearly equal atomic numbers (Sheriff & Hendrickson, 1987; Kitagawa et al., 1987).

The correction terms b0 and b00 are strongly wavelengthdependent. In favourable cases, b0 =b0 and b00 =b0 can be of the order of 10 whereas they are small fractions in X-ray anomalous scattering. In view of this pronounced anomalous effect in neutron scattering, Ramaseshan (1966) suggested that it could be used for structure solution. Subsequently, Singh & Ramaseshan (1968) proposed a two-wavelength method for unique structure analysis using neutron diffraction. The first part of the method is the determination of the positions of the anomalous scatterers from the estimated values of FQ. The method employed for estimating FQ is analogous to that using (2.4.4.9) except that data collected at two appropriate wavelengths are used instead of those from two isomorphous crystals. The second stage of the two-wavelength method involves phase evaluation. Referring to Fig. 2.4.3.2 and in a manner analogous to (2.4.3.5), we have

2.4.5. Anomalous scattering of neutrons and synchrotron radiation. The multiwavelength method The multiwavelength anomalous-scattering method (Ramaseshan, 1982) relies on the variation of dispersion-correction terms as a function of the wavelength used. The success of the method therefore depends upon the size of the correction terms and the availability of incident beams of comparable intensities at different appropriate wavelengths. Thus, although this method was used as early as 1957 (Ramaseshan et al., 1957) as an aid to structure solution employing characteristic X-rays, it is, as outlined below, ideally suited in structural work employing neutrons and synchrotron radiation. In principle, -radiation can also be used for phase determination (Raghavan, 1961; Moon, 1961) as the anomalous-scattering effects in -ray scattering could be very large; the wavelength is also easily tunable. However, the intensity obtainable for -rays is several orders lower than that obtainable from X-ray and neutron sources, and hence -ray anomalous scattering is of hardly any practical value in structural analysis.

where ¼
N

Q and subscript 1 refers to data collected at wavelength 1. Singh and Ramaseshan showed that cos 1 can also be determined when data are available at wavelength 1 and 2. We may define

sin

1

¼

2 2 FN1 ðþÞ
FN1 ð
Þ ; 00 4FN1 FQ1

Fm2 ¼ ½FN2 ðþÞ þ FN2 ð
Þ=2

ð2:4:5:2Þ

ð2:4:5:3Þ

and we have from (2.4.3.3), (2.4.3.4) and (2.4.5.3) FN ¼ ðFm2
FQ002 Þ1=2 :

ð2:4:5:4Þ

Then, cos

2.4.5.1. Neutron anomalous scattering

1

¼

2 2 2 002 2 Fm1
Fm2
½ðb21 þ b002 F 1 Þ
ðb2 þ b2 Þx þ Q1 ; 2ðb1
b2 ÞFN1 x FN1 ð2:4:5:5Þ

where x is the magnitude of the temperature-corrected geometrical part of FQ. 1 and hence
N1 can be calculated using (2.4.5.2) and (2.4.5.5).
N2 can also be obtained in a similar manner. During the decade that followed Ramaseshan’s suggestion, neutron anomalous scattering was used to solve half a dozen crystal structures, employing the multiple-wavelength methods as well as the methods developed for structure determination using X-ray anomalous scattering (Koetzle & Hamilton, 1975; Sikka & Rajagopal, 1975; Flook et al., 1977). It has also been demonstrated that measurable Bijvoet differences could be obtained, in favourable situations, in neutron diffraction patterns from protein crystals (Schoenborn, 1975). However, despite the early promise held by neutron anomalous scattering, the method has

Apart from the limitations introduced by experimental factors, such as the need for large crystals and the comparatively low flux of neutron beams, there are two fundamental reasons why neutrons are less suitable than X-rays for the ab initio determination of crystal structures. First, the neutron scattering lengths of different nuclei have comparable magnitudes whereas the atomic form factors for X-rays vary by two orders of magnitude. Therefore, Patterson techniques and the related heavy-atom method are much less suitable for use with neutron diffraction data than with X-ray data. Secondly, neutron scattering lengths could be positive or negative and hence, in general, the positivity criterion (Karle & Hauptman, 1950) or the squarability criterion (Sayre, 1952) does not hold good for nuclear density. Therefore,

293

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION not been as successful as might have been hoped. In addition to the need for large crystals, the main problem with using this method appears to be the time and expense involved in data collection (Koetzle & Hamilton, 1975).

Blow, D. M. & Rossmann, M. G. (1961). The single isomorphous replacement method. Acta Cryst. 14, 1195–1202. Blundell, T. L. & Johnson, L. N. (1976). Protein Crystallography. London: Academic Press. Bokhoven, C., Schoone, J. C. & Bijvoet, J. M. (1951). The Fourier synthesis of the crystal structure of strychnine sulphate pentahydrate. Acta Cryst. 4, 275–280. Bradley, A. J. & Rodgers, J. W. (1934). The crystal structure of the Heusler alloys. Proc. R. Soc. London Ser. A, 144, 340–359. Cannillo, E., Oberti, R. & Ungaretti, L. (1983). Phase extension and refinement by density modification in protein crystallography. Acta Cryst. A39, 68–74. Chacko, K. K. & Srinivasan, R. (1970). On the Fourier refinement of anomalous dispersion corrections in X-ray diffraction data. Z. Kristallogr. 131, 88–94. Collins, D. M. (1975). Efficiency in Fourier phase refinement for protein crystal structures. Acta Cryst. A31, 388–389. Cork, J. M. (1927). The crystal structure of some of the alums. Philos. Mag. 4, 688–698. Coster, D., Knol, K. S. & Prins, J. A. (1930). Unterschiede in der Intensita¨t der Ro¨ntgenstrahlenreflexion an den beiden 111-Flachen der Zinkblende. Z. Phys. 63, 345–369. Cromer, D. T. (1965). Anomalous dispersion corrections computed from self-consistent field relativistic Dirac–Slater wave functions. Acta Cryst. 18, 17–23. Cruickshank, D. W. J. & McDonald, W. S. (1967). Parameter errors in polar space groups caused by neglect of anomalous scattering. Acta Cryst. 23, 9–11. Cullis, A. F., Muirhead, H., Perutz, M. F., Rossmann, M. G. & North, A. C. T. (1961a). The structure of haemoglobin. VIII. A three-dimensional ˚ resolution: determination of the phase angles. Fourier synthesis at 5.5 A Proc. R. Soc. London Ser. A, 265, 15–38. Cullis, A. F., Muirhead, H., Perutz, M. F., Rossmann, M. G. & North, A. C. T. (1961b). The structure of haemoglobin. IX. A three-dimensional ˚ resolution: description of the structure. Proc. Fourier synthesis at 5.5 A R. Soc. London Ser. A, 265, 161–187. Dale, D., Hodgkin, D. C. & Venkatesan, K. (1963). The determination of the crystal structure of factor V 1a. In Crystallography and Crystal Perfection, edited by G. N. Ramachandran, pp. 237–242. New York, London: Academic Press. Dickerson, R. E., Kendrew, J. C. & Strandberg, B. E. (1961). The crystal structure of myoglobin: phase determination to a resolution of ˚ by the method of isomorphous replacement. Acta Cryst. 14, 1188– 2A 1195. Dickerson, R. E., Weinzierl, J. E. & Palmer, R. A. (1968). A least-squares refinement method for isomorphous replacement. Acta Cryst. B24, 997– 1003. Dodson, E., Evans, P. & French, S. (1975). The use of anomalous scattering in refining heavy atom parameters in proteins. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 423–436. Copenhagen: Munksgaard. Dodson, E. & Vijayan, M. (1971). The determination and refinement of heavy-atom parameters in protein heavy-atom derivatives. Some model calculations using acentric reflexions. Acta Cryst. B27, 2402–2411. Einstein, J. E. (1977). An improved method for combining isomorphous replacement and anomalous scattering diffraction data for macromolecular crystals. Acta Cryst. A33, 75–85. Flook, R. J., Freeman, H. C. & Scudder, M. L. (1977). An X-ray and neutron diffraction study of aqua(l-glutamato) cadmium(II) hydrate. Acta Cryst. B33, 801–809. Gilli, G. & Cruickshank, D. W. J. (1973). Effect of neglect of dispersion in centrosymmetric structures: results for OsO4. Acta Cryst. B29, 1983– 1985. Green, D. W., Ingram, V. M. & Perutz, M. F. (1954). The structure of haemoglobin. IV. Sign determination by the isomorphous replacement method. Proc. R. Soc. London Ser A, 225, 287–307. Green, E. A. (1979). A new statistical model for describing errors in isomorphous replacement data: the case of one derivative. Acta Cryst. A35, 351–359. Harker, D. (1956). The determination of the phases of the structure factors of non-centrosymmetric crystals by the method of double isomorphous replacement. Acta Cryst. 9, 1–9. Helliwell, J. R. (1984). Synchrotron X-radiation protein crystallography: instrumentation, methods and applications. Rep. Prog. Phys. 47, 1403– 1497.

2.4.5.2. Anomalous scattering of synchrotron radiation The most significant development in recent years in relation to anomalous scattering of X-rays has been the advent of synchrotron radiation (Helliwell, 1984). The advantage of using synchrotron radiation for making anomalous-scattering measurements essentially arises out of the tunability of the wavelength. Unlike the characteristic radiation from conventional X-ray sources, synchrotron radiation has a smooth spectrum and the wavelength to be used can be finely selected. Accurate measurements have shown that values in the neighbourhood of 30 electrons could be obtained in favourable cases for f 0 and f 00 (Templeton, Templeton, Phillips & Hodgson, 1980; Templeton, Templeton & Phizackerley, 1980; Templeton et al., 1982). Schemes for the optimization of the wavelengths to be used have also been suggested (Narayan & Ramaseshan, 1981). Interestingly, the anomalous differences obtainable using synchrotron radiation are comparable in magnitude to the isomorphous differences normally encountered in protein crystallography. Thus, the use of anomalous scattering at several wavelengths would obviously eliminate the need for employing many heavy-atom derivatives. The application of anomalous scattering of synchrotron radiation for macromolecular structure analysis began to yield encouraging results in the 1980s (Helliwell, 1985). Intensity measurements from macromolecular X-ray diffraction patterns using synchrotron radiation at first relied primarily upon oscillation photography (Arndt & Wonacott, 1977). This method is not particularly suitable for accurately evaluating anomalous differences. Much higher levels of accuracy began to be achieved with the use of position-sensitive detectors (Arndt, 1986). Anomalous scattering, in combination with such detectors, has developed into a major tool in macromolecular crystallography (see IT F, 2001). One of us (MV) acknowledges the support of the Department of Science & Technology, India.

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2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Srinivasan, R. & Chacko, K. K. (1970). On the determination of phases of a noncentrosymmetric crystal by the anomalous dispersion method. Z. Kristallogr. 131, 29–39. Sussman, J. L., Holbrook, S. R., Church, G. M. & Kim, S.-H. (1977). A structure-factor least-squares refinement procedure for macromolecular structures using constrained and restrained parameters. Acta Cryst. A33, 800–804. Templeton, D. H., Templeton, L. K., Phillips, J. C. & Hodgson, K. O. (1980). Anomalous scattering of X-rays by cesium and cobalt measured with synchrotron radiation. Acta Cryst. A36, 436–442. Templeton, L. K., Templeton, D. H. & Phizackerley, R. P. (1980). L3-edge anomalous scattering of X-rays by praseodymium and samarium. J. Am. Chem. Soc. 102, 1185–1186. Templeton, L. K., Templeton, D. H., Phizackerley, R. P. & Hodgson, K. O. (1982). L3-edge anomalous scattering by gadolinium and samarium measured at high resolution with synchrotron radiation. Acta Cryst. A38, 74–78. Terwilliger, T. C. & Eisenberg, D. (1983). Unbiased three-dimensional refinement of heavy-atom parameters by correlation of origin-removed Patterson functions. Acta Cryst. A39, 813–817. Ueki, T., Zalkin, A. & Templeton, D. H. (1966). Crystal structure of thorium nitrate pentahydrate by X-ray diffraction. Acta Cryst. 20, 836– 841. Vijayan, M. (1980a). On the Fourier refinement of protein structures. Acta Cryst. A36, 295–298. Vijayan, M. (1980b). Phase evaluation and some aspects of the Fourier refinement of macromolecules. In Computing in Crystallography, edited by R. Diamond, S. Ramaseshan & K. Venkatesan, pp. 19.01– 19.25. Bangalore: Indian Academy of Sciences. Vijayan, M. (1981). X-ray analysis of 2Zn insulin: some crystallographic problems. In Structural Studies on Molecules of Biological Interest, edited by G. Dodson, J. P. Glusker & D. Sayre, pp. 260–273. Oxford: Clarendon Press. Vijayan, M. (1987). Anomalous scattering methods. In Direct Methods, Macromolecular Crystallography and Crystallographic Statistics, edited by H. Schenk, A. J. C. Wilson & S. Parthasarathy, pp. 121– 139. Singapore: World Scientific Publishing Co. Pte. Ltd. Vos, A. (1975). Anomalous scattering and chemical crystallography. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 307–317. Copenhagen: Munksgaard. Wang, B. C. (1985). Resolution of phase ambiguity in macromolecular crystallography. Methods Enzymol. 115, 90–112. Watenpaugh, K. D., Sieker, L. C., Herriot, J. R. & Jensen, L. H. (1973). ˚ resolution. Refinement of the model of a protein: rubredoxin at 1.5 A Acta Cryst. B29, 943–956. Watenpaugh, K. D., Sieker, L. C. & Jensen, L. H. (1975). Anomalous scattering in protein structure analysis. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 393–405. Copenhagen: Munksgaard. Wilson, A. J. C. (1942). Determination of absolute from relative X-ray intensity data. Nature (London), 150, 151–152. Wilson, A. J. C. (1975). Effect of neglect of dispersion on apparent scale and temperature parameters. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 325–332. Copenhagen: Munksgaard. Zachariasen, W. H. (1965). Dispersion in quartz. Acta Cryst. 18, 714–716.

Ramaseshan, S. & Abrahams, S. C. (1975). Anomalous Scattering. Copenhagen: Munksgaard. Ramaseshan, S. & Venkatesan, K. (1957). The use of anomalous scattering without phase change in crystal structure analysis. Curr. Sci. 26, 352–353. Ramaseshan, S., Venkatesan, K. & Mani, N. V. (1957). The use of anomalous scattering for the determination of crystal structures – KMnO4. Proc. Indian Acad. Sci. 46, 95–111. Rango, C. de, Mauguen, Y. & Tsoucaris, G. (1975). Use of high-order probability laws in phase refinement and extension of protein structures. Acta Cryst. A31, 227–233. Robertson, J. M. (1936). An X-ray study of the phthalocyanines. Part II. Quantitative structure determination of the metal-free compound. J. Chem. Soc. pp. 1195–1209. Robertson, J. M. & Woodward, I. (1937). An X-ray study of the phthalocyanines. Part III. Quantitative structure determination of nickel phthalocyanine. J. Chem. Soc. pp. 219–230. Rossmann, M. G. (1960). The accurate determination of the position and shape of heavy-atom replacement groups in proteins. Acta Cryst. 13, 221–226. Rossmann, M. G. (1961). The position of anomalous scatterers in protein crystals. Acta Cryst. 14, 383–388. Rossmann, M. G. & Blow, D. M. (1961). The refinement of structures partially determined by the isomorphous replacement method. Acta Cryst. 14, 641–647. Sayre, D. (1952). The squaring method: a new method for phase determination. Acta Cryst. 5, 60–65. Sayre, D. (1974). Least-squares phase refinement. II. High-resolution phasing of a small protein. Acta Cryst. A30, 180–184. Schevitz, R. W., Podjarny, A. D., Zwick, M., Hughes, J. J. & Sigler, P. B. (1981). Improving and extending the phases of medium- and lowresolution macromolecular structure factors by density modification. Acta Cryst. A37, 669–677. Schoenborn, B. P. (1975). Phasing of neutron protein data by anomalous dispersion. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 407–416, Copenhagen: Munksgaard. Sheriff, S. & Hendrickson, W. A. (1987). Location of iron and sulfur atoms in myohemerythrin from anomalous-scattering measurements. Acta Cryst. B43, 209–212. Sikka, S. K. (1969). On the application of the symbolic addition procedure in neutron diffraction structure determination. Acta Cryst. A25, 539– 543. Sikka, S. K. & Rajagopal, H. (1975). Application of neutron anomalous dispersion in the structure determination of cadmium tartrate pentahydrate. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 503–514. Copenhagen: Munksgaard. Singh, A. K. & Ramaseshan, S. (1966). The determination of heavy atom positions in protein derivatives. Acta Cryst. 21, 279–280. Singh, A. K. & Ramaseshan, S. (1968). The use of neutron anomalous scattering in crystal structure analysis. I. Non-centrosymmetric structures. Acta Cryst. B24, 35–39. Srinivasan, R. (1972). Applications of X-ray anomalous scattering in structural studies. In Advances in Structure Research by Diffraction Methods, Vol. 4, edited by W. Hoppe & R. Mason, pp. 105–197. Braunschweig: Freidr. Vieweg & Sohn; and Oxford: Pergamon Press.

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2.5. Electron diffraction and electron microscopy in structure determination By J. M. Cowley,† J. C. H. Spence, M. Tanaka, B. K. Vainshtein,† B. B. Zvyagin,† P. A. Penczek and D. L. Dorset

dynamical diffraction effects as the basis for obtaining crystal structure information. The fact that dynamical diffraction is dependent on the relative phases of the diffracted waves then implies that relative phase information can be deduced from the diffraction intensities and the limitations of kinematical diffraction, such as Friedel’s law, do not apply. The most immediately practicable method for making use of this possibility is convergent-beam electron diffraction (CBED) as described in Section 2.5.3. A further important factor, determining the methods for observing electron diffraction, is that, being charged particles, electrons can be focused by electromagnetic lenses. Many of the resolution-limiting aberrations of cylindrical magnetic lenses have now been eliminated through the use of aberrationcorrection devices, so that for weakly scattering samples the ˚ by electronic and mechanical resolution is limited to about 1 A instabilities. This is more than sufficient to distinguish the individual rows of atoms, parallel to the incident beam, in the principal orientations of most crystalline phases. Thus ‘structure images’ can be obtained, sometimes showing direct representation of projections of crystal structures [see IT C (2004), Section 4.3.8]. However, the complications of dynamical scattering and of the coherent imaging processes are such that the image intensities vary strongly with crystal thickness and tilt, and with the defocus or other parameters of the imaging system, making the interpretation of images difficult except in special circumstances. Fortunately, computer programs are readily available whereby image intensities can be calculated for model structures [see IT C (2004), Section 4.3.6]. Hence the means exist for deriving the projection of the structure if only by a process of trial and error and not, as would be desirable, from a direct interpretation of the observations. The accuracy with which the projection of a structure can be deduced from an image, or series of images, improves as the resolution of the microscope improves but is not at all comparable with the accuracy attainable with X-ray diffraction methods. A particular virtue of high-resolution electron microscopy as a structural tool is that it may give information on individual small regions of the sample. Structures can be determined of ‘phases’ existing over distances of only a few unit cells and the defects and local disorders can be examined, one by one. The observation of electron-diffraction patterns forms an essential part of the technique of structure imaging in highresolution electron microscopy, because the diffraction patterns are used to align the crystals to appropriate axial orientations. More generally, for all electron microscopy of crystalline materials the image interpretation depends on knowledge of the diffraction conditions. Fortunately, the diffraction pattern and image of any specimen region can be obtained in rapid succession by a simple switching of lens currents. The ready comparison of the image and diffraction data has become an essential component of the electron microscopy of crystalline materials but has also been of fundamental importance for the development of electron-diffraction theory and techniques. The development of the nanodiffraction method in the field-emission scanning transmission electron microscope (STEM) has allowed microdiffraction patterns to be obtained from subnanometre-sized regions, and so has become the ideal tool for the structural analysis of the new microcrystalline phases important to nanoscience. The direct phasing of these coherent nanodiffraction patterns is an active field of research.

2.5.1. Foreword

By J. M. Cowley and J. C. H. Spence Given that electrons have wave properties and the wavelengths lie in a suitable range, the diffraction of electrons by matter is completely analogous to the diffraction of X-rays. While for X-rays the scattering function is the electron-density distribution, for electrons it is the potential distribution which is similarly peaked at the atomic sites. Hence, in principle, electron diffraction may be used as the basis for crystal structure determination. In practice it is used much less widely than X-ray diffraction for the determination of crystal structures but is receiving increasing attention as a means for obtaining structural information not readily accessible with X-ray- or neutron-diffraction techniques. Electrons having wavelengths comparable with those of the X-rays commonly used in diffraction experiments have energies of the order of 100 eV. For such electrons, the interactions with matter are so strong that they can penetrate only a few layers of atoms on the surfaces of solids. They are used extensively for the study of surface structures by low-energy electron diffraction (LEED) and associated techniques. These techniques are not covered in this series of volumes, which include the principles and practice of only those diffraction and imaging techniques making use of high-energy electrons, having energies in the range of 20 keV to 1 MeV or more, in transmission through thin specimens. For the most commonly used energy ranges of high-energy electrons, 100 to 400 keV, the wavelengths are about 50 times smaller than for X-rays. Hence the scattering angles are much smaller, of the order of 10
2 rad, the recording geometry is relatively simple and the diffraction pattern represents, to a useful first approximation, a planar section of reciprocal space. Extinction distances are hundreds of a˚ngstroms, which, when combined with typical lattice spacings, produces rocking-curve widths which are, unlike the X-ray case, a significant fraction of the Bragg angle. The elastic scattering of electrons by atoms is several orders of magnitude greater than for X-rays. This fact has profound consequences, which in some cases are highly favourable and in other cases are serious hindrances to structure analysis work. On the one hand it implies that electron-diffraction patterns can be obtained from very small single-crystal regions having thicknesses equal to only a few layers of atoms and, with recently developed techniques, having diameters equivalent to only a few interatomic distances. Hence single-crystal patterns can be obtained from microcrystalline phases. However, the strong scattering of electrons implies that the simple kinematical single-scattering approximation, on which most X-ray diffraction structure analysis is based, fails for electrons except for very thin crystals composed of light-atom materials. Strong dynamical diffraction effects occur for crystals ˚ thick, or less for heavy-atom materials. As which may be 100 A a consequence, the theory of dynamical diffraction for electrons has been well developed, particularly for the particular special diffracting conditions relevant to the transmission of fast electrons (see Chapter 5.2), and observations of dynamical diffraction effects are commonly made and quantitatively interpreted. The possibility has thus arisen of using the observation of † Deceased.

Copyright © 2010 International Union of Crystallography

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2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION The individual specimen regions giving single-crystal electrondiffraction patterns are, with few exceptions, so small that they can be seen only by use of an electron microscope. Hence, historically, it was only after electron microscopes were commonly available that the direct correlations of diffraction intensities with crystal size and shape could be made, and a proper basis was available for the development of the adequate dynamical diffraction theory. For the complete description of a diffraction pattern or image intensities obtained with electrons, it is necessary to include the effects of inelastic scattering as well as elastic scattering. In contrast to the X-ray diffraction case, the inelastic scattering does not produce just a broad and generally negligible background. The average energy loss for an inelastic scattering event is about 20 eV, which is small compared with the energy of about 100 keV for the incident electrons. The inelastically scattered electrons have a narrow angular distribution and are diffracted in much the same way as the incident or elastically scattered electrons in a crystal. They therefore produce a highly modulated contribution to the diffraction pattern, strongly peaked about the Bragg spot positions (see Chapter 4.3). Also, as a result of the inelastic scattering processes, including thermal diffuse scattering, an effective absorption function must be added in the calculation of intensities for elastically scattered electrons. The inelastic scattering processes in themselves give information about the specimen in that they provide a measure of the excitations of both the valence-shell and the inner-shell electrons of the solid. The inner-shell electron excitations are characteristic of the type of atom, so that microanalysis of small volumes of specimen material (a few hundreds or thousands of atoms) may be achieved by detecting either the energy losses of the transmitted electrons or the emission of the characteristic X-ray [see IT C (2004), Section 4.3.4]. The development of the annular dark field (ADF) mode in STEM provides a favourable detector geometry for microanalysis, in which the forward scattered beam may be passed to an electron energy-loss spectrometer (EELS) for spectral analysis, while scattering at larger angles is collected to form a simultaneous scanning image. The arrangement is particularly efficient because, using a magnetic sector dispersive spectrometer, electrons of all energy losses may be detected simultaneously (parallel detection). Fine structure on the EELS absorption edges is analysed in a manner analogous to soft X-ray absorption spectroscopy, but with a spatial resolution of a few nanometres. The spectra are obtained from points in the corresponding ADF image which can be identified with subnanometre accuracy. An adverse effect of the inelastic scattering processes, however, is that the transfer of energy to the specimen material results in radiation damage; this is a serious limitation of the application of electron-scattering methods to radiation-sensitive materials such as organic, biological and many inorganic compounds. The amount of radiation damage increases rapidly as the amount of information per unit volume, derived from the elastic scattering, is increased, i.e. as the microscope resolution is improved or as the specimen volume irradiated during a diffraction experiment is decreased. At the current limits of microscopic resolution, radiation damage is a significant factor even for the radiation-resistant materials such as semiconductors and alloys. In the historical development of electron-diffraction techniques the progress has depended to an important extent on the level of understanding of the dynamical diffraction processes and this understanding has followed, to a considerable degree, from the availability of electron microscopes. For the first 20 years of the development, with few exceptions, the lack of a precise knowledge of the specimen morphology meant that diffraction intensities were influenced to an unpredictable degree by dynamical scattering and the impression grew that electron-diffraction intensities could not meaningfully be interpreted.

It was the group in the Soviet Union, led initially by Dr Z. G. Pinsker and later by Dr B. K. Vainshtein and others, which showed that patterns from thin layers of a powder of microcrystals could be interpreted reliably by use of the kinematical approximation. The averaging over crystal orientation reduced the dynamical diffraction effects to the extent that practical structure analysis was feasible. The development of the techniques of using films of crystallites having strongly preferred orientations, to give patterns somewhat analogous to the X-ray rotation patterns, provided the basis for the collection of threedimensional diffraction data on which many structure analyses have been based [see Section 2.5.4 and IT C (2004), Section 4.3.5]. In recent years improvements in the techniques of specimen preparation and in the knowledge of the conditions under which dynamical diffraction effects become significant have allowed progress to be made with the use of high-energy electrondiffraction patterns from thin single crystals for crystal structure analysis. Particularly for crystals of light-atom materials, including biological and organic compounds, the methods of structure analysis developed for X-ray diffraction, including the direct methods (see Section 2.5.8), have been successfully applied in an increasing number of cases. Often it is possible to deduce some structural information from high-resolution electronmicroscope images and this information may be combined with that from the diffraction intensities to assist the structure analysis process [see IT C (2004), Section 4.3.8.8]. The determination of crystal symmetry by use of CBED (Section 2.5.3) and the accurate determination of structure amplitudes by use of methods depending on the observation of dynamical diffraction effects [IT C (2004), Section 4.3.7] came later, after the information on morphologies of crystals, and the precision electron optics associated with electron microscopes, became available. This powerful convergent-beam microdiffraction method has now been widely adopted as the preferred method for space-group determination of microphases, quasicrystals, incommensurate, twinned and other imperfectly crystalline structures. Advantage is taken of the fact that multiple scattering preserves information on the absence of inversion symmetry, while the use of an electron probe which is smaller than a mosaic block allows extinction-free structure-factor measurements to be made. Finally, an enhanced sensitivity to ionicity is obtained from electron-diffraction measurements of structure factors by the very large difference between electron scattering factors for atoms and those for ions at small angles. This section by M. Tanaka replaces the corresponding section by the late P. Goodman in previous editions, which researchers may also find useful. In spite of the problem of radiation damage, a great deal of progress has been made in the study of organic and biological materials by electron-scattering methods. In some respects these materials are very favourable because, with only light atoms present, the scattering from thin films can be treated using the kinematical approximation without serious error. Because of the problem of radiation damage, however, special techniques have been evolved to maximize the information on the required structural aspects with minimum irradiation of the specimen. Image-processing techniques have been evolved to take advantage of the redundancy of information from a periodic structure and the means have been devised for combining information from multiple images and diffraction data to reconstruct specimen structure in three dimensions. These techniques are outlined in Sections 2.5.5, 2.5.6 and 2.5.7. Section 2.5.6, written for the first and second editions by Boris Vainshtein, has been revised and extended for this third edition by Pawel Penczek. It deals with the general theory of three-dimensional reconstruction from projections and compares several popular methods. Section 2.5.7 describes the application of electron-microscope imaging to the structure analysis of proteins which cannot be crystallized,

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2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION and so addresses a crucial problem in structural biology. This is done by the remarkably successful method of single-particle image reconstruction, in which images of the same protein, lying in random orientations within a thin film of vitreous ice, are combined in the correct orientation to form a three-dimensional reconstructed charge-density map at nanometre or better resolution. The summation over many particles achieves the same radiation-damage-reduction effect as does crystallographic redundancy in protein crystallography. Finally, Section 2.5.8 describes experience with the application of numerical direct methods to the phase problem in electron diffraction. Although direct imaging ‘solves’ the phase problem, there are many practical problems in combining electron-microdiffraction intensities with corresponding high-resolution images of a structure over a large tilt range. In cases where multiple scattering can be minimized, some success has therefore been obtained using direct phasing methods, as reviewed in this section. For most inorganic materials the complications of many-beam dynamical diffraction processes prevent the direct application of these techniques of image analysis, which depend on having a linear relationship between the image intensity and the value of the projected potential distribution of the sample. The smaller sensitivities to radiation damage can, to some extent, remove the need for the application of such methods by allowing direct visualization of structure with ultra-high-resolution images and the use of microdiffraction techniques.

obtained. The resolution available is sufficient to distinguish neighbouring rows of adjacent atoms in the projected structures of thin crystals viewed in favourable orientations. It is therefore possible in many cases to obtain information on the structure of crystals and of crystal defects by direct inspection of electron micrographs. The electromagnetic electron lenses may also be used to form electron beams of very small diameter and very high intensity. In particular, by the use of cold field-emission electron guns, it is possible to obtain a current of 10
10 A in an electron beam of ˚ or less with a beam divergence of less than diameter 10 A 10
2 rad, i.e. a current density of 104 A cm
2 or more. The magnitudes of the electron scattering amplitudes then imply that detectable signals may be obtained in diffraction from assemblies of fewer than 102 atoms. On the other hand, electron beams may readily be collimated to better than 10
6 rad. The cross sections for inelastic scattering processes are, in general, less than for the elastic scattering of electrons, but signals may be obtained by the observation of electron energy losses, or the production of secondary radiations, which allow the analysis of chemical compositions or electronic excited states for regions ˚ or less in diameter. of the crystal 100 A On the other hand, the transfer to the sample of large amounts of energy through inelastic scattering processes produces radiation damage which may severely limit the applicability of the imaging and diffraction techniques, especially for biological and organic materials, unless the information is gathered from large specimen volumes with low incident electron beam densities. Structure analysis of crystals can be performed using electron diffraction in the same way as with X-ray or neutron diffraction. The mathematical expressions and the procedures are much the same. However, there are peculiarities of the electron-diffraction case which should be noted. (1) Structure analysis based on electron diffraction is possible for thin specimens for which the conditions for kinematical scattering are approached, e.g. for thin mosaic single-crystal specimens, for thin polycrystalline films having a preferred orientation of very small crystallites or for very extensive, very thin single crystals of biological molecules such as membranes one or a few molecules thick. (2) Dynamical diffraction effects are used explicitly in the determination of crystal symmetry (with no Friedel’s law limitations) and for the measurement of structure amplitudes with high accuracy. (3) For many radiation-resistant materials, the structures of crystals and of some molecules may be determined directly by imaging atom positions in projections of the crystal with a reso˚ or better. The information on atom positions is not lution of 2 A dependent on the periodicity of the crystal and so it is equally possible to determine the structures of individual crystal defects in favourable cases. (4) Techniques of microanalysis may be applied to the determination of the chemical composition of regions of diameter ˚ or less using the same instrument as for diffraction, so that 100 A the chemical information may be correlated directly with morphological and structural information. (5) Crystal-structure information may be derived from regions containing as few as 102 or 103 atoms, including very small crystals and single or multiple layers of atoms on surfaces. The material of this section is also reviewed in the text by Spence (2003).

2.5.2. Electron diffraction and electron microscopy1

By J. M. Cowley 2.5.2.1. Introduction The contributions of electron scattering to the study of the structures of crystalline solids are many and diverse. This section will deal only with the scattering of high-energy electrons (in the energy range of 104 to 106 eV) in transmission through thin samples of crystalline solids and the derivation of information on crystal structures from diffraction patterns and high-resolution images. The range of wavelengths considered is from about ˚ (12.2 pm) for 10 kV electrons to 0.0087 A ˚ (0.87 pm) for 0.122 A 1 MeV electrons. Given that the scattering amplitudes of atoms for electrons have much the same form and variation with ðsin
Þ= as for X-rays, it is apparent that the angular range for strong scattering of electrons will be of the order of 10
2 rad. Only under special circumstances, usually involving multiple elastic and inelastic scattering from very thick specimens, are scattering angles of more than 10
1 rad of importance. The strength of the interaction of electrons with matter is greater than that of X-rays by two or three orders of magnitude. The single-scattering, first Born approximation fails significantly for scattering from single heavy atoms. Diffracted beams from single crystals may attain intensities comparable with that of the ˚ , rather than 104 A ˚ incident beam for crystal thicknesses of 102 A or more. It follows that electrons may be used for the study of very thin samples, and that dynamical scattering effects, or the coherent interaction of multiply scattered electron waves, will modify the diffracted amplitudes in a significant way for all but very thin specimens containing only light atoms. The experimental techniques for electron scattering are largely determined by the possibility of focusing electron beams by use of strong axial magnetic fields, which act as electron lenses having focal lengths as short as 1 mm or less. Electron microscopes employing such lenses have been produced with resolutions ˚ . With such instruments, images showing indiapproaching 1 A vidual isolated atoms of moderately high atomic number may be

2.5.2.2. The interactions of electrons with matter (1) The elastic scattering of electrons results from the interaction of the charged electrons with the electrostatic potential distribution, ’ðrÞ, of the atoms or crystals. An incident electron of kinetic energy eW gains energy e’ðrÞ in the potential field. Alternatively it may be stated that an incident electron wave of

1

Questions related to this section may be addressed to Professor J. C. H. Spence (see list of contributing authors).

299

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION wavelength  ¼ h=mv is diffracted by a region of variable refractive index

(4) The experimentally important case of transmission of highenergy electrons through thin specimens is treated on the assumption of a plane wave incident in a direction almost perpendicular to an infinitely extended plane-parallel lamellar crystal, making use of the small-angle scattering approximation in which the forward-scattered wave is represented in the paraboloidal approximation to the sphere. The incident-beam direction, assumed to be almost parallel to the z axis, is unique and the z component of k is factored out to give

nðrÞ ¼ k=K0 ¼ f½W þ ’ðrÞ=Wg1=2 ’ 1 þ ’ðrÞ=2W: (2) The most important inelastic scattering processes are: (a) thermal diffuse scattering, with energy losses of the order of 2  10
2 eV, separable from the elastic scattering only with specially devised equipment; the angular distribution of thermal diffuse scattering shows variations with ðsin
Þ= which are much the same as for the X-ray case in the kinematical limit; (b) bulk plasmon excitation, or the excitation of collective energy states of the conduction electrons, giving energy losses of 3 to 30 eV and an angular range of scattering of 10
4 to 10
3 rad; (c) surface plasmons, or the excitation of collective energy states of the conduction electrons at discontinuities of the structure, with energy losses less than those for bulk plasmons and a similar angular range of scattering; (d) interband or intraband excitation of valence-shell electrons giving energy losses in the range of 1 to 102 eV and an angular range of scattering of 10
4 to 10
2 rad; (e) inner-shell excitations, with energy losses of 102 eV or more and an angular range of scattering of 10
3 to 10
2 rad, depending on the energy losses involved. (3) In the original treatment by Bethe (1928) of the elastic scattering of electrons by crystals, the Schro¨dinger equation is written for electrons in the periodic potential of the crystal; i.e. r2 ðrÞ þ K02 ½1 þ ’ðrÞ=W ðrÞ ¼ 0;

r2 þ 2k’ ¼ i2k

¼

ð2:5:2:1Þ

ðrÞ ¼

ð0Þ

Z ðrÞ þ ð=Þ

VðuÞ expf
2iu  rg du P Vh expf
2ih  rg;

The wavefunction ðrÞ within the integral is approximated by using successive terms of a Born series

ð2:5:2:2Þ

ðrÞ ¼

K0 is the wavevector in zero potential (outside the crystal) (magnitude 2=) and W is the accelerating voltage. The solutions of the equation are Bloch waves of the form P

Ch ðkÞ expf
iðk0 þ 2hÞ  rg;

where k0 is the incident wavevector in the crystal and h is a reciprocal-lattice vector. Substitution of (2.5.2.2) and (2.5.2.3) in (2.5.2.1) gives the dispersion equations ð


k2h ÞCh

þ

P0

Vh
g Cg ¼ 0:

ð1Þ

ð2:5:2:4Þ

g

Here  is the magnitude of the wavevector in a medium of constant potential V0 (the ‘inner potential’ of the crystal). The refractive index of the electron in the average crystal potential is then n ¼ =K ¼ ð1 þ V0 =WÞ

1=2

’ 1 þ V0 =2W:

ð0Þ

ðrÞ þ

ð1Þ

ð2Þ

ðrÞ þ

ðrÞ þ . . . :

ð2:5:2:8Þ

The first Born approximation is obtained by putting ðrÞ ¼ ð0Þ ðrÞ in the integral and subsequent terms ðnÞ ðrÞ are generated by putting ðn
1Þ ðrÞ in the integral. For an incident plane wave, ð0Þ ðrÞ ¼ expf
ik0  rg and for a point of observation at a large distance R ¼ r
r0 from the scattering object ðjRj  jr0 jÞ, the first Born approximation is generated as

ð2:5:2:3Þ

h

2

expf
ikjr
r0 jg 0 ’ðr Þ ðr0 Þ dr0 : jr
r0 j ð2:5:2:7Þ

R

h

ðrÞ ¼

ð2:5:2:6Þ

where k ¼ 2= and  ¼ 2me=h2 . [See Lynch & Moodie (1972), Portier & Gratias (1981), Tournarie (1962), and Chapter 5.2.] This equation is analogous to the time-dependent Schro¨dinger equation with z replacing t. Retention of the  signs on the righthand side is consistent with both and  being solutions, corresponding to propagation in opposite directions with respect to the z axis. The double-valued solution is of importance in consideration of reciprocity relationships which provide the basis for the description of some dynamical diffraction symmetries. (See Section 2.5.3.) (5) The integral form of the wave equation, commonly used for scattering problems, is written, for electron scattering, as

where ’ðrÞ ¼

@ ; @z

i expf
ik  Rg ðrÞ ¼ R

Z

’ðr0 Þ expfiq  r0 g dr0 ;

where q ¼ k
k0 or, putting u ¼ q=2 and collecting the preintegral terms into a parameter , R
ðuÞ ¼  ’ðrÞ expf2iu  rg dr:

ð2:5:2:9Þ

ð2:5:2:5Þ This is the Fourier-transform expression which is the basis for the kinematical scattering approximation. It is derived on the basis that all ðnÞ ðrÞ terms for n 6¼ 0 are very much smaller than ð0Þ ðrÞ and so is a weak scattering approximation. In this approximation, the scattered amplitude for an atom is related to the atomic structure amplitude, f ðuÞ, by the relationship, derived from (2.5.2.8),

Since V0 is positive and of the order of 10 V and W is 104 to 106 V, n
1 is positive and of the order of 10
4. Solution of equation (2.5.2.4) gives the Fourier coefficients ChðiÞ of the Bloch waves ðiÞ ðrÞ and application of the boundary conditions gives the amplitudes of individual Bloch waves (see Chapter 5.2).

300

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION literature. There is, however, a requirement for internal consistency within a particular analysis, independently of which set is adopted. Unfortunately, this requirement has not always been met and, in fact, it is only too easy at the outset of an analysis to make errors in this way. This problem might have come into prominence somewhat earlier were it not for the fact that, for centrosymmetric crystals (or indeed for centrosymmetric projections in the case of planar diffraction), only the signs used in the transmission and propagation functions can affect the results. It is not until the origin is set away from a centre of symmetry that there is a need to be consistent in every sign used. Signs for electron diffraction have been chosen from two points of view: (1) defining as positive the sign of the exponent in the structure-factor expression and (2) defining the forward propagating free-space wavefunction with a positive exponent. The second of these alternatives is the one which has been adopted in most solid-state and quantum-mechanical texts. The first, or standard crystallographic convention, is the one which could most easily be adopted by crystallographers accustomed to retaining a positive exponent in the structure-factor equation. This also represents a consistent International Tables usage. It is, however, realized that both conventions will continue to be used in crystallographic computations, and that there are by now a large number of operational programs in use. It is therefore recommended (a) that a particular sign usage be indicated as either standard crystallographic or alternative crystallographic to accord with Table 2.5.2.1, whenever there is a need for this to be explicit in publication, and (b) that either one or other of these systems be adhered to throughout an analysis in a self-consistent way, even in those cases where, as indicated above, some of the signs appear to have no effect on one particular conclusion.

Fig. 2.5.2.1. The variation with accelerating voltage of electrons of (a) the wavelength,  and (b) the quantity ½1 þ ðh2 =m20 c2 2 Þ ¼ c = which is proportional to the interaction constant  [equation (2.5.2.14)]. The limit is the Compton wavelength c (after Fujiwara, 1961).

expf
ik  rg f ðuÞ; ðrÞ ¼ expf
ik0  rg þ i R R f ðuÞ ¼ ’ðrÞ expf2iu  rg dr:

ð2:5:2:10Þ

For centrosymmetrical atom potential distributions, the f ðuÞ are real, positive and monotonically decreasing with juj. A measure of the extent of the validity of the first Born approximation is given by the fact that the effect of adding the higher-order terms of the Born series may be represented by replacing f ðuÞ in (2.5.2.10) by the complex quantities f ðuÞ ¼ jfj expfiðuÞg and for single heavy atoms the phase factor  may vary from 0.2 for juj ¼ 0 to 4 or 5 for large juj, as seen from the tables of IT C (2004, Section 4.3.3). (6) Relativistic effects produce appreciable variations of the parameters used above for the range of electron energies considered. The relativistic values are m ¼ m0 ð1
v2 =c2 Þ
1=2 ¼ m0 ð1
2 Þ
1=2 ; 2
1=2

 ¼ h½2m0 jejWð1 þ jejW=2m0 c Þ 2 1=2

¼ c ð1
 Þ =;

2.5.2.4. Scattering of electrons by crystals; approximations The forward-scattering approximation to the many-beam dynamical diffraction theory outlined in Chapter 5.2 provides the basis for the calculation of diffraction intensities and electronmicroscope image contrast for thin crystals. [See Cowley (1995), Chapter 5.2 and IT C (2004) Sections 4.3.6 and 4.3.8.] On the other hand, there are various approximations which provide relatively simple analytical expressions, are useful for the determination of diffraction geometry, and allow estimates to be made of the relative intensities in diffraction patterns and electron micrographs in favourable cases. (a) The kinematical approximation, derived in Section 2.5.2.2 from the first Born approximation, is analagous to the corresponding approximation of X-ray diffraction. It assumes that the scattering amplitudes are directly proportional to the threedimensional Fourier transform of the potential distribution, ’ðrÞ.

ð2:5:2:11Þ ð2:5:2:12Þ ð2:5:2:13Þ

VðuÞ ¼

˚, where c is the Compton wavelength, c ¼ h=m0 c ¼ 0:0242 A and

’ðrÞ expf2iu  rg dr;

ð2:5:2:15Þ

so that the potential distribution ’ðrÞ takes the place of the charge-density distribution, ðrÞ, relevant for X-ray scattering. The validity of the kinematical approximation as a basis for structure analysis is severely limited. For light-atom materials, such as organic compounds, it has been shown by Jap & Glaeser (1980) that the thickness for which the approximation gives reasonable accuracy for zone-axis patterns from single crystals is ˚ for 100 keV electrons and increases, of the order of 100 A approximately as 
1, for higher energies. The thickness limits quoted for polycrystalline samples, having crystallite dimensions smaller than the sample thickness, are usually greater (Vainshtein, 1956). For heavy-atom materials the approximation is more limited since it may fail significantly for single heavy atoms. (b) The phase-object approximation (POA), or high-voltage limit, is derived from the general many-beam dynamical

 ¼ 2me=h2 ¼ ð2m0 e=h2 Þðc =Þ ¼ 2=fW½1 þ ð1
2 Þ1=2 g:

R

ð2:5:2:14Þ

Values for these quantities are listed in IT C (2004, Section 4.3.2). The variations of  and  with accelerating voltage are illustrated in Fig. 2.5.2.1. For high voltages,  tends to a constant value, 2m0 ec =h2 ¼ e=h- c. 2.5.2.3. Recommended sign conventions There are two alternative sets of signs for the functions describing wave optics. Both sets have been widely used in the

301

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.2.1. Standard crystallographic and alternative crystallographic sign conventions for electron diffraction Standard

Alternative

Structure factors

exp½
iðk  r
!tÞ R ðrÞ exp½þ2iðu  rÞ dr R ðrÞ ¼
ðuÞ exp½
2iðu  rÞ du P VðhÞ ¼ ð1=Þ j fj ðhÞ expðþ2ih  rj Þ

exp½þiðk  r
!tÞ R ðrÞ exp½
2iðu  rÞ dr R
ðuÞ exp½þ2iðu  rÞ du P ð1=Þ j fj ðhÞ expð
2ih  rj Þ

Free-space wave Fourier transforming from real space to reciprocal space Fourier transforming from reciprocal space to real space Transmission function (real space)

exp½
i’ðx; yÞz

exp½þi’ðx; yÞz

Phenomenological absorption

’ðrÞ
iðrÞ

’ðrÞ þ iðrÞ

Propagation function P(h) (reciprocal space) within the crystal

expð
2i h zÞ

expðþ2i h zÞ

Iteration (reciprocal space)


nþ1 ðhÞ ¼ ½
n ðhÞ  PðhÞ  QðhÞ

Unitarity test (for no absorption)

TðhÞ ¼ QðhÞ  Q ð
hÞ ¼ ðhÞ

Propagation to the image plane-wave aberration function, where ðUÞ ¼ fU 2 þ 12 Cs 3 U 4 , U 2 ¼ u2 þ v2 and f is positive for overfocus

exp½i ðUÞ

exp½
i ðUÞ

 ¼ electron interaction constant ¼ 2me=h2 ; m ¼ (relativistic) electron mass;  ¼ electron wavelength; e ¼ (magnitude of) electron charge; h ¼ Planck’s constant; k ¼ 2=;  ¼ volume of the unit cell; u ¼ continuous reciprocal-space vector, components u, v; h ¼ discrete reciprocal-space coordinate; ’ðx; yÞ ¼ crystal potential averaged along beam direction (positive); z ¼ slice thickness; ðrÞ ¼ absorption potential [positive; typically  0:1’ðrÞ]; f ¼ defocus (defined as negative for underfocus); Cs ¼ spherical aberration coefficient; h ¼ excitation error relative to the incident-beam direction and defined as negative when the point h lies outside the Ewald sphere; fj ðhÞ ¼ atomic scattering factor for electrons, fe , related to the atomic scattering factor for X-rays, fX , by the Mott formula fe ¼ ðe=U 2 ÞðZ
fX Þ. QðhÞ ¼ Fourier transform of periodic slice transmission function.

diffraction expression, equation (5.2.13.1), Chapter 5.2, by assuming the Ewald sphere curvature to approach zero. Then the scattering by a thin sample can be expressed by multiplying the incoming wave amplitude by the transmission function

erating voltages due to relativistic effects (Watanabe et al., 1968), but they give incorrect results for the small-thickness limit. 2.5.2.5. Kinematical diffraction formulae

qðxyÞ ¼ expf
i’ðxyÞg;

ð2:5:2:16Þ

(1) Comparison with X-ray diffraction. The relations of realspace and reciprocal-space functions are analogous to those for X-ray diffraction [see equations (2.5.2.2), (2.5.2.10) and (2.5.2.15)]. For diffraction by crystals

R where ’ðxyÞ ¼ ’ðrÞ dz is the projection of the potential distribution of the sample in the z direction, the direction of the incident beam. The diffraction-pattern amplitudes are then given by two-dimensional Fourier transform of (2.5.2.16). This approximation is of particular value in relation to the electron microscopy of thin crystals. The thickness for its validity ˚ , depending for 100 keV electrons is within the range 10 to 50 A on the accuracy and spatial resolution involved, and increases with accelerating voltage approximately as 
1=2. In computational work, it provides the starting point for the multislice method of dynamical diffraction calculations (IT C, 2004, Section 4.3.6.1). (c) The two-beam approximation for dynamical diffraction of electrons assumes that only two beams, the incident beam and one diffracted beam (or two Bloch waves, each with two component amplitudes), exist in the crystal. This approximation has been adapted, notably by Hirsch et al. (1965), for use in the electron microscopy of inorganic materials. It forms a convenient basis for the study of defects in crystals having small unit cells (metals, semiconductors etc.) and provides good preliminary estimates for the determination of crystal thicknesses and structure amplitudes for orientations well removed from principal axes, and for electron energies up to 200– 500 keV, but it has decreasing validity, even for favourable cases, for higher energies. It has been used in the past as an ‘extinction correction’ for powder-pattern intensities (Vainshtein, 1956). (d) The Bethe second approximation, proposed by Bethe (1928) as a means for correcting the two-beam approximation for the effects of weakly excited beams, replaces the Fourier coefficients of potential by the ‘Bethe potentials’ Uh ¼ Vh
2k0 

X Vg  Vh
g g

2
k2g

:

’ðrÞ ¼

X Z

Vh ¼ ¼

Vh expf
2ih  rg;

h

’ðrÞ expf2’ih  rg dr

ð2:5:2:18Þ

1X f ðhÞ expf2ih  ri g;  i i

ð2:5:2:19Þ

where the integral of (2.5.2.18) and the summation of (2.5.2.19) are taken over one unit cell of volume (see Dawson et al., 1974). Important differences from the X-ray case arise because (a) the wavelength is relatively small so that the Ewald-sphere curvature is small in the reciprocal-space region of appreciable scattering amplitude; (b) the dimensions of the single-crystal regions giving appreciable scattering amplitudes are small so that the ‘shape transform’ regions of scattering power around the reciprocal-lattice points are relatively large; (c) the spread of wavelengths is small (10
5 or less, with no white-radiation background) and the degree of collimation is better (10
4 to 10
6) than for conventional X-ray sources. As a consequence of these factors, single-crystal diffraction patterns may show many simultaneous reflections, representing almost-planar sections of reciprocal space, and may show fine structure or intensity variations reflecting the crystal dimensions and shape. (2) Kinematical diffraction-pattern intensities are calculated in a manner analogous to that for X-rays except that (a) no polarization factor is included because of the smallangle scattering conditions; (b) integration over regions of scattering power around reciprocal-lattice points cannot be assumed unless appropriate experimental conditions are ensured. For a thin, flat, lamellar crystal of thickness H, the observed intensity is

ð2:5:2:17Þ

Use of these potentials has been shown to account well for the deviations of powder-pattern intensities from the predictions of two-beam theory (Horstmann & Meyer, 1965) and to predict accurately the extinctions of Kikuchi lines at particular accel-

302

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Ih =I0 ¼ jðVh =Þðsin  h HÞ=ð h Þj2 ;

(a) from plasmon and single-electron excitations, 0 is of the order of 0.1 V0 and h , for h 6¼ 0, is negligibly small; (b) from thermal diffuse scattering; h is of the order of 0.1 Vh and decreasing more slowly than Vh with scattering angle. Including absorption effects in (2.5.2.26) for the case h ¼ 0 gives

ð2:5:2:20Þ

where h is the excitation error for the h reflection and  is the unit-cell volume. For a single-crystal diffraction pattern obtained by rotating a crystal or from a uniformly bent crystal or for a mosaic crystal with a uniform distribution of orientations, the intensity is Ih ¼ I0

 2 jVh j2 Vc dh ; 42 2

I0 ¼ 12 expf
0 Hg½cosh h H þ cosð2Vh HÞ; Ih ¼ 12 expf
0 Hg½cosh h H
cosð2Vh HÞ:

ð2:5:2:21Þ The Borrmann effect is not very pronounced for electrons because h 0, but can be important for the imaging of defects in thick crystals (Hirsch et al., 1965; Hashimoto et al., 1961). Attempts to obtain analytical solutions for the dynamical diffraction equations for more than two beams have met with few successes. There are some situations of high symmetry, with incident beams in exact zone-axis orientations, for which the many-beam solution can closely approach equivalent two- or three-beam behaviour (Fukuhara, 1966). Explicit solutions for the three-beam case, which displays some aspects of many-beam character, have been obtained (Gjønnes & Høier, 1971; Hurley & Moodie, 1980).

where Vc is the crystal volume and dh is the lattice-plane spacing. For a polycrystalline sample of randomly oriented small crystals, the intensity per unit length of the diffraction ring is Ih ¼ I0

 2 jVh j2 Vc d2h Mh ; 82 2 L

ð2:5:2:22Þ

where Mh is the multiplicity factor for the h reflection and L is the camera length, or the distance from the specimen to the detector plane. The special cases of ‘oblique texture’ patterns from powder patterns having preferred orientations are treated in IT C (2004, Section 4.3.5). (3) Two-beam dynamical diffraction formulae: complex potentials including absorption. In the two-beam dynamical diffraction approximation, the intensities of the directly transmitted and diffracted beams for transmission through a crystal of thickness H, in the absence of absorption, are 
 2 1=2 2
1 2 2 Hð1 þ w Þ I0 ¼ ð1 þ w Þ w þ cos

h
 2 1=2 Hð1 þ w Þ Ih ¼ ð1 þ w2 Þ
1 sin2 ;

h

ð2:5:2:27Þ

2.5.2.6. Imaging with electrons Electron optics. Electrons may be focused by use of axially symmetric magnetic fields produced by electromagnetic lenses. The focal length of such a lens used as a projector lens (focal points outside the lens field) is given by fp
1 ¼

ð2:5:2:23Þ ð2:5:2:24Þ

e 8mWr

Z1

Hz2 ðzÞ dz;

ð2:5:2:28Þ


1

where Wr is the relativistically corrected accelerating voltage and Hz is the z component of the magnetic field. An expression in terms of experimental constants was given by Liebman (1955) as

where h is the extinction distance, h ¼ ð2jVh jÞ
1 , and w ¼ h h ¼ 
=ð2jVh jdh Þ;

1 A0 ðNIÞ2 ¼ ; f Wr ðS þ DÞ

ð2:5:2:25Þ

where 
is the deviation from the Bragg angle. For the case that h ¼ 0, with the incident beam at the Bragg angle, this reduces to the simple Pendello¨sung expression Ih ¼ 1
I0 ¼ sin2 f2jVh jHg:

ð2:5:2:29Þ

where A0 is a constant, NI is the number of ampere turns of the lens winding, S is the length of the gap between the magnet pole pieces and D is the bore of the pole pieces. Lenses of this type have irreducible aberrations, the most important of which for the paraxial conditions of electron microscopy is the third-order spherical aberration, coefficient Cs , giving a variation of focal length of Cs 2 for a beam at an angle  to the axis. Chromatic aberration, coefficient Cc , gives a spread of focal lengths

ð2:5:2:26Þ

The effects on the elastic Bragg scattering amplitudes of the inelastic or diffuse scattering may be introduced by adding an out-of-phase component to the structure amplitudes, so that for a centrosymmetric crystal, Vh becomes complex by addition of an imaginary component. Alternatively, an absorption function ðrÞ, having Fourier coefficients h, may be postulated so that Vh is replaced by Vh þ ih. The h are known as phenomenological absorption coefficients and their validity in many-beam diffraction has been demonstrated by, for example, Rez (1978). The magnitudes h depend on the nature of the experiment and the extent to which the various inelastically or diffusely scattered electrons are included in the measurements being made. If measurements are made of purely elastic scattering intensities for Bragg reflections or of image intensity variations due to the interaction of the sharp Bragg reflections only, the main contributions to the absorption coefficients are as follows (Radi, 1970):



W0 I þ2 f ¼ Cc I W0

 ð2:5:2:30Þ

for variations W0 and I of the accelerating voltage and lens currents, respectively. The objective lens of an electron microscope is the critical lens for the determination of image resolution and contrast. The action of this lens in a conventional transmission electron microscope (TEM) is described by use of the Abbe theory for coherent incident illumination transmitted through the object to produce a wavefunction 0 ðxyÞ (see Fig. 2.5.2.2). The amplitude distribution in the back focal plane of the objective lens is written

303

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION IðxyÞ ¼

RR

Gðf Þ  Hðu1 v1 Þ

 jF f
0 ðu
u1 ; v
v1 Þ  Tf ðu; vÞgj2 dðf Þ  du1 dv1 ; ð2:5:2:36Þ where F denotes the Fourier-transform operation. In the scanning transmission electron microscope (STEM), the objective lens focuses a small bright source of electrons on the object and directly transmitted or scattered electrons are detected to form an image as the incident beam is scanned over the object (see Fig. 2.5.2.2). Ideally the image amplitude can be related to that of the conventional transmission electron microscope by use of the ‘reciprocity relationship’ which refers to point sources and detectors for scalar radiation in scalar fields with elastic scattering processes only. It may be stated: ‘The amplitude at a point B due to a point source at A is identical to that which would be produced at A for the identical source placed at B’. For an axial point source, the amplitude distribution produced by the objective lens on the specimen is

Fig. 2.5.2.2. Diagram representing the critical components of a conventional transmission electron microscope (TEM) and a scanning transmission electron microscope (STEM). For the TEM, electrons from a source A illuminate the specimen and the objective lens forms an image of the transmitted electrons on the image plane, B. For the STEM, a source at B is imaged by the objective lens to form a small probe on the specimen and some part of the transmitted beam is collected by a detector at A.


0 ðu; vÞ  Tðu; vÞ;

ð2:5:2:31Þ

F ½Tðu; vÞ ¼ tðxyÞ:

where
0 ðu; vÞ is the Fourier transform of 0 ðx; yÞ and T(u, v) is the transfer function of the lens, consisting of an aperture function
Aðu; vÞ ¼

1 for ðu2 þ v2 Þ1=2  A 0 elsewhere

If this is translated by the scan to X, Y, the transmitted wave is 0 ðxyÞ

ð2:5:2:32Þ


ðuvÞ ¼ Qðu; vÞ  fTðuvÞ exp½2iðuX þ vYÞg;

IðX; YÞ ¼

 tðxyÞj2 :

ð2:5:2:40Þ

ð2:5:2:41Þ

which is identical to the result (2.5.2.35) for a plane incident wave in the conventional transmission electron microscope.

ð2:5:2:34Þ

2.5.2.7. Imaging of very thin and weakly scattering objects (a) The weak-phase-object approximation. For sufficiently thin objects, the effect of the object on the incident-beam amplitude may be represented by the transmission function (2.5.2.16) given by the phase-object approximation. If the fluctuations, ’ðxyÞ
’ , about the mean value of the projected potential are sufficiently small so that ½’ðxyÞ
’  1, it is possible to use the weakphase-object approximation (WPOA)

where tðxyÞ, given by Fourier transform of Tðu; vÞ, is the spread function. The image intensity is then 0 ðxyÞ

Hðu; vÞjQðu; vÞ  Tðu; vÞ

Iðx; yÞ ¼ jqðxyÞ  tðxyÞj2 ;

The image amplitude distribution, referred to the object coordinates, is given by Fourier transform of (2.5.2.31) as

IðxyÞ ¼ j ðxyÞj2 ¼ j

ð2:5:2:39Þ

If H(u, v) represents a small detector, approximated by a delta function, this becomes

v ¼ ðsin ’y Þ=:

 tðxyÞ;

R

 expf2iðuX þ vYÞgj2 du dv:

u ¼ ðsin ’x Þ=;

0 ðxyÞ

ð2:5:2:38Þ

and the image signal produced by a detector having a sensitivity function H(u, v) is

ð2:5:2:33Þ

and u, v are the reciprocal-space variables related to the scattering angles ’x, ’y by

ðxyÞ ¼

¼ qðxyÞ  tðx
X; y
YÞ:

The amplitude on the plane of observation following the specimen is then

and a phase function exp fi ðu; vÞg where the phase perturbation ðuvÞ due to lens defocus f and aberrations is usually approximated as  ðuvÞ ¼   f  ðu2 þ v2 Þ þ Cs 3 ðu2 þ v2 Þ2 ; 2

ð2:5:2:37Þ

ð2:5:2:35Þ

In practice the coherent imaging theory provides a good approximation but limitations of the coherence of the illumination have appreciable effects under high-resolution imaging conditions. The variation of focal lengths according to (2.5.2.30) is described by a function Gðf Þ. Illumination from a finite incoherent source gives a distribution of incident-beam angles Hðu1 ; v1 Þ. Then the image intensity is found by integrating incoherently over f and u1 ; v1 :

qðxyÞ ¼ expf
i’ðxyÞg ¼ 1
i’ðxyÞ;

ð2:5:2:42Þ

where ’ðxyÞ is referred to the average value, ’ . The assumption that only first-order terms in ’ðxyÞ need be considered is the equivalent of a single-scattering, or kinematical, approximation applied to the two-dimensional function, the projected potential

304

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION For bright-field STEM imaging with a very small detector placed axially in the central beam of the diffraction pattern (2.5.2.39) on the detector plane, the intensity, from (2.5.2.41), is given by (2.5.2.43). For a finite axially symmetric detector, described by DðuÞ, the image intensity is IðrÞ ¼ 1 þ 2’ðrÞ  fsðrÞ½dðrÞ  cðrÞ
cðrÞ½dðrÞ  sðrÞg; ð2:5:2:47Þ where dðrÞ is the Fourier transform of DðuÞ (Cowley & Au, 1978). For STEM with an annular dark-field detector which collects all electrons scattered outside the central spot of the diffraction pattern in the detector plane, it can be shown that, to a good approximation (valid except near the resolution limit)

Fig. 2.5.2.3. The functions ðUÞ, the phase factor for the transfer function of a lens given by equation (2.5.2.33), and sin ðUÞ for the Scherzer optimum defocus condition, relevant for weak phase objects, for which the minimum value of ðUÞ is
2=3.

IðrÞ ¼  2 ’2 ðrÞ  ½c2 ðrÞ þ s2 ðrÞ:

of (2.5.2.16). From (2.5.2.42), the image intensity (2.5.2.35) becomes IðxyÞ ¼ 1 þ 2’ðxyÞ  sðxyÞ;

Since c2 ðrÞ þ s2 ðrÞ ¼ jtðrÞj2 is the intensity distribution of the electron probe incident on the specimen, (2.5.2.48) is equivalent to the incoherent imaging of the function  2 ’2 ðrÞ. Within the range of validity of the WPOA or, in general, whenever the zero beam of the diffraction pattern is very much stronger than any diffracted beam, the general expression (2.5.2.36) for the modifications of image intensities due to limited coherence may be conveniently approximated. The effect of integrating over the variables f ; u1 ; v1, may be represented by multiplying the transfer function T(u, v) by so-called ‘envelope functions’ which involve the Fourier transforms of the functions Gðf Þ and Hðu1 ; v1 Þ. For example, if Gðf Þ is approximated by a Gaussian of width " (at e
1 of the maximum) centred at f0 and Hðu1 v1 Þ is a circular aperture function

ð2:5:2:43Þ

where the spread function s(xy) is the Fourier transform of the imaginary part of T(uv), namely AðuvÞ sin ðuvÞ. The optimum imaging condition is then found, following Scherzer (1949), by specifying that the defocus should be such that j sin j is close to unity for as large a range of U ¼ ðu2 þ v2 Þ1=2 as possible. This is so for a negative defocus such that ðuvÞ decreases to a minimum of about
2=3 before increasing to zero and higher as a result of the fourth-order term of (2.5.2.33) (see Fig. 2.5.2.3). This optimum, ‘Scherzer defocus’ value is given by d ¼0 du


for

Hðu1 v1 Þ ¼

¼
2=3

ð2:5:2:44Þ

x ¼

where  ¼ f0 ðu þ vÞ þ Cs 3 ðu3 þ v3 Þ þ i"2 2 ðu3 þ u2 v þ uv2 þ v3 Þ=2:

ð2:5:2:45Þ

˚ (200 keV), For example, for Cs ¼ 1 mm and  ¼ 2:51  10
2 A ˚ x ¼ 2:34 A. Within the limits of the WPOA, the image intensity can be written simply for a number of other imaging modes in terms of the Fourier transforms cðrÞ and sðrÞ of the real and imaginary parts of the objective-lens transfer function TðuÞ ¼ AðuÞ expfi ðuÞg, where r and u are two-dimensional vectors in real and reciprocal space, respectively. For dark-field TEM images, obtained by introducing a central stop to block out the central beam in the diffraction pattern in the back-focal plane of the objective lens, IðrÞ ¼ ½’ðrÞ  cðrÞ2 þ ½’ðrÞ  sðrÞ2 :

if u1 ; v1 < b otherwise,

expf
2 2 "2 ðu2 þ v2 Þ2 =4g  J1 ðBÞ=ðBÞ

The resolution limit is then taken as corresponding to the value of U ¼ 1:51Cs
1=4 
3=4 when sin becomes zero, before it begins to oscillate rapidly with U. The resolution limit is then 0:66Cs1=4 3=4 :

1 0

the transfer function T0 ðuvÞ for coherent radiation is multiplied by

or  1=2 f ¼
43 Cs  :

ð2:5:2:48Þ

ð2:5:2:49Þ

(b) The projected charge-density approximation. For very thin specimens composed of moderately heavy atoms, the WPOA is inadequate. Within the region of validity of the phase-object approximation (POA), more complicated relations analagous to (2.5.2.43) to (2.5.2.47) may be written. A simpler expression may be obtained by use of the two-dimensional form of Poisson’s equation, relating the projected potential distribution ’ðxyÞ to the projected charge-density distribution ðxyÞ. This is the PCDA (projected charge-density approximation) (Cowley & Moodie, 1960), IðxyÞ ¼ 1 þ 2f   ðxyÞ:

ð2:5:2:50Þ

ð2:5:2:46Þ This is valid for sufficiently small values of the defocus f, provided that the effects of the spherical aberration may be neglected, i.e. for image resolutions not too close to the Scherzer

Here, as in (2.5.2.42), ’ðrÞ should be taken to imply the difference from the mean potential value, ’ðrÞ
’ .

305

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION This relationship is satisfied for all h, k if a2 =b2 is an integer and

resolution limit (Lynch et al., 1975). The function ðxyÞ includes contributions from both the positive atomic nuclei and the negative electron clouds. For underfocus (f negative), single atoms give dark spots in the image. The contrast reverses with defocus.

f1
f2 ¼ 2na2 = and

2.5.2.8. Crystal structure imaging Cs1
Cs2 ¼ 4ma4 =3 ;

(a) Introduction. It follows from (2.5.2.43) and (2.5.2.42) that, within the severe limitations of validity of the WPOA or the PCDA, images of very thin crystals, viewed with the incident beam parallel to a principal axis, will show dark spots at the positions of rows of atoms parallel to the incident beam. Provided that the resolution limit is less than the projected distances ˚ ), the projection of the crystal strucbetween atom rows (1–3 A ture may be seen directly. In practice, theoretical and experimental results suggest that images may give a direct, although nonlinear, representation of the projected potential or charge-density distribution for thicknesses much greater than the thicknesses for validity of these ˚ for approximations, e.g. for thicknesses which may be 50 to 100 A ˚ resolutions and which increase for 100 keV electrons for 3 A comparable resolutions at higher voltage but decrease with improved resolutions. The use of high-resolution imaging as a means for determining the structures of crystals and for investigating the form of the defects in crystals in terms of the arrangement of the atoms has become a widely used and important branch of crystallography with applications in many areas of solid-state science. It must be emphasized, however, that image intensities are strongly dependent on the crystal thickness and orientation and also on the instrumental parameters (defocus, aberrations, alignment etc.). It is only when all of these parameters are correctly adjusted to lie within strictly defined limits that interpretation of images in terms of atom positions can be attempted [see IT C (2004, Section 4.3.8)]. Reliable interpretations of high-resolution images of crystals (‘crystal structure images’) may be made, under even the most favourable circumstances, only by the comparison of observed image intensities with intensities calculated by use of an adequate approximation to many-beam dynamical diffraction theory [see IT C (2004, Section 4.3.6)]. Most calculations for moderate or large unit cells are currently made by the multislice method based on formulation of the dynamical diffraction theory due to Cowley & Moodie (1957). For smaller unit cells, the matrix method based on the Bethe (1928) formulation is also frequently used (Hirsch et al., 1965). (b) Fourier images. For periodic objects in general, and crystals in particular, the amplitudes of the diffracted waves in the back focal plane are given from (2.5.2.31) by
0 ðhÞ  TðhÞ:

ð2:5:2:52Þ

where m, n are integers (Kuwabara, 1978). The relationship for f is an expression of the Fourier image phenomenon, namely that for a plane-wave incidence, the intensity distribution for the image of a periodic object repeats periodically with defocus (Cowley & Moodie, 1960). Hence it is often necessary to define the defocus value by observation of a nonperiodic component of the specimen such as a crystal edge (Spence et al., 1977). For the special case a2 ¼ b2, the image intensity is also reproduced exactly for f1
f2 ¼ ð2n þ 1Þa2 =;

ð2:5:2:53Þ

except that in this case the image is translated by a distance a=2 parallel to each of the axes. 2.5.2.9. Image resolution (1) Definition and measurement. The ‘resolution’ of an electron microscope or, more correctly, the ‘least resolvable distance’, is usually defined by reference to the transfer function for the coherent imaging of a weak phase object under the Scherzer optimum defocus condition (2.5.2.44). The resolution figure is taken as the inverse of the U value for which sin ðUÞ first crosses the axis and is given, as in (2.5.2.45), by x ¼ 0:66Cs1=4 3=4 :

ð2:5:2:45Þ

It is assumed that an objective aperture is used to eliminate the contribution to the image for U values greater than the first zero crossing, since for these contributions the relative phases are distorted by the rapid oscillations of sin ðUÞ and the corresponding detail of the image is not directly interpretable in terms of the projection of the potential distribution of the object. The resolution of the microscope in practice may be limited by the incoherent factors which have the effect of multiplying the WPOA transfer function by envelope functions as in (2.5.2.49). The resolution, as defined above, and the effects of the envelope functions may be determined by Fourier transform of the image of a suitable thin, weakly scattering amorphous specimen. The Fourier-transform operation may be carried out by use of an optical diffractometer. A more satisfactory practice is to digitize the image directly by use of a two-dimensional detector system in the microscope or from a photographic recording, and perform the Fourier transform numerically. For the optical diffractometer method, the intensity distribution obtained is given from (2.5.2.43) as a radially symmetric function of U,

ð2:5:2:51Þ

For rectangular unit cells of the projected unit cell, the vector h has components h=a and k=b. Then the set of amplitudes (2.5.2.34), and hence the image intensities, will be identical for two different sets of defocus and spherical aberration values f1 ; Cs1 and f2 ; Cs2 if, for an integer N,

IðUÞ ¼ jF IðxyÞj2

1 ðhÞ ¼ 2 ðhÞ ¼ 2N;

¼ ðUÞ þ 4 2 jðuÞj2  sin2 ðUÞ  E2 ðUÞ;

ð2:5:2:54Þ

i.e.  

2

2

h k þ a2 b2



where EðUÞ is the product of the envelope functions. In deriving (2.5.2.54) it has been assumed that: (a) the WPOA applies; (b) the optical transmission function of the photographic record is linearly related to the image intensity, IðxyÞ;

 2 2 1 h k2 ðf1
f2 Þ þ 3 2 þ 2 ðCs1
Cs2 Þ ¼ 2N: 2 a b

306

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION (c) the diffraction intensity jðUÞj2 is a radially symmetric, smoothly varying function such as is normally produced by a sufficiently large area of the image of an amorphous material; (d) there is no astigmatism present and no drift of the specimen; either of these factors would remove the radial symmetry. From the form of (2.5.2.54) and a preknowledge of jðUÞj2, the zero crossings of sin and the form of EðUÞ may be deduced. Analysis of a through-focus series of images provides more complete and reliable information. (2) Detail on a scale much smaller than the resolution of the electron microscope, as defined above, is commonly seen in electron micrographs, especially for crystalline samples. For example, lattice fringes, having the periodicity of the crystal ˚ in one direction, lattice planes, with spacings as small as 0.6 A have been observed using a microscope having a resolution of ˚ (Matsuda et al., 1978), and two-dimensionally periabout 2.5 A ˚ have been odic images showing detail on the scale of 0.5 to 1 A observed with a similar microscope (Hashimoto et al., 1977). Such observations are possible because (a) for periodic objects the diffraction amplitude
0 ðuvÞ in (2.5.2.31) is a set of delta functions which may be multiplied by the corresponding values of the transfer function that will allow strong interference effects between the diffracted beams and the zero beam, or between different diffracted beams; (b) the envelope functions for the WPOA, arising from incoherent imaging effects, do not apply for strongly scattering crystals; the more general expression (2.5.2.36) provides that the incoherent imaging factors will have much less effect on the interference of some sets of diffracted beams. The observation of finely spaced lattice fringes provides a measure of some important factors affecting the microscope performance, such as the presence of mechanical vibrations, electrical interference or thermal drift of the specimen. A measure of the fineness of the detail observable in this type of image may therefore be taken as a measure of ‘instrumental resolution’.

each incident-beam direction. The resulting CBED pattern has an intensity distribution IðuvÞ ¼

R

j
u1 v1 ðuvÞj2 du1 dv1 ;

ð2:5:2:55Þ

where
u1 v1 ðuvÞ is the Fourier transform of the exit wave at the specimen for an incident-beam direction u1 ; v1 . (c) Coherent illumination from a small bright source such as a field emission gun may be focused on the specimen to give an electron probe having an intensity distribution jtðxyÞj2 and a diameter equal to the STEM dark-field image resolution [equa˚ . The intensity distribution of the tion (2.5.2.47)] of a few A resulting microdiffraction pattern is then j
ðuvÞj2 ¼ j
0 ðuvÞ  TðuvÞj2 ;

ð2:5:2:56Þ

where
0 ðuvÞ is the Fourier transform of the exit wave at the specimen. Interference occurs between waves scattered from the various incident-beam directions. The diffraction pattern is thus an in-line hologram as envisaged by Gabor (1949). (d) Diffraction patterns may be obtained by using an optical diffractometer (or computer) to produce the Fourier transform squared of a small selected region of a recorded image. The optical diffraction-pattern intensity obtained under the ideal conditions specified under equation (2.5.2.54) is given, in the case of weak phase objects, by IðuvÞ ¼ ðuvÞ þ 4 2 jðuvÞj2  sin2 ðuvÞ  E2 ðuvÞ

ð2:5:2:57Þ

or, more generally, by IðuvÞ ¼ cðuvÞ þ j
ðuvÞ  TðuvÞ 
 ðuvÞ  T  ðuvÞj2 ; where
ðuvÞ is the Fourier transform of the wavefunction at the exit face of the specimen and c is a constant depending on the characteristics of the photographic recording medium.

2.5.2.10. Electron diffraction in electron microscopes Currently most electron-diffraction patterns are obtained in conjunction with images, in electron microscopes of one form or another, as follows. (a) Selected-area electron-diffraction (SAED) patterns are obtained by using intermediate and projector lenses to form an image of the diffraction pattern in the back-focal plane of the objective lens (Fig. 2.5.2.2). The area of the specimen from which the diffraction pattern is obtained is defined by inserting an aperture in the image plane of the objective lens. For parallel illumination of the specimen, sharp diffraction spots are produced by perfect crystals. A limitation to the area of the specimen from which the diffraction pattern can be obtained is imposed by the spherical aberration of the objective lens. For a diffracted beam scattered through an angle , the spread of positions in the object for which the diffracted beam passes through a small axial aperture in the image plane is Cs 3, e.g. for Cs ¼ 1 mm,  ¼ 5  10
2 rad (10,0,0 ˚ , so reflection from gold for 100 keV electrons), Cs 3 ¼ 1250 A ˚ is not that a selected-area diameter of less than about 2000 A feasible. For higher voltages, the minimum selected-area diameter decreases with 2 if the usual assumption is made that Cs increases for higher-voltage microscopes so that Cs  is a constant. (b) Convergent-beam electron-diffraction (CBED) patterns are obtained when an incident convergent beam is focused on the specimen, as in an STEM instrument or an STEM attachment for a conventional TEM instrument. For a large, effectively incoherent source, such as a conventional hot-filament electron gun, the intensities are added for

2.5.3. Point-group and space-group determination by convergentbeam electron diffraction

By M. Tanaka 2.5.3.1. Introduction Because the cross section for electron scattering is at least a thousand times greater than that for X-rays, and because multiple Bragg scattering preserves information on symmetry (such as the absence of inversion symmetry), electron diffraction is exquisitely sensitive to symmetry. The additional ability of modern electron-optical lenses to focus an electron probe down to nanometre dimensions, and so allow the study of nanocrystals too small for analysis by X-rays, has meant that the method of convergent-beam diffraction described here has now become the preferred method of symmetry determination for very small crystals, domains, twinned structures, quasicrystals, incommensurate structures and other imperfectly crystalline materials. Convergent-beam electron diffraction (CBED) originated with the experiments of Kossel & Mo¨llenstedt (1938). However, modern crystallographic investigations by CBED began with the studies performed by Goodman & Lehmpfuhl (1965) in a modified transmission electron microscope. They obtained CBED patterns by converging a conical electron beam with an angle of more than 10
3 rad on an ~30 nm diameter specimen area, which had uniform thickness and no bending. Instead of the

307

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION usual diffraction spots, diffraction discs (in Laue or transmission geometry) were produced. The diffraction intensity within a disc shows a specific symmetry, which enables one to determine the point groups and space groups of microcrystals. Unlike X-ray diffraction, the method is extremely sensitive to the presence or absence of inversion symmetry. The method corresponding to CBED in the field of light optics is the conoscope method. Using a conoscope, we can identify whether a crystal is isotropic, uniaxial or biaxial, and determine the optic axis and the sign of birefringence of a crystal. When CBED, a conoscope method using an electron beam, is utilized, more basic properties of a crystal – the crystal point group and space group – can be determined. Point- and space-group determinations are routinely also carried out by X-ray diffraction. This method, to which kinematical diffraction is applicable, cannot determine whether a crystal is polar or nonpolar unless anomalous absorption is utilized. As a result, the X-ray diffraction method can only identify 11 Laue groups among 32 point groups. CBED, based fully upon dynamical diffraction, can distinguish polar crystals from nonpolar crystals using only a nanometre-sized crystal, thus allowing the unique identification of all the point groups by inspecting the symmetries appearing in CBED discs. As pointed out above, an unambiguous experimental determination of crystal symmetry, in the case of X-ray diffraction, is usually not possible because of the apparent centrosymmetry of the diffraction pattern, even for noncentrosymmetric crystals. However, methods based on structure-factor and X-ray intensity statistics remain useful for the resolution of space-group ambiguities, and are routinely applied to structure determinations from X-ray data. These methods are described in Chapter 2.1 of this volume. In the field of materials science, correct space-group determination by CBED is often requested prior to X-ray or neutron structure refinement, in particular in the case of Rietveld refinements based on powder diffraction data. CBED can determine not only the point and space groups of crystals but also crystal structure parameters – lattice parameters, atom positions, Debye–Waller factors and low-order structure factors. The lattice parameters can be determined from submicron regions of thin crystals by using higher-order Laue zone (HOLZ) reflections with an accuracy of 1  10
4. Cherns et al. (1988) were the first to perform strain analysis of artificial multilayer materials using the large-angle technique (LACBED) (Tanaka et al., 1980). Since then, many strain measurements at interfaces of various multilayer materials have been successfully conducted. In recent years, strain analysis has been conducted using automatic analysis programs, which take account of dynamical diffraction effects (Kra¨mer et al., 2000). We refer to the book of Morniroli (2002), which carries many helpful figures, clear photographs and a comprehensive list of papers on this topic. Vincent et al. (1984a,b) first applied the CBED method to the determination of the atom positions of AuGeAs. They analysed the intensities of HOLZ reflections by applying a quasikinematical approximation. Tanaka & Tsuda (1990, 1991) and Tsuda & Tanaka (1995) refined the structural parameters of SrTiO3 by applying the dynamical theory of electron diffraction. The method was extended to the refinements of CdS, LaCrO3 and hexagonal BaTiO3 (Tsuda & Tanaka, 1999; Tsuda et al., 2002; Ogata et al., 2004). Rossouw et al. (1996) measured the order parameters of TiAl through a Bloch-wave analysis of HOLZ reflections in a CBED pattern. Midgley et al. (1996) refined two positional parameters of AuSn4 from the diffraction data obtained with a small convergence angle using multislice calculations. Low-order structure factors were first determined by Goodman & Lehmpfuhl (1967) for MgO. After much work on low-order structure-factor determination, Zuo & Spence determined the 200 and 400 structure factors of MgO in a very modern

way, by fitting energy-filtered patterns and many-beam dynamical calculations using a least-squares procedure. For the low-order structure-factor determinations, the excellent comprehensive review of Spence (1993) should be referred to. Saunders et al. (1995) succeeded in obtaining the deformation charge density of Si using the low-order crystal structure factors determined by CBED. For the reliable determination of the low-order X-ray crystal structure factors or the charge density of a crystal, accurate determination of the Debye–Waller factors is indispensable. Zuo et al. (1999) determined the bond-charge distribution in cuprite. Simultaneous determination of the Debye–Waller factors and the low-order structure factors using HOLZ and zerothorder Laue zone (ZOLZ) reflections was performed to determine the deformation charge density of LaCrO3 accurately (Tsuda et al., 2002). CBED can also be applied to the determination of lattice defects, dislocations (Cherns & Preston, 1986), stacking faults (Tanaka, 1986) and twins (Tanaka, 1986). Since this topic is beyond the scope of the present chapter, readers are referred to pages 156 to 205 of the book by Tanaka et al. (1994). We also mention the book by Spence & Zuo (1992), which deals with the whole topic of CBED, including the basic theory and a wealth of literature. 2.5.3.2. Point-group determination When an electron beam traverses a thin slab of crystal parallel to a zone axis, one can easily imagine that symmetries parallel to the zone axis should appear in the resulting CBED pattern. It is, however, more difficult to imagine what symmetries appear due to symmetries perpendicular to the incident beam. Goodman (1975) pioneered the clarification of CBED symmetries for the twofold rotation axis and mirror plane perpendicular to the incident beam, and the symmetry of an inversion centre, with the help of the reciprocity theorem of scattering theory. Tinnappel (1975) solved many CBED symmetries at various crystal settings with respect to the incident beam using a group-theoretical treatment. Buxton et al. (1976) also derived these results from first principles, and generalized them to produce a systematic method for the determination of the crystal point group. Tanaka, Saito & Sekii (1983) developed a method to determine the point group using simultaneously excited many-beam patterns. The point-group-determination method given by Buxton et al. (1976) is described with the aid of the description by Tanaka, Saito & Sekii (1983) in the following. 2.5.3.2.1. Symmetry elements of a specimen and diffraction groups Since CBED uses the Laue geometry, Buxton et al. (1976) assumed a perfectly crystalline specimen in the form of a parallelsided slab which is infinite in two dimensions. The symmetry elements of the specimen (as distinct from those of an infinite crystal) form ‘diffraction groups’, which are isomorphic to the point groups of the diperiodic plane figures and Shubnikov groups of coloured plane figures. The diffraction groups of a specimen are determined from the symmetries of CBED patterns taken at various orientations of the specimen. The crystal pointgroup of the specimen is identified by referring to Fig. 2.5.3.4, which gives the relation between diffraction groups and crystal point groups. A specimen that is parallel-sided and is infinitely extended in the x and y directions has ten symmetry elements. The symmetry elements consist of six two-dimensional symmetry elements and four three-dimensional ones. The operation of the former elements transforms an arbitrary coordinate (x, y, z) into (x0 , y0 , z), with z remaining the same. The operation of the latter transforms a coordinate (x, y, z) into (x0 , y0 , z0 ), where z0 6¼ z. A vertical mirror plane m and one-, two-, three-, four- and sixfold rotation axes that are parallel to the surface normal z are the two-

308

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.1. Two- and three-dimensional symmetry elements of an infinitely extended parallel-sided specimen Symbols in parentheses show CBED symmetries appearing in dark-field patterns. Two-dimensional symmetry elements

Three-dimensional symmetry elements

1

m0 (1R)

2

i (2R)

3

20 (m2, mR) 4 (4R)

4

Fig. 2.5.3.1. Four symmetry elements m0, i, 20 and 4 of an infinitely extended parallel-sided specimen.

5 6 m

dimensional symmetry elements. A horizontal mirror plane m0, an inversion centre i, a horizontal twofold rotation axis 20 and a fourfold rotary inversion 4 are the three-dimensional symmetry elements, and are shown in Fig. 2.5.3.1. The fourfold rotary inversion was not recognized as a symmetry element until the point groups of the diperiodic plane figures were considered (Buxton et al., 1976). Table 2.5.3.1 lists these symmetry elements, where the symbols in parentheses express symmetries of CBED patterns expected from three-dimensional symmetry elements. The diffraction groups are constructed by combining these symmetry elements (Table 2.5.3.2). Two-dimensional symmetry elements and their combinations are given in the top row of the table. The third symmetry m in parentheses is introduced automatically when the first two symmetry elements are combined. Three-dimensional symmetry elements are given in the first column. The equations given below the table indicate that no additional three-dimensional symmetry elements can appear by combination of two symmetry elements in the first column. As a result, 31 diffraction groups are produced by combining the elements in the first column with those in the top row. Diffraction groups in square brackets have already appeared ealier in the table. In the fourth row, three columns have two diffraction groups, which are produced when symmetry elements are combined at different orientations. In the last row, five columns are empty because a fourfold rotary inversion cannot coexist with threefold and sixfold rotation axes. In the last column, the number of independent diffraction groups in each row is given, the sum of the numbers being 31.

but do in a diffraction disc set at the Bragg condition, each of which we call a dark-field pattern (DP). The CBED symmetries obtained are illustrated in Fig. 2.5.3.2. A horizontal twofold rotation axis 20 , a horizontal mirror plane m0, an inversion centre i and a fourfold rotary inversion 4 produce symmetries mR (m2), 1R, 2R and 4R in DPs, respectively. Next we explain the symbols of the CBED symmetries. (1) Operation mR is shown in the left-hand part of Fig. 2.5.3.2(a), which implies successive operations of (a) a mirror m with respect to a twofold rotation axis, transforming an open circle beam (*) in reflection G into a beam (+) in reflection G0 and (b) rotation R of this beam by  about the centre point of disc G0 (or the exact Bragg position of reflection G0 ), resulting in position * in reflection G0 . The combination of the two operations is written as mR. When the twofold rotation axis is parallel to the diffraction vector G, two beams (*) in the left-hand part of the figure become one reflection G, and a mirror symmetry, whose mirror line is perpendicular to vector G and passes through the centre of disc G, appears between the two beams (the right-hand side figure of Fig. 2.5.3.2a). The mirror symmetry is labelled m2 after the twofold rotation axis. (2) Operation 1R (Fig. 2.5.3.2b) for a horizontal mirror plane is a combination of a rotation by 2 of a beam (*) about a zone axis O (symbol 1), which is equivalent to no rotation, and a rotation by  of the beam about the exact Bragg position or the centre of disc G. (3) Operation 2R is a rotation by  of a beam (*) in reflection G about a zone axis (symbol 2), which transforms the beam into a beam (+) in reflection
G, followed by a rotation by  of the beam (+) about the centre of disc
G, resulting in the beam (*) in disc
G (Fig. 2.5.3.2c). The symmetry is called translational symmetry after Goodman (1975) because the pattern of disc +G coincides with that of disc
G by a translation. It is emphasized that an inversion centre is identified by the test of translational symmetry about a pair of G dark-field patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair DP. (4) Operation 4R (Fig. 2.5.3.2d) can be understood in a similar manner. It is noted that regular letters are symmetries about a zone axis, while subscripts R represent symmetries about the exact Bragg position. We call a pattern that contains an exact Bragg position (if possible at the disc centre) a dark-field pattern. As far as CBED symmetries are concerned, we

2.5.3.2.2. Identification of three-dimensional symmetry elements It is difficult to imagine the symmetries in CBED patterns generated by the three-dimensional symmetry elements of the sample. The reason is that if a three-dimensional symmetry element is applied to a specimen, it turns it upside down, which is impractical in most experiments. The reciprocity theorem of scattering theory (Pogany & Turner, 1968) enables us to clarify the symmetries of CBED patterns expected from these threedimensional symmetry elements. A graphical method for obtaining CBED symmetries due to sample symmetry elements is described in the papers of Goodman (1975), Buxton et al. (1976) and Tanaka (1989). The CBED symmetries of the threedimensional symmetries do not appear in the zone-axis patterns,

Table 2.5.3.2. Symmetry elements of an infinitely extended parallel-sided specimen and diffraction groups 1

2

3

4

6

m

2m(m)

3m

4m(m)

6m(m)

1

1

2

3

4

6

m

2m(m)

3m

4m(m)

6m(m)

10

(m0 ) 1R (i) 2R

1R 2R

21R [21R]

31R 6R

41R [41R]

61R [61R]

m1R 2Rm(mR)

2m(m)1R [2m(m)1R]

3m1R 6Rm(mR)

4m(m)1R [4m(m)1R]

6m(m)1R [6m(m)1R]

10 4

(20 ) mR ð4 Þ 4R

mR

2mR(mR)

3mR

4mR(mR)

6mR(mR)

[4m(m)1R]

[6Rm(mR)]

4R

[41R]

[2Rm(mR)]

[2m(m)1R]

[3m1R]

[m1R]

[4R(m)mR]

[6Rm(mR)]

4Rm(mR)

[4Rm(mR)]

1R  2R = 2, 2R  2R = 1, mR  2R = m, 4R  2R = 4, 1R  mR = m  mR, 1R  4R = 4  1R, mR  4R = m  4R.

309

[4m(m)1R]

5 2

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.3. Illustration of symmetries appearing in dark-field patterns (DPs) and a pair of dark-field patterns (DP) for the combinations of symmetry elements.

specimen crystal has an inversion centre or not, because an inversion centre forms the lowest symmetry 1 in the BP. In conclusion, all the two-dimensional symmetry elements can be identified from the WP symmetries. Fig. 2.5.3.2. Illustration of symmetries appearing in dark-field patterns (DPs). (a) mR and m2; (b) 1R; (c) 2R; (d) 4R, originating from 20 , m0 , i and 4 , respectively.

2.5.3.2.4. Diffraction-group determination All the symmetry elements of the diffraction groups can be identified from the symmetries of a WP and DPs. But it is practical and convenient to use just the four patterns WP, BP, DP and DP to determine the diffraction group. The symmetries appearing in these four patterns are given for the 31 diffraction groups in Table 2.5.3.3 (Tanaka, Saito & Sekii, 1983), which is a detailed version of Table 2 of Buxton et al. (1976). All the possible symmetries of the DP and DP appearing at different crystal orientations are given in the present table. When a BP has a higher symmetry than the corresponding WP, the symmetry elements that produce the BP are given in parentheses in column II except only for the case of 4R. When two types of vertical mirror planes exist, these are distinguished by symbols mv and mv0 . Each of the two or three symmetries given in columns IV and V for many diffraction groups appears in a DP or DP in different directions. It is emphasized again that no two diffraction groups exhibit the same combination of BP, WP, DP and DP, which implies that the diffraction groups are uniquely determined from an inspection of these pattern symmetries. Fig. 2.5.3.3 illustrates the symmetries of the DP and DP appearing in Table 2.5.3.3, which greatly eases the cumbersome task of determining the symmetries. The first four patterns illustrate the symmetries appearing in a single DP and the others treat those in DPs. The pattern symmetries are written beneath the figures. The other symbols are the symmetries of a specimen. The crosses outside the diffraction discs designate the zone axis. The crosses inside the diffraction discs indicate the exact Bragg position. When the four patterns appearing in three photographs are taken and examined using Table 2.5.3.3 with the aid of Fig. 2.5.3.3, one diffraction group can be selected unambiguously. It is,

do not use the term dark-field pattern if a disc does not contain the exact Bragg position. The four three-dimensional symmetry elements are found to produce different symmetries in the DPs. These facts imply that these symmetry elements can be identified unambiguously from the symmetries of CBED patterns. 2.5.3.2.3. Identification of two-dimensional symmetry elements Two-dimensional symmetry elements that belong to a zone axis exhibit their symmetries in CBED patterns or zone-axis patterns (ZAPs) directly, even if dynamical diffraction takes place. A ZAP contains a bright-field pattern (BP) and a whole pattern (WP). The BP is the pattern appearing in the bright-field disc [the central or ‘direct’ (000) beam]. The WP is composed of the BP and the pattern formed by the surrounding diffraction discs, which are not exactly excited. The two-dimensional symmetry elements m, 1, 2, 3, 4 and 6 yield symmetry mv and one-, two-, three-, four- and sixfold rotation symmetries, respectively, in WPs, where the suffix v for mv is assigned to distinguish it from mirror symmetry m2 caused by a horizontal twofold rotation axis. It should be noted that a BP shows not only the zone-axis symmetry but also three-dimensional symmetries, indicating that the BP can have a higher symmetry than the symmetry of the corresponding WP. The symmetries of the BP due to threedimensional symmetry elements are obtained by moving the DPs to the zone axis. As a result, the three-dimensional symmetry elements m0, i, 20 and 4 produce, respectively, symmetries 1R, 1, m2 and 4 in the BP, instead of 1R, 2R, m2 and 4R in the DPs (Fig. 2.5.3.2). We mention that the BP cannot distinguish whether a

310

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.3. Symmetries of different patterns for diffraction and projection diffraction groups

Table 2.5.3.3 (cont.)

(II) Bright-field patterns (BPs); (III) whole patterns (WPs); (IV) dark-field patterns (DPs); and (V) dark-field patterns (DPs) for diffraction groups (I) and projection diffraction groups (VI). I

II

III

IV

V

VI

1

1

1

1

1

1R

1R

2 (1R)

1

2 = 1R

1

2

2

2

1

2

2R

1

1

1

2R

21R

2

2

2

21R

mR

m

1

1

1

(m2) m

mv

2mm

2mm

mv

mv

2RmmR

2mm1R

2mvmv0 mv

2mvmv0

1

1

1

mv

1

2

1

2 2mvmv0 mv

2mvmv0

2mvm2

1

1

2

m2

2mR(m2)

1

2

mv

2mv0 (mv)

1

2R

m2

2Rmv0 (m2)

mv

2RmR(mv)

2

21R

2mvm2

21Rmv0 (mv)

4

4

4

1

2

4R

4

2

1

2

41R

4

4

2

21R

4mRmR

4mm

4

1

2

m2

2mR(m2)

1

2

(4 + m2) 4mm 4RmmR

4mvmv0 4mm

4mvmv0 2mvmv0

(2mvmv0 + m2) 4mm1R

4mvmv0

4mvmv0

mv

2mv0 (mv)

1

2

m2

2mR(m2)

mv

2mv0 (mv)

2

21R

2mvm2

21Rmv0 (mv)

3

3

3

1

1

31R

6 (3 + 1R)

3

2

1

3mR

3m

3

1

1

(3 + m2) 3m

3mv

6mm

V

VI

1

2

6mm1R

m2

2mR(m2)

1

2

6RmmR

6mvmv0 3mv

6mvmv0 3mv

21R 6mm1R

6mvmv0

6mvmv0

mv

2mv0 (mv)

1

2R

m2

2Rmv0 (m2)

mv

2RmR(mv)

2

21R

2mvm2

21Rmv0 (mv)

m1R

however, noted that many diffraction groups are determined from a WP and BP pair without using a DP or DP (or from one photograph) or from a set of a WP, a BP and a DP without using a DP (or from two photographs). 2.5.3.2.5. Point-group determination Fig. 2.5.3.4 provides the relationship between the 31 diffraction groups for slabs and the 32 point groups for infinite crystals given by Buxton et al. (1976). When a diffraction group is determined, possible point groups are selected by consulting this figure. Each of the 11 high-symmetry diffraction groups corresponds to only one crystal point group. In this case, the point group is uniquely determined from the diffraction group. When more than one point group falls under a diffraction group, a different diffraction group has to be obtained for another zone axis. A point group is identified by finding a common point group among the point groups obtained for different zone axes. It is clear from the figure that high-symmetry zones should be used for quick determination of point groups because low-symmetry zone axes exhibit only a small portion of crystal symmetries in the CBED patterns. Furthermore, it should be noted that CBED cannot observe crystal symmetries oblique to an incident beam or horizontal three-, four- or sixfold rotation axes. The diffraction groups to be expected for different zone axes are given for all the point groups in Table 2.5.3.4 (Buxton et al., 1976). The table is useful for finding a suitable zone axis to distinguish candidate point groups expected in advance. We shall explain the point-group determination procedure using an Si crystal. Fig. 2.5.3.5(a) shows a [111] ZAP of the Si specimen. The BP has threefold rotation symmetry and mirror symmetry or symmetry 3mv, which are caused by the threefold rotation axis along the [111] direction and a vertical mirror plane. The WP has the same symmetry. Figs. 2.5.3.5(b) and (c) are 22 0 and 2 20 DPs, respectively. Both show symmetry m2 perpendicular to the reflection vector. This symmetry is caused by a twofold rotation axis parallel to the specimen surface. One DP coincides with the other upon translation. This translational or 2R symmetry indicates the existence of an inversion centre. By consulting Table 2.5.3.3, the diffraction group giving rise to these pattern symmetries is found to be 6RmmR. Fig. 2.5.3.4 shows that there are two point groups 3 m and m3 m causing diffraction group 6RmmR. Fig. 2.5.3.6 shows another ZAP, which shows symmetry 4mm in the BP and the WP. The point group which has fourfold rotation symmetry is not 3 m but m3 m. The point group of Si is thus determined to be m3 m.

2mm1R

41R

4mm1R

31R

3m1R

mR 3mv

m2

1

1

1 mv

3m1R

IV

6

6mm

mv1R

(2 + m2) 2mm

III

6mm

mR m2

[mv + m2 + (1R)] 2mRmR

II

6mRmR

(6 + m2)

mv m1R

I

3mv

mv

1

2

1

[3mv + m2 + (1R)]

mv1R 2mvm2

2.5.3.2.6. Projection diffraction groups HOLZ reflections appear as excess HOLZ rings far outside the ZOLZ reflection discs and as deficit lines in the ZOLZ discs. By ignoring these weak diffraction effects with components along the beam direction, we may obtain information about the symmetry of the sample as projected along the beam direction. Thus when HOLZ reflections are weak and no deficit HOLZ

1

6

6

6

1

2

6R

3

3

1

2R

61R

6

6

2

21R

61R

311

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.4. Relation between diffraction groups and crystal point groups (after Buxton et al., 1976).

lines are seen in the ZOLZ discs, the symmetry elements found from the CBED patterns are only those of the specimen projected along the zone axis. The projection of the specimen along the zone axis causes horizontal mirror symmetry m0, the corresponding CBED symmetry being 1R. When symmetry 1R is added to the 31 diffraction groups, ten projection diffraction groups having symmetry symbol 1R are derived as shown in column VI of Table 2.5.3.3. If only ZOLZ reflections are observed in CBED patterns, a projection diffraction group instead of a diffraction group is obtained, where only the pattern symmetries given in the rows of the diffraction groups having symmetry symbol 1R in Table 2.5.3.3 should be consulted. Two projection diffraction groups obtained from two different zone axes are the minimum needed to determine a crystal point group, because it is constructed by the three-dimensional combination of symmetry elements. It should be noted that if a diffraction group is determined carelessly from CBED patterns with no HOLZ lines, the wrong crystal point group is obtained.

simultaneously at the Bragg condition. If many such DPs are recorded (all simultaneously at the Bragg condition), many threedimensional symmetry elements can be identified from one photograph. Using a group-theoretical method, Tinnappel (1975) studied the symmetries appearing in simultaneously excited DPs for various combinations of crystal symmetry elements. Based upon his treatment, Tanaka, Saito & Sekii (1983) developed a method for determining diffraction groups using simultaneously excited symmetrical hexagonal six-beam, square four-beam and rectangular four-beam CBED patterns. All the CBED symmetries appearing in the symmetrical many-beam (SMB) patterns were derived by the graphical method used in the paper of Buxton et al. (1976). From an experimental viewpoint, it is advantageous that symmetry elements can be identified from one photograph. It was found that twenty diffraction groups can be identified from one SMB pattern, whereas ten diffraction groups can be determined by Buxton et al.’s method. An experimental comparison between the two methods was performed by Howe et al. (1986). SMB patterns are easily obtained by tilting a specimen crystal or the incident beam from a zone axis into an orientation to excite low-order reflections simultaneously. Fig. 2.5.3.7 illustrates the symmetries of the SMB patterns for all the diffraction groups except for the five groups 1, 1R, 2, 2R and 21R. For these groups, the two-beam method for exciting one reflection is satisfactory because many-beam excitation gives no more information than the two-beam case. In the six-beam and square four-beam cases,

2.5.3.2.7. Symmetrical many-beam method In the sections above, the point-group determination method established by Buxton et al. (1976) was described, where two- and three-dimensional symmetry elements were determined, respectively, from ZAPs and DPs. The Laue circle is defined as the intersection of the Ewald sphere with the ZOLZ, and all reflections on this circle are

312

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.4. Diffraction groups expected at various crystal orientations for 32 point groups This table is adapted from Buxton et al. (1976).

Zone-axis symmetries Point group

h111i

h100i

h110i

huv0i

huuwi

[uvw]

m3m 4 3m

6RmmR 3m

4mm1R 4RmmR

2mm1R m1R

2RmmR mR

2RmmR m

2R 1

432

3mR

4mRmR

2mRmR

mR

mR

1

Zone-axis symmetries Point group

h111i

h100i

huv0i

[uvw]

m3

6R

2mm1R

2RmmR

2R

23

3

2mRmR

mR

1

Point group

Zone-axis symmetries [0001] h112 0i

h11 00i

[uv.0]

[uu.w]

½uu:w

[uv.w]

6/mmm 6 m2

6mm1R 3m1R

2mm1R m1R

2mm1R 2mm

2RmmR m

2Rmm mR

2RmmR m

2R 1

6mm

6mm

m1R

m1R

mR

m

m

1

622

6mRmR

2mRmR

2mRmR

mR

mR

mR

1

Zone-axis symmetries Point group

[0001]

[uv.0]

6/m 6

61R

2RmmR

2R

31R

m

1

6

6

mR

1

Point group 3 m

[uv.w]

Zone-axis symmetries [0001] h112 0i

½uu:w

[uv.w]

6RmmR

2RmmR

2R

21R

3m

3m

1R

m

1

32

3mR

2

mR

1

Zone-axis symmetries Point group 3

[0001]

[uv.w]

6R

2R

3

3

1

Zone-axis symmetries Point group

[001]

h100i

h110i

[u0w]

[uv0]

[uuw]

[uvw]

4/mmm 4 2m

4mm1R

2mm1R

2mm1R

2RmmR

2RmmR

2RmmR

2R

4RmmR

2mRmR

m1R

mR

mR

m

1

4mm

4mm

m1R

m1R

m

mR

m

1

422

4mRmR

2mRmR

2mRmR

mR

mR

mR

1

Zone-axis symmetries Point group

[001]

[uv0]

[uvw]

4/m 4

41R

2RmmR

2R

4R

mR

1

4

4

mR

1

h100i

[u0w]

[uv0]

[uvw]

Zone-axis symmetries Point group

[001]

mmm

2mm1R

2mm1R

2RmmR

2RmmR

2R

mm2

2mm

m1R

m

mR

1

222

2mRmR

2mRmR

mR

mR

1

313

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.4 (cont.) Zone-axis symmetries Point group

[010]

[u0w]

2/m

21R

2RmmR

[uvw] 2R

m

1R

m

1

2

2

mR

1

Zone-axis symmetry Point group 1

[uvw]

1

1

2R

the CBED symmetries for the two crystal (or incident-beam) settings which excite respectively the +G and
G reflections are drawn because the vertical rotation axes create the SMB patterns at different incident-beam orientations. [This had already been experienced for the case of symmetry 2R (Goodman, 1975; Buxton et al., 1976).] In the rectangular four-beam case, the symmetries for four settings which excite the +G, +H,
G and
H reflections are shown. For the diffraction groups 3m, 3mR,

3m1R and 6RmmR, two different patterns are shown for the two crystal settings, which differ by /6 rad from each other about the zone axis. Similarly, for the diffraction group 4RmmR, two different patterns are shown for the two crystal settings, which differ by /4 rad. Illustrations of these different symmetries are given in Fig. 2.5.3.7. The combination of the vertical threefold axis and a horizontal mirror plane introduces a new CBED symmetry 3R. Similarly, the combination of the vertical sixfold rotation axis and an inversion centre introduces a new CBED symmetry 6R. There is an empirical and conventional technique for reproducing the symmetries of the SMB patterns which uses three operations of two-dimensional rotations, a vertical mirror at the centre of disc O and a rotation of  about the centre of a disc (1R) without involving the reciprocal process. For example, we may consider 3R between discs F and F 0 in Table 2.5.3.5 in the case of diffraction group 31R. Disc F 0 is rotated anticlockwise not about the zone axis but about the centre of disc O by 2/3 rad (symbol 3) to coincide with disc F, and followed by a rotation of  rad (symbol R) about the centre of disc F 0 , resulting in the correct symmetry seen in Fig. 2.5.3.7. When the symmetries appearing between different SMB patterns are considered, this technique assumes that the symmetry operations are conducted after discs  are superposed. Another assumption is that the vertical O and O  mirror plane perpendicular to the line connecting discs O and O acts at the centre of disc O when the symmetries between two SMB patterns are considered. As an example, symmetry 3R between discs S and S appearing in the two SMB patterns is reproduced by a threefold anticlockwise rotation of disc S about  ) and followed by a rotation of  rad the centre of disc O (or O (R) about the centre of disc S .

Fig. 2.5.3.5. CBED patterns of Si taken with the [111] incidence. (a) BP and WP show symmetry 3mv. (b) and (c) DPs show symmetry m2 and DP symmetry 2Rmv0 .

Fig. 2.5.3.6. CBED pattern of Si taken with the [100] incidence. The BP and WP show symmetry 4mm.

314

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.3.7. Illustration of symmetries appearing in hexagonal six-beam, square four-beam and rectangular four-beam dark-field patterns expected for all the diffraction groups except for 1, 1R, 2, 2R and 21R.

315

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.7 (cont.).

Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7 express the symmetries illustrated in Fig. 2.5.3.7 with the symmetry symbols for the hexagonal six-beam case, square four-beam case and rectangular fourbeam case, respectively. In the fourth rows of the tables the symmetries of zone-axis patterns (BP and WP) are listed because combined use of the zone-axis pattern and the SMB pattern is efficient for symmetry determination. In the fifth row, the symmetries of the SMB pattern are listed. In the following rows, the symmetries appearing between the two SMB patterns are listed because the SMB symmetries appear not only in an SMB pattern but also in the pairs of SMB patterns. That is, for each diffraction group, all the possible SMB symmetries appearing in a pair of symmetric six-beam patterns, two pairs AB and AC of the square four-beam patterns and three pairs AB, AC and AD of the rectangular four-beam patterns are listed, though such pairs are not always needed for the determination of the diffraction groups. It is noted that the symmetries in parentheses are the symmetries which add no new symmetries, even if they are present. In the last row, the point groups which cause the diffraction groups listed in the first row are given. By referring to Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7, the characteristic features of the SMB method are seen to be as follows. CBED symmetry m2 due to a horizontal twofold rotation axis can appear in every disc of an SMB pattern. Symmetry 1R due to a horizontal mirror plane, however, appears only in disc G or H of an SMB pattern. In the hexagonal six-beam case, an inversion centre i produces CBED symmetry 6R between discs S and S0 due to the combination of an inversion centre and a vertical threefold rotation axis (and/or of a horizontal mirror plane and a vertical sixfold rotation axis). This indicates that one hexagonal six-beam

pattern can reveal whether a specimen has an inversion centre or not, while the method of Buxton et al. (1976) requires two photographs for the inversion test. All the diffraction groups in Table 2.5.3.5 can be identified from one six-beam pattern except groups 3 and 6. Diffraction groups 3 and 6 cannot be distinguished from the hexagonal six-beam pattern because it is insensitive to the vertical axis. In the square four-beam case, fourfold rotary inversion 4 produces CBED symmetry 4R between discs F and F 0 in one SMB pattern, while Buxton et al.’s method requires four photographs to identify fourfold rotary inversion. Although an inversion centre itself does not exhibit any symmetry in the square four-beam pattern, it causes symmetry 1R due to the horizontal mirror plane produced by the combination of an inversion centre and the twofold rotation axis. Thus, symmetry 1R is an indication of the existence of an inversion centre in the square four-beam case. All of the seven diffraction groups in Table 2.5.3.6 can be identified from one square four-beam pattern. One rectangular four-beam pattern can distinguish all the diffraction groups in Table 2.5.3.7 except the groups m and 2mm. It is emphasized again that the inversion test can be carried out using one six-beam pattern or one square four-beam pattern. Fig. 2.5.3.8 shows CBED patterns taken from a [111] pyrite (FeS2) plate with an accelerating voltage of 100 kV. The space group of FeS2 is P21 =a3. The diffraction group of the plate is 6R due to a threefold rotation axis and an inversion centre. The zone-axis pattern of Fig. 2.5.3.8(a) shows threefold rotation symmetry in the BP and WP. The hexagonal six-beam pattern of Fig. 2.5.3.8(b) shows no symmetry higher than 1 in discs O, G, F and S but shows symmetry 6R between discs S and S0 , which

316

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.3.7 (cont.).

proves the existence of a threefold rotation axis and an inversion centre. The same symmetries are also seen in Fig. 2.5.3.8(c), , G  , F , S , F 0 and S 0 are excited. Table 2.5.3.5 where reflections O indicates that diffraction group 6R can be identified from only one hexagonal six-beam pattern, because no other diffraction groups give rise to the same symmetries in the six discs. When Buxton et al.’s method is used, three photographs or four patterns are necessary to identify diffraction group 6R (see Table 2.5.3.3). In addition, if the symmetries between Figs. 2.5.3.8(b) and (c) are  and symmetry 6R examined, symmetry 2R between discs G and G between discs F and F are found. All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.5. Fig. 2.5.3.9 shows CBED patterns taken from a [110] V3Si plate with an accelerating voltage of 80 kV. The space group of V3Si is Pm3n. The diffraction group of the plate is 2mm1R due to two vertical mirror planes and a horizontal mirror plane, a twofold rotation axis being produced at the intersection line of two perpendicular mirror planes. The zone-axis pattern of Fig. 2.5.3.9(a) shows symmetry 2mm in the BP and WP. The rectan-

gular four-beam pattern of Fig. 2.5.3.9(b) shows symmetry 1R in disc H due to the horizontal mirror plane and symmetry m2 in both discs S and F 0 due to the twofold rotation axes in the [001] and [110] directions, respectively. The same symmetries are also  , S0 and F are excited. seen in Fig. 2.5.3.9(c), where reflections H Table 2.5.3.7 implies that the diffraction group 2mm1R can be identified from only one rectangular four-beam pattern, because no other diffraction groups give rise to the same symmetries in the four discs. When Buxton et al.’s method is used, two photographs or three patterns are necessary to identify diffraction group 2mm1R (see Table 2.5.3.3). One can confirm the theoretically predicted symmetries between Fig. 2.5.3.9(b) and Fig. 2.5.3.9(c). All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.7. These experiments show that the SMB method is quite effective for determining the diffraction group of slabs. Buxton et al.’s method identifies two-dimensional symmetry elements in the first place using a zone-axis pattern, and three-dimensional symmetry elements using DPs. On the other hand, the SMB method primarily finds many three-dimensional symmetry elements in an

317

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.5. Symmetries of hexagonal six-beam CBED patterns for diffraction groups Projection diffraction group 31R

3m1R

61R

Diffraction group

3

31R

3mR

3m

3m1R

6

6R

61R

Two-dimensional symmetry

3

3

3

3m

3m

6

3

6

i

m0 , (i)

0

Three-dimensional symmetry Zone-axis pattern

Hexagonal six-beam pattern

A pair of symmetrical six-beam patterns

m

2

0

0

0

m , (2 )

Bright-field pattern

3

6

3m

3m

6mm

6

3

6

Whole-field pattern

3

3

3

3m

3m

6

3

6

O

1

1

1

m2

1

mv

m2

mv

1

1

1

G

1

1R

m2

1

1

mv

1R

1Rmv(m2)

1

1

1R

F S

1 1

1 1

m2 1

1 m2

1 1

1 1

1 m2

m2 1

1 1

1 1

1 1

FF 0

1

3R

1

1

1

mv

3R

3Rmv

1

1

3R

SS0

1

1

1

1

1

mv

1

mv

1

6R

6R

O

1

1R

m2

1

mv

1

mv1R

1Rm2

2

1

2(1R) 21R

G

1

1

1

mR

mv

1

mvmR

1

2

2R

F

1

1

1

1

mv

1

mv

1

1

6R

6R

S F 0 F S0 S

1

3R

1

1

mv

1

3Rmv

3R

1

1

3R

1

1

1

mR

1

1

mR

1

2

1

2

1

1 6

mR

1

1 1 43m, 3m

1 6 m2

mR

2

1

2

6

m3, 3

6/m

Point group

23, 3

432, 32

Projection diffraction group 6mm1R Diffraction group

6mRmR

6mm

Two-dimensional symmetry

6

6mm

Three-dimensional symmetry

20

Zone-axis pattern

Hexagonal six-beam pattern

A pair of symmetrical six-beam patterns

Point group

6RmmR

6mm1R

3m

6mm

i, (20 )

m0 , (i, 20 )

Bright-field pattern

6mm

6mm

3m

6mm

Whole-field pattern

6

6mm

3m

6mm

O

m2

mv

1

mv(m2)

mv(m2)

G

m2

mv

m2

mv

1Rmv(m2)

F

m2

1

m2

1

m2

S

m2

1

1

m2

m2

FF 0

1

mv

1

mv

3Rmv

SS0

1

mv

6R

6Rmv

6Rmv

O G

2m2 2mR

2mv0 2mv0

mv(m2) 2Rmv

1 2RmR

2(1R)mv0 (m2) 21Rmv0 (mR)

F

1

mv0

6Rmv

6R

6Rmv0

S F 0 F S0 S

1

mv0

mv

1

3Rmv0

2mR

2

1

mR

2mR

2mR

2

mR

1

2mR

622

6mm

m3m, 3 m

SMB pattern, and two-dimensional symmetry elements from a pair of SMB patterns, as shown in Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7. Therefore, the use of a ZAP and SMB patterns is the most efficient way to find as many crystal symmetry elements in a specimen as possible.

6/mmm

reflections forbidden by the lattice type are always absent, even if dynamical diffraction takes place. (This is true for all sample thicknesses and accelerating voltages.) By comparing the experimentally obtained absences and the extinction rules known for the lattice types [P, C (A, B), I, F and R], a lattice type may be identified for the crystal concerned.

2.5.3.3. Space-group determination 2.5.3.3.2. Identification of screw axes and glide planes There are three space-group symmetry elements of diperiodic plane figures: (1) a horizontal screw axis 201, (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g0. These are related to the point-group symmetry elements 20 , m and m0 of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten point-group symmetry elements form 80 space groups.)

2.5.3.3.1. Lattice-type determination When the point group of a specimen crystal is determined, the crystal axes may be found from a spot diffraction pattern recorded at a high-symmetry zone axis, using the orientations of the symmetry elements determined in the course of point-group determination. Integral-number indices are assigned to the spots of the diffraction patterns. The systematic absence of reflections indicates the lattice type of the crystal. It should be noted that

318

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.6. Symmetries of square four-beam CBED patterns for diffraction groups Projection diffraction group 41R

4mm1R

Diffraction group

4

4R

41R

4mRmR

4mm

4RmmR

4mm1R

Two-dimensional symmetry

4

(2) 4

4

4

4mm

4mm

m0 , (i, 4 )

20

(2mm) 4 , 20

Three-dimensional symmetry Zone-axis pattern

Bright-field pattern

4

4

4

4mm

4mm

4mm

4mm

Whole-field pattern

4

2

4

4

4mm

2mm

4mm

O

1

1

1

m2

mv

m2

mv

mv(m2)

G

1

1

1R

m2

mv

m2

mv

1Rmv(m2)

F

1

1

1

m2

1

1

m2

m2

FF 0

1

4R

4R

1

mv

4R

4Rmv

4Rmv

O

2

2

2(1R)

2m2

2mv0

2m2

2mv0

2(1R)mv0 (m2)

G

2

2

21R

2mR

2mv0

2mR

2mv0

21Rmv0 (mR)

FF 0

2

2

2

2mR

2

2

2mR

2mR

F

1

4R

4R

1

mv0

4R

4Rmv0

4Rmv0

OO0

4

4

4

4m2

4mv0 0

4mv

4m2

4mv0 0 (m2)

GG0

4

4R

41R

4mR

4mv0 0

4Rmv

4RmR

41Rmv0 0 (mR)

FS FS0

4 1

1 1

4 1R

4mR 1

4 mv0 0

mR mv

1 1

41Rmv0 0 (mR) 1Rmv0 0

4

4

4/m

432, 422

4mm

4 3m, 4 2m

Square four-beam pattern

Two pairs of square four-beam patterns

m0 , (i, 20 , 4 )

AB

AC

Point group

m3m, 4/mmm

Table 2.5.3.7. Symmetries of rectangular four-beam CBED patterns for diffraction groups Projection diffraction group m1R Diffraction group

mR

Two-dimensional symmetry

2mRmR

2mm

2RmmR

m

2

2mm

m

2mm1R 2mm

m0 , 20

20

20 , i

m0 , 20 , i

m

m

2mm

2mm

2mm

m

2mm

Whole-field pattern

1

m

m

2

2mm

m

2mm

O

1

1

1

1

1

1

1

G F

1 m2

1 1

1R m2

1 m2

1 1

1 m2

1R m2

S

1

1

1

m2

1

1

m2

OG OH  GH

m2 1

1 1

m2 1

m2 mR

mv mv

mv(m2) mv

mv(m2) mvmR

F F SS0

1 1

1 1

1 1R

1 1

mv mv

2Rmv mv

2Rmv mv1R

OGOH GH

1 mR

mv mv

mv mvmR

m2 mR

mv0 mv0

1 mR

mv0 (m2) mv0 mR

FF 0 SS

1

mv

mv1R

1

mv0

1

mv0 1R

1

mv

mv

1

mv0

2R

2Rmv0

OG OG

1

1

1R

2

2

1

2(1R)

GG F F 0 SS 0

1

1

1

2

2

2R

21R

AB

AC

AD

Point group

m1R

m Bright-field pattern

Rectangular four-beam pattern

Three pairs of rectangular four-beam patterns

m

20

Three-dimensional symmetry Zone-axis pattern

2mm1R

1

1

1

2mR

2

1

2mR

mR

1

mR

2mR

2

mR

2mR

2, 222, mm2, 4, 4 , 422, 4mm, 4 2m, 32, 6, 622, 6mm, 6 m2, 23, 432, 4 3m

m, mm2, 4mm, 4 2m, 3m, 6 , 6mm, 6 m2, 4 3m

mm2, 4mm, 42m, 6mm, 6 m2, 4 3m

222, 422, 4 2m, 622, 23, 432

mm2, 6 m2

2/m, mmm, 4/m, 4/mmm, 3 m, 6 =m, 6/mmm, m3, m3m

mmm, 4/mmm, m3, m3m, 6/mmm

The ordinary extinction rules for screw axes and glide planes hold only in the approximation of kinematical diffraction. The kinematically forbidden reflections caused by these symmetry elements appear owing to Umweganregung of dynamical diffraction. Extinction of intensity, however, does take place in these reflections at certain crystal orientations with respect to the

incident beam (i.e. in certain regions within a CBED disc). This dynamical extinction was first predicted by Cowley & Moodie (1959) and was discussed by Miyake et al. (1960) and Cowley et al. (1961). Goodman & Lehmpfuhl (1964) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie

319

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.8. CBED patterns of FeS2 taken with the [111] incidence. (a) Zone-axis pattern, (b) hexagonal six-beam pattern with excitation of reflection +G, (c) hexagonal six-beam pattern with excitation of reflection
G. Symmetry 6R is noted between discs S and S0 and discs F and F 0 .

(1965) discussed the dynamical extinction in a more general way considering not only ZOLZ reflections but also HOLZ reflections. They completely clarified the dynamical extinction rules by considering the exact cancellation which may occur along certain symmetry-related multiple-scattering paths. Based on the results of Gjønnes & Moodie (1965), Tanaka, Sekii & Nagasawa (1983) tabulated the dynamical extinctions expected at all the possible crystal orientations for all the space groups. These were later tabulated in a better form on pages 162 to 172 of the book by Tanaka & Terauchi (1985). Fig. 2.5.3.10(a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a b-glide plane exists perpendicular to the a axis and/or a 21 screw axis exists in the b direction. Let us consider an Umweganregung path a in the zeroth-order Laue zone to the 010 forbidden reflection and path b which is symmetric to path a with respect to axis k. Owing to the translation of one half of the lattice parameter b caused by the glide plane and/or the 21 screw axis, the following relations hold between the crystal structure factors:

Fðh; kÞ ¼ Fðh ; kÞ for k ¼ 2n; Fðh; kÞ ¼
Fðh ; kÞ for k ¼ 2n þ 1:

Since an Umweganregung path to the kinematically forbidden reflection 0k0 contains an odd number of reflections with odd k, the following relations hold: Fðh1 ; k1 ÞFðh2 ; k2 Þ . . . Fðhn ; kn Þ for path a ¼
Fðh 1 ; k1 ÞFðh 2 ; k2 Þ . . . Fðh n ; kn Þ for path b; ð2:5:3:2Þ where n P i¼1

hi ¼ 0;

n P

ki ¼ k ðk ¼ oddÞ

i¼1

and the functions including the excitation errors are omitted because only the cases in which the functions are the same for all the paths are considered. The excitation errors for paths a and b become the same when the projection of the Laue point along the zone axis concerned, L, lies on axis k. Since the two waves passing through paths a and b have the same amplitude but opposite signs, these waves are superposed on the 0k0 discs (k = odd) and cancel out, resulting in dark lines A in the forbidden discs, as shown in Fig. 2.5.3.10(b). The line A runs parallel to axis k passing through the projection point of the zone axis. In path c, the reflections are arranged in the reverse order to those in path b. When the 010 reflection is exactly excited, two paths a and c are symmetric with respect to the bisector m0 –m0 of

ð2:5:3:1Þ

That is, the structure factors of reflections hk0 and h k0 have the same phase for even k but have opposite phases for odd k.

320

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.3.9. CBED patterns of V3Si taken with the [110] incidence. (a) Zone-axis pattern, (b) rectangular four-beam pattern with excitation of reflections  , S and F . H, S and F, (c) rectangular four-beam pattern with excitation of reflections H

the 010 vector having the same excitation errors. The following equation holds:

FðhklÞ ¼ ð
1Þk Fðh klÞ for a 21 screw axis in the [010] direction, ð2:5:3:4Þ k FðhklÞ ¼ ð
1Þ Fðh klÞ for a b glide in the (100) plane.

Fðh1 ; k1 ÞFðh2 ; k2 Þ . . . Fðhn ; kn Þ for path a ¼
Fðh n ; kn ÞFðh n
1 ; kn
1 Þ . . . Fðh 1 ; k1 Þ for path c:

ð2:5:3:5Þ ð2:5:3:3Þ In the case of the glide plane, extinction lines A are still formed because two waves passing through paths a and b have opposite signs to each other according to equation (2.5.3.5), but extinction lines B are not produced because equation (2.5.3.4) holds only for the 21 screw axis. In the case of the 21 screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4), forming extinction lines B only. We call these lines A3 and B3 dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higher-order Laue zones. It was predicted by Gjønnes & Moodie (1965) that a horizontal glide plane g0 gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965). Table 2.5.3.8 summarizes the appearance of the dynamical extinction lines for the glide planes g and g0 and the 21 screw axis. The three space-group symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.

Since the waves passing through these paths have the same amplitude but opposite signs, these waves are superposed on the 010 discs and cancel out, resulting in dark line B in this disc, as shown in Fig. 2.5.3.10(b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zeroth-order Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A2 and B2 lines, subscript 2 indicating that the Umweganregung paths lie in the zeroth-order Laue zone. The dynamical extinction effect is analogous to interference phenomena in the Michelson interferometer. That is, the incident beam is split into two beams by Bragg reflections in a crystal. These beams take different paths, in which they suffer a relative phase shift of  and are finally superposed on a kinematically forbidden reflection to cancel out. When the paths include higher-order Laue zones, the glide plane produces only extinction lines A but the screw axis causes only extinction lines B. These facts are attributed to the different relations between structure factors for a 21 screw axis and a glide plane,

321

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.12. CBED pattern of FeS2 taken with the [110] electron-beam incidence. In the 001 and 001 discs, HOLZ lines are asymmetric with respect to extinction lines A2, indicating the existence of a 21 screw axis parallel to the c axis. In the 1 10 and 11 0 discs, HOLZ lines are symmetric with respect to extinction lines A2, indicating existence of a glide plane perpendicular to the c axis.

be noted that a relatively thick specimen area has to be selected to observe HOLZ lines in ZOLZ reflection discs. Fig. 2.5.3.11 shows CBED patterns taken from (a) thin and (b) thick areas of FeS2, whose space group is P21 =a3, at the 001 Bragg setting with the [100] electron-beam incidence. In the case of the thin specimen (Fig. 2.5.3.11a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the odd-order discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11b). The HOLZ lines are symmetric with respect to both A2 and B2 extinction lines. This fact proves the presence of the extinction lines A3 and B3, or both the c glide in the (010) plane and the 21 screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9. Fig. 2.5.3.12 shows a [110] zone-axis CBED pattern of FeS2. A2 extinction lines are seen in both the 001 and 1 10 discs. Fine HOLZ lines are symmetric with respect to the A2 extinction lines in the 1 10 disc but asymmetric about the A2 extinction line in the 001 disc, indicating formation of the A3 extinction line only in the 1 10 disc. This proves the existence of a 21 screw axis in the [001] direction and an a glide in the (001) plane. The appearance of HOLZ lines is easily changed by a change of a few hundred volts in the accelerating voltage. Steeds & Evans (1980) demonstrated for spinel changes in the appearance of HOLZ lines in the ZOLZ discs at accelerating voltages around 100 kV. Another practical method for distinguishing between glide planes and 21 screw axes is that reported by Steeds et al. (1978). The method is based on the fact that the extinction lines are observable even when a crystal is rotated with a glide plane kept parallel and with a 21 screw axis perpendicular to the incident

Fig. 2.5.3.10. Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a b-glide plane and a 21 screw axis. (a) Umweganregung paths a, b and c. (b) Dynamical extinction lines A are formed in forbidden reflections 0k0 (k = odd). Extinction line B perpendicular to the lines A is formed in the exactly excited 010 reflection.

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A3 and B3. It is, however, not easy to observe the extinction lines A3 and B3 because broad extinction lines A2 and B2 appear at the same time. The presence of the extinction lines A3 and B3 can be revealed by inspecting the symmetries of fine defect HOLZ lines appearing in the forbidden reflections instead of by direct observation of the lines A3 and B3 (Tanaka, Sekii & Nagasawa, 1983). That is, if HOLZ lines form lines A3 and B3, HOLZ lines are symmetric with respect to the extinction lines A2 and B2. If HOLZ lines do not form lines A3 and B3, HOLZ lines are asymmetric with respect to the extinction lines A2 and B2. When the HOLZ lines are symmetric about the extinction lines A2, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B2, a 21 screw axis exists. It should

Fig. 2.5.3.11. CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS2. (a) Dynamical extinction lines A2 and B2 are seen. (b) Extinction lines A3 and B3 as well as A2 and B2 are formed because HOLZ lines are symmetric about lines A2 and B2.

322

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.8. Dynamical extinction rules for an infinitely extended parallel-sided specimen Dynamical extinction lines Symmetry elements of plane-parallel specimen

Orientation to specimen surface

Two-dimensional (zeroth Laue zone) interaction

Three-dimensional (HOLZ) interaction

Glide planes

perpendicular: g

A2 and B2

A3

parallel: g0

—

intersection of A3 and B3

perpendicular: 21

—

—

parallel: 201

A2 and B2

B3

Twofold screw axis

beam. With reference to Fig. 2.5.3.10(a), extinction lines A3 produced by a glide plane remain even when the crystal is rotated with respect to axis h but the lines are destroyed by a rotation of the crystal about axis k. Extinction lines B3 originating from a 21 screw axis are not destroyed by a crystal rotation about axis k but the lines are destroyed by a rotation with respect to axis h.

suffix only), because the equivalent planes do not exist. The glide symbol in the [001] column for space group P4/mbm has only one suffix 1 or 2. The suffix distinguishes the equivalent glide planes lying in the x and y planes. The first suffix to distinguish the first and the second glide planes is not necessary because the space group has only one glide symbol b. When the index of the incident-beam direction is expressed with a symbol like [h0l] for point groups 2, m and 2/m, the index h or l can take a value of zero. That is, the extinction rules are applicable to the [100] and [001] electron-beam incidences. However, if columns for [100], [010] and [001] incidences are present, as in the case of point group mm2, [hk0], [0kl] and [h0l] incidences are only for nonzero h, k and l. The reflections in which the extinction lines appear are always perpendicular to the corresponding incident-beam directions ð0k0 l0 ? ½0kl; h0 k0 0 ? ½hk0; . . .Þ. The indices of the reflections in which extinction lines appear are odd if no remark is given. For c-glide planes of space groups R3c and R3 c and for d-glide planes, the reflections in which extinction lines appear are specified as 6n + 3 and 4n + 2 orders, respectively. The number of indistinguishable space groups was first counted by Tanaka, Sekii & Nagasawa (1983) but later corrections were made by Eades & Spence (1987). It was found that 177 space groups out of 230 can be identified using the extinction lines (Tanaka et al., 2002). Another reference for space-group determination is due to Eades (1988). The indistinguishable space-group sets using the extinction lines are listed in Table 2.5.3.10. Most of the sets are caused by the fact that CBED cannot identify 42, 31 (32) and 62 (64) screw axes. However, these sets can be rather easily distinguished in the ordinary way, that is, by observing how the intensities of the reflections which may be kinematically forbidden change when the crystal orientation is varied. If the axis concerned is a screw axis, kinematically forbidden reflections show a sudden decrease in intensity when an orientation change causes the loss of Umweganregung paths. If the axis is a rotation axis, the intensities of the reflections do not change conspicuously for such an orientation change. Using this test, each space group in the 23 sets can be identified except the pairs in parentheses and pairs (16) and (17) in Table 2.5.3.10 (see Eades, 1988). Tsuda et al. (2000) showed theoretically that the coherent CBED method can distinguish between space groups (I23 and I213) and between (I222 and I212121), which are indistinguishable pairs (16) and (17), respectively, in Table 2.5.3.10. The coherent CBED pattern is obtained in such a way that the convergence angle of the incident beam is set to a larger value than usual to make adjacent CBED discs overlap (Dowell & Goodman, 1973). When the focus point is displaced from the specimen, or a certain area is illuminated, sinusoidal interference fringes of the lattice spacing corresponding to the adjacent discs are formed in the overlapping regions if the probe size of the incident beam is smaller than the lattice spacing. (If the focus point of the incident beam is on the specimen, each overlapping region of the CBED discs shows uniform intensity.) Formation of the interference fringes was explained in detail first by Spence & Cowley (1978). Vine et al. (1992) showed distortion-free interference fringes from 6H-SiC and succeeded in observing the fringes with a shift of half a period due to a glide plane. Tsuda et al.’s method

2.5.3.3.3. Space-group determination We now describe a space-group determination method which uses the dynamical extinction lines caused by the horizontal screw axis 201 and the vertical glide plane g of an infinitely extended parallel-sided specimen. We do not use the extinction due to the glide plane g0 because observation of the extinction requires a laborious experiment. It should be noted that a vertical glide plane with a glide vector not parallel to the specimen surface cannot be a symmetry element of a specimen of finite thickness; however, the component of the glide vector perpendicular to the incident beam acts as a symmetry element g. (Which symmetry elements are observed by CBED is discussed in Section 2.5.3.3.5.) The 21, 41, 43, 61, 63 and 65 screw axes of crystal space groups that are set perpendicular to the incident beam act as a symmetry element 201 because two or three successive operations of 41, 43, 61, 63 and 65 screw axes make them equivalent to a 21 screw axis: (41)2 = (43)2 = (61)3 = (63)3 = (65)3 = 21. The 42, 31, 32, 62 and 64 screw axes that are set perpendicular to the incident beam do not produce dynamical extinction lines because the 42 screw axis acts as a twofold rotation axis due to the relation (42)2 = 2, the 31 and 32 screw axes give no specific symmetry in CBED patterns, and the 62 and 64 screw axes are equivalent to 31 and 32 screw axes due to the relations (62)2 = 32 and (64)2 = 31. Modifications of the dynamical extinction rules were investigated by Tanaka, Sekii & Nagasawa (1983) when more than one crystal symmetry element (that gives rise to dynamical extinction lines) coexists and when the symmetry elements are combined with various lattice types. Using these results, dynamical extinction lines A2, A3, B2 and B3 expected from all the possible crystal settings for all the space groups were tabulated. Table 2.5.3.9 shows all the dynamical extinction lines appearing in the kinematically forbidden reflections for all the possible crystal settings of all the space groups. The first column gives space groups. In each of the following pairs of columns, the lefthand column of the pair gives the reflection indices and the symmetry elements causing the extinction lines and the righthand column gives the types of the extinction lines. The (second) suffixes 1, 2 and 3 of a 21 screw axis in each column distinguish the first, the second and the third screw axis of the space group (as in the symbols 211 and 212 of space group P21212). The glide symbols in the [001] column for space group P4/nnc have two suffixes (n21 and n22). The first suffix 2 denotes the second glide plane of the space group. The second suffixes 1 and 2, which appear in the tetragonal and cubic systems, distinguish two equivalent glide planes which lie in the x and y planes. The equivalent planes are distinguished only for the cases of [100], [010] and [001] electronbeam incidences, for convenience. The c-glide planes of space group Pcc2 are distinguished with symbols c1 and c2 (the first

323

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.9. Dynamical extinction lines appearing in ZOLZ reflections for all crystal space groups except Nos. 1 and 2 Point groups 2, m, 2/m (second setting, unique axis b) Incident-beam direction Space group 3

P2

4

P21

5

C2

6

Pm

7

Pc

8

Cm

9

Cc

10

P2/m

11

P21/m

12

C2/m

13 14

15

[h0l] 0k0 21

A2

B2 B3

h0lo c

A2 A3

B2

he0lo c

A2 A3

B2

0k0 21

A2

B2 B3

P2/c

h0lo c

A2 A3

B2

P21/c

0k0 21

A2

B2 B3

h0lo

A2

B2

c

A3

he0lo c

A2 A3

C2/c

B2

Point group 222 Incident-beam direction Space group

[100]

[010]

[001]

[hk0]

16

P222

17

P2221

00l 21

A2

B2 B3

00l 21

A2

B2 B3

18

P21212

0k0 212

A2

B2 B3

h00 211

A2

B2 B3

h00 211 0k0 212

A2

B2 B3

19

P212121

0k0 212 00l 213

A2

B2 B3

h00 211 00l 213

A2

B2 B3

h00 211 0k0 212

A2

B2 B3

20

C2221

00l 21

A2

B2 B3

00l 21

A2

B2 B3

21

C222

22

F222

23

I222

24

I212121

00l 21

[0kl] A2

[h0l]

B2 B3

00l 213

A2

B2 B3

00l 21

A2

B2 B3

h00 211

A2

B2 B3

0k0 212

A2

B2 B3

h00 211

A2

B2 B3

0k0 212

A2

B2 B3

Point group mm2 Incident-beam direction Space group 25

Pmm2

26

Pmc21

27

Pcc2

28

Pma2

29

Pca21

[100]

[010]

00l c, 21

A2 A3

00l c2

A3

00l 21

B2 B3

00l 21 00l c1

B3

[001]

00l c, 21

[hk0] 00l 21

B3

[0kl] A2

B2 B3 0klo c1

A3

A2 A3

B2 B3

h00 a

A2 A3

B2

h00 a

A2 A3

B2

324

00l 21

A2

[h0l]

B2 B3

0klo c

A2 A3

A2 A3

B2

B2

h0lo c

A2 A3

B2

h0lo c2

A2 A3

B2

ho0l a

A2 A3

B2

ho0l a

A2 A3

B2

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Incident-beam direction Space group

[100]

[010]

[001]

30

00l c

0k0 n

A2 A3

B2

A3

00l n

00l n, 21

A2 A3

h00 n

A2 A3

B2

h00 a 0k0 b

A2 A3

B2

h00 a 0k0 n

A2 A3

B2

h00 n2 0k0 n1

A2 A3

B2

31

Pnc2

Pmn21

32

Pba2

33

Pna21

34

Pnn2

00l 21

B3

00l n2

A3

35

Cmm2 ba2

36

Cmc21 bn21

00l c, 21

A2 A3

Ccc2 nn2

00l c2

A3

37

B2 B3

B2 B3

A3

00l 21

B3

00l n, 21

A2 A3

00l n1

A3

00l 21 00l c1

B2 B3

[hk0]

0kl: k+l= 2n + 1 n 00l 21

00l 21

00l 21

B3

[0kl]

A2

A2

A2

[h0l] A2 A3

B2

B2 B3

B2 B3

A2 A3

B2

h0l: h+l= 2n + 1 n

A2 A3

B2

0kol b

A2 A3

B2

ho0l a

A2 A3

B2

0kl: k+l= 2n + 1 n

A2 A3

B2

ho0l a

A2 A3

B2

0kl: k+l= 2n + 1 n1

A2 A3

B2

h0l: h+l= 2n + 1 n2

A2 A3

B2

he0lo c

A2 A3

B2

he0lo c2

A2 A3

B2

ho0le a

A2 A3

B2

B2 B3

A3

h0lo c

0kelo c1

A2 A3

B2

0kolo b

A2 A3

B2

38

Amm2 nc21

39

Abm2 cc21

40

Ama2 nn21

h00 a

A2 A3

B2

41

Aba2 cn21

h00 a

A2 A3

B2

0kolo b

A2 A3

B2

ho0le a

A2 A3

B2

42

Fmm2

43

Fdd2 dd21

h00: h = 4n + 2 d2 0k0: k = 4n + 2 d1

A2 A3

B2

0kele: ke + le = 4n + 2 d1

A2 A3

B2

he0le: he + le = 4n + 2 d2

A2 A3

B2

0kolo b

A2 A3

B2

ho0lo a

A2 A3

B2

ho0lo a

A2 A3

B2

44

Imm2 nn21

45

Iba2 cc21

46

Ima2 nc21

00l: l = 4n + 2 d2

A3

00l: l = 4n + 2 d1

A3

Point group mmm Incident-beam direction Space group 47

P2/m2/m2/m

48

P2/n2/n2/n

49 50

51

P2/c2/c2/m P2/b2/a2/n

P21/m2/m2/a

[100] 00l n2 0k0 n3

[010]

A3

00l n1 h00 n3

[001] 0k0 n1 h00 n2

A3

00l c2

A3

00l c1

A3

0k0 n

A3

h00 n

A3

h00 21, a

A2 A3

0k0 b h00 a B2 B3

[hk0] hk0: h+k= 2n + 1 n3

A3

A3

h00 21

B3

325

[0kl] A2 A3

B2

[h0l]

0kl: k+l= 2n + 1 n1

A2 A3

B2

h0l: h+l= 2n + 1 n2

A2 A3

B2

0klo c1

A2 A3

B2

h0lo c2

A2 A3

B2

ho0l a

A2 A3

B2

hk0: h+k= 2n + 1 n

A2 A3

B2

0kol b

A2 A3

B2

hok0 a

A2 A3

B2

h00 21

A2

B2 B3

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Incident-beam direction Space group

[100]

[010]

[001]

52

00l n2

00l n1 h00 a

0k0 n1, 21

A2 A3

h00 n2

A3

h00 n

A3

P2/n21/n2/a

A3

0k0 21 53

P2/m2/n21/a

00l n, 21

A3

B3 A2 A3

B2 B3

h00 a

A3

00l 21 54

55

P21/c2/c2/a

P21/b21/a2/m

00l c2

A3

0k0 212

B3

[hk0] B2 B3

B3

00l c1

A3

h00 a, 21

A2 A3

h00 211

h00 21

B3

hok0 a

[0kl] A2 A3

B2

hok0 a

A2 A3

B2

00l 21

A2

B2 B3

hok0 a

A2 A3

B2

B2 B3 B3

0k0 b, 212

A2 A3

B2 B3

h00 a, 211 56

57

P21/c21/c2/n

P2/b21/c21/m

00l c2

A3

0k0 212, n

A2 A3

00l c, 212

A2 A3

0k0 211 58

P21/n21/n2/m

00l n2

60

P21/m21/m2/n

P21/b2/c21/n

A3

B2 B3

h00 211, n

A2 A3

B2 B3

00l 212

00l n1

A3

B3

h00 211

P21/b21/c21/a

P21/n21/m21/a

B3

0k0 b, 211

0k0 n1, 212 h00 n2, 211

A3

A2 A3

B2 B3

h00 n, 211

A2 A3

B2 B3

0k0 212 h00 211

00l c, 212

A2 A3

B2 B3

h00 n, 211

A2 A3

B2 B3

0k0 b

B3

211

B3

0k0 b, 212

00l

n

A3

00l c, 213

A2 A3

00l 213 0k0 212

64

C2/m2/c21/m C2/m2/c21/a

65

C2/m2/m2/m

66

C2/c2/c2/m

67

B2 B3

A2 A3

B2

00l 212

A2

B2 B3

B2 B3

B3

A3

h00

212 B2 B3

00l 213

B3

h00 a, 211

A2 A3

B2 B3

h00 211

00l n, 213 h00 a, 211

A2 A3

B2 B3

0k0 n, 212

B3

A2 A3

A2 A3

A2 A3

h00 211 63

hk0: h+k= 2n + 1 n

B3

A2 A3

0k0 212 62

B3

B2 B3

0k0 n, 212

0k0 61

0k0 212 h00 211

[h0l] A2 A3

B2

h0l: h+l= 2n + 1 n2

A2 A3

B2

0k0 21

A2

B2 B3

h0l: h+l= 2n + 1 n

A2 A3

B2

h0lo c2

A2 A3

B2

0klo c1

A2 A3

B2

h00 21

A2

B2 B3

0kol b

A2 A3

B2

ho0l a

A2 A3

B2

h00 211

A2

B2 B3

0k0 212

A2

B2 B3

0klo c1

A2 A3

B2

h0lo c2

A2 A3

B2

h00 211

A2

B2 B3

0k0 212

A2

B2 B3

0kol b

A2 A3

B2

h0lo c

A2 A3

B2

0k0 211

A2

B2 B3

B3

0k0 212 59

00l c1

0kl: k+l= 2n + 1 n1

0kl: k+l= 2n + 1 n1

A2 A3

B2

h0l: h+l= 2n + 1 n2

A2 A3

B2

h00 211

A2

B2 B3

0k0 212

A2

B2 B3

hk0: h+k= 2n + 1 n

A2 A3

B2

h00 211

A2

B2 B3

0k0 212

A2

B2 A3

hk0: h+k= 2n + 1 n

A2 A3

B2

0kol b

A2 A3

B2

h0lo c

A2 A3

B2

00l

A2

B2

h00

A2

B2

B3

211

B3

212

B2 B3

hok0 a

A2 A3

B2

0kol b

A2 A3

B3 B2

h0lo c

A2 A3

B2

00l 213

A2

B2 B3

h00 211

A2

B2 B3

0k0 212

A2

B3

B2 B3

B2 B3

hok0 a

A2 A3

B2

0kl: k+l= 2n + 1 n

A2 A3

B2

0k0 212

A2

B2 B3

00l 213

A2

B2 B3

h00 211

A2

B3

B2 B3

00l c, 21

A2 A3

B2 B3

00l 21

00l 21

A2

B2 B3

he0lo c

A2 A3

B2

B3

00l c, 21

A2 A3

B2 B3

00l 21

hoko0 a

A2 A3

B2

he0lo c

A2 A3

B2

B3

00l 21

A2

B2 B3 he0lo c2

A2 A3

B2

00l c2

A3

00l c1

0kelo c1

A3

C2/m2/m2/a

hoko0 a

326

A2 A3

B2

A2 A3

B2

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Incident-beam direction Space group

[100]

[010]

68

00l c2

00l c1

C2/c2/c2/a

69

F2/m2/m2/m

70

F2/d2/d2/d

00l: l= 4n + 2 d2 0k0: k= 4n + 2 d3

A3

A3

h00: h= 4n + 2 d3 00l: l= 4n + 2 d1

[001]

[hk0]

A3 0k0: k= 4n + 2 d1 h00: h= 4n + 2 d2

A3

A3

[0kl] A2 A3

B2

0kelo c1

A2 A3

B2

he0lo c2

A2 A3

B2

heke0: he + ke = 4n + 2 d3

A2 A3

B2

0kele: ke + le = 4n + 2 d1

A2 A3

B2

he0le: he + le = 4n + 2 d2

A2 A3

B2

0kolo b

A2 A3

B2

ho0lo a

A2 A3

B2

0kolo b

A2 A3

B2

ho0lo c

A2 A3

B2

71

I2/m2/m2/m

72

I2/b2/a2/m

73

I21/b21/c21/a

hoko0 a

A2 A3

B2

74

I21/m21/m21/a

hoko0 a

A2 A3

B2

Point groups 4; 4 ; 4=m Incident-beam direction Space group 75

P4

76

P41

77

P42

78

P43

79

I4

80

I41

81 82

P4 I 4

83

P4/m

[hk0] 00l 41

A2

B2 B3

00l 43

A2

B2 B3

84

P42/m

85

P4/n

hk0: h + k = 2n + 1 n

A2 A3

B2

86

P42/n

hk0: h + k = 2n + 1 n

A2 A3

B2

87

I4/m

88

I41/a

hoko0 a

A2 A3

B2

Point group 422 Incident-beam direction Space group

[hk0]

[0kl]

89

P422

90

P4212

91

P4122

00l 41

A2

B2 B3

92

P41212

00l 41

A2

B2 B3

93

P4222

94

P42212

95

P4322

00l 43

A2

B2 B3

96

P43212

00l 43

A2

B2 B3

97

I422

98

I4122

327

[h0l]

hoko0 a

h00 21

A2

B2 B3

h00 21

A2

B2 B3

h00 21

A2

B2 B3

h00 21

A2

B2 B3

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Point group 4mm. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column. Incident-beam direction Space group 99

P4mm

100

P4bm

101

P42cm

102

103 104

105 106

P42nm

P4cc P4nc

[001] h00 a2 0k0 b1

00l c2

A3

00l n2

A3

00l c12

A3

00l n2

A3

h00 n2 0k0 n1

h00 n2 0k0 n1

[110] A2 A3

A2 A3

A2 A3

P42bc

I4mm

108

I4cm

109

I41md

h00 a2 0k0 b1

hh0, h h0 d hh0, h h0 d

I41cd

A2 A3

[h0l]

B2

B2

B2

P42mc

107

110

[100]

B2

A2 A3

B2

A2 A3

B2

[hhl]

ho0l a

A2 A3

B2

h0lo c

A2 A3

B2

h0l: h + l = 2n + 1 n

A2 A3

B2

00l c2

h0lo c1

A2 A3

B2

hhlo c2

A2 A3

B2

A3

00l c

h0l: h + l = 2n + 1 n

A2 A3

B2

hhlo c

A2 A3

B2

A3

00l c

hhlo c

A2 A3

B2

A3

00l c

hhlo c

A2 A3

B2

A3

hhle: 2h + le = 4n + 2 d

A2 A3

B2

hhle: 2h + le = 4n + 2 d

A2 A3

B2

hhlo c

A2 A3

B2

hhlo c

A2 A3

B2

hhle: 2h + le = 4n + 2 d

A2 A3

B2

00l: l = 4n + 2 d

A3

00l: l = 4n + 2 d

A3

ho0l a

A2 A3

B2

ho0lo c

A2 A3

B2

ho0lo c

A2 A3

B2

Point group 4 2m. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column. Incident-beam direction Space group 111 P4 2m 112

[100]

[001]

[110]

P4 2c

00l c

113

P4 21 m

0k0 212

A2

B2 B3

h00 211 0k0 212

A2

B2 B3

114

P4 21 c

0k0 212

A2

B2 B3

h00 211 0k0 212

A2

B2 B3

115 116

P4 m2 P4 c2

117

P4 b2

118

P4 n2

119 120 121 122

00l c2

00l n2

00l c

[h0l]

A3

A3

A3

A3

0k0 21

A2

B2 B3

0k0 21

A2

B2 B3

h0lo c

A2 A3

B2

h00 a2 0k0 b1

A2 A3

B2

ho0l a

A2 A3

B2

h00 n2 0k0 n1

A2 A3

B2

h0l: h + l = 2n + 1 n

A2 A3

B2

ho0lo c

A2 A3

B2

I 4 m2 I 4 c2 I 4 2m I 4 2d

[hhl]

hh0, h h0 d

A2 A3

B2

00l: l = 4n + 2 d

A3

328

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Point group 4/mmm. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column. Incident-beam direction Space group

[100]

[001]

123

P4/mmm P4/m2/m2/m

124

P4/mcc P4/m2/c2/c

00l c12

A3

P4/nbm P4/n2/b2/m

0k0 n

A3

P4/nnc P4/n2/n2/c

0k0 n1 00l n22

P4/mbm P4/m21/b2/m

0k0 212

125

126

127

[110]

00l c2 h00 a2 0k0 b1 h00 n22 0k0 n21

A3

B3

h00 a2, 211

[h0l]

A3

A3

00l c

A3

A2 A3

A3

B2 B3

0k0 b1, 212 128

P4/mnc P4/m21/n2/c

00l n2

0k0 212 129

130

131 132 133

134

135

h00 n2, 211 0k0 n1, 212

A3

A2 A3

B2 B3

00l c

A3

B3

P4/nmm P4/n21/m2/m

0k0 n, 212

A2 A3

B2 B3

h00 211 0k0 212

P4/ncc P4/n21/c2/c

0k0 n, 212

A2 A3

B2 B3

h00 211 0k0 212

00l c12

A3

B3

B3

P42/mmc P42/m2/m2/c

00l c2

00l c

P42/mcm P42/m2/c2/m

00l c2

A3

P42/nbc P42/n2/b2/c

0k0 n

A3

P42/nnm P42/n2/n2/m

0k0 n1 00l n22

P42/mbc P42/m21/b2/c

0k0 212

h00 a2 0k0 b1 h00 n22 0k0 n21

A3

B3

h00 a2, 211

00l c

A3

A3

B2 B3

00l c

A3

A3

0k0 b1, 212 136

P42/mnm P42/m21/n2/m

00l n2

0k0 212 137

138

139

h00 n2, 211 0k0 n1, 212

A3

A2 A3

B2 B3

B3

P42/nmc P42/n21/m2/c

0k0 n, 212

A2 A3

B2 B3

h00 211 0k0 212

P42/ncm P42/n21/c2/m

0k0 n, 212

A2 A3

B2 B3

h00 211 0k0 212

00l c2

A3

B3

00l c

h0lo c1

A2 A3

B2

ho0l a

A2 A3

B2

h0l: h+l= 2n + 1 n2

A2 A3

B2

ho0l a

A2 A3

B2

0k0 21

A2

B2 B3

h0l: h+l= 2n + 1 n

A2 A3

B2

0k0 21

A2

B2 B3

0k0 21

A2

B2 B3

h0lo c1

A2 A3

B2

0k0 21

A2

B2 B3

A3

A3

A2 A3

[hhl]

A3

B3

I4/mmm I4/m2/m2/m

329

h0lo c

A2 A3

B2

ho0l a

A2 A3

B2

h0l: h+l= 2n + 1 n2

A2 A3

B2

ho0l a

A2 A3

B2

0k0 21

A2

B2 B3

h0l: h+l= 2n + 1 n

A2 A3

B2

0k0 21

A2

B2 B3

0k0 21

A2

B2 B3

h0lo c

A2 A3

B2

0k0 21

A2

B2 B3

hhlo c2

[hk0]

A2 A3

B2

hhlo c

A2 A3

B2

hhlo c

A2 A3

B2

hhlo c2

A2 A3

B2

hhlo c

A2 A3

B2

hhlo c

A2 A3

B2

hhlo c

A2 A3

B2

hhlo c

A2 A3

B2

hk0: h+k= 2n + 1 n

A2 A3

B2

hk0: h+k= 2n + 1 n1

A2 A3

B2

hk0: h+k= 2n + 1 n

A2 A3

B2

hk0: h+k= 2n + 1 n

A2 A3

B2

hk0: h+k= 2n + 1 n

A2 A3

B2

hk0: h+k= 2n + 1 n1

A2 A3

B2

hk0: h+k= 2n + 1 n

A2 A3

B2

hk0: h+k= 2n + 1 n

A2 A3

B2

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Incident-beam direction Space group

[100]

[001]

[110]

140

I4/mcm I4/m2/c2/m

141

I41/amd I41/a2/m2/d

hh0, h h0 d

A3

I41/acd I41/a2/c2/d

hh0, h h0 d

A3

142

[h0l]

[hhl]

ho0lo c 00l: l= 4n + 2 d h h0 a 00l: l= 4n + 2 d h h0 a

A2 A3

A3

A2 A3

B2

00l c

A2 A3

B2

00l: l = 6n + 3 c

A2 A3

B2

00l c

A2 A3

B2

00l: l = 6n + 3 c

A2 A3

B2

ho0lo c

A3

Point groups 3; 3 ; 32; 3m; 3 m

Space group

Incident-beam direction ½112 0 ½11 00

Nos. 143–155: no GM line 156

P3m1

157

P31m

158

P3c1

159

P31c

160

R3m

161

R3c

162

P3 1m P3 1c

163 164 165 166 167

00l c

00l c

A2 A3

A2 A3

B2

B2

P3 m1 P3 c1 R3 m R3 c

Point groups 6; 6 ; 6=m; 622; 6mm; 6 m2; 6=mmm

Space group

Incident-beam direction ½112 0 ½11 00

168

P6

169

P61

00l 61

A2

B2 B3

00l 61

A2

B2 B3

170

P65

00l 65

A2

B2 B3

00l 65

A2

B2 B3

171

P62

172

P64

173

P63

00l 63

A2

B2 B3

00l 63

A2

B2 B3

174

P6

175

P6/m

176

P63/m

00l 63

A2

B2 B3

00l 63

A2

B2 B3

330

[hk0]

B2 hhle: 2h + le = 4n + 2 d

A2 A3

B2

hok0 a

A2 A3

B2

hhle: 2h + le = 4n + 2 d

A2 A3

B2

hok0 a

A2 A3

B2

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Incident-beam direction ½112 0 ½11 00

Space group 177

P622

178

P6122

00l 61

A2

B2 B3

00l 61

A2

B2 B3

179

P6522

00l 65

A2

B2 B3

00l 65

A2

B2 B3

180

P6222

181

P6422

182

P6322

00l 63

A2

B2 B3

00l 63

A2

B2 B3

183

P6mm

184

P6cc

185

P63cm

186

P63mc

187

P6 m2 P6 c2

188

00l c2 00l 63 00l 63, c

190 191

P6/mmm

192

P6/mcc

193 194

A3

B3

00l 63, c

A2 A3

A2 A3

B2 B3

00l 63 00l c

P6 2m P6 2c

189

00l c1

A3

P63/mcm P63/mmc

00l c

A2 A3

00l c2

A3

00l 63 00l 63, c

A2 A3

B2 B3 B3

A2 A3

B2

B2

00l c1

A3

B3

00l 63, c

A2 A3

B2 B3

00l 63

B2 B3 B3

Point groups 23, m3 Incident-beam direction Space group 195

P23

196

F23

197

I23

198

P213

199

I213

200

Pm3 P2=m3

201

Pn3 P2=n3

202

Fm3 F2=m3

203

Fd3 F2=d3

204

Im3 I2=m3

205

Pa3 P21 =a3

[100] (cyclic)

00l 213 0k0 212

00l n2 0k0 n3

00l: l = 4n + 2 d2 0k0: k = 4n + 2 d3

00l c2, 213 0k0 212

206

Ia3 I21 =a3

[110] (cyclic)

A2

00l 21

A2

B2 B3

k h0 n

A2 A3

B2

A3

k h0: h + k = 4n + 2 d

A2 A3

B2

A3

00l 21 k h0 a k h0 a

A2

B2 B3

A2 A3

B2

A2 A3

B2

A2 A3

B2 B3

B2 B3 B3

00l 213

00l 213 h h0 a3 h h0 a3

331

A2

[hk0] (cyclic)

B2 B3

A2

B2 B3

A2 A3

B2

A2 A3

B2

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.9 (cont.) Point group 432 Incident-beam direction Space group 207

[hk0] (cyclic)

P432

208

P4232

209

F432

210

F4132

211

I432

212

P4332

00l 43

A2

B2 B3

213

P4132

00l 41

A2

B2 B3

214

I4132

Point group 4 3m Incident-beam direction Space group 215 P4 3m 217

F 4 3m I 4 3m

218

P4 3n

216

219

F 4 3c

220

I 4 3d

[100] (cyclic)

[110] (cyclic)

00l n

0kk, 0k k d

A2 A3

B2

[hhl] (cyclic)

A3

00l: l = 4n + 2 d

A3

hhlo n

A2 A3

B2

hoholo c

A2 A3

B2

hhle: 2h + le = 4n + 2 d

A2 A3

B2

Point group m3m Incident-beam direction Space group 221

Pm3m P4=m3 2=m

222

Pn3n P4=n3 2=n

223 224

[100] (cyclic)

00l n12 0k0 n13

A3

Pm3n P42 =m3 2=n Pn3m P42 =n3 2=m

00l n2 0k0 n3

225

Fm3m F4=m3 2=m

226

Fm3c F4=m3 2=c

227

Fd3m F41 =d3 2=m

00l: l = 4n + 2 d2 0k0: k = 4n + 2 d3

Fd3c F41 =d3 2=c

00l: l = 4n + 2 d2 0k0: k = 4n + 2 d3

228

[110] (cyclic)

229

Im3m I4=m3 2=m

230

Ia3d I41 =a3 2=d

0kk, 0k k d

[hk0] (cyclic)

00l n2

A3

00l n

A3

hk0: h + k = 2n + 1 n1

hk0: h + k = 2n + 1 n

A3

[hhl] (cyclic)

A2 A3

A2 A3

B2

hhlo n2

A2 A3

B2

hhlo n

A2 A3

B2

hoholo c

A2 A3

B2

B2

heke0: he + ke = 4n + 2 d

A2 A3

B2

A3

heke0: he + ke = 4n + 2 d

A2 A3

B2

hoholo c

A2 A3

B2

A3

hoko0 a

A2 A3

B2

hhle: 2h + le = 4n + 2 d

A2 A3

B2

A3

00l: l = 4n + 2 d h h0 a3

A3

332

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.10. Space-group sets indistinguishable by dynamical extinction lines (1) P3, (P31, P32)

(2) P312, (P3112, P3212)

(3) P321, (P3121, P3221)

(4) P6, (P62, P64)

(5) P622, (P6222, P6422)

(6) P63, (P61, P65)

(7) P6322, (P6122, P6522)

(8) P4, P42

(9) (P41, P43)

(10) P4/m, P42/m (13) P4212, P42212

(11) P4/n, P42/n (14) I4, I41

(12) P422, P4222 (15) I422, I4122

(16) I23, I213

(17) I222, I212121

(18) P432, P4232

(19) (P4132, P4332)

(20) I432, I4132

(21) F432, F4132

(22) (P4122, P4322)

(23) (P41212, P43212)

distinguishes the difference in the relative arrangements of twofold rotation axes and 21 screw axes along the [111] direction between the two space groups by examining the symmetry of intensity pairs appearing in the overlapping discs of a coherent [111] ZOLZ pattern. Saitoh, Tsuda et al. (2001) extended the method to distinguish the other ten indistinguishable space-group pairs. The method can distinguish between a space group which is composed of a principal rotation axis and a twofold rotation axis like P321 and a space group which is composed of a principal screw axis and a twofold rotation axis like P3121 (or P3221) by investigating the difference in the relative arrangements of the twofold rotation axis with respect to the principal axis. Table 2.5.3.11 shows the 12 space-group pairs which are distinguishable by applying the coherent CBED method. The pairs in parentheses form left- and right-handed space groups. Handedness or chirality may occur in space groups that do not possess mirror and/or inversion symmetry. The handedness of space groups is identified in such a way that the senses of two crystal axes are determined with the aid of kinematical structure-factor calculations and the sense of the third axis is determined with the aid of dynamical calculations. This method was used for quartz by Goodman & Secomb (1977) and Goodman & Johnson (1977) and for MnSi by Tanaka et al. (1985). We also mention that Taftø & Spence (1982) developed a simple but clever method without computation for determining the absolute polarity of the sphalerite structure utilizing multiplescattering effects on weak beams, which are almost independent of thickness. Because of the importance of structure in the field of semiconductor science, this method is conveniently used nowadays to determine polarity. It is worth mentioning that space groups that are indistinguishable by CBED (Table 2.5.3.10) do not appear frequently in real inorganic materials. The crystal data collected by Nowacki (1967) on 5572 different inorganic materials shows that the number of materials belonging to space groups among sets (2),

(3), (5), (7) and (11) in Table 2.5.3.10 is more than 15 but the number belonging to space groups among the other sets is less than ten. This implies that the probability of finding indistinguishable space groups is very low. 2.5.3.3.4. Dynamical extinction in HOLZ reflections Space-group determination as described in the previous sections is carried out using the extinction lines appearing in ZOLZ reflections. Vertical glide planes whose translation vectors are perpendicular to the specimen surface do not cause extinction lines in ZOLZ reflections but cause them in HOLZ reflections. (It is noted that the vertical glide planes with glide translations not parallel to the surface are not the symmetry elements of diperiodic plane figures.) Vertical glide planes whose translation vectors are parallel to the surface cause extinction lines in both ZOLZ and HOLZ reflections. Vertical screw axes are expected to form extinction lines in HOLZ reflections whose vectors are parallel to the screw axes. These reflections, however, cannot be observed by ordinary CBED. Thus, the extinction lines appearing in observable HOLZ reflections are used to identify not screw axes but glide planes. Examination of HOLZ extinction lines together with ZOLZ extinction lines is an efficient way to characterize the glide vectors and determine the space group. The dynamical extinction lines appearing in HOLZ reflections caused by the glide planes whose glide vectors are not only parallel but also not parallel to the specimen surface were tabulated by Nagasawa (1983) for various incident-beam orientations of all the space groups that have glide planes. The tabulated results appear on pages 214–225 of the book by Tanaka et al. (1988). Table 2.5.3.12 shows the results. The meanings of the letters used in the table are explained in Fig. 2.5.3.13. We consider a vertical glide plane with a glide vector perpendicular to the surface as is shown in Fig. 2.5.3.13(a). Letter A is given for cases in which the Ewald sphere intersects two circled-cross reflections in the first Laue zone as seen in Fig. 2.5.3.13(b), where black circles and circled crosses denote allowed reflections and kinematically forbidden but dynamically allowed reflections due to the glide plane, respectively. A* denotes cases in which the Ewald sphere intersects a circled-cross reflection on one side of the incident beam and a black-circled reflection on the other, as seen in Fig. 2.5.3.13(c). This case occurs only in space group P21 =a3 . Ah denotes cases in which the Ewald sphere intersects a circled-cross reflection on one side but does not intersect on the other, owing to the asymmetric arrangement of reflections with respect to the incident beam. The first column of Table 2.5.3.12 list the space groups and the following columns show the type of the extinction lines for possible incident-beam directions. In each pair of columns, the left-hand column gives the reflection indices of the extinction line and the symmetry elements causing the extinction and the righthand column gives the type of extinction. The first suffix 1, 2 or 3 of a glide symbol distinguishes the first, the second or the third glide plane of a space group. The second suffixes 1 and 2, which appear in the tetragonal and cubic systems, distinguish two equivalent glide planes which lie in the x and y planes. The suffix o of a reflection index implies that the index is odd-order. Figs.

Table 2.5.3.11. Space-group sets distinguishable by coherent CBED The space-group pairs in parentheses can not be distinguished by coherent CBED but can be distinguished by a handedness test. An asterisk (*) indicates the incidence at which the distinction is carried out by many-beam interference (Saitoh, Tsuda et al., 2001).

Space-group set (2) P312, (P3112, P3212) (3) P321, (P3121, P3221) (5) P622, (P6222, P6422) (7) P6322, (P6122, P6522) (12) P422, P4222

Incidence ½11 01 ½112 3 ½112 3 ½112 3 [321], [211], [112]*

(13) P4212, P42212

[211]

(15) I422, I4122

[111]

(16) I23, I213

[111]

(17) I222, I212121

[111]

(18) P432, P4232

[321], [211]*

(20) I432, I4132 (21) F432, F4132

[111] [432]

333

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION 2.5.3.14(a) and (b) were taken for FeS2, space group P21 =a3 , with actually observable symmetry elements because they do not form incident-beam directions of [100] and [110]. Inserts show a complete set of groups. It is of no importance to give the enlarged HOLZ patterns for ease of viewing. Extinction lines of relation between the 230 space groups of crystals and the 80 type A are seen in the hok0 HOLZ reflections in Fig. 2.5.3.14(a) space groups of diperiodic plane figures. Buxton et al.’s theory, due to the b-glide plane (equivalent to the a-glide plane in the which determines crystal point groups with the help of diperiodic space-group symbol) parallel to the (001) plane. An extinction plane figures, is very beautiful and successful. However, it is not line A* is seen in an hok0 HOLZ reflection in Fig. 2.5.3.14(b) due correct to state that CBED observes the symmetry elements of to the same glide plane as that of Fig. 2.5.3.14(a). It should be the diperiodic plane figures. The use of the groups of diperiodic noted that extinction lines in HOLZ reflections are better plane figures should be recognized as a convention for the sake of observed in thinner specimen areas than those suitable for the convenience. As a further example, horizontal screw axes and observation of the extinction lines in ZOLZ reflections, because horizontal glide planes must be located at the middle of a the profiles of HOLZ reflections are concentrated into small specimen to form symmetry elements of the diperiodic plane areas of CBED discs in thicker specimens. figures. However, those screw axes and glide planes which are not In summary, the use of not only ZOLZ, but also HOLZ located at the middle of a specimen do produce CBED symmeextinction lines is recommended for space-group determination. tries. Since we now know that CBED does not observe the symmetries of the diperiodic plane figures but observes those of a physical crystalline specimen, we can determine the corre2.5.3.3.5. Symmetry elements observed by CBED sponding infinite crystal symmetries more freely, by using our In the above sections, point-group and space-group determiknowledge of the symmetries of the sample concerned, guided nation methods were described following the theory of Buxton et but not restricted by the beautiful theory of Buxton et al. (1976). al. (1976). They assumed that the observable symmetry elements One point to note, for symmetry determination, is that one has to are those of an infinitely extended parallel-sided specimen or of be aware of spurious symmetries that appear for crystals of certain diperiodic plane figures. CBED patterns determine diffraction structure types (Tanaka et al., 1988, pp. 20–32 and 42–45) and groups. Crystal point groups are identified by consulting Fig. destroy the correct determination of the point and space groups. 2.5.3.4, which gives the relations between diffraction groups and Another point for precise symmetry determination is that one has crystal point groups. When the assumption made by Buxton et al. to be aware of how CBED symmetry is destroyed by a small (1976) is accepted in a strict sense, CBED symmetry m2 caused by breakdown of crystal symmetry (Tanaka et al., 1988 pp. 46–47). a twofold rotation axis oblique to the specimen surface, which is not a symmetry element of a diperiodic plane figure, ought not to 2.5.3.3.6. Examples of space-group determination be observed. However, the symmetry m2 due to a twofold rotation axis in the ½11 0 direction of an Si film with [100] surface A simple example of point-group determination has already normal has been clearly observed at [111] electron incidence been given for Si in Section 2.5.3.2.5. In this section, two exam(Tanaka et al., 1988, p. 33). This indicates that crystal symmetry ples of space-group determination for rutile and samarium seleelements oblique to the specimen surface are observable when nide are described, in which the point-group determination still the specimen is tilted. An important condition for CBED is that accounts for an important part. The examples look to be a little the top and bottom surfaces be parallel over the specimen area sophisticated but are a good exercise for those who want to illuminated by the incident beam. CBED observes the symmetry acquire experience in CBED space-group determination. The elements of a crystal to the extent that the boundary conditions at present determination is carried out by assuming the lattice the specimen surface do not break the symmetries of the CBED parameters to be known. patterns. Gjønnes & Gjønnes (1985) reported that the breaking Rutile (TiO2). The space group of rutile is well known to be P42/mnm. The lattice parameters are a = b = 0.459 nm and c = of CBED symmetry due to a surface oblique to the incident beam 0.296 nm. Fig. 2.5.3.15(a) shows a CBED pattern taken with the is practically negligible. [001] incidence at an accelerating voltage of 80 kV. Since no fine In Section 2.5.3.3 on space-group determination, space-group HOLZ lines appear in all the discs, projection diffraction groups symmetry elements of crystals which have glide and screw (column VI of Table 2.5.3.3) have to be applied to explain this components parallel to the specimen surface were considered to pattern. The projection (proj.) WP shows symmetry 4mm. The act as space-group symmetry elements of diperiodic plane figures projection diffraction group is found to be 4mm1R from Table by mitigating the strict application of the assumption of diperiodic plane figures. In fact, vertical glide planes with a glide vector 2.5.3.3. Thus, possible diffraction groups are 4mRmR, 4mm, not parallel to the specimen surface, which were dealt in Section 4RmmR and 4mm1R. Another CBED pattern at a second crystal 2.5.3.3.4, are not the symmetry elements of diperiodic plane orientation needs to be taken because Fig. 2.5.3.15(a) shows only figures. Ishizuka (1982) showed theoretically that a vertical glide projection symmetry. Figs. 2.5.3.15(b) and (c) show CBED patterns taken with the [101] incidence at an accelerating voltage plane with a vertical glide vector produces dynamical extinction of 100 kV. In Fig. 2.5.3.15(b), which is the central part of Fig. lines in HOLZ discs if the Laue zones are well separated. Tanaka 2.5.3.15(c), no HOLZ lines are seen. The symmetries of the et al. (1988, pp. 214–225) tabulated the extinction lines appearing in HOLZ discs caused by the vertical glide planes whose glide projection BP and projection WP are both 2mm. The projection vectors are not only parallel but also not parallel to the specimen diffraction group of the pattern is 2mm1R. The WP of Fig. 2.5.3.15(c) is seen to have one mirror symmetry m. The diffracsurface. Dynamical extinction lines caused by the glide planes tion groups which satisfy symmetry m are m, m1R and 2RmmR. with a glide vector not parallel to the surface have been Among these diffraction groups, the diffraction group whose demonstrated using FeS2 and MgAl2O4 (Tanaka et al., 1988, pp. projection becomes 2mm1R is only diffraction group 2RmmR. By 51–61). Vertical 21, 31, 32, . . . , 65 screw axes, which are not symmetry consulting Fig. 2.5.3.4, diffraction group 2RmmR obtained from elements of diperiodic plane figures, are expected to form Figs. 2.5.3.15(b) and (c) and one diffraction group 4mm1R among dynamical extinction lines in kinematically forbidden reflections diffraction groups 4mRmR, 4mm, 4RmmR and 4mm1R obtained that are located in the direction of the screw axes or of the surface from Fig. 2.5.3.15(a) commonly satisfy point group 4/mmm. Thus, normal. The extinction lines, however, are difficult to observe in the point group of rutile is determined to be 4/mmm. ordinary CBED. Thus, CBED does not observe all the symmetry Fig. 2.5.3.15(d) shows an ordinary diffraction pattern taken elements of the crystal space groups but observes many more with the [001] incidence at an accelerating voltage of 80 kV. With symmetry elements than those of the diperiodic plane figures. It is the help of the lattice parameters and the camera length, the clear now that it makes no sense to construct space groups using indices of the reflections are given as shown in the figure. There

334

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.12. Dynamical extinction lines appearing in HOLZ reflections for crystal space groups that have mirror and glide planes Point groups m, 2/m (second setting, unique axis b) Incident-beam direction Space group 6

Pm

7

Pc

8

Cm

9

Cc

10

P2/m

[u0w] h0lo c

Ah

he0lo c

Ah

11

P21/m

12

C2/m

13

P2/c

h0lo c

Ah

14

P21/c

h0lo c

Ah

15

C2/c

he0lo c

Ah

Point group mm2 Incident-beam direction Space group

[100]

[010]

25

Pmm2

26

Pmc21

h0lo c

A

27

Pcc2

h0lo c2

A

28

Pma2

ho0l a

A

29

Pca21

ho0l a

A

0klo c

30

Pnc2

h0lo c

A

0kl: k + l = 2n + 1 n

31

Pmn21

h0l: h + l = 2n + 1 n

A

32

Pba2

ho0l a

A

0kol b

33

Pna21

ho0l a

A

34

Pnn2

h0l: h + l = 2n + 1 n2

A

35

Cmm2 ba2

36

Cmc21 bn21

he0lo c

A

37

Ccc2 nn2

he0lo c2

A

[001]

[0vw]

h0lo c

A

0klo c1 h0lo c2

A

ho0l a

A

A

0klo c ho0l a

A

0klo c

A

0kl: k + l = 2n + 1 n h0lo c

A

0kl: k + l = 2n + 1 n

h0l: h + l = 2n + 1 n

A

A

0kol b ho0l a

A

0kol b

0kl: k + l = 2n + 1 n

A

0kl: k + l = 2n + 1 n ho0l a

A

0kl: k + l = 2n + 1 n1

A

0kl: k + l = 2n + 1 n1 h0l: h + l = 2n + 1 n2

A

he0lo c

A

0kelo c1 he0lo c2

A

0klo c1

0kelo c1

A

A

335

[u0w] h0lo c

Ah

h0lo c2

Ah

ho0l a

Ah

Ah

ho0l a

Ah

Ah

h0lo c

Ah

h0l: h + l = 2n + 1 n

Ah

Ah

ho0l a

Ah

0kl: k + l = 2n + 1 n

Ah

ho0l a

Ah

0kl: k + l = 2n + 1 n1

Ah

h0l: h + l = 2n + 1 n2

Ah

he0lo c

Ah

he0lo c2

Ah

0klo c1

0kelo c1

Ah

Ah

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Incident-beam direction Space group

[100]

[010]

38

Amm2 nc21

39

Abm2 cc21

40

Ama2 nn21

ho0le a

A

41

Aba2 cn21

ho0le a

A

0kolo b

42

Fmm2

43

Fdd2 dd21

he0le: he + le = 4n + 2 d2

A

44

Imm2 nn21

45

Iba2 cc21

ho0lo a

A

46

Ima2 nc21

ho0lo a

A

[001]

0kolo b

A

[0vw]

0kolo b

A

ho0le a

A

A

0kolo b ho0le a

A

0kolo b

0kele: ke + le = 4n + 2 d1

A

0kele: ke + le = 4n + 2 d1 he0le: he + le = 4n + 2 d2

A

0kolo b

A

0kolo b ho0lo a

A

ho0lo a

A

[u0w]

0kolo b

Ah ho0le a

Ah

Ah

ho0le a

Ah

0kele: ke + le = 4n + 2 d1

Ah

he0le: he + le = 4n + 2 d2

Ah

0kolo b

Ah

ho0lo a

Ah

ho0lo a

Ah

Point group mmm Incident-beam direction Space group

[100]

[010]

[001]

[uv0]

47

P2/m2/m2/m

48

P2/n2/n2/n

h0l: h+l= 2n + 1 n2 hk0: h+k= 2n + 1 n3

A

0kl: k+l= 2n + 1 n1 hk0: h+k= 2n + 1 n3

A

0kl: k+l= 2n + 1 n1 h0l: h+l= 2n + 1 n2

A

49

P2/c2/c2/m

h0lo c2

A

0klo c1

A

0klo c1 h0lo c2

A

50

P2/b2/a2/n

ho0l a hk0: h+k= 2n + 1 n

A

0kol b hk0: h+k= 2n + 1 n

A

0kol b ho0l a

A

51

P21/m2/m2/a

hok0 a

A

hok0 a

A

52

P2/n21/n2/a

h0l: h+l= 2n + 1 n2 hok0 a

A

0kl: k+l= 2n + 1 n1 hok0 a

A

0kl: k+l= 2n + 1 n1 h0l: h+l= 2n + 1 n2

53

P2/m2/n21/a

h0l: h+l= 2n + 1 n hok0 a

A

hok0 a

A

54

P21/c2/c2/a

h0lo c2 hok0 a

A

0klo c1 hok0 a

A

hk0: h+k= 2n + 1 n3

[0vw] Ah

hk0: h+k= 2n + 1 n

Ah

hok0 a

Ah

A

hok0 a

Ah

h0l: h+l= 2n + 1 n

A

hok0 a

Ah

0klo c1 h0lo c2

A

hok0 a

Ah

336

[u0w]

0kl: k+l= 2n + 1 n1

Ah

h0l: h+l= 2n + 1 n2

Ah

0klo c1

Ah

h0lo c2

Ah

0kol b

Ah

ho0l a

Ah

0kl: k+l= 2n + 1 n1

Ah

h0l: h+l= 2n + 1 n2

Ah

h0l: h+l= 2n + 1 n

Ah

h0lo c2

Ah

0klo c1

Ah

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Incident-beam direction Space group

[100]

55

P21/b21/a2/m

ho0l a

A

[010] 0kol b

A

[001] 0kol b ho0l a

A

[uv0]

56

P21/c21/c2/n

h0lo c2 hk0: h+k= 2n + 1 n

A

0klo c1 hk0: h+k= 2n + 1 n

A

0klo c1 h0lo c2

A

57

P2/b21/c21/m

h0lo c

A

0kol b

A

0kol b h0lo

58

P21/n21/n2/m

h0l: h+l= 2n + 1 n2

A

0kl: k+l= 2n + 1 n1

A

0kl: k+l= 2n + 1 n1 h0l: h+l= 2n + 1 n2

59

P21/m21/m2/n

hk0: h+k= 2n + 1 n

A

hk0: h+k= 2n + 1 n

A

60

P21/b2/c21/n

h0lo c hk0: h+k= 2n + 1 n

A

0kol b hk0: h+k= 2n + 1 n

A

0kol b h0lo c

61

P21/b21/c21/a

h0lo c hok0 a

A

0kol b hok0 a

A

62

P21/n21/m21/a

hok0 a

A

0kl: k+l= 2n + 1 n hok0 a

A

63

C2/m2/c21/m

he0lo c

A

64

C2/m2/c21/a

he0lo c hoko0 a

A

hoko0 a

[0vw]

[u0w]

0kol b

Ah

ho0l a

Ah

0klo c1

Ah

h0lo c2

Ah

A

0kol b

Ah

h0lo c

Ah

A

0kl: k+l= 2n + 1 n1

Ah

h0l: h+l= 2n + 1 n2

Ah

hk0: h+k= 2n + 1 n

Ah

c

hk0: h+k= 2n + 1 n

Ah

A

hk0: h+k= 2n + 1 n

Ah

0kol b

Ah

h0lo c

Ah

0kol b h0lo c

A

hok0 a

Ah

0kol b

Ah

h0lo c

Ah

0kl: k+l= 2n + 1 n

A

hok0 a

Ah

0kl: k+l= 2n + 1 n

Ah

he0lo c

A

he0lo c

Ah

A

he0lo c

A

he0lo c

Ah

0kelo c1 he0lo c2

A

65

C2/m2/m2/m

66

C2/c2/c2/m

he0lo c2

A

0kelo c1

A

67

C2/m2/m2/a

hoko0 a

A

hoko0 a

A

68

C2/c2/c2/a

he0lo c2 hoko0 a

A

0kelo c1 hoko0 a

A

0kelo c1 he0lo c2

69

F2/m2/m2/m

70

F2/d2/d2/d

he0le: he + le = 4n + 2 d2 heke0: he + ke = 4n + 2 d3

A

heke0: he + ke = 4n + 2 d3 0kele: ke + le = 4n + 2 d1

A

0kele: ke + le = 4n + 2 d1 he0le: he + le = 4n + 2 d2

71

I2/m2/m2/m

337

hoko0 a

Ah

0kelo c1

Ah

he0lo c2

Ah

hoko0 a

Ah

A

hoko0 a

Ah

0kelo c1

Ah

he0lo c2

Ah

A

heke0: he + ke = 4n + 2 d3

Ah

0kele: ke + le = 4n + 2 d1

Ah

he0le: he + le = 4n + 2 d2

Ah

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Incident-beam direction Space group

[100]

72

I2/b2/a2/m

ho0lo a

A

[010] 0kolo b

A

[001] 0kolo b ho0lo a

A

[uv0]

73

I21/b21/c21/a

ho0lo c hoko0 a

A

hoko0 a 0kolo b

A

0kolo b ho0lo c

A

74

I21/m21/m21/a

hoko0 a

A

hoko0 a

A

[0vw]

hoko0 a

Ah

hoko0 a

Ah

[u0w]

0kolo b

Ah

ho0lo a

Ah

0kolo b

Ah

ho0lo c

Ah

Point group 4/m Incident-beam direction Space group 83

[100], [110]

[uv0]

P4/m

84

P42/m

85

P4/n

hk0: h + k = 2n + 1 n

A

hk0: h + k = 2n + 1 n

Ah

86

P42/n

hk0: h + k = 2n + 1 n

A

hk0: h + k = 2n + 1 n

Ah

hoko0 a

A

hoko0 a

Ah

87

I4/m

88

I41/a

Point group 4mm. The symbol a in the column [u0w] is equivalent to the symbol b in the space groups of the first column. Incident-beam direction Space group

[100]

[001]

[110]

[u0w]

[uuw]

99

P4mm

100

P4bm

ho0l a2

A

0kol b1 ho0l a2

A

ho0l a

Ah

101

P42cm

h0lo c2

A

0klo c1 h0lo c2

A

h0lo c

Ah

102

P42nm

h0l: h + l = 2n + 1 n2

A

0kl: k + l = 2n + 1 n1 h0l: h + l = 2n + 1 n2

A

h0l: h + l = 2n + 1 n

Ah

103

P4cc

h0lo c12

A

0klo c11

A

hhlo c2

A

h0lo c1

Ah

hhlo c2

Ah

0kl: k + l = 2n + 1 n1 h0l: h + l = 2n + 1 n2 hhlo ; h hlo c hhlo ; h hlo c

A

hhlo c

A

h0l: h + l = 2n + 1 n

Ah

hhlo c

Ah

A

hhlo c

A

hhlo c

Ah

0kol b1 ho0l a2 hhlo ; h hlo c

A

hhlo c

A

hhlo c

Ah

h0lo c12 hhlo ; h hlo c2 104

P4nc

105

P42mc

106

P42bc

107

I4mm

h0l: h + l = 2n + 1 n2

ho0l a2

A

A

338

ho0l a

Ah

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Incident-beam direction Space group

[100]

108

I4cm

ho0lo c2

109

I41md

110

I41cd

[001] A

ho0lo c2

A

[110]

[u0w]

0kolo c1 ho0lo c2

A

hhle, h hle : 2h + le = 4n + 2 d

A

hhle: 2h + le = 4n + 2 d

A

0kolo c1 ho0lo c2 hhle, h hle : 2h + le = 4n + 2 d

A

hhle: 2h + le = 4n + 2 d

A

[uuw]

ho0lo c

Ah

ho0lo c

Ah

hhle: 2h + le = 4n + 2 d

Ah

hhle: 2h + le = 4n + 2 d

Ah

Point group 4 2m. The symbol a in the column [u0w] is equivalent to the symbol b in the space groups of the first column. Incident-beam direction Space group 111 P4 2m 112

P4 2c

113

P4 21 m P4 21 c

114

116

P4 m2 P4 c2

117

115

[100]

[001]

[110]

[u0w]

[uuw]

hhlo, h hlo c

A

hhlo c

A

hhlo c

Ah

hhlo, h hlo c

A

hhlo c

A

hhlo c

Ah

hhle: 2h + le = 4n + 2 d

Ah

h0lo c2

A

0klo c1 h0lo c2

A

h0lo c

Ah

P4 b2

ho0l a2

A

0kol b1 ho0l a2

A

ho0l a

Ah

118

P4 n2

h0l: h + l = 2n + 1 n2

A

0kl: k + l = 2n + 1 n1 h0l: h + l = 2n + 1 n2

A

h0l: h + l = 2n + 1 n

Ah

119

I 4 m2 I 4 c2

ho0lo c2

A

0kolo c1 ho0lo c2

A

ho0lo c

Ah

hhle, h hle : 2h + le = 4n + 2 d

A

120

121

I 4 2m

122

I 4 2d

hhle: 2h + le = 4n + 2 d

A

Point group 4/mmm. The symbol a in the column [u0w] is equivalent to the symbol b in the space groups of the first column. Incident-beam direction Space group

[100]

[001]

[110]

[u0w]

[uuw]

123

P4/mmm P4/m2/m2/m

124

P4/mcc P4/m2/c2/c

h0lo c12

A

0klo c11 h0lo c12 hhlo ; h hlo c2

A

hhlo c2

A

h0lo c1

Ah

125

P4/nbm P4/n2/b2/m

hk0: h+k= 2n + 1 n ho0l a2

A

0kol b1 ho0l a2

A

hk0: h+k= 2n + 1 n

A

ho0l a

Ah

339

hhlo c2

[uv0]

Ah

hk0: h+k= 2n + 1 n

Ah

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Incident-beam direction Space group

[100]

126

P4/nnc P4/n2/n2/c

hk0: h+k= 2n + 1 n1 h0l: h+l= 2n + 1 n22

A

[001] 0kl: k+l= 2n + 1 n21 h0l: h+l= 2n + 1 n22 hhlo ; h hlo c

A

[110]

127

P4/mbm P4/m21/b2/m

ho0l a2

A

0kol b1 ho0l a2

A

128

P4/mnc P4/m21/n2/c

h0l: h+l= 2n + 1 n2

A

0kl: k+l= 2n + 1 n1 h0l: h+l= 2n + 1 n2 hhlo ; h hlo c

A

129

P4/nmm P4/n21/m2/m

hk0: h+k= 2n + 1 n

A

130

P4/ncc P4/n21/c2/c

hk0: h+k= 2n + 1 n h0lo c12

A

131

P42/mmc P42/m2/m2/c

132

P42/mcm P42/m2/c2/m

h0lo c2

133

P42/nbc P42/n2/b2/c

134

hk0: h+k= 2n + 1 n1 hhlo c

[u0w] A

hhlo c

A

hk0: h+k= 2n + 1 n

A

0klo c11 h0lo c12 hhlo ; h hlo c2 hhlo ; h hlo c

A

hk0: h+k= 2n + 1 n hhlo c2

A

A

hhlo c

A

A

0klo c1 h0lo c2

A

hk0: h+k= 2n + 1 n ho0l a2

A

0kol b1 ho0l a2 hhlo ; h hlo c

A

hk0: h+k= 2n + 1 n hhlo c

P42/nnm P42/n2/n2/m

hk0: h+k= 2n + 1 n1 h0l: h+l= 2n + 1 n22

A

0kl: k+l= 2n + 1 n21 h0l: h+l= 2n + 1 n22

A

135

P42/mbc P42/m21/b2/c

ho0l a2

A

0kol b1 ho0l a2 hhlo ; h hlo c

A

136

P42/mnm P42/m21/n2/m

h0l: h+l= 2n + 1 n2

A

0kl: k+l= 2n + 1 n1 h0l: h+l= 2n + 1 n2

A

[uuw]

h0l: h+l= 2n + 1 n2

Ah

ho0l a

Ah

h0l: h+l= 2n + 1 n

Ah

h0lo c1

Ah

h0lo c

Ah

A

ho0l a

Ah

hk0: h+k= 2n + 1 n1

A

h0l: h+l= 2n + 1 n2

Ah

hhlo c

A

ho0l a

Ah

h0l: h+l= 2n + 1 n

Ah

340

[uv0]

hhlo c

Ah

hhlo c

Ah

hhlo c2

Ah

hhlo c

Ah

hhlo c

Ah

hhlo c

Ah

hk0: h+k= 2n + 1 n1

Ah

hk0: h+k= 2n + 1 n

Ah

hk0: h+k= 2n + 1 n

Ah

hk0: h+k= 2n + 1 n

Ah

hk0: h+k= 2n + 1 n1

Ah

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Incident-beam direction Space group

[100]

137

P42/nmc P42/n21/m2/c

hk0: h+k= 2n + 1 n

A

[001] hhlo ; h hlo c

A

[110] hhlo c hk0: h+k= 2n + 1 n

A

[u0w]

138

P42/ncm P42/n21/c2/m

hk0: h+k= 2n + 1 n h0lo c2

A

0klo c1 h0lo c2

A

hk0: h+k= 2n + 1 n

A

139

I4/mmm I4/m2/m2/m

140

I4/mcm I4/m2/c2/m

ho0lo c2

A

0kolo c1 ho0lo c2

A

141

I41/amd I41/a2/m2/d

hoko0 a

A

hhle ; h hle : 2h + le = 4n + 2 d

A

hoko0 a hhle: 2h + le = 4n + 2 d

A

142

I41/acd I41/a2/c2/d

hoko0 a ho0lo c2

A

0kolo c1 ho0lo c2 hhle ; h hle : 2h + le = 4n + 2 d

A

hoko0 a hhle: 2h + le = 4n + 2 d

A

[uuw]

[uv0]

hhlo c

h0lo c

Ah

ho0lo c

Ah

ho0lo c

Ah

Ah

hk0: h+k= 2n + 1 n

Ah

hk0: h+k= 2n + 1 n

Ah

hhle: 2h + le = 4n + 2 d

Ah

hoko0 a

Ah

hhle: 2h + le = 4n + 2 d

Ah

hoko0 a

Ah

Point groups 3m; 3 m Incident-beam direction Space group 156

½11 00

½112 0

[0001]

157

P31m

158

P3c1

hh 0lo ; 0hh lo ; h 0hlo c

A

159

P31c

hh2hlo ; h2hhlo ; 2hhhlo c

A

160

R3m

161

R3c

hh 0lo ; 0hh lo ; h 0hlo : h + lo = 3n c

Ah

162

P3 1m P3 1c

hh2hlo ; h2hhlo ; 2hhhlo c

A

hh 0lo ; 0hh lo ; h 0hlo c

A

hh 0lo c

hh 0lo ; 0hh lo ; h 0hlo : h + lo = 3n c

Ah

hh 0lo : h + lo = 3n c

163 164 165 166 167

½11 0w

½112 w

P3m1

P3 m1 P3 c1 R3 m R3 c

hh 0lo c hh2hlo c

Ah

hh 0lo : h + lo = 3n c

Ah

A

hh 0lo c

Ah

Ah

hh 0lo : h + lo = 3n c

Ah

A

hh2hlo c hh 0lo : h + lo = 3n c

hh2hlo c

hh 0lo c

A Ah

Ah

A

hh2hlo c

Ah

Point groups 6mm; 6 m2; 6=mmm Incident-beam direction Space group 183

P6mm

184

P6cc

hh 0lo ; 0hh lo ; h 0hlo c1 hh2hlo ; h2hhlo ; 2hhhlo c2

½11 00

½112 0

[0001] A

hh2hlo c2

341

A

hh 0lo c1

½11 0w

½112 w A

hh2hlo c2

Ah

hh 0lo c1

Ah

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Incident-beam direction Space group

½11 00 hh 0lo c

½112 0

185

P63cm

[0001] hh 0lo ; 0hh lo ; h 0hlo c

186

P63mc

hh2hlo ; h2hhlo ; 2hhhlo c

A

187

P6 m2 P6 c2

hh 0lo ; 0hh lo ; h 0hlo c

A

hh2hlo ; h2hhlo ; 2hhhlo c

A

hh2hlo c

A

hh 0lo ; 0hh lo ; h 0hlo c1 hh2hlo ; h2hhlo ; 2hhhlo c2 hh 0lo ; 0hh lo ; h 0hlo c

A

hh2hlo c2

A

hh2hlo ; h2hhlo ; 2hhhlo c

A

188

190

P6 2m P6 2c

191

P6/mmm

192

P6/mcc

193

P63/mcm

194

P63/mmc

189

A hh2hlo c

A

hh2hlo c hh 0lo c

A hh2hlo c

½112 w A

A

hh 0lo c

A

A

hh2hlo c

Ah

hh2hlo c2

Ah

hh2hlo c

Ah

hh 0lo c

Ah

hh 0lo c1

Ah

hh 0lo c

Ah

Ah

A

hh 0lo c1

½11 0w hh 0lo c

Ah

Point group m3 Incident-beam direction Space group 200

Pm3 P2=m3

201

Pn3 P2=n3

202

Fm3 F2=m3

203

Fd3 F2=d3

204

Im3 I2=m3

205

206

[100]

[110]

[uv0]

h0l: h + l = 2n + 1 n2 hk0: h + k = 2n + 1 n3

A

hk0: h + k = 2n + 1 n3

A

hk0: h + k = 2n + 1 n

Ah

he0le: he + le = 4n + 2 d2 heke0: he + ke = 4n + 2 d3

A

heke0: he + ke = 4n + 2 d3

A

heke0: he + ke = 4n + 2 d

Ah

Pa3 P21 =a3

h0lo c2 hok0 a3

A

hok0 a3

A*

hok0 a

Ah

Ia3 I21 =a3

ho0lo c2 hoko0 a3

A

hoko0 a3

A

hoko0 a

Ah

Point group 4 3m. The symbol a in the column [100] is equivalent to the symbol c in the space groups of the first column. Incident-beam direction Space group 215 P4 3m 217

F 4 3m I 4 3m

218

P4 3n

219

F 4 3c

220

I 4 3d

216

[100]

ho kk; ho k k n ho ko ko ; ho k o ko a he kk; he k k: 2k + he = 4n + 2 d

[110]

[uuw]

A

hhlo n

A

hhlo n

Ah

A

hoholo c

A

hoholo c

Ah

A

hhle: 2h + le = 4n + 2 d

A

hhle: 2h + le = 4n + 2 d

Ah

342

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.12 (cont.) Point group m3m. The symbol a in the column [100] is equivalent to the symbol c in the space groups of the first column. Incident-beam direction Space group 221

Pm3m P4=m3 2=m

222

Pn3n P4=n3 2=n

223

Pm3n P42 =m3 2=n

224

Pn3m P42 =n3 2=m

225

Fm3m F4=m3 2=m

226

[100]

[110]

[uv0]

h0l: h + l = 2n + 1 n12 hk0: h + k = 2n + 1 n13 ho kk; ho k k n2 ho kk; ho k k n

A

hk0: h + k = 2n + 1 n13 hhlo n2

A

A

hhlo n

A

h0l: h + l = 2n + 1 n2 hk0: h + k = 2n + 1 n3

A

hk0: h + k = 2n + 1 n3

A

Fm3c F4=m3 2=c

ho ko ko ; ho k o ko a

A

hoholo c

A

227

Fd3m F41 =d3 2=m

he0le: he + le = 4n + 2 d2 heke0: he + ke = 4n + 2 d3

A

heke0: he + ke = 4n + 2 d3

A

heke0: he + ke = 4n + 2 d

Ah

228

Fd3c F41 =d3 2=c

he0le: he + le = 4n + 2 d2 heke0: he + ke = 4n + 2 d3 ho ko ko ; ho k o ko a

A

hoholo c heke0: he + ke = 4n + 2 d3

A

heke0: he + ke = 4n + 2 d

229

Im3m I4=m3 2=m

230

Ia3d I41 =a3 2=d

hoko0 a3 ho0lo c2 he kk; he k k: 2k + he = 4n + 2 d

A

hhle: 2h + le = 4n + 2 d hoko0 a3

A

hoko0 a

are no kinematically forbidden reflections. Thus, the lattice type is determined to be primitive P. Possible space groups which satisfy point group 4/mmm and primitive lattice type P are those of Nos. 123–138 in Table 2.5.3.9. In Fig. 2.5.3.15(a), the dynamical extinction line A2 is seen in the 100 disc and in the equivalent 010 disc. By consulting Table 2.5.3.9, four space groups P4/mbm, P4/mnc, P42/mbc and P42/mnm are selected. In Fig. 2.5.3.15(b), the dynamical extinction line A2 is seen in the 010 disc but not in the 101 disc. Two space groups P4/mnc and P42/mnm are selected from the four. To distinguish the two space groups, it is found from Table 2.5.3.9 that a CBED pattern taken with the [110] electron incidence should be examined. Fig. 2.5.3.15(e) shows a CBED pattern taken with the [110] incidence at 100 kV, where the 001 reflection is exactly excited. The hh 1 reflections are kinematically allowed for space group P42/mnm but kinematically forbidden for P4/mnc. Since in the case of P4/mnc, no Umweganregung (multiple scattering) paths to the 001 reflection exist in the zeroth-order Laue zone, only the intensities of HOLZ lines, which are caused by Umweganregung via HOLZ reflections, are expected to appear in the 001 disc. If such Umweganregung is not practically excited, the 001 reflection must have no intensity. However, strong intensity produced by two-dimensional interaction is seen in the 001 disc of Fig. 2.5.3.15(e). This indicates that the reflection is an allowed reflection. Therefore, the space group of rutile is determined to be P42/mnm, which agrees with the space group already known.

hk0: h + k = 2n + 1 n1

[uuw]

hk0: h + k = 2n + 1 n

Ah

hhlo n2

Ah

hhlo n

Ah

hoholo c

Ah

Ah

hoholo c

Ah

Ah

hhle: 2h + le = 4n + 2 d

Ah

Ah

Samarium selenide (Sm3Se4). Sm3Se4 has the Th3P4 structure type with space group I 4 3d at high temperatures. The lattice parameters are a = b = c = 0.8885 nm. It was expected that Sm3Se4 would transform to an ordered state of electrons with two valences of +2 and +3 around 150 K. The determination of the space group of the material was conducted at 100 K and room temperature. The space groups at both temperatures were determined by CBED to be the same. The following experiments were performed at 100 K. Fig. 2.5.3.16(a) shows a CBED pattern taken with the [111] incidence at 80 kV, which clearly shows the first-order-Laue-zone reflections. The symmetry of the WP is seen to be 3m with the help of the enlarged insets. Possible diffraction groups are 3m, 3m1R and 6RmmR from Table 2.5.3.3. Fig. 2.5.3.16(b), which is the central part of Fig. 2.5.3.16(a), shows projection symmetry 3m, indicating that the projection diffraction group is 3m1R. Among the three groups 3m, 3m1R and 6RmmR, diffraction groups for which the projection diffraction group is 3m1R are 3m and 3m1R. Possible point groups are found to be 3m, 4 3m and 6 m2 from Fig. 2.5.3.4. Fig. 2.5.3.16(c) shows a CBED pattern taken with the [100] incidence at 80 kV. The WP is seen to have symmetry 2mm. Allowed diffraction groups are 2mm, 2mm1R and 4RmmR. Fig. 2.5.3.16(d), which is the central part of Fig. 2.5.3.16(c), shows projection WP symmetry 4mm, indicating that the projection diffraction group is 4mm1R. The diffraction group among the three groups 2mm, 2mm1R and 4RmmR whose projection diffraction group is 4mm1R is 4RmmR. Possible point groups are found to be 4 3m and 4 2m from Fig. 2.5.3.4. Thus, the point group

343

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.14. HOLZ CBED pattern of FeS2. (a) [100] incidence: type A dynamical extinction lines are seen clearly in the enlarged insets. (b) [110] incidence: a type A* dynamical extinction line is seen clearly in the enlarged insets.

which satisfies the results obtained at the two crystal orientations is 4 3m. Fig. 2.5.3.16(e) shows an ordinary diffraction pattern taken with the [100] incidence at 80 kV. With the help of the lattice parameters and the camera length, the indices of the reflections are given as shown in the figure. The reflections 0kl (k + l = 2n + 1) are found to be kinematically forbidden. Thus, the lattice type is determined to be I. The space groups having point group 4 3m and lattice type I are I 4 3m and I 4 3d from Table 2.5.3.9. Fig. 2.5.3.16(d) shows dynamical extinction lines A2 in the 033 disc and equivalent discs (also broad lines A2 in the 011 discs). Since the former space group does not give any dynamical extinction lines, the space group is determined to be I 4 3d. For confirmation, a CBED pattern which contains the second-order-Laue-zone reflections was taken (Fig. 2.5.3.16f ). Dynamical extinction lines A are seen in the 2,22,22 disc and the equivalent discs. This result also identifies the space group to be not I 4 3m but I 4 3d with the aid of Table 2.5.3.12. 2.5.3.4. Symmetry determination of incommensurate crystals 2.5.3.4.1. General remarks Incommensurately modulated crystals do not have threedimensional lattice periodicity. The crystals, however, recover lattice periodicity in a space higher than three dimensions. de Wolff (1974, 1977) showed that one-dimensional displacive and substitutionally modulated crystals can be described as a threedimensional section of a (3 + 1)-dimensional periodic crystal. Janner & Janssen (1980a,b) developed a more general approach for describing a modulated crystal with n modulations as (3 + n)dimensional periodic crystals (n = 1, 2, . . . ). Yamamoto (1982) derived a general structure-factor formula for n-dimensionally modulated crystals (n = 1, 2, . . . ), which holds for both displacive and substitutionally modulated crystals. Tables of the (3 + 1)dimensional space groups for one-dimensional incommensurately modulated crystals were given by de Wolff et al. (1981), where the wavevector of the modulation was assumed to lie in the

Fig. 2.5.3.13. Illustration of dynamical extinction lines appearing in HOLZ reflections due to glide planes. Black circles and circled crosses show kinematically allowed and kinematically forbidden reflections, respectively. (a) a glide in the (001) plane. (b) [100] incidence: dynamical extinction lines are formed in HOLZ reflections on both sides of the incident beam (type A). (c) [110] incidence: an extinction line is formed at a HOLZ reflection on one side of the incident beam because on the other side the Ewald sphere intersects an allowed HOLZ reflection (type A*). (d) An incidence between [100] and [110]: an extinction line is formed at a HOLZ reflection on one side of the incident beam because on the other side the Ewald sphere does not intersect a HOLZ reflection (type Ah).

344

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.3.15. CBED patterns of rutile. The procedures for identifying the symmetry are also shown. (a) [001] incidence at 80 kV: the projection (proj.) WP shows symmetry 4mm. (b, c) [101] incidence at 100 kV: the projection BP and projection WP show symmetry 2mm and the WP shows symmetry m (the point group is 4/mmm). (d) Spot diffraction pattern showing no extinction caused by the lattice type (lattice type P). (e) Near-[110] incidence at 100 kV to excite exactly the 001 reflection: no extinction lines in the 001 disc (space group P42/mnm).

c direction. Later, some corrections to the tables were made by Yamamoto et al. (1985). The analysis of incommensurately modulated crystals using (3 + 1)-dimensional space groups has become familiar in the field of X-ray structure analysis. Fung et al. (1980) applied the CBED method to the study of incommensurately modulated transition-metal dichalcogenides.

Steeds et al. (1985) applied the LACBED method (Tanaka et al., 1980) to the study of incommensurately modulated crystals of NiGe1
xPx. Tanaka et al. (1988, pp. 74–81) examined the symmetries of the incommensurate and fundamental reflections appearing in the CBED patterns obtained from the incommensurately modulated crystals of Sr2Nb2O7 and Mo8O23. Terauchi &

345

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.16. CBED patterns of Sm3Se4. The procedures for identifying the symmetry are also shown. (a, b) [111] incidence at 80 kV: the WP symmetry is 3m (a) and the projection (proj.) WP symmetry is 3m (b). (c, d) [100] incidence at 80 kV: the WP symmetry is 2mm (c) and the projection WP symmetry is 4mm (d). Dynamical extinction lines A2 and A3 are seen (d). The point group is determined to be 4 3m. (e) Spot diffraction pattern taken with the [100] incidence at 80 kV shows the absence of 0kl reflections. The lattice type is determined to be I. ( f ) [100] incidence at 100 kV: dynamical extinction lines A in HOLZ reflections confirm the existence of a glide plane. The space group is determined to be I 4 3d.

346

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.3.17. The (3 + 1)-dimensional description of one-dimensionally modulated crystals. Atoms are shown as strings along the fourth direction a4. (a) No modulation, shown as straight strings. (b) Displacive modulation, shown as wavy strings. (c) Amplitude modulation with varying-density strings.

Tanaka (1993) clarified theoretically the interrelation between the symmetries of CBED patterns and the (3 + 1)-dimensional point-group symbols for incommensurately modulated crystals and verified experimentally the theoretical results for Sr2Nb2O7 and Mo8O23. Terauchi et al. (1994) investigated dynamical extinction for the (3 + 1)-dimensional space groups. They clarified that approximate dynamical extinction lines appear in CBED

discs of the reflections caused by incommensurate modulations when the amplitudes of the incommensurate modulation waves are small. They tabulated the dynamical extinction lines appearing in the CBED discs for all the (3 + 1)-dimensional space groups of the incommensurately modulated crystals. The tables were stored in the British Library Document Supply Centre as Supplementary Publication No. SUP 71810 (65 pp.). They showed an example of the dynamical extinction lines obtained from Sr2Nb2O7. The point- and space-group determinations of the (3 + 1)-dimensional crystals are described compactly in the book by Tanaka et al. (1994, pp. 156–205). Fig. 2.5.3.17 illustrates (3 + 1)-dimensional descriptions of a crystal structure without modulation (a), a one-dimensional displacive modulated structure (b) and a one-dimensional substitutionally modulated structure (c). The arrows labelled a1– a3 (a, b and c) and a4 indicate the (3 + 1)-dimensional crystal axes. The horizontal line labelled R3 represents the three-dimensional space (external space). In the (3 + 1)-dimensional description, an atom is not located at a point as in the three-dimensional space, but extends as a string along the fourth direction a4 perpendicular to the three-dimensional space R3. The shaded parallelogram is a unit cell in the (3 + 1)-dimensional space. The unit cell contains two atom strings in this case. In the case of no modulations, the atoms are shown as straight strings, as shown in Fig. 2.5.3.17(a). For a displacive modulation, atoms are expressed by wavy strings periodic along the fourth direction a4 as shown in Fig. 2.5.3.17(b). The width of the atom strings indicates the spread of the atoms in R3. The atom positions of the modulated structure in R3 are given as a three-dimensional (R3) section of the atom strings in the (3 + 1)-dimensional space. A substitutional modulation, which is described by a modulation of the atom form factor, is expressed by atom strings with a density modulation along the direction a4 as shown in Fig. 2.5.3.17(c). The diffraction vector G is written as G ¼ h1 a þ h2 b þ h3 c þ h4 k; where a set of h1h2h3h4 is a (3 + 1)-dimensional reflection index, and a*, b* and c* are the reciprocal-lattice vectors of the reallattice vectors a, b and c of the average structure. The modulation vector k is written as k ¼ k1 a þ k2 b þ k3 c ; where one coefficient ki (i = 1–3) is an irrational number and the others are rational. Fig. 2.5.3.18(a) shows a diffraction pattern of a crystal with an incommensurate modulation wavevector k1a* (k2 and k3 = 0). Large and small black spots show the fundamental reflections and incommensurate reflections, respectively, only the first-order incommensurate reflections being shown. It

Fig. 2.5.3.18. (a) Schematic diffraction pattern from a modulated crystal. As an example, the wave number vector of modulation is assumed to be k1a*, k1 being an irrational number. Large and small spots denote fundamental and incommensurate reflections, respectively. (b) Incommensurate reflections are obtained by a projection of the Fourier transform of a (3 + 1)-dimensional periodic structure.

347

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION should be noted that the diffraction pattern of a modulated crystal is obtained by a projection of the Fourier transform of the (3 + 1)-dimensional periodic structure. Fig. 2.5.3.18(b) is assumed to be the Fourier transform of Fig. 2.5.3.17(b). Incommensurate reflections are obtained by a projection of the reciprocal-lattice points onto R3 . The displacive modulation is expressed by the atom displacement ui with x4. The structure factor Fðh1 h2 h3 h4 Þ for the (3 + 1)dimensional crystal with a displacive modulation is given by de Wolff (1974, 1977) as follows:

N P

Fðh1 h2 h3 h4 Þ ¼

¼1

F 0 ðh1 h2 h3 h4 Þ ¼

¼1





where x4 ¼


3  P dx4 ; exp 2i ðhi þ h4 ki Þui þ h4 x 4 i¼1

ð2:5:3:6Þ

where x 4 ¼ ðx1 þ n1 Þk1 þ ðx2 þ n2 Þk2 þ ðx3 þ n3 Þk3 :

f and x i (i = 1–3) are, respectively, the atom form factor and the ith component of the position of the th atom in the unit cell of the average structure. The symbol ui is the ith component of the displacement of the th atom. Since the atom in the (3 + 1)dimensional space is continuous along a4 and discrete along R3, the structure factor is expressed by summation in R3 and integration along a4 as seen in equation (2.5.3.6). The integration implies that the sum for the atoms with displacements is taken over the infinite number of unit cells of the average structure. That is, equation (2.5.3.6) is the structure factor for a unit cell with the lattice parameter of an infinite length in R3 along the direction of the modulation wavevector k. CBED patterns are obtained from a finite area of a specimen crystal. For the symmetry analysis of CBED patterns obtained from modulated structures, the effect of the finite size was considered by Terauchi & Tanaka (1993). The integration over a unit-cell length along a4 in equation (2.5.3.6) is rewritten in the following way with the summation over a finite number of threedimensional sections of the atom strings:

N P ¼1



n1

n2

n3


exp 2i

3 P

ðhi þ

h4 ki Þui

þ

h4 x 4

f expð2ih4 x4 Þ;

ð2:5:3:8Þ

P

i

ðxi þ ni Þki.

F 0 ðh1 h2 h3 h4 Þ N P ¼ f exp½2iðh1 x 1 þ h2 x 2 þ h3 x 3 Þ ¼1



P

expf2i½h1 u1 þ h2 u2 þ ðh3 þ h4 k3 Þu3 þ h4 x 4 g

n3

þ

N P ¼1



P

f exp½2ið
h1 x 1
h2 x 2 þ h3 x 3 Þ

expf2i½
h1 u1
h2 u2 þ ðh3 þ h4 k3 Þu3 þ h4 x 4 g;

n3

ð2:5:3:9Þ

f exp½2iðh1 x 1 þ h2 x 2 þ h3 x 3 Þ

XXX

PPP

2.5.3.4.2. Point-group determination The symmetries of the CBED patterns can be determined by examination of the symmetries of the structure factor F 0 ðh1 h2 h3 h4 Þ in equations (2.5.3.7) or (2.5.3.8). We consider a displacive modulated structure, which has a modulation wavevector k = k3c* and belongs to (3 + 1)-dimensional space group P2=m P 1 1 . This space-group symbol implies the following. (1) The modulation wavevector k exists inside the first Brillouin zone of the average structure (P). (2) The average structure belongs to space group P2/m, the twofold rotation axis being parallel to the c axis. (3) The symmetry subsymbol 1, which is written beneath symmetry symbol 2, indicates that the modulation wavevector k is transformed into itself by symmetry operation 2 of the average structure. The symmetry subsymbol beneath symmetry symbol m indicates that the wavevector k is transformed into
k by symmetry operation m. The modulated structure has a twofold rotation axis, which is common to the average structure, but does not have mirror symmetry, which is possessed by the average structure. For the twofold rotation axis (symbol 2) of this space group, the structure factor F 0 ðh1 h2 h3 h4 Þ is written as

0

F 0 ðh1 h2 h3 h4 Þ ¼

exp½2i ðh1 x1 þ h2 x2 þ h3 x3 Þ

n1 n2 n3

f exp½2iðh1 x 1 þ h2 x 2 þ h3 x 3 Þ

Z1

N P

 ;

i¼1

ð2:5:3:7Þ

where N1 < n1  N10 , N2 < n2  N20 and N3 < n3  N30 , N 0 ¼ ðN10
N1 ÞðN20
N2 ÞðN30
N3 Þ being the number of unit cells of the average structure included in a specimen volume from which CBED patterns are taken. The substitutional modulation arises from a periodic variation of the site-occupation probability of the atoms. This modulation is expressed by a modulation of the atom form factor f with x4. The structure factor F 0 ðh1 h2 h3 h4 Þ for a finite-size crystal is written as

348

where x4 ¼ ðx3 þ n3 Þk3. It is found from equation (2.5.3.9) that two structure factors F 0 ðh1 h2 h3 h4 Þ and F 0 ðh 1 h 2 h3 h4 Þ are the same when reflections h1 h2 h3 h4 and h 1 h 2 h3 h4 are equivalent with respect to the twofold rotation axis of the average structure. Thus, not only fundamental reflections (h4 = 0) from the average structure but also the satellite reflections (h4 6¼ 0) from the incommensurate structure show twofold rotation symmetry about the c* axis. For the mirror plane (symbol m), the structure factor is written in a similar manner to the case of the twofold rotation axis. It is found that F 0 ðh1 h2 h3 h4 Þ is not equal to F 0 ðh1 h2 h 3 h 4 Þ for the incommensurate reflections h4 6¼ 0. Hence, the incommensurate reflections do not show mirror symmetry with respect to the mirror plane of the average structure. For the fundamental reflections (h4 = 0), F 0 ðh1 h2 h3 h4 Þ is equal to F 0 ðh1 h2 h 3 h 4 Þ, indicating the existence of mirror symmetry. It should be noted that the mirror symmetry can be destroyed by the dynamical diffraction effect between the fundamental and incommensurate reflections. In most modulated structures, however, the amplitude of the modulation wave ui is not so large as to destroy the symmetry of the fundamental reflections. Thus, the fundamental

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.13. Wavevectors, point- and space-group symbols and CBED symmetries of one-dimensionally modulated crystals Wavevector transformation

Point-group symbol

k!k

1

k !
k

1

Symmetry of incommensurate reflection

Space-group symbol

Dynamical extinction lines

Same symmetry as average structure No symmetry

1, s (1/2), t (1/3), q (1/4), h (1/6) 1

Yes for s, q and h

reflections ought to show the symmetry of the average structure, while the incommensurate reflections lose the symmetry. The problem of the finite size of the illuminated area is discussed using equations (2.5.3.7) and (2.5.3.8) in a paper by Terauchi & Tanaka (1993) and in the book by Tanaka et al. (1994, pp. 156–205). The results are as follows: Even if the size and position of an illuminated specimen area are changed, the intensity distribution in a CBED pattern changes but the symmetry of the pattern does not. To obtain the symmetries of incommensurate crystals, it is not necessary to take CBED patterns from an area whose diameter is larger than the period of the modulated structure. The symmetries of the modulated structure can appear when more than one unit cell of the average structure is illuminated for displacive modulations. For substitutional modulations, a specimen volume that produces the average

No

atom form factor is needed, namely a volume of about 1 nm diameter area and 50 nm thick. Table 2.5.3.13 shows the point-group symmetries (third column) of the incommensurate reflections for the two pointgroup subsymbols. For symmetry subsymbol 1, both the fundamental and incommensurate reflections show the symmetries of the average structure. For symmetry subsymbol 1 , the fundamental reflections show the symmetries of the average structure but the incommensurate reflections do not have any symmetry. These facts imply that the symmetries of the incommensurate reflections are determined by the point group of the average structure and the modulation wavevector k. In other words, observation of the symmetries of the incommensurate reflections is not necessary for the determination of the point groups, although it can ascertain the point groups of the modulated crystals. An example of point-group determination is shown for the incommensurate phase of Sr2Nb2O7. Many materials of the A2B2O7 family undergo phase transformations from space group Cmcm to Cmc21 and further to P21 with decreasing temperature. An incommensurate phase appears between the Cmc21 phase and the P21 phase. Sr2Nb2O7 transforms at 488 K from the Cmc21 phase into the incommensurate phase with a modulation wavevector k = (12
)a* ( = 0.009–0.023) but does not transform into the P21 phase. The space group of Sr2Nb2O7 was reported C mc2 as P 1 s 1 1 (Yamamoto, 1988). (Since the space-group notation Cmc21 is broadly accepted, the direction of the modulation is . The taken as the a axis.) The point group of the phase is mm2 1 1 1 modulation wavevector k is transformed to
k by the mirror symmetry operation perpendicular to the a axis (m1 ) and by the twofold rotation symmetry operation about the c axis (21 ). The wavevector is transformed into itself by the mirror symmetry operation perpendicular to the b axis (m1 ). Fig. 2.5.3.19(a) shows a CBED pattern of the incommensurate phase of Sr2Nb2O7 taken with the [010] incidence at an accelerating voltage of 60 kV. The reflections indicated by arrowheads are the incommensurate reflections. Other reflections are the fundamental reflections. Since the pattern is produced by the interaction of the reflections in the zeroth-order Laue zone, symmetry operations (m1 ) and (21 ) act the same. These symmetries are confirmed by the fact that the fundamental reflections show mirror symmetry perpendicular to the a* axis (twofold rotation symmetry about the c* axis) but the incommensurate reflections do not. Fig. 2.5.3.19(b) shows a CBED pattern of the incommensurate phase of Sr2Nb2O7 taken with the [201] incidence at 60 kV. The reflections in the two rows indicated by arrowheads are the incommensurate reflections and the others are the fundamental reflections. Symmetry symbol (m1 ) implies that both the fundamental and incommensurate reflections display mirror symmetry perpendicular to the b* axis. Fig. 2.5.3.19(b) exactly exhibits the symmetry. 2.5.3.4.3. Space-group determination Table 2.5.3.13 shows the space-group symbols (fourth column) of the modulated crystals. When a glide (screw) component  4 between the modulation waves of two atom rows is 0, 1/2, 1/3, 1/4 or 1/6, symbol 1, s, t, q or h is given, respectively (de Wolff et al., 1981). Such glide components are allowed for point-group symmetry 1 but are not for point-group symmetry 1. Dynamical

Fig. 2.5.3.19. CBED patterns of the incommensurate phase of Sr2Nb2O7 taken at 60 kV. (a) [010] incidence: fundamental reflections show a mirror symmetry perpendicular to the a* axis but incommensurate reflections do not [symmetry (m1 )]. (b) [201] incidence: incommensurate reflections show mirror symmetry perpendicular to the b* axis [symmetry (m1 )]. The wave number vector of the modulation is k = (12
)a*.

349

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.20. (a) Mirror symmetry of modulation waves (m1 )  4 = 0. (b) Glide symmetry of modulation waves (ms )  4 = 12. The wave number vector of modulation is k3c*.

extinction occurs for glide components s, q and h but does not for glide component t. When the average structure does not have a glide component, dynamical extinction due to a glide component  4 appears in odd-order incommensurate reflections. When the average structure has a glide component, dynamical extinction due to a glide component  4 appears in incommensurate reflections with hi + h4 = 2n + 1, where hi and h4 are the reflection indices for the average structure and incommensurate structure, respectively. Details are given in the paper by Terauchi et al. (1994). Fig. 2.5.3.20(a) illustrates mirror symmetry (m1 ) between atom rows A and B, which is perpendicular to the b axis with no glide component ( 4 = 0). Here, the wave number vector of the modulation is assumed to be k = k3c* following the treatment of de Wolff et al. (1981). Fig. 2.5.3.20(b) illustrates glide symmetry (ms ) with a glide component  4 = 12. The structure factor Fðh1 h2 h3 h4 Þ is written for the glide plane (ms ) of an infinite incommensurate crystal as

Fig. 2.5.3.21. (a) Umweganregung paths a, b and c to the 0011 forbidden reflection. (b) Expected dynamical extinction lines are shown, the 0011 reflection being excited. The wave number vector of modulation is k3c*.

Thus, the following phase relations are obtained between the two structure factors: Fðh1 h2 h3 h4 Þ ¼ Fðh1 h 2 h3 h4 Þ for h4 even; Fðh1 h2 h3 h4 Þ ¼
Fðh1 h 2 h3 h4 Þ for h4 odd: ð2:5:3:11Þ

Fðh1 h2 h3 h4 Þ N P f exp½2iðh1 x 1 þ h2 x 2 þ h3 x 3 Þ ¼

These relations are analogous to the phase relations between the two structure factors for an ordinary three-dimensional crystal with a glide plane. The relations imply that dynamical extinction occurs for the glide planes and screw axes of the (3 + 1)dimensional crystal with an infinite dimension along the direction of the incommensurate modulation wavevector k. Terauchi et al. (1994) showed that approximate dynamical extinction occurs for an incommensurate crystal of finite dimension. Fig. 2.5.3.21(a) and (b) illustrate a spot diffraction pattern and a CBED pattern, respectively, expected from a modulated crystal with a (3 + 1)-dimensional space group PP2mm (k = k3c*) at the 1 s 1 [100] incidence. The large and small spots in Fig. 2.5.3.21(a) designate the fundamental (h4 = 0) and incommensurate reflections (h4 6¼ 0), respectively. The 00h3h4 (h4 = odd) reflections

¼1

R1  expf2i½h1 u1 þ h2 u2 þ ðh3 þ h4 k3 Þu3 þ h4 x 4 g dx4 0

þ expðh4 iÞ

N P ¼1

f exp½2iðh1 x 1
h2 x 2 þ h3 x 3 Þ

R1  expf2i½h1 u1
h2 u2 þ ðh3 þ h4 k3 Þu3 þ h4 x 4 g dx4 : 0

ð2:5:3:10Þ

350

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.3.22. Schematic diffraction pattern at the [001] incidence of Sr2Nb2O7. Umweganregung paths a and b via fundamental reflections to the 0001 incommensurate reflection. Large and small spots denote fundamental and incommensurate reflections, respectively. The wave number vector of modulation is k = (12
)a*.

shown by crosses are kinematically forbidden by the glide plane (ms ) perpendicular to the b axis. Umweganregung paths a, b and c in the ZOLZ to a kinematically forbidden reflection are drawn. The two paths a and b are geometrically equivalent with respect to the line m–m perpendicular to the b axis. Since every Umweganregung path to a kinematically forbidden reflection contains an odd number of F(0h2,ih3,ih4,i) with odd h4,i, the following equation is obtained. Fð0h2;1 h3;1 h4;1 ÞFð0h2;2 h3;2 h4;2 Þ . . . Fð0h2;n h3;n h4;n Þ for path a ¼
Fð0h 2;1 h3;1 h4;1 ÞFð0h 2;2 h3;2 h4;2 Þ . . . Fð0h 2;n h3;n h4;n Þ for path b;

ð2:5:3:12Þ

Pn Pn Pn where i¼1 h2;i ¼ 0, i¼1 h3;i ¼ h3 and i¼1 h4;i ¼ h4 (h4 = odd). When reflection 00h3h4 (h4 = odd) is exactly excited, the two paths a and c are symmetric with respect to the bisector m0 –m0 of the diffraction vector of the reflection and have the same excitation error. The waves passing through these paths have the same amplitude but different signs. Thus the following relation is obtained.

Fig. 2.5.3.23. Diffraction pattern of Sr2Nb2O7 taken with [001] incidence at 60 kV. (a) Spot diffraction pattern. Kinematically forbidden 0001 and 2001 incommensurate reflections exhibit definite intensity. (b) Zone-axis CBED pattern showing dynamical absence of 0001 and 2001 incommensurate reflections. (c) CBED pattern taken at an incidence with a small tilt from the zone axis to the b* direction. The kinematically forbidden incommensurate reflections have intensity due to incomplete cancellation of two waves through the Umweganregung paths. The wave number vector of modulation is k = (12
)a*.

Fð0h2;1 h3;1 h4;1 ÞFð0h2;2 h3;2 h4;2 Þ . . . Fð0h2;n h3;n h4;n Þ for path a ¼
Fð0h 2;n h3;n h4;n ÞFð0h 2;n
1 h3;n
1 h4;n
1 Þ . . . Fð0h 2;1 h3;1 h4;1 Þ for path c;

ð2:5:3:13Þ and small spots indicate the fundamental (h4 = 0) and incommensurate (h4 6¼ 0) reflections, respectively. Umweganregung paths a and b to the kinematically forbidden 0001 reflection via a fundamental reflection in the ZOLZ are drawn. Fig. 2.5.3.23(a) shows a spot diffraction pattern of the incommensurate phase of Sr2Nb2O7 taken with the [001] incidence at 60 kV. The incommensurate reflections in which dynamical extinction lines appear at this incidence are those with the indices h1,even00h4,odd because h3 = 0 and h1 + h2 = 2n due to the lattice type C of the average structure. The reflections in the four columns indicated by black arrowheads are incommensurate reflections. The reflections 0001, 0001 , 2001 and 2 001 designated by white arrowheads are kinematically forbidden but exhibit certain intensities, which are caused by multiple diffraction. Other reflections are fundamental reflections due to the average structure.

Pn Pn Pn where i¼1 h2;i ¼ 0, i¼1 h3;i ¼ h3 and i¼1 h4;i ¼ h4 (h4 = odd). Therefore, dynamical extinction occurs in kinematically forbidden reflections of incommensurate crystals. Fig. 2.5.3.21(b) schematically shows the extinction lines in odd-order incommensurate reflections, where the 0011 reflection is exactly excited. We consider the dynamical extinction from Sr2Nb2O7 whose c 1 space group is PC 1mc2  s 1 . The glide plane (s ) is perpendicular to the b axis with a glide vector (c + a4)/2. The wave number vector of the modulation is k = (12
)a*. (Since space-group notation Cmc21 is broadly accepted, the direction of the modulation is taken as the a axis.) The reflections h10h3h4 with h3 + h4 = 2n + 1 (n = integer) are kinematically forbidden. Fig. 2.5.3.22 shows a schematic diffraction pattern of Sr2Nb2O7 at the [001] incidence. The large

351

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.3.24. Three pairs of ZOLZ [(a), (c) and (e)] and HOLZ [(b), (d) and ( f )] CBED patterns taken at 60 kV from an area of Al74Mn20Si6 about 3 nm in diameter and about 10 nm thick (Tanaka, Terauchi, Suzuki et al., 1987). Symmetries are (a) 10mm, (b) 5m, (c) 6mm, (d) 3m, (e) 2mm and ( f ) 2mm.

Fig. 2.5.3.23(b) shows a CBED pattern corresponding to Fig. 2.5.3.23(a), taken from a specimen area 3 nm in diameter. The excitation errors of two Umweganregung paths a and b are the same at this electron incidence. The reflections 0001, 0001 , 2001 and 2 001 indicated by white arrowheads show no intensity. Dynamical extinction does not appear as a line in the present case because the width of the extinction line exceeds the disc size of the reflections. Fig. 2.5.3.23(c) shows a CBED pattern taken at an incidence slightly tilted toward the b* axis from that for Fig. 2.5.3.23(b) or the [001] zone-axis incidence. The excitation errors are no longer the same for the two Umweganregung paths. Thus, it is seen that the kinematically forbidden reflections indicated by white arrowheads have intensities due to incomplete cancellation of waves coming through different paths, which is an additional proof of the dynamical extinction.

fivefold rotational symmetries but with no translational symmetry. Mackay (1982) extended the tiling to three dimensions using acute and obtuse rhombohedra, which also resulted in the acquisition of local fivefold rotational symmetries and in a lack of translational symmetry. The three-dimensional space-filling method was later completed by Ogawa (1985). These studies, however, remained a matter of design or geometrical amusement until Shechtman et al. (1984) discovered an icosahedral symmetry presumably with long-range structural order in an alloy of Al6Mn (nominal composition) using electron diffraction. Since then, the term quasicrystalline order, a new class of structural order with no translational symmetry but long-range structural order, has been coined. Levine & Steinhardt (1984) showed that the quasilattice produces sharp diffraction patterns and succeeded in reproducing almost exactly the diffraction pattern obtained by Shechtman et al. (1984) using the Fourier transform of a quasiperiodic icosahedral lattice. When analysing X-ray and electrondiffraction data for a quasicrystal, the diffraction peaks can be successfully indexed by six independent vectors pointing to the vertices of an icosahedron. It was then found that the icosahedral quasicrystal can be described in terms of a regular crystal in six

2.5.3.5. Symmetry determination of quasicrystals 2.5.3.5.1. Icosahedral quasicrystals Penrose (1974) demonstrated that a two-dimensional plane can be tiled with thin and fat rhombi to give a pattern with local

352

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION Table 2.5.3.14. Diffraction groups and CBED symmetries for two icosahedral point groups Point group

Diffraction group

BP

WP

DP

DP

235

5mR

5m

5

1

1

m2

1

2 = 1R

1

mR (Projection) 5m1R

10mm

5m

mv1R 2mvm2

m3 5

10RmmR

(Projection) 10mm1R

10mm

10mm

5m

10mm

1

1

2R

m2

2Rm2

mv

2Rmv

2

21R

2mvm2

21Rmv

dimensions (e.g. Jaric´, 1988). A quasicrystal is produced by the intersection of the six-dimensional crystal with an embedded three-dimensional hyperplane (the cut-and-projection technique). Addition of several per cent of silicon to Al–Mn alloys caused a great increase in the degree of order of the quasicrystal. Bendersky & Kaufman (1986) prepared such a less-strained quasicrystalline Al71Mn23Si6 alloy and determined its point group. They obtained fairly good zone-axis CBED patterns that showed symmetries of 10mm, 6mm and 2mm in the ZOLZ discs and 5m, 3m and 2mm in HOLZ rings. From these results, they identified the point group to be centrosymmetric m3 5 . Figs. 2.5.3.24(a)–(f ) show three pairs of CBED patterns taken from an area about 100 nm thick and about 3 nm in diameter of an Al74Mn20Si6 quasicrystal at an accelerating voltage of 60 kV (Tanaka, Terauchi & Sekii, 1987). This quasicrystal was found to have much better ordering than Al71Mn23Si6. The fact that Kikuchi bands are clearly seen in the HOLZ patterns and the profiles of the bands are symmetric with respect to their

Fig. 2.5.3.26. CBED patterns of metastable Al70Ni15Fe15 taken from a 3 nm diameter area. (a) Electron incidence along the decagonal axis: symmetry 5m. (b) Electron incidence along direction A indicated in (a): symmetry m perpendicular to the decagonal axis. (c) Electron incidence along direction B indicated in (a): symmetry 2mm. This alloy is found to be noncentrosymmetric.

centre indicates (Figs. 2.5.3.24b, d and f) that the quasicrystal has sufficiently good quality or highly ordered atomic arrangements to perform reliable symmetry determination. Each pair of CBED patterns consists of a ZOLZ pattern and a HOLZ pattern. The former is produced solely by the interaction of ZOLZ reflections, showing distinct symmetries in several discs. The whole pattern of Fig. 2.5.3.24(a), formed by ZOLZ reflections, exhibits a tenfold rotation symmetry and two types of mirror symmetry, the resultant symmetry being expressed as

Fig. 2.5.3.25. CBED patterns of Al74Mn20Si6 taken with an electron incidence along the threefold axis. (a) Zone-axis pattern showing symmetry 3m. (b, c) DP showing translational symmetry or 2R, indicating that the quasicrystal is centrosymmetric.

353

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION disc +G agrees with that of disc
G when the former is superposed on the latter with a translation of
2G. This symmetry 2R directly proves that the quasicrystal is centrosymmetric, again confirming the point group as m3 5. The lattice type was found to be primitive and no dynamical extinction was observed. Thus, the space group of the alloy was determined to be Pm3 5 . Quasicrystals of Al–Mn alloys have been produced by the melt-quenching method and are thermodynamically metastable. Tsai et al. (1987) discovered a stable icosahedral phase in Al65Cu20Fe15. This alloy has larger grains and is much better quality with less phason strain than Al74Mn20Si6. The discovery of this alloy greatly accelerated the studies of icosahedral quasicrystals. It was found that the lattice type of this phase and of some other Al–Cu–TM (TM = transition metal) alloys is different from that of Al–Mn alloys. That is, Al–Cu–TM alloys display many additional spots in diffraction patterns of twofold rotation symmetry. The patterns were indexed either by all (six) even or all (six) odd, or by a face-centred (F) lattice. All the icosahedral quasicrystals known to date belong to the point group m3 5 ; none with the noncentrosymmetric point group 235 have been discovered.

Table 2.5.3.15. Pentagonal and decagonal point groups constructed by analogy with trigonal and hexagonal point groups This table is taken from Saito et al. (1992) with the permission of the Japan Society of Applied Physics.

Pentagonal

Decagonal

—

2.5.3.5.2. Decagonal quasicrystals The first decagonal quasicrystal was found by Bendersky (1985) in an alloy of Al–Mn using the electron-diffraction technique. This phase has periodic order parallel to the tenfold axis, like ordinary crystals, but has quasiperiodic long-range structural order perpendicular to the tenfold axis. The diffraction peaks were indexed by one vector parallel to the tenfold axis and four independent vectors pointing to the vertices of a decagon. Thus, the decagonal quasicrystal is described in terms of a regular crystal in five dimensions. Two space groups, P105/m and P105/mmc, have been proposed for the alloy by Bendersky (1986) and by Yamamoto & Ishihara (1988), respectively. However, owing to the low quality of the specimens, CBED examination of the alloy could not determine whether the point group is 10/m or 10/mmm. Furthermore, identification of the space-group symmetry was not possible because observation of dynamical extinction caused by the screw axis and/or the glide plane was difficult. The Al–M (M = Mn, Fe, Ru, Pt, Pd, . . . ) quasicrystals found at an early stage were thermodynamically metastable. Subsequently, thermodynamically stable decagonal phases were discovered in the ternary alloys Al65Cu15Co20 (Tsai et al., 1989a), Al65Cu20Co15 (He et al., 1988) and Al70Ni15Co15 (Tsai et al., 1989b). However, spacegroup determination was still difficult due to their poor quasicrystallinity. Tsai et al. (1989c) succeeded in producing a metastable but good-quality decagonal quasicrystal of Al70Ni15Fe15. This alloy was found to be the first decagonal quasicrystal that could tolerate symmetry determination using CBED. The space group was determined to be P10m2 by Saito et al. (1992). Fig. 2.5.3.26(a) shows a CBED pattern of Al70Ni15Fe15 taken with an incidence parallel to the fivefold axis (c axis). The pattern clearly exhibits fivefold rotation symmetry and a type of mirror symmetry, the total symmetry being 5m. The slowly varying intensity distribution in the discs indicates that the pattern is formed by the interaction between ZOLZ reflections. Thus, the projection approximation should be applied to the analysis of the pattern. Patterns that were related to Fig. 2.5.3.26(a) by an inversion were observed when the illuminated specimen area was changed, indicating the existence of inversion domains. Table 2.5.3.15 shows possible pentagonal and decagonal point groups, which are constructed by analogy with the trigonal and hexagonal point groups (Saito et al., 1992). It can be seen that the point groups that satisfy the observed symmetry 5m in the projection approximation are 52, 5m and 10m2. Point group 52 is a possibility because the horizontal

—

10mm. The whole pattern of Fig. 2.5.3.24(b), formed by HOLZ reflections, shows a fivefold rotation symmetry and a type of mirror plane, the resultant symmetry being 5m. Figs. 2.5.3.24(c) and (d) show symmetries 6mm and 3m, respectively. Figs. 2.5.3.24(e) and (f ) show symmetry 2mm. There are two icosahedral point groups, 235 and m3 5 (see Table 10.1.4.3 in IT A, 2005). The former is noncentrosymmetric with no mirror symmetry but the latter is centrosymmetric. Table 2.5.3.14 shows the diffraction groups expected from these point groups with the incident beam parallel to the fivefold or tenfold axis, and their symmetries appearing in the WP, BP, DP and DP. Projection diffraction groups and their symmetries, in which only the interaction between ZOLZ reflections is taken into account, are given in the second row of each pair. Diffraction groups obtained for the other incident-beam directions are omitted because they can be seen in Table 2.5.3.3. The whole-pattern symmetries observed for better-quality images of Al74Mn20Si6 have confirmed the result of Bendersky & Kaufman (1986), i.e. the point group m3 5 . Fig. 2.5.3.25(a) shows a zone-axis CBED pattern taken at an electron incidence along the threefold axis. Figs. 2.5.3.25(b) and (c) show DPs taken when tilting the incident beam to excite a low-order strong reflection. The pattern of

354

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.3.27. CBED patterns of metastable Al70Ni20Fe10 taken from a 3 nm diameter area. (a) Electron incidence along the decagonal axis: symmetry 10mm. (b) Electron incidence along direction A indicated in (a): symmetry 2mm. (c) Electron incidence along direction B indicated in (a): symmetry 2mm. (d) Reflections 00l (l = odd) show dynamical extinction lines. This alloy is determined to have the centrosymmetric space group P105/mmc.

twofold rotation axis is equivalent to the vertical mirror plane in the projection approximation. Figs. 2.5.3.26(b) and (c) were taken with beam incidences A and B, respectively, as denoted in Fig. 2.5.3.26(a). Mirror symmetry perpendicular to the c axis is seen in Fig. 2.5.3.26(b) and (c). Since the mirror symmetry requires a twofold rotation axis or a mirror plane perpendicular to the c axis, point groups 52 and 10m2 remain as possibilities. Fig. 2.5.3.26(c) exhibits symmetry 2mm. Mirror symmetry parallel to the c axis requires the existence of a mirror plane parallel to the axis (a twofold rotation axis is not possible because the fivefold rotation axis already exists.). Since the mirror plane does not exist in point group 52 but does exist in 10m2, the point group of the alloy is determined to be 10m2. Examination of the ordinary diffraction patterns of the alloy revealed that the lattice type is primitive with a periodicity of 0.4 nm in the c direction and no dynamical extinction was observed. Thus, the space group of Al70Ni15Fe15 was determined to be P10m2 (Saito et al., 1992) by full use of the potential of CBED. This is the first quasicrystal

with a noncentrosymmetric space group. High-resolution electron-microscope images revealed that the quasicrystal is composed of specific pentagonal atom clusters 2 nm in diameter (Tanaka et al., 1993). Dark-field microscopy revealed the existence of inversion domains with an antiphase shift of c/2, the polarity being perpendicular to the c direction (Tsuda et al., 1993). Quasicrystals of Al70Ni10+xFe20
x (0  x  10) were investigated by CBED and transmission electron microscopy (Tanaka et al., 1993). The change in space group takes place at x = 7.5 upon a sudden decrease of the size of the inversion domains or a rapid mixing of the atom clusters with positive and negative polarities. As a result, the average structure becomes centrosymmetric. A CBED pattern of Al70Ni20Fe10 taken at an incidence along the c axis shows tenfold rotation symmetry (Fig. 2.5.3.27a). CBED patterns taken at incidences A and B (shown in Fig. 2.5.3.27a) exhibit two mirror symmetries parallel and perpendicular to the c axis (Figs. 2.5.3.27b and c). Thus, the point group of this phase is

355

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION determined to be 10/mmm. Fig. 2.5.3.27(d) shows a CBED pattern taken by slightly tilting the incident beam to the c* direction from incidence A. Dynamical extinction lines (arrowheads) are seen in the odd-order reflections along the c* axis. This indicates the existence of a 105 screw axis and a c-glide plane. No other reflection absences were observed, implying the lattice type to be primitive. Therefore, the space group of Al70Ni20Fe10 is determined to be centrosymmetric P105/mmc. It was found that the alloys with 0  x  7.5 belong to the noncentrosymmetric space group P10m2 and those with 7.5 < x  15 belong to the centrosymmetric space group P105/mmc, keeping the specific polar structure of the basic clusters unchanged. Another phase was found in the same alloys with 15 < x  17. This phase showed the same CBED symmetries as the phase with 7.5 < x  15. The space group of the phase was also determined to be P105/mmc. However, high-angle annular dark-field (HAADF) observations of the phase with 15 < x  17 showed that each atom cluster has only one mirror plane of symmetry (Saitoh et al., 1997, 1999). This implies that the structure of the specific cluster is changed from that of the phase with 7.5 < x  15. The clusters are still polar but take ten different orientations, producing centrosymmetric tenfold rotation symmetry on average, which was confirmed by HAADF observations (Saitoh, Tanaka & Tsai, 2001). These three phases have been found for the similar alloys Al– M1–M2, where M1 = Ni and Cu, and M2 = Fe, Co, Rh and Ir (Tanaka et al., 1996). Subsequently, decagonal quasicrystals were found in Al–Pd–Mn, Zn–Mg–RE (RE = Dy, Er, Ho, Lu, Tm and Y) and other alloy systems (Steurer, 2004). There are seven point groups in the decagonal system (Table 2.5.3.15). However, only two point groups, 10m2 and 10/mmm, and two space groups, P10m2 and P105/mmc, are known reliably in real materials to date, though a few other point and space groups have been reported. For further crystallographic aspects of quasicrystals, the reader is referred to the comprehensive reviews of Tsai (2003) and Steurer (2004), and to a review of more theoretical aspects by Yamamoto (1996).

2.5.4. Electron-diffraction structure analysis (EDSA)

Special areas of EDSA application are: determination of unit cells; establishing orientational and other geometrical relationships between related crystalline phases; phase analysis on the basis of dhkl and Ihkl sets; analysis of the distribution of crystallite dimensions in a specimen and inner strains in crystallites as determined from line profiles; investigation of the surface structure of single crystals; structure analysis of crystals, including atomic position determination; precise determination of lattice potential distribution and chemical bonds between atoms; and investigation of crystals of biological origin in combination with electron microscopy (Vainshtein, 1964; Pinsker, 1953; Zvyagin, 1967; Pinsker et al., 1981; Dorset, 1976; Zvyagin et al., 1979). There are different kinds of electron diffraction (ED) depending on the experimental conditions: high-energy (HEED) (above 30–200 kV), low-energy (LEED) (10–600 V), transmission (THEED) and reflection (RHEED). In electron-diffraction studies use is made of special apparatus – electron-diffraction cameras in which the lens system located between the electron source and the specimen forms the primary electron beam, and the diffracted beams reach the detector without aberration distortions. In this case, high-resolution electron diffraction (HRED) is obtained. ED patterns may also be observed in electron microscopes by a selected-area method (SAD). Other types of electron diffraction are: MBD (microbeam), HDD (highdispersion), CBD (convergent-beam), SMBD (scanning-beam) and RMBD (rocking-beam) diffraction (see Sections 2.5.2 and 2.5.3). The recent development of electron diffractometry, based on direct intensity registration and measurement by scanning the diffraction pattern against a fixed detector (scintillator followed by photomultiplier), presents a new improved level of EDSA which provides higher precision and reliability of structural data (Avilov et al., 1999; Tsipursky & Drits, 1977; Zhukhlistov et al., 1997, 1998; Zvyagin et al., 1996). Electron-diffraction studies of the structure of molecules in vapours and gases is a large special field of research (Vilkov et al., 1978). See also Stereochemical Applications of Gas-Phase Electron Diffraction (1988). 2.5.4.2. The geometry of ED patterns ˚ or less. In HEED, the electron wavelength  is about 0.05 A The Ewald sphere with radius 
1 has a very small curvature and is approximated by a plane. The ED patterns are, therefore, considered as plane cross sections of the reciprocal lattice (RL) passing normal to the incident beam through the point 000, to scale L (Fig. 2.5.4.1). The basic formula is

2

By B. K. Vainshtein and B. B. Zvyagin

2.5.4.1. Introduction Electron-diffraction structure analysis (EDSA) (Vainshtein, 1964) based on electron diffraction (Pinsker, 1953) is used for the investigation of the atomic structure of matter together with X-ray and neutron diffraction analysis. The peculiarities of EDSA, as compared with X-ray structure analysis, are defined by a strong interaction of electrons with the substance and by a short wavelength . According to the Schro¨dinger equation (see Section 5.2.2) the electrons are scattered by the electrostatic field of an object. The values of the atomic scattering amplitudes, fe, are three orders higher than those of X-rays, fx, and neutrons, fn. Therefore, a very small quantity of a substance is sufficient to obtain a diffraction pattern. EDSA is used for the investigation of very thin single-crystal films, of ~5–50 nm polycrystalline and textured films, and of deposits of finely grained materials and surface layers of bulk specimens. The structures of many ionic crystals, crystal hydrates and hydro-oxides, various inorganic, organic, semiconducting and metallo-organic compounds, of various minerals, especially layer silicates, and of biological structures have been investigated by means of EDSA; it has also been used in the study of polymers, amorphous solids and liquids.

r ¼ jhjL; or rd ¼ L;

ð2:5:4:1Þ

where r is the distance from the pattern centre to the reflection, h is the reciprocal-space vector, d is the appropriate interplanar distance and L is the specimen-to-screen distance. The deviation of the Ewald sphere from a plane at distance h from the origin of the coordinates is h ¼ h2 =2. Owing to the small values of  and to the rapid decrease of fe depending on ðsin
Þ=, the diffracted beams are concentrated in a small angular interval ( 0:1 rad). Single-crystal ED patterns image one plane of the RL. They can be obtained from thin ideal crystalline plates, mosaic singlecrystal films or, in the RHEED case, from the faces of bulk single crystals. Point ED patterns can be obtained more easily owing to the following factors: the small size of the crystals (increase in the dimension of RL nodes) and mosaicity – the small spread of crystallite orientations in a specimen (tangential tension of the RL nodes). The crystal system, the parameters of the unit cell and the Laue symmetry are determined from point ED patterns; the probable space group is found from extinctions. Point ED patterns may be used for intensity measurements if the kinematic

2

Questions related to this section may be addressed to Dr D. L. Dorset (see list of contributing authors).

356

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.4.2. Triclinic reciprocal lattice. Points: open circles, projection net: black circles.

cos  00 ¼ sin

h

sin

k


cos

h

cos

k

cos ;

ð2:5:4:2cÞ

and by a system of parallel directions Fig. 2.5.4.1. Ewald spheres in reciprocal space. Dotted line: electrons, solid line: X-rays.

ph h þ pk k ¼ l;

approximation holds true or if the contributions of the dynamic and secondary scattering are not too large. The indexing of reflections and the unit-cell determination are carried out according to the formulae relating the RL to the DL (direct lattice) (Vainshtein, 1964; Pinsker, 1953; Zvyagin, 1967). Under electron-diffraction conditions crystals usually show a tendency to lie down on the substrate plane on the most developed face. Let us take this as (001). The vectors a and b are then parallel, while vector c is normal to this plane, and the RL points are considered as being disposed along direct lines parallel to the axis c with constant hk and variable l. The interpretation of the point patterns as respective RL planes is quite simple in the case of orthogonal lattices. If the lattice is triclinic or monoclinic the pattern of the crystal in the position with the face (001) normal to the incident beam does not have to contain hk0 reflections with nonzero h and k because, in general, the planes ab and a b do not coincide. However, the intersection traces of direct lines hk with the plane normal to them (plane ab) always form a net with periods ða sin Þ
1 ; ðb sin Þ
1 and angle  0 ¼ 


ð2:5:4:2aÞ

ð2:5:4:3Þ

from the ab plane. By changing the crystal orientation it is possible to obtain an image of the a b plane containing hk0 reflections, or of other RL planes, with the exception of planes making a small angle with the axis c. In the general case of an arbitrary crystal orientation, the pattern is considered as a plane section of the system of directions hk which makes an angle ’ with the plane ab, intersecting it along a direction [uv]. It is described by two periods along directions 0h, 0k; ða sin  cos


1 h Þ ; ðb sin 

cos


1 kÞ ;

ð2:5:4:4Þ

The angles h ; k are formed by directions 0h, 0k in the plane of the pattern with the plane ab. The coefficients ph ; pk depend on the unit-cell parameters, angle ’ and direction [uv]. These relations are used for the indexing of reflections revealed near the integer positions hkl in the pattern and for unit-cell calculations (Vainshtein, 1964; Zvyagin, 1967; Zvyagin et al., 1979). In RED patterns obtained with an incident beam nearly parallel to the plane ab one can reveal all the RL planes passing through c which become normal to the beam at different azimuthal orientations of the crystal. With the increase of the thickness of crystals (see below, Chapter 5.1) the scattering becomes dynamical and Kikuchi lines and bands appear. Kikuchi ED patterns are used for the estimation of the degree of perfection of the structure of the surface layers of single crystals for specimen orientation in HREM (IT C, 2004, Section 4.3.8). Patterns obtained with a convergent beam contain Kossel lines and are used for determining the symmetry of objects under investigation (see Section 2.5.3). Texture ED patterns are a widely used kind of ED pattern (Pinsker, 1953; Vainshtein, 1964; Zvyagin, 1967). Textured specimens are prepared by substance precipitation on the substrate, from solutions and suspensions, or from gas phase in vacuum. The microcrystals are found to be oriented with a common (developed) face parallel to the substrate, but they have random azimuthal orientations. Correspondingly, the RL also takes random azimuthal orientations, having c as the common axis, i.e. it is a rotational body of the point RL of a single crystal. Thus, the ED patterns from textures bear a resemblance, from the viewpoint of their geometry, to X-ray rotation patterns, but they are less complicated, since they represent a plane cross section of reciprocal space. If the crystallites are oriented by the plane (hkl), then the axis ½hkl is the texture axis. For the sake of simplicity, let us assume that the basic plane is the plane (001) containing the axes a and b, so that the texture axis is ½001, i.e. the axis c. The matrices of appropriate transformations will define a transition to the general case (see IT A, 2005). The RL directions hk ¼ constant, parallel to the texture axis, transform to cylindrical surfaces, the points with hkl ¼ constant are in planes perpendicular to the texture axis, while any ‘tilted’ lines transform to cones or hyperboloids of rotation. Each point hkl transforms to a ring lying on these surfaces. In practice, owing to a certain spread of c axes of single crystals, the rings are blurred into small band sections of a

(Fig. 2.5.4.2). The points hkl along these directions hk are at distances  ¼ ha cos  þ kb cos  þ lc

l ¼ 0; 1; 2; . . . :

ð2:5:4:2bÞ

with an angle  00 between them satisfying the relation

357

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION spherical surface with the centre at the point 000; the oblique cross section of such bands produces reflections in the form of arcs. The main interference curves for texture patterns are ellipses imaging oblique plane cross sections of the cylinders hk (Fig. 2.5.4.3). At the normal electron-beam incidence (tilting angle ’ ¼ 0 ) the ED pattern represents a cross section of cylinders perpendicular to the axis c, i.e. a system of rings. On tilting the specimen to an angle ’ with respect to its normal position (usually ’ ’ 60 ) the patterns image an oblique cross section of the cylindrical RL, and are called oblique-texture (OT) ED patterns. The ellipses ðhk ¼ constantÞ and layer lines ðl ¼ constantÞ for orthogonal lattices are the main characteristic lines of ED patterns along which the reflections are arranged. The shortcoming of oblique-texture ED patterns is the absence of reflections lying inside the cone formed by rotation of the straight line coming from the point 000 at an angle ð90
’Þ around the axis c and, in particular, of reflections 00l. However, at ’ <  60– 70 the set of reflections is usually sufficient for structural determination. For unit-cell determination and reflection indexing the values d (i.e. jhj) are used, and the reflection positions defined by the ellipses hk to which they belong and the values  are considered. The periods a ; b are obtained directly from h100 and h010 values. The period c , if it is normal to the plane a b (  being arbitrary), is calculated as

Fig. 2.5.4.3. Formation of ellipses on an electron-diffraction pattern from an oblique texture.

Obtaining xn ; yn one can calculate cn ¼ ½ðxn aÞ2 þ ðyn bÞ2 þ 2xn yn ab cos 1=2 :



c ¼ =l ¼

ðh2hkl




h2hk0 Þ1=2 =l:

ð2:5:4:5aÞ Since

For oblique-angled lattices d001 ¼ L=q sin ’; 

c ¼

½ðh2l1 þl

þ

h2l1
l




2h2l Þ=21=2 =l:

c ¼ ðc2n þ d2001 Þ1=2 :

ð2:5:4:5bÞ

The ,  values are then defined by the relations

In the general case of oblique-angled lattices the coaxial cylinders hk have radii

cos  ¼ ðxn a cos  þ yn bÞ=c; bhk ¼ ð1= sin Þ½ðh2 =a2 Þ þ ðk2 =b2 Þ
ð2hk cos =abÞ1=2

cos  ¼ ðxn a þ yn b cos Þ=c:

Because of the small particle dimensions in textured specimens, the kinematic approximation is more reliable for OT patterns, enabling a more precise calculation of the structure amplitudes from the intensities of reflections. Polycrystal ED patterns. In this case, the RL is a set of concentric spheres with radii hhkl . The ED pattern, like an X-ray powder pattern, is a set of rings with radii

ð2:5:4:7Þ

In OT patterns the bhk and  values are represented by the lengths of the small axes of the ellipses Bhk ¼ Lbhk and the distances of the reflections hkl from the line of small axes (equatorial line of the pattern) Dhkl ¼ L= sin ’ ¼ hp þ ks þ lq:

rhkl ¼ hhkl L:

ð2:5:4:8Þ

ð2:5:4:12Þ

2.5.4.3. Intensities of diffraction beams The intensities of scattering by a crystal are determined by the scattering amplitudes of atoms in the crystal, given by (see also Section 5.2.1)

Analysis of the Bhk values gives a, b, , while p, s and q are calculated from the Dhkl values. It is essential that the components of the normal projections cn of the axis c on the plane ab measured in the units of a and b are

feabs ðsÞ ¼ 4K

Z

’ðrÞr2

sin sr dr; sr

2me K ¼ 2 ; fe ¼ K
1 feabs ; h

xn ¼ ðc=aÞðcos 
cos  cos Þ= sin2  ¼
p=q; yn ¼ ðc=bÞðcos 
cos  cos Þ= sin2 

ð2:5:4:11Þ

ð2:5:4:6Þ

and it is always possible to use the measured or calculated values bhk in (2.5.4.5a) instead of hhk0, since  ¼ ðh2hkl
b2hk Þ1=2 :

ð2:5:4:10Þ

ð2:5:4:13Þ

ð2:5:4:9Þ where ’ðrÞ is the potential of an atom and s ¼ 4ðsin
Þ=. The absolute values of feabs have the dimensionality of length L. In EDSA it is convenient to use fe without K. The dimensionality of

¼
s=q:

358

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION ˚ 3 the value fe is [potential L3 ]. With the expression of fe in V A tion (2.5.4.20) is usually satisfied for textured and polycrystalline 2
1 ˚ K in (2.5.4.13) is 47.87 V A . specimens. But for mosaic single crystals as well, the kinematic The scattering atomic amplitudes fe ðsÞ differ from the respecapproximation limit is, in view of their real structure, substantive fx ðsÞ X-ray values in the following: while fx ð0Þ ¼ Z (electron tially wider than estimated by (2.5.4.20) for ideal crystals. The shell charge), the atomic amplitude at s ¼ 0 fulfillment of the kinematic law for scattering can be, to a greater or lesser extent, estimated by comparing the decrease of R averaged over definite experimental intensity Ih ½ðsin fe ð0Þ ¼ 4 ’ðrÞr2 dr ð2:5:4:14Þ P
Þ= 2 angular intervals, and sums fobs ½ðsin
Þ= calculated for the same angular intervals. is the ‘full potential’ of the atom. On average, fe ð0Þ ’ Z1=3 , but for For mosaic single-crystal films the integral intensity of reflection is small atomic numbers Z, owing to the peculiarities in the filling of the electron shells, fe ð0Þ exhibits within periods of the periodic 2 table of elements ‘reverse motion’, i.e. they decrease with Z 2 h tdh ’ 2h dh ; ð2:5:4:21Þ Ih ¼ j0 S increasing (Vainshtein, 1952, 1964). At large ðsin
Þ=, fe ’ Z.   The atomic amplitudes and, consequently, the reflection intensities, are recorded, in practice, up to values of (sin
)/ ’ 0.8– ˚
1, i.e. up to dmin ’ 0.4–0.6 A ˚. 1.2 A The structure amplitude hkl of a crystal is determined by the Fourier integral of the unit-cell potential (see Chapter 1.2), R hkl ¼ ’ðrÞ expf2iðr  hÞg dvr ;

for textures 2 tLp Ih ¼ j0 S ’ 2h p=R0 :  2R0 sin ’ 2  h

ð2:5:4:15Þ

ð2:5:4:22Þ



Here j0 is the incident electron-beam density, S is the irradiated specimen area, t is the thickness of the specimen,  is the average angular spread of mosaic blocks, R0 is the horizontal coordinate of the reflection in the diffraction pattern and p is the multiplicity factor. In the case of polycrystalline specimens the local intensity in the maximum of the ring reflection

where  is the unit-cell volume. The potential of the unit cell can be expressed by the potentials of the atoms of which it is composed: ’ðrÞ ¼

P

’at i ðr
ri Þ:

ð2:5:4:16Þ

cell; i

2 2  td pS Ih ¼ j0 S2 h h ’ 2h d2h p  4L

The thermal motion of atoms in a crystal is taken into account by the convolution of the potential of an atom at rest with the probability function wðrÞ describing the thermal motion: ’at ¼ ’at ðrÞ  wðrÞ:

is measured, where S is the measured area of the ring. The transition from kinematic to dynamic scattering occurs at critical thicknesses of crystals when A 1 (2.5.4.20). Mosaic or polycrystalline specimens then result in an uneven contribution of various crystallites to the intensity of the reflections. It is possible to introduce corrections to the experimental structure amplitudes of the first strong reflections most influenced by dynamic scattering by applying in simple cases the two-wave approximation (Blackman, 1939) or by taking into account multibeam theories (Fujimoto, 1959; Cowley, 1981; Avilov et al. 1984; see also Chapter 5.2). The application of kinematic scattering formulae to specimens of thin crystals (5–20 nm) or dynamic corrections to thicker specimens (20–50 nm) permits one to obtain reliability factors between the calculated calc and observed obs structure amplitudes of R = 5–15%, which is sufficient for structural determinations. With the use of electron-diffractometry techniques, reliability factors as small as R ¼ 2–3% have been reached and more detailed data on the distribution of the inner-crystalline potential field have been obtained, characterizing the state and bonds of atoms, including hydrogen (Zhukhlistov et al., 1997, 1998; Avilov et al., 1999). The applicability of kinematics formulae becomes poorer in the case of structures with many heavy atoms for which the atomic amplitudes also contain an imaginary component (Shoemaker & Glauber, 1952). The experimental intensity measurement is made by a photo method or by direct recording (Avilov, 1979). In some cases the amplitudes hkl can be determined from dynamic scattering patterns – the bands of equal thickness from a wedge-shaped crystal (Cowley, 1981), or from rocking curves.

ð2:5:4:17Þ

Accordingly, the atomic temperature factor of the atom in a crystal is feT ½ðsin
Þ= ¼ fe fT ¼ fe ½ðsin
Þ= expf
B½ðsin
Þ=2 g; ð2:5:4:18Þ where the Debye temperature factor is written for the case of isotropic thermal vibrations. Consequently, the structure amplitude is hkl ¼

P

feTi expf2iðhxi þ kyi þ lzi Þg:

ð2:5:4:19Þ

cell; i

This general expression is transformed (see IT I, 1952) according to the space group of a given crystal. To determine the structure amplitudes in EDSA experimentally, one has to use specimens satisfying the kinematic scattering condition, i.e. those consisting of extremely thin crystallites. The limit of the applicability of the kinematic approximation (Blackman, 1939; Vainshtein, 1964) can be estimated from the formula hh i t < 1; A ¼   

ð2:5:4:23Þ

ð2:5:4:20Þ

2.5.4.4. Structure analysis The unit cell is defined on the basis of the geometric theory of electron-diffraction patterns, and the space group from extinc-

where hh i is the averaged absolute value of h (see also Section 2.5.2). Since hh i are proportional to Z0:8, condition (2.5.4.20) is better fulfilled for crystals with light and medium atoms. Condi-

359

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION tions. It is also possible to use the method of converging beams (Section 2.5.3). The structural determination is based on experimental sets of values jhkl j2 or jhkl j (Vainshtein, 1964). The trial-and-error method may be used for the simplest structures. The main method of determination is the construction of the Patterson functions "

PðxyzÞ ¼

from the electron-density Fourier series in X-ray analysis. Owing to the peculiarities in the behaviour of the atomic amplitudes (2.5.4.13), which decrease more rapidly with increasing ðsin
Þ= compared with fx , the peaks of the atomic potential 1 ’at ðrÞ ¼ 2 2

#

hkl¼þ1 X

1 2000 þ 2 jhkl j2 cos 2ðhx þ ky þ lzÞ  hkl¼
1

and their analysis on the basis of heavy-atom methods, superposition methods and so on (see Chapter 2.3). Direct methods are also used (Dorset et al., 1979). Thus the phases of structure factors are calculated and assigned to the observed moduli

’ð0Þ ¼

ð2:5:4:25Þ

1X  expf
2iðhx þ ky þ lzÞg  h hkl

ð2:5:4:26aÞ

1X  expf
2iðhx þ kyÞg: S h hk0

ð2:5:4:26bÞ

or projections ’0 ðxyÞ ¼

The general formulae (2.5.4.26a) and (2.5.4.26b) transform, according to known rules, to the expressions for each space ˚ 3 and group (see IT I, 1952). If hkl are expressed in V A 3 2 ˚ and A ˚ , respectively, then the volume  or the cell area S in A the potential ’ is obtained directly in volts, while the projection ˚ . The amplitudes jhkl j are reduced to of the potential ’0 is in V A an absolute scale either according to a group of strong reflections P

jh jcalc ¼

P

jh jobs

1 22

Z

feT ðsÞ s2 ds  Z0:75 ;

r2 ’ðrÞ ¼
4e½ þ ðrÞ

ðrÞ:

ð2:5:4:27Þ

ð2:5:4:31Þ

ð2:5:4:32Þ

ð2:5:4:33Þ

This makes it possible to interrelate X-ray diffraction, EDSA and neutron-diffraction data. Thus for the atomic amplitudes

or using the Parseval equality X 1 Z 2 jh j ¼  h’ i ¼  feT ðsÞ s2 ds i 2 2 h¼
1 iðcellÞ þ1 X

sin sr 2 s ds sr

while in X-ray diffraction ð0Þ  Z1:2 . In such a way, in EDSA the light atoms are more easily revealed in the presence of heavy atoms than in X-ray diffraction, permitting, in particular, hydrogen atoms to be revealed directly without resorting to difference syntheses as in X-ray diffraction. Typical values of the atomic potential ’ð0Þ (which depend on thermal motion) in organic crystals are: H  35, C  165, O 215 V; in Al crystals 330 V, in Cu crystals 750 V. The EDSA method may be used for crystal structure determination, depending on the types of electron-diffraction patterns, for crystals containing up to several tens of atoms in the unit cell. The accuracy in determination of atomic coordinates in EDSA is ˚ on average. The precision of EDSA makes it about 0.01–0.005 A possible to determine accurately the potential distribution, to investigate atomic ionization, to obtain values for the potential between the atoms and, thereby, to obtain data on the nature of the chemical bond. If the positions in the cell are occupied only partly, then the measurement of ’i ð0Þ gives information on population percentage. There is a relationship between the nuclear distribution, electron density and the potential as given by the Poisson equation

The distribution of the potential in the unit cell, and, thereby, the arrangement in it of atoms (peaks of the potential) are revealed by the construction of three-dimensional Fourier series of the potential (see also Chapter 1.3) ’ðxyzÞ ¼

feT ðsÞ

are more ‘blurred’ and exhibit a larger half-width than the electron-density peaks at ðrÞ. On average, this half-width corresponds to the ‘resolution’ of an electron-diffraction pattern – ˚ or better. The potential in the maximum (‘peak about 0.5 A height’) does not depend as strongly on the atomic number as in X-ray analysis:

ð2:5:4:24Þ

h ¼ jh; obs j expficalc g:

Z

1

2

2

2

fe ðsÞ ¼ 4Ke½Z
fx ðsÞ s
2 ;

ð2:5:4:28Þ

ð2:5:4:34Þ

0

where Z is the nuclear charge and fx the X-ray atomic scattering amplitude, and for structure amplitudes

or Wilson’s statistical method h2 ½ðsin
Þ=i ¼

P i

2 feT ½ðsin
Þ=: i

hkl ¼ Ke½Zhkl
Fhkl jhj
2 ;

ð2:5:4:29Þ

where Fhkl is the X-ray structure amplitude of the electron density of a crystal and Zhkl is the amplitude of scattering from charges of nuclei in the cell taking into account their thermal motion. The values Zhkl can be calculated easily from neutrondiffraction data, since the charges of the nuclei are known and the experiment gives the parameters of their thermal motion. In connection with the development of high-resolution electron-microscopy methods (HREM) it has been found possible to combine the data from direct observations with EDSA methods. However, EDSA permits one to determine the

The term 000 defines the mean inner potential of a crystal, and is calculated from fe ð0Þ [(2.5.4.13), (2.5.4.19)] h’cr i ¼ 000 = ¼

1X fe ð0Þ: 

ð2:5:4:35Þ

ð2:5:4:30Þ

The Fourier series of the potential in EDSA possess some peculiarities (Vainshtein, 1954, 1964) which make them different

360

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION atomic positions to a greater accuracy, since practically the whole ˚ resolution is used and the of reciprocal space with 1.0–0.4 A three-dimensional arrangement of atoms is calculated. At the same time, in electron microscopy, owing to the peculiarities of electron optics and the necessity for an objective aperture, the image of the atoms in a crystal ’0 ðxÞ  AðxÞ is a convolution, with ˚ resolution. the aperture function blurring the image up to 1.5–2 A In practice, in TEM one obtains only the images of the heaviest atoms of an object. However, the possibility of obtaining a direct image of a structure with all the defects in the atomic arrangement is the undoubted merit of TEM.

diffraction – EDSA. This method makes use of information in reciprocal space – observation and measurement of electrondiffraction patterns and calculation from them of a twodimensional projection or three-dimensional structure of an object using the Fourier synthesis. To do this, one has to find the relative phases of the scattered beams. The wavefunction of an electron-microscopic image is written as I

0:

ð2:5:5:2Þ

Here 0 is the incident plane wave. When the wave is transmitted through an object, it interacts with the electrostatic potential ’ðrÞ [rðxyzÞ is the three-dimensional vector in the space of the object]; this process is described by the Schro¨dinger equation (Section 2.5.2.1). As a result, on the exit surface of an object the wave takes the form q 0 ðxÞ where q is the transmission function and x is the two-dimensional vector xðxyÞ. The diffraction of the wave q 0 is described by the two-dimensional Fourier operator:

2.5.5. Image reconstruction3

By B. K. Vainshtein

2.5.5.1. Introduction In many fields of physical measurements, instrumental and informative techniques, including electron microscopy and computational or analogue methods for processing and transforming signals from objects investigated, find a wide application in obtaining the most accurate structural data. The signal may be radiation from an object, or radiation transmitted through the object, or reflected by it, which is transformed and recorded by a detector. The image is the two-dimensional signal IðxyÞ on the observation plane recorded from the whole three-dimensional volume of the object, or from its surface, which provides information on its structure. In an object this information may change owing to transformation of the scattered wave inside an instrument. The real image JðxyÞ is composed of IðxyÞ and noise NðxyÞ from signal disturbances: JðxyÞ ¼ IðxyÞ þ NðxyÞ:

¼ F
1 T F q

R

F q ¼ QðuÞ ¼ qðxÞ exp½2iðxuÞ dx:

ð2:5:5:3Þ

Here, we assume the initial wave amplitude to be equal to unity and the initial phase to be zero, so that q 0 ¼ q, which defines, in this case, the wavefunction in the back focal plane of an objective lens with the reciprocal-space coordinates uðu; vÞ. The function Q is modified in reciprocal space by the lens transfer function TðuÞ. The scattered wave transformation into an image is described by the inverse Fourier operator F
1 TQ. The process of the diffraction F q 0 ¼ Q, as seen from (2.5.5.1), is the same in both TEM and EDSA. Thus, in TEM under the lens actions F
1 TQ the image formation from a diffraction pattern takes place with an account of the phases, but these phases are modified by the objective-lens transfer function. In EDSA, on the other hand, there is no distorting action of the transfer function and the ‘image’ is obtained by computing the operation F
1 Q. The computation of projections, images and Fourier transformation is made by discretization of two-dimensional functions on a two-dimensional network of points – pixels in real space xðxj ; yk Þ and in reciprocal space uðum ; vn Þ.

ð2:5:5:1Þ

Image-reconstruction methods are aimed at obtaining the most accurate information on the structure of the object; they are subdivided into two types (Picture Processing and Digital Filtering, 1975; Rozenfeld, 1969): (a) Image restoration – separation of IðxyÞ from the image by means of compensation of distortions introduced in it by an image-forming system as well as by an account of the available quantitative data reflecting its structure. (b) Image enhancement – maximum exclusion from the observed image JðxyÞ (2.5.5.1) of all its imperfections NðxyÞ from both accidental distortions in objects and various ‘noise’ in signals and detector, and obtaining IðxyÞ as the result. These two methods may be used separately or in combination. The image should be represented in the form convenient for perception and analysis, e.g. in digital form, in lines of equal density, in points of different density, in half-tones or colour form and using, if necessary, a change or reversal of contrast. Reconstructed images may be used for the three-dimensional reconstruction of the spatial structure of an object, e.g. of the density distribution in it (see Section 2.5.6). This section is connected with an application of the methods of image processing in transmission electron microscopy (TEM). In TEM (see Section 2.5.2), the source-emitted electrons are transmitted through an object and, with the aid of a system of lenses, form a two-dimensional image subject to processing. Another possibility for obtaining information on the structure of an object is structural analysis with the aid of electron

2.5.5.2. Thin weak phase objects at optimal defocus The intensity distribution IðxyÞ  j I j2 of an electron wave in the image plane depends not only on the coherent and inelastic scattering, but also on the instrumental functions. The electron wave transmitted through an object interacts with the electrostatic potential ’ðrÞ which is produced by the nuclei charges and the electronic shells of the atoms. The scattering and absorption of electrons depend on the structure and thickness of a specimen, and the atomic numbers of the atoms of which it is composed. If an object with the three-dimensional distribution of potential ’ðrÞ is sufficiently thin, then the interaction of a plane electron wave 0 with it can be described as the interaction with a twodimensional distribution of potential projection ’ðxÞ, Rb ’ðxÞ ¼ ’ðrÞ dz;

ð2:5:5:4Þ

0

where b is the specimen thickness. It should be noted that, unlike the three-dimensional function of potential ’ðrÞ with dimension ½M1=2 L3=2 T
1 , the two-dimensional function of potential projection ’ðxÞ has the potential-length dimension ½M1=2 L1=2 T
1  which, formally, coincides with the charge dimension. The transmission function, in the general case, has the form

3

Questions related to this section may be addressed to Dr P. A. Penczek (see list of contributing authors).

361

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION At high resolution, this method enables one to obtain an image of projections of the atomic structure of crystals and defects in the atomic arrangement – vacancies, replacements by foreign atoms, amorphous structures and so on; at resolution worse than atomic one obtains images of dislocations as continuous lines, inserted phases, inclusions etc. (Cowley, 1981). It is also possible to obtain images of thin biological crystals, individual molecules, biological macromolecules and their associations. Image restoration. In the case just considered (2.5.5.10), the projection of potential ’ðxyÞ, convoluted with the spread function, can be directly observed. In the general case (2.5.5.9), when the aperture becomes larger, the contribution to image formation is made by large values of spatial frequencies U, in which the function sin oscillates, changing its sign. Naturally, this distorts the image just in the region of appropriate high resolution. However, if one knows the form of the function sin (2.5.5.7), the true function ’ðxyÞ can be restored. This could be carried out experimentally if one were to place in the back focal plane of an objective lens a zone plate transmitting only one-sign regions of sin (Hoppe, 1971). In this case, the information on ’ðxyÞ is partly lost, but not distorted. To perform such a filtration in an electron microscope is a rather complicated task. Another method is used (Erickson & Klug, 1971). It consists of a Fourier transformation F
1 of the measured intensity distribution TQ (2.5.5.6) and division of this transform, according to (2.5.5.7a,b), by the phase function sin . This gives

Fig. 2.5.5.1. The function and two components of the Scherzer phase function sin ðUÞ and cos ðUÞ.

qðxÞ ¼ exp½
i’ðxÞ (2.5.2.42), and for weak phase objects the approximation ½’ 1 qðxÞ ¼ 1
i’ðxÞ

ð2:5:5:5Þ

is valid. In the back focal plane of the objective lens the wave has the form

TQ ¼ QðuvÞAðUÞ: sin

ð2:5:5:11aÞ

QðuvÞ  TðUÞ

ð2:5:5:6Þ

Then, the new Fourier transformation F QA yields (in the weakphase-object approximation) the true distribution

T ¼ AðUÞ expði UÞ

ð2:5:5:7aÞ

’ðxyÞ  aðxyÞ:

 ðUÞ ¼ f U 2 þ Cs 3 U 4 ; 2

ð2:5:5:7bÞ

The function sin depending on defocus f should be known to perform this procedure. The transfer function can also be found from an electron micrograph (Thon, 1966). It manifests itself in a circular image intensity modulation of an amorphous substrate or, if the specimen is crystalline, in the ‘noise’ component of the image. The analogue method (optical Fourier transformation for obtaining the image sin ) can be used (optical diffraction, see below); digitization and Fourier transformation can also be applied (Hoppe et al., 1973). The thin crystalline specimen implies that in the back focal objective lens plane the discrete kinematic amplitudes hk are arranged and, by the above method, they are corrected and released from phase distortions introduced by the function sin (see below) (Unwin & Henderson, 1975). For the three-dimensional reconstruction (see Section 2.5.6) it is necessary to have the projections of potential of the specimen tilted at different angles  to the beam direction (normal beam incidence corresponds to  ¼ 0). In this case, the defocus f changes linearly with increase of the distance l of specimen points from the rotation axis f ¼ f0 ð1 þ l sin Þ. Following the above procedure for passing on to reciprocal space and correction of sin , one can find ’ ðxyÞ (Henderson & Unwin, 1975).

where U ¼ ðu2 þ v2 Þ1=2 ; exp½i ðUÞ is the Scherzer phase function (Scherzer, 1949) of an objective lens (Fig. 2.5.5.1), AðUÞ is the aperture function, Cs the spherical aberration coefficient, and f the defocus value [(2.5.2.32)–(2.5.2.35)]. The bright-field image intensity (in object coordinates) is IðxyÞ ¼ j I ðxyÞ  tðxyÞj2 ;

ð2:5:5:8Þ

where t ¼ F
1 ½T. The phase function (2.5.5.7) depends on defocus, and for a weak phase object (Cowley, 1981) IðxyÞ ¼ 1 þ 2’ðxyÞ  sðxyÞ;

ð2:5:5:9Þ

where s ¼ F
1 ½AðUÞ sin , which includes only an imaginary part of function (2.5.5.6). While selecting defocus in such a way that under the Scherzer defocus conditions [(2.5.2.44), (2.5.2.45)] j sin j ’ 1, one could obtain IðxyÞ ¼ 1 þ 2’ðxyÞ  aðxyÞ:

ð2:5:5:11bÞ

2.5.5.3. An account of absorption

ð2:5:5:10Þ

Elastic interaction of an incident wave with a weak phase object is defined on its exit surface by the distribution of potential projection ’ðxyÞ; however, in the general case, the electron scattering amplitude is a complex one (Glauber & Schomaker, 1953). In such a way, the image itself has the phase and amplitude contrast. This may be taken into account if one considers not only the potential projection ’ðxyÞ, but also the ‘imaginary potential’

In this very simple case the image reflects directly the structure of the object – the two-dimensional distribution of the projection of the potential convoluted with the spread function a ¼ F
1 A. In this case, no image restoration is necessary. Contrast reversal may be achieved by a change of defocus.

362

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION ðxyÞ which describes phenomenologically the absorption in thin specimens. Then, instead of (2.5.5.5), the wave on the exit surface of a specimen can be written as qðxyÞ ¼ 1
i’ðxyÞ
ðxyÞ


j ðrÞ ¼

ð2:5:5:12Þ

ð2:5:5:13Þ

ð2:5:5:14Þ

and as the result, instead of (2.5.5.10), IðxyÞ ¼ 1 þ 2’ðxyÞ  F
1 ðsin Þ  aðUÞ
2ðxyÞ  F
1 ðcos Þ  aðUÞ:

ð2:5:5:16Þ

It turns out that only a few (bound and valence Bloch waves) have strong excitation amplitudes. Depending on the thickness of a crystal, only one of these waves or their linear combinations (Kambe, 1982) emerges on the exit surface. An electronmicroscopic image can be interpreted, at certain thicknesses, as an image of one of these waves [with a correction for the transfer function action (2.5.5.6), (2.5.5.7a,b)]; in this case, the identical images repeat with increasing thickness, while, at a certain thickness, the contrast reversal can be observed. Only the first Bloch wave which arises at small thickness, and also repeats with increasing thickness, corresponds to the projection of potential ’ðxyÞ, i.e. the atom projection distribution in a thin crystal layer. An image of other Bloch waves is defined by the function ’ðrÞ, but their maxima or minima do not coincide, in the general case, with the atomic positions and cannot be interpreted as the projection of potential. It is difficult to reconstruct ’ðxyÞ from these images, especially when the crystal is not ideal and contains imperfections. In these cases one resorts to computer modelling of images at different thicknesses and defocus values, and to comparison with an experimentally observed pattern. The imaging can be performed directly in an electron microscope not by a photo plate, but using fast-response detectors with digitized intensity output online. The computer contains the necessary algorithms for Fourier transformation, image calculation, transfer function computing, averaging, and correction for the observed and calculated data. This makes possible the interpretation of the pattern observed directly in experiment (Herrmann et al., 1980).

Usually,  is small, but it can, nevertheless, make a certain contribution to an image. In a sufficiently good linear approximation, it may be assumed that the real part cos of the phase function (2.5.5.7a) affects MðuvÞ, while ðxyÞ, as we know, is under the action of the imaginary part sin . Thus, instead of (2.5.5.6), one can write Qðexp i Þ ¼ ðuÞ
iðuÞ sin
MðuÞ cos ;

CHj expð
2ikHj  rÞ:

H

and in the back focal plane if  ¼ F ’ and M ¼ F  QðuvÞ ¼ ðuvÞ
iðuvÞ
MðuvÞ:

P

ð2:5:5:15Þ

The functions ’ðxyÞ and ðxyÞ can be separated by object imaging using the through-focus series method. In this case, using the Fourier transformation, one passes from the intensity distribution (2.5.5.15) in real space to reciprocal space. Now, at two different defocus values f1 and f2 [(2.5.5.6), (2.5.5.7a,b)] the values ðuÞ and MðuÞ can be found from the two linear equations (2.5.5.14). Using the inverse Fourier transformation, one can pass on again to real space which gives ’ðxÞ and ðxÞ (Schiske, 1968). In practice, it is possible to use several through-focus series and to solve a set of equations by the least-squares method. Another method for processing takes into account the simultaneous presence of noise NðxÞ and transfer function zeros (Kirkland et al., 1980). In this method the space frequencies corresponding to small values of the transfer function modulus are suppressed, while the regions where such a modulus is large are found to be reinforced.

2.5.5.5. Image enhancement The real electron-microscope image is subdivided into two components: JðxyÞ ¼ IðxyÞ þ NðxyÞ:

ð2:5:5:17Þ

The main of these, IðxyÞ, is a two-dimensional image of the ‘ideal’ object obtained in an electron microscope with instrumental functions inherent to it. However, in the process of object imaging and transfer of this information to the detector there are various sources of noise. In an electron microscope, these arise owing to emission-current and accelerating-voltage fluctuations, lens-supplying current (temporal fluctuations), or mechanical instabilities in a device, specimen or detector (spatial shifts). The two-dimensional detector (e.g. a photographic plate) has structural inhomogeneities affecting a response to the signal. In addition, the specimen is also unstable; during preparation or imaging it may change owing to chemical or some other transformations in its structure, thermal effects and so on. Biological specimens scatter electrons very weakly and their natural state is moist, while in the electron-microscope column they are under vacuum conditions. The methods of staining (negative or positive), e.g. of introducing into specimens substances containing heavy atoms, as well as the freeze-etching method, somewhat distort the structure of a specimen. Another source of structure perturbation is radiation damage, which can be eliminated at small radiation doses or by using the cryogenic technique. The structure of stained specimens is affected by stain graininess. We assume that all the deviations Ik ðxyÞ of a specimen image from the ‘ideal’ image Ik ðxyÞ are included in the noise term Nk ðxyÞ. The substrate may also be inhomogeneous. All kinds of perturbations cannot be separated and they appear on an electron microscope image as the full noise content NðxyÞ.

2.5.5.4. Thick crystals When the specimen thickness exceeds a certain critical value ˚ ), the kinematic approximation does not hold true (50–100 A and the scattering is dynamic. This means that on the exit surface of a specimen the wave is not defined as yet by the projection of R potential ’ðxyÞ ¼ ’ðrÞ dz (2.5.5.3), but one has to take into account the interaction of the incident wave 0 and of all the secondary waves arising in the whole volume of a specimen. The dynamic scattering calculation can be made by various methods. One is the multislice (or phase-grating) method based on a recurrent application of formulae (2.5.5.3) for n thin layers zi thick, and successive construction of the transmission functions qi (2.5.5.4), phase functions Qi ¼ F qi, and propagation function pk ¼ ½k=2iz exp½ikðx2 þ y2 Þ=2z (Cowley & Moodie, 1957). Another method – the scattering matrix method – is based on the solution of equations of the dynamic theory (Chapter 5.2). The emerging wave on the exit surface of a crystal is then found to diffract and experience the transfer function action [(2.5.5.6), (2.5.5.7a,b)]. The dynamic scattering in crystals may be interpreted using Bloch waves:

363

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION The image enhancement involves maximum noise suppression NðxyÞ and hence the most accurate separation of a useful signal IðxyÞ from the real image JðxyÞ (2.5.5.1). At the signal/noise ratio I=N ’ 1 such a separation appears to be rather complicated. But in some cases the real image reflects the structure sufficiently well, e.g. during the atomic structure imaging of some crystals ðI=N > 10Þ. In other cases, especially of biological specimen imaging, the noise N distorts substantially the image, (I/N) ~ 5– 10. Here one should use the methods of enhancement. This problem is usually solved by the methods of statistical processing of sets of images Jk ðk ¼ 1; . . . ; nÞ. If one assumes that the informative signal Ik ðxyÞ is always the same, then the noise error NðxyÞ may be reduced. The image enhancement methods are subdivided into two classes: (a) image averaging in real space xy; (b) Fourier analysis and filtration in reciprocal space. These methods can be used separately or in combination. The enhancement can be applied to both the original and the restored images; there are also methods of simultaneous restoration and enhancement. The image can be enhanced by analogue (mainly optical and photographic) methods or by computational methods for processing digitized functions in real and reciprocal space. The cases where the image has translational symmetry, rotational symmetry, and where the image is asymmetric will be considered. Periodic images. An image of the crystal structure with atomic or molecular resolution may be brought to self-alignment by a shift by a and b periods in a structure projection. This can be performed photographically by printing the shifted image on the same photographic paper or, vice versa, by shifting the paper (McLachlan, 1958). The Fourier filtration method for a periodic image Ip with noise N is based on the fact that in Fourier space the components F Ip and F N are separated. Let us carry out the Fourier transformation of the periodic signal Ip with the periods a, b and noise N:

decreased, since it is due, in part, to the noise. When the window w is sufficiently small, Ip in (2.5.5.19) represents the periodic distribution hIi (average over all the unit cells of the projection) included in Ip (2.5.5.18). Nevertheless, some error from noise in an image does exist, since with hk we also introduced into the inverse Fourier transformation the background transform values F
1 Nhk which are within the ‘windows’. This approach is realized by an analogue method [optical diffraction and filtering of electron micrographs in a laser beam (Klug & Berger, 1964)] and can also be carried out by computing. As an example, Fig. 2.5.5.2(b) shows an electron micrograph of the periodic structure of a two-dimensional protein crystal, while Fig. 2.5.5.2(c) represents optical diffraction from this layer. In order to dissect the aperiodic component F N in a diffraction plane, according to the scheme in Fig. 2.5.5.2(a), one places a mask with windows covering reciprocal-lattice points. After such a filtration, only the Ip component makes a contribution during the image formation by means of a lens, while the component F N diffracted by the background is delayed. As a result, an optical pattern of the periodic structure is obtained (Fig. 2.5.5.2d). Optical diffractometry also assists in determining the parameters of a two-dimensional lattice and its symmetry. Using the same method, one can separate the superimposed images of two-dimensional structures with different periodicity and in different orientation, the images of the ‘near’ and ‘far’ sides of tubular periodic structures with monomolecular walls (Klug & DeRosier, 1966; Kiselev et al., 1971), and so on. Computer filtering involves measuring the image optical density Jobs , digitization, and Fourier transformation (Crowther & Amos, 1971). The sampling distance usually corresponds to one-third of the image resolution. When periodic weak phase objects are investigated, the transformation (2.5.5.18) yields the Fourier coefficients. If necessary, we can immediately make corrections in them using the microscope transfer function according to (2.5.5.6), (2.5.5.7a,b) and (2.5.5.11a), and thereby obtain the true kinematic amplitudes hk. The inverse transformation (2.5.5.16) gives a projection of the structure (Unwin & Henderson, 1975; Henderson & Unwin, 1975). Sometimes, an observed image JðxÞ is ‘noised’ by the NðxÞ to a great extent. Then, one may combine data on real and reciprocal space to construct a sufficiently accurate image. In this case, the electron-diffraction pattern is measured and structurefactor moduli from diffraction reflection intensities Ihk; obs are obtained:

F J ¼ F ½Ip ðxyÞ þ NðxyÞ

R ¼ Ip ðxyÞ exp½2iðhx þ kyÞ dx dy þ F N P ¼ hk ðu
uhk Þ þ F N; 

ð2:5:5:18Þ



uhk ¼ ha þ kb : The left part of (2.5.5.18) represents the Fourier coefficients hk distributed discretely with periods a and b in the plane uðuvÞ. This is the two-dimensional reciprocal lattice. The right-hand side of (2.5.5.18) is the Fourier transform F N distributed continuously in the plane. Thus these parts are separated. Let us ‘cut out’ from distribution (2.5.5.18) only hk values using the ‘window’ function wðuvÞ. The window should match each of the real peaks hk which, owing to the finite dimensions of the initial periodic image, are not points, as this is written in an idealized form in (2.5.5.18) with the aid of  functions. In reality, the ‘windows’ may be squares of about a =10, b =10 in size, or a circle. Performing the Fourier transformation of product P (2.5.5.18) without F N, and set of windows wðuÞ ¼ wðuvÞ  h; k ðu
ha
kb Þ, we obtain:

jhk; obs j 

ð2:5:5:20Þ

At the same time, the structure factors hk; calc ¼ jhk; calc j expðihk; calc Þ

ð2:5:5:21Þ

are calculated from the processed structure projection image by means of the Fourier transformation. However, owing to poor image quality we take from these data only the values of phases hk since they are less sensitive to scattering density distortions than the moduli, and construct the Fourier synthesis

P JðxyÞ ¼ F
1 fwðuÞ h; k ðu
uh; k Þg

IðxyÞ ¼

h; k

¼ WðxyÞ  Ip ðxÞ;

pffiffiffiffiffiffiffiffiffiffiffiffi Ihk; obs :

P

jhk; obs j expðihk; calc Þ

hk

 exp½2iðhx þ kyÞ:

ð2:5:5:19Þ

the periodic component without the background, WðxyÞ ¼ F
1 wðuÞ. The zero coefficient 00 in (2.5.5.19) should be

ð2:5:5:22Þ

Here the possibilities of combining various methods open up, e.g. for obtaining the structure-factor moduli from X-ray

364

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.5.2. (a) Diagram of an optical diffractometer. D is the object (an electron micrograph), Mp is the diffraction plane and a mask that transmits only hk, Dp is the plane of the (filtered) image; (b) an electron micrograph of a crystalline layer of the protein phosphorylase b; (c) its optical diffraction pattern (the circles correspond to the windows in the mask that transmits only the hk diffracted beams from the periodic component of the image); (d) the filtered image. Parts (b)–(d) are based on the article by Kiselev et al. (1971).

diffraction, and phases from electron microscopy, and so on (Gurskaya et al., 1971). Images with point symmetry. If a projection of an object (and consequently, the object itself) has a rotational N-fold axis of symmetry, the structure coincides with itself on rotation through the angle 2=N. If the image is rotated through arbitrary angles and is aligned photographically with the initial image, then the best density coincidence will take place at a rotation through  ¼ ðk2=NÞ ðk ¼ 1; . . . ; NÞ which defines N. The pattern averaging over all the rotations will give the enhanced structure image with an ðNÞ1=2 times reduced background (Markham et al., 1963). Rotational filtering can be performed on the basis of the Fourier expansion of an image in polar coordinates over the angles (Crowther & Amos, 1971). Iðr; Þ ¼

þ 1 P

gn ðrÞ expðin’Þ:

The averaging of n images in real space gives

Ienh ¼ ð1=nÞ

n P

Jk ðxyÞ ¼ hIk iðxyÞ þ ð1=nÞ

P

Nk ðxyÞ: ð2:5:5:25Þ

k¼1

The signal/noise ratio on an average image is ðnÞ1=2 times enhanced. The degree of similarity and accuracy of superposition of two images with an account both of translational and angular shifts is estimated by a cross-correlation function4 of two selected images J1 and J2 (Frank, 1975, 1980). R kðx0 Þ ¼ J1  J2 ¼ J1 ðxÞ J2 ðx þ x0 Þ dx ¼ kI1 I2 þ kI1 N2 þ kI2 N1 þ kN1 N2 :

ð2:5:5:23Þ

ð2:5:5:26Þ

n¼
1

The value kðx0 Þ is the measure of image similarity, the x0 coordinate of the maximum indicates the shift of the images relative to each other. The first term of the resultant expression (2.5.5.26) is the cross-correlation function of noise-corrected images being compared, the second and third terms are approximately equal to zero, since the noise does not correlate with the signal; the last term is the autocorrelation function of the noise (Crame´r, 1954; Frank, 1975, 1980). The calculation of a correlation function is performed by means of Fourier transformation on the basis of the convolution theorem, since the Fourier transformation of the product of the Fourier transform of function J1 and the conjugated Fourier transform function J2 gives the cross-correlation function of the initial functions:

The integral over the radius from azimuthal components gn gives their power pn 

Ra

jgn j2 r dr;

ð2:5:5:24Þ

0

where a is the maximum radius of the particle. A set pn forms a spectrum, the least common multiple N of strong peaks defining the N-fold symmetry. The two-dimensional reconstructed image of a particle with rotational symmetry is defined by the synthesis (2.5.5.24) with n ¼ 0; N; 2N; 3N. Asymmetric images. In this case, a set of images is processed by computational or analogue methods. The initial selection of images involves the fulfilment of the maximum similarity condition.

4

At Ij ¼ Ik this is the autocorrelation function, an analogue of the Patterson function used in crystallography.

365

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION k ¼ F
1 ½F J1  F  J2 :

temperature factor. The method can be realized by computing or by optical diffraction.

ð2:5:5:27Þ

The probability density of samples for images has the form

2.5.6. Three-dimensional reconstruction5

By B. K. Vainshtein and P. A. Penczek

1 pffiffiffiffiffiffi ð 2Þn   n Z
1 X 2 ½Jk ðx þ xk Þ
JðxÞ dx :  exp 2 2

pðJ1 J2 . . . Jn Þ ¼

2.5.6.1. The object and its projection In electron microscopy (EM) and single-particle reconstruction, three-dimensional (3D) reconstruction methods are used for studying biological structures; that is, symmetric or asymmetric associations of biomacromolecules (muscles, spherical and rodlike viruses, bacteriophages, individual proteins and ribosomes) (Frank, 2006). The electron microscope is used to obtain parallelbeam two-dimensional (2D) projections ’2 ðx; sÞ (ray transform) of frozen hydrated 3D macromolecules ’3 ðrÞ suspended in random orientations (Fig. 2.5.6.1). The function ’2 ðxs Þ is the 2D projection of the 3D molecular electron distribution ’3 ðrÞ. One can also consider one-dimensional (1D) projections ’1 ðs; sÞ of multidimensional distributions; the set of these projections is called a Radon transform. For 2D distributions, ray and Radon transforms differ only in the notation. For (d > 2)-dimensional distributions the two are different: in a ray transform the integrals are calculated over straight lines and yield (d
1)-dimensional projections, while in Radon transforms the integrals are calculated over (d
1)-dimensional hyperplanes and yield 1D projections. In electron microscopy, Radon transforms are not directly measurable, but can be formed computationally and used in intermediate steps of the 3D reconstruction or in alignment procedures (Radermacher, 1994). Within the linear weak-phase-object approximation of the image-formation process in the microscope [see equation (2.5.2.42) in Section 2.5.2 of this chapter], 2D projections represent line integrals of the potential of the particle under examination convoluted with the point-spread function of the microscope, s, so, using (2.5.2.43),

ð2:5:5:28Þ Here J is the tentative image (as such, a certain ‘best’ image can first be selected, while at the repeated cycle an average image is obtained), Jk ðxÞ is the image investigated,  is the standard deviation of the normal distribution of noises and xk the relative shift of the image. This function is called a likelihood function; it has maxima relative to the parameters JðxÞ, xk ; . The average image and dispersion are n P JðxÞ ¼ ð1=nÞ ½Jk ðx
xk Þ; n P  2 ¼ ð1=nÞ ½Jk ðx
xk Þ
JðxÞ2 :

ð2:5:5:29Þ

This method is called the maximum-likelihood method (Crame´r, 1954; Kosykh et al., 1983). It is convenient to carry out the image alignment, in turn, with respect to translational and angular coordinates. If we start with an angular alignment we first use autocorrelation functions or power spectra, which have the maximum and the symmetry centre at the origin of the coordinates. The angular correlation maximum f ð
0 Þ ¼

R

fk ð


0 Þfe ð
Þ d


ð2:5:5:30Þ

R IðxyÞ ¼ 1 þ 2sðxyÞ  ’ðxyÞ ¼ 1 þ 2sðxyÞ  ’3 ðrÞ dz:

gives the mutual angle of rotation of two images. Then we carry out the translational alignment of rotationally aligned images using the translational correlation function (2.5.5.26) (Langer et al., 1970). In the iteration alignment method, the images are first translationally aligned and then an angular shift is determined in image space in polar coordinates with the centre at the point of the best translational alignment. After the angular alignment the whole procedure may be repeated (Steinkilberg & Schramm, 1980). The average image obtained may have false high-frequency components. They can be excluded by multiplying its Fourier components by some function and suppressing high-space frequencies, for instance by an ‘artificial temperature factor’ expf
Bjuj2 g. For a set of similar images the Fourier filtration method can also be used (Ottensmeyer et al., 1977). To do this, one should prepare from these images an artificial ‘two-dimensional crystal’, i.e. place them in the same orientation at the points of the twodimensional lattice with periods a, b. J¼

n P

Jk ðx
tp Þ; t ¼ p1 a þ p2 b:

ð2:5:6:1Þ Since R ’2 ðxs Þ ¼ ’3 ðrÞ d;

s ? x;

ð2:5:6:2Þ

IðxyÞ ¼ 1 þ 2sðxs Þ  ’2 ðxs Þ;

s ? x:

ð2:5:6:3Þ

we have

If we omit constant terms, we obtain R IðxyÞ ¼ sðxs Þ  ’2 ðxs Þ ¼ sðxs Þ  ’3 ðrÞ d;

s ? x:

ð2:5:6:4Þ

In this section, we will assume that all images were collected using the same defocus setting, so the point-spread function s is constant and does not depend on the projection direction s. Thus, we will concern ourselves with the inversion of the projection problem

ð2:5:5:31Þ

k¼1

’2 ðxs Þ ¼ The processing is then performed according to (2.5.5.18), (2.5.5.19); as a result one obtains hIðxyÞi with reduced background. Some translational and angular errors in the arrangement of the images at the artificial lattice points act as an artificial

5

R

’3 ðrÞ d;

s ? x:

ð2:5:6:5Þ

The original version of Section 2.5.6, written by the late B. K. Vainshtein, is here updated and expanded by P. A. Penczek.

366

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.6.1. A three-dimensional object ’3 and its two-dimensional projection ’2 . The electron beam penetrates the specimen in the direction of the z axis.

The projection direction is defined by a unit vector sð
; Þ and it is formed on the plane x perpendicular to s. The set of various projections ’2 ðxsi Þ ¼ ’2i ðxi Þ may be assigned by a discrete or continuous set of points on a unit sphere jj ¼ 1 (Fig. 2.5.6.2). In Fourier space, the relation between an object and its projection is referred to as the central section theorem: the Fourier transformation of projection ’2 of a 3D object ’3 is the central (i.e., passing though the origin of reciprocal space) 2D plane cross section of a 3D transform perpendicular to the projection vector (Bracewell, 1956; DeRosier & Klug, 1968; Crowther, Amos et al., 1970). In Cartesian coordinates, a 3D Fourier transform is

Fig. 2.5.6.2. The projection sphere and projection ’2 ðxs Þ of ’3 ðrÞ along s onto the plane s ? x. The case s ? Z represents orthoaxial projection. Points indicate an arbitrary distribution of projection directions s.

2.5.6.2. 3D reconstruction in the general case In the general case of the 3D reconstruction of ’3 ðrÞ from projections ’2 ðxs Þ, the projection vector s occupies arbitrary positions on the projection sphere (Fig. 2.5.6.2). First, let us consider the case of a 2D function 2 ðxÞ and its ray transform ’1 ðx; Þ. We introduce an operation of backprojection b, which is stretching along s each 1D projection ’1 ðx Þ (Fig. 2.5.6.3). When the result is integrated over the full angular range of projections ’1 ðx; Þ, we obtain the projection synthesis defined as

F ½’3 ðrÞ ¼ 3 ðu; v; wÞ

¼

RRR

’3 ðx; y; zÞ expf2iðux þ vy þ wzÞg dx dy dz: ð2:5:6:6Þ

The transform of projection ’2 ðx; yÞ along z is F ½’2 ðx; yÞ ¼ 3 ðu; v; 0Þ

¼

RRR

bðx; yÞ ¼

’3 ðx; y; zÞ expf2iðux þ vy þ 0zÞg dx dy dz

RRR ¼ ’ ðx; y; zÞ dz expf2iðux þ vyÞg dx dy RR 3 ¼ ’2 ðx; yÞ expf2iðux þ vyÞg dx dy ¼ 2 ðu; vÞ:

þ y sin ; Þ d :

ð2:5:6:9Þ

However, the backprojection operator is not the inverse of a 2D ray transform, as the resulting image b is blurred by the pointspread function ðx2 þ y2 Þ
1=2 (Vainshtein, 1971):

ð2:5:6:7Þ

us ? s:

’1 ðx cos

0

In the general case of projecting along the vector s, the central section theorem is F ½’2 ðxs Þ ¼ 3 ðus Þ;

R

bðx; yÞ ¼ ðx; yÞ  ðx2 þ y2 Þ
1=2 :

ð2:5:6:8Þ

ð2:5:6:10Þ

By noting that the Fourier transform of ðx2 þ y2 Þ
1=2 is ðu2 þ v2 Þ
1=2 and by using the convolution theorem F ½f  g ¼ F ½f F ½g, we obtain the ‘backprojection-filtering’ inversion formula:

From this theorem it follows that the inversion of the 3D ray transform is possible if there is a continuous set of projections ’s corresponding to the motion of the vector sð
; Þ over any continuous line connecting the opposite points on the unit sphere (Fig. 2.5.6.2) (Orlov’s condition: Orlov, 1976). This result is evidenced by the fact that in this case the cross sections F ½’2 ðxs Þ that are perpendicular to s in Fourier space continuously cover the whole Fourier space, i.e., they yield F ½’3 ðrÞ and thereby determine ’3 ðrÞ ¼ F
1 ½3 ðuÞ. In single-particle reconstruction, imaged objects are randomly and nonuniformly oriented on the substrate at different angles (Frank, 2006) and the distribution of their orientations is beyond our control; therefore, the practical impact of Orlov’s condition is limited. In fact, it is more useful to determine a posteriori, i.e., after the 3D reconstruction of the macromolecule is computed, how well the Fourier space was covered. This can be done by calculating the distribution of the 3D spectral signal-to-noise ratio (Penczek, 2002).

 ðx; yÞ ¼ bðx; yÞ  ðx2 þ y2 Þ1=2 ¼ F
1 jujF ½b ¼ Filtrationjuj ½Backprojectionð’1 Þ:

ð2:5:6:11Þ

The more commonly used ‘filtered-backprojection’ inversion is based on the 2D version of the central section theorem (2.5.6.8): F ½’1 ðx Þ ¼ 2 ðu Þ ¼ 2 ðR;
Þ;

ð2:5:6:12Þ

where F ½ 2  ¼ 2. With this in mind, 2 ðxÞ can be related to its ray transform by evaluating the Fourier transform of 2 in polar coordinates:

367

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.6.3. (a) Formation of a backprojection function; (b) projection synthesis (2.5.6.9) is a superposition of these functions.

2 ðxÞ ¼ ¼ ¼ ¼

R

2 ðuÞ expð
2iuxÞ du  R R1

2 ðR;
Þ expð
2ixsÞjRj dR d


0
1 R R1

F ½’1 ðx Þ expð
2ixsÞjRj dR d


0
1 R
1 

F

Fig. 2.5.6.4. Discretization in two dimensions (d = 2). The assumption is that the mass is located at the centre of the voxel.

the reconstruction problem as an algebraic problem (Section 2.5.6.4).

jRjF ½’1 ðx Þ d


0

¼ Backprojection½FiltrationjRj ð’1 Þ:

2.5.6.3. Discretization and interpolation In digital image processing, space is represented by a multidimensional discrete lattice. (It is sometimes expedient to use cylindrical or spherical coordinates, but these also have to be appropriately discretized.) The 2D projections ’2 ðxÞ are sampled on a Cartesian grid fka : k 2 Zd ; ð
K=2Þ  k  ðK=2Þg, where d is the dimensionality of the grid (d = 2 for projections, d = 3 for the reconstructed object), K 2 Zdþ is the size of the grid and a is the grid spacing (Fig. 2.5.6.4). In single-particle reconstruction, the units of a are usually a˚ngstroms and we also assume that the data were appropriately sampled, i.e., a  ð1=2umax Þ. Thus, the pixel size is less than or equal to the inverse of twice the maximum spatial frequency present in the data. Since the latter is not known in advance, a more practical rule is to select the pixel size at about one third of the expected resolution of the final structure, so that the adverse effects of interpolation are reduced. The input electron microscopy data (projections of the macromolecule) are discretized on a 2D Cartesian grid, but each projection has a particular orientation in polar coordinates. Except for a few cases (projection directions parallel to the axes of the coordinate system of the 3D structure), an interpolation is required to relate measured samples to the voxel (volume pixel) values on the 3D Cartesian grid of the reconstructed structure (Fig. 2.5.6.4). The step of backprojection can be visualized as a set of rays with base ad
1 extended from projections and the ray values being added to the intersected voxels on the grid of the reconstructed structure (Figs. 2.5.6.3 and 2.5.6.4). One can select schemes that aim at approximation of the physical reality of the data collection, for example to weight the contributions by the areas of the voxels intersected by the ray or by the lengths of the lines that transverse the voxel (Huesman et al., 1977). In order to reduce the time of calculations, in electron microscopy one usually assumes that all the mass is located at the centre of the voxel and the additional accuracy is achieved by application of tri- (or bi-)linear interpolation. The exception is the algebraic reconstruction technique with blobs algorithm (Marabini et al., 1998), where the voxels are represented by smooth spherically symmetric volume elements [for example, the Keiser–Bessel function (2.5.6.44)]. In real space, both the projection and backprojection steps can be implemented in two different ways: as voxel driven or as ray driven (Laurette et al., 2000). If we consider a projection, in the

ð2:5:6:13Þ

In three dimensions, the backprojection stretches each 2D projection ’2i ½x; sð
; Þi  along the projection direction sð
; Þi . A 3D synthesis is the integral over the hemisphere (Fig. 2.5.6.2) bðrÞ ¼

R

’2 ðx; !s Þ d!s ¼ ’3 ðrÞ  ðx2 þ y2 þ z2 Þ
1 :

ð2:5:6:14Þ

!

Thus, in three dimensions the image b obtained using the backprojection operator is blurred by the point-spread function 1=ðx2 þ y2 þ z2 Þ (Vainshtein, 1971). It is possible to derive inversion formulae analogous to (2.5.6.11) and (2.5.6.13). The inversion formulae demonstrate that it is possible to invert the ray transform for continuous functions and for a uniform distribution of projections. In electron microscopy, the projections are never distributed uniformly in three dimensions. Indeed, a uniform distribution is not even desirable, as only certain subsets of projection directions are required for the successful inversion of a 3D ray transform, as follows from the central section theorem (2.5.6.8). In practice, we always deal with sampled data and with discrete, random and nonuniform distributions of projection directions. Therefore, the inversion formulae can be considered only as a starting point for the development of the numerical (and practical!) reconstruction algorithms. According to (2.5.6.10) and (2.5.6.14), a simple backprojection step results in reconstruction that corresponds to a convolution of the original function with a point-spread function that depends only on the distribution of projections, but not on the structure itself. Taking into account the linearity of the backprojection operation, one has to conclude that for any practically encountered distribution of projections it should be possible to derive the corresponding point-spread function and then, using either deconvolution or Fourier filtration (with a ‘weighting function’), arrive at a good approximation of the structure. This observation forms the basis of the weighted backprojection algorithm (Section 2.5.6.5). Similarly, the central section theorem gives rise to direct Fourier inversion algorithms (Section 2.5.6.6). Nevertheless, since the data are discrete, the most straightforward methodology is to discretize and approach

368

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION voxel-driven approach the volume is scanned voxel by voxel. The nearest projection bin to the projection of each voxel is found, and the values in this bin and three neighbouring bins are increased by the corresponding voxel value multiplied by the weights calculated using bilinear interpolation. In the ray-driven approach, the volume is scanned along the projection rays. The value of the projection bin is increased by the values in the volume calculated in equidistant steps along the rays using trilinear interpolation. Because voxel- and ray-driven methods apply interpolation to projections or to voxels, respectively, the interpolation artifacts will be different in each case. Therefore, when calculating reconstructions using iterative algorithms that alternate between projection and backprojection steps, it is important to maintain consistency; that is, to use the same method for both steps. In either case, the computational complexity of each method is of the order of K3, although the voxel-driven approach is faster due to the smaller number of neighbouring points used in the interpolation. In the reconstruction methods based on the direct Fourier inversion of the 3D ray transform, the interpolation is performed in Fourier space. Unfortunately, it is difficult to design an accurate and fast interpolation scheme for the discrete Fourier space. Bilinear interpolation introduces local errors and when applied in real space it results in attenuation of high-frequency information. When applied in Fourier space, bilinear interpolation results in errors evenly spread over the whole frequency range, thus resulting in potentially severe nonlocal errors in real space. In order to eliminate this error it would be tempting to use interpolation based on Shannon’s sampling theorem [Shannon, 1949; reprinted in Proc. IEEE, (1998), 86, 447–457], which states that a properly sampled band-limited signal can be fully recovered from its discrete samples. For the signal represented by K3 equispaced Fourier samples 3hkl, the value of the Fourier transform 3 at the arbitrary location (u, v, w) is given by (Crowther, Amos et al., 1970) 3 ðu; v; wÞ ¼

K
1 P K
1 P K
1 P

3hkl wh ðuÞwk ðvÞwl ðwÞ;

independently along each of the three frequency axes (Crowther, DeRosier & Klug, 1970). Thus, in this case the solution to the problem of interpolation in Fourier space becomes a solution to the reconstruction problem. For a more general single-particle reconstruction application a moving Shannon window interpolation has been proposed (Lanzavecchia & Bellon, 1994, 1998). It is based on an attenuated version of the window function and in one dimension has the form 1 ðuÞ ¼

k¼m

K odd;

> > : sinfK½u
ðk=KÞg ; tanf½u
ðk=KÞg

K even:

ð2:5:6:18Þ

2.5.6.4. The algebraic and iterative methods The algebraic methods have been derived based on the observation that when the projection equation (2.5.6.5) is discretized, it forms a set of linear equations. Thus, pixels from all available N projections are placed (in an arbitrary order) in a n vector ’2jk ; n ¼ 1; . . . ; N ! f and the voxels of the 3D object in a vector ’3jkl ! g (in an order derived from the order of f by algebraic relations). Note we left the exact sizes of f and g undetermined, as the major advantage of algebraic methods is that we can include in the reconstruction only pixels located within an arbitrary support in two dimensions and this support can be different for each projection; similarly, the support of the object in three dimensions can be arbitrary. Thus, the number of elements in f and g are at most K2N and K3, respectively, with K2 being the number of pixels within chosen support. In the algebraic formulation the operation of projection is defined by the projection matrix P whose elements p are the interpolation weights. Their values are determined by the interpolation scheme used, but for the bi- and trilinear interpolations 0  p  1. The algebraic version of (2.5.6.5) is

ð2:5:6:15Þ

ð2:5:6:16Þ

In cryo-EM, samples of 3 are given at arbitrary 3D locations, as derived from Fourier transforms of 2D projection data (central section theorem) and one seeks to recover 3hkl on the 3D Cartesian grid. Upon the inverse Fourier transform, it will yield the reconstructed object. The problem can be solved as an overdetermined system of linear equations (Crowther, DeRosier & Klug, 1970). Indeed, if we write 3 and 3hkl as 1D arrays U3 and U3ðhklÞ , respectively (the former has K2  [number of projections] elements, while the latter has K3 elements), and we denote by W the appropriately dimensioned matrix of the interpolants, the least-squares solution is U3ðhklÞ ¼ ðWT WÞ
1 WT U3 :

sinf½u
ðk=KÞg sinfð=2Þ½u
ðk=KÞg

where n is the window size ðn KÞ and A is an integer that is even for K odd and vice versa. In multidimensional cases, a product of ’s from (2.5.6.18) is used. In the case of interpolation between equispaced samples of 2hk, excellent results have been reported for n = 11 (Lanzavecchia et al., 1996). However, the application to the reconstruction problem, i.e., to resampling of the nonuniformly distributed Fourier data onto a 3D Cartesian grid, does not yield satisfactory results. Although general conditions under which interpolation using (2.5.6.18) can be done are known (Clark et al., 1985), they are not met in practice and the results are at best nonexact. In addition, the relatively large window size required (n = 11) results in impractical calculation times.

where (Yuen & Fraser, 1979; Lanzavecchia & Bellon, 1994)

wk ðuÞ ¼

1k

 ðcosfð=2Þ½u
ðk=KÞgÞA ;

h¼0 k¼0 l¼0

8 sinfK½u
ðk=KÞg > > < sinf½u
ðk=KÞg ;

mþn
1 X

f ¼ Pg:

ð2:5:6:19Þ

Matrix P is rectangular and since in single-particle reconstruction the number of projections exceeds the linear size of the object (N  K) the system of equations is overdetermined. It can be solved in a least-squares sense: g~ ¼ ðPT PÞ
1 PT f;

ð2:5:6:20Þ

ð2:5:6:17Þ which yields a unique structure g~ that corresponds to the minimum of jPg
fj2. As in the case of direct inversion in Fourier space (2.5.6.17), the approach (2.5.6.20) is impractical because of the very large size of the projection matrix. Indeed, the size of P is K2N  K3, which for a modest number of projections N = 10 000 and image size K2 = 642 = 4096 yields a matrix of size ~(4  107)  (2  106)! Nevertheless, in some cases

The above method is impractical because of the large size of the matrix W. In some cases, when due to symmetries the projection data are distributed approximately evenly (as in the case of icosahedral structures), the problem can be solved to a good degree of accuracy by performing the interpolation (2.5.6.17)

369

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION the projections of the structure are orthoaxial, the full 3D reconstruction reduces to a series of independent 2D reconstructions, and it becomes possible to solve (2.5.6.19) by using the singular value decomposition (SVD) of the matrix P:

in (2.5.6.23) and by solving the problem for the structure g that in effect will be corrected for the CTF, LðgÞ ¼ ð1
Þ

P

jS P g
f  j2 þ jBgj2 :

ð2:5:6:25Þ



P ¼ URVT ;

ð2:5:6:21Þ Here we assumed that the projection data were grouped according to the defocus settings indexed by  and we introduced a Lagrange multiplier  whose value determines the smoothness of the solution g (Zhu et al., 1997). S represents the space-invariant point-spread function of the microscope for the th defocus group; thus S has a block-Toeplitz structure (Biemond et al., 1990) and S ¼ ST . If we assume that there is no astigmatism (which means that the point-spread function is rotationally invariant), S can be either applied to the projection of the structure or to the 3D structure itself. With these assumptions, (2.5.6.25) can be solved using the iterative scheme (Penczek et al., 1997)

where U and V contain eigenvectors of matrices PPT and PT P, respectively, and R is a diagonal matrix containing the first r nonnegative square roots of the eigenvalues of PPT (Golub & Van Loan, 1996). The solution is given by T g ¼ VR
1 l U f;

ð2:5:6:22Þ

where R
1 l is the inverse of the matrix containing the l  r largest elements of R with the remaining values set to zero. By selecting l appropriately we achieve a measure of regularization. The advantage of the approach is that for a given geometry the decomposition has to be calculated once; thus, the method becomes very efficient if the reconstruction has to be performed repeatedly for the same distribution of projections or if additional symmetries, such as helical, are taken into account (Holmes et al., 2003). In the general 3D case, a least-squares solution can be found using one of the iterative approaches that take advantage of the fact that the projection matrix is sparse. Indeed, if bilinear interpolation is used, a row of P will contain only about 4N nonzero elements (out of the total ~K2N). The main idea is that the matrix P is not explicitly calculated or stored; instead, its elements are calculated repeatedly during iterations as needed. In the simultaneous iterative reconstruction technique (SIRT) we find the minimum of LðgÞ ¼ jPg
fj2

P T T P giþ1 ¼ gi
 ð1
Þ ðS P P S gi Þ
ðST PT f  Þ    þ BT Bgi : ð2:5:6:26Þ The second sum can be precalculated and stored as a 3D volume in the computer memory. Thus, the input projections have to be read only once and are never accessed again during the course of the iterations, which eliminates the need to store them in the memory [as is also the case in (2.5.6.24)]. In addition, the product BTB is the 3D Laplacian, which can be applied efficiently without actually creating the matrix BTB. Finally, because of the large size of the matrix S it is more convenient to apply it in Fourier space and to modify the Fourier transform of the volume instead of using matrix multiplication or real-space convolution. Algorithms (2.5.6.24)–(2.5.6.26) are implemented in the SPIDER package (Frank et al., 1996) and (2.5.6.24) in the SPARX package (Hohn et al., 2007). The algebraic reconstruction technique (ART) predates SIRT; in the context of tomographic reconstructions it was proposed by Gordon and co-workers (Gordon et al., 1970) and later it was recognized as a version of Kaczmarz’s method for iteratively solving (2.5.6.23) (Kaczmarz, 1993). We write (2.5.6.19) as a set of systems of equations, each relating single pixels fn , n ¼ 1; . . . ; NK 2 in projections with the 3D structure,

ð2:5:6:23Þ

by selecting the initial 3D structure g0 (usually set to zero) and by iteratively updating its current approximation giþ1 using the gradient of the objective function (2.5.6.23) rLðgÞ, giþ1 ¼ gi
i PT ðPgi
fÞ ¼ gi
i ðPT Pgi
PT fÞ:

ð2:5:6:24Þ

Setting the relaxation parameter i ¼  ¼ constant yields Richardson’s algorithm (Gilbert, 1972). In single-particle reconstruction applications convergence (i.e., a solution that is not dominated by the noise) is usually reached in ~100 iterations. SIRT is extensively used in single-particle reconstruction (Frank et al., 1996) because it yields superior results over a wide range of experimental conditions (Penczek et al., 1992) and, when presented with angular gaps in the distribution of projections, produces the least disturbing artifacts. Note that strictly speaking, filtered backprojection and direct Fourier inversion algorithms are not applicable to data that do not cover the Fourier space fully. In addition, SIRT offers considerable flexibility in EM applications. First, it is possible to accelerate the convergence by adjusting relaxation parameters: setting i ¼ arg min 0 Lðgi
rLðgi ÞÞ results in a steepest decent algorithm. Secondly, even faster convergence (in ~10 iterations) is achieved by solving (2.5.6.23) using the conjugate-gradient method, but this requires addition of a regularizing term to (2.5.6.23) in order to prevent excessive enhancement of noise. Such a term has the form jBgj2 , where matrix B is a discrete approximation of a Laplacian or higher-order derivatives. Thirdly, it is possible to take into account the underfocus settings of the microscope by including the contrast transfer function (CTF) of the microscope

fn ¼ pTn g;

n ¼ 1; . . . ; NK 2 :

ð2:5:6:27Þ

Note (2.5.6.27) and (2.5.6.19) are equivalent, as f ¼ ½f1 f2 . . . fN T and P ¼ ½p1 p2 . . . pN T . With this notation and a relaxation parameter 0 <  < 2, ART comprises the following steps: (1) set i = 0 and the initial structure g0 ; (2) for n ¼ 1; . . . ; NK 2 , set giNK

2 þn

¼ giNK

2 þn
1


ðpTn giNK

2 þn
1


fn Þðpn =jpn jÞ; ð2:5:6:28Þ

(3) set i = i + 1; go to step (2). Although the mathematical forms of update equations in SIRT (2.5.6.24) and in ART (2.5.6.28) are very similar, there are profound differences between them. In SIRT, all voxels in the structure are corrected simultaneously after projections and backprojections of the current update of the structure are calculated. In ART, the projection/backprojection step (2.5.6.28) involves only correction with respect to an individual pixel in a single projection immediately followed by the update of the structure. This results in a much faster convergence of ART as compared to SIRT. Further acceleration can be achieved by

370

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION selecting the order in which pixels enter the correction step (2) in (2.5.6.28). It was observed that if a pixel is selected such that its projection direction is perpendicular to the projection direction of the previous pixel, the convergence is achieved faster (Hamaker & Solmon, 1978; Herman & Meyer, 1993). Interestingly, a random order works almost equally well (Natterer & Wu¨bbeling, 2001). In single-particle reconstruction, ART has been introduced in the form of ‘ART with blobs’ (Marabini et al., 1998) and is available in the Xmipp package (Sorzano et al., 2004). In this implementation, the reconstruction structure is represented by a linear combination of spherically symmetric, smooth, spatially limited basis functions, such as Kaiser–Bessel window functions (Lewitt, 1990, 1992; Matej & Lewitt, 1996). Introduction of blobs significantly reduces the number of iterations necessary to reach an acceptable solution (Marabini et al., 1998). The major advantage of iterative reconstruction methods is the ability to take advantage of a priori knowledge, i.e., any information about the protein structure that was not initially included in the data processing, and introduce it into the reconstruction process in the form of constraints. Examples of such constraints include similarity to the experimental (measured) data, positivity of the protein mass density (only valid in conjunction with the CTF correction), bounded spatial support etc. Formally, the process of enforcing selected constraints is best described in the framework of the projections onto convex sets (POCS) theory (Sezan & Stark, 1982; Youla & Webb, 1982; Sezan, 1992) introduced into EM by Carazo and co-workers (Carazo & Carrascosa, 1986, 1987; Carazo, 1992).

Fig. 2.5.6.5. Nonuniform distribution of projections. The projection weights for the reconstruction algorithms are chosen such that the backprojection integral becomes approximated by a Riemann sum and are equal to the angular length of an arc 
i (2.5.6.29). In Fourier space, projections of an object with real-space radius D form rectangles with width 1/D. In the exact filter backprojection reconstruction method, the weights are derived based on the amount the overlap between projections (2.5.6.29).

inversion of 3D ray transforms that is based on an intermediate step of converting 2D projection data to 1D projection data, as described in Section 2.5.6.6. In order to arrive at a workable solution, the weighting functions applicable to 2D projections are constructed based on an explicitly or implicitly formulated concept of the ‘local density’ of projections. This concept was introduced by Bracewell (Bracewell & Riddle, 1967), who suggested for a 2D case of nonuniformly distributed projections a heuristic weighting function,

2.5.6.5. Filtered backprojection The method of filtered backprojection (FBP) is based on inversion formulae (2.5.6.11) (in two dimensions) or (2.5.6.14) (in three dimensions). It comprises the following steps: (i) a Fourier transform of each projection is computed; (ii) Fourier transforms of projections are multiplied by filters that account for a particular distribution of projections in Fourier space; (iii) the filtered projections are inversely Fourier transformed; (iv) real-space backprojection of processed projections yields the reconstruction. The method is particularly attractive due to the fact that the reconstruction calculated using simple real-space backprojection can be made efficient if the filter function is easy to compute. In two dimensions with uniformly distributed projections the weighting function cðjRj;
Þ in Fourier space is the ‘ramp function’ jRj [(2.5.6.13)]. In two dimensions with nonuniformly distributed projections, when the analytical form of the distribution of projections is not known, an appropriate approximation to cðjRj;
Þ has to be found. A good choice is to select weights such that the backprojection integral becomes approximated by a Riemann sum (Penczek et al., 1996),

cðRj ;
i Þ ¼ P



l

Rj

exp
constantðj
i

l j mod Þ2

: ð2:5:6:30Þ

The weighting function (2.5.6.30) can be easily extended to three dimensions; however, it has a major disadvantage that for a uniform distribution of projections it does not approximate well the weighting function (2.5.6.29), which we consider optimal. Radermacher et al. (1986) proposed a derivation of a general weighting function using a deconvolution kernel calculated for a given (nonuniform) distribution of projections and, in modification of (2.5.6.14), a finite length of backprojection (Fig. 2.5.6.3). Such a ‘truncated’ backprojection is

1
iþ1

i
1 2 2 
i : ð2:5:6:29Þ ¼ Rj 4

cðjRj;
Þ ¼ jRj dR d
! cðRj ;
i Þ ¼ Rj

b^ i ðrÞ ¼ ’2i ðxs Þ  lðrÞ;

s?x

ð2:5:6:31Þ

with projection directions sð
; Þi and For a given set of angles the weights cðRj ;
i Þ are easily computed (Fig. 2.5.6.5). In three dimensions, the weighting (2.5.6.29) is applicable in a single-axis tilt data-collection geometry, where the 3D reconstruction can be calculated as a series of independent 2D reconstructions. In the general 3D case, the analogue of weighting (2.5.6.29) cannot be used, as the data are given in the form of 2D projections and it is not immediately apparent what fraction of the 3D Fourier volume is occupied by Fourier pixels in projections. However, the analogue of weighting (2.5.6.29) can be used in the inversion of 3D Radon transforms or in a direction

lðrÞ ¼ ðxs Þtðzs Þ;
1
ðD=2Þ  z  ðD=2Þ; tðzs Þ ¼ 0 otherwise;

ð2:5:6:32Þ

where D is the diameter of the object or the length of the backprojection l. By taking the Fourier transform of (2.5.6.31) and using the central section theorem (2.5.6.8), we obtain a 3D Fourier transform of the backprojected projection,

371

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION 3i ðu; v; wÞ ¼ 2i ðus ÞD sincðDws Þ;

ð2:5:6:33Þ

¼ Rmax il

and the 3D reconstruction is obtained by the inverse Fourier transform of the sum of contributions given by (2.5.6.33), F ½b^ ðrÞ ¼

P i

ð2:5:6:34Þ

oil ðRÞ ¼

zs and ws are variables in real and Fourier spaces, respectively, both extending in the direction of the projection direction sð
; Þi . In this analysis, the transfer function of the backprojection algorithm is obtained by setting 2i ðus Þ ¼ 1, that is by finding the response of the algorithm to the input composed of delta functions in real space. This yields the inversion formula for a general weighted backprojection algorithm: ’3 ðrÞ ¼ F ½b^ ðrÞcðu; v; wÞ;

1 : D sincðDw sÞ i

R > umax il

0

cðR;
i Þ ¼

:

ð2:5:6:39Þ

1þ

P

1 l;l6¼i

oil ðRÞ

:

ð2:5:6:40Þ

The weighting function (2.5.6.40) easily extends to three dimensions; however, the calculation of the overlap between central sections in three dimensions (represented by slabs) is more elaborate (Harauz & van Heel, 1986). The method is conceptually simple and computationally efficient. However, (2.5.6.40) does not approximate the correct weighting well for a uniform distribution of projections [i.e., it should yield cðR;
i Þ ¼ R]. This, as can be seen by integrating (2.5.6.39) over the whole angular range, is not the case. The exact filter backprojection reconstruction is implemented in the IMAGIC (van Heel et al., 1996), SPIDER (Frank et al., 1996) and SPARX (Hohn et al., 2007) packages. The 3D reconstruction methods based on filtered backprojection are commonly used in single-particle reconstruction. The reasons are: their versatility, ease of implementation, and – in comparison with iterative methods – good computational efficiency. Unlike in iterative methods, there are no parameters to adjust, although it has been noted that the results depend on the value of the diameter D of the structure in all three weighting functions [(2.5.6.30), (2.5.6.36) and (2.5.6.38)], so the performance of the reconstruction algorithm can be optimized for a particular data-collection geometry by changing the value of D (Paul et al., 2004). However, because computation of the weighting function involves calculation of pairwise distances between projections, the computational complexity is proportional to the square of the number of projections and for large data sets these methods become inefficient. It also has to be noted that the weighting functions (2.5.6.30), (2.5.6.36) and (2.5.6.40) remain approximations of the correct weighting function (2.5.6.29).

ð2:5:6:35Þ

ð2:5:6:36Þ

The general weighting function (2.5.6.36) is consistent with analytical solutions (2.5.6.11) and (2.5.6.14), as it can be shown that by assuming infinite support ðD ! 1Þ and continuous and uniform distribution of projection directions, in two dimensions one obtains cðu; vÞ ¼ ðu2 þ v2 Þ
1=2 (Radermacher, 2000). The derivation of (2.5.6.36) is based on the analysis of continuous functions and its direct application to discrete data results in reconstruction artifacts; therefore, Radermacher (1992) proposed attenuating the sinc functions in (2.5.6.36) by exponent functions with decay depending on the diameter of the structure D, or simply replacing the sinc functions by exponent functions. This, however, reduces the concept of the weighting function corresponding to the deconvolution to the concept of the weighting function representing the ‘local density’ of projections (2.5.6.30). The general weighted backprojection reconstruction is implemented in SPIDER (using exponent-based weighting functions) (Frank et al., 1996), Xmipp (Sorzano et al., 2004) and SPARX (Hohn et al., 2007). Harauz & van Heel (1986) proposed basing the calculation of the density of projections, thus the weighting function, on the overlap of Fourier transforms of projections in Fourier space. Although the concept is general, it can be easily approached in two dimensions. If the diameter of the object is D, the width of a Fourier transform of a projection is 2/D (Fig. 2.5.6.5), as follows from the central section theorem (2.5.6.8). Harauz and van Heel postulated that the weighting should be inversely proportional to the sum of geometrical overlaps between a given central section and the remaining central sections. For a pair of projections il, this overlap is oil ðRÞ ¼ T½DR sinð
i

l Þ;

1
ðR=Rmax 0  R < umax il Þ il

In effect, the weighting function, called by the authors an ‘exact filter’, is

with the weighting function given by cðu; v; wÞ ¼ P

ð2:5:6:38Þ

Thus, the overlap function becomes (

2i ðus ÞD sincðDws Þ:

2 : D sinð
i

l Þ

2.5.6.6. Direct Fourier inversion Direct Fourier methods are based on the central section theorem (2.5.6.8). A set of the 2D Fourier transforms of projections yields an approximation to 3 on a nonuniform 3D grid, and a subsequent numerical 3D inverse Fourier transform gives an approximation to ’3. If the 3D inverse Fourier transform could be realized by means of the 3D inverse fast Fourier transform (FFT), one would have a very fast reconstruction algorithm. Unfortunately, the preprocessing step yields 3 on a nonuniform grid. In effect, the 3D inverse FFT is not applicable and an additional step of recovering samples of 3 on a uniform grid from the available samples on a nonuniform grid is necessary. One possibility is to resample the nonuniformly sampled version of 3 onto a 3D Cartesian grid by some form of interpolation. For example, Grigorieff used a modified trilinear interpolation scheme in the FREALIGN package (Grigorieff, 1998). Simple interpolation methods have been found to give inaccurate results, although more sophisticated interpolation schemes can go a long way to improve the accuracy (Lanza-

ð2:5:6:37Þ

where T represents the triangle function (selected by the authors because it can be calculated efficiently). Also, owing to the Friedel symmetry of central sections, the angles in (2.5.6.37) are restricted such that 0 
i

l  ð=2Þ. In this formulation, the overlap is limited to the maximum frequency,

372

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION oversampling is set to two, although smaller factors can also yield good results. Second, we need a window function F ½w whose support in Fourier space is ‘small’. In order to assure good computational efficiency of the convolution step (2.5.6.41), in GDFR this support was set to six Fourier voxels. In addition, in order to prevent division by zeros in (2.5.6.43), the weighting function w ¼ F
1 ½F ½w must be positive within the support of the reconstructed object. A recommendable window function is the separable, bell-shaped Kaiser–Bessel window (O’Sullivan, 1985; Jackson et al., 1991; Schomberg & Timmer, 1995) (Fig. 2.5.6.6a): 8 d Y I fDs ½1
ðu =s Þ2 1=2 g > 0 v v v < ; 2sv F ½wKB ðuÞ ¼ v¼1 > : 0;

u 2 ½
s; s; u2 = ½
s; s; ð2:5:6:44Þ

where I0 is the zero-order modified Bessel function and  = 1.25 is a parameter. The weighting function associated with F ½wKB  is h   i 2 1=2 d sinh Ds 1
fx =½ðD=2Þ g Y v v : ð2:5:6:45Þ wKB ðxÞ ¼  1=2 Dsv 1
fxv =½ðD=2Þ2 g ¼1 Finally, the gridding weights c are chosen such that in discrete implementation of (2.5.6.41) and (2.5.6.42) we obtain a Riemann sum approximating the respective integral (Schomberg, 2002). The Voronoi diagram (Okabe et al., 2000) of the sampling points provides a good partitioning of sampling space, so the gridding weights c are computed by constructing a Voronoi diagram of the grid points and by choosing the weights as the volumes of the Voronoi cells (Fig. 2.5.6.7a). In cryo-EM, the number of projections, thus the number of sampling points in Fourier space, is extremely large. Thus, although the calculation of the gridding weights via the 3D Voronoi diagram of the nonuniformly spaced grid points for F ½’3  would lead to an accurate direct Fourier method, the method would be very slow and would require excessive computer memory. To circumvent this problem, in GDFR (Penczek et al., 2004) the 2D reverse gridding method is used to compute the Fourier transform of each projection on a 2D polar grid. In this way, F ½’3  is obtained on a 3D spherical grid, where the grid points are located both on centred spheres and on straight lines through the origin. Accordingly, it becomes possible to partition the sampled region into suitable sampling cells via the computation of a 2D Voronoi diagram on a unit sphere, rather than a 3D Voronoi diagram in Euclidean space (Fig. 2.5.6.7b). This significantly reduces the memory requirements and improves the computational efficiency of the method, particularly when a fast Oðn log nÞ algorithm for computing the Voronoi diagram on the unit sphere is employed (Renka, 1997). The gridding weights in GDFR correspond to the weighting functions in filtered backprojection algorithms. Moreover, setting the gridding weights to be equal to angular regions on the unit sphere directly corresponds to the equating of weighting function (2.5.6.29) in 2D filtered backprojection to the length of an arc on a unitary circle. Thus, both methods yield nearly optimum projection density functions. The reversed gridding method is obtained by reversing the sequence of steps (1)–(3) of the gridding method: (1) the input image is divided by the weighting function ’2 =w; (2) the image is padded with zeros and 2D FFT is used to compute F ½’2 =w; (3) gridding is used to compute F ½w  F ½’2 =w on an arbitrary nonuniform grid.

Fig. 2.5.6.6. Kaiser–Bessel window function used in the gridding reconstruction algorithm GDFR. (a) In Fourier space, the window function is effectively zero outside the support of six Fourier pixels. (b) In real space, the zeros of the window function are beyond the radius of the reconstructed object.

vecchia et al., 1993). Unfortunately, the recommended window size makes them impractical for most applications. The most accurate Fourier reconstruction methods are those that employ nonuniform Fourier transforms, particularly the 3D gridding method (O’Sullivan, 1985; Schomberg & Timmer, 1995). The gridding-based direct Fourier reconstruction algorithm (GDFR) (Penczek et al., 2004) was developed specifically for single-particle reconstruction. It comprises three steps: (1) The first step, known as ‘gridding’, involves calculating for the Fourier transform of each projection the convolution P i

F ½w  ðcF ½’2i Þ;

ð2:5:6:41Þ

where c are ‘gridding weights’ designed to compensate for the nonuniform distribution of the grid points and F ½w is an appropriately chosen convolution kernel. After processing all projections, this step yields samples of F ½w  F ½’3  on a Cartesian grid. (2) 3D inverse FFT is used to compute  w’3 ¼ F
1 F ½w  F ½’3 

ð2:5:6:42Þ

on a Cartesian grid. (3) The weights are removed using division: ’3 ¼ w’3 =w:

ð2:5:6:43Þ

The method involves a number of parameters. First, we need to decide the oversampling factor for the padding with zeros of input projections before FFTs are computed. In GDFR, the

373

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.6.7. Partitions of the sampling Fourier space using Voronoi diagrams in direct inversion algorithms. (a) Central section of a 3D Fourier volume with sampling points originating from 2D Fourier transforms of projections. Although 2D projections are sampled on a uniform (Cartesian) grid, the arbitrary rotations of projections in 3D space yields a nonuniform distribution of points in three dimensions. In effect, the 3D reconstruction by direct inversion using 3D FFT is not possible. (b) Voronoi diagram on a sphere in the GDFR algorithm. Using the reverse gridding method, the 2D Fourier transforms of projections are resampled onto 1D central lines using a constant angular step. In 3D Fourier space, they are located on central sections and their angular directions are evenly distributed on grand circles. However, since central sections have nonuniform distributions, the distribution of angular directions (sampling points on the unitary sphere) is also nonuniform and effectively random.

Fig. 2.5.6.8. (a) Plot of correlation coefficients (CCs) calculated between Fourier transforms of the reconstructed structure and the original phantom as a function of the magnitude of the spatial frequency using five reconstruction algorithms. GDFR: gridding direct Fourier reconstruction with Voronoi weights; GD3D: gridding reconstruction with simplified gridding weights; WBP1: general weighted backprojection with exponentbased weighting function; WBP2: exact filter weighted backprojection; SIRT: simultaneous iterative reconstruction algorithm. Noise-free projection data were computed in Fourier space using the reverse gridding method. (b) Rescaled version of the low-frequency range of (a). Note the different scales in (a) and (b). The horizontal axis is scaled in absolute frequency units with 0.5 equal to the Nyquist frequency.

Note the reverse gridding does not require explicit gridding weights, as in the third step they are constant (Penczek et al., 2004). In GDFR, the third step of the reversed gridding results in a set of 1D Fourier central lines F ½’1l  calculated using a constant angular step. Clearly, upon inverse Fourier transform they amount to a Radon transform ’1 ðu; Þ of the projection and, upon repeating the process for all available projections, they yield a Radon transform ’1 ðu; hÞ of a 3D object ’3 (albeit nonuniformly sampled with respect to Eulerian angles). Thus GDFR, in addition to being a method of inverting a 3D ray transform, is also a highly accurate method of inverting a 3D Radon transform. GDFR is implemented in the SPIDER package (Frank et al., 1996). The results of a comparison of selected reconstruction algorithms are shown in Fig. 2.5.6.8. The tests were performed using simulated data with the projections of a phantom 3D structure calculated using the inverse gridding method (Penczek et al., 2004). The GDFR yields a virtually perfect reconstruction that agrees with the phantom over the whole frequency range (with the exception of the highest frequencies, but these cannot be reproduced due to geometrical limitations). GD3D is also a gridding-based reconstruction algorithm, in which the gridding weights are calculated directly from contributions of the

weighting function to the 3D Fourier pixels and are applied to the voxels in the 3D Fourier volume (Jackson et al., 1991). This yields an approximation to the ‘local density’ of contributing projections. The simplified gridding weights in GD3D result in deterioration of the reconstruction in the low- and intermediatefrequency ranges. The general weighted backprojection with exponent-based weighting function (WBP1) performs quite well, although reproduction in the low-frequency range is inferior. Similar artifacts are present in the reconstruction using the exact filter weighted backprojection (WBP2), which, in addition, performs disappointingly at higher frequencies. This relatively poor performance is attributed to the nonoptimal weighting schemes used in both methods. It is also interesting to note that the backprojection step is identical in both algorithms; they only differ by the weighting function. The significant difference in their performance attests to the importance of good weighting schemes for high-quality 3D reconstructions from nonuniformly distributed projections. SIRT yields a reconstruction that in the low- and intermediate-frequency ranges matches in quality the reconstruction obtained with GDFR. However, there is a significant loss of quality at high frequencies. This is due to the

374

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION inferior linear interpolation scheme used in SIRT (this impediment is shared with WBP1 and WBP2). [The dip of the SIRT fidelity curve at a spatial frequency of 0.42 in Fig. 2.5.6.8(a) is due to an inconsistency between the interpolation methods used internally in SIRT to generate intermediate projections of the object to be reconstructed and the Fourier-space-based method used to generate the test data.] In terms of computational efficiency, gridding-based algorithms outperform weighted backprojection algorithms by a small factor while SIRT is approximately ten times slower (depending on the number of iterations used).

be ambiguous and the wrong symmetry can be chosen (Egelman & Stasiak, 1988). Further complications exist when the filament does not have a precisely defined helical symmetry, such as F-actin, which has a random variable twist (Egelman et al., 1982). To address these problems, Egelman developed a real-space refinement method for the reconstruction of helical filaments that is capable of determining the helical symmetry of an unknown structure (Egelman, 2000). In this approach, the Fourier–Bessel reconstruction algorithm is replaced by a general reconstruction algorithm discussed in Section 2.5.6.6 with the real-space projections supplied as input. Thus, the symmetry is not enforced within the reconstruction algorithm; instead, it is determined and imposed subsequently by real-space averaging. The presence of point-group symmetry in the structure means that any projection yields as many symmetry-related (but differently oriented) copies of itself as the number of symmetry operations in the group. For example, for three dimensions, each projection enters the reconstruction process at six different projection directions, while for icosahedral symmetry (I ) the number is sixty! This high multiplicity makes it possible to write a reconstruction program that explicitly takes into account the symmetry and can perform the task much faster than generic algorithms. Such programs are usually part of dedicated software packages that were specifically designed for the determination of icosahedral structures (Fuller et al., 1996; Lawton & Prasad, 1996; Liang et al., 2002). These icosahedral reconstruction programs are heavily indebted to a set of FORTRAN routines written by R. A. Crowther at the Medical Research Council Laboratory of Molecular Biology (MRC LMB) and perform the 3D reconstruction using the Fourier–Bessel transform strategy outlined above [(2.5.6.46)–(2.5.6.49)] (Crowther, Amos et al., 1970; Crowther & Amos, 1972). The general reconstruction methods (algebraic, filtered backprojection, direct Fourier inversion) easily accommodate symmetries in the data. In major single-particle reconstruction packages, reconstruction programs are implemented such that the point-group symmetry is a parameter of the program. The symmetry operation is internally taken into account during calculation of the weighting function and it is also applied to the set of Eulerian angles assigned to each projection so the multiple copies are implicitly created and processed. This approach results in an extended time of calculations, but it is entirely general. In direct Fourier inversion algorithms the numerical inaccuracy of the symmetrization performed in Fourier space will result in nonsymmetric artifacts in real space. Thus, in SPIDER (Frank et al., 1996) an additional real-space symmetrization is performed after the reconstruction is completed. Finally, it has to be noted that although it might be tempting to calculate a 3D reconstruction without enforcement of symmetry and to symmetrize the resulting structure, this approach is incorrect. This fact can be seen from the way weighting functions are constructed: weighting functions calculated with symmetries taken into account are not equal to the weighting function calculated for a unique set of projections and subsequently symmetrized.

2.5.6.7. 3D reconstruction of symmetric objects Many objects imaged by EM have symmetries; the two types that are often met are helical (phage tails, F-actin, microtubules, myosin thick filaments, and bacterial pili and flagella) and pointgroup symmetries (many macromolecular assemblies, virus capsids). If the object has helical symmetry, it is convenient to use cylindrical coordinates and a dedicated reconstruction algorithm; particularly for the initial analysis of the data that has to be done in Fourier space. Diffraction from such structures with c periodicity and scattering density ’ðr; ; zÞ is defined by the Fourier–Bessel transform: Z1 Z2 Zl h  i ðR;
; ZÞ ¼ exp in
þ ’ðr; ; zÞ 2
1 þ1 X

0

0

0

 Jn ð2rRÞ exp½
iðn þ 2zZÞr dr d h  X i ¼ : Gn ðR; ZÞ exp in
þ 2 n

dz ð2:5:6:46Þ

The inverse transform has the form ðr; ; zÞ ¼

PR

gn ðr; ZÞ expðin Þ expð2izZÞ dZ;

ð2:5:6:47Þ

n

so that gn and Gn are the mutual Bessel transforms Gn ðR; ZÞ ¼

R1

gn ðr; ZÞ Jn ð2rRÞ2r dr

ð2:5:6:48Þ

0

and gn ðr; ZÞ ¼

R1 Gn ðR; ZÞ Jn ð2rRÞ2R dR:

ð2:5:6:49Þ

0

Owing to helical symmetry, (2.5.6.48) and (2.5.6.49) contain only those of the Bessel functions that satisfy the selection rule (Cochran et al., 1952) l ¼ mp þ ðnq=NÞ;

ð2:5:6:50Þ

2.5.7. Single-particle reconstruction

By P. A. Penczek

where N, q and p are the helix symmetry parameters, m ¼ 0; 1; 2; . . .. Each layer l is practically determined by the single function Jn with the lowest n; the contributions of other functions are neglected. Thus, the Fourier transform of one projection of a helical structure, with an account of symmetry and phases, gives the 3D transform (2.5.6.49). However, biological specimens tend to be flexible and disordered, and exact helical symmetry is rarely observed. A possible approach to dealing with flexibility is to computationally straighten filaments (Egelman, 1986), but this has the potential for introducing artifacts. Another difficulty with helical analysis is that the indexing of a pattern can

2.5.7.1. Formation of projection images in single-particle reconstruction Cryo-electron microscopy (cryo-EM) in combination with the single-particle approach is a new method of structure determination for large macromolecular assemblies. Currently, resolution ˚ can be reached routinely, although in a in the range 10 to 30 A number of pilot studies it has been possible to obtain structures at ˚ . Theoretically, electron microscopy can yield data 4 to 8 A

375

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION  k l exceeding atomic resolution, but the difficulties in overcoming mi mj i6¼j ¼ 0; k,  l = S, B) and also uncorrelated with the very low signal-to-noise ratio (SNR) and low contrast in the the signal ( di mki ¼ 0; k = S, B). Model (2.5.7.1) is semidata, combined with the adverse effects of the contrast transfer empirical in that, unlike in the standard model, we have two function (CTF) of the microscope, hamper progress in fulfilling contributions to the noise. Although in principle amorphous ice the potential of the technique. However, in recent years, cryo-EM should not be affected by the CTF, so the term mS should be has proven its power in the structure determination of large absorbed into mB, in practice the buffer in which the protein is macromolecular assemblies and machines which are too large purified is not pure water and it is possible to observe CTF effects by imaging frozen buffer alone. Moreover, if a thin support and complex for the more traditional techniques of structural carbon is used, it will be a source of very strong CTF-affected biology, i.e., X-ray crystallography and NMR spectroscopy. noise also included in mB. Single-particle reconstruction is based on the assumption that In Fourier space, (2.5.7.1) is written by taking advantage of the a protein exists in solution in multiple copies of the same basic structure. Unlike in crystallography, no ordering of the structure central section theorem [equation (2.5.6.8) of Section 2.5.6]: the within a crystal grid is required; the enhancement of the SNR is Fourier transform of a projection is extracted as a Fourier plane achieved by bringing projection images of different (but strucuv of a rotated Fourier transform of a 3D object: turally identical) proteins into register and averaging them. This n o is why the technique is sometimes called ‘crystallography without Dn ðuÞ ¼ CTFðu; fn ; qÞEn ðuÞ ½FðTvÞ uz ¼0 þMnS ðuÞ þ MnB ðuÞ: crystals’. Within the linear weak-phase-object approximation of the ð2:5:7:3Þ image formation process in the microscope [see equation (2.5.2.43) in Section 2.5.2], 2D projections represent line integrals The capital letters denote Fourier transforms of objects of the Coulomb potential of the particle under examination appearing in (2.5.7.1) while CTF (a Fourier transform of s) convoluted with the point-spread function of the microscope, s, as depends, among other parameters that are set very accurately introduced in Section 2.5.1. In addition, we have to consider the (such as the accelerating voltage of the microscope), on the translation t of the projection in the plane of micrograph, defocus setting fn and the amplitude contrast ratio 0  q < 1 suppression of high-frequency information by the envelope B S that reflects the presence of the amplitude contrast that is due to function E of the microscope, and two additive noises m and m . the removal of widely scattered electrons [the real term in The first one is a coloured background noise, while the second is (2.5.5.14)]. For the range of frequency considered, q is assumed to attributed to the residual scattering by the solvent or the be constant and the CTF is written in terms of the phase supporting thin layer of carbon, if used, assumed to be white and perturbation function [given by equation (2.5.2.33)] as affected by the transfer function of the microscope in the same way as the imaged protein. In order to have the image formation

model correspond more closely to the physical reality of data CTFðu; f Þ ¼ ½1 þ 2qðq
1Þ
1=2 ð1
qÞ sin½ ðjuj; f Þ  collection, we write equation (2.5.6.4) from Section 2.5.6 such
q cos½ ðjuj; f Þ that the projection operation is always realized in the z direction

 ¼ sin ðjuj; f Þ
arctan½q=ð1
qÞ ; of the coordinate system (corresponding to the direction of propagation of the electron beam), while the molecule is rotated ð2:5:7:4Þ arbitrarily by three Eulerian angles: dn ðxÞ ¼ sn ðxÞ  en ðxÞ  n ¼ 1; . . . ; N:

R

f ðTn rÞ dz þ mSn ðxÞ þ mBn ðxÞ;

where for simplicity we assumed no astigmatism. Finally, the rotationally averaged power spectrum of the observed image, calculated as the expectation value of its squared Fourier intensities (2.5.7.3), is given by

ð2:5:7:1Þ

3

 Pd ðuÞ ¼ CTF2 ðuÞ E2 ðuÞ Pf ðuÞ þ PS ðuÞ þ PB ðuÞ;

Here f 2 Rn represents the three-dimensional (3D) electron 2 density of the imaged macromolecule and d 2 Rn is the nth observed two-dimensional (2D) projection image. The total number of projection images N depends on the structure determination project, and can vary from a few hundred to hundreds of thousands. Further, e is the inverse Fourier transform of the envelope function, x ¼ ½x yT is a vector of coordinates in the plane of projections, r ¼ ½rx ry rz 1T is a vector of coordinates associated with nth macromolecule, T is the 4  4 transformation matrix given by 

R TðR; tÞ ¼ 0

 t ; 1

2 3 x 4 z 5 ¼ Tr; 1

ð2:5:7:5Þ

where u ¼ juj is the modulus of spatial frequency. 2.5.7.2. Structure determination in single-particle reconstruction The goal of single-particle reconstruction is to determine the 3D electron-density map f of a biological macromolecule such that its projections agree in a least-squares sense with a large number of collected 2D electron-microscopy projection images, 2 dn 2 Rn (n = 1, 2, . . . , N), of isolated (single) particles with random and unknown orientations. Thus, we seek a least-squares solution to the problem stated by (2.5.7.1) [or, equivalently, in Fourier space, to (2.5.7.3)]. This is formally written as a nonlinear optimization problem (Yang et al., 2005),

ð2:5:7:2Þ

with t ¼ ½tx ty 1T being the shift vector of translation of the object (and its projection) in the xy plane (translation in z is irrelevant due to the projection operation) and Rð ;
; ’Þ is the 3  3 rotation matrix specified by three Eulerian angles. As in Section 2.5.6, two of the angles define the direction of projection sð
; ’Þ, while the third angle results in rotation of the projection image in the plane of the formed image xy; changing this angle does not provide any additional information about the structure f. Both types of noise are assumed to be mutually uncorrelated and independent between projection images (i.e.,

min

n ;
n ;’n ;txn ;tyn ;f ;fn ;q;...

12

N  P 

Lð

n ;
n ; ’n ; txn ; tyn ; f ; fn ; q; . . .Þ

2 R sn ðxÞ  en ðxÞ  f ðTn rÞ dz
dn ðxÞ :

n¼1

ð2:5:7:6Þ The factor of 12 is included merely for convenience. The objective function in (2.5.7.6) is clearly nonlinear due to the coupling

376

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION or less directly. (vii) Visualization and interpretation of the resulting 3D electron-density map is the last step; it often involves docking of X-ray structures of molecules into EM density maps in order to reveal the arrangement of known molecules within the EM envelope (Fig. 2.5.7.1). As within the weak-phase-object approximation of the image formation in EM the relation between densities in collected images and the 3D electron density of the imaged macromolecule is linear [(2.5.7.1)], all data-processing methods employed in the structure determination project should be linear, so the densities in the cryo-EM 3D model can be interpreted in terms of the electron density of the protein. In the actual single-particle project not all the steps have to be executed in the order outlined above. The technique has proved to be particularly useful in studies of functional complexes of proteins whose base state is known to a certain resolution or even of functional complexes whose atomic (X-ray crystallographic) structure is known. In these cases, steps (iv) and (v) can be omitted and the structure of the functional complex (for examples with ligands bound to it) can be relatively easily determined using the native structure as a starting point for step (vi). In addition to difficulties with obtaining good cryo-EM data, the technique is computationally intensive. The reason is that in order to obtain a sufficient SNR in the 3D structure, processing of hundreds of thousands of EM projection images of the molecule might be necessary. For each, five orientation parameters have to be determined, and this is in addition to determination of the image-formation parameters required for the optimization of correlation searches. In effect, it is not unusual for single-particle projects to consume weeks of the computer time of multiprocessing clusters. This also explains why the knowledge of the base structure simplifies the work to a large degree: when it is known, initial values of the orientation parameters can be easily established, reducing not only the computational time, but also possibilities of errors in the structure-determination process.

Fig. 2.5.7.1. Typical steps performed in a single-particle cryo-EM structure determination project.

between the orientation parameters n ;
n ; ’n ; txn ; tyn (n = 1, 2, . . . , N) and the 3D density f. The parameters in (2.5.7.6) to be determined can be separated into two groups. (1) The orientation parameters n ;
n ; ’n ; txn ; tyn that have to be determined entirely by solving (2.5.7.6) and for which there are no initial guesses, and the structure f itself, for which we may or may not have an initial guess. The number of parameters in this group is very large: n3 + 5m. Note that in single-particle reconstruction, the number of projection data m is far greater than the linear size of the data in pixels, i.e., m  n. (2) Various parameters which we will broadly call the parameters of the image formation model (2.5.7.1)–(2.5.7.4): the defocus settings of the microscope fn, the amplitude contrast ratio q and, if analytical forms of the envelope function E, the power spectrum of the background noise M, or the structure F are adopted, the parameters of these equations. Some of the parameters in the second group are usually known very accurately or can be estimated from micrograph data before one attempts to solve (2.5.7.6) (see Section 2.5.7.4), but they can also be refined during the structure determination process [for the method for correcting the defocus settings, see Mouche et al. (2001)]. Owing to the very large number of parameters in (2.5.7.6) and the nonlinearities present, one almost never attempts to solve the problem directly. Instead, structure determination using the single-particle technique involves several steps. (i) The macromolecular complex is prepared with a purity of at least 90%. (ii) The sample is flash-frozen in liquid ethane. Alternatively, cryonegative stain techniques or traditional negative stain methods can be used. (iii) Pictures of the macromolecular complexes are taken. (iv) Exhaustive analysis of 2D particle images aimed at increasing the SNR of the data and evaluation of the homogeneity of the sample is performed. (v) An initial low-resolution model of the structure is established using either experimental techniques or computational methods. (vi) The initial structure is refined in order to increase the resolution using an enlarged data set. Only in this step does one attempt to minimize (2.5.7.6) more

2.5.7.3. Electron microscopy and data digitization The electron microscope is a phase imaging system; i.e., in order to create contrast in images, they have to be underfocused. Owing to the particular form of the CTF of the microscope [(2.5.7.4)], not only the amplitudes of the image in Fourier space are modified, but information in some ranges of spatial frequencies is set to zero and some phases have reversed sign. Therefore, in order to obtain possibly uniform coverage of Fourier space, the standard practice is to take pictures using different defocus settings and merge them computationally in order to fill gaps in Fourier space. The problem is compounded by the relation between underfocus and the envelope function of the microscope. Far-from-focus images have high contrast, but the envelope function has a relatively steep fall-off limiting the range of useful spatial frequencies. Conversely, close-to-focus images have little contrast, but the envelope function is decreasing, slowly extending useful information to high spatial frequencies. In effect, it is easier to process computationally far-from-focus data and to obtain accurate alignment of particles, but the results have severely limited resolution. Processing of close-to-focus data is challenging and results tend to be less accurate, but there is the potential to obtain high-resolution information. The experimental techniques of initial structure determination (random conical tilt, tomography) require collection of tilt data. This is facilitated by dedicated microscope stages that can be rotated inside the microscope column yielding additional views of the same field. However, collection of high-quality tilt images is difficult. The quality of tilted images tends to be adversely affected by charging and drift effects. Moreover, as the stage is tilted the effective ice thickness increases (inversely proportionally to the cosine of the tilt angle, so at 60 the factor is two) and the contrast of the images decreases correspondingly. Finally,

377

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION the defocus in tilted micrographs varies depending on the position in the field, often forcing users to restrict the particle selection only to regions in the vicinity of the tilt axis. However, tilting establishes geometrical relations between different projections of the same particle, unambiguously allowing for robust determination of an initial 3D model and the handedness of the quaternary structure of the complex. Electron microscope images can be either recorded on the film and subsequently converted to digital format, or they can be recorded using a charge-coupled device (CCD) camera in a digital format directly on a microscope. In either case, it is necessary to select the magnification of the microscope and the eventual pixel size of the digitized data before the data-collection session. High magnification can potentially yield high-resolution data, but at the same time it decreases the yield of particles. Lower magnification values can be used when images are recorded on film, which does not attenuate high spatial frequencies to the same extent as CCD cameras tend to do. The pixel size has to be adjusted according to the expected resolution of the final structure. Although it is tempting to adopt a small pixel size (in the hope of achieving high resolution of the results), in most cases this is counterproductive, as it results in very large computer files that are difficult to handle and in excessively long data-processing times. Theoretically, the optimum pixel size is tied to the maximum frequency present in the data by Shannon’s sampling theorem, which states that no information is lost if the signal is sampled at twice the maximum frequency present in the signal, and no additional information is gained by sampling using higher frequency. Thus, if the expected ˚ , it should be sufficient to use a pixel size (on resolution is 12 A ˚ . In practice, various image-processing the specimen scale) of 6 A operations performed during alignment of the data and 3D reconstruction of the complex significantly lower the range of useful frequencies. This is because in currently available singleparticle reconstruction software packages rather unsophisticated interpolation schemes are employed, which were selected mainly for the speed of calculations. Therefore, it is advisable to oversample the data by a factor of 1.5 or even 3.0. For an expected ˚ this corresponds to pixel sizes of 4 and 2 A ˚, resolution of 12 A respectively. The windowed particles have to be normalized to adjust the image densities to a common framework of reference. The reason for this step is that microscopy conditions are never exactly the same and also within the same micrograph field the background densities can vary by a significant margin due to uneven ice thickness and other factors. A sensible approach to normalization is to assume that the statistical distribution of noise in areas surrounding particles should be the same (Boisset et al., 1993). Hence a large portion of one of the micrographs from the processed set is selected and a reference histogram of its pixel values is generated. Next, assuming a linear transformation of pixel values, the two parameters of this transformation are found in such a way that the histogram of the transformed pixel values surrounding the particle optimally matches the reference histogram using 2 statistics as a discrepancy measure.

The method of averaged overlapping periodograms (Welch, 1967) is commonly used in EM to calculate the power spectrum. It is designed to improve the statistical properties of the estimate by taking advantage of the fact that when K identically distributed independent measurements are averaged, the variance of the average is decreased with respect to the individual variance by the ratio 1/K. Thus, instead of calculating a periodogram (squared moduli of the discrete Fourier transform) of the entire micrograph field, one subdivides it into much smaller windows, calculates their periodograms and averages them. Typically, one would chose a window size of 512  512 pixels and an overlap of 50%, which will result in the reduction of the variance of the estimate to few percent with respect to the variance of the periodogram of the entire field (Fernandez et al., 1997; Zhu et al., 1997). Further reduction of the variance is achieved by rotational averaging of the 2D power-spectrum estimate. The resulting onedimensional (1D) profile is finally used in the third step of our procedure. For a set of micrographs the power spectra can be evaluated either visually or computationally in an automated fashion. Of main concern are the presence of Thon rings, the astigmatism and the extent to which Thon rings can be detected. Although in principle astigmatic data could be used in subsequent analysis (in fact, astigmatism could be considered advantageous, as particles from the same micrograph would contain complementary information in Fourier space), in practice they are discarded as currently there is no software that can process astigmatic data efficiently. The extent of Thon rings indicates the ‘resolution’ of the data, i.e., the maximum frequency to which information in the data can be present. A number of well established programs can assist the user in the calculation of power spectra and automated estimation of defocus and astigmatism (Huang et al., 2003; Mindell & Grigorieff, 2003; Sander et al., 2003; Mallick et al., 2005). Given the analytical form of the CTF [(2.5.7.4)], the problem is solved by a robust fitting of the CTF parameters such that the analytical form of the CTF matches the power spectrum of the micrograph. Usually, the steps employed are: (1) robust estimation of the power spectrum; (2) calculation of the rotational average of the power spectrum; (3) subtraction from this rotational average of the slowly decreasing background [roughly corresponding to PB in (2.5.7.5)]; (4) fitting of the defocus value fn using known settings of the microscope (voltage, spherical aberration constant, . . . ) and usually assuming a constant and known value of the amplitude contrast ratio q (for cryo-EM data, q should be in the range 0.02–0.10); and (5) using the established defocus value fn, analysis of the 2D power spectrum and fitting of the astigmatism amplitude and angle while refining the defocus. As long as the defocus value is not too small and there are at least two detectable zeros of the CTF, all available programs give very good and comparable results. In some single-particle packages, the automated calculation of defocus is integrated with the estimation of additional characteristics of the image-formation parameters that are required for advanced application of a Wiener filter [(2.5.7.18)] (Saad et al., 2001; Huang et al., 2003), i.e., the power spectra of two noise distributions PS and PB and the envelope function of the microscope En for each micrograph. A possible approach is to select slowly varying functions and fit their parameters to match the estimates of PS, PB and Pd obtained from the data. Finally, it is necessary to have a description of the 1D rotationally averaged power spectrum of the complex Pf . One possibility is to carry out X-ray solution scattering experiments (Gabashvili et al., 2000; Saad et al., 2001) that yield a 1D power spectrum of the complex in solution. However, these experiments require large amounts of purified sample and the accuracy of the results in terms of the overall fall-off of the power spectrum can be disputed. For the purpose of cryo-EM, a simple approximation of the protein power spectrum by analytical functions is satisfactory.

2.5.7.4. Assessment of the data quality and estimation of the image formation parameters The initial assessment of the quality of the micrographs is usually performed during the data collection and in most cases before the micrographs are digitized. The micrographs are examined visually and those that have noticeable drift, astigmatisms, noticeable contamination or simply too low a number of particles to justify further analysis are simply discarded. After digitization of the accepted micrographs, the first step is estimation of the power spectrum, which will be examined for the presence of Thon rings (thus confirming that the micrograph is indeed usable) and astigmatism.

378

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION 2.5.7.5. 2D data analysis – particle picking

straightforward cross-correlation with a generic shape (a Gaussian function, a low-passed circle) (Frank & Wagenknecht, 1984) to matched filters with large number of templates and parameters derived from the image-formation model of the micrograph (CTF and envelope functions) (Huang & Penczek, 2004; Sigworth, 2004). The motivation is clear: given an ideal object and imageformation parameters, it should be only a matter of sheer computer power and user’s patience to have all particles matching the template selected. In practice, the problem is much more challenging and the success rate of template-based methods does not necessarily exceed the success rate of carefully tuned ad hoc methods. One of the difficulties with the application of correlation techniques to the particle-picking problem is the unevenness of micrographs, which is caused by uneven illumination by the electron beam and, to a much larger degree, by the uneven thickness of the ice layer and, when used, the supporting carbon. A possible remedy is to calculate a ‘locally normalized’ crosscorrelation function, in which the total variance of the micrograph is replaced by the local variance of the micrograph calculated within a window of n pixels centred on the current location l. This method has a fast implementation in Fourier space (van Heel, 1982; Roseman, 2003). A faster method is to just apply a high-pass filtration of the micrograph using a high-pass Gaussian ˚
1, where p is the pixel Fourier filter with a half-width (1/np) A size. This simple step will all but eliminate the unevenness of the micrograph background. The main difficulty with the correlation technique is the computational complexity of the problem arising from the very large number of templates that have to be considered. The particles in the micrograph are projections of a 3D object with arbitrary in-plane rotations. In effect, to perform an exhaustive search, it is necessary to sample quasi-uniformly three Eulerian angles [equation (2.5.7.17) with  ¼ 
]. For example, a very crude angular step of 
¼ 10 results in ~13 000 2D templates! A reduction in the number of templates can be achieved either using clustering techniques (Huang & Penczek, 2004; Wong et al., 2004) or by exploring the eigenstructure of the whole set of templates (Sigworth, 2004).

Depending on the properties of the imaged complex and the magnification used, a single micrograph can yield from a few to thousands of individual particle projections. The first step of the data processing is identification of particle projections in micrographs and their selection. The particles have to be windowed (boxed) using a window size exceeding the particle size by a 30– 50% margin. Thus, for example, in order to determine the ˚ to structure of a 550 kDa complex that has a diameter of ~120 A ˚ ˚ 12 A resolution, it is appropriate to choose a pixel size of 3 A and a window size of 60 pixels. The selection of particles is a labour-intensive process; however, the quality of selected particle projections is a major factor in the subsequent steps of analysis and the inclusion of too many imperfect images may preclude successful determination of the 3D structure. There are three possible approaches: (1) manual selection; (2) semi-automated selection; (3) fully automated selection. In the early stages of analysis, particularly when little is known about the shape of the protein and the distribution of projection views, the manual approach is preferable. The researcher displays the micrograph on a computer screen (usually preprocessed by Fourier filtration and contrast-adjusted for better visibility of the protein) and interactively identifies locations of particle views. A trained and careful operator can yield much better results than automated approaches. The main risk is in inherent bias of a human operator – there is a tendency to focus on more familiar and more easily visible particle projections, omitting less frequently appearing orientations and in effect jeopardizing successful structure determination. In semiautomated approaches, an initial step in which putative particle projections in a micrograph are chosen is performed by a computer, all candidates are windowed and the user screens a gallery of possible particles instead of the full micrograph. Algorithms that perform the initial identification of particle views range from very simple (for example a band-pass filtration of a micrograph with subsequent selection of peaks that are no closer to each other than half of the expected particle size) to sophisticated nonlinear noise-suppression methods [for details on various algorithms see the Special Issue of the Journal of Structural Biology (Zhu et al., 2004)]. Since the human operator will be responsible for the ultimate decision, preference is given to the faster method. In most cases, semi-automated methods are implemented within a framework of a user-friendly graphical user interface that can greatly facilitate the work. Fully automated methods are currently actively under development but, curiously, even for proteins whose high-resolution structure is known, the success rate cannot match that of a human operator (Zhu et al., 2004). The automated procedures can be divided into three groups: (1) those that rely on ad hoc steps of denoising and contrast enhancement followed by the search for regions of known size that emerge above the background level (Adiga et al., 2004); (2) those that extract orientation-independent statistical features from regions of the micrograph that may contain particles and proceed with classification (Lata et al., 1995; Hall & Patwardhan, 2004); and (3) those that employ templates, i.e., either class averages of particles selected from micrographs or projections of a known 3D structure of the complex (Huang & Penczek, 2004; Sigworth, 2004). The advantage of the first two approaches is that they do not require template images, i.e., since they are based on a very broadly defined description of particles (general size, shape or abstract features derived from examples of typical particles), they are applicable in cases when no 3D structure of the complex is available. Methods from the second category usually require a training session for the algorithm to construct a set of weights for the predefined features. The methods that take advantage of the availability of templates vary greatly in complexity from

2.5.7.6. 2D alignment of EM images Alignment of pairs of 2D images is a fundamental step in single-particle reconstruction. It is aimed at bringing into register various particle projections by determining three orientation parameters (rotation angles and x and y translations) and is employed in 2D alignment of large sets of 2D noisy data and in 3D structure-refinement algorithms. The computational efficiency and numerical accuracy of this step are deciding factors in achieving high-quality structural results in an acceptable time. All 2D alignment methods considered are aimed at finding transformation parameters such that the least-squares discrepancy between two images f and g is minimized, R f ðxÞ
gðTxÞ 2 dx ! min;

ð2:5:7:7Þ

where x ¼ ½x y 1T is a vector containing the coordinates. T is the transformation matrix given by 2

cos 
sin  Tð; x; yÞ ¼ 4 sin  cos  0 0

3 tx ty 5; 1

ð2:5:7:8Þ

and is dependent on three transformation parameters: rotation angle  and two translations tx and ty . It has to be noted that a minimum of (2.5.7.7) can be found rapidly using the fast Fourier

379

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION transform (FFT) algorithm if only the xy translation is sought (2D FFT), or if only the rotation angle is needed (1D FFT). 2D alignment methods can be divided into three classes: (1) those that employ exhaustive searches in order to find three orientation parameters; (2) those that perform exhaustive searches by using either simplifications (separate searches for translation and rotation parameters) (Penczek et al., 1992) or by taking advantage of invariant image representations (Schatz & van Heel, 1990; Frank et al., 1992 and the following discussion; Schatz & van Heel, 1992; Marabini & Carazo, 1996); or finally (3) those that are aimed at improvement of previously determined parameters and employ local searches. In practice, as the windowed particles are approximately centred, the search for translation parameters can be restricted to relatively small values. A very efficient algorithm that takes advantage of the geometry is based on resampling to polar coordinates of the area of the image that roughly corresponds to the particle size. The resampling is done around centres placed on pixels located within a distance from the image centre that corresponds to a preset maximum translation (Joyeux & Penczek, 2002) (Fig. 2.5.7.2). For each translation, a 1D rotational crosscorrelation function in polar coordinates is calculated. Overall, the alignment method based on resampling to polar coordinates comprises the following steps: (1) the image is resampled to polar coordinates; (2) 1D FFTs of various lengths are calculated, appropriately weighted and padded with zeros to equalize their lengths; (3) complex multiplications with 1D Fourier transforms of the similarly processed referenced image are calculated; (4) the inverse 1D FFT is calculated and the position of the maximum is found. The last step yields the rotation angle. Steps (1)–(4) are repeated with the image that is being aligned shifted to account for translations. In addition, the rotation angle for one of the images being mirrored is efficiently calculated in parallel with step (3) by repeating the multiplication with the 1D Fourier transforms of the reference image complex conjugated. This additional check is a necessity in the analysis of single-particle data sets, as usually one can expect on average half of the images to be mirrored versions of the other half in the data set. Overall, the method is very accurate, because only data under the circular mask enter the calculation. For a set of N images containing the same object in various orientations and corrupted by an additive noise, the problem of alignment would be relatively simple. For proteins that have strong preferred orientation and particularly when a staining technique is used for grid preparation, this is certainly the case. In the procedure called reference-based alignment, one of the images that appears ‘typical’ is selected and used as a reference to align the remaining images. After all available images are aligned their average is calculated and used as a reference in a repeated alignment of all images. The process is iterated until the orientations of the images stabilize (Frank et al., 1982). More formally, Frank et al. (1988) proposed the definition of a set of N images fk , k = 1, . . . , N, aligned if a set of transformations Tk, k = 1, . . . , N, (rotation angles and translations) is found such that all pairs of images are mutually brought into register, so the expression

L1 ðff g; fTgÞ ¼

N
1 P

Fig. 2.5.7.2. The geometrical constraints of the 2D alignment problem. (a) The reference 2D particle is placed within a square image frame n  n pixels and its size is such that it can be bounded by a circle with a radius r no larger than 0.9n. (b) The particle projection, the size of which is bounded by the same radius as the reference view, can be located within a circle centred on discrete locations within the image frame, such that the maximum translation is k = (n/2)
r. The number of possible translations is (2k + 1)2. Reprinted from Joyeux & Penczek (2002) with permission from Elsevier.

leads to the conclusion that if the minimum of L1 could be found, a set of diverse images could be aligned; moreover, upon alignment similar images would have similar orientation and subsequent classification of such an aligned data set would reveal subsets of similar images. A practical method of minimizing, called a reference-free alignment, was proposed by Penczek et al. (1992) by showing that minimization of L1 is equivalent to maximization of L2 ðff g; fTgÞ ¼

k¼1

¼

  fk ðTk xÞ
f 2 ; k

ð2:5:7:10Þ

where   f k¼

N X 1 f ðT xÞ N
1 l¼1;l6¼k l l

ð2:5:7:11Þ

is the partial average of the set of images calculated with the exclusion of the kth image. The method is based on the observation that given a set of approximately aligned images, it should be possible to minimize L2 by sequentially correcting alignments of individual images using the cross-correlation function between each image and the average of the remaining ones. On each step, depending whether the orientation of the image changes or not, (2.5.7.10) will decrease or remain constant. The outcome of the reference-free alignment algorithm is an aligned set of N images, so all particles that have similar shapes will have similar orientations. Thus, it is natural (and because of the alignment possible) to divide the data set into classes of images that have similar shapes and orientations, i.e., to cluster them. A number of well known clustering algorithms have been adopted for EM applications (Frank, 1990). The general purpose of clustering is to organize objects (in the case of EM, images) into classes whose members are similar to each other, while dissimilar to objects from other classes. Reference-free alignment with subsequent clustering works well as long as all particles share the same overall shape (i.e., the very low frequency component), as is the case for ribosomes. However, some molecules yield projections that have quite different shapes, as for example is the case for barrel-like proteins GroEL (Roseman et al., 1996) with rectangular views and circular end views or flat and rectangular hemocyanin (Boisset et al., 1995). In this case, the reference-free alignment tends to be unstable, as (2.5.7.10) has multiple local minima, which in practice means that the global average of the whole data set can vary significantly depending on the initiation of the procedure. In general, reference-free alignment is an ‘alignment first, classifi-

N   P fk ðTk xÞ
fl ðTl xÞ2

k¼1 l¼kþ1 N
1 P

N
1 P

 N     P fk ðTk xÞ2 þfl ðTl xÞ2
2fk ðTk xÞfl ðTl xÞ

k¼1 l¼kþ1

ð2:5:7:9Þ

is minimized. Although there is no simple way to minimize L1, the interesting observation is that there is no requirement of the images to represent the same particle, not even a similar one. This

380

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION cation second’ approach. It is possible to reverse this order by using invariants with the supporting rationale that once approximately homogeneous classes of images were found, it should be easy to align them subsequently as within each class they will share the same motif. A practical approach to reference-free alignment known as alignment by classification (Dube et al., 1993) is based on the observation that for a very large data set and centred particles one can expect that although the in-plane rotation is arbitrary, there is a high chance that at least some of the similar images will be in the same rotational orientation. Therefore, in this approach the images are first (approximately) centred, then subjected to classification, and subsequently aligned. In its simplest form, the multireference alignment belongs to the class of supervised classification methods: given a set of templates (i.e., reference images; these can be selected unprocessed particle projections, or class averages that resulted from preceding analysis, or projections of a previously determined EM structure, or projections of an X-ray crystallographic structure), each of the images from the available data sets is compared (using a selected discrepancy measure) with all templates and assigned to the class represented by the most similar one. Equally often multireference alignment is understood as a form of unsupervised classification, more precisely K-means classification, even if the description is not formalized in terms of the latter. Given a number of initial 2D templates, the images are compared with all templates and assigned to the most similar one. New templates are calculated by averaging images assigned to their predecessors and the whole procedure is repeated until a stable solution is reached.

Fig. 2.5.7.3. Principle of random conical tilt reconstruction. A tilt pair of images of the same grid area is collected. By aligning the particle images in the untilted micrograph (left), the Eulerian angles of their counterparts in the tilted micrograph (right) are established. The particle images from the tilted micrograph are used for 3D reconstruction of the molecule (bottom). The set of projections form a cone in Fourier space; information within the cone remains undetermined.

2.5.7.7. Initial determination of 3D structure using tilt experiments The 2D analysis of projection images provides insight into the behaviour of the protein on the grid in terms of the structural consistency and the number and shape of projection images. In order to obtain 3D information, it is necessary to find geometrical relations between different observed 2D images. The most robust and historically the earliest approach is based on tilt experiments. By tilting the stage in the microscope and acquiring additional pictures of the same area of the grid it is possible to collect projection images of the same molecule with some of the required Eulerian angles determined accurately by the setting of the goniometer of the microscope. In random conical tilt (RCT) reconstruction (Radermacher et al., 1987), two micrographs of the same specimen area are collected: the first one is recorded at a tilt angle of ~50 while the second one is recorded at 0 (Fig. 2.5.7.3). If particles have preferred orientation on the support carbon film (or within the amorphous ice layer, if no carbon support is used), the projections of particles in the tilted micrographs form a conical tilt series. Since in-plane rotations of particles are random, the azimuthal angles of the projections of tilted particles are also randomly distributed; hence the name of the method. The untilted image is required for two reasons: (i) the particle projections from the untilted image are classified, thus a subset corresponding to possibly identical images can be selected ensuring that the projections originated from similar and similarly oriented structures; and (ii) the in-plane rotation angle found during alignment corresponds to the azimuthal angles in three dimensions (one of the three Eulerian angles needed). The second Eulerian angle, the tilt, is either taken from the microscope setting of the goniometer or calculated based on geometrical relations between tilted and untilted micrographs. The third Eulerian angle corresponds to the angle of the tilt axis of the microscope stage and is also calculated using the geometrical relations between two micrographs. In addition, it is necessary to centre the particle projections selected from tilted micrographs; although various correlation-based schemes have been proposed,

the problem is difficult as the tilt data tend to be very noisy and have very low contrast. Given three Eulerian angles and centred tilted projections, a 3D reconstruction is calculated. There are numerous advantages of the RCT method. (i) Assuming the sign of the tilt angle is read correctly (it can be confirmed by analysing the defocus gradient in the tilted micrographs), the method yields a correct hand of the structure. (ii) With the exception of the in-plane rotation of untilted projections, which can be found relatively easily using alignment procedures, the remaining parameters are determined by the experimental settings. Even if they are not extremely accurate, the possibility of a gross error is eliminated, which positively distinguishes the method from the ab initio computational approaches that use only untilted data. (iii) The computational analysis is entirely done using the untilted data, which have high contrast. (iv) The RCT method is often the only method of obtaining 3D information if the molecule has strongly preferential orientation and only one view is observed in untilted micrographs. The main disadvantage is that the conical projection series leaves a significant portion of the Fourier space undetermined. This follows from the central section theorem [equation (2.5.6.8) of Section 2.5.6]: as the tilt angle is less than 90 , the undetermined region can be thought to form a cone in three dimensions and is referred to as the missing cone. The problem can be overcome if the molecule has more than one preferred orientation. Subsets of particles that have similar untilted appearance (as determined by clustering) are processed independently and for each a separate 3D structure is calculated. If the preferred orientations are sufficiently different, i.e., the orientations of the original particles in three dimensions are sufficiently different in terms of their angles with respect to the z axis, the 3D structures can be aligned and merged, all but eliminating the problem of the missing cone and yielding a robust, if resolution-limited, initial model of the molecule (Penczek et al.,

381

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION 1994). It should be noted that RCT by itself almost never results in a high-resolution 3D model of the molecule. This is due to a variety of reasons, the main ones being the already mentioned poor quality of high-tilt data and difficulties with the collection of large numbers of high-quality tilted micrographs (they are often marred by drift). In cases when the molecule does not have well defined preferred orientations, it is possible to use electron tomography to obtain the initial model. In this method, a single-axis tilt series of projection images of the same specimen area is collected using an angular step of ~2 and a maximum tilt angle not exceeding 60 (Crowther, DeRosier & Klug, 1970). The single-axis tilt datacollection geometry yields worse coverage of the Fourier space than the RCT method, leaving missing wedges uncovered (Penczek & Frank, 2006). This results in severe artifacts in real space, which make smaller objects virtually unrecognizable. The situation can be largely rectified using so-called double-axis tomography, in which a second single-axis tilt series of data are collected after rotating the specimen grid in-plane by 90

(Penczek et al., 1995). This reduces the undetermined region to a missing pyramid and makes the resolution almost isotropic in the xy plane. The tomographic projection data have to be aligned. This is done using either correlation techniques that enforce pairwise alignment of images (Frank & Mcewen, 1992) or by taking advantage of fiducial markers and enforcing their consistency with respect to a 3D model (Lawrence, 1992; Penczek et al., 1995). In the application to single-particle work, it is possible to use locations of protein in the micrographs as markers. After the 3D reconstruction is calculated, regions collecting individual molecules are windowed from the volume and all molecules are aligned in three dimensions (Walz et al., 1997). While generally robust, the procedure is labour- and computer-intensive. Unlike RCT, where only two exposures of the same field are required, electron tomography may require over one hundred images, raising serious concerns about radiation damage. Moreover, most of the data have to be collected at high tilt angle, thus are of lower quality. Particularly troublesome is alignment of 3D molecules deteriorated by the missing wedge/pyramid artifacts, with the directions of artifacts different for each object. However, when successful, electron tomography yields a very good initial model of the molecule, free from missing-Fourier-space-related artifacts and with defined handedness.

Fourier transform of each projection intersects itself (or rather the symmetry-related copies of itself) 37 times with the exception of degenerate cases of projections in directions of one of three symmetry axes, in which cases the number of common lines is less. Thus, it is possible to find the orientation of a single projection with respect to the chosen system of symmetry axes. For asymmetric objects a set of three projections that do not intersect along the same line (which would correspond to the single-axis tilt geometry) uniquely determine their respective orientations (with the exception of the overall rotation, which remains arbitrary, and the handedness of the solution, which remains undetermined). Indeed, three projections span three common lines, and each common line yields two angles: for each of the intersecting sections it is the angle between the x axis in the system of coordinates of this section and the common line in the plane of this section. Thus, we have a total of six angles. At the same time, by arbitrarily setting the orientation of the first projection in 3D space to three Eulerian angles equal to zero (or the corresponding rotation matrix R1 = I), we need to determine two rotation matrices R2 and R3 (or two sets of three Eulerian angles) for the remaining two projections, respectively. So, given six in-plane angles we have to find a solution for six Eulerian angles. Let the angle of the common line between the ith and jth projection in the plane of the ith projection be ij and the corresponding unit vector in the plane of ith projection be

2.5.7.8. Ab initio 3D structure determination using computational methods The experiment-based methods of initial 3D structure determination (RCT and electron tomography) are quite powerful, but rather challenging to employ in practice. Particularly frustrating is the fact that a large volume of difficult-to-record tilt data have to be collected, even though they cannot be used for subsequent high-resolution work. Therefore, whenever possible, preference is given to computational methods in which 3D geometrical relations between particle projections are established using various mathematical approaches using only untilted data. The most straightforward approach and historically the earliest is based on the central section theorem [equation (2.5.6.8) of Section 2.5.6]: because Fourier transforms of 2D projections of a 3D object are the central section of the 3D Fourier transform of this object, it is a straightforward consequence that Fourier transforms of any two projections intersect along a line, henceforth called a common line. (Two trivial exceptions are the case of projections in the same direction, in which their Fourier transforms coincide with possible differences in in-plane rotation, and the case of projections in opposite directions, in which they are mirror versions of each other.) This fact was originally used by Crowther, DeRosier & Klug (1970) to solve the structure of viruses with icosahedral (60-fold) symmetry. In this case, the

Equations (2.5.7.13) have two solutions corresponding to two different hands of the molecule [for details see Farrow & Ottensmeyer (1992)]. The common-lines method works very well in the absence of noise. However, even a modest amount of noise can yield quite erroneous results or no results at all. The reason is that the solution of (2.5.7.13) is highly nonlinear with respect to the six given angles ij and small errors in the location of peaks in crosscorrelation functions can lead to quite large discrepancies from the correct solution. The main difficulty with the application of the common-lines method is that the analytical solution in the form of (2.5.7.13) exists only for three projections (Goncharov et al., 1987). As a working approach to ab initio structure determination, the common-lines method has been implemented under the name of angular reconstitution in IMAGIC (van Heel, 1987a; van Heel et al., 1996). In order to reduce sensitivity to noise, the method is applied not to individual projection images, but to class averages resulting from the multireference alignment of input data (van Heel et al., 2000). In order to overcome the problem of the lack of solution for the larger than three number of projections, the user has to begin with selection of three judiciously chosen class averages, obtain the solution using (2.5.7.13) and subsequently include (angle) additional class averages using a brute-force approach, in which the Eulerian angles of the new projection are

nij ¼ ½ cos ij


sin ij

0 T ;

i; j ¼ 1; 2; 3;

i 6¼ j; ð2:5:7:12Þ

where we added the third coordinate for convenience. The orientations of unit vectors nij in 3D space have to be related by rotation matrices; for example, vector n21 (the direction of the common line between the first and second projection in the plane of the second projection) should coincide with vector n12 (the direction of the same common line, but in the plane of the first projection) upon rotation by (unknown) matrix R2. All possible relations are R2 n21 ¼ n12 R2 n23 ¼ R3 n32 R3 n31 ¼ n13 :

382

ð2:5:7:13Þ

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION calculated using a similarity measure based on common lines with the already-angled set serving as a reference. Some of the disadvantages of the angular reconstitution were addressed in the common-lines-based method for determining orientations for N > 3 particle projections simultaneously (Penczek et al., 1996). In this method, the problem is formulated in terms of minimization of the variance of the 3D structure, as expressed in terms of common-lines discrepancy between N projections. In a sense, the design of the method is the exact opposite of the ‘standard’ common-lines approach: instead of trying to the determine the Eulerian angles (rotation matrices Ri) based on angles ij of common lines in the planes of the projections, one assumes that rotation matrices Ri are known, finds the set of angles ij of common lines and computes the overall discrepancy along these lines. For a pair of projections i and j, the in-plane angles of common lines are found by solving the system of equations Ri nij ¼ Rj nji

For structures that have reasonably high symmetry and for those for which it is possible to collect high-quality EM data, it is sometimes possible to determine the initial structure using the 3D projection alignment method, which will be described in the next section. However, the approach is extremely computationally intensive and it is virtually impossible to try the method repeatedly to verify that the approach converges to more-or-less the same 3D structure, as is recommended for other ab initio methods described in this section. When the method is successful it is quite powerful, as an intermediate resolution structure can be obtained without going through intermediate and quite laborious steps of analysis of the data. A word of caution is warranted: with the direct method, unless there is external evidence that the obtained structure is correct, it is possible to obtain a selfconsistent but entirely incorrect model of the molecule. In the absence of reliable objective measures of the correctness of the structure, one can apply common sense in order to spot definitely improbable 3D maps. Given the mass of the complex it is possible to calculate the corresponding volume, and thus the threshold at which the map should be examined (Section 2.5.7.11). If at this threshold the mass density is discontinuous or there are pieces of mass surrounding the structures, the map is most likely to be incorrect. Similarly, strong directional artifacts appearing as streaks permeating the structure indicate that either the collected projection images are dominated by one or two views of the structure or that the angular assignment is incorrect. In addition, the 3D map should be centred in the window box; although the centring is not strictly speaking a mathematical requirement for a successful reconstruction, all single-particle structure-determination software packages take advantage of the fact that for centred objects orientation searches are easier to perform. So, if the map is not centred it is a clear indication of the failure of the procedure. Finally, for symmetric structures there should be no large pieces of mass on the symmetry axes.

ð2:5:7:14Þ

for ij and ji . The discrepancy minimized in the method is the variance of the 3D structure that, by analogy to the 2D case (2.5.7.10) and (2.5.7.11) is Lcl ðfFg; fRgÞ ¼

M P N   P Fk ðum ; ; Rk Þ
hF ik 2 u2m ukl ; m¼0 k¼1

h F ik ¼

N X 1 F ðu ; ; Rl Þ; N
1 l¼1;l6¼k l m

ð2:5:7:15Þ

where Lcl is written in Fourier 3D polar coordinates and Fk ðum ; ; Rk Þ is the Fourier transform of the kth projection in 2D polar coordinates ðum ; Þ with the orientation in 3D Fourier space given by the rotation matrix Rk. All Fourier planes Fk are considered to have zero thickness, so all discrepancies are calculated only along common lines and the ‘partial average’ hF ik is in fact an arrangement of N
1 Fourier planes in 3D space. An approximation to u2m ukl is calculated by equating the values of kl to the areas of the Voronoi diagram cells constructed on a unit sphere for points of intersection of common lines with this unit sphere (see Section 2.5.6.6). Generally the method performs very well, particularly if the projection images cover 3D angular space evenly. Some macromolecules, particularly those that have an elongated barrel-like shape, will have a strongly preferential orientation with respect to the direction of electron beam showing only what are often called ‘side views’, i.e., projections perpendicular to rotation along one axis corresponding to single-axis tilt datacollection geometry. These orthoaxial projections form singleaxis tilt reconstruction geometry. In this case, Fourier transforms of all projections share only one common line, the line coinciding with the rotation axis, and clearly the common-lines-based method is not applicable to the ab initio structure determination. To cope with this situation, a method termed Sidewinder was developed (Pullan et al., 2006). It is based on the observation that a Fourier transform of a finite object with a diameter D can be considered to have a nonzero thickness 1/D (Fig. 2.5.6.5). Thus, if the angle between two central sections of the 3D Fourier transform of this object, as derived from 2D Fourier transforms of its projections, is not too large, then these two sections will share information in Fourier space that is proportional to the amount of overlap of the two ‘slabs’ in Fourier space. Using this observation, the general idea employed in Sidewinder is to calculate pairwise cross-correlation coefficients (CCCs) between class averages of side views and to use this information to deduce the values of the azimuthal Eulerian angles using the Monte Carlo minimization method (Fishman, 1995).

2.5.7.9. Refinement of a 3D structure Given an initial low-resolution model of the 3D structure and the data set of 2D projection images of the complex that have Fourier-space information extending beyond the resolution of the model, it is possible to refine the structure such that the full extent of the resolution information in the data will be utilized. In some cases, it is also possible to use as an initial structure in the refinement procedure a structure of a homologous protein, thus avoiding the process of ab initio structure determination altogether. The goal of the refinement is to find such orientation parameters for each of the particle projections for which (2.5.7.6) is minimized. There exist various implementations of the structure-refinement strategy and they can be roughly divided into those that perform exhaustive searches for all five orientation parameters (two translations and three Eulerian angles per 2D projection image) and those that perform local searches, usually by employing gradient information. Finally, the strategies may differ in how the correction for the CTF is implemented. The original 3D projection-matching strategy (Penczek et al., 1994) is based on the observation that given an ideal structure f and the necessary parameters of the CTF and image-formation model, it is straightforward to find five orientation parameters for each projection image. One begins with the determination of the sufficient angular step: assuming the structure is properly sampled at the Nyquist frequency and has a real-space radius of r voxels, the angular step is given by 
ffi arctanð1=rÞ:

ð2:5:7:16Þ

Next, keeping in mind that projection directions are parametrized by two Eulerian angles ð’;
Þ, one generates a set of projection directions quasi-uniformly distributed over half a unit

383

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION

Fig. 2.5.7.4. Schematic of the 3D projection-alignment procedure. Fig. 2.5.7.5. Schematic of 3D projection alignment with CTF correction performed on the level of 3D maps reconstructed from projection images sorted into groups that share similar defocus settings.

sphere (or, in the case of a symmetric structure, over an asymmetric subunit) by taking fixed steps along the altitude or tilt angle
and a number of samples azimuthally in proportion to sin
(Penczek et al., 1994). So, for a chosen constant increment 
and given
angle the increment of the ’ angle varies according to ’ ¼ 
=jsin
j:

Fmerged

P CTFk SSNRk Fk ¼Pk ; 2 k CTFk SSNRk þ 1

ð2:5:7:18Þ

ð2:5:7:17Þ where SSNRk is the spectral signal-to-noise ratio estimated for each defocus group using (2.5.7.22). Subsequently, the resolution of the merged volume is estimated by merging the half-volumes into two half-merged volumes using (2.5.7.18) and comparing them using (2.5.7.19). Next, the merged volume is filtered using (2.5.7.25) and the structure is centred so that its centre of mass is placed at the centre of the volume in which it is embedded. The 3D projection-matching approach works very well during the initial stages of the refinement as it constitutes a very efficient approach to an exhaustive search for orientation parameters of all projection data. Once the orientation parameters are known to a degree of accuracy, it is straightforward to modify the procedure such that only subsets of reference projections are generated at a time and projection images are compared only with reference projections within a specified angular distance from their angular direction established during previous iteration. This modification speeds up the procedure significantly and makes it possible to refine structures to very high resolution by using a very small angular step 
. Another possible modification is to introduce an additional step of 2D alignment of the projection data that share the same angular direction (Ludtke et al., 1999). The advantage is that this can correct possible errors of alignment to the projection of a limited-resolution reference structure and also, to an extent, reduces the danger of bias from artifacts in the reference structure. Finally, it is also possible to incorporate into the refinement strategy a correction for the envelope function of the microscope (Ludtke et al., 1999). The 3D projection-matching strategy is widely popular and most EM software packages have implementations of various versions of basic strategies, as outlined above (Frank et al., 1996; Ludtke et al., 1999; Hohn et al., 2007).

If all three Eulerian angles are to be sampled, as is necessary in some applications, then is sampled uniformly in steps of 
. In order to find the orientation parameters of projection images, one step of projection matching is performed. The reference structure is projected in all directions given by (2.5.7.17), yielding a set of reference images. Next, for each projection image, 2D cross-correlation functions with all reference images are calculated using one of the methods described in Section 2.5.7.6 and the overall maximum yields the translation, the in-plane rotation angle, the number of the most similar reference image (thus the remaining two Eulerian angles) and information about whether the image should be mirrored. Given this, a new 3D structure can be calculated using a 3D reconstruction algorithm (see Section 2.5.6). This simple protocol constitutes the core of 3D projection alignment (Fig. 2.5.7.4). In a simple implementation of the 3D projection-matching procedure, all projection data are assembled into defocus groups, i.e., groups of projection images that have similar defocus settings (Frank et al., 2000). During refinement, for each defocus group the reference volume is multiplied by the CTF with the appropriate defocus value, one step of projection matching is performed and a refined structure is reconstructed for this group (Fig. 2.5.7.5). In addition, the within-group resolution is estimated using the Fourier shell correlation (FSC) approach (2.5.7.19) applied to two volumes calculated from two subsets of projection images randomly split into halves. After all defocus groups have been processed, the individual refined volumes are merged in Fourier space with a CTF correction using Wiener-filter methodology (Penczek et al., 1997),

384

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION A possible improvement over the 3D projection-matching procedure can be achieved by working in transformed spaces in which the distinction between orientation search and 3D reconstruction is removed: (1) spherical harmonics (Provencher & Vogel, 1988; Vogel & Provencher, 1988), which have found applications exclusively in the determination of icosahedral structures (Yin et al., 2001, 2003); (2) Radon transform (Radermacher, 1994), with selected applications in the determination of asymmetric particles (Ruiz et al., 2003); or (3) Fourier transform, implemented in the FREALIGN package (Grigorieff, 2007). In FREALIGN, the transformation between the arbitrarily oriented Fourier 2D central section and the 3D Fourier Euclidean grid is implemented using trilinear interpolation that includes ad hoc correction for the CTF effects. In high-resolution structurerefinement mode, the program uses a gradient-based Powell optimization algorithm (Powell, 1973), thus overcoming the main deficiency of 3D projection-matching algorithms. A unified approach to direct minimization of (2.5.7.6) was proposed by Yang et al. (2005) and is implemented in the SPARX package as the YNP method (Hohn et al., 2007). The premise of the YNP method is that the orientation parameters are approximately known (thus the initial 3D map) and both the orientation parameters and the density map are updated simultaneously in a gradient-based optimization scheme. In the YNP method, the derivatives with respect to the density distribution are calculated analytically and the derivatives with respect to orientation parameters are calculated using finite difference approximations. The YNP method is very efficient and its major advantage is that it avoids many problems associated with approximate solutions inherent in methods that work in transform spaces. The projection/backprojection operations are carried out rapidly using linear interpolation, which due to sufficient oversampling of the data does not have a significant adverse impact on the solution. Moreover, because the density map f is updated simultaneously with the orientation parameters, the computationally demanding separate step of 3D reconstruction is eliminated.

function of spatial frequency (Penczek, 2002). The ‘resolution’ of the reconstruction is reported as a spatial frequency limit beyond which the SSNR drops below a selected level, for example below one. The FSC is evaluated by taking advantage of the large number of single-particle images: the total data set is randomly split into halves; for each subset a 3D reconstruction is calculated (in two dimensions, a simple average); and two maps f and g are compared in Fourier space, FSCðf ; g; uÞ

Pnr  jkun k
rj" Fðun ÞG ðun Þ ¼ nhP ih io1=2 : nr Fðun Þ 2 Pnr Gðun Þ 2 jkun k
rj" jkun k
rj" ð2:5:7:19Þ

In (2.5.7.19), 2" is a preselected ring/shell   thickness, the un form a uniform grid in Fourier space, u ¼ un  is the magnitude of the spatial frequency and nr is the number of Fourier voxels in the shell corresponding to frequency u. The FSC yields a 1D curve of correlation coefficients as a function of u. Note that the FSC is insensitive to linear transformations of the densities of the objects. An FSC curve everywhere close to one reflects strong similarity between f and g; an FSC curve with values close to zero indicates the lack of similarity between f and g. Particularly convenient for the interpretation of the results in terms of ‘resolution’ is the relation between the FSC and the SSNR, which is easily derived by taking the expectation of (2.5.7.19) under the assumption that both f and g are sums of the same signal and different realizations of the noise, which are uncorrelated with the signal and between them (Saxton, 1978): SSNR : SSNR þ 1

E½FSC ffi

ð2:5:7:20Þ

By solving (2.5.7.20) for SSNR we obtain

2.5.7.10. Resolution estimation and analysis of errors in singleparticle reconstruction The development of resolution measures in EM was greatly influenced by earlier work in X-ray crystallography. In EM, the problem is somewhat more difficult as, unlike in crystallography, both the amplitude and the phase information in the data are affected by alignment procedures (which we consider distant analogues of phase-extension methods in crystallography). Therefore, resolution measures in EM reflect the self-consistency of the results; however, as the data are subject to alignment, there is a significant risk of introducing artifacts resulting from the alignment of the noise component in the data. Ultimately, these artifacts will unduly ‘improve’ the resolution of the map. The resolution measures used in EM fall into two categories: measures based on averaging of Fourier transforms of individual images and measures based on comparisons of averages calculated for subsets of the data. In the first group, we have the Q-factor (van Heel & Hollenberg, 1980; Kessel et al., 1985) and the spectral signal-to-noise ratio (SSNR) introduced for the 2D case by Unser and co-workers (Unser et al., 1987), and for the 3D case for a class of reconstruction algorithms data are based on direct Fourier inversion by Penczek (Penczek, 2002). The second group of measures includes the differential phase residual (DPR) (Frank et al., 1981) and the Fourier ring correlation (FRC) (Saxton & Baumeister, 1982). A marked advantage of these measures is that they are equally well applicable to 2D or 3D data. In the latter case, the volumes resulting from 3D reconstruction algorithms take the place of the 2D averages. The resolution measures used in single-particle reconstruction are designed to evaluate the SSNR in the reconstruction as a

SSNR ¼

FSC ; 1
FSC

ð2:5:7:21Þ

which, taking into account that the FSC was calculated from the data set split into halves, has to be modified to (Unser et al., 1987)  SSNR ¼ 2

 FSC : 1
FSC

ð2:5:7:22Þ

In order to calculate the FSC that corresponds to a given SSNR, one inverts (2.5.7.22) to FSC ¼

SSNR : SSNR þ 2

ð2:5:7:23Þ

Equations (2.5.7.21) and (2.5.7.22) serve as a basis for various ‘resolution criteria’ used in EM. The often-used 3 criterion (van Heel, 1987b) equates resolution with the point at which the FSC is larger than zero at a 3 level, where  is the expected standard deviation of the FSC that has an expected value of zero, in essence finding a frequency for which the SSNR is significantly larger than zero. The 3 criterion has a distinct disadvantage of reporting the resolution at a frequency at which there is no significant signal, while tempting the user to interpret the detail in the map at this resolution. Moreover, as the FSC approaches zero, its relative error increases, so the curve oscillates widely

385

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION around the zero level increasing the chance of selecting an incorrect resolution point. In other criteria one tries to equate the resolution with the frequency at which noise begins to dominate the signal. A good choice of the cut-off level is SSNR = 1.0, a level at which the power of the signal in the reconstruction is equal to the power of the noise. According to (2.5.7.22), this corresponds to FSC = 0.333. Another often-used cut-off level is FSC = 0.5, at which the SSNR in the reconstruction is 2.0 (Bo¨ttcher et al., 1997; Conway et al., 1997; Penczek, 1998). The main reason behind the determination of the resolution of the EM maps is the necessary step of low-pass filtration of the results before the interpretation of the map is attempted. In order to avoid mistakes, particularly the danger of overinterpretation, one has to remove from the map unreliable Fourier coefficients. Inclusion of Fourier coefficients with a low SNR will result in the creation of spurious details and artifacts in the map. Thus, the optimal filtration should be based on the SSNR distribution in the map and the solution is given by a Wiener filter: WðuÞ ¼

SSNRðuÞ : SSNRðuÞ þ 1

elements of secondary structure become directly visible. Maps at resolution lower than that we will call intermediate resolution, as at this scale of detail one can only determine a general arrangement of subunits. However, it is good to realize that there is a huge difference between the amount and reliability of information derived from a map of the same complex determined ˚ as compared to a map determined at 30 A ˚ resolution. at 10 A ˚ resolution Similarly, very large complexes determined at 50 A will yield more information than very small complexes deter˚ resolution. On the other hand, even intermediatemined at 15 A resolution EM maps provide extremely valuable information if they can be placed in the context of other structural work. The single-particle structure can be also investigated within a context of a more complex system using other, lower-resolution techniques, for example electron tomography. In this case, by using docking approaches one can determine the distribution, orientation and general arrangement of smaller cryo-EM determined complexes within larger subcellular systems. On a different scale of resolution, it is quite common to have structures of some domains or even of the entire complexes determined to atomic resolution by X-ray crystallography. Again, by using docking techniques it is possible to determine whether the conformation of the EM structure differs from that determined by X-ray crystallography or to map subunits and domains of the larger complex by fitting available atomic resolution structures. The basic mode of visualization of cryo-EM maps is surface representation. The first step involves the choice of an appropriate threshold level for the displayed surface, particularly when the scaling of the cryo-EM data is arbitrary. A good guide is provided by the total molecular mass of the complex: given a ˚ , an average protein density d = 1.36  pixel size of p A ˚
3 and the total molecular mass of the complex M Da, 10
24 g A the number of voxels Nv occupied by the complex is

ð2:5:7:24Þ

Based on the relation of FSC to SSNR (2.5.7.22), we can write (2.5.7.24) as WðuÞ ¼

2FSCðuÞ : FSCðuÞ þ 1

ð2:5:7:25Þ

In practice, because of the irregular shape of typical FSC curves (particularly for small values of FSC) it is preferable to approximate the shape of the Wiener filter (2.5.7.25) by one of the standard low-pass filters, such as Butterworth (Gonzalez & Woods, 2002) or hyperbolic tangent (Basokur, 1998). The FRC/FSC methodology can be used to compare a noisecorrupted map with a noise-free ideal version of the same object. In single-particle reconstruction this situation emerges when an X-ray crystallographic structure of either the entire EMdetermined structure or of some of its domains is available (Penczek et al., 1999). In this case, we assume that in (2.5.7.19) f represents a sum of the signal and additive uncorrelated noise and g represents the noise-free signal, so is straightforward to calculate the expectation of (2.5.7.19) in order to obtain the relation between the cross-resolution (CRC) and the SSNR: 

SSNR E½CRC ffi SSNR þ 1

  Nv ¼ M= p3 dNA ;

where NA is the Avogadro’s number (6.02  1023 atoms mole
1). Based on that, one can find the threshold that for a given structure encompasses the determined number of voxels Nv [appropriate functions are implemented in SPIDER (Agrawal et al., 1996; Frank et al., 1996) and SPARX (Hohn et al., 2007)]. At a sufficiently high resolution, cryo-EM maps can be analysed in the same manner as X-ray crystallographic maps and using the same graphical/analytical packages (‘backbone tracing’) (Jones et al., 1991) (Fig. 2.5.7.6). The complexity of cryo-EM maps of large macromolecular assemblies combined with their limited resolution invites attempts to automate some of the steps of analysis in an attempt to make the results more robust and less dependent on the researcher’s bias. A good example of semi-automated analysis is ˚ cryo-EM map of the nucleic acid–protein separation in a 11.5 A the 70S E. coli ribosome (Spahn et al., 2000). In the procedure, the (continuous-valued) densities were analysed making use of (i) the difference in scattering density between protein and nucleic acids; (ii) continuity constraints that the image of any nucleic acid molecule must obey and (iii) knowledge of the molecular volumes of all proteins. As a result, it was possible to reproduce boundary assignments between ribosomal RNA (rRNA) and proteins made from higher-resolution X-ray maps of the ribosomal subunits with a high degree of accuracy, and allowed plausible predictions to be made for the placements of proteins and RNA components as yet unassigned. One of the conclusions derived from this separation was that the 23S rRNA is solely responsible for the catalysis of peptide-bond formation; thus, the ribosome is a ribozyme. The same conclusion was reached independently in the studies of the X-ray crystallographic structure of the 70S ribosome (Nissen et al., 2000). The method by Spahn et al. cannot be easily extended to other

1=2 :

ð2:5:7:26Þ

Thus SSNR ¼

CRC2 : 1
CRC2

ð2:5:7:28Þ

ð2:5:7:27Þ

Interestingly, for the same SSNR cut-off levels, corresponding values of CRC are higher than those for FSC. For example, for SSNR = 1, CRC = 0.71, while FSC = 0.33. For SSNR = 2, CRC = 0.82, while FSC = 0.5. 2.5.7.11. Analysis of 3D cryo-EM maps The amount of structural information that can be derived from a structure of a macromolecular complex determined by cryo-EM depends on two factors: the resolution of the map and the availability of additional structural information about the system. Generally, we will refer to complexes at a resolution better than ˚ as high-resolution structures, as at this resolution the 7A

386

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION

Fig. 2.5.7.7. Hrs (blue) embedded into the membrane (yellow) of an early endosome. Two functional domains have been docked into the cryo-EM ˚ resolution, and are shown density map of hexameric Hrs, determined to 16 A as ribbons coloured by secondary structure (Pullan et al., 2006). The structures of the VHS and FYVE domains have been determined crystallographically (Mao et al., 2000) and are docked into the EM density map of the Hrs. The knowledge of the location of the FYVE and VHS domains, which are reported to bind to PI(3)P molecules found within the endosomal membrane (Kutateladze et al., 1999), has guided the hypothetical placement of Hrs within the endosomal membrane. The immersion of the end caps of Hrs into the endosomal membrane demonstrates an ‘end-on’ binding model of the Hrs particles with the membrane. According to this model, either end cap can embed into the membrane, allowing the other end cap to carry out other essential protein trafficking functions, or to embed into another membrane, thus preventing fusion of membranes during early endosomal fusion.

Fig. 2.5.7.6. A high-resolution cryo-EM map allows backbone tracing. (a) Cryo-EM map of the cricket paralysis virus (CrPV) IRES RNA in complex ˚ resolution. The map is with the yeast 80S ribosome determined at 7.3 A shown from the L1 protuberance side with the ribosomal 40S subunit in yellow, the 60S subunit in blue and the CrPV IRES in magenta. Landmarks for the 40S subunit: b, body; h, head; p, platform. Landmarks for the 60S subunit: CP, central protuberance; L1, L1 protuberance. PKIII denotes helix PKIII of the CrPV IRES RNA and SL the two stem loops present in the secondary structure of the RNA. (b) Structure of the CrPV IRES RNA. Based on the cryo-EM map and additional biochemical knowledge, the complete chain of the RNA is found (189 nucleotides could be traced). The IRES molecular model is shown as a coloured ribbon docked into the cryoEM density (grey mesh). PK and P denote the individual helical elements of the IRES and SL denotes the stem loops (Schu¨ler et al., 2006).

In docking of X-ray crystallographic structures into EM maps, the first step is the conversion of atomic coordinates from X-ray molecular models, as given in Protein Data Bank (PDB) files, into an electron-density map in a way that would mimic the physical image formation process. Although sophisticated methods of computational emulation of the image-formation process in the electron microscope are available, very simple approaches to conversion yield quite satisfactory results at the resolution of the EM results. The most common one is to assume that the Coulomb potential of an atom is proportional to its atomic number and add these atomic numbers within a Euclidean grid with a cell size ˚ . The atomic coordinates of atoms equal to the EM pixel size in A are interpolated within the grid using trilinear interpolation. After such conversion, the X-ray map can be handled using the general image-processing tools of a single-particle software package. Initial orientation (or orientations, if the general placement is not immediately visually apparent) of the X-ray map can be easily performed manually within any number of graphical packages, for example Chimera (Pettersen et al., 2004). The initial six orientation parameters (three translations and three Eulerian angles) are next transferred to the EM package (for details see Baldwin & Penczek, 2007) and the manual docking is refined using correlation techniques (Fig. 2.5.7.7). Similarly, the handedness of the EM map can be established or confirmed by performing fitting of the X-ray determined structure to two EM maps that differ by their hand. Docking of EM maps into the broader cellular context of structures determined by electron tomography can provide information about the distribution of complexes and their interactions within the cell. Conceptually, the approach is very similar to that of particle picking, i.e., template matching, with the main difference being that calculations are performed in three instead of two dimensions. Given a 3D structure of a single-particle EM complex, a set of 3D templates is prepared by rotating the

macromolecules that comprise only protein and generally it is very difficult to delineate at intermediate resolution subunits of large macromolecular assemblies, automatically or not, in the absence of independent knowledge about their shape. The reason is that both the density and shape of the subunits are affected by the limited resolution differently depending on their spatial context. In general, subunits that are isolated and located on the surface or protruding from the structure will have relatively lower density while at the same time their overall shape will be better preserved and easier to discern. Subunits located inside the structure and surrounded by other structural elements, while having higher density, are more difficult to recognize, as they fuse with the surrounding mass densities. Therefore, it is difficult to provide a general method that could cope with the problem of automated mass-density analysis. As most cryo-EM structures are determined at intermediate resolution, the most common mode of analysis is either to compare the map with the available X-ray crystallographic structures of its domains or to consider the result in the context of larger, subcellular structures obtained by electron tomography. In both cases correlation techniques are used extensively to obtain objective results or to validate the results obtained by manual fitting.

387

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION template around its centre of mass using the quasi-uniformly distributed three Eulerian angles [equation (2.5.7.17) with  ¼ 
]. However, in application to tomography the angular step 
can be relatively large, resulting in a much smaller number of templates than in two dimensions, the reason being the rather low resolution of typical electron tomograms (not exceeding ˚ ). Next, a brute-force 3D cross-correlation search with all 50 A templates is performed (Frangakis et al., 2002). After windowing out 3D subvolumes containing putative complexes, subsequent averaging and classification can be performed. Cryo-EM is a unique structural technique in its ability to detect conformational variability of large molecular assemblies within one sample that may contain a mixture of complexes in various conformational states. In addition to the expected conformational heterogeneity of the assemblies, due to fluctuations of the structure around the ground state one can expect to capture molecules in different functional states, especially if the binding of a ligand induces a conformational change in the macromolecular assembly. Therefore, a data set of images from an EM experiment must be interpreted as a mixture of projections from similar but not identical structures. The analysis of the extent of the resulting variability requires the calculation of the real-space distribution of 3D variance/covariance in macromolecules reconstructed from a set of their projections. The problem is difficult, as there is no clear relation between the variance in sets of projections that have the same angular direction and the variance of the 3D structure calculated from these projections. Penczek, Chao et al. (2006) proposed calculating the variance in the 3D mass distribution of the structure using a statistical bootstrap resampling technique, in which a new set of projections is selected with replacements from the available whole set of N projections. In the new set, some of the original projections will appear more than once, while others will be omitted. This selection process is repeated a number of times and for each new set of projections the corresponding 3D volume is calculated. Next, the voxel-by-voxel bootstrap variance B2 of the resulting set of volumes is calculated. The target variance is obtained using a relationship between the variance of arithmetic means for sampling with replacements and the sample variance,

 2 ¼ NB2 :

2.5.8. Direct phase determination in electron crystallography

By D. L. Dorset

2.5.8.1. Problems with ‘traditional’ phasing techniques The concept of using experimental electron-diffraction intensities for quantitative crystal structure analyses has already been presented in Section 2.5.4. Another aspect of quantitative structure analysis, employing high-resolution images, has been presented in Sections 2.5.5 to 2.5.7. That is to say, electron micrographs can be regarded as an independent source of crystallographic phases. Before direct methods (Chapter 2.2) were developed as the standard technique for structure determination in small-molecule X-ray crystallography, there were two principal approaches to solving the crystallographic phase problem. First, ‘trial and error’ was used, finding some means to construct a reasonable model for the crystal structure a priori, e.g. by matching symmetry properties shared by the point group of the molecule or atomic cluster and the unit-cell space group. Secondly, the autocorrelation function of the crystal, known as the Patterson function (Chapter 2.3), was calculated (by the direct Fourier transform of the available intensity data) to locate salient interatomic vectors within the unit cell. The same techniques had been used for electron-diffraction structure analysis (nowadays known as electron crystallography). In fact, advocacy of the first method persists. Because of the perturbations of diffracted intensities by multiple-beam dynamical scattering (Chapter 5.2), it has often been suggested that trial and error be used to construct the scattering model for the unit crystal in order to test its convergence to observed data after simulation of the scattering events through the crystal. This indirect approach assumes that no information about the crystal structure can be obtained directly from observed intensity data. Under more favourable scattering conditions nearer to the kinematical approximation, i.e. for experimental data from thin crystals made up of light atoms, trial-and-error modelling, simultaneously minimizing an atom–atom nonbonded potential function with the crystallographic residual, has enjoyed widespread use in electron crystallography, especially for the determination of linear polymer structures (Brisse, 1989; Pe´rez & Chanzy, 1989). Interpretation of Patterson maps has also been important for structure analysis in electron crystallography. Applications have been discussed by Vainshtein (1964), Zvyagin (1967) and Dorset (1994a). In face of the dynamical scattering effects for electron scattering from heavy-atom crystals realized later (e.g. Cowley & Moodie, 1959), attempts had also been made to modify this autocorrelation function by using a power series in jFh j to sharpen the peaks (Cowley, 1956). (Here Fh h, replacing the notation for the kinematical electron-diffraction structure factor employed in Section 2.5.4.) More recently, Vincent and coworkers have selected first-order-Laue-zone data from inorganics to minimize the effect of dynamical scattering on the interpretability of their Patterson maps (Vincent & Exelby, 1991, 1993; Vincent & Midgley, 1994). Vainshtein & Klechkovskaya (1993) have also reported use of the Patterson function to solve the crystal structure of a lead soap from texture electron-diffraction intensity data. It is apparent that trial-and-error techniques are most appropriate for ab initio structure analysis when the underlying crystal structures are reasonably easy to model. The requisite positioning of molecular (or atomic) groups within the unit cell may be facilitated by finding atoms that fit a special symmetry position [see IT A (2005)]. Alternatively, it is helpful to know the molecular orientation within the unit cell (e.g. provided by the Patterson function) to allow the model to be positioned for a conformational or translational search. [Examples would include

ð2:5:7:29Þ

The estimated structure-variance map can be used for (i) detection of different functional states (for example, those characterized by binding of a ligand) and subsequent classification of the data set into homogeneous groups (Penczek, Frank & Spahn, 2006a), (ii) analysis of the significance of small details in 3D reconstructions, (iii) analysis of the significance of details in difference maps, and (iv) docking of known structural domains into EM density maps. The bootstrap technique also leads to the analysis of conformational modes of macromolecular complexes, and this is due to the fact that the covariance matrix of the structure can be directly calculated from the bootstrap volumes. The covariance matrix obtained this way would be very large. One possibility is to calculate only correlation coefficients between regions of interest that have large variance (Penczek, Chao et al., 2006). Another possibility is to use the iterative Lanczos technique (Parlett, 1980) and calculate eigenvolumes directly from bootstrap volumes without forming the covariance matrix. These eigenvolumes are related to conformational modes of the molecule, as captured by the projection data of the sample (Penczek, Frank & Spahn, 2006b). Thus, this direct relation to the actual cryo-EM projection data positively distinguishes this approach from other techniques in which conformations are postulated based on flexible models of the EM map (Ming et al., 2002; Mitra et al., 2005).

388

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION the polymer-structure analyses cited above, as well as the layerpacking analysis of some phospholipids (Dorset, 1987).] While attempts at ab initio modelling of three-dimensional crystal structures, by searching an n-dimensional parameter space and seeking a global internal energy minimum, has remained an active research area, most success so far seems to have been realized with the prediction of two-dimensional layers (Scaringe, 1992). In general, for complicated unit cells, determination of a structure by trial and error is very difficult unless adequate constraints can be placed on the search. Although Patterson techniques have been very useful in electron crystallography, there are also inherent difficulties in their use, particularly for locating heavy atoms. As will be appreciated from comparison of scattering-factor tables for X-rays [IT C (2004), Chapter 6.1] with those for electrons [IT C (2004), Chapter 4.3], the relative values of the electron form factors are more compressed with respect to atomic number than are those for X-ray scattering. As discussed in Chapter 2.3, it is desirable thatPthe ratio of summed scattering-factor terms, P r ¼ heavy Z2 = light Z2 , where Z is the scattering-factor value at sin
= ¼ 0, be near unity. A practical comparison would be the value of r for copper (dl-alaninate) solved from electrondiffraction data by Vainshtein et al. (1971). For electron diffraction, r ¼ 0:47 compared to the value 2.36 for X-ray diffraction. Orientation of salient structural features, such as chains and rings, would be equally useful for light-atom moieties in electron or X-ray crystallography with Patterson techniques. As structures become more complicated, interpretation of Patterson maps becomes more and more difficult unless an automated search can be carried out against a known structural fragment (Chapter 2.3).

GðsÞ ¼ FðsÞ expð2is  r0 Þ ¼ jFðsÞj exp½iðs þ 2is  r0 Þ: In addition to the crystallographic phases s, it will, therefore, be necessary to find the additional phase-shift term 2is  r0 that will access an allowed unit-cell origin. Such origin searches are carried out automatically by some commercial image-averaging computer-software packages. In addition to applications to thin protein crystals (e.g. Henderson et al., 1990; Jap et al., 1991; Ku¨hlbrandt et al., 1994), there are numerous examples of molecular crystals that have ˚ , many of which have been been imaged to a resolution of 3–4 A discussed by Fryer (1993). For -delocalized compounds, which are the most stable in the electron beam against radiation ˚ resolution) have been obtained at damage, the best results (2 A 500 kV from copper perchlorophthalocyanine epitaxically crystallized onto KCl. As shown by Uyeda et al. (1978–1979), the averaged images clearly depict the positions of the heavy Cu and Cl atoms, while the positions of the light atoms in the organic residue are not resolved. (The utility of image-derived phases as a basis set for phase extension will be discussed below.) A number ˚ of aromatic polymer crystals have also been imaged to about 3 A resolution, as reviewed recently (Tsuji, 1989; Dorset, 1994b). Aliphatic molecular crystals are much more difficult to study because of their increased radiation sensitivity. Nevertheless, monolamellar crystals of the paraffin n-C44H90 have been imaged ˚ resolution with a liquid-helium cryomicroscope (Zemlin to 2.5 A et al., 1985). Similar images have been obtained at room temperature from polyethylene (Revol & Manley, 1986) and also a number of other aliphatic polymer crystals (Revol, 1991). As noted by J. M. Cowley and J. C. H. Spence in Section 2.5.1, dynamical scattering can pose a significant barrier to the direct interpretation of high-resolution images from many inorganic materials. Nevertheless, with adequate control of experimental conditions (limiting crystal thickness, use of high-voltage electrons) some progress has been made. Pan & Crozier (1993) have ˚ images from zeolites in terms of the phasedescribed 2.0 A grating approximation. A three-dimensional structural study has been carried out on an aluminosilicate by Wenk et al. (1992) with thin samples that conform to the weak-phase-object approximation at the 800 kV used for the imaging experiment. Heavy and light (e.g. oxygen) atoms were located in the micrographs in good agreement with an X-ray crystal structure. Heavy-atom positions from electron microscopic and X-ray structure analyses have also been favourably compared for two heavy-metal oxides (Hovmo¨ller et al., 1984; Li & Hovmo¨ller, 1988).

2.5.8.2. Direct phase determination from electron micrographs The ‘direct method’ most familiar to the electron microscopist is the high-resolution electron micrograph of a crystalline lattice. Retrieval of an average structure from such a micrograph assumes that the experimental image conforms adequately to the ‘weak phase object’ approximation, as discussed in Section 2.5.5. If this is so, the use of image-averaging techniques, e.g. Fourier filtration or correlational alignment, will allow the unit-cell contents to be visualized after the electron-microscope phase contrast transfer function is deconvoluted from the average image, also discussed in Section 2.5.5. Image analyses can also be extended to three dimensions, as discussed in Section 2.5.6, basically by employing tomographic reconstruction techniques to combine information from the several tilt projections taken from the crystalline object. The potential distribution of the unit cell to the resolution of the imaging experiment can then be used, via the Fourier transform, to obtain crystallographic phases for the electron-diffraction amplitudes recorded at the same resolution. This method for phase determination has been the mainstay of protein electron crystallography. Once a set of phases is obtained from the Fourier transform of the deconvoluted image, they must, however, be referred to an allowed crystallographic origin. For many crystallographic space groups, this choice of origin may coincide with the location of a major symmetry element in the unit cell [see IT A (2005)]. Hence, since the Fourier transform of translation is a phase term, if an image shift ½ðr þ r0 Þ is required to translate the origin of the repeating mass unit ’ðrÞ from the arbitrary position in the image to a specific site allowed by the space group,

2.5.8.3. Probabilistic estimate of phase invariant sums Conventional direct phasing techniques, as commonly employed in X-ray crystallography (e.g. see Chapter 2.2), have also been used for ab initio electron-crystallographic analyses. As in X-ray crystallography, probabilistic estimates of a linear combination of phases (Hauptman & Karle, 1953; Hauptman, 1972) are made after normalized structure factors are calculated via electron form factors, i.e. jE2h j ¼ Iobs ="

P

fi2 ; where hjEj2 i ¼ 1:000:

i

(Here, an overall temperature factor can be found from a Wilson plot. Because of multiple scattering, the value of B may be found ˚ 2.) The phase invariant sums occasionally to lie close to 0.0 A

gðrÞ ¼ ’ðrÞ  ðr þ r0 Þ ¼ ’ðr þ r0 Þ;

¼ h1 þ h2 þ h3 þ . . .

where the operation ‘’ denotes convolution. The Fourier transform of this shifted density function will be

389

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION complicated and an enantiomorph-defining reflection must be added. In the evaluation of phase-invariant sums above a certain probability threshold, phase values are determined algebraically after origin (and enantiomorph) definition until a large enough set is obtained to permit calculation of an interpretable potential map (i.e. where atomic positions can be seen). There may be a few invariant phase sums above this threshold probability value which are incorrectly predicted, leading either to false phase assignments or at least to phase assignments inconsistent with those found from other invariants. A small number of such errors can generally be tolerated. Another problem arises when an insufficient quantity of new phase values is assigned directly from the phase invariants after the origin-defining phases are defined. This difficulty may occur for small data sets, for example. If this is the case, it is possible that a new reflection of proper index parity can be used to define the origin. Alternatively, n ¼ a; b; c . . . algebraic unknowns can be used to establish the phase linkage among certain reflections. If the structure is centrosymmetric, and when enough reflections are given at least symbolic phase assignments, 2n maps are calculated and the correct structure is identified by inspection of the potential maps. When all goes well in this so-called ‘symbolic addition’ procedure, the symbols are uniquely determined and there is no need to calculate more than a single map. If algebraic values are retained for certain phases because of limited vectorial connections in the data set, then a few maps may need to be generated so that the correct structure can be identified using the chemical knowledge of the investigator. The atomic positions identified can then be used to calculate phases for all observed data (via the structure-factor calculation) and the structure can be refined by Fourier (or, sometimes, least-squares) techniques to minimize the crystallographic R factor. The first actual application of direct phasing techniques to experimental electron-diffraction data, based on symbolic addition procedures, was to two methylene subcell structures (an n-paraffin and a phospholipid; Dorset & Hauptman, 1976). Since then, evaluation of phase invariants has led to numerous other structures. For example, early texture electron-diffraction data sets obtained in Moscow (Vainshtein, 1964) were shown to be suitable for direct analysis. The structure of diketopiperazine (Dorset, 1991a) was determined from these electron-diffraction data (Vainshtein, 1955) when directly determined phases allowed computation of potential maps such as the one shown in Fig. 2.5.8.1. Bond distances and angles are in good agreement with the X-ray structure, particularly after least-squares refinement (Dorset & McCourt, 1994a). In addition, the structures of urea (Dorset, 1991b), using data published by Lobachev & Vainshtein (1961), paraelectric thiourea (Dorset, 1991b), using data published by Dvoryankin & Vainshtein (1960), and three mineral structures (Dorset, 1992a), from data published by Zvyagin (1967), have been determined, all using the original texture (or mosaic single-crystal) diffraction data. The most recent determination based on such texture diffraction data is that of basic copper chloride (Voronova & Vainshtein, 1958; Dorset, 1994c). Symbolic addition has also been used to assign phases to selected-area diffraction data. The crystal structure of boric acid (Cowley, 1953) has been redetermined, adding an independent low-temperature analysis (Dorset, 1992b). Additionally, a direct structure analysis has been reported for graphite, based on highvoltage intensity data (Ogawa et al., 1994). Two-dimensional data from several polymer structures have also been analysed successfully (Dorset, 1992c) as have three-dimensional intensity data (Dorset, 1991c,d; Dorset & McCourt, 1993). Phase information from electron micrographs has also been used to aid phase determination by symbolic addition. Examples include the epitaxically oriented paraffins n-hexatriacontane (Dorset & Zemlin, 1990), n-tritriacontane (Dorset & Zhang, 1991) and a 1:1 solid solution of n-C32H66/n-C36H74 (Dorset,

Fig. 2.5.8.1. Potential map for diketopiperazine ([001] projection) after a direct phase determination with texture electron-diffraction intensity data obtained originally by Vainshtein (1955).

can be particularly effective for structure analysis. Of particular importance historically have been the 2 -triple invariants where h1 þ h2 þ h3 ¼ 0 and h1 6¼ h2 6¼ h3 . The probability of predicting ¼ 0 is directly related to the value of A ¼ ð23 =23=2 ÞjEh1 Eh2 Eh3 j; PN where h ¼ j¼1 Zjn and Z is the value of the scattering factor at sin
= ¼ 0. Thus, the values of the phases are related to the measured structure factors, just as they are found to be in X-ray crystallography. The normalization described above imposes the point-atom structure (compensating for the fall-off of an approximately Gaussian form factor) often assumed in deriving the joint probability distributions. Especially for van der Waals structures, the constraint of positivity also holds in electron crystallography. (It is also quite useful for charged atoms so long as the reflections are not measured at very low angles.) Other useful phase invariant sums are the 1 triples, where h1 ¼ h2 ¼
1=2h3 , and the quartets, where h1 þ h2 þ h3 þ h4 ¼ 0 and h1 6¼ h2 6¼ h3 6¼ h4 . The prediction of a correct phase for an invariant is related in each case to the normalized structurefactor magnitudes. The procedure for phase determination, therefore, is identical to the one used in X-ray crystallography (see Chapter 2.2). Using vectorial combinations of Miller indices, one generates triple and quartet invariants from available measured data and ranks them according to parameters such as A, defined above, which, as shown in Chapter 2.2, are arguments of the Cochran formula. The invariants are thus listed in order of their reliability. This, in fact, generates a set of simultaneous equations in crystallographic phase. In order to begin solving these equations, it is permissible to define arbitrarily the phase values of a limited number of reflections (three for a three-dimensional primitive Punit cell) for reflections with Miller-index parity hkl 6¼ ggg and i hi ki li 6¼ ggg, where g is an even number. This defines the origin of a unit cell. For noncentrosymmetric unit cells, the condition for defining the origin, which depends on the space group, is somewhat more

390

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION ˚ 1990a). Similarly, lamellar electron-diffraction data to ca 3 A by symbolic addition. This procedure may allow determination of resolution from epitaxically oriented phospholipids have been a large enough basis phase set to produce an interpretable map. phased by analysis of 1 and 2 -triplet invariants (Dorset, 1990b, An alternative procedure is to use an automatic version of the ˚ resolution 1991e,f), in one case combined with values from a 6 A tangent formula in a multisolution process. This procedure is electron-microscope image (Dorset et al., 1990, 1993). Most described in Chapter 2.2. After origin definition, enough algerecently, such data have been used to determine the layer packing braic unknowns are defined (two values if centrosymmetric and of a phospholipid binary solid solution (Dorset, 1994d). four values, cycling through phase quadrants, if noncentrosymAn ab initio direct phase analysis was carried out with zonal metric) to access as many of the unknown phases as possible. electron-diffraction data from copper perchlorophthalocyanine. These are used to generate a number of trial phase sets and the ˚ thick sample collected at Using intensities from a ca 100 A likelihood of identifying the correct solution is based on the use 1.2 MeV, the best map from a phase set with symbolic unknowns of some figure of merit. retrieves the positions of all the heavy atoms, equivalent to the Multisolution approaches employing the tangent formula results of the best images (Uyeda et al., 1978–1979). Using these include MULTAN (Germain et al., 1971), QTAN (Langs & positions to calculate an initial phase set, the positions of the DeTitta, 1975) and RANTAN (Yao, 1981). RANTAN is a version remaining light C, N atoms were found by Fourier refinement so of MULTAN that allows for a larger initial random phase set that the final bond distances and angles were in good agreement (with suitable control of weights in the tangent formula). QTAN with those from X-ray structures of similar compounds (Dorset et utilizes the hest definition, where al., 1991). A similar analysis has been carried out for the ( )1=2 perbromo analogue (Dorset et al., 1992). Although dynamical X XX I1 ðAh; k ÞI1 ðA0h; k Þ scattering and secondary scattering significantly perturb the 2 0 Ah; k þ 2 Ah; k Ah; k ; hest ¼ observed intensity data, the total molecular structure can be I0 ðAh; k ÞI0 ðA0h; k Þ k k 6¼ k0 visualized after a Fourier refinement. Most recently, a threedimensional structure determination was reported for C60 buckminsterfullerene based on symbolic addition with results most in for evaluating the phase variance. (Here I0, I1 are modified Bessel accord with a rotationally disordered molecular packing (Dorset functions.) After multiple solutions are generated, it is desirable & McCourt, 1994b). to locate the structurally most relevant phase sets by some figure of merit. There are many that have been suggested (Chapter 2.2). 2.5.8.4. The tangent formula The most useful figure of merit in QTAN has been the NQEST (De Titta et al., 1975) estimate of negative quartet invariants (see Given a triple phase relationship Chapter 2.2). More recently, this has been superseded by the minimal function (Hauptman, 1993): h ’ k þ h
k ; P where h, k and h
k form a vector sum, it is often possible to find a more reliable estimate of h when all the possible vectorial contributions to it within the observed data set kr are considered as an average, viz:

RðÞ ¼

For actual phase determination, this can be formalized as follows. After calculating normalized structure-factor magnitudes jEh j from the observed jFh j to generate all possible phase triples within a reasonably high Ah threshold, new phase values can be estimated after origin definition by use of the tangent formula (Karle & Hauptman, 1956): P k Wh jEk jjEh
k j sinðk þ h
k Þ : tan h ¼ P r kr Wh jEk jjEh
k j cosðk þ h
k Þ The reliability of the phase estimate depends on the variance Vðh Þ, which is directly related to the magnitude of h, i.e.  P kr

2 Ah; k cosðk þ h
k Þ

þ

 P

Ah; k ðcos h; k
th; k Þ2 P ; h; k Ah; k

where th; k ¼ I1 ðAh; k Þ=I0 ðAh; k Þ and h; k ¼ h þ k þ 
h
k . In the first application (Dorset et al., 1979) of multisolution phasing to electron-diffraction data (using the program QTAN), n-beam dynamical structure factors generated for cytosine and disodium 4-oxypyrimidine-2-sulfinate were used to assess the effect of increasing crystal thickness and electron accelerating voltage on the success of the structure determination. At 100 kV ˚ thick were usable for data collection and at samples at least 80 A ˚ – 1000 kV this sample thickness limit could be pushed to 300 A ˚ if a partial structure were accepted for later or, perhaps, 610 A Fourier refinement. NQEST identified the best phase solutions. Later QTAN was used to evaluate the effect of elastic crystal bend on the structure analysis of cytosine (Moss & Dorset, 1982). In actual experimental applications, two forms of thiourea were investigated with QTAN (Dorset, 1992d), using published three-dimensional electron-diffraction intensities (Dvoryankin & Vainshtein, 1960, 1962). Analysis of the centrosymmetric paraelectric structure yielded results equivalent to those found earlier by symbolic addition (Dorset, 1991b). Analysis of the noncentrosymmetric ferroelectric form was also quite successful (Dorset, 1992d). In both cases, the correct structure was found at the lowest value of NQEST. Re-analysis of the diketopiperazine structure with QTAN also found the correct structure (Dorset & McCourt, 1994a) within the four lowest values of NQEST, but not the one at the lowest value. The effectiveness of this figure of merit became more questionable when QTAN was used to solve the noncentrosymmetric crystal structure of a polymer (Dorset, McCourt, Kopp et al., 1994). The solution could not be found readily when NQEST was used but was easily identified when the minimal function RðÞ was employed instead. MULTAN has been used to phase simulated data from copper ˚ perchlorophthalocyanine (Fan et al., 1985). Within the 2 A

h ’ hk þ h
k ikr :

2h ¼

h; k

2 Ah; k sinðk þ h
k Þ ;

kr

Ah; k is identical to the A value defined in the previous section. In the initial stages of phase determination h is replaced by an expectation value E until enough phases are available to permit its calculation. The phase solutions indicated by the tangent formula can thus be ranked according to the phase variance and the determination of phases can be made symbolically from the most probable triple-product relationships. This procedure is equivalent to the one described above for the evaluation of phase-invariant sums

391

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION resolution of the electron-microscope image, if one seeks phases for diffraction data in reciprocal-space regions where the objective lens phase contrast transfer function jCðsÞj  0:2, the method proves to be successful. The method is also quite effective for ˚ to 1 A ˚ diffraction resolution, where the phase extension from 2 A low-angle data serve as a large initial phase set for the tangent formula. However, no useful results were found from an ab initio phase determination carried out solely with the electrondiffraction structure-factor magnitudes. Similar results were obtained when RANTAN was used to phase experimental data from this compound (Fan et al., 1991), i.e. the multisolution approach worked well for phase extension but not for ab initio phase determination. Additional tests were subsequently carried out with QTAN on an experimental hk0 electron-diffraction data set collected at 1200 kV (Dorset, McCourt, Fryer et al., 1994). Again, ab initio phase determination is not possible by this technique. However, if a basis set was constructed from the ˚ image, a correct solution could be Fourier transform of a 2.4 A found, but not at the lowest value of NQEST. This figure of merit was useful, however, when the basis set was taken from the symbolic addition determination mentioned in the previous section.

implied (Sayre, 1980). In X-ray crystallography this relationship has not been used very often, despite its accuracy. Part of the reason for this is that it requires relatively high resolution data for it to be useful. It can also fail for structures comprised of different atomic species. Since, relative to X-ray scattering factors, electron scattering factors span a narrower range of magnitudes at sin
= ¼ 0, it might be thought that the Sayre equation would be particularly useful in electron crystallography. In fact, Liu et al. (1988) were able to extend phases for simulated data from copper per˚ and chlorophthalocyanine starting at the image resolution of 2 A ˚ limit of an electron-diffraction data set. This reaching the 1 A ˚ basis set obtained from analysis has been improved with a 2.4 A the Fourier transform of an electron micrograph of this material ˚ limit of a 1200 kV electronat 500 kV and extended to the 1.0 A diffraction pattern (Dorset et al., 1995). Using the partial phase sets for zonal diffraction data from several polymers by symbolic addition (see above), the Sayre equation has been useful for extending into the whole hk0 set, often with great accuracy. The size of the basis set is critical but the connectivity to access all reflections is more so. Fan and co-workers have had considerable success with the analysis of incommensurately modulated structures. The average structure (basis set) is found by highresolution electron microscopy and the ‘superlattice’ reflections, corresponding to the incommensurate modulation, are assigned phases in hyperspace by the Sayre convolution. Examples include a high Tc superconductor (Mo et al., 1992) and the mineral ankangite (Xiang et al., 1990). Phases of regular inorganic crystals have also been extended from the electron micrograph to the electron-diffraction resolution by this technique (Hu et al., 1992). In an investigation of how direct methods might be used for phase extension in protein electron crystallography, lowresolution phases from two proteins, bacteriorhodopsin (Henderson et al., 1986) and halorhodopsin (Havelka et al., 1993) were extended to higher resolution with the Sayre equation (Dorset et al., 1995). For the noncentrosymmetric bacter˚ basis set was used, whereas a iorhodopsin hk0 projection a 10 A ˚ 15 A set was accepted for the centrosymmetric halorhodopsin ˚ resolution were projection. In both cases, extensions to 6 A reasonably successful. For bacteriorhodopsin, for which data ˚ , problems with the extension were were available to 3.5 A ˚ , corresponding to a minimum in a plot of encountered near 5 A average intensity versus resolution. Suggestions were made on how a multisolution procedure might be successful beyond this point.

2.5.8.5. Density modification Another method of phase determination, which is best suited to refining or extending a partial phase set, is the Hoppe– Gassmann density modification procedure (Hoppe & Gassmann, 1968; Gassmann & Zechmeister, 1972; Gassmann, 1976). The procedure is very simple but also very computer-intensive. Starting with a small set of (phased) Fh , an initial potential map ’ðrÞ is calculated by Fourier transformation. This map is then modified by some real-space function, which restricts peak sizes to a maximum value and removes all negative density regions. The modified map ’0 ðrÞ is then Fourier-transformed to produce a set of phased structure factors. Phase values are accepted via another modification function in reciprocal space, e.g. Ecalc =Eobs p, where p is a threshold quantity. The new set is then transformed to obtain a new ’ðrÞ and the phase refinement continues iteratively until the phase solution converges (judged by lower crystallographic R values). The application of density modification procedures to electron-crystallographic problems was assessed by Ishizuka et al. (1982), who used simulated data from copper perchlorophthalocyanine within the resolution of the electron-microscope image. The method was useful for finding phase values in reciprocal-space regions where the transfer function jCðsÞj  0:2. As a technique for phase extension, density modification was acceptable for test cases where the resolution was extended from ˚ , or 2.01 to 1.21 A ˚ , but it was not very satisfactory for 1.67 to 1.0 A ˚ . There appear to a resolution enhancement from 2.5 to 1.67 A have been no tests of this method yet with experimental data. However, the philosophy of this technique will be met again below in the description of the maximum entropy and likelihood procedure.

2.5.8.7. Maximum entropy and likelihood Maximum entropy has been applied to electron crystallography in several ways. In the sense that images are optimized, the entropy term S¼


Pi ln Pi ;

i

P where Pi ¼ pi = i pi and pi is a pixel density, has been evaluated for various test electron-microscope images. For crystals, the true projected potential distribution function is thought to have the maximum value of S. If the phase contrast transfer function used to obtain a micrograph is unknown, test images (i.e. trial potential maps) can be calculated for different values of ftrial. The value that corresponds to the maximum entropy would be near the true defocus. In this way, the actual objective lens transfer function can be found for a single image (Li, 1991) in addition to the other techniques suggested by this group. Another use of the maximum-entropy concept is to guide the progress of a direct phase determination (Bricogne & Gilmore, 1990; Gilmore et al., 1990). Suppose that there is a small set H of known phases h2H (corresponding either to origin

2.5.8.6. Convolution techniques One of the first relationships ever derived for phase determination is the Sayre (1952) equation: Fh ¼

P


X FF ; V k k h
k

which is a simple convolution of phased structure factors multiplied by a function of the atomic scattering factors. For structures with nonoverlapping atoms, consisting of one atomic species, it is an exact expression. Although the convolution term resembles part of the tangent formula above, no statistical averaging is

392

2.5. ELECTRON DIFFRACTION AND ELECTRON MICROSCOPY IN STRUCTURE DETERMINATION definition, or the Fourier transform of an electron micrograph, or both) with associated unitary structure-factor amplitudes jUh2H j. [The unitary structure factor is defined as jUh j ¼ jEh j=ðNÞ1=2 .] As usual, the task is to expand into the unknown phase set K to solve the crystal structure. From Bayes’ theorem, the procedure is based on an operation where pðmapjdataÞ / pðmapÞpðdatajmapÞ. This means that the probability of successfully deriving a potential map, given diffraction data, is estimated. This so-called posterior probability is approximately proportional to the product of the probability of generating the map (known as the prior) and the probability of generating the data, given the map (known as the likelihood). The latter probability consults the observed data and can be used as a figure of merit. Beginning with the basis set H, a trial map is generated from the limited number of phased structure factors. As discussed above, the map can be immediately improved by removing all negative density. The map can be improved further if its entropy is maximized using the equation given above for S. This produces the so-called maximum-entropy prior qME ðXÞ. So far, it has been assumed that all jUh2K j ¼ 0. If large reflections from the K set are now added and their phase values are permuted, then a number of new maps can be generated and their entropies can be maximized as before. This creates a phasing ‘tree’ with many possible solutions; individual branch points can have further reflections added via permutations to produce further sub-branches, and so on. Obviously, some figure of merit is needed to ‘prune’ the tree, i.e. to find likely paths to a solution. The desired figure of merit is the likelihood LðHÞ. First a quantity

potential distribution or its Fourier transform is represented significantly in the recorded signal. It would be a mistake, however, to presume that these data ever conform strictly to the kinematical approximation, for there is always some deviation from this ideal scattering condition that can affect the structure analysis. Despite this fact, some direct phasing procedures have been particularly ‘robust’, even when multiple scattering perturbations to the data are quite obvious (e.g. as evidenced by large crystallographic residuals). The most effective direct phasing procedures seem to be those based on the 2 triple invariants. These phase relationships will not only include the symbolic addition procedure, as it is normally carried out, but also the tangent formula and the Sayre equation (since it is well known that this convolution can be used to derive the functional form of the three-phase invariant). The strict ordering of jEh j magnitudes is, therefore, not critically important so long as there are no major changes from large to small values (or vice versa). This was demonstrated in direct phase determinations of simulated n-beam dynamical diffraction data from a sulfur-containing polymer (Dorset & McCourt, 1992). Nevertheless, there is a point where measured data cannot ˚ -thick epitaxically be used. For example, intensities from ca 100 A oriented copper perchlorophthalocyanine crystals become less and less representative of the unit-cell transform at lower electron-beam energies (Tivol et al., 1993) and, accordingly, the success of the phase determination is compromised (Dorset, McCourt, Fryer et al., 1994). The similarity between the Sayre convolution and the interactions of structure-factor terms in, e.g., the multislice formulation of n-beam dynamical scattering was noted by Moodie (1965). It is interesting to note that dynamical scattering interactions observed by direct excitation of 2 and 1 triples in convergent-beam diffraction experiments can actually be exploited to determine crystallographic phases to very high precision (Spence & Zuo, 1992, pp. 56–63). While the evaluation of positive quartet invariant sums (see Chapter 2.2) seems to be almost as favourable in the electron diffraction case as is the evaluation of 2 triples, negative quartet invariants seem to be particularly sensitive to dynamical diffraction. If dynamical scattering can be modelled crudely by a convolutional smearing of the diffraction intensities, then the lowest structure-factor amplitudes, and hence the estimates of lowest jEh j values, will be the ones most compromised. Since the negative quartet relationships require an accurate prediction of small ‘cross-term’ jEh j values, multiple scattering can, therefore, limit the efficacy of this invariant for phase determination. In initial work, negative quartets have been mostly employed in the NQEST figure of merit, and analyses (Dorset, McCourt, Fryer et al., 1994; Dorset & McCourt, 1994a) have shown how the degradation of weak kinematical jEh j terms effectively reduced its effectiveness for locating correct structure solutions via the tangent formula, even though the tangent formula itself (based on triple phase estimates) was quite effective for phase determination. Substitution of the minimal function RðÞ for NQEST seems to have overcome this difficulty. [It should be pointed out, though, that only the 2 -triple contribution to RðÞ is considered.] Structure refinement is another area where the effects of dynamical scattering are also problematic. For example, in the analysis of the paraelectric thiourea structure (Dorset, 1991b) from published texture diffraction data (Dvoryankin & Vainshtein, 1960), it was virtually impossible to find a chemically reasonable structure geometry by Fourier refinement, even though the direct phase determination itself was quite successful. The best structure was found only when higher-angle intensities (i.e. those least affected by dynamical scattering) were used to generate the potential map. Later analyses on heavy-atomcontaining organics (Dorset et al., 1992) found that the lowest kinematical R-factor value did not correspond to the chemically correct structure geometry. This observation was also made in the

h ¼ 2NR exp½
Nðr2 þ R2 ÞIo ð2NrRÞ; where r ¼ jME Uh j (the calculated unitary structure factors) and R ¼ jo Uh j (the observed unitary structure factors), is defined. From this one can calculate LðHÞ ¼

P

ln h :

h62H

The null hypothesis LðHo Þ can also be calculated from the above when r ¼ 0, so that the likelihood gain LLg ¼ LðHÞ
LðHo Þ ranks the nodes of the phasing tree in order of the best solutions. Applications have been made to experimental electroncrystallographic data. A small-molecule structure starting with phases from an electron micrograph and extending to electrondiffraction resolution has been reported (Dong et al., 1992). Other experimental electron-diffraction data sets used in other direct phasing approaches (see above) also have been assigned phases by this technique (Gilmore, Shankland & Bricogne, 1993). These include intensities from diketopiperazine and basic copper chloride. An application of this procedure in protein structure analysis has been published by Gilmore et al. (1992) and Gilmore, ˚ phases, it was Shankland & Fryer (1993). Starting with 15 A possible to extend phases for bacteriorhodopsin to the limits of the electron-diffraction pattern, apparently with greater accuracy than possible with the Sayre equation (see above). 2.5.8.8. Influence of multiple scattering on direct electroncrystallographic structure analysis The aim of electron-crystallographic data collection is to minimize the effect of dynamical scattering, so that the unit-cell

393

2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION Bo¨ttcher, B., Wynne, S. A. & Crowther, R. A. (1997). Determination of the fold of the core protein of hepatitis B virus by electron cryomicroscopy. Nature (London), 386, 88–91. Bracewell, R. N. (1956). Strip integration in radio astronomy. Austr. J. Phys. 9, 198–217. Bracewell, R. N. & Riddle, A. C. (1967) Inversion of fan-beam scans in radio astronomy. Astrophys. J. 150, 427–434. Bricogne, G. & Gilmore, C. J. (1990). A multisolution method of phase determination by combined maximization of entropy and likelihood. I. Theory, algorithms and strategy. Acta Cryst. A46, 284–297. Brisse, F. (1989). Electron diffraction of synthetic polymers: the model compound approach to polymer structure. J. Electron Microsc. Tech. 11, 272–279. Buxton, B., Eades, J. A., Steeds, J. W. & Rackham, G. M. (1976). The symmetry of electron diffraction zone axis patterns. Philos. Trans. R. Soc. London Ser. A, 181, 171–193. Carazo, J. M. (1992). The fidelity of 3D reconstruction from incomplete data and the use of restoration methods. In Electron Tomography, edited by J. Frank, pp. 117–166. New York: Plenum. Carazo, J. M. & Carrascosa, J. L. (1986). Information recovery in missing angular data cases: an approach by the convex projections method in three dimensions. J. Microsc. 145, 23–43. Carazo, J. M. & Carrascosa, J. L. (1987). Restoration of direct Fourier three-dimensional reconstructions of crystalline specimens by the method of convex projections. J. Microsc. 145, 159–177. Cherns, D., Kiely, C. J. & Preston, A. R. (1988). Electron diffraction studies of strain in epitaxial bicrystals and multilayers. Ultramicroscopy, 24, 355–370. Cherns, D. & Preston, A. R. (1986). Convergent beam diffraction studies of crystal defects. Proc. XI Int. Congr. Electron Microsc., Kyoto, Japan, p. 721. Clark, J. J., Palmer, M. R. & Lawrence, P. D. (1985). A transformation method for the reconstruction of functions from nonuniformly spaced samples. IEEE Trans. Acoust. Speech Signal Process. 33, 1151–1165. Cochran, W., Crick, F. H. C. & Vand, V. (1952). The structure of synthetic polypeptides. 1. The transform of atoms on a helix. Acta Cryst. 5, 581– 586. Conway, J. F., Cheng, N., Zlotnick, A., Wingfield, P. T., Stahl, S. J. & Steven, A. C. (1997). Visualization of a 4-helix bundle in the hepatitis B virus capsid by cryo-electron microscopy. Nature (London), 386, 91–94. Cowley, J. M. (1953). Structure analysis of single crystals by electron diffraction. II. Disordered boric acid structure. Acta Cryst. 6, 522–529. Cowley, J. M. (1956). A modified Patterson function. Acta Cryst. 9, 397– 398. Cowley, J. M. (1961). Diffraction intensities from bent crystals. Acta Cryst. 14, 920–927. Cowley, J. M. (1981). Diffraction Physics, 2nd ed. Amsterdam: NorthHolland. Cowley, J. M. (1995). Diffraction Physics, 3rd ed. Amsterdam: NorthHolland. Cowley, J. M. & Au, A. Y. (1978). Image signals and detector configurations for STEM. In Scanning Electron Microscopy, Vol. 1, pp. 53–60. AFM O’Hare: SEM Inc. Cowley, J. M. & Moodie, A. F. (1957). The scattering of electrons by atoms and crystals. I. A new theoretical approach. Acta Cryst. 10, 609–619. Cowley, J. M. & Moodie, A. F. (1959). The scattering of electrons by atoms and crystals. III. Single-crystal diffraction patterns. Acta Cryst. 12, 360– 367. Cowley, J. M. & Moodie, A. F. (1960). Fourier images. IV. The phase grating. Proc. Phys. Soc. London, 76, 378–384. Cowley, J. M., Moodie, A. F., Miyake, S., Takagi, S. & Fujimoto, F. (1961). The extinction rules for reflections in symmetrical electron diffraction spot patterns. Acta Cryst. 14, 87–88. Cowley, J. M., Rees, A. L. G. & Spink, J. A. (1951). Secondary elastic scattering in electron diffraction. Proc. Phys. Soc. London Sect. A, 64, 609–619. Crame´r, H. (1954). Mathematical Methods of Statistics. Princeton University Press. Crowther, R. A. & Amos, L. A. (1971). Harmonic analysis of electron microscope images with rotational symmetry. J. Mol. Biol. 60, 123–130. Crowther, R. A. & Amos, L. A. (1972). Three-dimensional image reconstructions of some small spherical viruses. Cold Spring Harbor Symp. Quant. Biol. 36, 489–494. Crowther, R. A., Amos, L. A., Finch, J. T., DeRosier, D. J. & Klug, A. (1970). Three dimensional reconstruction of spherical viruses by

least-squares refinement of diketopiperazine (Dorset & McCourt, 1994a). It is obvious that, if a global minimum is sought for the crystallographic residual, then dynamical structure factors, rather than kinematical values, should be compared to the observed values (Dorset et al., 1992). Ways of integrating such calculations into the refinement process have been suggested (Sha et al., 1993). Otherwise one must constrain the refinement to chemically reasonable bonding geometry in a search for a local R-factor minimum. Corrections for such deviations from the kinematical approximation are complicated by the presence of other possible data perturbations, especially if microareas are being sampled, e.g. in typical selected-area diffraction experiments. Significant complications can arise from the diffraction incoherence observed from elastically deformed crystals (Cowley, 1961) as well as secondary scattering (Cowley et al., 1951). These complications were also considered for the larger (e.g. millimetre diameter) areas sampled in an electron-diffraction camera when recording texture diffraction patterns (Turner & Cowley, 1969), but, because of the crystallite distributions, it is sometimes found that the two-beam dynamical approximation is useful (accounting for a number of successful structure analyses carried out in Moscow). MT is grateful to Professor J. C. H. Spence for critical reading of the manuscript of Section 2.5.3, for correcting the English and for helpful comments, and to Professor M. Terauchi and Dr K. Tsuda of Tohoku University and Dr K. Saitoh of Nagoya University for useful discussions.

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3.1. Distances, angles, and their standard uncertainties By D. E. Sands

3.1.1. Introduction A crystal structure analysis provides information from which it is possible to compute distances between atoms, angles between interatomic vectors, and the uncertainties in these quantities. In Cartesian coordinate systems, these geometric computations require the Pythagorean theorem and elementary trigonometry. The natural coordinate systems of crystals, though, are determined by symmetry, and only in special cases are the basis vectors (or coordinate axes) of these systems constrained to be of equal lengths or mutually perpendicular. It is possible, of course, to transform the positional parameters of the atoms to a Cartesian system and perform the subsequent calculations with the transformed coordinates. Along with the coordinates, the transformations must be applied to anisotropic thermal factors, variance–covariance matrices and other important quantities. Moreover, leaving the natural coordinate system of the crystal sacrifices the simplified relationships imposed by translational and point symmetry; for example, if an atom has fractional coordinates x1, x2 , x3 , an equivalent atom will be at 1 þ x1 , x2 , x3 , etc. Fortunately, formulation of the calculations in generalized rectilinear coordinate systems is straightforward, and readily adapted to computer languages (Section 3.1.12 illustrates the use of Fortran for such calculations). The techniques for these computations are those of tensor analysis, which provides a compact and elegant notation. While an effort will be made to be self-sufficient in this chapter, some proficiency in vector algebra is assumed, and the reader not familiar with the basics of tensor analysis should refer to Chapter 1.1 and Sands (1982a).

Subscripts are used for quantities that transform the same way as the basis vectors ai ; such quantities are said to transform covariantly. Superscripts denote quantities that transform the same way as coordinates xi ; these quantities are said to transform contravariantly (Sands, 1982a). Equation (3.1.2.4) is in a form convenient for computer evaluation, with indices i and j taking successively all values from 1 to 3. The matrix form of (3.1.2.4) is useful both for symbolic manipulation and for computation,

3.1.2. Scalar product The scalar product of vectors u and v is defined as

The length of v is, therefore, given by

u
v ¼ uv cos ’;

u
v ¼ uT gv;

ð3:1:2:6Þ

where the superscript italic T following a matrix symbol indicates a transpose. Written out in full, (3.1.2.6) is 0

g11 u
v ¼ ðu1 u2 u3 Þ@ g21 g31

g12 g22 g32

10 1 1 g13 v g23 A@ v2 A: g33 v3

ð3:1:2:7Þ

If u is the column vector with components u1 ; u2 ; u3, uT is the corresponding row vector shown in (3.1.2.7).

3.1.3. Length of a vector By (3.1.2.1), the scalar product of a vector with itself is v
v ¼ ðvÞ2 :

v ¼ ðvi v j gij Þ1=2 :

ð3:1:3:1Þ

ð3:1:3:2Þ

ð3:1:2:1Þ Computation of lengths in a generalized rectilinear coordinate system is thus simply a matter of evaluating the double summation vi v j gij and taking the square root.

where u and v are the lengths of the vectors and ’ is the angle between them. In terms of components, u
v ¼ ðui ai Þ
ðv j aj Þ

ð3:1:2:2Þ

3.1.4. Angle between two vectors

i j

ð3:1:2:3Þ

i j

ð3:1:2:4Þ

By (3.1.2.1) and (3.1.2.4), the angle ’ between vectors u and v is given by

u
v ¼ u v ai
aj u
v ¼ u v gij :

’ ¼ cos1 ½ui v j gij =ðuvÞ: In all equations in this chapter, the convention is followed that summation is implied over an index that is repeated once as a subscript and once as a superscript in an expression; thus, the right-hand side of (3.1.2.4) implies the sum of nine terms

An even more concise expression of equations such as (3.1.4.1) is possible by making use of the ability of the metric tensor g to convert components from contravariant to covariant (Sands, 1982a). Thus,

u1 v1 g11 þ u1 v2 g12 þ . . . þ u3 v3 g33 :

vi ¼ gij v j ; The gij in (3.1.2.4) are the components of the metric tensor [see Chapter 1.1 and Sands (1982a)] gij ¼ ai
aj :

uj ¼ gij ui ;

ð3:1:4:2Þ

and (3.1.2.4) may be written succinctly as u
v ¼ ui vi

ð3:1:2:5Þ

Copyright © 2010 International Union of Crystallography

ð3:1:4:1Þ

404

ð3:1:4:3Þ

3.1. DISTANCES, ANGLES, AND THEIR STANDARD UNCERTAINTIES or

"ijk ¼ 0; i

u
v ¼ ui v :

"ijk ¼ 0; if j ¼ i or k ¼ i or k ¼ j:

ð3:1:6:3Þ

ð3:1:4:4Þ If the indices are all different,

With this notation, the angle calculation of (3.1.4.1) becomes "ijk ¼ PV; 1

1

i

i

’ ¼ cos ½u vi =ðuvÞ ¼ cos ½ui v =ðuvÞ:

"ijk ¼ PV 

ð3:1:6:4Þ

ð3:1:4:5Þ for even permutations of ijk (123, 231, or 312), and

The summations in (3.1.4.3), (3.1.4.4) and (3.1.4.5) include only three terms, and are thus equivalent in numerical effort to the computation in a Cartesian system, in which the metric tensor is represented by the unit matrix and there is no numerical distinction between covariant components and contravariant components. Appreciation of the elegance of tensor formulations may be enhanced by noting that corresponding to the metric tensor g with components gij there is a contravariant metric tensor g with components gij ¼ ai
a j :

"ijk ¼ PV;

The ai are contravariant basis vectors, known to crystallographers as reciprocal axes. Expressions parallel to (3.1.4.2) may be written, in which g plays the role of converting covariant components to contravariant components. These tensors thus express mathematically the crystallographic notions of crystal space and reciprocal space [see Chapter 1.1 and Sands (1982a)].

3.1.7. Components of vector product As is shown in Sands (1982a), the components of the vector product u ^ v are given by u ^ v ¼ "ijk ui v j ak ;

ð3:1:7:1Þ

where again ak is a reciprocal basis vector (some writers use a ; b ; c to represent the reciprocal axes). A special case of (3.1.7.1) is

3.1.5. Vector product The scalar product defined in Section 3.1.2 is one multiplicative operation of two vectors that may be defined; another is the vector product, which is denoted as u ^ v (or u  v or [uv]). The vector product of vectors u and v is defined as a vector of length uv sin ’, where ’ is the angle between the vectors, and of direction perpendicular to both u and v in the sense that u, v and u ^ v form a right-handed system; u ^ v is generated by rotating u into v and advancing in the direction of a right-handed screw. The magnitude of u ^ v, given by

ai ^ aj ¼ "ijk ak ;

ð3:1:7:2Þ

which may be taken as a defining equation for the reciprocal basis vectors. Similarly, ai ^ a j ¼ "ijk ak ;

ð3:1:5:1Þ

ð3:1:7:3Þ

which completes the characterization of the dual vector system with basis vectors ai and a j obeying

is equal to the area of the parallelogram defined by u and v. It follows from the definition that u ^ v ¼ v ^ u:

ð3:1:6:5Þ

for odd permutations (132, 213, or 321). Here, P ¼ þ1 for righthanded axes, P ¼ 1 for left-handed axes, V is the unit-cell volume, and V  ¼ 1=V is the volume of the reciprocal cell defined by the reciprocal basis vectors ai ; a j ; ak. A discussion of the properties of the permutation tensors may be found in Sands (1982a). In right-handed Cartesian systems, where P ¼ 1, and V ¼ V  ¼ 1, the permutation tensors are equivalent to the permutation symbols denoted by eijk.

ð3:1:4:6Þ

ju ^ vj ¼ uv sin ’

"ijk ¼ PV 

ai
a j ¼
ij :

ð3:1:5:2Þ

ð3:1:7:4Þ

In (3.1.7.4),
ij is the Kronecker delta, which equals 1 if i ¼ j, 0 if i 6¼ j. The relationships between these quantities are explored at some length in Sands (1982a).

3.1.6. Permutation tensors Many relationships involving vector products may be expressed compactly and conveniently in terms of the permutation tensors, defined as "ijk ¼ ai
aj ^ ak "

ijk

i

j

k

¼a
a ^a :

3.1.8. Some vector relationships The results developed above lead to several useful relationships between vectors; for derivations, see Sands (1982a).

ð3:1:6:1Þ

3.1.8.1. Triple vector product

ð3:1:6:2Þ

Since ai
aj ^ ak represents the volume of the parallelepiped defined by vectors ai ; aj ; ak, it follows that "ijk vanishes if any two indices are equal to each other. The same argument applies, of course, to "ijk. That is,

405

u ^ ðv ^ wÞ ¼ ðu
wÞv  ðu
vÞw

ð3:1:8:1Þ

ðu ^ vÞ ^ w ¼ ðv
wÞu þ ðu
wÞv:

ð3:1:8:2Þ

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING 3.1.8.2. Scalar product of vector products

x
h¼1

ðu ^ vÞ
ðw ^ zÞ ¼ ðu
wÞðv
zÞ  ðu
zÞðv
wÞ:

ð3:1:8:3Þ

in which the vector h has coordinates h ¼ ð1=u1 ; 1=v2 ; 1=w3 Þ:

A derivation of this result may be found also in Shmueli (1974).

ð3:1:8:4Þ

ðu ^ vÞ ^ ðw ^ zÞ ¼ ðu
v ^ zÞw  ðu
v ^ wÞz:

ð3:1:8:5Þ

h ¼ 1=d:

Among several ways of characterizing a plane in a general rectilinear coordinate system is a description in terms of the coordinates of three non-collinear points that lie in the plane. If the points are U, V and W, lying at the ends of vectors u, v and w, the vectors u  v, v  w and w  u are in the plane. The vector ð3:1:9:1Þ

is normal to the plane. Expansion of (3.1.9.1) yields z ¼ ðu ^ vÞ þ ðv ^ wÞ þ ðw ^ uÞ:

l ¼ ð1  p
hÞ=h;

ð3:1:9:2Þ

ð3:1:9:3Þ

 ¼ cos1 ½h
h0 =ðhh0 Þ:

If now x is any vector from the origin to the plane, x  u is in the plane, and ðx  uÞ
z ¼ 0:

ð3:1:9:12Þ

where a negative sign indicates that the point is on the opposite side of the plane from the origin. The dihedral angle  between planes with normals h and h0 is

Making use of (3.1.7.1), z ¼ "ijk ðu j vk þ v j wk þ w j uk Þai :

ð3:1:9:11Þ

If the indices hi are relatively prime integers, the theory of numbers tells us that the Diophantine equation (3.1.9.8) has solutions xi that are integers. Points whose contravariant components are integers are lattice points, and such a plane passes through an infinite number of lattice points and is called a lattice plane. Thus, the hi for lattice planes are the familiar Miller indices of crystallography. Calculations involving planes become quite manageable when the normal vector h is introduced. Thus, the distance l from a point P with coordinates pi to a plane characterized by h is

3.1.9. Planes

z ¼ ðu  vÞ ^ ðv  wÞ

From (3.1.9.2),

 ¼ cos1 ½ðu ^ vÞ
ðv ^ wÞ=ju ^ vjjv ^ wj: ð3:1:9:5Þ

th ¼ t  ðt
hÞh=h2 j k

j

k

j k

i j

k

"ijk x ðu v þ v w þ w u Þ ¼ "ijk u v w :

ð3:1:9:6Þ

uh ¼ u  ðu
hÞh=h :

ð3:1:9:16Þ

The angle between th and uh represents a torsion about h (Sands, 1982b). Another approach to the torsion angle, which gives equivalent results (Shmueli, 1974), is to compute the angle between t ^ h and u ^ h using (3.1.8.3).

ð3:1:9:7Þ 3.1.10. Variance–covariance matrices Refinement of a crystal structure yields both the parameters that describe the structure and estimates of the uncertainties of those parameters. Refinement by the method of least squares minimizes a weighted sum of squares of residuals. In the matrix notation of Hamilton’s classic book (Hamilton, 1964), values of

which may be written xi hi ¼ 1

ð3:1:9:15Þ 2

If, in particular, the points are on the coordinate axes, their designations are ½u1 ; 0; 0, ½0; v2 ; 0 and ½0; 0; w3 , and (3.1.9.6) becomes x1 =u1 þ x2 =v2 þ x3 =w3 ¼ 1;

ð3:1:9:14Þ

A similar calculation gives angles of torsion. Let th and uh be, respectively, the projections of vectors t and u onto the plane with normal h.

Rearrangement of (3.1.9.4) with x
z on the left and u
z on the right, and using (3.1.9.3) for z on the left leads to i

ð3:1:9:13Þ

A variation of (3.1.9.13) expresses  in terms of vector u in the first plane, vector w in the second plane, and vector v, the intersection of the planes, as (Shmueli, 1974)

ð3:1:9:4Þ

u
z ¼ u
v ^ w:

ð3:1:9:10Þ

That is, the covariant components of h are given by the reciprocals of the intercepts of the plane on the axes. The vector h is normal to the plane it describes (Sands, 1982a) and the length of h is the reciprocal of the distance d of the plane from the origin; i.e.,

3.1.8.3. Vector product of vector products ðu ^ vÞ ^ ðw ^ zÞ ¼ ðu
w ^ zÞv  ðv
w ^ zÞu

ð3:1:9:9Þ

ð3:1:9:8Þ

or

406

3.1. DISTANCES, ANGLES, AND THEIR STANDARD UNCERTAINTIES meters i of atoms A and B, respectively, we define M AA, M AB , M BA and M BB as 3  3 matrices with ijth elements covðxiA ; xAj Þ, covðxiA ; xBj Þ, covðxiB ; xAj Þ and covðxiB ; xBj Þ, respectively. If atom B0 is generated from atom B by symmetry operator S, such that

the m parameters to be determined are expressed by the m  1 column vector X given by X ¼ ðAT PAÞ1 AT PF;

ð3:1:10:1Þ

where F is an n  1 matrix representing the observations (structure factors or squares of structure factors), P is an n  n weight matrix that is proportional to the variance–covariance matrix of the observed F, A is an n  m design matrix consisting of the derivatives of each element of F with respect to each of the parameters and AT is the transpose of A. The variance– covariance matrix of the parameters is then given by M ¼ V T PVðAT PAÞ1 =ðn  mÞ:

xB0 ¼ SxB xiB0

@f @f  ð f Þ ¼ i j covðxi ; x j Þ; @x @x

M AB0 ¼ M AB ST

@f1 @f2 covðxi ; x j Þ: @xi @x j

ð3:1:10:8Þ

M B0 A ¼ SM BA

ð3:1:10:9Þ T

M B0 B0 ¼ SM BB S :

ð3:1:10:10Þ

If symmetry operator S is applied to both atoms A and B to generate atoms A0 and B0 , the corresponding matrices may be expressed by the matrix equation

ð3:1:10:3Þ

where, as usual, we are using the summation convention and summing over all parameters included in f. A generalization of (3.1.10.3) for two functions is covð f1 ; f2 Þ ¼

ð3:1:10:7Þ

it is shown in Sands (1966) that the variance–covariance matrices involving atom B0 are

ð3:1:10:2Þ

Here, V is the n  1 matrix of residuals, consisting of the differences between the observed and calculated values of the elements of F. Since V T PV=ðn  mÞ is just a single number, M is proportional to the inverse least-squares matrix ðAT PAÞ1 . Once the variance–covariance matrix of the parameters is known, the variances and covariances of any quantities derived from these parameters can be computed. The variance of a single function f is given by 2

¼

ð3:1:10:6Þ

Sij xBj ;




M A0 A0 M B0 A0

M A0 B0 M B0 B0




¼

SM AA ST SM BA ST

 SM AB ST : SM BB ST

ð3:1:10:11Þ

ð3:1:10:4Þ If G is a matrix that transforms to a new set of axes,

[The covariance of two quantities is, of course, just the variance if the two quantities are the same. For an elementary discussion of statistical covariance and correlation, see Sands (1977).] Equation (3.1.10.4) may now be extended to any number of functions (Sands, 1966); the k  k variance–covariance matrix C of k functions of m parameters is given in terms of the m  m variance–covariance matrix of the parameters by C ¼ DMDT ;

a0 ¼ Ga;

ð3:1:10:12Þ

the transformed variance–covariance matrix of the atomic parameters is

ð3:1:10:5Þ

in which the ijth element of the k  m matrix D is the derivative of function fi with respect to parameter j. Element CII (no summation implied over I) is the variance of function fI , and CIJ is the covariance of functions fI and fJ . The calculation of C must, of course, include the contributions of all sources of error, so M in (3.1.10.5) should include the variances and covariances of the unit-cell dimensions and of any other relevant parameters with non-negligible uncertainties. It may be easier, in some cases, to carry out calculations of variances and covariances in steps. For example, the variance– covariance matrix of a set of distances may be computed and then other quantities may be determined as functions of the distances. It is imperative that all nonvanishing covariances be included in every stage of the calculation; only in special cases are the covariances negligible, and often they are large enough to affect the results seriously (Sands, 1977). These principles may be used to explore the effects of symmetry or of transformations on the variance–covariance matrices of atomic parameters and derived quantities. Using the notation of Sands (1966), with xiA and xiB the positional para-

M 0 ¼ ðGT Þ1 MG1 :

ð3:1:10:13Þ

To apply these formulae to calculations of the errors and covariances of interatomic distances and angles, consider the triangle of atoms A, B, C with edges l1 ¼ AB, l2 ¼ BC, l3 ¼ CA, and angles 1, 2 , 3 at A, B, C, respectively. If the atoms are not related by symmetry,

 2 ðl1 Þ ¼ lT1 gðM AA  M AB  M BA þ M BB Þgl1 =l12 covðl1 ; l2 Þ ¼

lT1 gðM AB

ð3:1:10:14Þ

 M AC  M BB þ M BC Þgl2 =l1 l2 : ð3:1:10:15Þ

If atom B is generated from atom A by symmetry matrix S, the results, as derived in Sands (1966), are

407

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING variances and covariances of dihedral angles, of best planes, of torsion angles, and of other molecular parameters.

 2 ðl1 Þ ¼ l T1 gðM AA  SM AA  M AA ST þ SM AA ST Þgl 1 =l12 2

 ðl2 Þ ¼

ð3:1:10:16Þ

l T2 gðSM AA ST

T

 M AC S  SM AC þ M CC Þgl2 =l22

 2 ðl3 Þ ¼ l T3 gðM AA  M AC  M CA þ M CC Þgl3 =l32

3.1.11. Mean values ð3:1:10:17Þ

The weighted mean of a set of quantities Xi is hXi ¼

ð3:1:10:18Þ

covðl1 ; l2 Þ ¼ l T1 gðM AA ST  SM AA ST  M AC þ SM AC Þgl2 =l1 l2

covðl2 ; l3 Þ ¼

 2 ðhXiÞ ¼ wT Mw=ð ð3:1:10:21Þ

P

wi Þ2 :

w ¼ M 1 v;

ð3:1:11:1Þ

ð3:1:11:2Þ

ð3:1:11:3Þ

where the components of vector v are all equal (vi ¼ vj for all i and j); since (3.1.11.1) and (3.1.11.2) require only relative weights, we can assign vi ¼ 1 for all i. Placing these weights in (3.1.11.2) yields

 2 ð1 Þ ¼ ½cos2 2  2 ðl1 Þ  2 cos 2 covðl1 ; l2 Þ þ 2 cos 2 cos 3 covðl1 ; l3 Þ þ  2 ðl2 Þ  2 cos 3 covðl2 ; l3 Þ

 2 ðhXiÞ ¼ 1=

P

wi :

ð3:1:11:4Þ

ð3:1:10:22Þ

covð1 ; 2 Þ ¼ ½cos 1 cos 2  2 ðl1 Þ þ ðcos 2 cos 3  cos 1 Þ covðl1 ; l2 Þ

For the case of uncorrelated Xi , the weights are inversely proportional to the corresponding variances

þ ðcos 1 cos 3  cos 2 Þ covðl1 ; l3 Þ

wi ¼ 1= 2 ðXi Þ:

 cos 3  2 ðl2 Þ þ ð1 þ cos2 3 Þ covðl2 ; l3 Þ  cos 3  2 ðl3 Þ=ðl12 sin 1 sin 2 Þ ð3:1:10:23Þ covð1 ; l1 Þ ¼ ½ cos 2  2 ðl1 Þ þ covðl1 ; l2 Þ  cos 3 covðl1 ; l3 Þðl2 =l1 l3 sin 1 Þ

wi ;

Minimization of  2 ðhXiÞ leads to weights given by

In equations (3.1.10.14)–(3.1.10.21), li is a column vector with components the differences of the coordinates of the atoms connected by the vector. Representative formulae involving the angles 1, 2 , 3 are

þ cos2 3  2 ðl3 Þðl2 =l1 l3 sin 1 Þ2

P

ð3:1:10:20Þ

l T2 gðSM AA

þ M CA þ SM AC  M CC Þgl3 =l2 l3 :

wi Xi =

where the weights are typically chosen to minimize the variance of hXi. The variance may be computed from the variance– covariance matrix M of the Xi by

ð3:1:10:19Þ

covðl1 ; l3 Þ ¼ l T1 gðM AA þ SM AA þ M AC  SM AC Þgl3 =l1 l3

P

ð3:1:11:5Þ

For the case of two correlated variables, wi ¼ 1=½ 2 ðXi Þ  covðX1 ; X2 Þ:

ð3:1:10:24Þ

ð3:1:11:6Þ

2

covð1 ; l2 Þ ¼ ½ cos 2 covðl1 ; l2 Þ þ  ðl2 Þ  cos 3 covðl2 ; l3 Þðl2 =l1 l3 sin 1 Þ:

ð3:1:10:25Þ

Derivation and discussion of these equations may be found in Sands (1966, 1982b). The presence of systematic errors in the experimental data often results in these formulae producing estimates of the standard uncertainties of molecular dimensions that are too small; it has been suggested that such error estimates should be multiplied by 1.5 to make them more realistic (Taylor & Kennard, 1983). It is essential also that averages be computed only of similar quantities, and interatomic distances corresponding to different bond orders or different environments may not represent the same physical quantities; that is, there are reasons for the discrepancies, and averaging may obscure important information. Another source of error in molecular geometry parameters determined from crystallographic measurements is thermal motion, and distances should be corrected for such effects before making comparisons (Busing & Levy, 1964; Johnson, 1970, 1980). A discussion of the appropriateness of weighted and unweighted means may be found in Taylor & Kennard (1985), which suggests that the unweighted mean might even be preferable if environmental effects are large.

If any of the angles approach 0 or 180 , the denominators in (3.1.10.22)–(3.1.10.25) will become very small, necessitating highprecision arithmetic. Indeterminacies resulting from special relationships between atomic positions may require rederivation of the equations for variances and covariances, to take the relationships into account explicitly and avoid the indeterminacies. A true symmetry condition requiring, for example, a linear bond should cause little problem, and the corresponding variance will be zero. It is the indeterminacies not originating from crystal symmetry that demand caution, in recognizing them and in coping with them correctly. A general expression for the variance of a dihedral angle, in terms of the variances and covariances of the coordinates of the four atoms, is (Shmueli, 1974)  2 ðÞ ¼

XX @ @ j i j cov½xðkÞ ; xðnÞ ; i @x @x ðkÞ k n ðnÞ

ð3:1:10:26Þ

3.1.12. Computation It has been mentioned that the tensor formulation used in this chapter is particularly amenable to machine computation. As a simple illustration of this point, the following Fortran program will compute the lengths of vectors X and Y and the angle between them.

where, in addition to the usual tensor summation over i and j from 1 to 3, summation must be carried out over the four atoms (i.e., k and n vary from 1 to 4). Special cases of (3.1.10.26), corresponding to various levels of approximation of diagonal matrices and isotropic errors, are given in Shmueli (1974). Formulae in dyadic notation are given in Waser (1973) for the

408

3.1. DISTANCES, ANGLES, AND THEIR STANDARD UNCERTAINTIES References

DIMENSION X(3),Y(3),G(3,3),SUM(3) READ (5,10)(X(I),I = 1,3)

Busing, W. R. & Levy, H. A. (1964). Effect of thermal motion on the estimation of bond lengths. Acta Cryst. 17, 142–146. Hamilton, W. C. (1964). Statistics in Physical Science. New York: Ronald Press. Johnson, C. K. (1970). The effect of thermal motion on interatomic distances and angles. In Crystallographic Computing, edited by F. R. Ahmed, pp. 220–226. Copenhagen: Munksgaard. Johnson, C. K. (1980). Thermal motion analysis. In Computing in Crystallography, edited by R. Diamond, S. Ramaseshan & K. Venkatesan, pp. 14.01–14.19. Bangalore: Indian Academy of Sciences. Sands, D. E. (1966). Transformations of variance–covariance tensors. Acta Cryst. 21, 868–872. Sands, D. E. (1977). Correlation and covariance. J. Chem. Educ. 54, 90– 94. Sands, D. E. (1982a). Vectors and Tensors in Crystallography. Reading: Addison Wesley. Reprinted (1995) Dover Publications. Sands, D. E. (1982b). Molecular geometry. In Computational Crystallography, edited by D. Sayre, pp. 421–429. Oxford: Clarendon Press. Shmueli, U. (1974). On the standard deviation of a dihedral angle. Acta Cryst. A30, 848–849. Taylor, R. & Kennard, O. (1983). The estimation of average molecular dimensions from crystallographic data. Acta Cryst. B39, 517–525. Taylor, R. & Kennard, O. (1985). The estimation of average molecular dimensions. 2. Hypothesis testing with weighted and unweighted means. Acta Cryst. A41, 85–89. Waser, J. (1973). Dyadics and the variances and covariances of molecular parameters, including those of best planes. Acta Cryst. A29, 621–631.

READ (5,10)(Y(I),I = 1,3) READ (5,10)((G(I,J),J = 1,3),I = 1,3) 10 FORMAT (3F10.5) DO 20 I = 1,3 20 SUM(I) ¼ 0 DO 30 I = 1,3 DO 30 J = 1,3 SUM(1) ¼ SUM(1) + X(I)  X(J)  G(I,J) SUM(2) ¼ SUM(2) + Y(I)  Y(J)  G(I,J) SUM(3) ¼ SUM(3) + X(I)  Y(J)  G(I,J) 30 CONTINUE DIST1 ¼ SQRT(SUM(1)) DIST2 ¼ SQRT(SUM(2)) ANGLE ¼ 57.296  ACOS(SUM(3)/(DIST1  DIST2)) WRITE (6,10) DIST1,DIST2,ANGLE END

409

references

International Tables for Crystallography (2010). Vol. B, Chapter 3.2, pp. 410–417.

3.2. The least-squares plane By R. E. Marsh and V. Schomaker†

times go astray. As to the propagation of errors, numerous treatments have been given, but none that we have seen is altogether satisfactory. We refer all vectors and matrices to Cartesian axes, because that is the most convenient in calculation. However, a more elegant formulation can be written in terms of general axes [e.g., as in Shmueli (1981)]. The notation is troublesome. Indices are needed for atom number and Cartesian direction, and the exponent 2 is needed as well, which is difficult if there are superscript indices. The best way seems to be to write all the indices as subscripts and distinguish among them by context – i, j, 1, 2, 3 for directions; k, l, p (and sometimes K, . . . ) for atoms. In any case, atom first then direction if there are two subscripts; direction, if only one index for a vector component, but atom (in this section at least) if for a weight or a vector. And
d1 , e.g., for the standard uncertainty of the distance of atom 1 from a plane. For simplicity in practice, we use Cartesian coordinates throughout. The first task is to find the plane, which we write as

3.2.1. Introduction By way of introduction, we remark that in earlier days of crystal structure analysis, before the advent of high-speed computers and routine three-dimensional analyses, molecular planarity was often assumed so that atom coordinates along the direction of projection could be estimated from two-dimensional data [see, e.g., Robertson (1948)]. Today, the usual aim in deriving the coefficients of a plane is to investigate the degree of planarity of a group of atoms as found in a full, three-dimensional structure determination. We further note that, for such purposes, a crystallographer will often be served just as well by establishing the plane in an almost arbitrary fashion as by resorting to the most elaborate, nit-picking and pretentious least-squares treatment. The approximate plane and the associated perpendicular distances of the atoms from it will be all he needs as scaffolding for his geometrical and structural imagination; reasonable common sense will take the place of explicit attention to error estimates. Nevertheless, we think it appropriate to lay out in some detail the derivation of the ‘best’ plane, in a least-squares sense, through a group of atoms and of the standard uncertainties associated with this plane. We see two cases: (1) The weights of the atoms in question are considered to be isotropic and uncorrelated (i.e. the weight matrix for the positions of all the atoms is diagonal, when written in terms of Cartesian axes, and for each atom the three diagonal elements are equal). In such cases the weights may have little or nothing to do with estimates of random error in the atom positions (they may have been assigned merely for convenience or convention), and, therefore, no one should feel that the treatment is proper in respect to the theory of errors. Nevertheless, it may be desired to incorporate the error estimates (variances) of the atom positions into the results of such calculations, whereupon these variances (which may be anisotropic, with correlation between atoms) need to be propagated. In this case the distinction between weights (or their inverses) and variances must be kept very clear. (2) The weights are anisotropic and are presumably derived from a variance–covariance matrix, which may include correlation terms between different atoms; the objective is to achieve a truly proper Gaussian least-squares result.

0 ¼ m
r  d  mT r  d; where r is here the vector from the origin to any point on the plane (but usually represents the measured position of an atom), m is a unit vector parallel to the normal from the origin to the plane, d is the length of the normal, and m and r are the column representations of m and r. The least-squares condition is to find the stationary values of S  ½wk ðmT rk  dÞ2  subject to mT m ¼ 1, with rk , k ¼ 1; . . . ; n, the vector from the origin to atom k and with weights, wk , isotropic and without interatomic correlations for the n atoms of the plane. We also write S as S  ½wðmT r  dÞ2 , the subscript for atom number being implicit in the Gaussian summations ð½. . .Þ over all atoms, as it is also in the angle-bracket notation for the weighted average over all atoms, for example in hri – the weighted centroid of the groups of atoms – just below. First solve for d, the origin-to-plane distance. 1 @S ¼ ½wðmT r  dÞ ¼ 0; 2 @d d ¼ ½wmT r=½w  mT hri: 0¼

3.2.2. Least-squares plane based on uncorrelated, isotropic weights This is surely the most common situation; it is not often that one will wish to take the trouble, or be presumptive enough, to assign anisotropic or correlated weights to the various atoms. And one will sometimes, perhaps even often, not be genuinely interested in the hypothesis that the atoms actually are rigorously coplanar; for instance, one might be interested in examining the best plane through such a patently nonplanar molecule as cyclohexane. Moreover, the calculation is simple enough, given the availability of computers and programs, as to be a practical realization of the off-the-cuff treatment suggested in our opening paragraph. The problem of deriving the plane’s coefficients is intrinsically nonlinear in the way first discussed by Schomaker et al. (1959; SWMB). Any formulation other than as an eigenvalue– eigenvector problem (SWMB), as far as we can tell, will some-

Then S  ½wðmT r  dÞ2  ¼ ½wfmT ðr  hriÞg2   ½wðmT sÞ2   mT ½wssT m  mT Am: Here sk  rk  hri is the vector from the centroid to atom k. Then solve for m. This is the eigenvalue problem – to diagonalize A (bear in mind that Aij is just ½wsi sj ) by rotating the coordinate axes, i.e., to find the 3  3 arrays M and L, L diagonal, to satisfy MT AM ¼ L;

A and M are symmetric; the columns m of M are the direction cosines of, and the diagonal elements of L are the sums of

† Deceased.

Copyright © 2010 International Union of Crystallography

MT M ¼ I:

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3.2. THE LEAST-SQUARES PLANE weighted squares of residuals from, the best, worst and intermediate planes, as discussed by SWMB.

mi  "i ¼ i3 =ðL33  Lii Þ

3.2.2.1. Error propagation

¼ ½wðsi 3 þ s3 i Þ=ð½ws23   ½ws2i Þ; dc  "3 ¼ ½w3 =½w  h3 i;

Waser et al. (1973; WMC) carefully discussed how the random errors of measurement of the atom positions propagate into the derived quantities in the foregoing determination of a leastsquares plane. This section presents an extension of their discussion. To begin, however, we first show how standard firstorder perturbation theory conveniently describes the propagation of error into M and L when the positions rk of the atoms are incremented by the amounts rk  k and the corresponding quantities sk  rk  hri (the vectors from the centroid to the atoms) by the amounts k ; ðs ! s þ Þ; k  k  hi. (The need to account for the variation in position of the centroid, i.e. to distinguish between k and k , was overlooked by WMC.) The consequent increments in A; M and L are

and also d ¼ "1 hr1 i þ "2 hr2 i þ "3 : These results have simple interpretations. The changes in direction of the plane normal (the mi ) are rotations, described by "1 and "2 , in response to changes in moments acting against effective torsion force constants. For "2 , for example, the contribution of atom k to the total relevant moment, about direction 1, is wk sk3 sk2 (wk sk3 the ‘force’ and sk2 the lever arm), and its nominally first-order change has two parts, wk sk2 3 from the change in force and wk sk3 2 from the change in lever arm; the resisting torsion constant is ½ws22   ½ws23 , which, reflection will show, is qualitatively reasonable if not quantitatively obvious. The perpendicular displacement of the plane from the original centroid hri is "3, but there are two further contributions to d, the change in distance from origin to plane along the plane normal, that arise from the two components of out-of-plane rotation of the plane about its centroid. Note that "3 is not given by ½w3 =½w ¼ ½wð3  h3 iÞ=½w, which vanishes identically. There is a further, somewhat delicate point: If the group of atoms is indeed essentially coplanar, the sk3 are of the same order of magnitude as the ki, unlike the ski, i 6¼ 3, which are in general about as big as the lateral extent of the group. It is then appropriate to drop all terms in i or i ; i 6¼ 3, and, in the denominators, the terms in s2k3 . The covariances of the derived quantities (by covariances we mean here both variances and covariances) can now be written out rather compactly by extending the implicit designation of atom numbers to double sums, the first of each of two similar factors referring P to the first atom P index and the second to the second, e.g., kl wwðsi sj Þ . . .  kl wk wl ðski sij Þ . . .. Note that the various covariances, i.e. the averages over the presumed population of random errors of replicated measurements, are indicated by overlines, angle brackets having been pre-empted for averages over sets of atoms.

A ¼ ½wsT  þ ½wsT   ; M ¼ M; L  : Here the columns of  are expressed as linear combinations of the columns of M. Note also that both perturbations,  and , which are the adjustments to the orientations and associated eigenvalues of the principal planes, will depend on the reduced coordinates s and the perturbing influences  by way of , which in turn depends only on the reduced coordinates and the reduced shifts k. In contrast, d ¼ ðmT hriÞ ¼ ðmT Þhri þ mT hi; the change in the origin-to-plane distance for the plane defined by the column vectors m of M, depends on the hri and hi directly as well as on the s and  by way of the m: The first-order results arising from the standard conditions, MT M ¼ I; L diagonal, and MT AM ¼ L, are T þ  ¼ 0;  diagonal;

covðmi ; mj Þ  "i "j P kl wwðsi sj 3 3 þ s3 s3 i j þ si s3 3 j þ s3 sj i 3 Þ ; i; j ¼ 1; 2 ¼ f½wðs23  s2i Þgf½wðs23  s2j Þg P wwðski 3 3 þ sk3 i 3 Þ covðmi ; dc Þ  "i "3 ¼ kl ; i; j ¼ 1; 2 f½wðs23  s2i Þg½w P ww 
2 ðdc Þ  "23 ¼ kl 2 3 3 ½w

and T MT AM þ MT M þ MT AM ¼ T L þ L þ MT M ¼ : Stated in terms of the matrix components ij and ij , the first condition is ij ¼ ji, hence ii ¼ 0; i; j ¼ 1; 2; 3, and the second is ij ¼ 0; i 6¼ j. With these results the third condition then reads jj ¼ ðMT MÞjj ;


2 ðdÞ  hðdÞ2 i ¼ hr1 i2 "21 þ hr2 i2 "22 þ "23 þ 2hr1 ihr2 i"1 "2 þ 2hr1 i"1 "3 þ 2hr2 i"2 "3 :

j ¼ 1; 2; 3

ij ¼ ðMT MÞij =ðLjj  Lii Þ; i 6¼ j;

i ¼ 1; 2; k  rk ;

i; j ¼ 1; 2; 3: Interatomic covariance (e.g., k3 l3 ; k 6¼ l) thus presents no formal difficulty, although actual computation may be tedious. Nonzero covariance for the ’s may arise explicitly from interatomic covariance (e.g., ki lj ; k 6¼ l) of the errors in the atomic positions rk, and it will always arise implicitly because hi in k ¼ k  hi includes all the k and therefore has nonzero covariance with all of them and with itself, even if there is no interatomic covariance among the i ’s. If both types of interatomic covariance (explicit and implicit) are negligible, the " covariances simplify a great deal, the double

All this is analogous to the usual first-order perturbation theory, as, for example, in elementary quantum mechanics. Now rotate to the coordinates defined by WMC, with axes parallel to the original eigenvectors ½M ¼ I; Aij ¼ Lij ij ; ðMT MÞij ¼ ij , restrict attention to the best plane ðM13  m1 ¼ 0, M23  m2 ¼ 0; M33  m3 ¼ 1Þ, and define "T as ðm1 ; m2 ; dc Þ, keeping in mind m3 ¼ 33 ¼ 0; dc itself, the original plane-tocentroid distance, of course vanishes. One then finds

411

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING summations reducing to single summations. [The formal expression for
2 ðdÞ does not change, so it will not be repeated.]

ki pi ¼ wk ki2 =½w  wp pi2 =½w þ ½w2 i2 =½w2 ; 2ki

covðmi ; mj Þ  "i "j ¼

k3 p3 ¼

½w2 ðsi sj 23 þ s23 i j þ si s3 3 j þ s3 sj i 3  ; i; j ¼ 1; 2 f½wðs23  s2i Þgf½wðs23  s2j Þg

covðmi ; dc Þ  "i "3 ¼

½w2 ðsi 3 3 þ s3 i 3 Þ ; f½wðs23  s2i Þg½w

¼ ð1 

2wk =½wÞki2

k 6¼ p; i ¼ 1; 2 þ ½w2 i2 =½w2 ;

i ¼ 1; 2

2 wp p3 =½w;

and i; j ¼ 1; 2 2 2 2 2k3 ¼ k3  wk k3 =½w ¼ k3 ð1  wk =½wÞ:

½w2 32 
ðdc Þ  "23 ¼ : ½w2 2

Here the difference between the correct covariance values and the values obtained on ignoring the variation in hri may be important if the number of defining atoms is small, say, 5 or 4 or, in the extreme, 3.

When the variances are the same for  as for  (i.e. i j ¼ i j , all i, j) and the covariances all vanish ði j ¼ 0; i 6¼ jÞ, the "i "j simplify further. If the variances are also isotropic ði2 ¼ j2 ¼
2 , all i, j), there is still further simplification to

There are two cases, as has been pointed out, e.g., by Ito (1982). (1) The atom (atom K) was not included in the specification of the plane.

2 2


2 ðmi Þ  "2i ¼

½w
ðs2i þ s23 Þ ; f½wðs23  s2i Þg2 2 2 2

3.2.2.2. The standard uncertainty of the distance from an atom to the plane

i ¼ 1; 2

dK ¼ mT ðrK  hriÞ ¼ rk3  hr3 i dK ¼ K3 þ sK1 "1 þ sK2 "2  "3


2 ðdc Þ  "23 ¼ ½w
=½w

½w2
2 s1 s2  f½wðs23  s21 Þgf½wðs23  s22 Þg ½w2
2 si  ; i ¼ 1; 2: covðmi ; dc Þ  "i "3 ¼ f½wðs23  s2i Þg½w

covðm1 ; m2 Þ  "1 "2 ¼

2
d2K ¼ K3 þ s2K1 "21 þ s2K2 "22 þ "23

þ 2sK1 sK2 "1 "2  2sK1 "1 "3  2sK2 "2 "3 þ 2sK1 K3 "1 þ 2sK2 K3 "2  2K3 "3 : In the isotropic, ‘no-correlation’ case the last three terms, i.e. the terms in "i K3 , are all negligible or zero. 2 and the appropriate "i "j values In either case the value for K3 from the least-squares-plane calculation need to be inserted. (2) Atom K was included in the specification of the plane. The expression for
d2K remains the same, but the averages in it may be importantly different. For example, consider a plane defined by only three atoms, one of overwhelmingly great w at (0, 0, 0), one at (1, 0, 0) and one at (0, 1, 0). The centroid is at (0, 0, 0) and we take K ¼ 2, i.e.
d2 is the item of interest. Of course, it is obvious without calculation that the standard uncertainties vanish for the distances of the three atoms from the plane they alone define; the purpose here is only to show, in one case for one of the atoms, that the calculation gives the same result, partly, it will be seen, because the change in orientation of the plane is taken into account. If the only varia2 ¼
2, one has tion in the atom positions is described by 23 s21 ¼ 1; "3 ¼ "2 ¼ 0; "1 ¼ 23 , and K3 "1 ¼
2 . The nonvanishing terms in the desired variance are then

If allowance is made for the difference in definition between "3 and d, these expressions are equivalent to the ones (equations 7–9) given by WMC, who, however, do not appear to have been aware of the distinction between  and  and the possible consequences thereof. If, finally, w1 for each atom is taken equal to its 2j ¼
2, all j, there is still further simplification.


2 ðmi Þ  "2i ¼

½wðs23 þ s2i Þ ; f½wðs23  s2i Þg2

i ¼ 1; 2


2 ðdc Þ  "23 ¼ ½w=½w2 ¼ 1=½w ½ws1 s2  covðm1 ; m2 Þ  "1 "2 ¼ 2 f½wðs3  s21 Þgf½wðs23  s22 Þg ½wsi  ; i ¼ 1; 2: covðmi ; dc Þ  "i "3 ¼ f½wðs23  s2i Þg½w

2 þ s221 "21 þ 2s21 23 "1
d22 ¼ 23

For the earlier, more general expressions for the components of ""T it is still necessary to find ki lj and kiP l3 in terms of ki lj, with ki  ski ¼ ðrki  hrki iÞ ¼ ki  hi i ¼ l wl ðki  li Þ=½w. ki pj ¼

P

¼ ð1 þ 1  2Þ
2 ¼ 0: If, however, the problem concerns the same plane and a fourth atom at position ð1; 0; r43 Þ, not included in the specification of the plane and uncertain only in respect to r43 (which is arbitrary) with 2 43 ¼
2 (the same mean-square variation in direction 3 as for atom 2) and 43 23 ¼ 0, the calculation for
d24 runs the same as before, except for the third term:

wl wq ðki  li Þðpj  qj Þ=½w2

l;q

¼ ðki  hi iÞðpj  hj iÞ P ki p3 ¼ wl ðki  li Þp3 =½w ¼ ki p3  hi ip3 : l


d24 ¼ ð1 þ 1  0Þ
2 ¼ 2
2 : In the isotropic, ‘no-correlation’ case, for example, these reduce to

412

3.2. THE LEAST-SQUARES PLANE Our first example of this has already been given.3 A second, which actually inspired our first, is to be found in Hamilton (1964; example 5-8-1, p. 177), who discusses a rectangle of four points with identical error ellipsoids elongated in the long direction of the rectangle. The unweighted best line bisects the pattern along this direction, but the weighted best line is parallel to the short direction, if the elongation of the ellipsoids is sufficient. A third example (it is severely specialized so that a precise result can be attained without calculation) has three atoms ABC arranged like a C2 mm molecule with bond angle 90. The central atom, B, has overwhelming isotropic weight; A and C have parallel extremely elongated error ellipsoids, aligned parallel to the A—B bond. The unweighted best line passes through B parallel to A


C; the weighted best line passes through B and through C. Our last example is of a plane defined by a number of atoms of which one lies a considerable distance above the plane and at a distance from the normal through the centroid, but with the downward semi-axis of its extremely elongated prolate error ellipsoid intersecting that normal before it intersects the plane. If this atom is moved at right angles to the plane and further away from it, the centroid normal tips toward the atom, whereas it would tip away if the atom’s weight function were isotropic or if the calculation were the usual one and in effect constrained the adjusted position of the atom to move at right angles to the plane. The lead notion here – that the observed points are to be adjusted individually to fit a curve (surface) of required type exactly, rather than that the curve should simply be constructed to fit the observed points as well as possible in the sense of minimizing the weighted sum of squares of the distances along some preordained direction (perhaps normal to the plane, but perhaps as in ordinary curve fitting parallel to the y axis) – we first learned from the book by Deming (1943), Statistical Adjustment of Data, but it is to be found in Whittaker & Robinson (1929), Arley & Buch (1950), Hamilton (1964, our most used reference), and doubtless widely throughout the least-squares literature. It has recently been strongly emphasized by Lybanon (1984), who gives a number of modern references. It is the only prescription that properly satisfies the least-squares conditions, whereas (a) and other analogous prescriptions are only arbitrary regressions, in (a) a normal regression onto the plane.4 We have explored the problem of least-squares adjustment of observed positions subject to anisotropic weights with the help of three Fortran programs, one for the straight line and two for the plane. In the first plane program an approximate plane is derived, coordinates are rotated as in WMC (1973), and the parameters of the plane are adjusted and the atoms moved onto it, either normally or in full accord with the least-squares condition, but in either case subject to independent anisotropic weight matrices. The other plane program, described in Appendix A3.2.1, proceeds somewhat more directly, with the help of the method of Lagrange multipliers. However, neither program has been brought to a satisfactory state for the calculation of the variances and covariances of the derived quantities.

Extreme examples of this kind show clearly enough that variation in the direction of the plane normal or in the normal component of the centroid position will sometimes be important, the remarks to the contrary by Shmueli (1981) and, for the centroid, the omission by WMC notwithstanding. If only a few atoms are used to define the plane (e.g., three or, as is often the case, a very few more), both the covariance with the centroid position and uncertainty in the direction of the normal are likely to be important. The uncertainty in the normal may still be important, even if a goodly number of atoms are used to define the plane, whenever the test atom lies near or beyond the edge of the lateral domain defined by the other atoms.

3.2.3. The proper least-squares plane, with Gaussian weights If it is desired to weight the points to be fitted by a plane in the sense of Gaussian least squares, then two items different from what we have seen in the crystallographic literature have to be brought into view: (1) the weights may be anisotropic and include interatomic correlations, because the error matrix of the atom coordinates may in general be anisotropic and include interatomic correlations; and (2) it has to be considered that the atoms truly lie on a plane and that their observed positions are to be adjusted to lie precisely on that plane, whatever its precise position may turn out to be and no matter what the direction, in response to the anisotropic weighting, of their approach to the plane. An important consequence of (1), the nondiagonal character of the weight matrix, even with Cartesian coordinates, is that the problem is no longer an ordinary eigenvalue problem as treated by SWMB (1959),1 not even if there is no interatomic correlation and the anisotropy is the same for each atom. On this last case the contrary remark in SWMB at the beginning of the footnote, p. 601, is incorrect, and the treatments in terms of the eigenvector–eigenvalue problem by Hamilton (1961, 1964, pp. 174–177) and Ito (1981a)2 evade us. At best the problem is still not a genuine eigenvalue problem if the anisotropies of the atoms are not all alike. Hypothesis (2), of perfect planarity, may be hard to swallow. It has to be tested, and for any set of atoms the conclusion may be that they probably do not lie on a plane. But if the hypothesis is provisionally adopted (and it has to be decided beforehand which of the following alternatives is to be followed), the adjusted positions are obtained by moving the atoms onto the plane (a) along paths normal to the plane, or (b) along the proper paths of ‘least resistance’ – that is, paths with, in general, both normal and lateral components differently directed for each atom so as to minimize the appropriately weighted quadratic form of differences between the observed and adjusted coordinates. The lateral motions (and the anisotropic weights that induce them) may change the relative weights of different atoms in accordance with the orientation of the plane; change the perpendicular distance of origin-to-plane; and change the orientation of the plane in ways that may at first give surprise.

3.2.3.1. Formulation and solution of the general Gaussian plane We conclude with an outline for a complete feasible solution, including interatomic weight-matrix elements. Consider atoms at observed vector positions rk, k ¼ 1; . . . ; n, designated in the following equations by R, an n-by-3 array, with Rki ¼ r ki ; the corresponding adjusted positions denoted by the array Ra ; n constraints (each adjusted position ra – ‘a’ for ‘adjusted’ – must be

1

A simple two-dimensional problem illustrates the point. A regular polygon of n atoms is to define a ‘best’ line (always a central line). If the error matrix (the same for each atom) is isotropic, the weighted sum of squares of deviations from the line is independent of its orientation for n > 2, i.e. the problem is a degenerate eigenvalue problem, with two equal eigenvalues. However, if the error ellipsoids are not isotropic and are all oriented radially or all tangentially (these are merely the two orientations tried), the sum has n/2 equal minima for even n and 2 equal minima for odd n, in the one- range of possible orientations of the line. Possibly similar peculiarities might be imagined if the anisotropic weights were more complicated (e.g., ‘star’ shaped) than can be described by a nonsingular matrix, or by any matrix. Such are of course excluded here. 2 Ito observes that his method fails when there are only three points to define the plane, his least-squares normal equations becoming singular. But the situation is worse: his equations are singular for any number of points, if the points fit a plane exactly.

3

See first footnote1. Ito’s second method (Ito, 1981b), of ‘substitution’, is also a regression, essentially like the regression along z at fixed x and y used long ago by Clews & Cochran (1949, p. 52) and like the regressions of y on fixed x that – despite the fact that both x and y are afflicted with random errors – are commonly taught or practised in schools, universities and laboratories nearly 200 years after Gauss, to the extent that Deming, Lybanon and other followers of Gauss have so far had rather little influence. Kalantar’s (1987) short note is a welcome but still rare exception. 4

413

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING on the plane), and 3n þ 3 adjustable parameters (3n Ra components and the 3 components of the vector b of reciprocal intercepts of the plane), so that the problem has n  3 degrees of freedom. The 3n-by-3n weight matrix P may be anisotropic for the separate atoms, and may include interatomic elements; it is symmetric in the sense P kilj ¼ P ljki, but P kilj will not in general be equal to P kjli. The array K denotes n Lagrange multipliers, one for each atom and unrelated to the ’s of Section 3.2.2; m and d still represent the direction cosines of the plane normal and the perpendicular origin-to-plane distance. For a linear least-squares problem we know (see, e.g., Hamilton, 1964, p. 143) that the proper weight matrix is the reciprocal of the atomic error matrix ðP ¼ M 1 Þ;5 note that ‘M’ is unrelated to the ‘M’ of Section 3.2.2. The least-squares sum is now

further mention of this nonlinearity in Appendix A3.2.1) that have to be solved for b and K: Ra ¼ R þ MKb ¼ R þ M bK

0 ¼ F  ðKMKÞb þ KR 0 ¼ G  ðbM bÞK  ð1  bRÞ: The best way of handling these equations, at least if it is desired to find their solutions both for Ra , K, and b and for the error matrix of these derived quantities in terms of the error matrix for R, seems to be to find an approximate solution by one means or another, to make sure that it is the desired solution, if (as may be) there is ambiguity, and to shift the origin to a point far enough from the plane and close enough to the centroid normal to avoid the difficulties discussed by SWMB. Then linearize the first and second equations by differentiation and solve the results first iteratively to fit the current residuals F0 and G0 and then for nominally infinitesimal increments b and K. In effect, one deals with equations QX ¼ Y, where Q is the ðn þ 3Þ  ðn þ 3Þ matrix of coefficients of the following set of equations, X is the ðn þ 3Þdimensional vector of increments b and K, and Y is the vector either of the first terms or of the second terms on the right-hand side of the equations.

S ¼ ðR  Ra ÞPðR  Ra Þ; and the augmented sum for applying the method of Lagrange multipliers is
¼ S=2  KbRa : Here the summation brackets ð½. . .Þ have been replaced by matrix products, but the simplified notation is not matrix notation. It has to be regarded as only a useful mnemonic in which all the indices and their clutter have been suppressed in view of their inconveniently large number for some of the arrays. If the dimensions of each are kept in mind, it is easy to recall the indices, and if the indices are resupplied at any point, it is not difficult to discover what is really meant by any of the expressions or whether evaluations have to be executed in a particular order. The conditions to be satisfied are

ðKMKÞb þ ðMKb þ KM b þ RÞK ¼ F0  KR ðM bK þ bMK þ RÞb þ ðbM bÞK ¼ G0  bR: When X becomes the vector " of errors in b and K as induced by errors   R in the measured atom positions, these equations become, in proper matrix notation, Q" ¼ Z, with solution " ¼ Q1 Z, where Z is the ðn þ 3Þ-dimensional vector of components, first of b then of K. The covariance matrix ""T , from which all the covariances of b, Ra , and R (including for the latter any atoms that were not included for the plane) can be derived, is then given by

@
¼ PðR  Ra Þ þ Kb; @Ra @
0¼ ¼ KRa ; @b 1 ¼ bRa : 0¼

""T ¼ Q1 ZT ZT ðQ1 ÞT : This is not as simple as the familiar expression for propagation of errors for a least-squares problem solved without the use of Lagrange multipliers, i.e. ""T ¼ B1 , where B is the matrix of the usual normal equations, both because B ¼ BT is no longer valid and because the middle factor ZT ZT is no longer equal to B1. It is easy to verify that K consists of a set of coefficients for combining the n row vectors of M b, in the expression for Ra , into corrections to R such that each adjusted position lies exactly on the plane:

That the partial derivatives of S=2 should be represented by PðR  Ra Þ depends upon the above-mentioned symmetry of P. Note that each of the n unit elements of 1 expresses the condition that its ra should indeed lie on the plane, and that Kb is just the same as bK. The perpendicular origin-to-plane distance and the direction cosines of the plane normal are d2 ¼ 1=bT b and pffiffiffiffiffiffiffi ffi m ¼ b= bT b. On multiplication by M the first condition solves for Ra , and that expression combined separately with the second condition and with the third gives highly nonlinear equations (there is

1

bRa ¼ bR þ bM bðbM bÞ ð1  bRÞ

¼ bR þ 1  bR ¼ 1:

5

Is this statement firm for a nonlinear problem? We use it, assuming that at convergence the problem has become effectively linear. But in fact this will depend on how great the nonlinearity is, in comparison with the random errors (variances) that eventually have to be considered. Another caveat may be in order in regard to our limited knowledge of Gauss’s second derivation of the method of least squares, the one he preferred [see Whittaker & Robinson (1929)] and which establishes for a linear system that the best linear combination of a set of observations, afflicted by random errors, for estimating any arbitrary derived quantity – best in the sense of being unbiased and having minimal mean-square error – is given by the method of least squares with the weight matrix set equal to the inverse error matrix of the observations. Hamilton, and Whittaker & Robinson, prove this only for the case that the derived parameters are not constrained, whereas here they are. We believe, however, that the best choice of weights is a question concerning only the observations, and that it cannot be affected by the method used for minimizing S subject to any constraints, whether by eliminating some of the parameters by invoking the constraints directly or by the use of Lagrange multipliers.

At the same time one can see how the lateral shifts occur in response to the anisotropy of M, especially if, now, only the anisotropic case without interatomic correlations is considered. For atom k write b in terms of its components along the principal axes of Mk, associated with the eigenvalues ; and ; the shifts are then proportional to M b2 ; M

b2 and M b2 , each along its principal axis, and the sums of the contributions of these shift components to the total displacement along the plane normal or along either of two orthogonal directions in the plane can readily be visualized. In effect Mk is the compliance matrix for these shifts of atom k. Similarly, it can be seen that in the isotropic case with interatomic correlations a pair of equally weighted atoms

414

3.2. THE LEAST-SQUARES PLANE located, for example, at some distance apart and at about the same distance from the plane, will have different shifts (and different influences on d and m) depending on whether the covariance between the errors in the perpendicular components of their observed positions relative to the plane is small, or, if large, whether it is positive or is negative. If the covariance is large and positive, the adjusted positions will both be pulled toward the plane, strongly resisting, however, the apparent necessity that both reach the plane by moving by different amounts; in consequence, there will be a strong tendency for the plane to tilt toward the more distant atom, and possibly even away from the nearer one. But if the covariance is large and negative, the situation is reversed: the more distant atom can readily move more than the nearer one, while it is very difficult to move them together; the upshot is then that the plane will move to meet the original midpoint of the two atoms while they tilt about that midpoint to accommodate the plane. It is attractive to solve our problem with the ‘normal’ formulation of the plane, mr ¼ d, and so directly avoid the problems that arise for d  0. The solution in terms of the reciprocal intercepts b has been given first, however, because reducing by two (d and a Lagrange multiplier) the number of parameters to be solved for may more than make up for the nuisance of having to move the origin. The solution in terms of d follows. The augmented variation function is now

K ¼ ðmM mÞ1 ðd1  mRÞ  ¼ mKR þ mKMKm; which readily follow from the previous equations. As in the ‘intercepts’ solution the linearized expression for the increments in ; K; d and m can be used together with the equation for Ra to obtain all the covariances needed in the treatment described in Section 3.2.2. 3.2.3.2. Concluding remarks Proper tests of statistical significance of this or that aspect of a least-squares plane can be made if the plane has been based on a proper weight matrix as discussed in Section 3.2.3; if it can be agreed that the random errors of observation are normally distributed; and if an agreeable test (null) hypothesis can be formulated. For example, one may ask for the probability that a degree of fit of the observed positions to the least-squares plane at least as poor as the fit that was found might occur if the atoms in truth lie precisely on a plane. The 2 test answers this question: a table of probabilities displayed as a function of 2 and  provides the answer. Here 2 is just our minimized S ¼ bKMPMKb ¼ bKMKb;


¼ ðR  Ra ÞPðR  Ra Þ=2  KðmRa  d1Þ þ mm=2;

and  ¼ nobservations  nadjusted parameters  nconstraints

the term in the new Lagrange multiplier, , and the term in d1 having been added to the previous expression. The 1, an n-vector of 1’s, is needed to express the presence of n terms in the K sum. There are then five equations to be satisfied – actually n þ 1 þ 3n þ 3 þ 1 ¼ 4n þ 5 ordinary equations – in the 3n Ra components, the n K’s, the 3 m components, , and d  4n þ 5 unknowns in all, as required. These equations are as follows:

¼ 3n  ðn þ 3Þ  n ¼ n  3; is the number of degrees of freedom for the problem of the plane (erroneously cited in at least one widely used crystallographic system of programs as 3n  3). There will not usually be any reason to believe that the atoms are exactly coplanar in any case; nevertheless, this test may well give a satisfying indication of whether or not the atoms are, in the investigator’s judgment, essentially coplanar. It must be emphasized that 2 as calculated in Section 3.2.3 will include proper allowance for uncertainties in the d and orientation of the plane with greater reliability than the estimates of Section 3.2.2, which are based on nominally arbitrary weights. Both, however, will allow for the large variations in d and tilt that can arise in either case if n is small. Some of the earlier, less complete discussions of this problem have been mentioned in Section 3.2.2. Among the problems not considered here are ones of fitting more than one plane to a set of observed positions, e.g. of two planes fitted to three sets of atoms associated, respectively, with the first plane, the second plane, and both planes, and of the angle between the two planes. For the atoms common to both planes there will be a fundamental point of difference between existing programs (in which, in effect, the positions of the atoms in common are inconsistently adjusted to one position on the first plane and, in general, a different position on the second) and what we would advocate as the proper procedure of requiring the adjusted positions of such atoms to lie on the line of intersection of the two planes. As to the dihedral angle there is a difficulty, noted by WMC (1973, p. 2705), that the usual formulation of
2 ð 0 Þ in terms of the cosine of the dihedral angle reduces to 0/0 at 0 ¼ 0. However, this variance is obviously well defined if the plane normals and their covariances are well defined. The essential difficulty lies with the ambiguity in the direction of the line of intersection of the planes in the limit of zero dihedral angle. For the torsion angle about a line defined by two atoms, there should be no such difficulty. It seems likely that for the twoplane problem proposed above, the issue that decides whether

mRa ¼ d1 mm ¼ 1



 @
¼ PðR  Ra Þ þ Km 0¼ @Ra @
0¼ ¼ KRa þ m @m @
¼ K1: 0¼ @d

As before, multiply the third equation by M and solve for Ra . Then substitute the result into the first and fourth equations to obtain mR þ mM mK ¼ d1; mm ¼ 1;

KR þ KMKm ¼ m; K1 ¼ 0

as the n þ 5 mostly nonlinear equations to be solved for m, K, d and  by linearizing (differentiation), solving for increments, and iterating, in the pattern described more fully above. An approximate solution for m and d has first to be obtained somehow, perhaps by the method of SWMB (with isotropic uncorrelated weights), checked for suitability, and extended to a full complement of first approximations by

415

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING    @
1  bT r 1  bT r rþ T Mb : 0 ¼ T ¼ ½ra  ¼ @b bT Mb b Mb

the dihedral angle will behave like the standard dihedral angle or, instead, like the torsion angle, will be found to be whether or not two or more atoms are common to both planes. All that we have tried to bring out about the covariances of derived quantities involving the plane requires that the covariances of the experimental atom positions (reduced in our formulations to Cartesian coordinates) be included. However, such covariances of derived quantities are often not available in practice, and are usually left unused even if they are. The need to use the covariances, not just the variances, has been obvious from the beginning. It has been emphasized in another context by Schomaker & Marsh (1983) and much more strongly and generally by Waser (1973), whose pleading seems to have been generally ignored, by now, for about thirty years.

This last equation, FðbÞ ¼ 0, is to be solved for b. It is highly nonlinear: FðbÞ ¼ Oðb3 Þ=Oðb4 Þ. One can proceed to a first approximation by writing 0 ¼ ½ð1  bT rÞr; i.e., ½rr
b ¼ ½r, in dyadic notation. ½M ¼ I, all atoms; 1  bT r ¼ 0 in the multiplier of Mb=ðbT MbÞ: A linear equation in b, this approximation usually works well.6 We have also used the iterative Frazer, Duncan & Collar eigenvalue solution as described by SWMB (1959), which works even when the plane passes exactly through the origin. To continue the solution of the nonlinear equations, by linearizing and iterating, write FðbÞ ¼ 0 in the form

APPENDIX A3.2.1 Consider n atoms at observed vector positions r (expressed in Cartesians), n constraints (each adjusted position ra – ‘a’ for ‘adjusted’ – must be on the plane) and 3n þ 3 adjustable parameters (3n ra components and the 3 components of the vector a of reciprocal intercepts of the plane), so that the problem has n  3 degrees of freedom. The weight matrices P may be differently anisotropic for each atom, but there are no interatomic correlations. As before, square brackets, ‘½. . .’, represent the Gaussian sum over all atoms, usually suppressing the atom indices. We also write , not the  of Section 3.2.2, for the Lagrange multipliers (one for each atom); m for the direction cosines of the plane normal; and d for the perpendicular origin-to-plane distance. As before, Pk is the reciprocal of the atomic error matrix: 1 Pk ¼ M1 k (correspondingly, P  M Þ but ‘M’ is no longer the ‘M’ of Section 3.2.2. The appropriate least-squares sum is

0 ¼ ½r þ 2 Mb   @ @ 2  r þ 2Mb þ  M b þ ½ðr þ MbÞ0 ; @b @b solve for b, reset b to b þ b, etc., until the desired degree of convergence of jbj=jbj toward zero has been attained. By @=@b is meant the partial derivative of the above expression for  with respect to b, as detailed in the next paragraph. In the Fortran program DDLELSP (double precision Deming Lagrange, with error estimates, least-squares plane, written to explore this solution) the preceding equation is recast as 

Bb  2 M þ ðr þ 2MbÞ

 @ b @b

 Y ¼ ½ðr þ MbÞ0 ;

S ¼ ½ðr  ra ÞT Pðr  ra Þ with and the augmented sum for applying the method of Lagrange multipliers is

@ ¼ rT =bT Mb  ½2ð1  bT rÞ=ðbT MbÞ2  @b ¼ ðrT þ 2bT MÞ=bT Mb:


¼ S=2  ½aT ra :

The usual goodness of fit, GOF2 in DDLELSP, evaluates to


is to be minimized with respect to variations of the adjusted atom positions rka and plane reciprocal intercepts bi, leading to the equations

1=2 1=2  Smin 1 2 T ½ b MPMb GOF2 ¼ ¼ n3 n3  1=2 1 ½2 bT Mb ¼ n3   1=2 1 ð1  bT rÞ2 ¼ : n3 bT Mb 

@ ¼ Pðr  ra Þ þ b and @rTa @
0¼ ¼ ½ra ; @b T 0¼

subject to the planepffiffiffiffiffiffiffi conditions bT ra ¼ 1, each atom, with ffi d2 ¼ 1=ðbT bÞ, m ¼ b= bT b. These equations are nonlinear. A convenient solution runs as follows: first multiply the first equation by M and solve for the adjusted atom positions in terms of the Lagrange multipliers  and the reciprocal intercepts b of the plane; then multiply that result by bT applying the plane conditions, and solve for the ’s ra ¼ r þ Mb;

6 We do not fully understand the curious situation of this equation. It arises immediately if the isotropic problem is formulated as one of minimizing ½ð1  bT rÞ2  by varying b, and it fails then [SWMB (1959) referred to it as ‘an incorrect method’], as it obviously must – observe the denominator – if the plane passes too close to the origin. However, it fails in other circumstances also. The main point about it is perhaps that it is linear in b and is obtained as the supposedly exact and unique solution of the isotropic problem, whereas the problem has no unique solution but three solutions instead (SWMB, 1959). From the point of view of Gaussian least squares, the essential fault in minimizing Slin ¼ ½ð1  bT rÞ2  may be that the apparently simple weighting function in it, i.e. the identity, is actually complicated and unreasonable. In terms of distance deviations from the plane, we have Slin ¼ ½wðd  mT rÞ2 , with w ¼ bT b ¼ d2 . Prudence requires that the origin be shifted to a point sufficiently far from the plane and close enough to the centroid normal to avoid the difficulties discussed by SWMB. Note that for the one-dimensional problem of fitting a constant to a set of measurements of a single entity the Deming–Lagrange treatment with the condition 1 ¼ cxa and weights w reduces immediately to the standard result 1=c ¼ ½wx=½w.

M  P1

1 ¼ bT r þ bT Mb;  ¼ ð1  bT rÞ=ðbT MbÞ: Next insert these values for the ’s and ra ’s into the second equation:

416

3.2. THE LEAST-SQUARES PLANE This is only an approximation, because the residuals 1  bT r are not the differences between the observations and appropriate linear functions of the parameters, nor are their variances (the bT Mb’s) independent of the parameters (or, in turn, the errors in the observations). We ask also about the perpendicular distances, e, of atoms to plane and the mean-square deviation ðeÞ2 to be expected in e.

Deming, W. E. (1943). Statistical Adjustment of Data. New York: John Wiley. Hamilton, W. C. (1961). On the least-squares plane through a set of points. Acta Cryst. 14, 185–189. Hamilton, W. C. (1964). Statistics in Physical Science. New York: Ronald Press. Ito, T. (1981a). Least-squares refinement of the best-plane parameters. Acta Cryst. A37, 621–624. Ito, T. (1981b). On the least-squares plane through a group of atoms. Sci. Pap. Inst. Phys. Chem. Res. Saitama, 75, 55–58. Ito, T. (1982). On the estimated standard deviation of the atom-to-plane distance. Acta Cryst. A38, 869–870. Kalantar, A. H. (1987). Slopes of straight lines when neither axis is errorfree. J. Chem. Educ. 64, 28–29. Lybanon, M. (1984). A better least-squares method when both variables have uncertainties. Am. J. Phys. 52, 22–26. Robertson, J. M. (1948). Bond-length variations in aromatic systems. Acta Cryst. 1, 101–109. Schomaker, V. & Marsh, R. E. (1983). On evaluating the standard deviation of Ueq. Acta Cryst. A39, 819–820. Schomaker, V., Waser, J., Marsh, R. E. & Bergman, G. (1959). To fit a plane or a line to a set of points by least squares. Acta Cryst. 12, 600– 604. Shmueli, U. (1981). On the statistics of atomic deviations from the ‘best’ molecular plane. Acta Cryst. A37, 249–251. Waser, J. (1973). Dyadics and variances and covariances of molecular parameters, including those of best planes. Acta Cryst. A29, 621–631. Waser, J., Marsh, R. E. & Cordes, A. W. (1973). Variances and covariances for best-plane parameters including dihedral angles. Acta Cryst. B29, 2703–2708. Whittaker, E. T. & Robinson, G. (1929). The Calculus of Observations. London: Blackie.

pffiffiffiffiffiffiffiffi e ¼ ð1  bT rÞ= bT b ¼ dð1  bT rÞ e ¼ dðbT  þ rT "Þ þ Oð2 Þ: Here  and " are the errors in r and b. Neglecting ‘Oð2 Þ’ then leads to ðeÞ2 ¼ d2 ðbT T b þ 2bT "T r þ rT ""T rÞ: We have T ¼ M ¼ P1, but "T and ""T perhaps still need to be evaluated. References Arley, N. & Buch, K. R. (1950). Introduction to the Theory of Probability and Statistics. New York: John Wiley and London: Chapman and Hall. Clews, C. J. B. & Cochran, W. (1949). The structures of pyrimidines and purines. III. An X-ray investigation of hydrogen bonding in aminopyrimidines. Acta Cryst. 2, 46–57.

417

references

International Tables for Crystallography (2010). Vol. B, Chapter 3.3, pp. 418–448.

3.3. Molecular modelling and graphics By R. Diamond and L. M. D. Cranswick

A third form, suitable only for rhombohedral cells, is 0 1 p þ 2q p
q p
q aB C M ¼ @ p
q p þ 2q p
q A 3 p
q p
q p þ 2q 01 2 1 1 1 11 þ

Bp q p q p qC C B 1 B1 1 1 2 1 1C C
þ
M
1 ¼ B C 3a B Bp q p q p qC @1 1 1 1 1 2A

þ p q p q p q

3.3.1. Graphics

By R. Diamond

3.3.1.1. Coordinate systems, notation and standards 3.3.1.1.1. Cartesian and crystallographic coordinates It is usual, for purposes of molecular modelling and of computer graphics, to adopt a Cartesian coordinate system using mutually perpendicular axes in a right-handed system using the a˚ngstro¨m unit or the nanometre as the unit of distance along such axes, and largely to ignore the existence of crystallographic coordinates expressed as fractions of unit-cell edges. Transformations between the two are thus associated, usually, with the input and output stages of any software concerned with modelling and graphics, and it will be assumed after this section that all coordinates are Cartesian using the chosen unit of distance as the unit of coordinates. For a discussion of coordinate transformations and rotations without making this assumption see Chapter 1.1 in which formulations using co- and contravariant forms are presented. The relationship between these systems may be written

in which pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼  1 þ 2 cos
q ¼  1
cos
; which preserves the equivalence of axes. Here the chiralities of the Cartesian and crystallographic axes are the same if p is chosen positive, and different otherwise, and the two sets of axes coincide in projection along the triad if q is chosen positive and are  out of phase otherwise.

x ¼ M
1 X

X ¼ Mx

3.3.1.1.2. Homogeneous coordinates Homogeneous coordinates have found wide application in computer graphics. For some equipment their use is essential, and they are of value analytically even if the available hardware does not require their use. Homogeneous coordinates employ four quantities, X, Y, Z and W, to define the position of a point, rather than three. The fourth coordinate has a scaling function so that it is the quantity X/W (as delivered to the display hardware) which controls the left– right positioning of the point within the picture. A point with jX=Wj < 1 is in the picture, normally, and those with jX=Wj > 1 are outside it, but see Section 3.3.1.3.5. There are many reasons why homogeneous coordinates may be adopted, among them the following: (i) X, Y, Z and W may be held as integers, thus enabling fast arithmetic whilst offering much of the flexibility of floating-point working. A single W value may be common to a whole array of X, Y, Z values. (ii) Perspective transformations can be implemented without the need for any division. Only high-speed matrix multiplication using integer arithmetic is necessary, provided only that the drawing hardware can provide displacements proportional to the ratio of two signals, X and W or Y and W. Rotation, translation, scaling and the application of perspective are all effected by operations of the same form, namely multiplication of a fourvector by a 4  4 matrix. The hardware may thus be kept relatively simple since only one type of operation needs to be provided for. (iii) Since kX, kY, kZ, kW represents the same point as X, Y, Z, W, the hardware may be arranged to maximize resolution without risk of integer overflow. For analytical purposes it is convenient to regard homogeneous transformations in terms of partitioned matrices    M V X ; U N W

in which X and x are position vectors in direct space, written as column vectors, with x expressed in crystallographic fractional coordinates (dimensionless) and X in Cartesian coordinates (dimension of length). There are two forms of M in common use. The first of these sets the first component of X parallel to a and the third parallel to c and is 0 1 a’= sin
0 0 B C M ¼ @ aðcos 
cos
cos Þ= sin
b sin
0 A a cos 

0

b cos


sin
=a’

0

B M
1 ¼ @ ðcos
cos 
cos Þ=b’ sin
ðcos
cos 
cos Þ=c’ sin
in which ’¼

c

1=b sin

1=c tan


0

1

C 0 A 1=c

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
cos2

cos2 
cos2  þ 2 cos
cos  cos 

¼ sin
sin  sin   : ’ is equal to the volume of the unit cell divided by abc, and is unchanged by cyclic permutation of
,  and  and of
,  and   . The Cartesian and crystallographic axes have the same chirality if the positive square root is taken. The second form sets the first component of X parallel to a and the third component of X parallel to c and is 0 1 a b cos  c cos  B C M ¼ @ 0 b sin  cðcos

cos  cos Þ= sin  A 0

0 1=a

B M
1 ¼ @ 0 0

0
1=a tan  1=b sin  0

c’= sin  ðcos
cos 
cos Þ=a’ sin 

1

C ðcos  cos 
cos
Þ=b’ sin  A: sin =c’

Copyright © 2010 International Union of Crystallography

418

3.3. MOLECULAR MODELLING AND GRAPHICS where M is a 3  3 matrix, V and X are three-element column vectors, U is a three-element row vector and N and W are scalars. Matrices and vectors which are equivalent under the considerations of (iii) above will be related by the sign ’ in what follows. Hardware systems which use true floating-point representations have less need of homogeneous coordinates and for these N and W may normally be set to unity.

(PHIGS) (Brown, 1985; Abi-Ezzi & Bunshaft, 1986) since this allows hierarchical segmentation of picture content to exist in both the applications software and the graphics device in a related manner, which GKS does not. Some graphics devices now available support this type of working and its exploitation indicates the choice of PHIGS. Furthermore, Fortran implementations of GKS and GKS-3D require points to be stored in arrays dimensioned as X(N), Y(N), Z(N) which may be equivalenced (in the Fortran sense) to XYZ(N, 3) but not to XYZ(3, N), which may not be convenient. PHIGS also became an International Standard in 1988: American National Standards Institute, American National Standard for Information Processing Systems – Computer Graphics – Programmer’s Hierarchical Graphics System (PHIGS) Functional Description, Archive File Format, Clear-Text Encoding of Archive File (1988). PHIGS has also been extended to support the capability of raster-graphics machines to represent reflections, shadows, see-through effects etc. in a version known as PHIGS+ (van Dam, 1988). Increasingly, manufacturers of graphics equipment are orienting their products towards one or other of these standards. While these standards are not the subject of this chapter it is recommended that they be studied before investing in equipment. In addition to these standards, related standards are evolving under the auspices of the ISO for defining images in a file-storage, or metafile, form, and for the interface between the deviceindependent and device-dependent parts of a graphics package. Arnold & Bono (1988) describe the ANSI and ISO Computer Graphics Metafile standard which provides for the definition of (two-dimensional) images. The definition of three-dimensional scenes requires the use of (PHIGS) archive files.

3.3.1.1.3. Notation In this chapter the conventions of matrix algebra will be adhered to except where it is convenient to show operations on elements of vectors, matrices and tensors, where a subscript notation will be used with a modified summation convention in which summation is over lower-case subscripts only. Thus the equation xI ¼ AIj Xj is to be read ‘For any I, xI is AIjXj summed over j’. Subscripts using the letter i or later in the alphabet will relate to the usual three dimensions and imply a three-term summation. Subscripts a to h are not necessarily so limited, and, in particular, the subscript a is used to imply summation over atoms of which there may be an arbitrary number. We shall use the superscript T to denote a transpose, and also use the Kronecker delta, IJ , which is 1 if I = J and zero otherwise, and the tensor "IJK which is 1 if I, J and K are a cyclic permutation of 1, 2, 3,
1 if an anticyclic permutation, and zero otherwise. "IJK ¼ ðI
JÞðJ
KÞðK
IÞ=2

1  I; J; K  3:

A useful identity is then "iJK "iLM ¼ JL KM
JM KL :

3.3.1.2. Orthogonal (or rotation) matrices It is a basic requirement for any graphics or molecularmodelling system to be able to control and manipulate the orientation of the structures involved and this is achieved using orthogonal matrices which are the subject of these sections.

Single modulus signs surrounding the symbol for a square matrix denote its determinant, and around a vector denote its length. The symbol ’ is defined in the previous section. 3.3.1.1.4. Standards The sections of this chapter concerned with graphics are primarily concerned with the mathematical aspects of graphics programming as they confront the applications programmer. The implementations outlined in Section 3.3.3 have all, so far as the author is aware, been developed ab initio by their inventors to deal with these aspects using their own and unrelated techniques and protocols. It is clear, however, that standards are now emerging, and it is to be hoped that future developments in applications software will handle the graphics aspects through one or other of these standards. First among these standards is the Graphical Kernel System, GKS, defined in American National Standards Institute, American National Standard for Information Processing Systems – Computer Graphics – Graphical Kernel System (GKS) Functional Description (1985) and described and illustrated by Hopgood et al. (1986) and Enderle et al. (1984). GKS became a full International Standards Organization (ISO) standard in 1985, and its purpose is to standardize the interface between application software and the graphics system, thus enhancing portability of software. Specifications for Fortran, Pascal and Ada formulations are at an advanced stage of development. Its value to crystallographers is limited by the fact that it is only two-dimensional. A three-dimensional extension known as GKS-3D, defined in International Standards Organisation, International Standard Information Processing Systems – Computer Graphics – Graphical Kernel System for Three Dimensions (GKS-3D), Functional Description (1988) became an ISO standard in 1988. Perhaps of greatest interest to crystallographers, however, is the Programmers’ Hierarchical Interactive Graphics System

3.3.1.2.1. General form If a vector v is expressed in terms of its components resolved onto an axial set of vectors X, Y, Z which are of unit length and mutually perpendicular and right handed in the sense that ðX  YÞ  Z ¼ þ1, and if these components are vI , and if a second set of axes X0 , Y0, Z0 is similarly established, with the same origin and chirality, and if v has components v0I on these axes then v0I ¼ aIj vj ; in which aIJ is the cosine of the angle between the ith primed axis and the jth unprimed axis. Evidently the elements aIJ comprise a matrix R, such that any row represents one of the primed axial vectors, such as X0 , expressed as components on the unprimed axes, and each column represents one of the unprimed axial vectors expressed as components on the primed axes. It follows that RT ¼ R
1 since elements of the product RT R are scalar products among perpendicular unit vectors. A real matrix whose transpose equals its inverse is said to be orthogonal. Since X, Y and Z can simultaneously be superimposed on X0 , Y0 and Z0 without deformation or change of scale the relationship is one of rotation, and orthogonal matrices are often referred to as rotation matrices. The operation of replacing the vector v by Rv corresponds to rotating the axes from the unprimed to the primed set with v itself unchanged. Equally, the same operation corresponds to retaining fixed axes and rotating the vector in the opposite sense. The second interpretation is the one more frequently helpful since conceptually it corresponds more closely

419

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING to rotational operations on objects, and it is primarily in this sense that the following is written. If three vectors u, v and w form the edges of a parallelepiped, then its volume V is

which is commonly employed in four-circle diffractometers for which ’ ¼
’1 ,  ¼ ’2 and ! ¼
’3 . In terms of the fixed-axes– moving-object conceptualization this corresponds to a rotation ’1 about Z followed by ’2 about Y followed by ’3 about Z. In the familiar diffractometer example, when  ¼ 0 the ’ and ! axes are both vertical and equivalent. If ’ is altered first, then the  axis is still in the direction of a fixed Y axis, but if ! is altered first it is not. Since all angles are to be rotations about fixed axes to describe a rotating object it follows that it is ’ rather than ! which corresponds to ’1. In general, when rotating parts are mounted on rotating parts the rotation closest to the moved object must be applied first, forming the right-most factor in any multiple transformation, with the rotation closest to the fixed part as the left-most factor, assuming data supplied as column vectors on the right. Given an orthogonal matrix, in either numerical or analytical form, it may be required to discover  and the axis of rotation, or to factorize it as a product of primitives. From the first form we see that the vector

V ¼ u  ðv  wÞ ¼ "ijk ui vj wk and if these vectors are transformed by the matrix R as above, then the transformed volume V 0 is V 0 ¼ "lmn u0l v0m w0n ¼ "lmn ali amj ank ui vj wk : But the determinant of R is given by jRj"IJK ¼ "lmn alI amJ anK so that V 0 ¼ jRjV and the determinant of R must therefore be +1 for a transformation which is a pure rotation. Nevertheless orthogonal matrices with determinant
1 exist though these do not describe a pure rotation. They may always be described as the product of a pure rotation and inversion through the origin and are referred to here as improper rotations. In what follows all references to orthogonal matrices refer to those with positive determinant only, unless stated otherwise. An important general form of an orthogonal matrix in three dimensions was derived as equation (1.1.4.32) and is 0

l2 þ ðm2 þ n2 Þ cos  @ R ¼ lmð1
cos Þ þ n sin  nlð1
cos Þ
m sin 

lmð1
cos Þ
n sin  m2 þ ðn2 þ l2 Þ cos  mnð1
cos Þ þ l sin 

vI ¼ "Ijk ajk ; consisting of the antisymmetric part of R, has elements
2 sin  times the direction cosines l, m, n, which establishes the direction immediately, and normalization using l2 þ m2 þ n2 ¼ 1 determines sin . Furthermore, the trace is 1 þ 2 cos  so that the quadrant of  is also fixed. This method fails, however, if the matrix is symmetrical, which occurs if  ¼ . In this case only the direction of the axis is required, which is given by

1 nlð1
cos Þ þ m sin  mnð1
cos Þ
l sin  A n2 þ ðl2 þ m2 Þ cos 

l : m : n ¼ ða23 Þ
1 : ða31 Þ
1 : ða12 Þ
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for nonzero elements, or l ¼ 12ða11 þ 1Þ etc., with the signs chosen to satisfy a12 ¼ 2lm etc. The Eulerian form may be factorized by noting that tan ’1 ¼
a32 =a31 ; tan ’3 ¼ a23 =a13 ; cos ’2 ¼ a33 . There is then freedom to choose the sign of sin ’2, but the choice then fixes the quadrants of ’1 and ’3 through the elements in the last row and column, and the primitives may then be constructed. These expressions for ’1 and ’3 fail if sin ’2 ¼ 0, in which case the rotation reduces to a primitive rotation about Z with angle ð’1 þ ’3 Þ; ’2 ¼ 0, or ð’3
’1 Þ; ’2 ¼ . Eulerian angles are unlikely to be the best choice of primitive angles unless they are directly related to the parameters of a system, as with the diffractometer. It is often more important that the changes to primitive angles should be quasi-linearly related to  for any small rotations, which is not the case with Eulerian angles when the required rotation axis is close to the X axis. In such a case linearized techniques for solving for the primitive angles will fail. Furthermore, if the required rotation is about Z only ð’1 þ ’3 Þ is determinate. Quasi-linear relationships between  and the primitive rotations arise if the primitives are one each about X, Y and Z. Any order of the three factors may be chosen, but the choice must then be adhered to since these factors do not commute. For sufficiently small rotations the primitive rotations are then l, m and n, whilst for larger  linearized iterative techniques for finding the primitive rotations are likely to be convergent and well conditioned. The three-dimensional space of the angles ’1, ’2 and ’3 in either case is nonlinearly related to . In the Eulerian case the worst nonlinearities occur at the origin of ’-space. Equally severe nonlinearities occur in the quasi-linear case also but are 90 away from the origin and less likely to be troublesome. Neither of the foregoing general forms of orthogonal matrix has ideally convenient properties. The first is inconvenient because it uses four nonequivalent variables l, m, n and , with a linking equation involving l, m and n, so that they cannot be treated as independent variables for analytical purposes. The

or RIJ ¼ ð1
cos ÞlI lJ þ IJ cos 
"IJk lk sin ; in which l, m and n are the direction cosines of the axis of rotation (which are the same when referred to either set of axes under either interpretation) and  is the angle of rotation. In this form, and with R operating on column vectors on the right, the sign of  is such that, when viewed along the rotation axis from the origin towards the point lmn, the object is rotated clockwise for positive  with a fixed right-handed axial system. If, under the same viewing conditions, the axes are to be rotated clockwise through  with the object fixed then the components of vectors in the object, on the new axes, are given by R with the same lmn and with  negated. This is the transpose of R, and if R is constructed from a product, as below, then each factor matrix in the product must be transposed and their order reversed to achieve this. Note that if, for a given rotation, the viewing direction from the origin is reversed, l, m, n and  are all reversed and the matrix is unchanged. Any rotation about a reference axis such that two of the direction cosines are zero is termed a primitive rotation, and it is frequently a requirement to generate or to interpret a general rotation as a product of primitive rotations. A second important general form is based on Eulerian angles and is the product of three such primitives. It is 10 10 cos ’2 0 sin ’2 cos ’1
sin ’1 cos ’3
sin ’3 0 CB CB B R ¼ @ sin ’3 cos ’3 0 A@ 0 1 0 A@ sin ’1 cos ’1 0 0 1
sin ’2 0 cos ’2 0 0 0 1 ðcos ’3 cos ’2 cos ’1
ðcos ’3 cos ’2 sin ’1 cos ’3 sin ’2 B
sin ’ sin ’ Þ C þ sin ’3 cos ’1 Þ 3 1 B C B C B C ¼ B ðsin ’3 cos ’2 cos ’1 ð
sin ’3 cos ’2 sin ’1 sin ’3 sin ’2 C B C B þ cos ’3 sin ’1 Þ C þ cos ’3 cos ’1 Þ @ A 0


sin ’2 cos ’1

sin ’2 sin ’1

0

1

C 0A 1

cos ’2

420

3.3. MOLECULAR MODELLING AND GRAPHICS second form (the product of primitives) is not ideal because the three angles, though independent, are not equivalent, the nonequivalence arising from the noncommutation of the primitive factors. In the remainder of this section we give two further forms of orthogonal matrix which each use three variables which are independent and strictly equivalent, and a third form using four whose squares sum to unity. The first of these is based on the diagonal and uses the three independent variables p, q, r, from which we construct the auxiliary variables pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ¼  1 þ p
q
r; Q ¼  1
p þ q
r ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼  1
p
q þ r; S ¼  1 þ p þ q þ r ; then

0

p

B R ¼ @ 12½PQ þ RS 1 2½PR
QS

1 2½PQ


RS

1 2½QR

q þ PS

Note that if r1 and r2 are parallel this reduces to the formula for the tangent of the sum of two angles, and that if r1  r2 ¼ 1 the combined rotation is always 180 . Note, too, that reversing the order of application of the rotations reverses only the vector product. If three rotations r1, r2 and r3 are applied successively, r1 first, then their combined rotation is r ¼ ½r3 ð1
r1  r2 Þ þ r2 ð1 þ r3  r1 Þ þ r1 ð1
r3  r2 Þ þ r 3  r2 þ r 3  r1 þ r 2  r 1   ½1
r1  r2
r2  r3
r3  r1
r3  ðr2  r1 Þ
1 : Note the irregular pattern of signs in the numerator. Similar ideas, using a vector of magnitude sinð=2Þ, are developed in Aharonov et al. (1977). The third form of orthogonal matrix uses four variables, , ,

and , which comprise a four-dimensional vector q, such that  ¼ ls, ¼ ms, ¼ ns with s ¼ sinð=2Þ and  ¼ cosð=2Þ. In terms of these variables

1

1 2½PR þ QS C 1 2½QR
PS A

r

0

ð2
2
2 þ  2 Þ R¼@ 2ð  þ Þ 2ð
Þ

is orthogonal with positive determinant for any of the sixteen sign combinations. The signs of P, Q, R and S are, respectively, the signs of the direction cosines of the rotation axis and of sin . ffi pffiffiffiffiffiffiffiffiffiffiffiffi Using also T ¼ 4
S2, which may be deemed positive without loss of generality,

Two further matrices S and T may be defined (Diamond, 1988), 1 0 1 0



 


 B

 B

 C 
 C B C; C S¼B @

 A and T ¼ @
  A 






l ¼ P=T; m ¼ Q=T; n ¼ R=T; sin  ¼ ST=2; cos  ¼ 1
T 2 =2 ¼ S2 =2
1: Although p, q and r are independent, the point [pqr] is bound, by the requirement that P, Q, R and S be real, to lie within a tetrahedron whose vertices are the points [111], ½11
1
, ½1
11
 and ½1
1
1, corresponding to the identity and to 180 rotations about each of the axes. The facts that the identity occurs at a vertex of the feasible region and that ð1
cos Þ, rather than sin , is linear on p, q and r in this vicinity make this form suitable only for substantial rotations. The second form consists in defining a rotation vector r with components u, v, w such that u ¼ lt, v ¼ mt, w ¼ nt with t ¼ tanð=2Þ and r  r ¼ t2 . Then the matrix 1 0 1 þ u2
v2
w2 2ðuv
wÞ 2ðuw þ vÞ C B 1 þ t2 1 þ t2 1 þ t2 C B C B 2ðuv þ wÞ 1
u2 þ v2
w2 2ðvw
uÞ C B R¼B C 2 2 2 C B 1 þ t 1 þ t 1 þ t C B @ 2 2 2A 2ðuw
vÞ 2ðvw þ uÞ 1
u
v þw 1 þ t2 1 þ t2 1 þ t2 2
1 RIJ ¼ ð1 þ t Þ ½IJ ð1
uk uk Þ þ 2ðuI uJ
"IJl ul Þ

which are themselves orthogonal (though S has determinant
1) and which have the property that   R 0 2 S ¼ 0T 1 so that, for example, if homogeneous coordinates are being employed (Section 3.3.1.1.2) 10 1 10 0 01 0



 x x



 CB y C CB

 B y0 C B

   CB C B 0C¼B CB @ z A @

 A@

 A@ z A w w 

 

 is a rotation of (x, y, z, w) through the angle  about the axis (l, m, n). With suitably pipelined hardware this forms an efficient means of applying rotations since the ‘overhead’ of establishing S is so trivial. T has the property that the rotation vector q arising from a concatenation of n rotations is q ¼ T n T n
1 . . . T 1 q0 ;

is orthogonal and the variables u, v, w are independent, equivalent and unbounded, and, unlike the previous form, small rotations are quasi-linear on these variables. As examples, r = [100] gives 90 about X, r = [111] gives 120 about [111]. R then transforms a vector d according to Rd ¼ d þ

1 2ð
Þ 2ð þ Þ A: 2ð
Þ ð
2 þ 2
2 þ  2 Þ 2ð þ Þ ð
2
2 þ 2 þ  2 Þ

in which qT0 is the vector (0, 0, 0, 1) which defines a null rotation. This equation may be used as a basis for factorizing a given rotation into a concatenation of rotations about designated axes (Diamond, 1990a). Finally, an exact rotation of the vector d may be obtained without using matrices at all by writing 1 P d ¼ dn

2 fðr  dÞ þ ½r  ðr  dÞg: 1 þ t2

Multiplying two such matrices together allows us to establish the manner in which the rotation vectors r1 and r2 combine. r þ r1 þ r2  r1 r¼ 2 1
r2  r 1

0

in which 1 dn ¼ ðh  dn
1 Þ n

for a rotation r1 followed by r2, so that rotations expressed in terms of rotation angles and axes may be compounded into a single such rotation without the need to form and decompose a product matrix.

and d0 is the initial position which is to be rotated. Here h is a vector with direction cosines l, m and n, and magnitude equal to

421

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING   @AIJ the required rotation angle in radians (Diamond, 1966). This ¼
"IJk lk ; method is particularly efficient when jhj 1 or when the number @ ¼0 of vectors to be transformed is small since the overhead of establishing R is eliminated and the process is simple to program. therefore It is the three-dimensional analogue of the power series for sin      @Aij @E and cos  and has the same convergence properties. ¼ 2Wa R x ðA R x
Xia Þ @ ¼0 @ ¼0 jk ka il lm ma 3.3.1.2.2. Measurement of rotations and strains from ¼ 2"ijl Rjk Mki ll : coordinates Given the coordinates of a molecular fragment it is often a For this to vanish for all possible rotation axes l the vector requirement to relate the fragment to its image in some standard gL ¼ "ijL Rjk Mki orientation by a transformation which may be required to be a pure rotation, or may be required to be a combination of rotation must vanish, i.e. at the end of the iteration R must be such that and strain. Of the methods reviewed in this section all except (iv) the matrix are concerned with pure rotation, ignoring any strain that may be present, and give the best rigid-body superposition. In all these NJI ¼ RJk MkI methods, unless inhomogeneous strain is being considered, the best possible superposition is obtained if the centroids of the two is symmetrical. The vector g represents the couple exerted on the images are first brought into coincidence by translation and rotating body by forces 2WA ðRIj xjA
XIA Þ acting at the atoms. treated as the origin. Choosing Methods (i) to (v) seek transformations which perform the lL ¼ gL =jgj superposition and impose on these, in various ways, the requirements of orthogonality for the rotational part. All these gives the greatest j@E=@j¼0 and ð@E=@Þ vanishes when methods therefore need some defence against indeterminacy that arises in the general transformation if one or both of the frag"ijk Nji lk tan  ¼ ments is planar, and, if improper rotations are to be excluded, Npq ðlp lq
pq Þ need a defence against these also if the fragment and its image are of opposite chirality. Methods (vi) and (vii) pay no attention in which N is constructed from the current R matrix. A is then to the general transformation (which defines the general constructed from l and this  and AR replaces R. The process is superposition) and work with variables which are intrinsically iterative because a couple about some new axis can appear when rotational in character, and always produce an orthogonal rotation about g eliminates the couple about g. transformation with positive determinant, with no degeneracy Note that for each rotation axis l there are two values of , arising from planar fragments which need no special attention. differing by , which reduce jgj to zero, corresponding to Even collinear atoms cause no problem, the superposition being maximum and minimum values of E. The minimum is that which performed correctly but with an arbitrary rotation about the makes length of the line being present in the result. These methods are therefore to be preferred over the earlier ones unless the purpose @2 E ¼ 2ðtr N
li Nij lj Þ of the operation is to detect differences of chirality, although this, @2 too, can be detected with a simple test. In this review we adopt the same notation for all the methods positive. Adding  to  alters R and N and negates this quantity. which, unavoidably, means that symbols are used in ways which Note, too, that the process is essentially characterized as that differ from the original publications. We use the symbol x for the which makes the product RM symmetrical with R orthogonal. We vector set which is to be rotated and X for the vector set whose return to this point in (iii). orientation is not to be altered, and write the residuals as (ii) Kabsch’s method (Kabsch, 1976, 1978) minimizes E with respect to the nine elements of D, subject to the six constraints eIA ¼ DIj xjA
XIA DkI DkJ
IJ ¼ 0IJ ; and, by choice of origin, by using an auxiliary function Wa xIa ¼ Wa XIa ¼ 0I F ¼ Lij ðDki Dkj
ij Þ for weights W. The quadratic residual to be minimized is in which L is symmetric containing six Lagrange multipliers. The E ¼ Wa eia eia Lagrangian function and we define the matrix MIJ ¼ Wa xIa XJa and use l for the direction cosines of the rotation axis. (i) McLachlan’s first method (McLachlan, 1972, 1982) is iterative and conceptually the simplest. It sets

G¼EþF then has minima with respect to the elements of D at locations which are dependent, inter alia, on the elements of L. By suitably choosing L a minimum of G may be brought into coincidence with the constrained minimum of E. A minimum of G occurs where

DIJ ¼ AIk RkJ in which A and R are both orthogonal with R being a current estimate of D and A being an adjustment which, at the beginning of each cycle, has a zero angle associated with it. One iterative cycle estimates a nontrivial A, after which the product AR replaces R.

@G ¼ 2DIk ðSJk þ LJk Þ
2MJI ¼ 0IJ @DIJ and the 9  9 matrix

AIJ ¼ ð1
cos ÞlI lJ þ IJ cos 
"IJk lk sin 

@2 G ¼ 2MI ðSJK þ LJK Þ @DMK @DIJ

and

422

3.3. MOLECULAR MODELLING AND GRAPHICS D ¼ RT

is positive definite, block diagonal, and has

T 2 ¼ DT D

SJK ¼ Wa xJa xKa

T ¼ ðDT DÞ1=2 which is symmetrical. Thus L must be chosen so as to make the symmetric matrix ðS þ LÞ such that

R ¼ DðDT DÞ
1=2 :

DðS þ LÞT ¼ M T

Furthermore, the solution for D is D ¼ M T S
1

with D orthogonal, or RN ¼ M T with R replacing D since we are now confined to the orthogonal case, and N is symmetric and positive definite. (iii) Comparison of the Kabsch and McLachlan methods. Using the initials of these authors as subscripts, we have seen that the Kabsch solution involves solving RWK N WK ¼ M

(in the notation of Kabsch), so that R ¼ M T S
1 ðS
1 MM T S
1 Þ
1=2 which may be compared with the results of the previous paragraph. Although this R matrix by itself (i.e. applied without T) does not produce the best rotational superposition (i.e. smallest E), it is the one which exactly superposes the only three vectors in x whose mutual dispositions are conserved, on their equivalents in X, so that the rotation so found is arguably the best defined one. Alternatives based on D ¼ TR, D
1 ¼ RT, D
1 ¼ TR are all easily developed, and these ideas are developed by Diamond (1976a) to include nonhomogeneous strains also. (v) McLachlan’s second method. This method (McLachlan, 1979) is based on the properties of the 6  6 matrix   0 M MT 0

T

for an orthogonal matrix RWK given that N WK is symmetrical and positive definite. Thus MM T ¼ N TWK RTWK RWK N WK ¼ N 2WK and RWK ¼ M T ðMM T Þ
1=2 : By comparison, the McLachlan treatment leads to an orthogonal R matrix satisfying RADM ¼ N ADM M
1

and is immune to singularity of M. If p and q are threedimensional vectors such that ðpT ; qT Þ is an eigenvector of this matrix then        0 M p Mq p ¼ ¼ : MT 0 q MT p q

in which N ADM is also symmetric and positive definite, which similarly leads to RADM ¼ ðM T MÞ1=2 M
1 : These seemingly different expressions for RWK and RADM are, in fact, equal, as the following shows

If q is negated the second equality is maintained provided  is also negated. Therefore an orthogonal 6  6 matrix   H H K
K


1 T
1 RWK ¼ RADM R
1 ADM RWK ¼ RADM MN ADM M N WK ;

therefore (consisting of 3  3 partitions) exists for which  T     0 M H H K H KT ¼ MT 0 K
K 0 H T
K T

RTWK RWK ¼ I
1 T T
1 T
1 ¼ N
1 WK MN ADM M RADM RADM MN ADM M N WK :

Multiplying on both sides by N WK gives

0
K



in which K is diagonal and contains non-negative eigenvalues. The reverse transformation shows that

T 2 N 2WK ¼ ðMN
1 ADM M Þ ;

M ¼ 2HKK T and since both N matrices are positive definite and multiplying the eigenvectors together gives

T N WK ¼ MN
1 ADM M

H T H ¼ K T K ¼ 12I ¼ HH T ¼ KK T :

and conversely Therefore

N ADM ¼ M T N
1 WK M;

2KH T M ¼ 4KH T HKK T ¼ 2KKK T ;

therefore T

RWK ¼ M M

T
1

N ADM M


1

but 2KH T is orthogonal and 2KKK T is symmetrical, therefore [by paragraphs (i) and (iii) above] 2KH T is the required rotation. Similarly, forming

¼ RADM :

(iv) Diamond’s first method. This method (Diamond, 1976a) differs from the previous ones in that the transformation D is allowed to be a general transformation which is then factorized into the product of an orthogonal matrix R and a symmetrical matrix T. The transformation of x to fit X is thus interpreted as the combination of homogeneous strain and pure rotation in which x is subjected to strain and the result is rotated.

M T ¼ 2KKH T 2M T HK
1 H T ¼ 4KKH T HK
1 H T ¼ 2KH T corresponds to the Kabsch formulation [paragraphs (ii) and (iii)] since 2HK
1 H T is symmetrical and the same rotation, 2KH T , appears.

423

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING R ¼ 2ll T
I:

Note that the determinant of the orthogonal matrix so found is twice the product of the determinants of H and of K, and since the positive eigenvalues are collected into K it follows that the sign of the determinant of M is the same as the sign of the determinant of the resulting orthogonal matrix. If this is negative it means that the best superposition is obtained if one vector set is inverted and that x and X are of opposite chirality. Expanding the expression for E, the weighted sum of squares of errors, for an orthogonal transformation shows that this is least when the trace of the product RM is greatest. In this treatment

Note that MacKay’s residual E is quadratic in r. E therefore has one minimum and no maximum, and the minimum is reached on the first cycle of least squares. By contrast, the objective function E that is minimized in methods (i), (ii), (v) and (vii) has one minimum, one maximum and two saddle points in the space of the vector r, as shown in (vii). It may be shown (Diamond, 1989) that if MacKay’s solution vector r is denoted by rM and that given by the other methods [except (iv)] by rO then

trðRMÞ ¼ trð2KH T  2HKK T Þ ¼ trð2KKK T Þ ¼ trðKÞ:

rM ¼ rO
A
1 BrO

Hence, if the eigenvalues in K and
K are arranged in decreasing order of modulus, and if the determinant of M is negative, then exchanging the third and sixth columns of   H H K
K

in which A and B are real symmetric, positive semi-definite. A is positive definite unless all the individual vector sums ðX þ xÞ are parallel, as can happen when the best rotation is 180 . Thus the MacKay method only gives the same result as the other methods if: (a) the initial orientation is optimal, for then rO ¼ 0, or (b) perfect fitting is possible, for then B ¼ 0, or (c) all the residual vectors (after fitting by rO ) are parallel to rO, for then B is singular such that BrO ¼ 0. In general, jrM j  jrO j. rO may be found by iterating rM , but x must be replaced by Rx on each iteration. (vii) Diamond’s second method. This is closely related to MacKay’s method, but uses a four-dimensional vector q with components , , and  in which , and are the direction cosines of the rotation axis multiplied by sinð=2Þ and  is cosð=2Þ. In terms of such a vector Diamond (1988) showed that

produces a product 2KH T with positive determinant (i.e. a proper rotation) at minimum cost in residual. Similarly, if M is singular and one or more eigenvalues in K vanishes it is necessary only to complete an orthonormal set of eigenvectors such that the determinants of H and K have the same sign. (vi) MacKay’s method. MacKay (1984) was the first to consider the rotational superposition problem in terms of the vector r of Section 3.3.1.2.1. Using quaternion algebra he showed that if a vector x is rotated to X ¼ Rx then ðX
xÞ ¼ r  ðX þ xÞ;

E ¼ E0
2qT Pq

where jrj ¼ tanð=2Þ and the direction of r is the axis of rotation, as may also be shown from elementary considerations. MacKay then solves this for the vector r by least squares given the vector pairs X and x. The individual errors are

in which E is the weighted sum of squares of coordinate differences, as before, E0 is its value before any rotation is applied and P is the matrix   Q V : P¼ VT 0

eIA ¼ "Ijk rj ðXkA þ xkA Þ
ðXIA
xIA Þ and E ¼ Wa eia eia :

The rotation matrix R corresponding to the vector q is then the last of the forms for R given in Section 3.3.1.2.1. The minimum E is therefore E0 minus twice the largest eigenvalue of P since qT q ¼ 1, and a stationary value of E occurs when q is any of the four eigenvectors of P. E thus has a maximum, a minimum and two saddle points, in general, and its value may be determined before any coordinates are transformed. Diamond also showed that the orientations giving these stationary values are related by the operations of 222 symmetry. Equivalent results have also been obtained by Kearsley (1989). As an alternative to solving a 4  4 eigenproblem, Diamond also showed that the vector r, as in MacKay’s solution, may be obtained by iterating

Setting @E=@rP ¼ 0P gives Wa "iPk "ilm rl ðXka þ xka ÞðXma þ xma Þ ¼ Wa "iPk ðXka þ xka ÞðXia
xia Þ which reduces to 2V ¼
ðQ þ Q0 Þr in which Q ¼ M þ M T
2I tr M Q0 ¼ S þ S0
Iðtr S þ tr S0 Þ VI ¼ "Ijk Mjk


0 ¼ E0 =2

SIJ ¼ Wa xIa xJa

rn ¼ ð
n I
QÞ
1 V

S0IJ

¼ Wa XIa XJa :


nþ1 ¼

Thus a direct solution for r is obtained, r ¼
2ðQ0 þ QÞ
1 V;

V  rn þ
n r2n 1 þ r2n

which has the property that if X and x are exactly superposable then
0 is the exact solution and no iteration is necessary. If X and x are similar but not exactly superposable then a small number of iterations may be required to reach a stable r vector, though the matrix Q0 is not required. As in MacKay’s solution, ð
I
QÞ is singular at the end of the iteration if the required rotation is 180 , but the MacKay and Diamond methods both have the advantage that improper rotations are never generated by these means, and methods based on P and q rather than Q and r are trouble-free

the elements of which are u, v and w, and may be used to construct the orthogonal matrix as in Section 3.3.1.2.1. Q þ Q0 may be obtained directly from X þ x. If the requisite rotation is 180 , ðQ0 þ QÞ is singular and cannot be inverted. In this case any row or column of the adjoint of ðQ0 þ QÞ is a vector in the direction of the axis. Normalizing this vector to unity, giving l, gives the requisite orthogonal matrix as

424

3.3. MOLECULAR MODELLING AND GRAPHICS 

with R orthogonal and T symmetrical gives

for 180 rotations. The iterative loop in this method does not require Rx to be redetermined on each cycle. Finally, it may be shown that if p1 ; p2 ; p3 ; p4 are the eigenvalues of P arranged in descending order and

T ¼ ðM T MÞ1=2 ;

The rotation so found is the one which exactly superposes those three mutually perpendicular directions which remain mutually perpendicular under the transformation M. T
I is then the strain tensor of an unrotated body. Writing M ¼ TR, T ¼ ðMM T Þ1=2 , R ¼ ðMM T Þ
1=2 M may also be useful, in which T
I is the strain tensor of a rotated body. See also Section 3.3.1.2.2 (iv).

p1
p2
p3 þ p4 is negative, then a closer superposition may be obtained by reversing the chirality of one of the vector sets, and the R matrix constructed from q4 optimally superimposes Rx onto
X, the enantiomer of X (Diamond, 1990b). These ideas lend themselves to the problem of the multiple simultaneous superposition of many structures, which is a nonlinear problem with the possibility of multiple solutions [Diamond (1992); see also Kearsley (1990) and Shapiro et al. (1992)], and they can provide for cluster analysis based on structural similarity to find subsets of similar structures within an ensemble of structures, such as may arise from NMR calculations (Diamond, 1995).

3.3.1.2.4. Eigenvalues and eigenvectors of orthogonal matrices If R is the orthogonal matrix given in Section 3.3.1.2.1 in terms of the direction cosines l, m and n of the axis of rotation, then it is clear that (l, m, n) is an eigenvector of R with eigenvalue unity because 0 1 0 1 l l R@ m A ¼ @ m A: n n

3.3.1.2.3. Orthogonalization of impure rotations There are several ways of deriving a strictly orthogonal matrix from a given approximately orthogonal matrix, among them the following. (i) The Gram–Schmidt process. This is probably the simplest and the easiest to compute. If the given matrix consists of three column vectors v1, v2 and v3 (later referred to as primers) which are to be replaced by three column vectors u1, u2 and u3 then the process is

Consideration of the determinant jR
Ij ¼ 0 shows that the sum of the three eigenvalues is 1 þ 2 cos  and that their product is unity. Hence the three eigenvalues are 1, ei and e
i . Since R is real, its product with any real vector is also real, yet its product with an eigenvector must, in general, be complex. Thus the eigenvectors must themselves be complex. The remaining two eigenvectors u may be found using the results of Section 3.3.1.2.1 (q.v.) according to

u1 ¼ v1 =jv1 j u2 ¼ v2
ðu1  v2 Þu1 u2 ¼ u2 =ju2 j

Ru ¼ u þ

2 1  it ; fðr  uÞ þ ½r  ðr  uÞg ¼ uei ¼ u 1 þ t2 1 it

which is solved by any vector of the form

u3 ¼ v3
ðu1  v3 Þu1
ðu2  v3 Þu2 u3 ¼ u3 =ju3 j:

u ¼ l  v il  ðl  vÞ for any real vector v, where l is the normalized axis vector, lt ¼ r, jlj ¼ 1, t ¼ tanð=2Þ. Eigenvectors for the two eigenvalues may have unrelated v vectors though the sign choices are coupled. If the vector v is rotated about l through an angle ’ the corresponding vector u is multiplied by e
i’ and remains an eigenvector. Using superscript signs to denote the sign of  in the eigenvalue with which each vector is associated, the matrix

As successive vectors are established, each vector v has subtracted from it its components in the directions of established vectors, and the remainder is normalized. The method will fail at the normalization step if the vectors v are not linearly independent. Otherwise, the process may be extended to any number of dimensions. The weakness of the method is that, though u1 differs from v1 only in scale, uN may differ grossly from vN as the various columns are not treated equivalently. (ii) A preferable method which treats all vectors equivalently is to iteratively replace the matrix M by 12ðM þ M T
1 Þ. Defining the residual matrix E as

U ¼ ðl; uþ ; u
Þ has the properties that

then on each iteration E is replaced by E2 ðMM T Þ
1 =4

and

0

1 U U ¼ @0 0

and convergence necessarily ensues. (iii) A third method resolves M into its symmetric and antisymmetric parts A ¼ 12ðM
M T Þ;

0

1 RU ¼ U @ 0 0

E ¼ MM T
I;

S ¼ 12ðM þ M T Þ;

R ¼ MðM T MÞ
1=2 :

T

M ¼SþA

0 ei 0

0 2jl  vþ j2 0

1 0 0 A e
i 1 0 A 0 2jl  v
j2

which places restrictions on v if this is to be the identity. Note that the 23 element vanishes even in the absence of any relationship between vþ and v
. A convenient form for U, symmetrical in the elements of l, is obtained by setting vþ ¼ v
¼ ½111 and is

and constructs an orthogonal matrix for which only S is altered. A determines l, m, n and  as shown in Section 3.3.1.2.1, and from these a new S may be constructed. (iv) A fourth method is to treat the general matrix M as a combination of pure strain and pure rotation. Setting

0 l fðm
nÞ
i½lðl þ m þ nÞ
1g=d U ¼ @ m fðn
lÞ
i½mðl þ m þ nÞ
1g=d n fðl
mÞ
i½nðl þ m þ nÞ
1g=d

M ¼ RT

425

1 fðm
nÞ þ i½lðl þ m þ nÞ
1g=d fðn
lÞ þ i½mðl þ m þ nÞ
1g=d A fðl
mÞ þ i½nðl þ m þ nÞ
1g=d

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING in which the normalizing denominator is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ 2 1
lm
mn
nl:

3.3.1.3.2. Translation The transformation    NI V X 0T

N

W

 ¼ 

3.3.1.3. Projection transformations and spaces ’

In the following section we address the question of the relationship between the coordinates of a molecular model and the corresponding coordinates on the screen of the graphics device. A good introduction to this topic is given by Newman & Sproull (1973), and Foley et al. (1990) give a comprehensive account of the field, including recent developments, especially those arising from the development of raster-graphics technologies.

XN þ VW



’ NW  X=W þ V=N



X þ VW=N



W

1

evidently corresponds to the addition of the vector VW=N to the components of X or of V=N to the components of X=W. (I is the identity.) Displacements may thus be affected by expressing the required displacement vector in homogeneous coordinates with any suitable choice of N (commonly, N ¼ W), with V scaled to correspond to this choice, and loading the 4  4 transformation matrix as indicated above.

3.3.1.3.1. Definitions Typically, the coordinates, X, of points in an object to be drawn are held in homogeneous Cartesian form as described in Section 3.3.1.1.2. Such coordinates are said to be in data space or world coordinates and this coordinate system is generally a constant aspect of the problem. In order to view these data in convenient ways such coordinates may be subjected to a 4  4 viewing transformation T, affecting orientation, scale etc., the resulting coordinates TX being then in display space. Here, and throughout what follows, we treat position vectors as columns with transformation matrices as factors on the left, though some writers do the reverse. In general, only some portion of display space which lies inside a certain frustum of a pyramid is required to fall within the picture. The pyramid may be thought of as having the observer’s eye at its vertex, with a rectangular base corresponding to the picture area. This volume is called a window. A transformation, U, which takes display-space coordinates as input and generates vectors (X, Y, Z, W) for which X=W and Y=W ¼ 1 for points on the left, right, top and bottom boundaries of the window and for which Z=W takes particular values on the front and back planes of the window, is said to be a windowing transformation. In machines for which Z=W controls intensity depth cueing, the range of Z=W corresponding to the window is likely to be 0 to 1 rather than
1 to 1. Coordinates obtained by multiplying displayspace coordinates by U are termed picture-space coordinates. Mathematically, U is a 4  4 matrix like any other, but functionally it is special. Declaring a transformation to be a windowing transformation implies that only resulting points having jXj; jYj < W and positive Z < W are to be plotted. Machines with clipping hardware to truncate lines which run out of the picture perform clipping on the output from the windowing transformation. Finally, the picture has to be drawn in some rectangular portion of the screen which is allocated for the purpose. Such an area is termed a viewport and is defined in terms of screen coordinates which are defined absolutely for the hardware in question as n for full-screen deflection, where n is declared by the manufacturer. Screen coordinates are obtained from picture coordinates with a viewport transformation, V.1 To summarize, screen coordinates, S, are given by

3.3.1.3.3. Rotation Rotation about the origin is achieved by        NR 0 X NRX RX ’ ; ¼ 0T N W NW W in which R is an orthogonal 3  3 matrix. R necessarily has elements not exceeding one in modulus. For machines using integer arithmetic, therefore, N would be chosen large enough (usually half the largest possible integer) for the product NR to be well represented in the available word length. Characteristically, N affects resolution but not scale. 3.3.1.3.4. Scale The transformation        SNI 0 X SNX SX ¼ ’ 0T N W NW W scales the vector (X, W) by the factor S. For integer working and jSj < 1, N should be set to the largest representable integer. For jSj > 1 the product SN should be the largest representable integer, N being reduced accordingly. 3.3.1.3.5. Windowing and perspective It is necessary at this point to relate the discussion to the axial system inherent in the graphics device employed. One common system adopts X horizontal and to the right when viewing the screen, Y vertically upwards in the plane of the screen, and Z normal to X and Y with +Z into the screen. This is, unfortunately, a left-handed system in that ðX  YÞ  Z is negative. Since it is usual in crystallographic work to use right-handed axial systems it is necessary to incorporate a matrix of the form 0 1 W 0 0 0 B 0 W 0 0 C B C @ 0 0
W 0 A 0 0 0 W either as the left-most factor in the matrix T or as the right-most factor in the windowing transformation U (see Section 3.3.1.3.1). The latter choice is to be preferred and is adopted here. The former choice leads to complications if transformations in display space will be required. Display-space coordinates are necessarily referred to this axial system. Let L, R, T, B, N and F be the left, right, top, bottom, near and far boundaries of the windowed volume ðL < R; T > B; N < FÞ, S be the Z coordinate of the screen, and C, D and E be the coordinates of the observer’s eye position, all ten of these parameters being referred to the origin of display space as origin, which may be anywhere in relation to the hardware. L, R, T and B are to be evaluated in the screen plane. All ten parameters may be referred

1 In recent years it has become increasingly common, especially in twodimensional work, to apply the term ‘window’ to what is here called a viewport, but in this chapter we use these terms in the manner described in the text.

426

3.3. MOLECULAR MODELLING AND GRAPHICS

Fig. 3.3.1.1. The relationship between display-space coordinates (X, Y, Z, W) and picture-space coordinates (x, y, z, w) derived from them by the window transformation, U. (a) Display space (in X, Z projection) showing a square object P, Q, R, S for display viewed from the position (C, D, E, V). The bold trapezium is the window (volume) and the bold line is the viewport portion of the screen. The points P, Q, R and S must be plotted at p, q, r and s to give the correct impression of the object. (b) Picture space (in x, z projection). The window is mapped to a rectangle and all sight lines are parallel to the z axis, but the object P, Q, R, S is no longer square. The distribution of p, q, r and s is identical in the two cases. Note that z/w values are not linear on Z/W, and that the origin of picture space arises at the midpoint of the near clipping plane, regardless of the location of the origin of display space. The figure is accurately to scale for coincident viewport positions. The words ‘Left clipping plane’, if part of the scene in display space, would currently be obscured, but would come into view if the eye moved to the right, increasing C, as the left clipping plane would pivot about the point L/V in the screen plane.

which provides for jx=wj and jy=wj to be unity on the picture boundaries, which is usually a requirement of the clipping hardware, and for 0 < z=w < 1, zero being for the near-plane boundary. Even though z/w is not linear on Z/W, straight lines and planes in display space transform to straight lines and planes in picture space, the nonlinearity affecting only distances. Thus vector-drawing machines are not disadvantaged by the introduction of perspective. Note that the dimensionality of X/W must equal that of S/V and that this may be regarded as length or as a pure number, but that in either case x/w is dimensionless, consistent with the stipulation that the picture boundaries be defined by the pure number 1. The above matrix is U and is suited to left-handed hardware systems. Note that only the last column of U (the translational part) is sensitive to the location of the origin of display space and that if the eye is on the normal to the picture centre then C ¼ 12ðR þ LÞ, D ¼ 12ðT þ BÞ and simplifications result. If C, D and E can be continuously monitored then dynamic parallax as well as perspective may be obtained (Diamond et al., 1982). If data space is referred to right-handed axes, the viewing transformation T involves only proper rotations and the hardware uses a left-handed axial system then elements in the third column of U should be negated, as explained in the opening paragraph. To provide for orthographic projection, multiply every element of U by
K=E and then let E !
1, choosing some positive K

to their own fourth coordinate, V, meaning that the point (X, Y, Z, W) in display space will be on the left boundary of the picture if X=W ¼ L=V when Z=W ¼ S=V. V may be freely chosen so that all eleven quantities and all elements of U suit the word length of the machine. These relationships are illustrated in Fig. 3.3.1.1. Since   XV YV ZV ; ; ;V ; ðX; Y; Z; WÞ ’ W W W XV=W is a display-space coordinate on the same scale as the window parameters. This must be plotted on the screen at an X coordinate (on the scale of the window parameters) which is the weighted mean of XV=W and C, the weights being ðS
EÞ and ðZV=W
SÞ, respectively. This provides perspective because the weighted mean is at the point where the straight line from ðX; Y; Z; WÞ to the eye intersects the screen. This then has to be mapped into the L-to-R interval, so that picture-space coordinates ðx; y; z; wÞ are given by 1 0 2ðS
EÞV x B C B ðR
LÞ B C B B C B ByC B 0 B C B B C¼B B C B B C B BzC B 0 @ A @ 0

w

0

0 2ðS
EÞV ðT
BÞ 0 0

ð2C
R
LÞV ðR
LÞ ð2D
T
BÞV ðT
BÞ ðF
EÞV ðF
NÞ V

10 1 ðR þ LÞE
2SC X CB C ðR
LÞ CB C B C ðT þ BÞE
2SD C CB Y C CB C ðT
BÞ CB C CB C B C
NðF
EÞ C CB Z C A@ A ðF
NÞ
E W

427

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING in which U 000 is the matrix U obtained by setting ðC; D; E; VÞ to correspond to the point midway between the viewer’s eyes and 1 0 1 0 c=ðS
EÞ
cS=ðS
EÞV C B C B0 1 0 0 C B S¼B C C B0 0 1 0 A @ 0 0 0 1 1 0 V 0 cV=ðS
EÞ
cS=ðS
EÞ C B C B0 V 0 0 C B ’B C C B0 0 V 0 A @

to suit the word length of the machine [see Section 3.3.1.1.2 (iii)]. The result is 1 0 2KV
KðR þ LÞ 0 0 B ðR
LÞ ðR
LÞ C C B B 2KV
KðT þ BÞ C C B 0 0 C B U0 ’ B ðT
BÞ ðT
BÞ C; C B B KV
KN C C B 0 0 @ ðF
NÞ ðF
NÞ A 0 0 0 K which is the orthographic window. It may be convenient in some applications to separate the functions of windowing and the application of perspective, and to write

0

where U and U 0 are as above and P is a perspective transformation given by 0 1 S
E 0 C
SC=V B 0 S
E D
SD=V C B C P ¼ ðU 0 Þ
1 U ’ B C; @ 0 0 F
E þ N
NF=V A 0 0 V
E

0

which involves F and N but not R, L, T or B. In this form the action of P may be thought of as compressing distant parts of display space prior to an orthographic projection by U 0 into picture space. Other factorizations of U are possible, for example

with

0

2KV T
B 0 0

B B B P ’B B @ 0


KðR þ LÞ 1 ðR
LÞ C C C
KðT þ BÞ C C 0 ðT
BÞ C C C KVðN
EÞðF
EÞ KNðF
EÞ C C E2 ðF
NÞ EðF
NÞ A 0

0 S
E

0

0 C

0

S
E

D

0

0


E

0

0

V


SC=V

1

V

0

0

1

with  ¼ c=ðS
EÞ. The main difference is in the resulting Z value, which only affects depth cueing and z clipping. The X translation which arises if S 6¼ 0 is also suppressed, but this is not likely to be noticeable.  is often treated as a constant, such as sin 3 . The distinction in principle between the true S and the rotational approximation is that with the true S the eye moves relative to the screen and the displayed object, whereas with the approximation the eye and the screen are moved relative to the displayed object, in going from one view to the other. Strobing of left and right images may conveniently be accomplished with an electro-optic liquid-crystal shutter as described by Harris et al. (1985). The shutter is switched by the display itself, thus solving the synchronization problem in a manner free of inertia. A further discussion of stereopairs may be found in Johnson (1970) and in Thomas (1993), the second of which generalizes the treatment to allow for the possible presence of an optical system.

U ¼ U 00 P 0

0

0

in which (c, 0, 0, V) is the position of the right eye relative to the mean eye position, and the left-eye view is obtained by negating c. Stereo is often approximated by introducing a rotation about the Y axis of  sin
1 ½c=ðS
EÞ to the views or sin
1 ½2c=ðS
EÞ to one of them. The first corresponds to 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 1
 0  0 B 0 1 0 0C B C pffiffiffiffiffiffiffiffiffiffiffiffiffi S¼B C 2 @
 0 1
 0A

U ¼ U 0 P;

0 2KV BR
L B B B B 0 00 U ’B B B B B 0 @

0

K

C
SD=V C C C; C 0 A
E

3.3.1.3.7. Viewports The window transformation of the previous two sections has been constructed to yield picture coordinates (X, Y, Z, W) (formerly called x, y, z, w) such that a point having X=W or Y=W ¼ 1 is on the boundary of the picture, and the clipping hardware operates on this basis. However, the edges of the picture need not be at the edges of the screen and a viewport transformation, V, is therefore needed to position the picture in the requisite part of the screen. 0 1 ðr
lÞ=2 0 0 ðr þ lÞ=2 C B 0 ðt
bÞ=2 0 ðt þ bÞ=2 C B V¼B C; @ A 0 0 n 0 0 0 0 n

which renders P 0 independent of all six boundary planes, but U 00 is no longer independent of E. It is not possible to factorize U so that the left factor is a function only of the boundary planes and the right factor a function only of eye and screen positions. Note that as E !
1, U 00 ! U 0 , P and P 0 !
IE ’ I. 3.3.1.3.6. Stereoviews Assuming that left- and right-eye views are to be presented through the same viewport (next section) or that their viewports are to be superimposed by an external optical system, e.g. Ortony mirrors, then stereopairs are obtained by using appropriate eye coordinates in the U matrix of the previous section. However, U may be factorized according to

where r, l, t and b are now the right, left, top and bottom boundaries of the picture area, or viewport, expressed in screen coordinates, and n is the full-screen deflection value. Thus a point with X=W ¼ 1 in picture space plots on the screen with an X coordinate which is a fraction r=n of full-screen deflection to the

U ¼ U 000 S

428

3.3. MOLECULAR MODELLING AND GRAPHICS right. Z=W is unchanged by V and is used only to control intensity in a technique known as depth cueing. It is necessary, of course, to arrange for the aspect ratio of the viewport, ðr
lÞ=ðt
bÞ, to equal that of the window otherwise distortions are introduced.

If a rotation is to be about a point   V N then

  0 NR 0 NI NI V T ¼ T T 0 0T 0 N N 0   NR V
RV ’ T 0T N 0

3.3.1.3.8. Compound transformations In this section we consider the viewing transformation T of Section 3.3.1.3.1 and its construction in terms of translation, rotation and scaling, Sections 3.3.1.3.2–4. We use T 0 to denote a new transformation in terms of the prevailing transformation T. We note first that any 4  4 matrix of the form   UR V ; 0T W



or T0 ¼ T ’T

with U a scalar, may be factorized according to      UR V UI 0 UI V UR ’ 0T W 0T W 0T U 0T and also that multiplying



UR 0T

V W

0 U





 0  NR 0 NI V NI 0T N 0 0T N 0T   NR V
RV 0T


V T N


V N



N

according to whether R and V are both defined in display space or both in data space. If the rotation is defined in display space and the position of the centre of rotation is defined in data space, then the first form of T 0 must be used, in which V is first computed from     V U ¼T N W



by an isotropic scaling matrix, a rotation, or a translation, either on the left or on the right, yields a product matrix of the same form, and its inverse   WRT
RT V 0T U

for a rotation centre at



U W



in data space. For continuous rotations defined in display space it is usually worthwhile to bring the centre of rotation to the origin of display space and leave it there, i.e. to omit the left-most factor in the first expression for T 0 . Incremental rotations can then be made by further rotational factors on the left without further attention to V. When continuous rotations are implemented by repeated multiplication of T by a rotation matrix, say thirty times a second for a minute or so, the orthogonality of the top-left partition of T may become degraded by accumulation of round-off error and this should be corrected occasionally by one of the methods of Section 3.3.1.2.3. It is sometimes a requirement, depending on hardware capabilities, to effect a transformation in display space when access to data space is all that is readily available. In such a case

is also of this form, i.e. any combination of these three operations in any order may be reduced by the above factorization to a rotation about the original origin, a translation (which defines a new origin) and an expansion or contraction about the new origin, applied in that order. If   NR 0 0T N is a rotation matrix as in Section 3.3.1.3.3, its application produces a rotation about an axis through the origin defined only in the space in which it is applied. For example, if 0 1 cos  sin  0 B C R ¼ @
sin  cos  0 A; 0 0 1      X NR 0 X T0 ¼T T W 0 N W

T 0 ¼ T 1 T ¼ TT 2 ; where T 1 is the required alteration to the prevailing viewing transformation T and T 2 is the data-space equivalent,   
1   UR V U1 R1 V1 UR V
1 T2 ¼ T T1T ¼ 0T W 0T W 0T W1   T T UU1 R R1 R R ðU1 R1 V þ WV1
W 1 VÞ : ’ 0T UW 1

rotates the image about the z axis of data space, whatever the prevailing viewing transformation, T. Forming     NR 0 X T 0T N W

An important special case is when T 1 is to effect a rotation about the origin of display space without change of scale, so that V1 ¼ 0; U1 ¼ W1 ¼ W, for then   URT R1 R RT ðR1
IÞV T2 ’ : U 0T

rotates the image about the z axis of display space, i.e. the normal to the tube face under the usual conventions, whatever the prevailing T. Furthermore, if this rotation is to appear to be about some chosen position in the picture, e.g. the centre, then the window transformation U, Section 3.3.1.3.5, must place the origin of display space there by setting F > S ¼ R þ L ¼ T þ B ¼ 0 > N, in the notation of that section.

If r is the required axis of rotation of R1 in display space then the axis of rotation of RT R1 R in data space is s ¼ RT r since RT R1 Rs ¼ s. This gives a particularly simple result if R1 is to be a

429

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING 1 0 R
L RþL primitive rotation for then s is the relevant row of R, and RT R1 R 0 0 B 2 can be constructed directly from this and the required angle of 2 C C B B T
B T þ BC rotation. 0
1 C: B 0 0 U ’B 2 2 C C B 3.3.1.3.9. Inverse transformations @ 0 0 F
N N A It is frequently a requirement to be able to identify a feature or 0 0 0 V position in data space from its position on the screen. Facilities for identifying an existing feature on the screen are in many Each of these inverse matrices may be suitably scaled to suit the instances provided by the manufacturer as a ‘hit’ function which word length of the machine [Section 3.3.1.1.2 (iii)]. correlates the position indicated on the screen by the user (with a Having determined the end points of one sight line in data tablet or light pen) with the action of drawing and flags the space the viewing transformation T may then be changed and the corresponding item in the drawing internally as having been hit. required position marked again through the screen in the new In other instances it may be necessary to be able to indicate a orientation. Each such operation generates a pair of points in position in data space independently of any drawn feature and data space, expressed in homogeneous form, with a variety of this may be done by setting two or more nonparallel sight lines values for the fourth coordinate. Each such point must then be through the displayed volume and finding their best point of converted to three dimensions in the form (X/W, Y/W, Z/W), and intersection in data space. for each sight line any (three-dimensional) point pA on the line In Section 3.3.1.3.1 the relationship between data-space and the direction qA of the line are established. For each sight line coordinates and screen-space coordinates was given as a rank 2 projector matrix MA of order 3 is formed as S ¼ VUTX; M A ¼ I
qA qTA =qTA qA hence data-space coordinates are given by and the best point of intersection of the sight lines is given by X ¼ T
1 U
1 V
1 S:



P

A line of sight through the displayed volume passing through the point

a

  x y

1 x x By yC C S¼B @o nA n n in screen-space coordinates, as in Section 3.3.1.3.7, from which the corresponding two points in data space may be obtained using 0

V
1

0

0

2n 0 t
b 0 1 0

0

1
ðr þ lÞ ðr
lÞ C C
ðt þ bÞ C C C ðt
bÞ C C A 0 1

0 and 0

U
1

R
L B 2ðS
EÞ B B B 0 B B ’B B B 0 B B @ 0

0 T
B 2ðS
EÞ 0 0

 M a pa ;

a

3.3.1.3.10. The three-axis joystick The three-axis joystick is a device which depends on compound transformations for its exploitation. As it is usually mounted it consists of a vertical shaft, mounted at its lower end, which can rotate about its own length (the Y axis of display space, Section 3.3.1.3.1), its angular setting, ’, being measured by a shaft encoder in its mounting. At the top of this shaft is a knee-joint coupling to a second shaft. The first angle ’ is set to zero when the second shaft is in some selected direction, e.g. normal to the screen and towards the viewer, and goes positive if the second shaft is moved clockwise when seen from above. The knee joint itself contains a shaft encoder, providing an angle, , which may be set to zero when the second shaft is horizontal and goes positive when its free end is raised. A knob on the tip of the second shaft can then rotate about an axis along the second shaft, driving a third shaft encoder providing an angle . The device may then be used to produce a rotation of the object on the screen about an axis parallel to the second shaft through an angle given by the knob. The necessary transformation is then

0

0

P

to which three-vector a fourth coordinate of unity may be applied.

on the screen is the line joining the two position vectors

2n Br
l B B B ’B 0 B B @ 0


1  Ma


CðF
NÞ ðF
EÞðN
EÞ
DðF
NÞ ðF
EÞðN
EÞ
EðF
NÞ ðF
EÞðN
EÞ
VðF
NÞ ðF
EÞðN
EÞ

cos ’

0


sin ’

B R¼@ 0 1 0 sin ’ 0 cos ’ 0 cos 
sin  B  @ sin  cos 

1

ðR þ LÞðN
EÞ
2CðN
SÞ C 2ðN
EÞðS
EÞ C ðT þ BÞðN
EÞ
2DðN
SÞ C C C 2ðN
EÞðS
EÞ C C N C C C ðN
EÞ C A V ðN
EÞ

0 B @

in the notation of Section 3.3.1.3.5, and T
1 was given in Section 3.3.1.3.8. If orthographic projection is being used ðE ¼
1Þ then U
1 simplifies to

which is

430

0

10

1

0

0

CB A@ 0 cos 0
sin 10 0 1 0 CB 0 A@ 0 cos

0
sin

0

cos

0

1

cos ’

0

sin ’

0

1


sin ’

0

C 0 A cos ’

1

sin

sin cos

1 C A 1 C A

0

2

c

2

s ’ þ ð1
c

2

3.3. MOLECULAR MODELLING AND GRAPHICS 0

2

s ’Þc


1 B 0 S ¼B @ 0 0


s c s’ð1
cÞ
c c’s

B B
s c s’ð1
cÞ þ c c’s @
c2 s’c’ð1
cÞ
s s

s2 þ c2 c s c c’ð1
cÞ
c s’s
c2 s’c’ð1
cÞ þ s s

1

and

C s c c’ð1
cÞ þ c s’s C A c2 c2 ’ þ ð1
c2 c2 ’Þc



in which cos and sin are abbreviated to c and s, which is the standard form with l ¼
cos sin ’, m ¼ sin , n ¼ cos cos ’.

S

M
1 0T

0 1



0


1 B 0 ¼B @ 0 0

1

C C 0A 1 1 2

0 1 0 0

0 0
1 0

1 0 1 C 2 b C: 0 A 1

The two sections under this heading are concerned only with the graphical aspects of conformational changes. Determination of such changes is considered under Section 3.3.2.2. 3.3.1.4.1. Rotation about a bond It is a common requirement in molecular modelling to be able to rotate part of a molecule relative to the remainder about a bond between two atoms. If four atoms are bonded 1–2–3–4 then the dihedral angle in the bond 2–3 is zero if the four atoms are cis planar, and a rotation in the 2–3 bond is, by convention (IUPAC–IUB Commission on Biochemical Nomenclature, 1970), positive if, when looking along the 2–3 bond, the far end rotates clockwise relative to the near end. This is valid for either viewing direction. This sign convention, when applied to the R matrix of Section 3.3.1.2.1, leads to the following statement. If one of the two atoms is selected as the near atom and the direction cosines are those of the vector from the near atom to the far atom, and if the matrix is to rotate material attached to the far atom (with the reference axes fixed), then a positive rotation in the foregoing sense is generated by a positive . Rotation about a bond normally involves compounding R with translations in the manner of Section 3.3.1.3.8.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 þ z2 .

3.3.1.4.2. Stacked transformations A flexible molecule may require to be drawn in any of a number of conformations which are related to one another by, for example, rotations about single bonds, changes of bond angles or changes of bond lengths, all of which changes may be brought about by the application of suitable homogeneous transformations during the drawing of the molecule (Section 3.3.1.3.8). With suitable organization, this may be done without necessarily altering the coordinates of the atoms in the coordinate list, only the transformations being manipulated during drawing. The use of transformations in the manner shown below is straightforward for simply connected structures or structures containing only rigid rings. Flexible rings may be similarly handled provided that the matrices employed are consistent with the consequential constraints as described in Section 3.3.2.2.1, though this requirement may make real-time folding of flexible rings difficult. Any simply connected structure may be organized as a tree with a node at each branch point and with an arbitrary number of sites of conformational change between one node and the next. We shall call such sites and their associated matrices ‘conformons’. The technique then depends on the stacking technique in which matrices are stored and later recovered in the reverse order of their storage.

3.3.1.3.12. Symmetry In Section 3.3.1.1.1 it was pointed out that it is usual to express coordinates for graphical purposes in Cartesian coordinates in a˚ngstro¨m units or nanometres. Symmetry, however, is best expressed in crystallographic fractional coordinates. If a molecule, with Cartesian coordinates, is being displayed, and a symmetry-related neighbour is also to be displayed, then the data-space coordinates must be multiplied by    
1   W T M 0 W
T M 0 ; S 0T W 0T 1 0T W 0T 1 where T W

0 1



0

3.3.1.4. Modelling transformations

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with c ¼ 1
ðx2 þ y2 Þ2 is about an axis in the xy plane (i.e. the screen face) normal to ðx; yÞ and with sin  ¼ x2 þ y2 . Applied repetitively this gives a quadratic velocity characteristic. Similarly, if an atom at ðx; y; z; wÞ in display space is to be brought onto the z axis by a rotation with its axis in the xy plane the necessary matrix, in homogeneous form, is 1 0 2 x z þ y2 r
xyðr
zÞ
x 0 C B x2 þ y2 x2 þ y2 C B C B C B
xyðr
zÞ x2 r þ y2 z C B 2
y 0 2 2 C B x þ y2 x þy C B @ x y z 0A 0 0 0 r



M 0T



0 0
1 0

S comprises a proper or improper rotational partition, S, in the upper-left 3  3 in the sense that MSM
1 is orthogonal, and with the associated fractional lattice translation in the last column, with the last row always consisting of three zeros and 1 at the 4, 4 position. See IT A (2005, Chapters 5.2 and 8.1) for a fuller discussion of symmetry using augmented (i.e. 4  4) matrices.

3.3.1.3.11. Other useful rotations If rotations in display space are to be controlled by trackerball or tablet then there are two measures available, an x and a y, which can define an axis of rotation in the plane of the screen and an angle . If x and y are suitably scaled coordinates of a pen on a tablet then the rotation 0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 y þ x2 c
xyð1
cÞ x x2 þ y2 C B x2 þ y2 x2 þ y2 C B B
xyð1
cÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C x2 þ y2 c C B y x2 þ y2 C B A @ x2 þ y2 x2 þ y2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x x2 þ y2
y x2 þ y2 c

with r ¼

0 1 0 0



are the data-space coordinates of the crystallographic origin, M and M
1 are as in Section 3.3.1.1.1 and S is a crystallographic symmetry operator in homogeneous coordinates, expressed relative to the same crystallographic origin. For example, in P21 with the origin on the screw dyad along b,

431

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING One begins at some reference point deemed to be fixed in data space and at this point one stacks the prevailing viewing transformation. From this reference point one advances through the molecule along the structural tree and as each conformon is encountered its matrix is calculated. The product of the prevailing matrix with the conformon matrix is formed and stacked, and this product becomes the prevailing matrix. This product is constructed with the conformon matrix as a factor on the right, i.e. in data space as defined in Section 3.3.1.3.1, and is calculated using the coordinates of the molecule in their unmodified form, i.e. before any shape changes are brought about. This progression leads eventually to an extremity of the tree. At this point drawing is commenced using the prevailing matrix and working backwards towards the fixed root, unstacking (or ‘popping’) a matrix as each conformon is passed until a node is reached, which, in general, will occur only part way back to the root. On reaching such a node drawing is suspended and one advances along the newly found branch as before, stacking matrices, until another extremity is reached when drawing towards the root is resumed. This alternation of stacking matrices while moving away from the root and drawing and unstacking matrices while moving towards the root is continued until the whole tree is traversed. This process is illustrated schematically in Fig. 3.3.1.2 for a simple tree with one node, numbered 1, and three conformons at a, b and c. One enters the tree with a current viewing transformation T and progresses upwards from the fixed lower extremity. When the conformon at a is encountered, T is stacked and the product TM a is formed. Continuing up the tree, at node 1 either branch may be chosen; we choose the left and, on reaching b, TM a is stacked and TM a M b is formed. On reaching the tip drawing down to b is done with this transformation, TM a is then unstacked and drawing continues with this matrix until node 1 is reached. The other branch is then followed to c whereupon TM a is again stacked and the product TM a M c is formed. From the tip down as far as c is drawn with this matrix, whereupon TM a is unstacked and drawing continues down to a, where T is unstacked before drawing the section nearest the root. With this organization the matrices associated with b and c are unaffected by changes in the conformation at a, notwithstanding the fact that changes at a alter the direction of the axis of rotation at b or c. Two other approaches are also possible. One of these is to start at the tip of the left branch, replace the coordinates of atoms between b and the tip by M b X, and later replace all coordinates between the tip and a by M a X, with a similar treatment for the other branch. The advantage of this is that no storage is required for stacked matrices, but the disadvantage is that atoms near the tips of the tree have to be reprocessed for every conformon. It also modifies the stored coordinates, which may or may not be desirable. The second alternative is to draw upwards from the root using T until a is reached, then using TM a until b is reached, then using TM 0b M a to the tip, but in this formulation M 0b must be based on the geometry that exists at b after the transformation M a has been applied to this region of the molecule, i.e. M 0b is characteristic of the final conformation rather than the initial one.

Fig. 3.3.1.2. Schematic representation of a simple branched-chain molecule with a stationary root and two extremities. The positions marked a, b and c are the loci of possible conformational change, here called conformons, and there is a single, numbered branch point.

colour and intensity to be displayed. Many computer terminals have one bit per pixel (said to be ‘single-plane’ systems) and these are essentially monochrome and have no grey scale. Fourplane systems are cheap and popular and commonly provide 4-bit by 4-bit look-up tables between the pixel memory and the monitor with one such table for each of the colours red, green and blue. If these tables are each loaded identically then 16 levels of monochrome grey scale are available, but if they are loaded differently 16 different colours are available simultaneously chosen from a total of 4096 possibilities. Four-plane systems are adequate for many applications where colour is used for coding, but are inadequate if colour is intended also to provide realism, where brilliance and saturation must be varied as well as hue. For these applications eight-plane systems are commonly used which permit 256 colours chosen from 16 million using three look-up tables, though the limitations of these can also be felt and full colour is only regarded as being available in 24-plane systems. Raster-graphics devices are ideal for drawing objects represented by opaque surfaces which can be endowed with realistic reflecting properties (Max, 1984) and they have been successfully used to give effects of transparency. They are also capable of representing shadows, though these are generally difficult to calculate (see Section 3.3.1.5.5). Many devices of this type provide vectorization, area fill and anti-aliasing. Vectorization provides automatically for the loading of relevant pixels on a straight line between specified points. Area fill automatically fills any irregular pre-defined polygon on the screen with a uniform colour with the user specifying only the colour and one point within the polygon. Anti-aliasing is the term used for a technique which softens the staircase effect that may be seen on a line which runs at a small angle to a vertical or horizontal row of pixels. The main drawback with this type of equipment is that it is slow compared to vector machines. Only relatively simple objects can be displayed with smooth rotation in real time as transformed coordinates have to be converted to pixel addresses and the previous frame needs to be deleted with each new frame unless it is known that each new frame will specify every pixel. However, the technology is advancing rapidly and these restrictions are already disappearing. Vector machines, on the other hand, are specialized to drawing straight lines between specified points by driving the electron beam along such lines. No time is wasted on blank areas of the screen. Dots may be drawn with arbitrary coordinates, in any order, but areas, if they are to be filled, must be done with a ruling technique which is very seldom done. Images produced by vector

3.3.1.5. Drawing techniques 3.3.1.5.1. Types of hardware There are two main types of graphical hardware in use for interactive work, in addition to plotters used for batch work. These main types are raster and vector. In raster equipment the screen is scanned as in television, with a grid of points, called pixels, addressed sequentially as the scan proceeds. Associated with each pixel is a word of memory, usually containing something in the range of 1 to 24 bits per pixel, which controls the

432

3.3. MOLECULAR MODELLING AND GRAPHICS machines are naturally transparent in that foreground does not obscure background, which makes them ideal for seeing into representations of molecular structure.

points to locate points on the contour. For vector-graphics applications it is expedient to connect such points with straight lines; some equipment may be capable of connecting them with splines though this is burdensome or impossible if real-time rotation of the scene is required. Precalculation of splines stored as short vectors is always possible if the proliferation of vectors is acceptable. For efficient drawing it is necessary for the line segments of a contour to be end-to-end connected, which means that it is necessary to contour by following contours wherever they go and not by scanning the grid. Algorithms which function in this way have been given by Heap & Pink (1969) and Diamond (1982a). Contouring by grid scanning followed by line connection by the methods of the previous section would be possible but less efficient. Further contouring methods are described by Sutcliffe (1980) and Cockrell (1983). For raster-graphics devices there is little disadvantage in using curved contours though many raster devices now have vectorizing hardware for loading a line of pixels given only the end points. For these devices well shaped contours may be computed readily, using only linear arithmetic and a grid-scanning approach (Gossling, 1967). Others have colour-coded each pixel according to the density, which provides a contoured visual impression without performing contouring (Hubbard, 1983).

3.3.1.5.2. Optimization of line drawings A line drawing consisting of n line segments may be specified by anything from ðn þ 1Þ to 2n position vectors depending on whether the lines are end-to-end connected or independent. Appreciable gains in both processing time and storage requirements may be made in complicated drawings by arranging for line segments to be end-to-end connected as far as possible, and an algorithm for doing this is outlined below. For further details see Diamond (1984a). Supposing that a list of nodal points (atoms if a covalent skeleton is being drawn) exists within a computer with each node appearing only once and that the line segments to be drawn between them are already determined, then at each node there are, generally, both forward and backward connections, forward connections being those to nodes further down the list. A quantity D is calculated at each node which is the number of forward connections minus the number of backward connections. At the commencement of drawing, the first connected node in the list must have a positive D, the last must have a negative D, the sum of all D values must be zero and the sum of the positive ones is the number of strokes required to draw the drawing, a ‘stroke’ being a sequence of end-to-end connected line segments drawn without interruption. The total number of position vectors required to specify the drawing is then the number of nodes plus the number of strokes plus the number of rings minus one. Drawing should then be done by scanning the list of nodes from the top looking for a positive D (usually found at the first node), commencing a stroke at this node and decrementing its D value by 1. This stroke is continued from node to node using the specified connections until a negative D is encountered, at which point the stroke is terminated and the D value at the terminating node is incremented by 1. This is done even though this terminating node may also possess some forward connections, as the total number of strokes required is not minimized by keeping a stroke going as far as possible, but by terminating a stroke as soon as it reaches a node at which some stroke is bound to terminate. The next stroke is initiated by resuming the scan for positive D values at the point in the node list where the previous stroke began. If this scan encounters a zero D value at a node which has not hitherto been drawn to, or drawn from, then the node concerned is isolated and not connected to any other, and such nodes may require to be drawn with some special symbol. The expression already given for the number of vectors required is valid in the presence of isolated nodes if drawing an isolated node is allowed one position vector, this vector not being counted as a stroke. The number of strokes generated by this algorithm is sensitive to the order in which the nodes are listed, but if this resembles a natural order then the number of strokes generated is usually close to the minimum, which is half the number of nodes having an odd number of connections. For example, the letter E has six nodes, four of which have an odd number of connections, so it may be drawn with two strokes.

3.3.1.5.4. Representation of surfaces by dots Connolly (Langridge et al., 1981; Connolly, 1983a,b) represents surfaces by placing dots on the surface with an approximately uniform superficial density. Connolly’s algorithm was developed to display solvent-accessible surfaces of macromolecules and provides for curved concave portions where surface atoms meet. Pearl & Honegger (1983) have developed a similar algorithm, based on a grid, which generates only convex portions which meet in cusps, but is faster to compute than the Connolly surface. Bash et al. (1983) have produced a van der Waals surface algorithm fast enough to permit real-time changes to the structure without tearing the surface. It has become customary to use a dot representation to display computed surfaces, such as the surface at a van der Waals radius from atomic centres, and to use lines to represent experimentally determined surfaces, especially density contours. 3.3.1.5.5. Representation of surfaces by shading Many techniques have been developed, mainly for rastergraphics devices, for representing molecular surfaces and these have been very well reviewed by Max (1984). The simplest technique in this class consists in representing each atom by a uniform disc, or high polygon, which can be colour-coded and area-filled by the firmware of the device. If such atoms are sorted on their z coordinate and drawn in order, furthest ones first, so that nearer ones partly or completely overwrite the further ones then the result is a simple representation of the molecule as seen from the front. This technique is fast and has its uses when a rapid schematic is all that is required. In one sense it is wasteful to process distant atoms when they are going to be overwritten by foreground atoms, but front-to-back processing requires the boundaries of visible parts of partially obscured atoms near the front to be determined before they can be painted or, alternatively, every pixel must be tested before loading to see if it is already loaded. Not only does this approach give a uniform rendering over the whole area of one atom, it also gives a boundary between overlapping atoms with almost equal z values which completes the circle of the nearer atom, though it should be an arc of an ellipse when the atoms are drawn with radii exceeding their covalent radii. Greater realism is achieved by establishing a z buffer, which is an additional area of memory with one word per pixel, in which is stored the z value of the currently loaded feature in each pixel.

3.3.1.5.3. Representation of surfaces by lines The commonest means of representing surfaces, especially contour surfaces, is to consider evenly spaced serial sections and to perform two-dimensional contouring on each section. Repeating this on serial sections in two other orientations then provides a good representation of the surface in three dimensions when all such contours are displayed. The density is normally cited on a grid with submultiples of a, b and c as grid vectors, inverse linear interpolation being used between adjacent grid

433

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING 2ðS
EÞðF
NÞðP
CÞ Treatments which take account of the sphericity are then possible
¼ and correct arcs of intersection for interpenetrating spheres and ðF
EÞðN
EÞðR
LÞ more complicated entities arise naturally through loading a colour value into a pixel only if the z coordinate is less than that and similarly for y0 and . of the currently loaded value. This z buffer and the associated x, y coordinates should be in picture space or screen space rather 3.3.1.5.6. Advanced hidden-line and hidden-surface algorithms than display space since only after the application of perspective Hidden surfaces may be handled quite generally with the can points with the same x/w and y/w coordinates obscure one z-buffer technique described in the previous section but this another. technique becomes very inefficient with very complicated scenes. It is usual in such systems to vary the intensity of colour within Faster techniques have been developed to handle computations one atom by darkening it towards the circumference on the basis in real time (e.g. 25 frames s
1) on raster machines when both the of the z coordinate. Some systems augment this impression of viewpoint and parts of the environment are moving and sphericity by highlighting. The simplest form of highlighting is an substantial complexity is required. These techniques generally extension of the uniform disc treatment in which additional, represent surfaces by a number of points in the surface, brighter discs, possibly off centre, are associated with each atom. connected by lines to form panels. Many algorithms require these More general highlighting (Phong, 1975) is computed from four panels to be planar and some require them to be triangular. Of unit vectors, these being the normal to the surface, the direction those that permit polygonal panels, most require the polygons to to a light source, the direction to the viewer and the normalized be convex with no re-entrant angles. Yet others are limited to vector sum of these last two. Intensity levels may then be set as cases where the objects themselves are convex. Some can handle the sum of three terms: a constant, a term proportional to the interpenetrating surfaces, others exclude these. Some make scalar product of the first two vectors (if positive) and a term enormous gains in efficiency if the objects in the scene are proportional to a high power of the scalar product of the first and separable by the insertion of planes between them and degrade last vectors; the higher the power the glossier the surface appears to lower efficiency if required, for example, to draw a chain. Some to be. This final term normally adds a white term, rather than the are especially suited to vector machines and others to raster surface colour, supposing the light source to be white. machines, the latter capitalizing on the finite resolution of such Shadows may also be rendered to give even greater realism. In systems. In all of these the basic entities for consideration are addition to the z buffer and (x, y) frame buffer a second z buffer entire panels or edges, and in some cases vertices, point-by-point 0 0 0 for z values associated with x and y is also required. These treatment of the entire surface being avoided until after all 0 0 0 coordinates are then related by x ¼ x þ
z, y ¼ y þ z, z = z. decisions are made concerning what is or is not visible. 0 0 The second buffer is a ray buffer since x y are the coordinates All of these algorithms strive to derive economies from the with which an illuminating ray passing through (xyz) passes notion of ‘coherence’. The fact that, in a cine context, one frame 0 0 0 through the z = 0 plane, and z , stored at x , y , records the depth is likely to be similar to the previous frame is referred to as ‘frame at which this ray encounters material. Thus any two pixels coherence’. In raster scans line coherence also exists, and other 0 ðx1 y1 z1 Þ and ðx2 y2 z2 Þ are on the same illuminating ray if their x kinds of coherence can also be identified. The presence of any 0 0 and y values are equal and the one with smaller z shadows the form of coherence may enable the computation to be concerned other. Processing a pixel at ðx1 y1 z1 Þ therefore involves first primarily with changes in the situation, rather than with the determining its visibility on the basis of the z buffer, as before, totality of the situation so that, for example, computation is 0 then, whether or not it is visible, setting z1 ¼ z1 and considering required where one edge crosses in front of another, but only 0 0 0 0 the value of z currently stored at x y , which we call z2 . trivial actions are involved so long as scan lines encounter the 0 0 If z1 < z2 then x1 y1 z1 is in light and must be loaded accordingly. projections of edges in the same order. 0 From z2 we find the previously processed pixel ðx2 y2 z2 Þ which is The choice of technique from among many possibilities may now in shade and which was in light when originally processed, so even depend on the viewpoint if the scene has a statistical that the colour value stored at x2 y2 needs to be altered unless the anisotropy. For example, the depiction of a city seen from a pixel at x2 y2 is now ðx2 y2 z3 Þ with z3 < z2 , in which case the pixel viewpoint near ground level involves many hidden surfaces. ðx2 y2 z2 Þ which has now become shadowed by ðx1 y1 z1 Þ has, in the Distant buildings may be hidden many times over. The same meantime, been obscured by ðx2 y2 z3 Þ which is not shadowed by scene depicted from an aerial viewpoint shows many more 0 ðx1 y1 z1 Þ and no change is therefore needed. In either event z1 surfaces and fewer overlaps. This difference may swing the 0 then replaces z2. balance of advantage between an algorithm which sorts first on z 0 0 If z1 > z2 then ðx1 y1 z1 Þ, if visible, is in shade and must be or one which leaves that till last. 0 coloured accordingly, and in this case z2 is not superseded. These advanced techniques have, so far, found little applicaThis shadowing scheme corresponds to illumination by a light tion in crystallography, but this may change. Ten such techniques source at infinity in picture space or, equivalently, with a z are critically reviewed and compared by Sutherland et al. (1974), coordinate equal to that of the eye in display space. For its and three of these are described in detail by Newman & Sproull implementation x, y and z may be in any convenient coordinate (1973). system, e.g. pixel addresses, but if x and y are expressed with the range
1 to 1 and z with the range 0 to 1 corresponding to the window then they may be identified as the quantities x/w, y/w and 3.3.2. Molecular modelling, problems and approaches z/w of picture space (Section 3.3.1.3.1). By R. Diamond If, in the notation of Section 3.3.1.3.5, the light source is placed at (P, Q, E, V) in display space and a ray leaves it in the direction (p, q, r, V) then This section is concerned with software techniques which permit a set of atomic coordinates for a molecule to be generated ab initio, or to be modified, by reference to some chosen criterion, p 2ðS
EÞ 2ðS
EÞðP
CÞ 2C
R
L usually the electron density. Software that can change the shape þ þ ; x0 ¼  r ðR
LÞ ðN
EÞðR
LÞ R
L of a molecule must be cognizant of the connectivity of the molecule and the bonding characteristics of atom types. It must also have means of regaining good stereochemistry if current which varies only with beam direction, coordinates are poor in this respect, or of performing its

434

3.3. MOLECULAR MODELLING AND GRAPHICS manipulations in ways which conserve essential stereochemical features. Approaches to some of these problems are outlined below. Many of the issues involved, including the topics outlined in Section 3.3.1.4 above, have been excellently reviewed by Hermans (1985), though with little reference to graphical aspects, and a comprehensive treatment of modelling methods based on energies is given by Burkert & Allinger (1982).

however, it may be exceedingly efficient. Both proteins and nucleic acids are of a class which permits their logical connectivity to be specified entirely by list ordering, and the software described in Section 3.3.3.2.6 uses no connectivity tables for this purpose. The ordering rules concerned are given by Diamond (1976b). Drawing connectivity needs explicit specification in such a case; this may be done using only one 16-bit integer per atom, which may be stored as part of the atom list without the need of a separate table. This integer consists of two signed bytes which act as relative pointers in the list, positive pointers implying draw-to, negative pointers implying move-to. As each atom is encountered during drawing the right byte is read and utilized, and the two bytes are swapped before proceeding. This allows up to two bonds drawn to an atom and two bonds drawn from it, four in all, with a minimum of storage (Diamond, 1984a). Brandenburg et al. (1981) handle drawing connectivity by enlarging the molecular list with duplicate atoms such that each is connected to the next in the list, but moves and draws still need to be distinguished. Levitt (1971) has developed a syntax for specifying structural connectivity implicitly from a list structure which is very general, though designed with biopolymers in mind, and the work of Katz & Levinthal (1972) includes something similar.

3.3.2.1. Connectivity It is necessary to distinguish three different kinds of connectivity, namely structural, logical and drawing connectivities. Structural connectivity consists of the specification of the chemical bonding of the molecule and, as such, is an absolute property of the molecule. Logical connectivity consists of the specification of what part or parts of a molecule are moved, and in what way, if some stereochemical feature is altered. Logical and structural connectivity are closely related and in simple cases coincide, but the distinction is apparent, for example, if the puckering of a five-membered ring is being modelled by permitting folding of the pentagon about a line connecting nonadjacent corners. This line is then a logical connection between two moving parts, but it is not a feature of the structural connectivity. Drawing connectivity consists of a specification of the lines to be drawn to represent the molecule and often coincides with the structural connectivity. However, stylized drawings, such as those showing the
-carbon atoms of a protein, require to be drawn with lines which are features neither of the logical nor of the structural connectivity.

3.3.2.2. Modelling methods Fundamental to the design of any software for molecular modelling are the choices of modelling criteria, and of parameterization. Criteria which may be adopted might include the fitting of electron density, the minimization of an energy estimate or the matching of complementary surfaces between a pair of molecules. Parameterizations which may be adopted include the use of Cartesian coordinates of atoms as independent variables, or of internal coordinates, such as dihedral angles, as independent variables with atomic positions being dependent on these. Systems designed to suit energetic criteria usually use Cartesian coordinates since all aspects of the structure, including bond lengths, must be treated as variables and be allowed to contribute to the energy estimate. Systems designed to fit a model to observed electron density, however, may adequately meet the stereochemical requirements of modelling on either parameterization, and examples of both types appear below. Inputs to modelling systems vary widely. Systems intended for use mainly with proteins or other polymeric structures usually work with a library of monomers which the software may develop into a polymer. Systems intended for smaller molecules usually develop the molecular structure atom-by-atom rather than a residue at a time, and systems of this kind require a very general form of input. They may accept a list of atom types and coordinates if measurement and display of a known molecule is the objective, or they may accept ‘sketch-pad’ input in the form of a hand-drawn two-dimensional sketch of the type conventional in chemistry, if the objective is the design of a molecule. Sketch-pad input is a feature of some systems with quantum-mechanical capabilities.

3.3.2.1.1. Connectivity tables The simplest means of storing connectivity information is by means of tables in which, for each atom, a list of indices of other atoms to which it is connected is stored. This approach is quite general; it may serve any type of molecular structure and permits structures to be traversed in a variety of ways. In this form, however, it is extravagant on storage because every connection is stored twice, once at each of the nodes it connects. It may, however, provide the starting material for the algorithm of Section 3.3.1.5.2 and its generality may justify its expense. From such a list, lists of bonds, bond angles and dihedral angles may readily be derived in which each entry points to two, three or four atoms in the atom list. Lists of these three types form the basis of procedures which adjust the shape of a molecule to reduce its estimated potential energy (Levitt & Lifson, 1969; Levitt, 1974), and of search-and-retrieval techniques (Allen et al., 1979). Katz & Levinthal (1972) discuss the explicit specification of structural connectivity in terms of a tree structure in which, for each atom, is stored a single pointer to the connected atom nearer to the root, virtual atoms being used to allow ring structures to be treated as trees. An algorithm is also presented which allows such a tree specification to be redetermined if an atom in the tree is newly chosen as the root atom or if the tree itself is modified. Cohen et al. (1981) have developed methods of handling connectivity in complicated fused- and bridged-ring systems.

3.3.2.2.1. Methods based on conformational variables Suppose that t represents a vector from the current position of an atom in the model to a target position then (see Section 3.3.1.1.3), to first order, the observational equations are

3.3.2.1.2. Implied connectivity In cases where software is required to deal only with a certain class of molecule, it may be possible to exploit the characteristics of that class to define an ordering for lists of atoms such that connectivity is implied by the ordering of items in the list. Such an ordering may successfully define one of the three types of connectivity defined in Section 3.3.2.1 but it is unlikely to be able to meet the needs of all three simultaneously. It may also be at a disadvantage when required to deal with structures not part of the class for which it is designed. Within these limitations,

tIA ¼ DIpA p þ vIA in which h represents changes to conformational variables which may include dihedral angles, bond angles, bond lengths, and parameters determining overall position and orientation of the molecule as a whole. If every such parameter is included the model acquires 3n degrees of freedom for n atoms, in which case

435

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING

DIPA

0 R0



in which E are the eigenvectors of M 1 having positive eigenvalues, E0 are those having zero eigenvalues, and R and R0 are arbitrary orthogonal matrices. Then  T  T  T A1 A1 A1 V M h1 ¼ ðA B Þ 1 1 1 1 T T B1 B1 BT1  T  T  A1 A1 M 1 A1 0 h1 ¼ 0 0 BT1 AT1 h1 ¼ ðAT1 M 1 A1 Þ
1 AT1 V1

@r ¼ IA ¼ "Ijk njP ðrkA
rkP Þ @P

in which nP is a unit vector defining the axis of rotation for an angular variable P, rA and rP are position vectors of the atom A and the site of the parameter P, and vA represents a residual vector.  ¼ via via is minimized by h ¼ M
1 V

in which the matrix to be inverted is positive definite. A1 , however, is rectangular so that multiplying on the left by A1 does not necessarily serve to determine h 1, but we may write   u h1 ¼ ðA1 B1 Þ 1 w1

in which MPQ ¼ DiPa DiQa VP ¼ DiPa tia : More generally, if  represents any scalar quantity which is to be minimized, e.g. an energy, then

giving 

1 @ 2 @P 1 @2  ¼ : 2 @P @Q

VP ¼
MPQ



R ðA1 B1 Þ ¼ ðE E0 Þ 0

the methods of the next section are more appropriate, but if bond lengths and some or all bond angles are being treated as constants then the above equation becomes the basis of the treatment.

  T AT1 A1 M 1 A1 V ¼ 1 T 0 B1 u1 ¼

0 0



u1



w1

ðAT1 M 1 A1 Þ
1 AT1 V1

and w1 is indeterminate and free to adopt any value. We therefore adopt

It is beyond the scope of this chapter to review the methods available for evaluating h from these equations. Difficulties may arise from two sources: (i) Inversion of M may be difficult if M is large or ill conditioned and impossible if M is singular. (ii) Successful evaluation of M
1 V will not minimize  in one step if t is not linearly dependent on h or, equivalently, @2 =@P @Q is not constant, and substantial changes h are involved. Iteration is then necessary. Difficulties of the first kind may be overcome by gradient methods, for example the conjugate gradient method without searches if M is available or with searches if it is not available, or they may be overcome by methods based on eigenvalue decompositions. If nonlinearity is serious less dependence should be placed on M and gradient methods using searches are more valuable. In this connection Diamond (1966) introduced a sliding filter technique which produced rapid convergence in extreme conditions of nonlinearity. These topics have been reviewed elsewhere (Diamond, 1981b, 1984b) and are the subject of many textbooks (Walsh, 1975; Gill et al., 1981; Luenberger, 1984). Warme et al. (1972) have developed a similar system using dihedral angles as variables and a variety of alternative optimization algorithms. The modelling of flexible rings or lengths of chain with two or more fixed parts is sometimes held to be a difficulty in methods using conformational variables, although a simple two-stage solution does exist. The principle involved is the sectioning of the space of the variables into two orthogonal subspaces of which the first is used to satisfy the constraints and the second is used to perform the optimization subject to those constraints. The algebra of the method may be outlined as follows, and is given in more detail by Diamond (1971, 1980a, 1981a). Parametric shifts h1 which satisfy the constraints are solutions of

h1 ¼ A1 u1 ¼ A1 ðAT1 M 1 A1 Þ
1 AT1 V; which is the smallest vector of parametric shifts which will satisfy the constraints, and allow w1 to be determined by the remaining observational equations in which the target vectors, t, are now modified to t2 according to t2 ¼ t
D 2 h 1 ; D2 and t2 being the derivatives and effective target vectors for the unconstrained atoms. We then solve V2 ¼ M 2 h 2 in which h 2 is required to be of the form h2 ¼ B1 w1 giving h 2 ¼ B1 ðBT1 M 2 B1 Þ
1 BT1 V2 and apply the total shifts H ¼ h1 þ h2 to obtain a structure which is optimized within the restrictions imposed by the constraints. It may happen that BT1 M 2 B1 is itself singular because there are insufficient data in the vector t2 to control the structure and the parametric shifts contained in h2 fully. In this event the same process may be applied again, basing the solution for h 2 on  T A2 BT1 M 2 B1 ðA2 B2 Þ BT2

V1 ¼ M 1 h 1

so that the vectors in B2 represent the degrees of freedom which remain uncommitted. This method of application of constraints by subspace sectioning may be nested to any depth and is completely general. A valid matrix A1 may be found from M 1 by using the fact that the columns of M 1 are all linear combinations of the columns of

in which V1 and M 1 depend only on the target vectors, t1 , of the atoms with constrained positions and on the corresponding derivatives. We then find a partitioned orthogonal matrix ðA1 B1 Þ satisfying

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3.3. MOLECULAR MODELLING AND GRAPHICS E and are void of any contribution from E0 . It follows that A1 may be found by using the columns of M 1 as priming vectors in the Gram–Schmidt process [Section 3.3.1.2.3 (i)] until the normalizing step involves division by zero. A1 is then complete if all the columns of M 1 have been tried. ðA1 B1 Þ may then be completed by using arbitrary vectors as primers. Manipulation of a ring of n atoms may be achieved by treating it as a chain of ðn þ 2Þ atoms [having ðn þ 1Þ bond lengths, n bond angles and ðn
1Þ dihedral angles] in which atom 1 is required to coincide with atom ðn þ 1Þ and atom 2 with ðn þ 2Þ. t1 then contains two vectors, namely the lack-of-closure vectors at these points, and is typically zero. A1 is then found to have five columns corresponding to the five degrees of freedom of two points of fixed separation; h1 contains only zeros if the ring is initially closed, and contains ring-closure corrections if, through nonlinearity or otherwise, the ring has opened. B1 contains ðp
5Þ columns if the chain of ðn þ 2Þ points has p variable parameters. It follows, if bond lengths and bond angles are treated as constants, that the seven-membered ring is the smallest ring which is flexible, that the six-membered ring (if it can be closed with the given bond angles) has no flexibility (though it may have discrete alternatives) and that it may be impossible to close a fivemembered ring. Therefore some variation of bond angles and/or bond lengths is essential for the modelling of flexible five- and six-membered rings. Treating the ring as a chain of ðn þ 1Þ atoms is less satisfactory as there is then no control over the bond angle at the point of ring closure. A useful concept for the modelling of flexible five-membered rings with near-constant bond angles is the concept of the pseudorotation angle P, and amplitude m, for which the jth dihedral angle is given by   4j : j ¼ m cos P þ 5

sparse matrix W 1=2 D, the dense matrix DT WD, and its inverse, being never calculated. The method is extremely efficient in annealing a model structure for which an initial position for every atom is available, especially if the required shifts are within the quasi-linear region, but is less effective when large dihedral-angle changes are involved or when many atoms are to be placed purely by interpolation between a few others for which target positions are available. Interbond angles are controlled by assigning d values to second-nearest-neighbour distances; this is effective except for bond angles near 180 so that, in particular, planar groups require an out-of-plane dummy atom to be included which has no target position of its own but does have target values of distances between itself and atoms in the planar group. The method requires a d value to be supplied for every type of nearest- and next-nearest-neighbour distance in the structure, of which there are many, together with W values which are the inverse variances of the distances concerned as assessed by surveys of the corresponding distances in small-molecule structures, or from estimates of their accuracy, or from estimates of accuracy of the target positions. Hermans & McQueen (1974) published a similar method which differs in that it moves only one atom at a time, in the environment of its neighbours, these being considered fixed while the central atom is under consideration. This is inefficient in the sense that in any one cycle one atom moves only a small fraction (3%) of the distance it will ultimately be required to move, but individual atom cycles are so cheap and simple that many cycles can be afforded. The method was selected for inclusion in Frodo by Jones (1978) (Section 3.3.3.2.7) and is an integral part of the GRIP system (Tsernoglou et al., 1977; Girling et al., 1980) (Section 3.3.3.2.2) for which it was designed.

P4 This formulation has the property j¼0 j ¼ 0, which is not exactly true; nevertheless, j values measured from observed conformations comply with this formulation within a degree or so (Altona & Sundaralingam, 1972). Software specialized to the handling of condensed ring systems has been developed by van der Lieth et al. (1984) (Section 3.3.3.3.1) and by Cohen et al. (1981) (Section 3.3.3.3.2).

3.3.2.2.3. Approaches to the problem of multiple minima Modelling methods which operate by minimizing an objective function of the coordinates (whether conformational or positional) suffer from the fact that any realistic objective function representing the potential energy of the structure is likely to have many minima in the space of the variables for any but the simplest problems. No general system has yet been devised that can ensure that the global minimum is always found in such cases, but we indicate here two approaches to this problem. The first approach uses dynamics to escape from potentialenergy minima. Molecular-mechanics simulations allow each atom to possess momentum as well as position and integrate the equations of motion, conserving the total energy. By progressively removing energy from the simulation by scaling down the momentum vectors some potential-energy minimum may be found. Conversely, a minimization of potential energy which has led to a minimum thought not to be the global minimum may be continued by introducing atomic momenta sufficient to overcome potential-energy barriers between minima, and subsequently attenuate the momenta again. In this way a number of minima may be found (Levitt & Warshel, 1975). It is equivalent to initializing a potential-energy minimization from a number of different conformations but it has the property that the minima so found are separated by energy barriers for which an upper limit is known so that the possibility exists of exploring transition pathways. A second approach is described by Purisima & Scheraga (1986). If the objective function to be minimized can be expressed in terms of interatomic distances, and if each atom is given coordinates in a space of n
1 dimensions for n atoms, then a starting structure may be postulated for which the interatomic distances all take their ideal values and the objective function is then necessarily at an absolute minimum. This multidimensional structure is then projected into a space of fewer dimensions, within which it is again optimized with respect to the

3.3.2.2.2. Methods based on positional coordinates Modelling methods in which atomic coordinates are the independent variables are mathematically simpler than those using angular variables especially if the function to be minimized is a quadratic function of interatomic distances or of distances between atoms and fixed points. The method of Dodson et al. (1976) is representative of this class and it may be outlined as follows. If d is a column vector containing ideal values of the scalar distances from atoms to fixed target points or to other atoms, and if l is a column vector containing the prevailing values of these quantities obtained from the model, then d ¼ l þ Ddx þ " in which the column matrix dx contains alterations to the atomic coordinates, " contains residual discrepancies and D is a large sparse rectangular matrix containing values of @l=@x, of which there are not more than six nonzero values on any row, consisting of direction cosines of the line of which l is the length. " T W " is then minimized by setting DT W ðd
lÞ ¼ DT WDdx; which they solve by the method of conjugate gradients without searches. This places reliance on the linearity of the observational equations (Diamond, 1984b). It also works entirely with the

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3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING objective function. The dimensionality of the model is thus progressively reduced until a three-dimensional model is attained at a low energy. This means that the minimum so attained in three dimensions is approached from beneath, having previously possessed a lower value in a higher-dimensional space. This, in itself, does not guarantee that the three-dimensional minimumenergy structure so found is at the global minimum, but it is not affected by energy barriers between minima in the same way, and it does appear to reach very low minima, and frequently the global one. Because it is formulated entirely in terms of interatomic distances it offers great promise for modelling molecules on the basis of data from nuclear magnetic resonance.

building and computer graphics are now covered by Chapters 17.1 and 17.2 in International Tables for Crystallography Volume F (2001). 3.3.3.1. Systems for the display and modification of retrieved data One of the earliest systems designed for information retrieval and display was that described by Meyer (1970, 1971) which used television raster technology and enabled the contents of the Brookhaven Protein Data Bank (Meyer, 1974; Bernstein et al., 1977) to be studied visually by remote users. It also enabled a rigid two-ring molecule to be solved from packing considerations alone (Hass et al., 1975; Willoughby et al., 1974). Frames for display were written digitally on a disk and the display rate was synchronized to the disk rotation. With the reduction in the cost of core storage, contemporary systems use large frame buffer memories thus avoiding synchronization problems and permitting much richer detail than was possible in 1970. A majority of the systems in this section use raster techniques which preclude real-time rotation except for relatively simple drawings, though GRAMPS is an exception (O’Donnell & Olson, 1981; Olson, 1982) (Section 3.3.3.1.4).

3.3.3. Implementations

By R. Diamond

In this section the salient characteristics of a number of systems are described. Regrettably, it cannot claim to be a complete guide to all existing systems, but it probably describes a fairly representative sample. Some of these systems have arisen in academia and these are freely described. Some have arisen in or been adopted by companies which now market them, and these are described by reference to the original publications. Other marketed systems for which originators’ published descriptions have not been found are not described. Yet other systems have been developed, for example, by companies within the chemical and pharmaceutical industries for their own use, and these have generally not been described in what follows since it is assumed that they are not generally available, even where published descriptions exist. Software concerned especially with molecular dynamics has not been included unless it also provides static modelling capability, since this is a rapidly growing field and it has been considered to be beyond the intended scope of this chapter. Systems for which outline descriptions have already been given (Levitt & Lifson, 1969; Levitt, 1974; Diamond, 1966; Warme et al., 1972; Dodson et al., 1976; Hermans & McQueen, 1974) are not discussed further. For some of the earliest work Levinthal (1966) still makes interesting reading and Feldmann (1976) is still an excellent review of the technical issues involved. The issues have not changed, the algorithms there described are still valuable, only the manner of their implementation has moved on as hardware has developed. A further review of the computer generation of illustrations has been given by Johnson (1980). Excellent bibliographies relating to these sections have been given by Morffew (1983, 1984), which together contain over 250 references including their titles. The following material is divided into three sections. The first is concerned primarily with display rather than modelling though some of these systems can modify a model, the second is concerned with molecular modelling with reference to electron density and can develop a model ab initio, and the third is concerned with modelling with reference to other criteria. Where software names are known to be acronyms constructed from initial letters, or where the original authors have used capitals, the names are capitalized here. Otherwise names are lower case with an initial capital. While it is recognised that many of the systems here described are now of mainly historical interest, most were retained for the second edition, some were updated and some new paragraphs were added. For the third edition a whole new section, Section 3.3.4, has been added, which provides an overview of recent developments relevant in particular to small and medium-sized structures. Recent developments in the macromolecular field of model

3.3.3.1.1. ORTEP This program, the Oak Ridge Thermal Ellipsoid Program, due to Johnson (1970, 1976) was developed originally for the preparation of line drawings on paper though versions have since been developed to suit raster devices with interactive capability. The program draws molecules in correct perspective with each atom represented by an ellipsoid which is the equi-probability surface for the atomic centre, as determined by anisotropic temperature factor refinement, the principal axes of which are displayed. Bonds are represented by cylindrical rods connecting the atoms which in the drawing are tapered by the perspective. In line-drawing versions the problem of hidden-line suppression is solved analytically, whereas the later versions for raster devices draw the furthest elements of the picture first and either overwrite these with nearer features of the scene if area painting is being done or use the nearer features as erase templates if line drawings are being made. 3.3.3.1.2. Feldmann’s system R. J. Feldmann and co-workers (Feldmann, 1983) at the National Institutes of Health, Bethesda, Maryland, USA, were among the first to develop a suite of programs to display molecular structure using colour raster-graphics techniques. Their system draws with coloured shaded spheres, usually with one sphere to represent each atom, but alternatively the spheres may represent larger moieties like amino acids or whole proteins if lower-resolution representations are required. These workers have made very effective use of colour. Conventionally, oxygens have been modelled in red, but this system allows charged oxygens to be red and uncharged ones to be pink, with a similar treatment in blue for charged and uncharged nitrogens. By such means they have been able to give immediacy to the hydrophobic and electrostatic properties of molecular surfaces, and have used these characteristics effectively in studies of the binding possibilities of benzamidine derivatives to trypsin (Feldmann et al., 1978). The algorithm developed by Porter (1978) for shading spheres to be darkened near their peripheries also computes the proper appearance of the line of intersection of two spheres wherever interpenetration occurs, in contrast to some simpler systems which draw a complete disc for whichever sphere is forward of the other. Provided that all opaque spheres are drawn first, the system is also capable of representing transparent spheres by darkening the colour of the existing background inside, and especially near the edge of, discs representing transparent foreground spheres.

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3.3. MOLECULAR MODELLING AND GRAPHICS Other systems that produce space-filling pictures of a similar general character have been produced by Motherwell (1978), by Sundaram & Radhakrishnan (1979) and by Lesk (next section).

comparisons, and it is able to make modifications to structures. Objects for display may be molecular or nonmolecular, the former having an atomic substructure and the latter consisting of a vector list which may not be subdivided into referrable components. Map fitting with the current version has been reported.

3.3.3.1.3. Lesk & Hardman software The complexity of macromolecules is a formidable obstacle to perceiving the basic features of their construction. Stylized drawings to facilitate this perception were pioneered by A. Rossmann and B. Furugren, and developed further by Richardson (1977, 1981, 1985) and by Lesk and Hardman, whose software is capable of mixing several styles of representation, among them the style of cylinders for
-helices, arrows for -strands and ribbons for less-organized regions, or the creasedribbon technique for the whole chain, or a ball-and-stick representation of atoms and bonds, or space-filling spheres. All these styles are available simultaneously in a single picture with depth cueing, colour and shading, and hidden-feature suppression as appropriate. It is also able to show a stylized drawing of a complete molecule together with a magnified part of it in a more detailed style. See Lesk & Hardman (1982, 1985) and Lesk (1991).

3.3.3.1.8. PLUTO PLUTO was developed by Motherwell (1978) at the Cambridge Crystallographic Data Centre (CCDC) for the display of molecular structures and crystal-packing diagrams, including an option for space-filling model style with shadowing. The emphasis was on a free format command and data structure, and the ability to produce ball-and-spoke drawings with line shadowing suitable for reproduction in journal publication. Many variant versions have been produced, with essentially the 1978 functionality, its popularity deriving from its ease of use and the provision of default options for establishing connectivity using standard bonding radii. It was distributed as part of the CCDC software associated with the Cambridge Structural Database, with an interface for reading entries from the database. In 1993 Motherwell and others at the CCDC added an interactive menu and introduced colour and PostScript output. New features were introduced to allow interactive examination of intermolecular contacts, particularly hydrogen-bonded networks, and sections through packing diagrams (Cambridge Structural Database, 1994).

3.3.3.1.4. GRAMPS This system, due to O’Donnell & Olson (O’Donnell & Olson, 1981; Olson, 1982) provides a high-level graphics language and its associated interpretive software. It provides a general means of defining objects, drawable by line drawings, in such a way that these may be logically connected in groups or trees using a simple command language. Such a system may, for example, define a subunit protein of an icosahedral virus and define icosahedral symmetry, in such a way that modification of one subunit is expressed simultaneously in all subunits whilst preserving the symmetry, and simultaneously allowing the entire virus particle to be rotated. Such logical and functional relationships are established by the user through the medium of the GRAMPS language at run time, and a great diversity of such relationships may be created. The system is thus not limited to any particular type of structure, such as linear polymers, and has proved extremely effective as a means of providing animation for the production of cine film depicting viral and other structures. GRAMPS runs on all Silicon Graphics workstations under IRIX 4.0 or above.

3.3.3.1.9. MDKINO This system, due to Swanson et al. (1989), provides for the extraction and visualization of selected regions from moleculardynamics simulations. It permits stereo viewing, interactive geometric interrogation and both forwards and backwards display of motion. 3.3.3.2. Molecular-modelling systems based on electron density Systems described in this section require real-time rotation of complicated transparent scenes and all used vector-graphics technology in their original implementations for that reason, though many are now available for raster machines. In every case the graphics are the means of communication between the user and software possessing high functionality, capable of building a representation of a molecule ab initio and to alter it, change its shape and position it optimally in relation to an electron-density map, with due attention being paid to stereochemical considerations, by one or more of several approaches.

3.3.3.1.5. Takenaka & Sasada’s system Takenaka & Sasada (1980) have described a system for the manipulation and display of molecular structures, including packing environments in the crystal, using a minicomputer loosely coupled to a mainframe. Their system is also capable of model building by the addition of groups of one or more atoms with a facility for monitoring interaction distances while doing so.

3.3.3.2.1. CHEMGRAF Katz & Levinthal (1972) have developed a powerful modelling and display system for macromolecules known as CHEMGRAF. This system permits the definition of many atom types which includes bonding specifications, so that, for example, four types of carbon atom are included in the basic list and others may be added. A molecular fragment with an unsatisfied valency (by which it might later be attached to another such group) would have that feature represented by a ‘vanishing virtual’ atom which removes the need for any organizational distinction between such fragments and complete molecules. Fragments, such as aminoacid residues, may be assembled from atoms, and molecules may be assembled from atoms and/or from such fragments invoked by name, by the superposition and elimination of the relevant vanishing virtuals. The assembly process includes the development of a connectivity tree for the molecule and provision is made for the ‘turning’ or reconstruction of such trees if the combination of such fragments redefines the root atom of one or more of the fragments. The system also provides for ring closure. Model building initially uses fixed bond lengths and angles,

3.3.3.1.6. MIDAS This system, due to Langridge and co-workers (Gallo et al., 1983; Ferrin et al., 1984) is primarily concerned with the display of existing structures rather than with the establishment of new ones, but it may modify such structures by bond rotations under manual control. It is of particular value in the study of molecular interactions since two or more molecules may be manipulated simultaneously and independently. Visual docking of molecules is greatly facilitated by the display of van der Waals surfaces, which may be computed in real time so that the turning of a bond in the underlying structure does not tear the surface (Bash et al., 1983). 3.3.3.1.7. Insight This system, originally due to Dayringer et al. (1986), has a functionality similar to MIDAS. It has been replaced by Insight II (version 2.3.5 or later). It appears to be well suited to the study of intermolecular relationships in docking and in structural

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3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING varying only the dihedral angles in single bonds, but has a library of preformed rings which could not otherwise be modelled on the simple basis. The results of such modelling may then be subjected to an energy-minimization routine using a steepest-descent method in the space of the dihedral angles and referring to the Lennard–Jones potential for nonbonded atom pairs. Atoms are first sorted into contiguous cubes so that all neighbours of any atom may be found by searching not more than 27 cubes. The system is also capable of modelling by reference to electron density either by the translation and rotation of molecular fragments and the rotation of rotatable bonds within them or by the automatic linking of peaks in an electron-density map which ˚ , which is an important aid to are separated by less than, say, 1.8 A interpretation when the resolution is sufficiently high.

3.3.3.2.4. MMS-X The Molecular Modelling System-X (MMS-X) is a system of purpose-built hardware developed by Barry, Marshall and others at Washington University, St Louis. Associated with it are several sets of software. The St Louis software consists of a suite of programs rather than one large one and provides for the construction of a polymer chain in helical segments which may be adjusted bodily to fit the electron density, and internally also if the map requires this too. Nonhelical segments are built helically initially and unwound by user-controlled rotations in single bonds. The fitting is done to visual criteria. An example of the use of this system is given by Lederer et al. (1981). Miller et al. (1981) have described an alternative software system for the same equipment. Functions invoked from a keyboard allow dials to be coupled to dihedral angles in the structure. Their system communicates with a mainframe computer which can deliver small blocks of electron density to be contoured and stored locally by the graphics system; this provides freedom of choice of contour level at run time. An example of the use of this system is given by Abad-Zapatero et al. (1980).

3.3.3.2.2. GRIP This system, developed by Professor F. P. Brooks, Dr W. V. Wright and associates at the University of North Carolina, Chapel Hill, NC, USA, was designed for biopolymers and was the first to enable a protein electron-density map to be interpreted ab initio without the aid of mechanical models (Tsernoglou et al., 1977). Girling et al. (1980) give a more recent example of its use. In its 1975 version GRIP is a three-machine system. Centrally there is a minicomputer which receives inputs from the user and controls a vector display with high-speed matrix-multiplication capability. The third machine is a mainframe computer with highspeed communication with the minicomputer. The system develops a polymer chain from a library of monomers and manipulates it through bond rotations or free rotations of fragments explicitly specified by the user, with the aid of dials which may be coupled to bonds for the purpose. Bond rotations in the main chain either rotate part of the molecule relative to the remainder, which may have undesirable longrange effects, or the scope of the rotation is artificially limited with consequential discontinuities arising in the chain. Such discontinuities are removed or alleviated by the mainframe computer using the method of Hermans & McQueen (1974), which treats atomic position vectors, rather than bond rotations, as independent variables. The system made pioneering use of a two-axis joystick to control orientation and a three-translation joystick to control position.

3.3.3.2.5. Texas A&M University system This system (Morimoto & Meyer, 1976), a development of Meyer’s earlier system (Section 3.3.3.1), uses vector-graphics technology and a minicomputer and is free of the timing restrictions of the earlier system. The system allows control dials to be dynamically coupled by software to rotations or translations of parts of the structure, thus permitting the reshaping or repositioning of the model to suit an electron-density map which may be contoured and managed by the minicomputer host. The system may be used to impose idealized geometry, such as planar peptides in proteins, or it may work with nonidealized coordinates. The system was successfully applied to the structures of rubredoxin and the extracellular nuclease of Staphylococcus aureus (Collins et al., 1975) and to binding studies of sulfonamides to carbonic anhydrase (Vedani & Meyer, 1984). In addition, two of the first proteins to be constructed without the aid of a ‘Richards’ Box’ were modelled on this system: monoclinic lysozyme in 1976 (Hogle et al., 1981) and arabinose binding protein in 1978 (Gilliland & Quiocho, 1981). 3.3.3.2.6. Bilder This system (Diamond, 1980b, 1981a, 1982b) runs on a minicomputer independent of any mainframe. It builds a polymer chain from a library of residues and adapts it by internal rotations and overall positioning in much the same way as previous systems described in this section. Like them, it can provide usercontrolled bond rotations, but its distinctive feature is that it has an optimizer within the minicomputer which will determine optimal combinations of bond rotations needed to meet the user’s declared objectives. Such objectives are normally target positions for atoms set by the user by visual reference to the density, using the method of Section 3.3.1.3.9, but they may include target values for angles. These latter may either declare a required shape that is to take precedence over positional requirements, which are then achieved as closely as the declared shape allows, or they may be in least-squares competition with the positional requests. The optimizer also recognizes the constraints imposed by chain continuity and enables an internal section of the main chain to be modified without breaking its connection to the rest of the molecule. Similar techniques also allow ring systems to adopt various conformations, by bond rotation, without breaking the ring, simultaneously permitting the ring to have target positions. The optimizer is unperturbed by underdetermined situations, providing a minimum-disturbance result in such cases. All these properties of the optimizer are generated without recourse to any ‘special cases’ by a general-

3.3.3.2.3. Barry, Denson & North’s systems These systems (Barry & North, 1971; North et al., 1981; North, 1982) are examples of pioneering work done with minicomputers before purpose-built graphics installations became widespread; examples of their use are given by Ford et al. (1974), Potterton et al. (1983) and Dodson et al. (1982). They have the ability to develop a polymer chain in sections of several residues, each of which may subsequently be adjusted to remove any misfit errors where the sections overlap. Manipulations are by rotation and translation of sections and by bond rotations within sections. These movements are directly controlled by the user, who may simultaneously observe on the screen the agreement with electron density, or calculated estimates of potential or interaction energy, or a volume integral of the product of observed and model densities, or predicted shifts of proton magnetic resonance spectra. Thus models which are optimal by various criteria may be constructed, but there is no optimizer directly controlling the rotational adjustments which are determined by the user. One of the earliest applications of them (Beddell, 1970) was in the fitting of substrate molecules to the active site of lysozyme using difference electron densities; however, the systems also permitted the enzyme–substrate interaction to be studied simultaneously and to be taken into account in adjusting the model.

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3.3. MOLECULAR MODELLING AND GRAPHICS ization of the subspace section technique which was used to maintain chain continuity in a ‘real-space-refinement’ program (Diamond, 1971). This is based entirely on the rank of the normal matrix that arises during optimization, which may serve to satisfy a constraint such as chain continuity or ring closure and simultaneously to establish what degrees of freedom remain to be controlled by other criteria. In Bilder this is achieved without establishing eigenvalues or eigenvectors. The method is described in outline in Section 3.3.2.2.1 and in detail by Diamond (1980a, 1981a). The angular variables used normally comprise all single bonds but may include others, such as the peptide bond with or without a target of 180 . Thus this bond may be completely rigid, elastic, or completely free. Any interbond angles may also be parameterized but at some cost in storage. The normal mode of working is to develop a single chain for the entire length of the molecule, but if cumulative error makes fitting difficult a fresh chain may be started at any stage. Bilder may itself reconnect such chains at a later stage. Construction and manipulation operates on a few residues at a time within the context of a polymer chain, but any or all of the rest of the molecule, or other molecules, may be displayed simultaneously. Contouring is done in advance to produce a directoried file of contoured bricks of space, each brick containing up to 20 independently switchable elements which need not all be from the same map. Choice of contour level and displayed volume is thus instantaneous within the choices prepared. The system is menu driven from a tablet, only file assignments and the like requiring the keyboard, and it offers dynamic parallax as an aid to 3D perception (Diamond et al., 1982). Bloomer et al. (1978), Phillips (1980) and Evans et al. (1981) give examples of its use.

small rigid groupings by direct reference to electron density in the manner of Diamond (1971) but without the maintenance of chain continuity, which is subsequently reintroduced by regularization. Horjales and Cambillau (Cambillau & Horjales, 1987; Cambillau et al., 1984) have also provided a development of Frodo which allows the optimization of the interaction of a ligand and a substrate with both molecules being treated as flexible. 3.3.3.2.8. Guide Brandenburg et al. (1981) have described a system which enables representations of macromolecules to be modified with reference to electron density. Such modifications include rotation about single bonds under manual control, or the movement with six degrees of freedom, also under manual control, of any part or parts of the molecule relative to the remainder. The latter operation may necessitate subsequent regularization of the structure if the moved and unmoved parts are chemically connected, and this is done as a separate operation on a different machine. The system also has the capability of displaying several molecules and of manually superimposing these on each other for comparison purposes. 3.3.3.2.9. HYDRA This program, due to Hubbard (1985) (and, more recently, to Molecular Simulations) has several functional parts, referred to as ‘heads’, which all use the same data structure. The addition of further heads may be accomplished, knowing the data structure, without the need to know anything of the internal workings of existing heads. The program contains extensive features for the display, analysis and modelling of molecular structure with particular emphasis on proteins. Display options include dotted surfaces, molecular skeletons, protein cartoons and a variety of van der Waals, ball-and-stick, and other raster-graphics display techniques such as ray tracing and shaded molecular surfaces. Protein analysis features include the analysis of hydrogen bonding, and of secondary and domain structure, as well as computational assessment of deviations from accepted protein structural characteristics such as abnormal main-chain or side-chain conformations and solvent exposure of hydrophobic amino acids. A full set of protein modelling facilities are provided including homology modelling and the ‘docking’ of substrate molecules. The program contains extensive tools for interactive modelling of structures from NMR or X-ray crystallographic data, and provides interfaces to molecular-mechanics and dynamics calculations. There are also database searching facilities to analyse and compare features of protein structure, and it is well suited to the making of cine films.

3.3.3.2.7. Frodo This system, due to Jones (Jones, 1978, 1982, 1985; Jones & Liljas, 1984), in its original implementation was a three-machine system comprising graphics display, minicomputer and mainframe, though more recent implementations combine the last two functions in a ‘midi’. Its capabilities are similar to those of Bilder described above, but its approach to stereochemical questions is very different. Where Bilder does not allow an atom to be moved out of context (unless it comprises a ‘chain’ of one atom) Frodo will permit an atom or group belonging to a chain to be moved independently of the other members of the chain and then offers regularization procedures based on the method of Hermans & McQueen (1974) to regain good stereochemistry. During this regularization, selected atoms may be fixed, remaining atoms then adjusting to these. A peptide, for example, may be inverted by moving the carbonyl oxygen across the peptide and fixing it, relying on the remaining atoms to rearrange themselves. (Bilder would do the equivalent operation by cutting the chain nearby, turning the peptide explicitly, reconnecting the chain and optimizing to regain chain continuity.) The Frodo approach is easy to use especially when large displacements of an existing structure are called for, but requires that ideal values be specified for all bond lengths, angles and fixed dihedrals since the system may need to regain such values in a distorted situation. Bilder, in contrast, never changes such features and so need not know their ideal values. Frodo may work either with consecutive residues of a polymer chain, useful for initial building, or with a volume centred on a chosen position, which is ideal for adjusting interacting side chains which are close in space but remote in sequence. In recent implementations Frodo can handle maps both in density grid form and in contour form and permits online contouring. It has also been developed (Jones & Liljas, 1984) to allow the automatic adjustment of the position and orientation of

3.3.3.2.10. O Jones et al. (1991) have developed a modelling system for proteins with a radically different approach to any of the foregoing, in that they begin by reducing the available electrondensity map to a skeletal representation (Greer, 1974; Williams, 1982) which consists of a line running through the density close to its maximal values, this being the basis of a chain trace. Provisional
-carbon positions are also estimated at this stage. A database of known structures is then scanned for pentapeptides which may be superimposed on five successive positions in the chain trace, the best fit so found being taken to provide coordinates for the three central residues of the developing model. The process advances by three residues at each step, the first and last residues of the pentapeptide being used only to ensure that the central residues are built in a manner compatible with what precedes and follows. The process ensures that conformations so built are free from improbable conformations, and the whole forms an adequate

441

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING starting structure for molecular-dynamics procedures, even though some imperfect geometry is to be expected where each tripeptide joins the next.

the most stable one from twelve possibilities for a four-ring system. The program is a development of similar work by Cohen (1971) in which the molecule was defined in terms of a tree structure and an optimizer based on search techniques rather than gradient vectors was used. The method included van der Waals terms and hence estimated energy differences between stereoisomers in condensed ring systems arising from steric hindrance.

3.3.3.3. Molecular-modelling systems based on other criteria Systems described within this section mostly have some form of energy minimization as their objective but some are purely geometrical. The optimization of molecules through empirical force fields has been reviewed by Allinger (1976), Burkert & Allinger (1982) and Boyd & Lipkowitz (1982). Some of these systems are in the academic domain, others are commercial. Most have capabilities exceeding the features referred to here and, of necessity, the list cannot be complete. No attempt at comparative evaluations is attempted or implied.

3.3.3.3.3. CHARMM This system, due to Brooks et al. (1983), is primarily concerned with molecular dynamics but it includes the capability of modelbuilding proteins and nucleic acids from sequence information and values of internal coordinates (bond lengths, bond angles and dihedral angles). The resulting structure (or a given structure) may then be optimized by minimizing an empirical energy function which may include electrostatic and hydrogen-bonding terms as well as the usual van der Waals energy and a Hookean treatment of the covalent skeleton. Hydrogen atoms need not be handled explicitly, groups such as —CH2— being treated as single pseudo atoms, and this may be advisable for large structures. For small or medium proteins hydrogens may be treated explicitly and their initial positions may be determined by CHARMM if they are not otherwise known.

3.3.3.3.1. Molbuild, Rings, PRXBLD and MM2/MMP2 Liljefors (1983) has described a system for constructing representations of organic molecules. The system develops the molecule with plausible geometry and satisfied valencies at all stages of the development with explicit recognition of lone pairs and the various possible hybridization states. Growth is generally by substitution in which a substituent and the atom it is to replace are both nominated from the screen. The bond which is reconstructed in a substitution is generally a single bond. Double and triple bonds are introduced by the substitution of moieties containing them. Atom types may be changed after incorporation in the growing molecule, so that although the menu of substituents includes —CH3 but not —NH2 the latter may be obtained by incorporating —CH3, then changing C to N and one of the hydrogens to a lone pair. Facilities are also provided for cyclization and acyclization. van der Lieth et al. (1984) have described an extension to this that is specialized to the construction of fused-ring systems. It permits the joining of rings by fusion of a bond, in which two adjacent atoms in one ring are superposed on two in another. It also permits the construction of spiro links in which one atom is common to two rings, or the construction of bridges, or the polymerization of ring systems to form, for example, oligosaccharides. Again the satisfaction of valencies is maintained during building and the geometry of the result is governed by superposition of relevant atoms in the moieties involved. PRXBLD is a molecular-model-building program which accepts two-dimensional molecular drawings in a manner similar to Script (Section 3.3.3.3.2) and constructs approximate threedimensional coordinates from these. It is the model-building component of SECS (Simulation and Evaluation of Chemical Synthesis) (Wipke et al., 1977; Wipke & Dyott, 1974; Wipke, 1974). See also Anderson (1984). All three of these programs produce output which is acceptable as input to MM2(82)/MMP2 which are developments of Allinger’s geometrical optimization based on molecular mechanics (Allinger, 1976).

3.3.3.3.4. Commercial systems A number of very powerful molecular-modelling systems are now available commercially and we mention a few of these here. Typically, each consists of a suite of programs sharing a common data structure so that the components of a system may be acquired selectively. The Chem-X system, from Chemical Design Ltd, enables models to be developed from sketch-pad input, provides for their geometrical optimization and interfaces the result to Gaussian80 for quantum-mechanical calculations. MACCS, from Molecular Design Ltd, and related software (Allinger, 1976; Wipke et al., 1977; Potenzone et al., 1977) has similar features and also has extensive database-maintenance facilities including data on chemical reactions. Sybyl, from Tripos Associates (van Opdenbosch et al., 1985), also builds from sketches with a standard fragment library, and provides interfaces to quantum-mechanical routines, to various databases and to MACCS. Insight II (Section 3.3.3.1.7) is available from Biosym and GRAMPS (Section 3.3.3.1.4) is available from T. J. O’Donnell Associates.

3.3.4. Graphics software for the display of small and mediumsized molecules

By L. M. D. Cranswick

3.3.3.3.2. Script This system, described by Cohen et al. (1981), is specialized for fused-ring systems, especially steroids, but is not limited to these classes. The system allows the user to draw on the screen (with a light pen or equivalent) a two-dimensional representation of a molecule using single lines for single bonds, double lines for double bonds, and wedges to indicate out-of-plane substituents. The software can then enumerate the possible distinct conformers, each of which is expected to be near an energy minimum on the conformational potential surface. Each conformer may then be annealed to reach an energy minimum using an energy estimate based on bond lengths, bond angles, torsion angles and van der Waals, electrostatic and hydrogen-bonding terms. An example is given of the identification of an unusual conformer as

3.3.4.1. Introduction In the age of the Internet, a wide variety of software can be easily obtained for the display of small and medium-sized molecules. An obvious question to ask is ‘which software is the best?’. It can be ‘best’ to try all the relevant available software quickly before settling on two to three programs that are found to be most suitable. Not relying on just a single program can be important for visual and numeric cross validation of the resulting structural plot, as bugs (if present) can be quite subtle in their effect, but moderately easy to find by comparing plots made by different programs. Most software can import crystallographic data in some format. The IUCr CIF format [see International Tables for

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3.3. MOLECULAR MODELLING AND GRAPHICS Table 3.3.4.1. Functionality of software for crystal structure display

Program ATOMS Balls&Sticks BALSAC Cameron CaRIne Crystallographica CrystalMaker Crystal Studio CrystMol Diamond DrawXTL FpStudio GRETEP Mercury MolXtl OLEX ORTEP-III ORTEP-3 for Windows ORTEX/Oscail X PEANUT Platon/Pluton PowderCell PRJMS SCHAKAL STRUPLO STRUPLO for Windows STRUVIR VENUS XmLmctep X-Seed Xtal-3D XtalDraw

Ball and stick Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Yes Yes Yes Yes Yes

ADPs

MSDA

Yes

Polyhedral display

Cartesian coordinates

Yes Yes

Yes

Comparison/ overlay of multiple structures

Extended structures/ topology analysis

Magnetic structures

Incommensurate structures

Yes

Yes Yes Yes Yes Yes Yes

Yes Yes Yes Yes Yes Yes Yes

Yes Yes

Yes Yes Yes Yes

Yes Yes Yes Yes Yes

Yes

Yes Yes Yes Yes

Yes Yes Yes Yes Yes Yes

Yes

Yes

Yes

Yes

Yes Yes Yes Yes

Yes Yes Yes Yes Yes

Yes Yes Yes Yes

Yes Yes

Yes

Yes Yes Yes

Yes

Crystallography Volume G (2005)] is slowly becoming a standard in this regard, displacing the single-crystal SHELX INS format, which has been a de-facto standard file format for much crystallographic data exchange. Entering crystallographic data by hand is slow and often introduces errors via typographical mistakes. Such mistakes can be minimized by importing structures using a known file type, or reformatting using a text editor or spreadsheet program into a known file type. A variety of software programs can be used for translating crystallographic structure files; however, the output (especially the handling of the symmetry operators and the space group) should be carefully checked. The CCP14 website (Cockroft & Stephenson, 2005) lists a variety of programs that can be used for this, of which a specialist program is Cryscon (Dowty, 2005).

3.3.4.2.2. Anisotropic displacement parameters A subset of the programs that display ball-and-stick structures can also display surfaces related to anisotropic displacement parameters (ADPs) (also known colloquially as ‘thermals’, ‘anisotropic thermal ellipsoids’ or ‘ORTEPS’). By default, most programs display the ellipsoid surfaces at a probability of 50% and normally allow this value to be changed to values between 1 and 99%. Programs that can draw ADPs include ATOMS, Cameron, Crystallographica, CrystalMaker, Crystal Studio, CrystMol, Diamond, DrawXTL, FpStudio, GRETEP, MolXtl, ORTEP-III, ORTEP-3 for Windows, ORTEX, PEANUT, Platon, VENUS, XmLmctep, X-Seed and XtalDraw. 3.3.4.2.3. Mean-square displacement amplitude When a more thorough investigation of the ADPs would be informative (Hummel, Raselli & Bu¨rgi, 1990), PEANUT can be used for plotting the mean-square displacement amplitude (MSDA), root-mean-square displacements (RMSDs) and difference surfaces. MSDA ‘peanuts’ can be displayed where the ADPs are non-positive-definite and the ellipsoids cannot be drawn. ORTEP-3 for Windows also has an option for plotting MSDAs. Care should be taken to ensure the resulting display is correct.

3.3.4.2. Types of crystal structure display and functionality The following information was current at the time of writing, but most software is continually changing with the insertion of new features. Thus occasional checks for updated functionality can be useful. Most software distributions include an ‘updates’ file containing new features and bug fixes. Detailed information on the software referred to in this section, including functionality, authorship, source and availability, is given in Tables 3.3.4.1 and 3.3.4.2.

3.3.4.2.4. Polyhedral display A method for understanding inorganic and intermetallic structures is the use of coordination polyhedra. The faces defined by the outer coordinated atoms generate a polyhedral object that is displayed instead of the individual atoms. This can aid in understanding the structures of polymeric inorganic materials involving both simple and complex tilt systems, and distorted

3.3.4.2.1. Ball and stick This is one of the most fundamental methods of displaying a crystal structure and almost all software supports this. The exceptions are STRUVIR and STRUPLO for Windows [a port of STRUVIR incorporating a graphical user interface (GUI)], which are both optimized for the polyhedral display of crystal structures.

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3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING Table 3.3.4.2. Availability of software for crystal structure display Program

Company or contact person

Contact and/or web address

ATOMS

Shape Software

521 Hidden Valley Road, Kingsport, TN 37663, USA. http://www.shapesoftware.com/

Reference

Status

Balls&Sticks

Tadashi Ozawa

http://www.toycrate.org/

BALSAC

Klaus Hermann

Fritz-Haber-Institut der MPG, Berlin. http://w3.rz-berlin.mpg.de/~hermann/hermann/balpam.html [Copyright (1991–2004) Klaus Hermann. All rights reserved.]

Cameron

David Watkin

http://www.xtl.ox.ac.uk/crystals.html

CaRIne

ESM Software

http://pro.wanadoo.fr/carine.crystallography/ [Copyright (1989–2005) C. Boudias and D. Monceau.]

Crystallographica

Oxford Cryosystems Ltd

http://www.crystallographica.co.uk/

CrystalMaker

CrystalMaker Software Ltd

5 Begbroke Science Park, Sandy Lane, Yarnton, OX5 1PF, UK. http://www.crystalmaker.co.uk/ (CrystalMaker 6 for Mac OS X)

Commercial

Crystal Studio

CrystalSoft & Crystal Systems Co. Ltd

PO Box 7006, Wattle Park, VIC 3128, Australia. http://www.crystalsoftcorp.com/ [Copyright (1999–2005) Crystal Systems Co., Ltd]

Commercial

CrystMol

David Duchamp

6209 Litchfield Lane, Kalamazoo, MI 49009-9159, USA. http://www.crystmol.com/

Commercial

Diamond

Crystal Impact

K. Brandenburg & H. Putz, Crystal Impact GbR, Postfach 1251, D-53002 Bonn, Germany. http://www.crystalimpact.com/

Commercial

DrawXTL

Larry Finger

http://lwfinger.net/drawxtl/

[4]

Freeware

FpStudio

Juan Rodriguez-Carvajal

http://www.ill.fr/dif/soft/fp/

[5]

Freeware

GRETEP

Jean Laugier

J. Laugier and B. Bochu, ENSP/Laboratoire des Mate´riaux et du Ge´nie Physique, BP 46, 38042 Saint Martin d’He`res, France. http://www.ccp14.ac.uk/tutorial/lmgp/

Mercury

Cambridge Crystallographic Data Centre

http://www.ccdc.cam.ac.uk/

[6]

Freeware

MolXtl

Dennis W. Bennett

http://www.uwm.edu/Dept/Chemistry/molxtl/

[7]

Freeware

OLEX

Oleg Dolomanov

http://www.ccp14.ac.uk/ccp/web-mirrors/lcells/ olex_index.htm

[8]

Freeware

ORTEP-III

Carroll Johnson

http://www.ornl.gov/ortep/ortep.html

[9]

Freeware

ORTEP-3 for Windows

Louis Farrugia

http://www.chem.gla.ac.uk/~louis/software/ortep3/

[10]

Freeware

ORTEX/Oscail X

Patrick McArdle

http://www.nuigalway.ie/cryst/software.htm

[11], [12]

Freeware

PEANUT

Hans-Beat Bu¨rgi

Laboratorium fu¨r chemische und mineralogische Kristallographie, Universita¨t Bern, Freiestrasse 3, CH-3012 Bern, Switzerland. E-mail: [email protected]

[13]

Freeware

Platon/Pluton

Anthony Spek

http://www.cryst.chem.uu.nl/platon/

[14], [15]

Freeware

PowderCell

Gert Nolze

http://www.ccp14.ac.uk/ccp/web-mirrors/powdcell/a_v/v_1/ powder/e_cell.html

[16]

Freeware

PRJMS

Akiji Yamamoto

http://quasi.nims.go.jp/yamamoto/

[17]

Freeware

SCHAKAL

Egbert Keller

http://www.krist.uni-freiburg.de/ki/Mitarbeiter/Keller/ schakal.html

[18]

Freeware

STRUPLO

Reinhadt Fischer

http://www.brass.uni-bremen.de/

[19]

Freeware Freeware

Commercial [1]

Freeware Commercial

[2]

Freeware Commercial

[3]

Commercial

Freeware

STRUPLO for Windows

Louis Farrugia

http://www.chem.gla.ac.uk/~louis/software/struplo/

[19], [20]

STRUVIR

Armel Le Bail

http://www.cristal.org/vrml/struvir.html

[19], [20]

Freeware

VENUS

Fujio Izumi

http://homepage.mac.com/fujioizumi/visualization/VENUS.html

[21], [22]

Freeware

XmLmctep

Alain Soyer

http://www.lmcp.jussieu.fr/~soyer/Lmctep_en.html

[23]

Freeware

X-Seed

Len Barbour

http://x-seed.net/

[24], [25]

Commercial

Xtal-3D

Alan Hewat

http://barns.ill.fr/dif/xtal-3d.html

[26]

Freeware

XtalDraw

Bob Downs

http://www.geo.arizona.edu/xtal/xtaldraw/xtaldraw.html

[27]

Freeware

References: [1] Ozawa & Kang (2004); [2] Watkin et al. (1996); [3] Siegrist (1997); [4] Finger et al. (2007); [5] Chapon & Rodriguez-Carvajal (2005); [6] Bruno et al. (2002); [7] Bennett (2004); [8] Dolomanov et al. (2003); [9] Burnett & Johnson (1996); [10] Farrugia (1997); [11] McArdle (1994); [12] McArdle et al. (2004); [13] Hummel, Hauser & Bu¨rgi (1990); [14] Spek (1998); [15] Spek (2003); [16] Kraus & Nolze (1996); [17] Yamamoto (1982); [18] Keller (1999); [19] Fischer (1985); [20] Le Bail (1996); [21] Izumi & Dilanian (2002); [22] Izumi (2004); [23] Soyer (1993); [24] Barbour (2001); [25] Atwood & Barbour (2003); [26] Hewat (2002); [27] Downs & Hall-Wallace (2003).

coordination structures. Polyhedral display is provided by the programs ATOMS, Balls&Sticks, CaRIne, Crystallographica, CrystalMaker, Crystal Studio, Diamond, DrawXTL, STRUPLO, STRUPLO for Windows (a port of STRUVIR incorporating a GUI), STRUVIR, VENUS, Xtal-3D and XtalDraw. Nearly all the

programs display polyhedra automatically after the user has defined (i) a central atom, (ii) the coordinated atoms and (iii) minimum (often a default of near zero) and maximum bond distances. One exception is Balls&Sticks, where graphical point and click of the mouse is used to define the polyhedra.

444

3.3. MOLECULAR MODELLING AND GRAPHICS 3.3.4.2.5. Cartesian coordinates Importing Cartesian coordinates can allow the display of incommensurate and quasicrystal structures if the refinement software has this as an output option. Using Cartesian coordinates can sometimes be more convenient for the slight modification of structures for the display of distortions or individual molecules. A structure defined as triclinic with space group P1 and a cubic cell with edges of unit length would also work for importing a structure or molecule originally defined in a Cartesian frame of reference.

Altona, C. & Sundaralingam, M. (1972). Conformational analysis of the sugar ring in nucleosides and nucleotides. A new description using the concept of pseudorotation. J. Am. Chem. Soc. 94, 8205–8212. American National Standards Institute, American National Standard for Information Processing Systems – Computer Graphics – Graphical Kernel System (GKS) Functional Description (1985). ISO 7942, ISO Central Secretariat, Geneva, Switzerland. American National Standards Institute, American National Standard for Information Processing Systems – Computer Graphics – Programmer’s Hierarchical Graphics System (PHIGS) Functional Description, Archive File Format, Clear-Text Encoding of Archive File (1988). ANSI X3.144–1988. ANSI, New York, USA. Anderson, S. (1984). Graphical representation of molecules and substructure-search queries in MACCS. J. Mol. Graphics, 2, 83–90. Arnold, D. B. & Bono, P. R. (1988). CGM and CGI: Metafile and Interface Standards for Computer Graphics. Berlin: Springer-Verlag. Atwood, J. L. & Barbour, L. J. (2003). Molecular graphics: from science to art. Cryst. Growth Des. 3, 3. Barbour, L. J. (2001). X-Seed – A software tool for supramolecular crystallography. J. Supramol. Chem. 1, 189. Barry, C. D. & North, A. C. T. (1971). The use of a computer-controlled display system in the study of molecular conformations. Cold Spring Harbour Symp. Quant. Biol. 36, 577–584. Bash, P. A., Pattabiraman, N., Huang, C., Ferrin, T. E. & Langridge, R. (1983). Van der Waals surfaces in molecular modelling: implementation with real-time computer graphics. Science, 222, 1325–1327. Beddell, C. J. (1970). An X-ray Crystallographic Study of the Activity of Lysozyme. DPhil thesis, University of Oxford, England. Bennett, D. W. (2004). MolXtl: molecular graphics for small-molecule crystallography. J. Appl. Cryst. 37, 1038. Bernstein, F. C., Koetzle, T. F., Williams, G. J. B., Meyer, E. F., Brice, M. D., Rodgers, J. R., Kennard, O., Shimanouchi, T. & Tasumi, M. (1977). The Protein Data Bank: a computer-based archival file for macromolecular structures. J. Mol. Biol. 112, 535–542. Bloomer, A. C., Champness, J. N., Bricogne, G., Staden, R. & Klug, A. ˚ resolution showing (1978). Protein disk of tobacco mosaic virus at 2.8 A the interactions within and between subunits. Nature (London), 276, 362–368. Boyd, D. B. & Lipkowitz, K. B. (1982). Molecular mechanics, the method and its underlying philosophy. J. Chem. Educ. 59, 269–274. Brandenburg, N. P., Dempsey, S., Dijkstra, B. W., Lijk, L. J. & Hol, W. G. J. (1981). An interactive graphics system for comparing and model building of macromolecules. J. Appl. Cryst. 14, 274–279. Brooks, B. R., Brucolleri, R. E., Olafson, B. D., States, D. J., Swaminathan, S. & Karplus, M. (1983). CHARMM: a program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem. 4, 187–217. Brown, M. D. (1985). Understanding PHIGS. Template, Megatek Corp., San Diego, California, USA. Bruno, I. J., Cole, J. C., Edgington, P. R., Kessler, M., Macrae, C. F., McCabe, P., Pearson, J. & Taylor, R. (2002). New software for searching the Cambridge Structural Database and visualizing crystal structures. Acta Cryst. B58, 389–397. Burkert, U. & Allinger, N. L. (1982). Molecular Mechanics. ACS Monogr. No. 177. Burnett, M. N. & Johnson, C. K. (1996). ORTEP-III: Oak Ridge thermal ellipsoid plot program for crystal structure illustrations. Report ORNL6895. Oak Ridge National Laboratory, Tennessee, USA. Cambillau, C. & Horjales, E. (1987). TOM: a FRODO subpackage for protein-ligand fitting with interactive energy minimization. J. Mol. Graphics, 5, 174–177. Cambillau, C., Horjales, E. & Jones, T. A. (1984). TOM, a display program for fitting ligands into protein receptors and performing interactive energy minimization. J. Mol. Graphics, 2, 53–54. Cambridge Structural Database (1994). Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge, England. Chapon, L. C. & Rodriguez-Carvajal, J. (2005). FpStudio. Rutherford Appleton Laboratory, UK, and Laboratoire Le´on Brillouin, Saclay, France. Cockcroft, J. K. & Stephenson, R. (2005). EPSRC-funded Collaborative Computational Project Number 14 for Single Crystal and Powder Diffraction (CCP14), http://www.ccp14.ac.uk/. Cockrell, P. R. (1983). A new general purpose method for large volume production of contour charts. Comput. Graphics Forum, 2, 35–47.

3.3.4.2.6. Comparing or overlaying crystal structures The graphical comparison of crystallographic structures can be useful and time-saving for comparison of polymorphs or a chemically similar series of small-molecule structures. One program that can perform this function is CrystMol, where multiple molecular structures can be compared using a point and click menu or via the CrystMol scripting system. RMS differences are also listed. Superposition of structures is discussed in Section 3.3.1.2.2. 3.3.4.2.7. Extended structures and topology analysis Currently, the only available program that rigorously analyses extended structures (involving overlapping or interpenetrating molecules) is OLEX. For graphical viewing of extended structures OLEX displays particular fragments in a single colour. GRETEP also has this display functionality, making it useful for viewing extended structures. 3.3.4.2.8. Magnetic crystal structure display The software listed in Tables 3.3.4.1 and 3.3.4.2 includes programs that can display graphics representing magnetic vectors without necessarily having the ability to understand magnetic symmetry. Programs that can display magnetic structures include ATOMS, CrystalMaker, Diamond, DrawXTL, FpStudio, VENUS and Xtal-3D. 3.3.4.2.9. Incommensurate crystal structures PRJMS and FpStudio are currently the only programs that can plot modulated structures in three-dimensional space; FpStudio is currently restricted to incommensurate magnetic structures. However, importing Cartesian coordinates can be used to display incommensurate structures when incommensurate refinement software can output coordinates in this format. BALSAC provides a good example of software which by default uses Cartesian coordinates, from which plots of incommensurate and quasicrystal structures can be generated.

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Morimoto, C. N. & Meyer, E. F. (1976). Information retrieval, computer graphics, and remote computing. In Crystallographic Computing Techniques, edited by F. R. Ahmed, K. Huml & B. Sedlacek, pp. 488–496. Copenhagen: Munksgaard. Motherwell, W. D. S. (1978). Pluto – a program for displaying molecular and crystal structures. Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge, England. Newman, W. M. & Sproull, R. F. (1973). Principles of Inter-active Computer Graphics. New York: McGraw-Hill. North, A. C. T. (1982). Use of interactive computer graphics in studying molecular structures and interactions. Chem. Ind. pp. 221–225. North, A. C. T., Denson, A. K., Evans, A. C., Ford, L. O. & Willoughby, T. V. (1981). The use of an interactive computer graphics system in the study of protein conformations. In Biomolecular Structure, Conformation, Function and Evolution, Vol. 1, edited by R. Srinivasan, pp. 59–72. Oxford: Pergamon Press. O’Donnell, T. J. & Olson, A. J. (1981). GRAMPS – a graphics language interpreter for real-time, interactive, three-dimensional picture editing and animation. Comput. Graphics, 15, 133–142. Olson, A. J. (1982). GRAMPS: a high level graphics interpreter for expanding graphics utilization. In Computational Crystallography, edited by D. Sayre, pp. 326–336. Oxford University Press. Opdenbosch, N. van, Cramer, R. III & Giarrusso, F. F. (1985). Sybyl, the integrated molecular modelling system. J. Mol. Graphics, 3, 110– 111. Ozawa, T. C. & Kang, S. J. (2004). Balls&Sticks: easy-to-use structure visualization and animation program. J. Appl. Cryst. 37, 679. Pearl, L. H. & Honegger, A. (1983). Generation of molecular surfaces for graphic display. J. Mol. Graphics, 1, 9–12, C2. Phillips, S. E. V. (1980). Structure and refinement of oxymyoglobin at ˚ resolution. J. Mol. Biol. 142, 531–554. 1.6 A Phong, B. T. (1975). Illumination for computer generated images. Commun. ACM, 18, 311–317. Porter, T. K. (1978). Spherical shading. Comput. Graphics, 12, 282–285. Potenzone, R., Cavicchi, E., Weintraub, H. J. R. & Hopfinger, A. J. (1977). Molecular mechanics and the CAMSEQ processor. Comput. Chem. 1, 187–194. Potterton, E. A., Geddes, A. J. & North, A. C. T. (1983). Attempts to design inhibitors of dihydrofolate reductase using interactive computer graphics with real time energy calculations. In Chemistry and Biology of Pteridines, edited by J. A. Blair, pp. 299–303. Berlin, New York: Walter de Gruyter. Purisima, E. O. & Scheraga, H. A. (1986). An approach to the multipleminima problem by relaxing dimensionality. Proc. Natl Acad. Sci. USA, 83, 2782–2786. Richardson, J. S. (1977). -Sheet topology and the relatedness of proteins. Nature (London), 268, 495–500. Richardson, J. S. (1981). The anatomy and taxonomy of protein structure. Adv. Protein Chem. 34, 167–339. Richardson, J. S. (1985). Schematic drawings of protein structures. In Methods in Enzymology, Vol. 115, Diffraction Methods for Biological Molecules, Part B, edited by H. W. Wyckoff, C. H. W. Hirs & S. N. Timasheff, pp. 359–380. Orlando: Academic Press. Shapiro, A., Botha, J. D., Pastore, A. & Lesk, A. M. (1992). A method for multiple superposition of structures. Acta Cryst. A48, 11–14.

448

references

International Tables for Crystallography (2010). Vol. B, Chapter 3.4, pp. 449–457.

3.4. Accelerated convergence treatment of R
n lattice sums By D. E. Williams† ˚ the Coulombic lattice at the rather large summation limit of 20 A ˚ sum has not converged and is incorrect by about 8%. The 20 A sum included 832 molecules and 2494 individual distances. At various smaller summation limits the truncation error fluctuates wildly and can be either positive or negative. Note that the results shown in the table always refer to summation over whole molecules, that is, over neutral charge units. If the Coulombic summation is not carried out over neutral charge units the truncation error is even larger. These considerations support the conclusion that accelerated-convergence treatment of the Coulombic lattice sum should be regarded as mandatory. Table 3.4.2.2 gives an example of the convergence behaviour of the untreated ðn ¼ 6Þ dispersion sum for benzene. In obtaining this sum it is not necessary to consider whole molecules as in the Coulombic case. The exclusion of atoms (or sites) in the portions of molecules outside the summation limit greatly reduces the number of terms to be considered. At the ˚ , 439 benzene molecules and 22 049 summation limit of 20 A individual distances are considered; the dispersion-sum truncation error is 0.4%. Thus, if sufficient computer time is available it may be possible to obtain a moderately accurate dispersion sum without the use of accelerated convergence. However, as shown below, the use of accelerated convergence will greatly speed up the calculation, and is in practice necessary if higher accuracy is required.

3.4.1. Introduction The electrostatic energy of an ionic crystal is often represented by taking a pairwise sum between charge sites interacting via Coulomb’s law (the n ¼ 1 sum). The individual terms may be positive or negative, depending on whether the pair of sites have charges of the same or different signs. The Coulombic energy is very long-range, and it is well known that convergence of the Coulombic lattice-energy sum is extremely slow. For simple structure types Madelung constants have been calculated which represent the Coulombic energy in terms of the cubic lattice constant or a nearest-neighbour distance. Glasser & Zucker (1980) give tables of Madelung constants and review the subject giving references dating back to 1884. If the ionic crystal structure is not of a simple type usually no Madelung constant will be available and the Coulombic energy must be obtained for the specific crystal structure being considered. In carrying out this calculation, accelerated-convergence treatment of the Coulombic lattice sum is indispensable to achieve accuracy with a reasonable amount of computational effort. A model of a molecular crystal may include partial net atomic charges or other charge sites such as lone-pair electrons. The ðn ¼ 1Þ sum also applies between these site charges. The dispersion energy of ionic or molecular crystals may be represented by an ðn ¼ 6Þ sum over atomic sites, with possible inclusion of ðn ¼ 8; 10; . . .Þ terms for higher accuracy. The dispersion-energy sum has somewhat better convergence properties than the Coulombic sum. Nevertheless, acceleratedconvergence treatment of the dispersion sum is strongly recommended since its use can yield at least an order of magnitude improvement in accuracy for a given calculation effort. The repulsion energy between nonbonded atoms in a crystal may be represented by an exponential function of short range, or possibly by an ðn ¼ 12Þ function of short range. The convergence of the repulsion energy is fast and no accelerated-convergence treatment is normally required.

3.4.3. Preliminary description of the method Ewald (1921) developed a method which modified the mathematical representation of the Coulombic lattice sum to improve the rate of convergence. This method was based on partially transforming the lattice sum into reciprocal space. Bertaut (1952) presented another method for derivation of the Ewald result which used the concept of the crystallographic structure factor. His formula extended the Ewald treatment to a composite lattice with more than one atom per lattice point. Nijboer & DeWette (1957) developed a general Fourier transform method for the evaluation of R
n sums in simple lattices. Williams (1971) extended this treatment to a composite lattice and gave general formulae for the R
n sums for any crystal. A review article, on which this chapter is based, appeared later (Williams, 1989a,b). Consider a function, W(R), which is unity at R ¼ 0 and smoothly declines to zero as R approaches infinity. If each term of the lattice sum is multiplied by W(R), the rate of convergence is increased. However, the rate of convergence of the remainder of

3.4.2. Definition and behaviour of the direct-space sum This pairwise sum is taken between atoms (or sites) in the reference unit cell and all other atoms (or sites) in the crystal, excluding the self terms. Thus, the second atom (or site) is taken to range over the entire crystal, with elimination of self-energy terms. If Vn represents an energy, each atom is assigned one half of the pair energy. Therefore, the energy per unit cell is one cells Pcell allP 0

Vn ¼ ð1=2Þ

j

Table 3.4.2.1. Untreated lattice-sum results for the Coulombic energy (n = 1) of ˚ ); the lattice constant is taken as 5.628 A ˚ sodium chloride (kJ mol
1, A

Qjk R
n jk ;

k

Truncation limit

where Qjk is a given coefficient, Rjk is an interatomic distance, and the prime on the second sum indicates that self terms are omitted. In the case of the Coulombic sum, n ¼ 1 and Qjk ¼ qj qk is the product of the site charges. Table 3.4.2.1 gives an example of the convergence behaviour of the untreated ðn ¼ 1Þ Coulombic sum for sodium chloride. Even † Deceased. Questions related to this chapter may be addressed to Dr Bill Smith, Molecular Simulation Group, Computational Science and Engineering Department, CCLRC Daresbury Laboratory, Daresbury, Warrington WA4 4AD, United Kingdom.

Copyright © 2010 International Union of Crystallography

Number of terms

Calculated energy

6.0

23

67


696.933

8.0

59

175


597.371

10.0

108

322


915.152

12.0 14.0

201 277

601 829


773.475
796.248

16.0

426

1276


826.502

18.0

587

1759


658.995

20.0

832

2494

Converged value

449

Number of molecules


794.619
862.825

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING Table 3.4.2.2. Untreated lattice-sum results for the dispersion energy (n = 6) of ˚) crystalline benzene (kJ mol
1, A Truncation limit

Number of molecules

6.0

Number of terms

26

524

XðxÞ ¼ x1 a1 þ x2 a2 þ x3 a3 ;

Calculated energy

where x1 ; x2 ; x3 are the fractional cell coordinates of X. A lattice vector in direct space is defined as


69.227

8.0

51

1313


76.007

10.0

77

2631


78.179

12.0

126

4718


79.241

14.0 16.0

177 265

7531 11274


79.726
80.013

18.0

344

15904


80.178

20.0

439

22049


80.295

XðdÞ ¼ d1 a1 þ d2 a2 þ d3 a3 ; where d1 ; d2 ; d3 are integers (specifying particular values of x1 ; x2 ; x3 ) designating a lattice point. Vd is the direct-cell volume which is equal to a1  a2  a3. A general point in the direct lattice is X(x); the contents of the lattice are by definition identical as the components of x are increased or decreased by integer amounts. The reciprocal-lattice vectors are defined by the relations


80.589

Converged value

the original sum, which contains the difference terms, is not increased. Vn ¼ ð1=2Þ

one cells Pcell allP 0 j

aj  bk ¼ 1

j¼k

¼0

j 6¼ k:

Qjk R
n jk WðRÞ

k

one cells Pcell allP 0

þ ð1=2Þ

j

A general vector in reciprocal space H(r) is defined as

Qjk R
n jk ½1
WðRÞ:

k

HðrÞ ¼ r1 b1 þ r2 b2 þ r3 b3 : In the accelerated-convergence method the difference terms are expressed as an integral of the product of two functions. According to Parseval’s theorem (described below) this integral is equal to an integral of the product of the two Fourier transforms of the functions. Finally, the integral over the Fourier transforms of the functions is converted to a sum in reciprocal (or Fourier-transform) space. The choice of the convergence function W(R) is not unique; an obvious requirement is that the relevant Fourier transforms must exist and have correct limiting behaviour. Nijboer and DeWette suggested using the incomplete gamma function for W(R). More recently, Fortuin (1977) showed that this choice of convergence function leads to optimal convergence of the sums in both direct and reciprocal space:

A reciprocal-lattice vector H(h) is defined by the integer triplet h1 ; h2 ; h3 (specifying particular values of r1 ; r2 ; r3 ) so that HðhÞ ¼ h1 b1 þ h2 b2 þ h3 b3 : In other sections of this volume a shortened notation h is used for the reciprocal-lattice vector. In this section the symbol H(h) is used to indicate that it is a particular value of H(r). The three-dimensional Fourier transform gðtÞ of a function f ðxÞ is defined by gðtÞ ¼ FT3 ½ f ðxÞ ¼

2

R

f ðxÞ expð2
ix  tÞ dx:

2

WðRÞ ¼
ðn=2;
w R Þ=
ðn=2Þ; 2

The Fourier transform of the set of points defining the direct lattice is the set of points defining the reciprocal lattice, scaled by the direct-cell volume. It is useful for our purpose to express the lattice transform in terms of the Dirac delta function ðx
xo Þ which is defined so that for any function f ðxÞ

2

where
ðn=2Þ and
ðn=2;
w R Þ are the gamma function and the incomplete gamma function, respectively:
ðn=2;
w2 R2 Þ ¼

R1

tðn=2Þ
1 expð
tÞ dt f ðxo Þ ¼


w2 R2

and

R

ðx
xo Þf ðxÞ dx:

We then write
ðn=2Þ ¼
ðn=2; 0Þ:

FT3 f

P

½XðxÞ
XðdÞg ¼ Vd
1

d

The complement of the incomplete gamma function is

P

½HðrÞ
HðhÞ:

h

First consider the lattice sum over the direct-lattice points X(d), relative to a particular point XðxÞ ¼ R, with omission of the origin lattice point.

ðn=2;
w2 R2 Þ ¼
ðn=2Þ

ðn=2;
w2 R2 Þ:

S0 ðn; RÞ ¼

P

jXðdÞ
Rj
n :

d6¼0

3.4.4. Preliminary derivation to obtain a formula which accelerates the convergence of an R
n sum over lattice points X(d)

The special case with R ¼ 0 will also be needed: S0 ðn; 0Þ ¼

The three-dimensional direct-space crystal lattice is specified by the origin vectors a1, a2 and a3 . A general vector in direct space is defined as

P d6¼0

450

jXðdÞj
n :

3.4. ACCELERATED CONVERGENCE TREATMENT OF R
N LATTICE SUMS Now define a sum of Dirac delta functions f 0 ½XðdÞ ¼

P

FT3 ½ðn=2;
w2 jX
Rj2 ÞjX
Rj
n  ¼
n
ð3=2Þ jHjn
3
½ð
n=2Þ

½XðxÞ
XðdÞ:

d6¼0

þ ð3=2Þ;
w
2 jHj2  expð2
iH  RÞ:

Then S0 can be represented as an integral S0 ðn; RÞ ¼

R

Evaluation of the two Fourier transforms in the first term gives R P ½
ðn=2Þ
1 Vd
1 ½HðhÞ
H
n
ð3=2Þ jHjn
3

f 0 ½XðdÞjX
Rj
n dX;

h


½ð
n=2Þ þ ð3=2Þ;
w
2 jHj2  expð2
iH  RÞ dH:

in which a term is contributed to S0 whenever the direct-space vector X coincides with the lattice vector X(d), except for d ¼ 0. Now apply the convergence function to S0 :

Because of the presence of the Dirac delta function in each integral, we can convert the integrals with h unequal to zero into a sum

R S0 ðn; RÞ ¼ ½
ðn=2Þ
1 f 0 ½XðdÞjX
Rj
n 
ðn=2;
w2 jX
Rj2 Þ dX R þ ½
ðn=2Þ
1 f 0 ½XðdÞjX
Rj
n 2

½
ðn=2Þ
1 Vd
1
n
ð3=2Þ

h6¼0

The h ¼ 0 term needs to be evaluated in the limit. Clearly, the complex exponential goes to unity. If n is greater than 3 the limit of the indeterminate form infinity/infinity is needed:

The first integral is shown here only for the purpose of giving a consistent representation of S0 ; in fact, the first integral will be reconverted back into a sum and evaluated in direct space. The second integral will be transformed to reciprocal space using Parseval’s theorem [see, for example, Arfken (1970)], which states that f ðXÞg ðXÞ dX ¼

R


½ð
n=2Þ þ ð3=2Þ;
w
2 jHj2  jHj!0 jHj3
n R1 ð
n=2Þþð1=2Þ expð
tÞ dt
w
2 jHj2 t ¼ lim : 3
n jHj!0 jHj lim

FT3 ½ f ðXÞFT3 ½g ðXÞ dH:

Considering only the second integral in the formula for S0 and explicitly introducing the d ¼ 0 term we have

The limit can be found by L’Hospital’s rule [see, for example, Widder (1961)] which states that if f ðxÞ and gðxÞ both approach infinity as x approaches a constant, c, and the limit of the ratio of the first derivatives f 0 ðxÞ and g0 ðxÞ exists, that limit is also true for the limit of the ratio of the functions:

R ½
ðn=2Þ
1 f ½XðdÞjXðdÞ
Rj
n ðn=2;
w2 jX
Rj2 Þ dX R
½
ðn=2Þ
1 ðXÞjRj
n ðn=2;
w2 jRj2 Þ dX;

lim

x!c

where the unprimed f includes the h ¼ 0 term which was earlier omitted from f 0 : f ðXÞ ¼

P

jHðhÞjn
3


½ð
n=2Þ þ ð3=2Þ;
w
2 jHðhÞj2  exp½2
iHðhÞ  R:

2

 ðn=2;
w jX
Rj Þ dX:

R

P

f ðxÞ f 0 ðxÞ ¼ lim 0 : gðxÞ x!c g ðxÞ

To differentiate the definite integral function, Leibnitz’s formula may be used [see, for example, Arfken (1970)]. This formula states that

½XðxÞ
XðdÞ:

d

Using Parseval’s theorem, and evaluating the origin term, we have

d dx

ZhðxÞ f ðt; xÞ dt ¼ gðxÞ

R ½
ðn=2Þ
1 FT3 ff ½XðdÞgFT3

ZhðxÞ

df ðt; xÞ dt dx

gðxÞ

þ f ½hðxÞ

 ½jXðdÞ
Rj
n ðn=2;
w2 jX
Rj2 Þ dH

dhðxÞ dgðxÞ
f ½gðxÞ : dx dx


½
ðn=2Þ
1 jRj
n ðn=2;
w2 jRj2 Þ: In our case, x becomes jHj; f becomes tð
n=2Þþð1=2Þ expð
tÞ which is independent of jHj; g becomes
w
2 jHj2 ; and h is infinite. Thus only the last term of Leibnitz’s formula is nonzero and we obtain for the ratio of the first derivatives

The Fourier transform of the complement of the incomplete gamma function divided by jXjn is (Nijboer & DeWette, 1957)


ð
w
2 jHj2 Þð
n=2Þþð1=2Þ expð

w2 jHj2 Þ2
w
2 jHj jHj!0 ð3
nÞjHj2
n ð
n=2Þþð3=2Þ n
3 ¼
w ½2=ðn
3Þ;

FT3 ½ðn=2;
w2 jXj2 ÞjXj
n 

lim

¼
n
ð3=2Þ jHjn
3
½ð
n=2Þ þ ð3=2Þ;
w
2 jHj2 :

If there is a change of origin and the point ðX
RÞ is used instead of X the transform is

so that the limiting value for the h ¼ 0 term for n greater than 3 is

451

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING and the third group had d not zero and j ¼ k. (A possible fourth group with d ¼ 0 and j ¼ k is omitted, as defined.)

þ½
ðn=2Þ
1 Vd
1
n=2 wn
3 ½2=ðn
3Þ: The final result for S0 is S0 ðn; RÞ ¼ ½
ðn=2Þ
1

P

Vðn; Rj Þ ¼ ð1=2Þ

P

Qjk jRk
Rj j
n

j6¼k

P P þ ð1=2Þ Qjk S0 ðn; jRj
Rk jÞ þ ð1=2Þ Qjj S0 ðn; 0Þ:

jXðdÞ
Rj
n

d6¼0

j

j6¼k


ðn=2;
w2 jXðdÞ
Rj2 Þ
½
ðn=2Þ
1 jRj
n ðn=2;
w2 jRj2 Þ P þ ½
ðn=2Þ
1 Vd
1
n
ð3=2Þ jHðhÞjn
3

By expanding this expression we obtain

h6¼0


½ð
n=2Þ þ ð3=2Þ;
w
2 jHðhÞj2 

Vðn; Rj Þ ¼ ð1=2Þ

 exp½2
iHðhÞ  R

(

þ ½
ðn=2Þ
1 Vd
1
n=2 wn
3 2ðn
3Þ
1 :

P

Qjk jRk
Rj j
n

þ ½1=2
ðn=2Þ

P

Qjk

j6¼k

The significance of the terms is as follows. The first term represents the convergence-accelerated direct sum, which does not include the origin term; the next term, also in direct space, corrects for the remainder resulting from the subtraction of the origin term; the third term comes from Parseval’s theorem and is a sum over the nonzero h reciprocal-lattice points; and the last term is the reciprocal-lattice h ¼ 0 term. If R ¼ 0 the second term becomes an indeterminate form 0/0. The limit can be found with use of L’Hospital’s rule again, this time for the 0/0 form. We need the limit of f ðxÞ=gðxÞ, where f ðRÞ ¼ ðn=2;
w2 R2 Þ and gðRÞ ¼ Rn . To differentiate the incomplete gamma function, we can again use Leibnitz’s formula. In this case only the second term of the formula is nonzero and we obtain for the ratio of the first derivatives

ð1Þ

j6¼k

P

jRk þ XðdÞ
Rj j
n

d6¼0

) 2

2


ðn=2;
w jRk þ XðdÞ
Rj j Þ (
½1=2
ðn=2Þ

P

ð2Þ

Qjk jRk
Rj j
n

j6¼k

)

 ðn=2;
w2 jRk
Rj j2 Þ

ð3Þ

( þ ½1=2
ðn=2ÞVd
1
n
ð3=2Þ

P

Qjk

j6¼k

P

jHðhÞjn
3

h6¼0


½ð
n=2Þ þ ð3=2Þ;
w
2 jHðhÞj2  )  exp½2
iHðhÞ  ðRk
Rj Þ

2
n=2 wn jRjn
1 expð

w2 jRj2 Þ ; njRjn
1

ð4Þ

þ ½1=2
ðn=2ÞVd
1
n=2 wn
3 2ðn
3Þ
1

Qjk

þ ½1=2
ðn=2Þ

P

Qjj

j


½
ðn=2Þ
1 2
n=2 wn n
1 : 2

P

jXðdÞj
n

d6¼0

) 2


ðn=2;
w jXðdÞj Þ Therefore, the value of the sum when R ¼ 0 is S0 ðn; 0Þ ¼ ½
ðn=2Þ


1 P


½1=
ðn=2Þ
n=2 wn n
1 jXðdÞj
n
ðn=2;
w2 jXðdÞj2 Þ


½
ðn=2Þ
1 2
n=2 wn n
1 þ ½
ðn=2Þ
1 Vd
1
n
ð3=2Þ

P

ð7Þ

Qjj P j

P

Qjj


2

jHðhÞjn
3

h6¼0

)

jHðhÞjn
3 2


½ð
n=2Þ þ ð3=2Þ;
w jHðhÞj 


½ð
n=2Þ þ ð3=2Þ;
w
2 jHðhÞj2 

þ ½1=2
ðn=2ÞVd
1
n=2 wn
3 2ðn
3Þ
1

þ ½
ðn=2Þ
1 Vd
1
n=2 wn
3 2ðn
3Þ
1 :

ð8Þ P

Qjj :

ð9Þ

j

This expression for V has nine terms, which are numbered on the right-hand side. Term (3) can be expressed in terms of
rather than :

3.4.5. Extension of the method to a composite lattice Define a general lattice sum over direct-space points Rj which interact with pairwise coefficients Qjk, where Qjk ¼ Qkj : PP0 P Vðn; Rj Þ ¼ ð1=2Þ Qjk jRk þ XðdÞ
Rj j
n ; k

P

þ ½1=2
ðn=2ÞVd
1
n
ð3=2Þ

h6¼0

j

ð6Þ

j

(

d6¼0

ð5Þ

j6¼k

(

so that the limiting value for this term as jRj approaches zero is

P

ð3Þ ¼
ð1=2Þ

d

P

Qjk jRk
Rj j
n

j6¼k

þ ½1=
ðn=2Þ

P

Qjk jRk
Rj j
n
ðn=2;
w2 jRk
Rj j2 Þ:

j6¼k

where the prime indicates that when d ¼ 0 the self-terms with j ¼ k are omitted. For convenience the terms may be divided into three groups: the first group of terms has d ¼ 0, where j is unequal to k; the second group has d not zero and j not equal to k;

It is seen that cancellation occurs with term (1) so that

452

3.4. ACCELERATED CONVERGENCE TREATMENT OF R
N LATTICE SUMS ð1Þ þ ð3Þ ¼ ½1=
ðn=2Þ

P

The integral representation of the term (5) is

Qjk jRk
Rj j
n

j6¼k

½1=2
ðn=2ÞVd
1
n
ð3=2Þ


ðn=2;
w2 jRk
Rj j2 Þ;

P

R Qjk ð0ÞjHjn
3

j6¼k


½ð
n=2Þ þ ð3=2Þ;
w
2 jHj2   exp½2
iH  ðRk
Rj Þ dH

which is the d ¼ 0, j unequal to k portion of the treated directlattice sum. The d unequal to 0, j unequal to k portion corresponds to term (2) and the d unequal to 0, j ¼ k portion corresponds to term (6). The direct-lattice terms may be consolidated as ð1Þþð2Þþð3Þþð6Þ ¼ ½1=2
ðn=2Þ

P P0 j

Qjk

k

P

and for term (9) is ½1=2
ðn=2ÞVd
1
n
ð3=2Þ

P

R Qjj ð0ÞjHjn
3

j

jRk þ XðdÞ
Rj j
n


½ð
n=2Þ þ ð3=2Þ;
w
2 jHj2  dH:

d 2

2


½n=2;
w jRk þ XðdÞ
Rj j : Combining these two sums of integrals into one integral sum gives

Now let us combine terms (4) and (8), carrying out the h summation first: ð4Þ þ ð8Þ ¼ ½1=2
ðn=2ÞVd
1
n
ð3=2Þ

P

R ½1=2
ðn=2ÞVd
1
n
ð3=2Þ ð0ÞjHjn
3 PP 
½ð
n=2Þ þ ð3=2Þ;
w
2 jHj2  Qjk

jHðhÞjn
3

h
2

j


½ð
n=2Þ þ ð3=2Þ;
w jHðhÞj  PP  Qjk exp½2
iHðhÞ  ðRk
Rj Þ: j

k

For n ¼ 1, suppose qj are net atomic charges so that the geometric combining law holds for Qjk ¼ qj qk . Then the double sum over j and k can be factored so that the limit that needs to be considered is

Terms (5) and (9) may be combined:

ð5Þ þ ð9Þ ¼ ½
ðn=2Þ


1

Vd
1
n=2 wn
3 ðn


3Þ


1

P

Qij þ

j

P

!


P

Qjk :

lim

j6¼k

¼ ½1=2
ðn=2Þ

P P0 j

k

Qjk

P

jRk þ XðdÞ
Rj j
n

ð2Þ

jHj2

:


1

k

ð3Þ

k


3Þ



  P 2
i qk R3k expð2
iH3 R3k Þ

2


½ð
n=2Þ þ ð3=2Þ;
w jHðhÞj  PP  Qjk exp½2
iHðhÞ  ðRk
Rj Þ þ ½
ðn=2Þ

qj expð
2
iH  Rj Þ

It is seen that in addition to cell neutrality the product of the first derivatives of the sums must exist. These sums are

P þ ½1=2
ðn=2ÞVd
1
n
ð3=2Þ jHðhÞjn
3

Vd
1
n=2 wn
3 ðn

j

ð1Þ

j


1


P

d2 ðuvÞ d2 v d2 u du dv : ¼ u þ v þ2 dx2 dx2 dx2 dx dx


ðn=2;
w2 jRk þ XðdÞ
Rj j2 Þ P
½1=
ðn=2Þ
n=2 wn n
1 Qjj

j

qk expð2
iH  Rk Þ

If the unit cell does not have a net charge, the sum over the q’s goes to zero in the limit and this is a 0/0 indeterminate form. Let jHj approach zero along the polar axis so that H  Rk ¼ H3 R3k , where subscript 3 indicates components along the polar axis. To find the limit with L’Hospital’s rule the numerator and denominator are differentiated twice with respect to H3. Represent the numerator of the limit by the product ðuvÞ and note that

d

h
2

k

jHj!0

The final formula is shown below. The significance of the four terms is: (1) the treated direct-lattice sum; (2) a correction for the difference resulting from the removal of the origin term in direct space; (3) the reciprocal-lattice sum, except h ¼ 0; and (4) the h ¼ 0 term of the reciprocal-lattice sum. Vðn; Rj Þ

k

 exp½2
iH  ðRk
Rj Þ dH:

2

PP j

! Qjk :

and

ð4Þ

"

k


2
i

P

# qj R3j expð
2
iH3 R3j Þ ;

j

3.4.6. The case of n ¼ 1 (Coulombic lattice energy) As taken above, the limit of the reciprocal-lattice h ¼ 0 term of S0 ðn; RÞ or S0 ðn; 0Þ existed only if n was greater than 3. The corresponding contributions to Vðn; Rj Þ were terms (5) and (9) of Section 3.4.5. To extend the method to n ¼ 1 we will show in this section that these h ¼ 0 terms vanish if conditions of unit-cell neutrality and zero dipole moment are satisfied.

which vanish if the unit P cell has no dipole moment in the polar direction, that is, if qj R3j ¼ 0. Since the second derivative of the denominator is a constant, the desired limit is zero under the specified conditions. Now the polar direction can be chosen arbitrarily, so the unit cell must not have a dipole moment in any direction for the limit of the numerator to be zero. Thus we have the formula for the Coulombic lattice sum

453

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING Vð1; Rj Þ ¼ ½1=2
ð1=2Þ

P P0 j

Qjk

k

P

R Sn ¼ ½1=2Vd
ðn=2Þ jXj
n

jRk þ XðdÞ
Rj j
1

d

 ½PðXÞ
PðXÞðXÞ
ðn=2;
w2 jXj2 Þ dX R þ ½1=2Vd
ðn=2Þ jXj
n


ð1=2;
w jRk þ XðdÞ
Rj j2 Þ P þ ½1=2
ð1=2ÞVd
1

1=2 jHðhÞj
2 h PP 
ð1=2;
w
2 jHðhÞj2 Þ Qjk 2

j

 ½PðXÞ
PðXÞðXÞðn=2;
w2 jXj2 Þ dX:

k

The first integral is shown only for a consistent representation; actually it will be reconverted to a sum and evaluated in direct space. The first part of the second integral will be evaluated with Parseval’s theorem and the second part in the limit as jXj approaches zero:

 exp½2
iHðhÞ  ðRk
Rj Þ P
½1=
ð1=2Þ
1=2 w q2j ; j

R ½1=2Vd
ðn=2Þ FT3 ½PðXÞ

which holds on conditions that the unit cell be electrically neutral and have no dipole moment. If the unit cell has a dipole moment, the limiting value discussed above depends on the direction of H. For methods of obtaining the Coulombic lattice sum where the unit cell does have a dipole moment, the reader is referred to the literature (DeWette & Schacher, 1964; Cummins et al., 1976; Bertaut, 1978; Massidda, 1978).

 FT3 ½jXj
n ðn=2;
w2 jXj2 Þ dH
lim ½1=2Vd
ðn=2Þ½Pð0ÞjXj
n ðn=2;
w2 jXj2 Þ: X!0

The first Fourier transform (of the Patterson function) is the set of amplitudes of the structure factors and the second Fourier transform has already been discussed above; the method for obtaining the limit (for n equal to or greater than 1) was also discussed above. The result obtained is

3.4.7. The cases of n ¼ 2 and n ¼ 3 If n ¼ 2 the denominator considered for the limit in the preceding section is linear in |H| so that only one differentiation is needed to obtain P the limit by L’Hospital’s method. Since a term of the type qj expð2
iH  Rj Þ is always a factor, the requirement that the unit cell have no dipole moment canPbe relaxed. For n ¼ 2 the zero-charge condition is still required: qj ¼ 0. When n ¼ 3 the expression becomes determinate and no differentiation is required to obtain a limit. In addition, factoring the Qjk sums the only remaining into qj sums is not necessary so that PP requirement for this term to be zero is Qjk ¼ 0, which is a further relaxation beyond the requirement of cell neutrality.

R ½1=2Vd
ðn=2Þ
n
ð3=2Þ jF½HðhÞj2 jHjn
3 
½ð
n=2Þ þ ð3=2Þ;
w
2 jHj2  dH
½1=2Vd
ðn=2ÞjFð0Þj2 2
n=2 wn n
1 : The integral can be converted into a sum, since jF½HðhÞj is nonzero only at the reciprocal-lattice points: ½1=2Vd
ðn=2Þ
n
ð3=2Þ

P

jF½HðhÞj2 jHðhÞjn
3

h


½ð
n=2Þ þ ð3=2Þ;
w
2 jHðhÞj2 : The term with HðhÞ ¼ 0 is evaluated in the limit, for n greater than 3, as

3.4.8. Derivation of the accelerated convergence formula via the Patterson function The structure factor with generalized coefficients qj is defined by F½HðhÞ ¼

P

½
ðn=2Þ
1 Vd
1
n=2 wn
3 ðn
3Þ
1 jFð0Þj2 : PP qj qk, this term is identical with the third Since jFð0Þj2 ¼ term of Vðn; Rj Þ as derived earlier. The case of n ¼ 1 is handled in the same way as previously discussed, where the limit of this term is zero provided the unit cell has no net charge or dipole moment.

qj exp½2
iHðhÞ  Rj :

j

The corresponding Patterson function is defined by PðXÞ ¼ Vd
1

P

jF½HðhÞj2 exp½2
iHðhÞ  X:

3.4.9. Evaluation of the incomplete gamma function

h

The incomplete gamma function may be expressed in terms of commonly available functions such as the exponential integral and the complement of the error function. The definition of the exponential integral is

The physical interpretation of the Patterson function is that it is nonzero only at the intersite vector points Rk þ XðhÞ
Rj. If the origin point is removed, the lattice sum may be expressed as an integral over the Patterson function. This origin point in the Patterson function corresponds to intersite vectors with j ¼ k and HðhÞ ¼ 0:

E1 ðx2 Þ ¼

R1

t
1 expð
tÞ dt ¼
ð0; x2 Þ:

x2

R Sn ¼ ð1=2Vd Þ jXj
n ½PðXÞ
PðXÞðXÞ dX:

The definition of the complement of the error function is erfcðxÞ ¼

Using the incomplete gamma function as a convergence function, this formula expands into two integrals

R1 x

454

expð
t2 Þ dt ¼

1=2
ð1=2; x2 Þ:

3.4. ACCELERATED CONVERGENCE TREATMENT OF R
N LATTICE SUMS P Numerical approximations to these functions are given, for qj sin½2
HðhÞ  Rj : BðhÞ ¼ example, by Hastings (1955). The recursion formula for the j incomplete gamma function (Davis, 1972)
ðn þ 1; x2 Þ ¼ n
ðn; x2 Þ þ x2n expð
x2 Þ

The formulae below describe Vðn; Rj Þ in terms of T0 ; T1 and T2 ; the distance jRk þ XðdÞ
Rj j is simply represented by Rjkd.

may be used to obtain working formulae starting from the special values of
ð0; x2 Þ and
ð1=2; x2 Þ which are defined above. Also we note that
ð1; x2 Þ ¼ expð
x2 Þ.

Vð1; Rj Þ ¼ ð1=2Þ

P P0 j

qj qk

k

P

þ ð1=2
Vd Þ 3.4.10. Summation over the asymmetric unit and elimination of intramolecular energy terms Let us consider the case where the unit cell contains Z molecules which are related by Z symmetry operations, and it is desired to include only intermolecular distances in the summation. In the direct sum (1) the indices j and k will then run only over the asymmetric unit, and all terms with d ¼ 0 are eliminated. The calculated energy refers then to one molecule (or mole) rather than to one unit cell. The correction term (2) also refers to one molecule according to the range of j and k. Since the reciprocallattice sum refers to the entire unit cell, terms (3) and (4) need to be divided by Z to refer the energy to one molecule. Both the direct and reciprocal sums must be corrected for the elimination of intramolecular terms. Using the convergence function WðRÞ, we have Vðn; Rj Þ ¼

Vð2; Rj Þ ¼ ð1=2Þ

j

k

þ ð
=2Vd Þ

T2 ðhÞjHðhÞj
2 expð
b2 Þ
wT0

h6¼0

Qjk

P

P P0 j

P

T2 ðhÞjHðhÞj
1 erfcðbÞ
ð
=2Þw2 T0

Qjk

P

2 R
2 jkd expð
a Þ

d

P

k

þ ð
=Vd Þ

R
1 jkd erfcðaÞ

d

h6¼0

Vð3; Rj Þ ¼ ð1=2Þ


1=2 R
3 expð
a2 Þ jkd ½erfcðaÞ þ 2a


d

T2 ðhÞE1 ðb2 Þ
ð2
=3Þw3 T0

h6¼0

Vð4; Rj Þ ¼ ð1=2Þ

P P0 j

k

þ ð
5=2 =Vd Þ

Qjk P

P

2 2 R
4 jkd ð1 þ a Þ expð
a Þ

d

T2 ðhÞjHðhÞj

h6¼0

 ½

erfcðbÞ þ b
1 expð
b2 Þ þ ð
2 =Vd ÞwT1
ð
2 =4Þw4 T0 P P0 P Vð5; Rj Þ ¼ ð1=2Þ Qjk R
5 jkd 1=2

P

P jRj
n WðRÞ þ jRj
n WðRÞ inter intra P
n P þ jRj ½1
WðRÞ þ jRj
n ½1
WðRÞ: inter

P P0

P

j

k

d
1=2

 ½erfcðaÞ þ 2
að1 þ 2a2 =3Þ expð
a2 Þ P þ ð2
3 =3Vd Þ T2 ðhÞjHðhÞj2

intra

As mentioned above, the second summation term, which is the intramolecular term in direct space, is simply left out of the calculation. When using the accelerated-convergence method the third and fourth summation terms are always obtained, evaluated in reciprocal space. The undesired inclusion of the intramolecular term (fourth term above) in the reciprocal-space sum may be compensated for by explicit subtraction of this term from the sum.

h6¼0

 ½b


2

expð
b2 Þ
E1 ðb2 Þ

þ ð2
2 =3Vd Þw2 T1
ð4
2 =15Þw5 T0 P P0 P 2 4 2 Vð6; Rj Þ ¼ ð1=2Þ Qjk R
6 jkd ½1 þ a þ ða =2Þ expð
a Þ j

k

9=2

þ ð
=3Vd Þ

P

d

T2 ðhÞjHðhÞj3

h6¼0

 ½
1=2 erfcðbÞ þ ½ð1=2b3 Þ
ð1=bÞ expð
b2 Þ 3.4.11. Reference formulae for particular values of n 2
w2 jRk þ
Rj j2 P and In this section let a2 ¼P PXðdÞ P b ¼ 2
2 2 Qjk ¼
w jHðhÞj P P. Let T0 ¼ Qjj ¼ qj ; T1 ¼ T0 þ 2 jP > k Qjk . If the geometric mean combining law holds, T1 ¼ ð qj Þ2 ; let

þ ð
3 =6Vd Þw3 T1
ð
3 =12Þw6 T0 P P0 P Vð7; Rj Þ ¼ ð1=2Þ Qjk R
7 jkd j

j

T2 ðhÞ ¼

PP

Qjk exp½2
iHðhÞ  ðRk
Rj Þ PP ¼ T0 þ 2 Qjk cos½2
HðhÞ  ðRk
Rj Þ: j

k

d

 ½erfcðaÞ þ 2

1=2 a½1 þ ð2=3Þa2 þ ð4=15Þa4   expð
a2 Þ P þ ð2
5 =15Vd Þ T2 ðhÞjHðhÞj4

k

h6¼0

 ½ð
b

j>k


2


4

þ b Þ expð
b2 Þ þ E1 ðb2 Þ

þ ð2
3 =15Vd Þw4 T1
ð8
3 =105Þw7 T0 P P0 P Vð8; Rj Þ ¼ ð1=2Þ Qjk R
8 jkd

Then

j

2   P   2 T2 ðhÞ ¼ jFðhÞj ¼  qj exp½2
iHðhÞ  Rj  ¼ AðhÞ2 þ BðhÞ2 ;  j 

k 2

d 4

 ½1 þ a þ ða =2Þ þ ða6 =6Þ expð
a2 Þ P þ ð2
13=2 =45Vd Þ T2 ðhÞjHðhÞj5 h6¼0 1=2

 ½

erfcðbÞ þ ½ð1=bÞ
ð1=2b3 Þ þ ð3=4b5 Þ

where AðhÞ ¼

P

 expð
b2 Þ þ ð
4 =30Vd Þw5 T1
ð
4 =48Þw8 T0

qj cos½2
HðhÞ  Rj 

j

and

455

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING Table 3.4.12.1. Accelerated-convergence results for the Coulombic sum (n = 1) ˚ ): the direct sum plus the constant term of sodium chloride (kJ mol
1, A Limit 6.0

w = 0.1

w = 0.15

w = 0.2

w = 0.3

w = 0.4


779.087


838.145


860.393


924.275


1125.372
1125.372

8.0


818.549


860.194


863.764


924.282

10.0


865.323


862.818


863.811


924.282

12.0 14.0


861.183
862.717


862.824
862.828


863.811


862.828

16.0


862.792

18.0


862.810

20.0


862.825

Table 3.4.12.3. Accelerated-convergence results for the dispersion sum (n = 6) ˚ ); the figures shown are the direct-lattice of crystalline benzene (kJ mol
1, A sum plus the two constant terms Limit

˚ ) for the Coulombic Table 3.4.12.2. The reciprocal-lattice results (kJ mol
1, A sum (n = 1) of sodium chloride Limit

w = 0.1

w = 0.15

w = 0.2

w = 0.3

w = 0.4

0.0


277.872


416.806


555.742


833.613


1111.483

0.4

0.000

0.003

0.5 0.6

0.986

0.003

0.986

0.7

j

Qjk

k

P

w = 0.4

6.0


73.452


77.761


79.651


61.866

76.645

8.0


79.029


80.374


80.256


61.870

76.645

10.0 12.0


80.217
80.527


80.571
80.585


80.265
80.265


61.870

14.0


80.578


80.585

16.0


80.588

18.0


80.589

20.0


80.589

w = 0.1

w = 0.15

w = 0.2

w = 0.3

261.042


5.547


16.706


32.326


43.681

80.947

61.451 61.457

261.042 262.542

0.3

0.000


0.004


0.321


16.792


117.106

0.000


0.004

61.457

262.542

0.4 0.5


0.324
0.324


18.656
18.716


152.651
155.940

61.451

w = 0.4

262.547

0.6


18.719


157.102

262.547

0.7


18.719


157.227

R
9 jkd

0.8


157.233

0.9


157.234

1.0


157.234

d

nation spheres is þ6905:928=a which actually has the wrong sign for a stable crystal. The third coordination sphere again makes a negative contribution. Each ion is surrounded by eight ions of opposite sign at a distance of a=31=2. The energy contribution is
ð1=2Þð16Þð31=2 =aÞð1389:3654Þ ¼
19251:612=a, now giving a total so far of
12345:684=a. In the fourth coordination sphere each ion is surrounded by six others of the same sign at a distance of a. The energy contribution is +(1/2)(12)(1/a)(1389.3654) = +8336.19/a to yield a total of
4009:491=a. It is seen immediately by examining the numbers that the Coulombic sum is converging very slowly in direct space. Madelung (1918) devised a method for accurate evaluation of the sodium chloride lattice sum. However, his method is not generally applicable for more complex lattice structures. Evjen (1932) emphasized the importance of summing over a neutral domain, and replaced the sum with an integral outside of the first few shells of nearest neighbours. But the method of Ewald remained as the only completely general and accurate method of evaluating the Coulombic sum for a general lattice. Although it was derived in a somewhat different way, Ewald’s method is equivalent to accelerated convergence for the special case of n ¼ 1. In the reciprocal lattice of sodium chloride only points with indices (hkl) all even or all odd are permitted by the face-centred symmetry. The reciprocal cell has edge length 1=a and the reciprocal-axis directions coincide with the direct-lattice axis directions. The closest points to the origin are the eight (111) forms at a distance of ð1=aÞ=31=2. For sodium chloride this distance is ˚
1. 0.3078 A Table 3.4.12.1 shows the effect of convergence acceleration on the direct-space portion of the ðn ¼ 1ÞPsum for the sodium chloride structure. The constant term
w q2j is included in the values given. This constant term is always large if w is not zero; for instance, when w ¼ 0:1 this term is
277:872 (Table 3.4.12.2). For w ¼ 0:1 the reciprocal-lattice sum is zero to six figures. Thus, only the direct sum (plus the constant term) is needed, evaluated ˚ in direct space, to obtain six-figure accuracy. As out to 20 A shown in Table 3.4.2.1 above, the same summation effort without the use of accelerated convergence gave 8% error, or only

h6¼0

 ½ðb
2
b
4 þ 2b
6 Þ expð
b2 Þ
E1 ðb2 Þ þ ð8
4 =315Vd Þw6 T1
ð16
4 =945Þw9 T0 P P0 P Vð10; Rj Þ ¼ ð1=2Þ Qjk R
10 jkd k 2

w = 0.3

˚ ) for the dispersion Table 3.4.12.4. The reciprocal-lattice results (kJ mol
1, A sum (n = 6) of crystalline benzene

 ½erfcðaÞ þ 2

1=2 a½1 þ ð2=3Þa2 þ ð4=15Þa4 þ ð8=105Þa6  expð
a2 Þ P þ ð4
7 =315Vd Þ T2 ðhÞjHðhÞj6

j

w = 0.2

0.0

0.9

P P0

w = 0.15

Limit

0.8

Vð9; Rj Þ ¼ ð1=2Þ

w = 0.1

d 4

 ½1 þ a þ ða =2Þ þ ða6 =6Þ þ ða8 =24Þ expð
a2 Þ P þ ð
17=2 =315Vd Þ T2 ðhÞjHðhÞj7 h6¼0

 ½
1=2 erfcðbÞ þ ½ð
1=bÞ þ ð1=2b3 Þ
ð3=4b5 Þ þ ð15=8b7 Þ expð
b2 Þ þ ð
5 =168Vd Þw7 T1
ð
5 =240Þw10 T0 :

3.4.12. Numerical illustrations Consider the case of the sodium chloride crystal structure (a facecentred cubic structure) as a simple example for evaluation of the Coulombic sum. The sodium ion can be taken at the origin, and the chloride ion halfway along an edge of the unit cell. The results can easily be generalized for this structure type by using the unitcell edge length, a, as a scaling constant. First, consider the nearest neighbours. Each sodium and each chloride ion is surrounded by six ions of opposite sign at a distance of a=2. The Coulombic energy for the first coordination sphere is
ð1=2Þð12Þð2=aÞð1389:3654Þ ¼
16672:385=a kJ mol
1. Table 3.4.2.1 shows that the converged value of the lattice energy is
4855:979=a. Thus the nearest-neighbour energy is over three times more negative than the total lattice energy. In the second coordination sphere each ion is surrounded by 12 similar ions at a distance of a=21=2. The energy contribution of the second sphere is þð1=2Þð24Þð21=2 =aÞð1389:3654Þ ¼ þ23578:313=a. Thus, major cancellation occurs and the net energy for the first two coordi-

456

3.4. ACCELERATED CONVERGENCE TREATMENT OF R
N LATTICE SUMS optimum value of w for the situation of a particular crystal structure, program and computer.

Table 3.4.12.5. Approximate time (s) required to evaluate the dispersion sum (n = 6) for crystalline benzene within 0.001 kJ mol
1 truncation error w

Direct terms

Reciprocal terms

0.0

˚ summation limit) (not yet converged at 20 A

0.1

15904

0

Time, direct 77

Time, reciprocal

Total time >107

0

References

77

0.15

4718

34

23

6

29

0.2

2631

78

13

14

27

0.3 0.4

1313 524

246 804

7 3

46 149

53 152

Arfken, G. (1970). Mathematical Methods for Physicists, 2nd ed. New York: Academic Press. Bertaut, E. F. (1952). L’e´nergie e´lectrostatique de re´seaux ioniques. J. Phys. (Paris), 13, 499–505. Bertaut, E. F. (1978). The equivalent charge concept and its application to the electrostatic energy of charges and multipoles. J. Phys. (Paris), 39, 1331–1348. Busing, W. R. (1981). WMIN, a computer program to model molecules and crystals in terms of potential energy functions. Oak Ridge National Laboratory Report ORNL-5747. Oak Ridge, Tennessee 37830, USA. Cummins, P. G., Dunmur, D. A., Munn, R. W. & Newham, R. J. (1976). Applications of the Ewald method. I. Calculation of multipole lattice sums. Acta Cryst. A32, 847–853. Davis, P. J. (1972). Gamma function and related functions. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz & I. A. Stegun, pp. 260–262. London, New York: John Wiley. [Reprint, with corrections of 1964 Natl Bur. Stand. publication.] DeWette, F. W. & Schacher, G. E. (1964). Internal field in general dipole lattices. Phys. Rev. 137, A78–A91. Evjen, H. M. (1932). The stability of certain heteropolar crystals. Phys. Rev. 39, 675–694. Ewald, P. P. (1921). Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. (Leipzig), 64, 253–287. Fortuin, C. M. (1977). Note on the calculation of electrostatic lattice potentials. Physica (Utrecht), 86A, 574–586. Glasser, M. L. & Zucker, I. J. (1980). Lattice sums. Theor. Chem. Adv. Perspect. 5, 67–139. Hastings, C. Jr (1955). Approximations for digital computers. New Jersey: Princeton University Press. Madelung, E. (1918). Das elektrische Feld in Systemen von regelma¨ssig angeordneten Punktladungen. Phys. Z. 19, 524–532. Massidda, V. (1978). Electrostatic energy in ionic crystals by the planewise summation method. Physica (Utrecht), 95B, 317–334. Nijboer, B. R. A. & DeWette, F. W. (1957). On the calculation of lattice sums. Physica (Utrecht), 23, 309–321. Pietila, L.-O. & Rasmussen, K. (1984). A program for calculation of crystal conformations of flexible molecules using convergence acceleration. J. Comput. Chem. 5, 252–260. Widder, D. V. (1961). Advanced Calculus, 2nd ed. New York: PrenticeHall. Williams, D. E. (1971). Accelerated convergence of crystal lattice potential sums. Acta Cryst. A27, 452–455. Williams, D. E. (1984). PCK83, a crystal molecular packing analysis program. Quantum Chemistry Program Exchange, Department of Chemistry, Indiana University, Bloomington, Indiana 47405, USA. Williams, D. E. (1989a). Accelerated convergence treatment of R
n lattice sums. Crystallogr. Rev. 2, 3–23. Williams, D. E. (1989b). Accelerated convergence treatment of R
n lattice sums. Corrections. Crystallogr. Rev. 2, 163–166.

slightly better than one-figure accuracy. The acceleratedconvergence technique therefore yielded nearly five orders of magnitude improvement in accuracy, even without evaluation of the reciprocal-lattice sum. The column showing w ¼ 0:15 shows an example of how the reciprocal-lattice sum can also be neglected if lower accuracy is required. Table 3.4.12.2 shows that the reciprocal-lattice sum is still only 0.003. But now the direct-lattice sum only needs to be ˚ , with further savings in calculation effort. evaluated out to 14 A For w values larger than 0.15 the reciprocal sum is needed. For ˚
1 to obtain sixw ¼ 0:4 this sum must be evaluated out to 0.8 A figure accuracy. Table 3.4.12.3 illustrates an
1 application for the ðn ¼ 6Þ ˚ five figures of accuracy can be dispersion sum. When w ¼ 0:1 A obtained without consideration of the reciprocal sum. The direct ˚ . If w ¼ 0:15, better than four-figure sum is required out to 18 A accuracy can still be obtained without evaluating the reciprocallattice sum. In this case, the direct lattice needs to be summed ˚ , and there is a saving of an order of magnitude in the only to 12 A length of the calculation. As with the Coulombic sum, if w is greater than 0.15 the reciprocal-lattice summation is needed; Table 3.4.12.4 shows the values. The time required to obtain a lattice sum of given accuracy will vary depending on the particular structure considered and of course on the computer and program which are used. An example of timing for the benzene dispersion sum is given in Table 3.4.12.5 for the PCK83 program (Williams, 1984) running on a VAX-11/750 computer. In this particular case direct terms were evaluated at a rate of about 200 terms s
1 and reciprocal terms, being a sum themselves, were evaluated at a slower rate of about 5 terms s
1. Table 3.4.12.5 shows the time required for evaluation of the dispersion sum using various values of the convergence constant, w. The timing figures show that there is an optimum choice for w; for the PCK83 program the optimum value indicated is 0.15– ˚
1. In the program of Pietila & Rasmussen (1984) values in 0.2 A ˚
1 are also suggested. For the WMIN the range 0.15–0.2 A ˚
1 is program (Busing, 1981) a slightly higher value of 0.25 A suggested. Trial calculations can be used to determine the

457

references

International Tables for Crystallography (2010). Vol. B, Chapter 3.5, pp. 458–481.

3.5. Extensions of the Ewald method for Coulomb interactions in crystals By T. A. Darden

crystallization, the lowest-free-energy polymorph at given temperature and pressure may not be the likeliest to form. The kinetics of growth of microcrystals may largely determine which low-energy polymorph appears (Dunitz, 2003). However, it is generally agreed that accurate calculation of the relative free energy of polymorphs is a prerequisite for predicting crystal structures. To assess progress towards solving this latter problem, a series of blind tests of crystal-structure prediction has been undertaken (Day, Motherwell, Ammon et al., 2005). The results of these tests have highlighted the need for continued improvements in sampling methods and intermolecular energy potentials. Since extensive sampling of the crystal-structure parameters is necessary [between 104 and 105 starting structures, each followed by parameter minimizations (Price & Price, 2005)], there is a tradeoff in the computational cost versus accuracy of the intermolecular energy functions used. Calculating the work of transforming between polymorphs is yet more ambitious in terms of sampling. Consequently empirical force fields are likely to be needed for the near term at least. In the remainder of the introduction we outline some of the approaches to empirical potentials used in the calculation of the lattice energy, and then, motivated by these developments, discuss techniques for efficient summation of the electrostatic and other slow-decaying interaction terms that occur in these potential functions. Methodological developments in the intermolecular force fields used in crystal-structure prediction from early times to the present state of the art have been reviewed (Price & Price, 2005). Until recently, these force fields were made up of atom–atom interactions. The earliest involved only repulsion and dispersion, usually in the ‘exp-6’ form

3.5.1. Introduction High-precision single-crystal X-ray structural analysis of small organic molecules, yielding the space group, the unit-cell parameters and the fractional coordinates of the atoms making up the molecule(s) in the asymmetric unit, has become a routine matter as long as crystals of sufficient quality can be obtained. The thermodynamic stability of the crystal, as described by the enthalpy of sublimation
Hsub , can also be determined experimentally (although not always to high precision). Theoretical models for calculating intermolecular interaction energies can be used to connect the crystal structure to the molar enthalpy of sublimation using the relationship
Hsub ð0 KÞ ¼
Elattice ; where the lattice or packing energy Elattice is the total (molar) intermolecular interaction energy between all the molecules in the crystal, which are treated as rigid entities with zero-point energies of intra- and intermolecular vibrations neglected. Connection to experimentally accessible heats of sublimation at higher temperatures involves thermodynamic corrections. Methods for calculating thermodynamic quantities of solids are discussed in Gavezzotti (2002a) and (in more detail) in Frenkel & Smit (2002). Thus, given a parameterized intermolecular potential-energy function, or if computationally affordable a first-principles approach such as density-functional theory (or preferably, when it becomes feasible for crystals, a good-quality post-Hartree–Fock potential-energy surface that describes dispersion interactions), one can sum the intermolecular energies to obtain the lattice energy as a function of the above parameters defining the crystal structure. If such an energy function is used together with a method for systematic search of the crystal-structure parameters, one could in principle predict the minimum-lattice-energy crystal structure for a rigid organic molecule. To extend this approach to flexible molecules one would need to minimize the sum of the intramolecular energy plus the lattice energy. If the experimental crystal structure corresponds to the thermodynamic minimumenergy structure (i.e. it is not a metastable state determined by crystal-growth kinetics), one could in principle predict the experimental crystal structure of an organic compound through this minimization protocol. Moreover, one could ideally predict the additional metastable forms of the crystal. Prediction of the structure of crystals of an organic molecule from its molecular structure is a difficult problem that has been compared to the protein folding problem (Dunitz, 2003; Dunitz & Scheraga, 2004). Like the protein folding problem, a solution of the crystal prediction problem has significant practical ramifications. A compound is often polymorphic, that is it has more than one crystal structure, and it may be difficult to characterize the conditions under which a particular crystal structure is formed (Dunitz & Bernstein, 1995). Polymorphs may have very different physical properties. An obvious example is diamond versus graphite, but other commercially important examples include food additives, various solid forms of explosives and the bioavailability of various forms of a drug such as ritonavir (Chemburkar et al., 2000). A method for predicting the possible crystal structures of the compound, and ideally for predicting the dominant crystal structure given the experimental conditions, would thus be very valuable. Note that due to the subtleties of Copyright © 2010 International Union of Crystallography

MN MN U MN ¼ Urep þ Udisp ¼

P

Aij expð
Bij rij Þ
Ci;j r
6 ij ;

i2M;j2N

where UMN denotes the intermolecular potential energy between molecules M and N and rij is the distance between atoms i 2 M and j 2 N. Sometimes the exponential form in the above equation is replaced by a simpler power law, as in the Lennard–Jones potential. As was pointed out by Dunitz (2003), in comparison with more sophisticated force fields, this repulsion–dispersion form readily allows analysis of the significance of particular atom–atom interactions, since the interactions are short-ranged and thus can be localized. That is, the r
6 form of the attractive dispersion energy means that interaction energies are halved for every 12% increase in distance. In contrast, introduction of longrange Coulombic interactions not only entails subtleties in lattice summation (the subject of this contribution), but greatly complicates the assignment of ‘key’ atom–atom interactions. Gavezzotti and Fillipini systematically explored the use of the exp-6 potential in fitting organic crystal structures with and without hydrogen-bond interactions (Gavezzotti & Fillipini, 1994). They were surprisingly successful in accounting for weak hydrogen bonding in this way, but selective use of point charges improved the directionality of the potential. Earlier, Williams derived exp-6 parameters for the atoms C, H, N, O, Cl, F and polar H for use in organic crystal structures, but found it necessary (Williams & Cox, 1984) to supplement these with selected point charges, both atomic and at off-atom sites. Price and

458

3.5. EXTENSIONS OF THE EWALD METHOD co-workers have used a more sophisticated electrostatic model based on distributed multipole analysis (DMA) with multipoles up to hexadecapole order positioned at atom centres (Stone, 1996). A similar approach has been adopted by Mooji et al., with the addition of dipole polarizabilities to account for intermolecular induction energy (Mooji et al., 1999). The AMOEBA force field has similar capabilities (Ren & Ponder, 2003). Recently, a systematic study (Day et al., 2004; Day, Motherwell & Jones, 2005) of the effect of the electrostatic model on crystalstructure prediction for rigid organic molecules was carried out. The exp-6 parameters from the W99 force field (Williams, 2001) were used in conjunction with point charges derived either from a bond-increment model or electrostatic potential (ESP) fit, or with a DMA model as used by Price and co-workers. Importantly, they found a systematic and significant improvement in prediction success going from the simple bond-increment model to the ESP charges to point multipoles. Similar results were found by Broderson et al. (2003). This is a positive indicator for the use of theory in attacking this difficult prediction problem. On the other hand, Broderson et al. found that more sophisticated electrostatic models did not help in predicting the packing of flexible molecules. As discussed by Price & Price (2005), crystal-packing energies of flexible molecules are more problematic for at least two reasons. First the intramolecular energy must be fully compatible with the intermolecular energy in order that energy deformation from the gas-phase minimum be on the same scale as the concomitant gain in packing energy. Secondly, the conformational dependence of the electrostatic model must be accurately accounted for, or any gains from a more sophisticated model are immediately negated. Existing force fields are unable to meet these challenges, which may explain the lower success rate for predicting crystal structures of flexible molecules. Price has developed a methodology for flexible molecules wherein intramolecular conformations are systematically explored. For each conformer studied, the gas-phase intramolecular energy and DMA electrostatic model are calculated using quantum-chemical methods. The optimal packing of that conformer is then obtained by treating it as a rigid molecule, and the resulting minimum lattice energy is added to its intramolecular energy to give its total energy. The total energy for that conformer is then compared with that of the other conformers to predict the optimal conformation and packing of the flexible molecule. This approach has proved successful in a number of cases, for example with the successful prediction of a novel second polymorph of 1-hydroxy-7-azabenzotriazole (Nowell et al., 2006). The above electrostatic models are all derived from gas-phase quantum-chemical calculations. A different line of development has been the parameterization of electrostatic models from condensed-phase experimental or theoretical electron densities, which should better represent the polarization due to the crystal environment. This is accomplished using aspherical multipole refinement of experimental X-ray charge densities (Destro et al., 2000) or of theoretical charge densities from quantum-chemical calculations carried out under the space-group symmetry (Dovesi et al., 2005). Crystal structure analysis is usually carried out assuming the independent-atom model (IAM), where the individual atomic densities are modelled as those of the spherically symmetric isolated atoms, with deviations in the experimental density being fitted to anisotropic temperature factors modelling the thermal motion of the nuclei. However, the IAM breaks down for data of sufficiently high quality, and the residual or deformation density is then fitted by a linear combination of atom-centred density basis elements resembling the well known Slater atomic basis sets from quantum chemistry (i.e. solid harmonics times a decaying exponential in the distance r from the atom). Technical details are provided in Coppens (1997) and Chapter 1.2 of this volume. The theoretical density can be similarly approached as a sum of pro-molecule density (superposition of isolated atomic densities in the molecular geometry) plus

deformation density modelled by a sum of atomic Slater-like density elements. Until recently, the theoretical densities were thought to be of lower quality than the experimental densities, but continued improvements in the level of theory that can be applied have resulted in much closer agreement (Coppens & Volkov, 2004). Coppens and co-workers as well as others (Spackman et al., 1988) have applied these modelled densities in studies of intermolecular electrostatic energies. The Slater-like functions are typically difficult to work with in this context, so multipole approximations are used except at close range, where numerical integration is used instead (Volkov et al., 2004). A similar density-based approach using gas-phase theoretical density was developed previously (Gavezzotti, 2002b). Gavezzotti splits the density into numerical ‘pixels’ (small cubes of volume) within which the density is considered constant, allowing in principle exact numerical calculation of intermolecular electrostatic interactions. He sums the pixel–pixel interactions discretely, using a hierarchical approach to increase efficiency. The electrostatic interactions are supplemented by induction and repulsion terms, also modelled by pixel-based interactions. These developments, in particular the large size of the positive and negative components of the electrostatic interactions treated in terms of actual electron density, have caused a reassessment of the idea that intermolecular interactions can be accurately decomposed even in principle into localized atom–atom interactions (Dunitz & Gavezzotti, 2005). Cisneros et al. have also begun implementing an intermolecular potential based on the explicit modelling of gas-phase theoretical electron density, using the methodology of auxiliary basis set fitting (Cisneros et al., 2005). The density is expanded into atom and off-atom sites using Gaussian density functions similar to those used as atomic basis sets in quantum chemical codes. More recently, other components of the potential such as induction, charge transfer and non-isotropic site–site repulsion based on density overlap have been implemented (Piquemal et al., 2006) and efficient calculation of terms in periodic boundary conditions has also been implemented (Cisneros et al., 2006), generalizing previous developments for point multipoles (Sagui et al., 2004). Summarizing the above developments, it appears that accurate evaluation of crystal lattice energies necessitates the use of complex electrostatic interactions beyond the spherical-atom approximation, and ultimately including an account of penetration effects through explicit modelling of electron density. The electrostatic lattice energies are differences between large positive and negative contributions that are long-ranged, necessitating an accurate treatment of their lattice summation. Dispersion interactions, modelled by higher-order inverse powers of distance, are short-ranged but their calculation can be greatly accelerated through lattice-summation techniques. Ewald summation in ideal infinite crystals, for interactions that depend on inverse powers of site–site distance, is covered in Chapter 3.4 of this volume, where there is also a numerical demonstration of the advantages of the Ewald approach over direct numerical summation. In this contribution we focus on the lattice-summation problem for large but finite crystals, highlighting the role of summation order and leading to a derivation of the surface term and polarization response. We also consider extensions to Coulombic interactions between Gaussian density elements, both spherical and Hermite (multipolar), and methods to further accelerate the lattice summation to (almost) linear scaling. These developments are organized into three sections. The first section treats lattice summation of point charges. We begin with definitions and an explanation of the origin of the conditional convergence problem for these sums, anticipating the nature of the results to follow. Following that we provide for the benefit of less mathematically oriented readers the standard physics-based derivation of the Ewald sum for point charges, minus any

459

3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING accounting for the order of convergence or resulting surface term. This is followed by a more careful derivation focused on the order of summation, including results for higher-order inverse powers as well as the Coulomb interactions, finishing with a discussion of the surface term, the polarization response to an external dielectric and the pressure calculation. Although pitched at a higher mathematical level, we have tried to provide many details in the derivations unavailable elsewhere for the benefit of the reader. The alert reader will note that the key identity, equation (3.5.2.19), that separates 1/r n into short-ranged and long-ranged components, is the same as that used Chapter 3.4 of this volume, highlighting the essential similarity of our approach. In the second section we generalize these point-charge results to lattice sums of interacting spherical Gaussian and Hermite Gaussian-based densities. The third and final section provides a discussion of two methods that greatly increase the numerical efficiency of these lattice sums in the case of large unit cells by utilizing the fast Fourier transform. Again we have tried to provide mathematical details not available elsewhere.

C¼

[

Un :

n

A point charge qi in U at position ri interacts with other unitcell charges qj ; j 6¼ i at positions rj as well as with all of their periodic images qjn at positions rjn ¼ rj þ n for all lattice vectors n. It also interacts with its own periodic images qin at rin for all nonzero lattice vectors n, that is all n ¼ n1 a1 þ n2 a2 þ n3 a3 with ðn1 ; n2 ; n3 Þ 6¼ ð0; 0; 0Þ. Suppose the unit cell U is made up of N point charges q1 ; . . . ; qN at positions r1 ; . . . ; rN. Then the electrostatic energy of U, given by the sum of the Coulombic interactions of charges in U with each other and with the rest of C, is written as

EðrfNg Þ ¼ Eðr1 ; . . . ; rN Þ N X N X0 X ¼ 12 n

i¼1 j¼1

qi qj ; jri
rj
nj

ð3:5:2:3Þ

3.5.2. Lattice sums of point charges We begin by discussing an idealized infinite crystal C, made up of point charges.

where here and below rfNg denotes the set of points fr1 ; . . . ; rN g and where the outer sum on the right is over the general lattice vectors n, the prime indicating that terms with i = j and n = 0 are omitted. The treatment of this infinite sum requires some care. The PN energy diverges unless the unit cell is neutral (i.e. unless i¼1 qi ¼ 0). When U is neutral, the sum, if carried out in a naive fashion, can still converge quite slowly (see Chapter 3.4). In addition, the convergence is conditional, that is the resulting energy can vary depending on the order in which the sum is carried out! To better understand these phenomena, it is helpful to re-express the above energy. Let E(U, U) denote the Coulomb interaction energy of U with itself and EðU; U n Þ; n 6¼ 0 the Coulomb interaction energy of U with U n , that is

3.5.2.1. Basic quantities The periodicity of an idealized crystal C is specified by lattice basis vectors a1, a2 and a3 , which are linearly independent as vectors and thus form a basis for the usual vector space of points in three dimensions. The conjugate reciprocal vectors a
are defined by the relations a
 a ¼ 
 (the Kronecker delta) for
;  ¼ 1; 2; 3. Thus for example a1 is given by a1 = ða2  a3 Þ=½a1  ða2  a3 Þ . General lattice vectors n are given by integral combinations of the lattice basis vectors: n ¼ n1 a1 þ n2 a2 þ n3 a3 ;

ð3:5:2:1Þ EðU; UÞ ¼

where n1 ; n2 ; n3 are integers. General reciprocal-lattice vectors m are given by integral combinations of the reciprocal-lattice basis vectors:

X qi qj jr
rj j iK

expð
r t Þ dt

Z1


1=2

for some K > 0. Although the sum can thus be evaluated directly, it can be significantly accelerated using the same Ewald directand reciprocal-space decomposition. For r 6¼ 0, using equation (3.5.2.19) we have

and so lim

1

r!0 jrj

p

½1
fp ð
1=2 jrjÞ

2 ¼ lim r!0 ðp=2Þ

Z
1=2 t

p
1

X

2 2

expð
r t Þ dt

n 0

p=2

¼

2
: pðp=2Þ

ð3:5:2:22Þ

n

Note that gp ðxÞ is a smooth function of x for x 6¼ 0, and that its derivatives are uniformly bounded for x bounded away from 0. To show this, substitute s = xt in equation (3.5.2.21) to write gp ðxÞ ¼ 2

R1

t2
p expð
x2 t2 Þ dt:

In the case that p = 6, gp ðxÞ is smooth with uniformly bounded derivatives for all x, but this is unfortunately not true for all p. However, from the above expression we can show that for p > 3, gp ðxÞ is bounded and continuous as x ! 0, whereas for 1  p  3 gp ðxÞ ! 1 as x ! 0. Thus for p > 3 we see that for any x > 0, gp ðxÞ < gp ð0Þ ¼ 2=ðp
3Þ. However for 1  p  3, for any R > 1 if R x > 0 sufficiently Rsmall expð
x2 R2 Þ > 12; thus gp ðxÞ > R R 2 1 t 2
p expð
x2 t2 Þ dt > 1 t 2
p dt  lnðRÞ and thus gp ðxÞ is unbounded as x ! 0. Despite this latter result, the integral of gp ðrÞ over 3) direct space. Three types of solids are commonly included in the class of aperiodic structures: (i) incommensurately modulated structures (IMSs), where structural parameters of any kind (coordinates, occupancies, orientations of extended molecules or subunits in a structure, atomic displacement parameters) deviate periodically from the average (‘basic’) structure and where the modulation period is incommensurate compared to that of the basic structure; (ii) composite structures (CSs), which consist of two (or more) intergrown incommensurate substructures with mutual interactions giving rise to mutual (incommensurate) modulations; and (iii) quasicrystals (QCs), which might simply be viewed as made up by n ( 2) different tiles – analogous to the elementary cell in crystals – which are arranged according to specific matching rules and which are decorated by atoms or atomic clusters. Most common are icosahedral (i-type) quasicrystals with 3D aperiodic order (3D ‘tiles’) and decagonal quasicrystals (d-phases) with 2D aperiodicity and one unique axis along which the QC is periodically ordered. The basic ‘crystallography’ of aperiodic crystals is given in Chapter 4.6 of this volume; see also the review articles by Janssen & Janner (1987) and van Smaalen (2004) (and further references therein). We only point out some of the aspects here to provide the background for a short discussion about disorder diffuse scattering in aperiodic crystals. As outlined in Section 4.6.1, a d-dimensional (dD) ideal aperiodic crystal can be defined as a dD irrational section of an nD (‘hyper’-)crystal with nD crystal symmetry. Corresponding to the section of the nD hypercrystal with the dD (d = 1, 2, 3) direct physical (= ‘external’ or ‘parallel’) space we have a projection of the (weighted) nD reciprocal hyperlattice onto the dD reciprocal physical space. The occurrence of Bragg reflections as a signature

k

hjFj2 i ¼

PP k

k0

fk fk0 f j0 ½H  ðrk
rk0 Þ


j0 ðH  rk Þj0 ðH  rk0 Þg:

ð4:2:5:95Þ

j0 ðzÞ is the zeroth order of the spherical Bessel functions and describes an atom k uniformly distributed over a shell of radius rk . In practice the molecules perform finite librations about the main orientation. The structure factor may then be found by the method of symmetry-adapted functions [see, e.g., Press (1973), Press & Hu¨ller (1973), Dolling et al. (1979), Prandl (1981, and references therein)]. hFi ¼

P k

fk 4


þ P P

i j ðH  rk ÞC ðkÞ  Y ð; ’Þ:

ð4:2:5:96Þ

 ¼


j ðzÞ is the th order of spherical Bessel functions, the coefficients C ðkÞ  characterize the angular distribution of rk and Yð; ’Þ are the spherical harmonics where jHj; ; ’ denote polar coordinates of H. The general case of an arbitrary crystal, site and molecular symmetry and the case of several symmetrically equivalent orientationally disordered molecules per unit cell are treated by Prandl (1981); an example is given by Hohlwein et al. (1986). As mentioned above, cubic plastic crystals are common and therefore mostly studied up to now. The expression for hFi may then be formulated as an expansion in cubic harmonics K ð; ’Þ: P PP  0 ðkÞ hFi ¼ fk 4
i j ðH  rk ÞC K ð; ’Þ; ð4:2:5:97Þ k





0 where C are modified expansion coefficients. Taking into account isotropic centre-of-mass translational displacements, which are not correlated with the librations, we obtain

hF 0 i ¼ hFi expf
16H 2 hU 2 ig:

ð4:2:5:98Þ

U is the mean-square translational displacement of the molecule. Correlations between translational and vibrational displacements are treated by Press et al. (1979).

526

4.2. DISORDER DIFFUSE SCATTERING OF X-RAYS AND NEUTRONS of an ordered (in at least an averaged sense) aperiodic crystal form a countable dense pattern of the projected reciprocal (hyper-)lattice vectors. Disorder phenomena, i.e. deviations from the periodicity in higher-dimensional direct (hyper-)space, are thus related to diffuse phenomena in reciprocal (hyper-)space projected down to the reciprocal physical space. Therefore only fluctuations from the aperiodic order infer diffuse scattering in its true sense, which might be hard to discern in a dense pattern of discrete (Bragg) reflections. As a consequence, in the higherdimensional description of aperiodic crystals the atoms must be replaced by ‘hyperatoms’ or ‘atomic surfaces’, which are extended along (n
d) dimensions. If, for example, an IMS exhibits a modulation in only one direction (1D IMS), we have n = 2 and a 1D atomic surface (which is a continuous modulation function extended along the 1D ‘internal’ or ‘perpendicular’ subspace). The physical subspace is spanned by 1 (and +2 ‘unaffected’) dimensions. In a second example, a decagonal QC with aperiodic order in two dimensions is described by n = 4 (plus the remaining coordinate along the unaffected periodic direction) and d = 2, i.e. by 2D atomic surfaces. The atomic surfaces are, as shown in Chapter 4.6, continuous n-dimensional objects for IMSs and CSs, and discrete ones in the case of QCs. In addition to the ‘hyperlattice’ fluctuations within the physical (parallel) subspace we have therefore an additional quality of disorder phenomena related to positions and shape and size fluctuations of the atomic surfaces within the perpendicular subspace. Therefore, one has to consider disordering effects in aperiodic crystals related to fluctuations in the (external) physical subspace as well as in the internal subspace, but one should bear in mind that the two types are often coupled. Disordering due to displacements along directions within the physical subspace are – in the context of aperiodic crystals – commonly termed ‘phononlike’, in contrast to ‘phason-like’ disorder related to displacements parallel to directions within the internal subspace. The term phason originates from a particular type of (dynamical) fluctuations of an IMS (see below).

matrix structure. On the other hand, maxima superimposed on the diffuse planes reflect the influence of the matrix structure on the chain system. The thickness of the diffuse planes depends on the degree of short- or long-range order along the unique direction (cf. Section 4.2.5.3). This might be either a consequence of faults in the surrounding matrix which interrupt the (longitudinal) coherence of the chains (Rosshirt et al., 1991) or, intrinsically, due to the degree of misfit between the periods of the mutually incommensurate structures of the host and guest structures. If this misfit becomes large, e.g. as a consequence of different thermal expansion coefficients of the two substructures, the modulation is only preserved within small domains and one observes a series of diffuse satellite planes (Weber et al., 2000). Equivalent considerations relate to composite systems made up from stacks of planar molecules (e.g. van Smaalen et al., 1998) or an intergrowth of layer-like substructures where the modulation is mainly due to a one-dimensional stacking along the normal to the layers. Correspondingly, quite extended diffuse streaks occur at positions of the satellites in reciprocal space. A further discussion of the more complicated disorder diffuse scattering of higher- (than one-) dimensionally modulated IMSs and CSs and the corresponding diffuse patterns in reciprocal space is beyond the scope of this chapter. We refer to Section 4.6.3 and the references cited therein. For examples, see also Petricek et al. (1991). 4.2.6.3. Quasicrystals As in the case of conventional crystals, there is no unique theory of diffuse scattering by quasicrystals. Chemical disorder, phonon- and phason-like displacive disorder, topological glasslike disorder and domain disorder exist. The term domain covers those with an aperiodic structure as well as periodic approximant domains, which are also known as approximant phases (cf. Section 4.6.3). Approximant phases exhibit local atomic clusters which do not differ significantly from those of related aperiodic QCs. In d-phases, disorder diffuse scattering occurs which is related to the periodic direction. Reviews of disorder diffuse scattering from quasicrystals are given, e.g., by Steurer & Frey (1998) and Estermann & Steurer (1998). Chemical disorder. Many of the quasicrystals are ternary intermetallic phases consisting of atomic clusters where two components are transition metals (TMs) with only a small difference in Z (the number of electrons). Most of them exhibit a certain amount of substitutional (chemical) disorder, in particular with respect to the distribution of the TMs. This behaviour is equally true in the approximant phases. Chemical short-range order between the TMs, if any, is largely uninvestigated because conventional X-ray diffraction and electron-microscopy methods are not sensitive enough to provide significant contrast between the TMs. If the X-ray form-factor difference f is small, X-ray patterns do not show diffuse scattering, whereas an analogous neutron pattern could reveal a diffuse component due to the different scattering contrast b of the TMs (where the b’s are the neutron scattering lengths). In practice, the chemical disorder phenomena might be more complex as the compositional stability of the QCs is often rather extended and the atomic distribution in a sample is not always structurally homogeneous. Depending on the crystal-growth process, microstructures may occur with locally coexisting small QC domains with fluctuating chemical content and, moreover, coexisting QC and approximant domains. Chemical disorder might also be caused by phasonic disorder. Phonon-type (static or dynamic) displacements, fluctuations or straining relate to the physical subspace coordinates giving rise to continuous local distortions of the atomic structure. Phason-type disorder describes, as indicated above, discontinuous atomic jumps between different sites in an aperiodic structure. Qualitatively, dominant phonon- or phason-related scattering may be separated by analysing the dependence of the diffracted inten-

4.2.6.1. Incommensurately modulated structures Quite generally, small domains with periodic or aperiodic structures give rise to broadening of reflections. Diffuse satellite scattering is due to the limited coherence length of a modulation wave within a crystal or due to short-range order of twin or nanodomains (with an ‘internal’ modulated structure) embedded in a periodic matrix structure. This type of diffuse scattering can be treated by the rules outlined in Section 4.2.3.5. Structural fluctuations with limited correlation lengths are, for example, responsible for diffuse scattering in 1D organic conductors (Pouget, 2004). In some cases, modulated structures are intermediate phases within a limited temperature range between a high- and low-temperature phase (e.g. quartz) where the structural change is driven by a dynamical instability (a soft mode). In addition to (inelastic) soft-mode diffuse scattering, dynamic fluctuations of phase and amplitude of the modulation wave give rise to diffuse intensity. In particular, the phase fluctuations, phasons, give rise to low-frequency scattering which might be observable as an additional diffuse contribution around condensing reflections. 4.2.6.2. Composite structures In the case of CSs, diffuse scattering relates to interactions between the component substructures which are responsible for mutual modulations: each substructure becomes modulated with the period of the other. If one of the substructures is lowdimensional, for example chain-like structural elements embedded in tubes of a host structure, one observes diffuse planes if direct interchain correlations are mostly absent. The diffuse planes are, however, not only due to the included subsystem, but also due to corresponding modulations of the

527

4. DIFFUSE SCATTERING AND RELATED TOPICS sities on the components of the scattering vector in the external and the internal subspace, He (Qe) and Hi (Qi), respectively (cf. above and also Section 4.6.3). There are different kinds of phason-like disorder, including random phason fluctuations, phason-type modulations and phason straining, and also shape and size fluctuations of the hyperatomic surfaces. If displacing an atomic surface (hyperatom) parallel to an internal space component by any kind of phason fluctuation which is equivalent to an infinitesimal rotation of the external (physical) space, it might happen that the hyperatom no longer intersects the physical space. Then an empty site is created, which is compensated by a (real) atom ‘appearing’ at a different position in physical space. In addition, there might even be a change of atomic species as a hyperatom may be chemically different at different sites of the atomic surface. Randomly distributed phason strains are responsible for Bragg-peak broadening and Huang-type diffuse scattering close to Bragg peaks. There are well developed theories based on the elastic theory of icosahedral (e.g. Jaric & Nelson, 1988) or decagonal (Lei et al., 1999) quasicrystals that also include phonon–phason coupling. An example of the quantitative analysis of diffuse scattering by an i-phase (Al–Pd–Mn) is given by de Boissieu et al. (1995) and by a d-phase (Al–Ni–Fe) by Weidner et al. (2004). Dislocations in quasicrystals have partly phonon- and partly phason-like character; a discussion of the specific dislocation-related diffuse scattering will not be given here (cf. Section 4.6.3). Arcs and rings of localized diffuse scattering are observed in various i-phases and could be modelled in terms of some short-range ‘glass-like’ ordering of icosahedral clusters (Goldman et al., 1988; Gibbons & Kelton, 1993). Phason-related diffuse scattering phenomena are discussed in more detail in Section 4.6.3. Depending on the exact stoichiometry and the growth conditions, d-phases very often show intergrown domain structures where the internal atomic structure of an individual domain varies between a (periodic) approximant structure, more or less strained aperiodic domains, or transient aperiodic variants (Frey & Weidner, 2002). Apart from finite size effects of domains of any kind which cause peak broadening, there are complex diffraction patterns of satellite reflections, diffuse maxima and diffuse streaking in d-Al–Ni–Co and other d-phases (Weidner et al., 2001). In various d-phases one also observes, in addition to the Bragg layers, prominent diffuse layers perpendicular to the unique ‘periodic’ axis. They correspond to n-fold (n = 2, 4, 8) superperiods along this direction. In different d-phases there is a gradual change from almost completely diffuse planes to layers of superstructure reflections (satellites). The picture of a kind of stacking disorder of aperiodic layers does not match such observations. An explanation is rather due to 1D columns of atomic icosahedral clusters along the unique direction (Steurer & Frey, 1998). The temperature behaviour of these diffuse layers was studied by in situ neutron diffraction (Frey & Weidner, 2003), which shed some light on the complicated order/disorder phase transitions in d-Al–Ni–Co.

ated, which is then subjected to coherent light (from a laser) to produce a diffraction image. Problems arise for strong scatterers requiring very large holes. These problems were overcome in the second method, where the mask is replaced by a computer image with intensities proportional to the scattering power for each pixel. With the advent of more and more powerful computers these methods are now replaced by complete computer simulations, both to set up the disorder model and to calculate the diffraction pattern. 4.2.7.2. Simulation programs Having established a disordered crystal with the types and positions of all atoms involved (a configuration), e.g. by using one of the methods described below, computer programs employing fast-Fourier-transform techniques can be used to calculate the diffraction pattern, which may be compared with the observation. It has to be borne in mind, however, that there is still a large gap between a real crystal with its ~1023 atoms and the one simulated by the computer with only several thousand atoms. This means that very long range correlations can not be included and have to be treated in an average manner. Furthermore, the limited size of the simulated crystal leads to termination effects, giving rise to considerable noise in the calculated diffraction pattern. Butler & Welberry (1992) have introduced a technique to avoid this problem in their program DIFFUSE by dividing the simulated crystal into smaller ‘lots’. For each lot the intensity is calculated and then the intensities of all lots are summed up incoherently, which finally results in a smooth intensity distribution. Note, however, that in this case long-range correlations are restricted to even smaller values. To overcome this problem Boysen (1997) has proposed a method for suppressing the subsidiary maxima by multiplying the scattering density of the model crystal by a suitably designed weighting function simulating the effect of the instrumental resolution function. A very versatile computer program, DISCUS, which allows not only the calculation of the scattering intensities but also allows the model structures to be built up in various ways, has been designed by Proffen & Neder (1999). It contains modules for reverse Monte Carlo (RMC, see below) simulation, the calculation of powder patterns, RMC-type refinement of pairdistribution functions (PDFs, see below) and many other useful tools for analysing disorder diffuse scattering. Other computer programs have been developed to calculate diffuse scattering, some for specific tasks, such as SERENA (Micu & Smith, 1995), which uses a collection of atomic configurations calculated from a molecular dynamics simulation of molecular crystals. 4.2.7.3. Modelling procedures Several well established methods can be used to create the simulated crystal on the computer. With such a crystal at hand, it is possible to calculate various thermodynamical properties and study the effect of specific parameters of the underlying model. By Fourier transformation, one obtains the diffraction pattern of the total scattering, i.e. the diffuse intensities and the Bragg peaks. The latter may lead to difficulties due to the termination effects mentioned in Section 4.2.7.2. These may be circumvented by excluding regions around the Bragg peaks (note, however, that in this case valuable information about the disorder may be lost), by subtracting the average structure or by using the approximation of Boysen (1997) mentioned above. A major advantage of such modelling procedures is that realistic physical models are introduced at the beginning, providing further insight into the pair interactions of the system, which can only be obtained a posteriori from the correlation parameters or fluctuation wave amplitudes derived from one of the methods described in Section 4.2.5.

4.2.7. Computer simulations and modelling 4.2.7.1. Introduction The various analytical expressions given in Section 4.2.5 are mostly rather complex, so their application is often restricted to relatively simple systems. Even in these cases analytical solutions are often not available. For larger displacements the approximations that use an expansion of the exponentials [e.g. equation (4.2.3.27)] are no longer valid. Hence there is a need for alternative approaches to tackling more complicated disorder models. In the past, optical transforms (e.g. Harburn et al., 1974, 1975) and the videographic method (Rahman, 1993) were developed for this purpose. In the first method, a 2D mask with holes with sizes that represent the scattering power of the atoms is gener-

528

4.2. DISORDER DIFFUSE SCATTERING OF X-RAYS AND NEUTRONS where I are the differences between the observed and calculated intensities and w are the appropriate weights. The problem with this approach is that the necessary differentials I/pi must be calculated numerically by performing full MC simulations with parameters pi and pi + pi at each iteration step, which presents a formidable task even for the fastest modern computers. (2) Reverse MC calculations. Application of the direct MC method may be very time consuming, as many MC simulations are necessary to arrive at a final configuration. To overcome this problem, the so-called reverse Monte Carlo (RMC) method has been developed, originally for liquids and glasses (McGreevy & Pusztai, 1988) and later for disordered single crystals (Nield et al., 1995). RMC is a model-free approach, i.e. without the need to define a proper interaction Hamiltonian. Otherwise it proceeds in a very similar way to direct MC analysis. A starting configuration is set up and random ‘moves’ of the atoms are carried out. The only difference is that acceptance or non-acceptance of a move is based on the agreement between observed and calculated diffraction intensities, i.e. equation (4.2.7.2) is replaced by

4.2.7.3.1. Molecular dynamics Molecular dynamics (MD) techniques have been developed to study the dynamics of a system. They may also be used to study static disorder problems (by taking time averages or snapshots), but they are particularly useful in the case of dynamic disorder, e.g. diffusing atoms in superionic conductors. The principle is to set up a certain configuration of atoms with assumed interatomic potentials ij(rij) and subject them to Newton’s equation of motion, Fi ðtÞ ¼ mi

d2 ri ðtÞ ; dt2

ð4:2:7:1Þ

where the force Fi(t) is calculated from the gradient of . The equations are solved approximately by replacing the differential dt by a small but finite time step t to find new positions ri(t + t). This is repeated until an equilibrium configuration is found. MD techniques are quite useful if only short-range interactions are effective, even allowing the transfer of potential parameters between different systems, but are less reliable in the presence of significant long-range interactions.

P ¼ expf
2 =2g;

4.2.7.3.2. Monte Carlo calculations Monte Carlo (MC) methods appear to be more suited to the study of static disorder and many examples of their application can be found in the literature. Different variants allowing simulations and refinements have been applied: (1) Direct MC simulation. In this method, introduced by Metropolis et al. (1953), a starting configuration is again set up in accordance with the known average structure and other crystal, chemical and physical information if available, and an appropriate interaction potential is chosen. The choice of this potential can be a quite delicate task. On the one hand, it should be as simple as possible to allow as large a simulation box size as can reasonably be handled by the computer’s capacity. On the other hand, it must be detailed enough to include all the relevant interaction parameters of the system. Frequently used interaction potentials are a pseudo spin Ising Hamiltonian, where the ‘spins’ can either be binary (e.g. atom types or molecular orientations) or continuous (displacements), and harmonic springs for the displacements. Having set up the starting configuration and defined the Hamiltonian, one proceeds by choosing a specific site at random and changing its parameters (occupancies and positions) by a random amount (a ‘move’). The energy of this new configuration is calculated and compared with the old one. If the difference E is negative, the move is accepted. If it is positive it is accepted with a probability P ¼ expf
E=kB Tg=½1 þ expf
E=kB Tg;

ð4:2:7:4Þ

where 2 is defined in equation (4.2.7.3) and 2 ¼ 2new
2old . Usually only a single MC run is necessary to arrive at a configuration with a diffraction pattern consistent with the observation. A drawback of the RMC technique is that usually only one configuration is found satisfying the observed diffraction pattern. In fact, it is possible that different configurations may produce the same or very similar intensity distributions (see e.g. Welberry & Butler, 1994). In this context, it has to be borne in mind that the diffuse scattering contains information about two-body correlations only, while the physical reason for a particular disordered structure may also be influenced by three- and manybody correlations. Such many-body correlations can easily be incorporated in the direct MC method. Moreover, RMC is susceptible to fitting artefacts (noise) in the diffraction pattern. To avoid unreasonable atomic configurations, restrictions such as a limiting nearest-neighbour approach can be built in. Weak constraints, e.g. ranges of interatomic distances or bond angles (i.e. three-body correlations) can be introduced by adding further terms in the definition of 2 [equation (4.2.7.3)]. In the same way, multiple data sets (e.g. neutron and X-ray data, EXAFS measurements etc.) may be incorporated. For more details and critical reviews of this method see e.g. McGreevy (2001) and Welberry & Proffen (1998). Many applications of this method can be found in the literature, mainly for powder diffraction, but also for single-crystal data. (3) Simulated annealing and evolutionary algorithms. A general problem with MC methods is that they may easily converge to some local minima without having found the global one. One way to reduce such risks is to start with a high probability in (4.2.7.2) by using a large ‘temperature’ T, i.e. initially allowing many ‘false’ moves before gradually reducing T during the course of the simulation cycles. This is called ‘simulated annealing’, although it has nothing to do with real annealing. In the RMC method one may introduce a weighting parameter similar to T in (4.2.7.2). Another effective way to find the global minimum and also to accelerate the optimization of the energy parameters of a given MC model is by using the principles of evolutionary theory: selection, recombination and mutation. Two such evolutionary (or genetic) algorithms have been proposed: the differential evolution (DE) algorithm (Weber & Bu¨rgi, 2002) and the cooperative evolution (CE) algorithm (Weber, 2005). A single parameter of the model is called a gene and a set of genes is called a chromosome, p being the genotype of an individual. First a population consisting of several individuals is built up and the corresponding diffraction patterns are calculated and compared

ð4:2:7:2Þ

where T is a temperature and kB is Boltzmann’s constant. Then another site is chosen at random and the process is repeated again and again until an equilibrium configuration is found, i.e. until the energies fluctuate around some average value. After this, the corresponding intensity distribution is calculated and compared with the observed one. The parameters of the Hamiltonian may then be modified to improve the agreement between the calculated and observed diffraction patterns. This rather cumbersome trial-and-error method may be used to study the influence of the various parameters of the model. Further details of this method together with some illustrative examples may be found in Welberry & Butler (1994). In an attempt to automate the adjustment of the model parameters, Welberry et al. (1998) have used a least-squares procedure to minimize P 2 ¼ wðIÞ2 ; ð4:2:7:3Þ

529

4. DIFFUSE SCATTERING AND RELATED TOPICS with the observation. The fitness of each individual is quantified by 2. Children are then created by choosing one parent individual and calculating the second one from three randomly chosen individuals according to p0c ¼ pc þ fm ðpa
pb Þ;

scattering, or by designing special algorithms, e.g. by swapping two atoms at the same time (Proffen & Welberry, 1997). Moreover, possible traps and corrections like local minima, termination errors, instrumental resolution, statistical noise, inelasticity etc. must be carefully considered. All this means that the analytical methods outlined in Section 4.2.5 keep their value and should be preferred wherever possible. The newly emerging technique of using full quantummechanical ab initio calculations for structure predictions along with MD simulations may also be applied to disordered systems. The limited currently available computer power, however, restricts this possibility to rather simple systems and small simulation box sizes, but, with the expected further increase of computer capacities, this may open up new perspectives for the future.

ð4:2:7:5Þ

where fm governs the mutations. The chromosome of the child is obtained by combining the genes of the two parents governed by a crossover (or recombination) constant fr. If its fitness is higher than that of the parent, it replaces it. This procedure is repeated until some convergence criterion is reached. This DE method is still rather time-consuming on the computer. Therefore, the CE technique was introduced, where only one crystal is built up during the refinement. Here a large population is created spanning a large but reasonable parameter space. Individuals are selected at random to decide upon acceptance or rejection of an MC move via their own energy criteria. Then 2 is calculated and according to its positive or negative change a gratification or penalty weight is given to that individual. It may live as long as this weight is positive, otherwise it is replaced by a new individual calculated from (4.2.7.5). This way, useful individuals live longer to act on the same crystal, while unsuccessful ones are eliminated early. Note that the recombination operation is not used in this technique. (4) The PDF method. The pair-distribution function (PDF) has long been used for the analysis of liquids, glasses and amorphous substances (Warren, 1969), but has recently regained considerable interest for the analysis of crystalline substances as well (Egami, 1990; Billinge & Egami, 1993; Egami, 2004). The PDF is obtained by a Fourier transformation of the total (Bragg plus diffuse) scattering in a powder pattern, R 0 GðrÞ ¼ 0 þ 12 H½SðHÞ
1 sinð2
HÞ dH: ð4:2:7:6Þ

4.2.8. Experimental techniques and data evaluation Single-crystal and powder diffractometry are used in diffuse scattering work. Conventional and more sophisticated special techniques and instruments are now available at synchrotron facilities and modern neutron reactor and spallation sources. The full merit of the dedicated machines may be assessed by inspecting the corresponding handbooks, which are available upon request from the facilities. In the following, some common important aspects that should be considered when planning and performing a diffuse-scattering experiment are summarized and a short overview of the techniques is given. Methodological aspects of diffuse scattering at low angles, i.e. small-angle-scattering techniques, and high-resolution single-crystal diffractometers are excluded. Instruments of the latter type are used when diffuse intensities beneath Bragg reflections or reflection profiles and tails must be analysed to study long-range distortion fields around single defects or small defect aggregates. In the case of small defect concentrations, the crystal structure remains almost perfect and the dynamical theory of diffraction is more appropriate. This topic is beyond the scope of this chapter.

This is nothing other than the van Hove correlation function (4.2.2.2) or the related Patterson function (4.2.2.5) averaged spherically and taken at t = 0 (a snapshot) and is given by P 
2 ci bi 4
r0 GðrÞ; ð4:2:7:7Þ GPDF ðrÞ ¼ 4
r½ðrÞ
0  ¼ where 0 is the average number density and P ðrÞ ¼ ð1=NÞ ðbi bi0 =hbi2 Þðr
rii0 Þ:

4.2.8.1. Single-crystal techniques In general, diffuse scattering is weak in comparison with Bragg scattering, and is anisotropically and inhomogeneously distributed in reciprocal space. The origin may be a static phenomenon or a dynamic process, giving rise to elastic or inelastic (quasielastic) diffuse scattering, respectively. If the disorder problem relates to more than one structural element, different parts of the diffuse scattering may show different behaviour in reciprocal space and/or on an energy scale. Therefore, before starting an experiment, some principal aspects should be considered: Is there need for X-ray and/or neutron methods? What is the optimum wavelength or energy (band), or does a ‘white’ technique offer advantages? Can focusing techniques be used without too strong a loss of resolution and what are the best scanning procedures? How can the background be minimized? Has the detector a low intrinsic noise and a high dynamic range? On undertaking an investigation of a disorder problem by an analysis of the diffuse scattering, an overall picture should first be recorded by X-ray diffraction. Several sections through reciprocal space help to define the problem. For this purpose ‘oldfashioned’ film methods may be used, where the classical film is now commonly replaced by an imaging plate (IP) or a chargecoupled device (CCD) camera (see below). Clearly, short crystalto-detector distances provide larger sections and avoid long exposure times, but suffer from spatial resolution. Distorted sections through the reciprocal lattice, such as produced by the Weissenberg method, may be transformed into a form suitable for easy interpretation (Welberry, 1983). The transformation of diffuse data measured using an IP or CCD requires suitable

ð4:2:7:8Þ

The  functions are then convoluted with a normalized Gaussian to account for (harmonic) thermal motion. Parameter refinement proceeds in a similar way to the RMC technique. First a model is built, initially within just one unit cell and with periodic boundary conditions, then its PDF is calculated, compared with the observed one and further improved using, for example, MC simulated annealing. The model is then enlarged to include longer-range correlations. Owing to the small size of the models, this technique is much faster than the RMC method. It is essential, however, that data are measured up to very high H values to minimize truncation errors in (4.2.7.6). 4.2.7.3.3. General remarks All of the different modelling techniques mentioned in this section have their specific merits and limitations and have contributed much to our understanding of disorder in crystalline materials following the interpretation of the corresponding diffuse scattering. It should be borne in mind, however, that application of these methods is still far from being routine work and it requires a lot of intuition to ensure that the final model is physically and chemically reasonable. In particular, it must always be ensured that the average structure remains consistent with that derived from the Bragg reflections alone. This may be done by keeping the Bragg reflections, i.e. by analysing the total

530

4.2. DISORDER DIFFUSE SCATTERING OF X-RAYS AND NEUTRONS software for the specific type of detector and is not always routinely available (Estermann & Steurer, 1998). Standing-crystal techniques in combination with monochromatic radiation, usually called monochromatic Laue techniques (see, e.g., Flack, 1970), save exposure time, which is particularly interesting for ‘in-house’ laboratory diffuse X-ray work. The Noromosic technique (Jagodzinski & Korekawa, 1973) is characterized by a convergent monochromatic beam which simulates an oscillation photograph over a small angular range. Heavily overexposed images, with respect to Bragg scattering, allow for sampling of diffuse intensity if a crystal is oriented in such a way that there is a well defined section between the Ewald sphere and the diffuse phenomenon under consideration. By combining single Noromosic photographs, Weissenberg patterns can be simulated. This relatively tedious method is often unavoidable because the heavily overexposed Bragg peaks obscure weak diffuse phenomena. Furthermore, standing pictures at distinct crystal settings in comparison with conventional continuous recording are frequently sufficient in diffuse scattering work and save time. Long-exposure Weissenberg photographs are therefore not equivalent to a smaller set of standing photographs. In this context it should be mentioned that a layer-line screen has not only the simple function of a selecting diaphragm, but the gap width determines the resolution volume within which diffuse intensity is collected (Welberry, 1983). For further discussion of questions of resolution see below. A comparison of Weissenberg and diffractometer methods for the measurement of diffuse scattering is given by Welberry & Glazer (1985). It should be pointed out, however, that diffractometer methods at synchrotron sources become more widely used if only very small (micrometre-sized) single-crystal specimens have to be used to study a disorder problem. The basic arguments for using neutron-diffraction methods were given in Section 4.2.2.2: (i) the different interactions of X-rays and neutrons with matter; (ii) the lower absorption of neutrons, in particular when using longer (> 0.15 nm) wavelengths; and (iii) the matching of the energy of thermal neutrons with that of the phonons that contribute to the TDS background, and, in consequence, to separate it by a ‘purely’ elastic measurement. A comparative consideration of synchrotron- and neutron-related diffuse work on disordered alloys is given by Schweika (1998). Specific aspects of neutron diffraction and instruments are discussed at the end of this section. Intensities: As mentioned above, diffuse intensities are usually weaker by several orders of magnitude than Bragg data. Therefore intense radiation sources are needed. Even a modern X-ray tube is a stronger source, defined by the flux density from the anode (number of photons cm
2 s
1), than modern neutron sources. For this reason most experimental work which can be performed with X-rays should be. In home laboratories the intense characteristic spectrum of an X-ray tube is commonly used. At synchrotron storage rings any wavelength from a certain range can be selected. The extremely high brilliance (number of quanta cm
2, sr
1, s
1 and wavelength interval) of modern synchrotron sources is, however, unnecessary in the case of slowly varying diffuse phenomena. In these cases, an experimental setup at a laboratory rotating anode is competitive and often even superior if specimens with sufficient size are available. Various aspects of diffuse X-ray work at a synchrotron facility are discussed by Matsubara & Georgopoulos (1985), Oshima & Harada (1986) and Ohshima et al. (1986). Diffuse neutrondiffraction work can only be performed on a high-flux reactor or on a powerful spallation source. Highly efficient monochromator systems are needed when using a crystal diffractometer. Time-offlight (TOF) neutron diffractometers at pulsed (spallation) neutron sources are equivalent to conventional diffractometers at reactors (Windsor, 1982). The merits of diffuse neutron work at pulsed sources have been discussed by Nield & Keen (2001).

Wavelength: The choice of an optimum wavelength is important with respect to the problem to be solved. For example, point defects cause diffuse scattering to fall off with increasing scattering vector; short-range ordering between clusters causes broad peaks corresponding to large d spacings; lattice-relaxation processes induce a broadening of the interferences; static modulation waves with long periods give rise to satellite scattering close to Bragg peaks. In all these cases, a long wavelength is preferable due to the higher resolution. The use of a long wavelength is also profitable when the main diffuse contributions can be recorded within an Ewald sphere as small as the Bragg cutoff of the sample at H ’ 1/(2dmax). ‘Contamination’ by Bragg scattering can thus be avoided. This is also advantageous from a different point of view: because the contribution of thermal diffuse scattering increases with increasing scattering vector H, the relative amount of this component becomes negligibly small within the first reciprocal cell. However, one has to take care with the absorption of long-wavelength neutrons. On the other hand, short wavelengths are needed where atomic displacements play the dominant role. If diffuse peaks in large portions of the reciprocal space, or diffuse streaks or planes, have to be recorded up to high values of the scattering vector in order to decide between different structural disorder models, hard X-rays or hot neutrons are needed. For example, highenergy X-rays (65 keV) provided at a synchrotron source were used by Welberry et al. (2003) to study diffuse diffraction from ceramic materials and allowed studies with better d resolution. The 3 dependence of the scattered intensity, in the framework of the kinematical theory, is a crucial point for exposure or dataacquisition times. Moreover, the accuracy with which an experiment can be carried out suffers from a short wavelength: generally, momentum as well as energy resolution are lower. For a quantitative estimate detailed considerations of resolution in reciprocal space (and energy) are needed. A specific wavelength aspect concerns the method of (X-ray) anomalous dispersion, which may also be used in diffusescattering work. It allows the contrast and identification of certain elements in a disordered structure. Even small concentrations of impurity atoms or defects as low as 10
6 can be determined by this method if the impurity atoms are located at specific sites, e.g. in domain boundaries, or if certain other defect structures exist with characteristic diffuse scattering in reciprocal space. The (weak) diffuse scattering can then be contrasted by tuning the wavelength across an absorption edge of the particular atomic species. To avoid strong fluorescence background scattering such an experiment is usually performed at different wavelengths (energies) at the ‘low absorption side’ close to an edge. The merit of this method was demonstrated by Berthold & Jagodzinski (1990), who analysed diffuse streaks due to boundaries between lamellar domains in an albite feldspar. Similar element contrast can be achieved in neutron diffuse work if using specimens with different isotopes, e.g. H–D exchange. Monochromacy: In classical crystal diffractometry, monochromatic radiation is used in order to eliminate broadening effects due to the wavelength distribution. Focusing monochromators and other focusing devices (guides, mirrors) help to overcome the lack of luminosity. A focusing technique is very helpful for deciding between geometrical broadening and ‘true’ diffuseness. In a method that is used with some success, the sample is placed in a monochromatic divergent beam with its selected axis lying in the scattering plane of the monochromator (Jagodzinski, 1968). The specimen is fully embedded in the incident beam, which is focused onto a 2D detector. Using this procedure the influence of the sample size is suppressed in one dimension. In white-beam (neutron) time-of-flight diffractometry the time resolution is the counterpart to the wavelength resolution. This is discussed in some detail in the textbook of Keen & Nield (2004).

531

4. DIFFUSE SCATTERING AND RELATED TOPICS Detectors: Valuable developments with a view to diffusescattering work are multidetectors (see, e.g., Haubold, 1975) and position-sensitive detectors for X-rays (Arndt, 1986a,b) and neutrons (Convert & Forsyth, 1983). A classical ‘2D area detector’ is the photographic film – nowadays of less importance – which has to be ‘read out’ by microdensitometer scanning. Important progress in recording diffuse X-ray data has been made by the availability of multiwire detectors, IPs and CCD cameras (Estermann & Steurer, 1998). For 2D position-sensitive (proportional) counters problems may arise from inhomogeneities of the wire array as well as the limited dynamic range when a Bragg reflection is accidentally recorded. IP systems have the major advantage of a larger dynamic range, 105–106, compared to 102–103 for an X-ray film. IPs can also be used in diffuse experiments carried out with hard (65 keV) X-rays (Welberry et al., 2003). IPs are also available for neutron work, where the necessary transformation of the detected neutrons into light signals is provided by special neutron-absorbing converter foils (Niimura et al., 2003). The problem of an intrinsic sensitivity to gamma radiation may be overcome by protection with a thin sheet of lead. IPs allow data collection either in plane geometry or in simple rotation or Weissenberg geometry, both in combination with low- and high-temperature devices. CCD detectors are well suited for X-ray diffuse scattering, and when used in combination with converter foils they can also be used for the detection of neutrons. A basic prerequisite is a low intrinsic noise, which can be achieved by cooling the CCD with liquid nitrogen. With these detectors, extended diffuse data sets can be collected by rotating the sample in distinct narrow steps around a spindle axis over several  settings and subsequent oscillations over a small angular range. Thus large parts of the reciprocal space can be recorded. An example is given by Campbell et al. (2004), who studied subtle defect structures in the microporous framework material mordenite. Linear position-sensitive detectors are mainly used in powder work, but can also be used for recording diffuse scattering by single crystals. By combining a linear position-sensitive detector and the TOF method, a whole area in reciprocal space is accessible simultaneously (Niimura et al., 1982; Niimura, 1986). Diffuse data recorded with IPs or CCDs are commonly evaluated by commercial software supplied by the manufacturer. These software packages include corrections for extracted Bragg data, but no special software tools for treating extended (diffuse) data are commonly available. One particular problem relates to the background definition for diffuse IP data. Even after subtraction of electronic noise, there remains considerable uncertainty about the amount of true background scattering from the sample as recorded by an IP scanner. The definitions of errors and error maps remain doubtful as long as the true conversion rate between the captured neutron or photon versus the recorded optical signal is unknown. Absorption: Special attention must be paid to absorption phenomena, in particular when (in the X-ray case) an absorption edge of an element of the sample is close to the wavelength. Then strong fluorescence scattering may completely obscure weak diffuse-scattering phenomena. In comparison with X-rays, the generally lower absorption coefficients of neutrons make absolute measurements easier. This also allows the use of larger sample volumes, which is not true in the X-ray case. Moreover, the question of sample environment is less serious in the neutron case than in the X-ray case. However, the availability of hard X-rays at a synchrotron source (Butler et al., 2000; Welberry et al., 2003) makes the X-ray absorption problem less serious: irregularly shaped specimens without special surface treatment could be used. There is also no need for complicated absorption corrections and the separation of fluorescence background is rather simple. Extinction: An extinction problem does not generally exist in diffuse-scattering work.

Background: An essential prerequisite for a diffuse-scattering experiment is the careful suppression of background scattering. Incoherent X-ray scattering by a sample gives continuous blackening in the case of fluorescence, or scattering at high 2 angles owing to Compton scattering or ‘incoherent’ inelastic effects. Protecting the image plate with a thin Al or Ni foil is of some help against fluorescence, but also attenuates the diffuse intensity. Obviously, energy-dispersive counter methods are highly efficient in this case (see below). Air scattering produces a background at low 2 angles which may easily be avoided by special slit systems and evacuation of the camera. In X-ray or neutron diffractometer measurements, incoherent and multiple scattering contribute to a background which varies only slowly with 2 and can be subtracted by linear interpolation or fitting a smooth curve, or can even be calculated quantitatively and then subtracted. In neutron diffraction there are rare cases when monoisotopic and ‘zero-nuclear-spin’ samples are available and, consequently, the corresponding incoherent scattering part vanishes completely. In some cases, a separation of coherent and incoherent neutron scattering is possible by polarization analysis (Gerlach et al., 1982). An ‘empty’ scan can take care of instrumental background contributions. Evacuation or controlledatmosphere studies need a chamber, which may give rise to spurious scattering. This can be avoided if no part of the vacuum chamber is hit by the primary beam. The problem is less serious in neutron work. The mounting of the specimen, e.g., on a silica fibre with cement, poorly aligned collimators and beam catchers are further sources. Sometimes a specimen has to be enclosed in a capillary, which will always be hit by the incident beam. Careful and tedious experimental work is necessary in the case of lowand high-temperature (or -pressure) investigations, which have to be carried out in many disorder problems. While the experimental situation is again less serious in neutron scattering, there are problems with scattering from walls and containers in X-ray work. Because TDS dominates at high temperatures and in the presence of a static disorder problem, a quantitative separation can rarely be carried out in the case of high experimental background. Calculation and subtraction of the TDS is possible in principle, but difficult in practice. If the disorder problem in which one is interested in does not change with temperature, a low-temperature experiment can be carried out. Another way to get rid off TDS, at least partly, is by using a neutron diffractometer with an additional analyser set to zero-energy transfer. Resolution: A quantitative analysis of diffuse-scattering data is essential for reaching a definite decision about a disorder model, but – in many cases – it is cumbersome. By comparing the calculated and corrected experimental data the magnitudes of the parameters of the structural disorder model may be derived. A careful analysis of the data requires, therefore, after separation of the background (see above), corrections for polarization (in the X-ray case), absorption (in conventional X-ray work) and resolution. Detailed considerations of instrumental resolution are necessary; the resolution depends, in addition to other factors, on the scattering angle and implies intensity corrections analogous to the Lorentz factor used in structure analysis from sharp Bragg reflections. Resolution is conveniently described by the function RðH
H0 Þ, which is defined as the probability of detecting a photon or neutron with momentum transfer hH ¼ hðk
k0 Þ when the instrument is set to measure H0. This function R depends on the instrumental parameters (such as the collimations, the mosaic spread of monochromator and the scattering angle) and the spectral width of the source. Fig. 4.2.8.1 shows a schematic sketch of a diffractometer setting. Detailed considerations of resolution volume in X-ray and neutron diffractometry are given by Sparks & Borie (1966) and by Cooper & Nathans (1968a,b), respectively. If a triple-axis (neutron) instrument is used, for example in a purely elastic configuration, the set of instrumental parameters includes the mosaicity of the analyser

532

4.2. DISORDER DIFFUSE SCATTERING OF X-RAYS AND NEUTRONS The intensity variation of diffuse peaks with 2 measured with a single detector was studied in detail by Yessik et al. (1973). In principle, all special cases are included there. In practice, however, some important simplifications can be made if d=d
is either very broad or very sharp compared with R, i.e. for Bragg peaks, sharp streaks, ‘thin’ diffuse layers or extended 3D diffuse peaks (Boysen & Adlhart, 1987). In the latter case, the cross section d=d
may be treated as nearly constant over the resolution volume so that the corresponding ‘Lorentz’ factor is independent of 2: L3D ¼ 1:

ð4:2:8:2Þ

For a diffuse plane within the scattering plane with very small thickness and slowly varying cross section within the plane, one derives for a point measurement in the plane 0

L2D; k ¼ ð 12 þ 22 þ 422v sin2 Þ
1=2 ;

ð4:2:8:3Þ

exhibiting an explicit dependence on  ( 01 , 2 and 2v determine an effective vertical divergence before the sample, the divergence before the detector and the vertical mosaic spread of the sample, respectively). In the case of relaxed vertical collimations 01 ; 2 2v 0

L2D; k ¼ ð 12 þ 22 Þ
1=2 ;

ð4:2:8:3aÞ

i.e. again independent of . Scanning across the diffuse layer in a direction perpendicular to it one obtains an integrated intensity which is also independent of 2. This is even true if approximations other than Gaussians are used. If, on the other hand, an equivalent diffuse plane is positioned perpendicular to the scattering plane, the equivalent expression for L2D; ? of a point measurement is given by

Fig. 4.2.8.1. Schematic sketch of a diffractometer setting.

and the collimations between the analyser and detector. In general, Gaussians are assumed to parameterize the mosaic distributions and transmission functions. Sophisticated resolution-correction programs are usually provided at any standard instrument for carrying out experiments at synchrotron and neutron facilities. The general assumption of Gaussians is not too serious in the X-ray case (Iizumi, 1973). Restrictions are due to absorption, which makes the profiles asymmetric. Box-like functions are considered to be better for the spectral distribution or for large apertures (Boysen & Adlhart, 1987). These questions are treated in some detail by Klug & Alexander (1954). The main features, however, may also be derived by the Gaussian approximation. In practice, the function R may be obtained either by calculation from the known instrumental parameters or by measuring Bragg peaks from a perfect unstrained crystal. In the latter case, the intensity profile is given solely by the resolution function. Normalization with the Bragg intensities is also useful in order to place the diffuse-scattering intensity on an absolute scale. In conventional single-crystal diffractometry the measured intensity is given by the convolution product of d=d
with R, Z d IðH0 Þ ¼ ðHÞ½RðH
H0 Þ dH; ð4:2:8:1Þ d


L2D; ? ’ ½422H sin2  cos2

þ 22 sin2 ð
Þ

0

þ 12 sin2 ð þ Þ þ 400 sin2  sin2
4 001 sin  sin

sinð þ Þ;

ð4:2:8:4Þ

where gives the angle between the vector H0 and the line of intersection between the diffuse plane and the scattering plane. The coefficients 2H, 2 , 01 , 001 and  are either instrumental parameters or functions of them, defining horizontal collimations and mosaic spreads. In the case of a sharp X-ray line (produced, for example, by filtering) the last two terms in equation (4.2.8.4) vanish. The use of integrated intensities from individual scans perpendicular to the diffuse plane, now carried out within the scattering plane, again gives a Lorentz factor independent of 2. In the third fundamental special case, diffuse streaking along one reciprocal direction within the scattering plane (with a narrow cross section and slowly varying intensity along the streak), the Lorentz factor for a point measurement may be expressed by the product

where d=d
describes the scattering cross section for the disorder problem. In a more accurate form the mosaicity of the sample has to be included: Z d ðH
KÞ½ðKÞRðH
H0 Þ dH dðKÞ IðH0 Þ ¼ d
Z d 0 0 0 ðH Þ½R ðH
H0 Þ dH0 : ¼ ð4:2:8:1aÞ d


L1D; k ’ L2D; k L2D; ? ;

ð4:2:8:5Þ

where now defines the angle between the streak and H0 . The integrated intensity taken from an H scan perpendicular to the streak has to be corrected by a Lorentz factor which is equal to L2D; k [equation (4.2.8.3)]. In the case of a diffuse streak perpendicular to the scattering plane, a relatively complicated equation holds for the corresponding Lorentz factor (Boysen & Adlhart, 1987). Again, simpler expressions hold for integrated intensities from H scans perpendicular to the streaks. Such scans may be performed in the radial direction (corresponding to a –2 scan),

R R0 ðH0
H0 Þ ¼ ðKÞRðH0 þ K
H0 Þ dðKÞ. The mosaic block distribution around a most probable vector K0 is described by ðKÞ: K ¼ K
K0 ; H0 ¼ H
K. In (4.2.8.1) all factors independent of 2 are neglected. All intensity expressions have to be calculated from equations (4.2.8.1) or (4.2.8.1a).

L1D; ?; rad ¼ ð422H þ 22 þ 021 Þ
1=2 ð1= sin Þ;

533

ð4:2:8:6Þ

4. DIFFUSE SCATTERING AND RELATED TOPICS or perpendicular to the radial direction (within the scattering plane) (corresponding to an ! scan),

Munich) for common solid-state investigations. Information about all these instruments can be found in the respective handbooks or on the websites of the facilities.

L1D; ?; per ¼ ð 22 þ 21 þ 400 tan2 
4 001 tan Þ
1=2 ð1= cos Þ:

4.2.8.2. Powders and polycrystals

ð4:2:8:7Þ

The diffuse background in powder diagrams also contains valuable information about disorder. Only in very simple cases can a model be deduced from a powder pattern alone, but a refinement of a known disorder model can favourably be carried out, e.g. the temperature dependence may be studied. On account of the intensity integration, the ratio of diffuse intensity to Bragg intensity is enhanced in a powder pattern. Moreover, a powder pattern contains, in principle, all the information about the sample and might thus reveal more than single-crystal work. However, in powder-diffractometer experiments preferred orientations and textures could lead to a complete misidentification of the problem. Single-crystal experiments are generally preferable in this respect. Nevertheless, high-resolution powder investigations may give quick supporting information, e.g. about superlattice peaks, split reflections, lattice strains, domain-size effects, lattice-constant changes related to a disorder effect etc. Evaluation of diffuse-scattering data from powder diffraction follows the same theoretical formulae developed for the determination of the radial distribution function for glasses and liquids. The final formula for random distributions may be given as (Fender, 1973) P IDp ¼ fhjFðHÞj2 i
jhFðHÞij2 g si sinð2
Hri Þ=ð2
Hri Þ: ð4:2:8:9Þ

Note that only the radial scan yields a simple  dependence ð 1= sin Þ. From these considerations it is recommended that integrated intensities from scans perpendicular to a diffuse plane or a diffuse streak should be used in order to extract the disorder cross sections. For other scan directions, which make an angle with the intersection line (diffuse plane) or with a streak, the L factors are simply L2D; ? = sin and L1D; ? = sin , respectively. One point should be emphasized: since in a usual experiment with a single counter the integration is performed over an angle ! via a general ! : ðg2Þ scan, an additional correction factor arises: !=H ¼ sinð þ Þ=ðk0 sin 2Þ:

ð4:2:8:8Þ

is the angle between H0 and the scan direction H ; and g ¼ ðtan þ tan Þ=ð2 tan Þ defines the coupling ratio between the rotation of the crystal around a vertical axis and the rotation of the detector shaft. The so-called 1:2 and !-scan techniques are most frequently used, where ¼ 0 and 90 , respectively. White-beam techniques: Techniques for the measurement of diffuse scattering using a white spectrum are common in neutron diffraction. Owing to the relatively low velocity of thermal or cold neutrons, TOF methods in combination with time-resolving detector systems placed at a fixed angle 2 allow for a simultaneous recording along a radial direction through the origin of reciprocal space (see, e.g., Turberfield, 1970; Bauer et al., 1975). The scan range is limited by the Ewald spheres corresponding to max and min , respectively. With several such detector systems placed at different angles or a 2D detector several scans may be carried out simultaneously during one neutron pulse. An analogue of neutron TOF diffractometry in the X-ray case is a combination of a white source of X-rays and an energydispersive detector. This technique, which has been known in principle for a long time, suffered from relatively weak white sources. With the development of high-power X-ray generators and synchrotron sources this method has now become highly interesting. Its use in diffuse-scattering work (in particular, the effects on resolution) is discussed by Harada et al. (1984). Some examples of neutron instruments dedicated to diffuse scattering are the diffractometers D7 at the Institut Laue– Langevin (ILL, Grenoble), DNS at the Forschungsneutronenquelle Heinz Maier-Liebnitz (FRM-II, Munich), G4-4 at the Laboratoire Le´on Brillouin (LLB, Saclay) or SXD at ISIS, Rutherford Appleton Laboratory (RAL, UK). The instrument DNS uses a polarization analysis for diffuse scattering (Schweika & Bo¨ni, 2001; Schweika, 2003) and the SXD diffractometer is a TOF instrument using a (pulsed) white beam (Keen & Nield, 2004). All these instruments are equipped with banks of detectors. The single-crystal diffractometer D19 at the ILL, equipped with a multiwire area detector, is also suitable for collecting diffuse data. The flat-cone machine E2 at the Hahn–Meitner Institut (HMI, Berlin) is equipped with a bank of area detectors, has an option to record higher-order layers and can also be operated in an ‘elastic’ mode with a multicrystal analyser. The instrument D10 at the ILL is a versatile instrument which can be operated as a low-background two-axis or three-axis diffractometer with several further options. Neutron diffractometers that have recently become operational are BIX-3 at the Japan Atomic Energy Research Institute (JAERI, Japan), LADI at the ILL, where neutron-sensitive IPs are used for macromolecular work (for a comparison see Niimura et al., 2003), and RESI at the Forschungsneutronenquelle Heinz Maier-Liebnitz (FRM-II,

i

si represents the number of atoms at distance ri from the origin. An equivalent expression for a substitutional binary alloy is P IDp ¼ ð1
Þfj f2 ðHÞ
f2 ðHÞj2 g si sinð2
Hri Þ=ð2
Hri Þ: i

ð4:2:8:10Þ A quantitative calculation of a diffuse background is also helpful in combination with Rietveld’s method (1969) for refining an averaged structure by fitting Bragg data. In particular, for highly anisotropic diffuse phenomena characteristic asymmetric line shapes occur. The calculation of these line shapes is treated in the literature, mostly neglecting the instrumental resolution (see, e.g., Warren, 1941; Wilson, 1949; Jones, 1949; and de Courville-Brenasin et al., 1981). This is not justified if the variation of the diffuse intensity becomes comparable with that of the resolution function, as is often the case in neutron diffraction. The instrumental resolution may be incorporated using the resolution function of a powder instrument (Caglioti et al., 1958). A detailed analysis of diffuse peaks is given by Yessik et al. (1973) and the equivalent considerations for diffuse planes and streaks are discussed by Boysen (1985). The case of 3D random disorder (incoherent neutron scattering, monotonous Laue scattering, averaged TDS, multiple scattering or short-range-order modulations) is treated by Sabine & Clarke (1977). In polycrystalline samples the cross section has to be averaged over all orientations: Z dp 0 nc d 0 0 0 ðH ÞR ðjH j
jH0 jÞ dH0 ; ðH Þ ¼ 2 ð4:2:8:11Þ d
d
H where nc is number of crystallites in the sample; this averaged cross section enters the relevant expressions for the convolution product with the resolution function. A general intensity expression may be written as (Boysen, 1985) P In ðH 0 Þ ¼ P mðTÞAn n ðH 0 ; TÞ: ð4:2:8:12Þ T

534

4.2. DISORDER DIFFUSE SCATTERING OF X-RAYS AND NEUTRONS (d) Slowly varying diffuse scattering in three dimensions 3 = constant. Consequently, the intensity is directly proportional to the cross section. The characteristic functions 0, 1 and 2 are shown in Fig. 4.2.8.2 for equal values of T and D. Note the relative peak shifts and the high-angle tail.

4.2.8.3. Total diffraction pattern As mentioned in Section 4.2.7.3.2, the atomic pair-distribution function (PDF), which is classically used for the analysis of the atomic distributions in liquids, melts or amorphous samples, can also be used to gain an understanding of disorder in crystals. The PDF is the Fourier transform of the total scattering. The measurement of total scattering is basically similar to recording X-ray or neutron powder patterns. The success of the method depends, however, decisively on various factors: (i) The availability of a large data set, i.e. reliable intensities up to high H values, in order to get rid of truncation ripples, which heavily influence the interpretation. Currently, values of Hmax of more ˚
1 can be achieved either with synchrotron X-rays [at than 7 A the European Synchrotron Radiation Facility (ESRF) or Cornell High Energy Synchrotron Source (CHESS)] or with neutrons from reactors (e.g. instrument D4 at the ILL) and spallation sources [at ISIS or at LANSCE (instrument NPDF)]. (ii) High H resolution. (iii) High intensities, in particular at high H values. (iv) Low background of any kind which does not originate from the sample. High-quality intensities are therefore to be extracted from the raw data by taking care of an adequate absorption correction, correction of multiple-scattering effects, separation of inelastically scattered radiation (e.g. Compton scattering) and careful subtraction of ‘diffuse’ background which is not the ‘true’ diffuse scattering from the sample. These conditions are in practice rather demanding. A further detailed discussion is beyond the scope of this chapter, but a more thorough discussion is given, e.g., by Egami (2004), and some examples are given by Egami & Billinge (2003).

Fig. 4.2.8.2. Line profiles in powder diffraction for diffuse peaks (full line), continuous streaks (dot–dash lines) and continuous planes (broken lines). For explanation see text.

P denotes a scaling factor that depends on the instrumental luminosity, T is the shortest distance to the origin of the reciprocal lattice, mðTÞ is the corresponding symmetry-induced multiplicity, An contains the structure factor of the structural units and the type of disorder, and n describes the characteristic modulation of the diffuse phenomenon of dimension n in the powder pattern. These expressions are given below with the assumption of Gaussian line shapes of width D for the narrow extension(s). The formulae depend on a factor M ¼ A1=2 ð4k21
H02 Þ=ð32 ln 2Þ, where A1=2 describes the dependence of the Bragg peaks on the instrumental parameters U, V and W (see Caglioti et al., 1958), A21=2 ¼ U tan2  þ V tan  þ W;

ð4:2:8:13Þ

and k1 ¼ 1=. (a) Isotropic diffuse peak around T 0 ¼ ½2
ðM 2 þ D2 Þ
1=2 ð1=T 2 Þ  expf
ðH 0
TÞ2 =2ðM2 þ D2 Þg:

ð4:2:8:14Þ References

The moduli jH0 j and T enter the exponential, i.e. the variation of d=d
along jH0 j is essential. For broad diffuse peaks ðM  DÞ the angular dependence is due to 1=T 2, i.e. proportional to 1= sin2 . This result is valid for diffuse peaks of any shape. (b) Diffuse streak R1 2 1 ¼ ½2
ðM2 þ D2 Þ
1=2 ðT þ H 2 Þ
1=2

Adlhart, W. (1981). Diffraction by crystals with planar domains. Acta Cryst. A37, 794–801. Amoro´s, J. L. & Amoro´s, M. (1968). Molecular Crystals: Their Transforms and Diffuse Scattering. New York: John Wiley. Arndt, U. W. (1986a). X-ray position sensitive detectors. J. Appl. Cryst. 19, 145–163. Arndt, U. W. (1986b). The collection of single crystal diffraction data with area detectors. J. Phys. (Paris) Colloq. 47(C5), 1–6. Aroyo, M. I., Boysen, H. & Perez-Mato, J. M. (2002). Inelastic neutron scattering selection rules for phonons: application to leucite phase transitions. Appl. Phys. A, 74, S1043–S1048. Axe, J. D. (1980). Debye–Waller factors for incommensurate structures. Phys. Rev. B, 21, 4181–4190. Bardhan, P. & Cohen, J. B. (1976). X-ray diffraction study of short-rangeorder structure in a disordered Au3Cu alloy. Acta Cryst. A32, 597–614. Bauer, G., Seitz, E. & Just, W. (1975). Elastic diffuse scattering of neutrons as a tool for investigation of non-magnetic point defects. J. Appl. Cryst. 8, 162–175. Bauer, G. S. (1979). Diffuse elastic neutron scattering from nonmagnetic materials. In Treatise on Materials Science and Technology, Vol. 15, edited by G. Kostorz, pp. 291–336. New York: Academic Press. Berthold, T. & Jagodzinski, H. (1990). Analysis of the distribution of impurities in crystals by anomalous X-ray scattering. Z. Kristallogr. 193, 85–100. Bessie`re, M., Lefebvre, S. & Calvayrac, Y. (1983). X-ray diffraction study of short-range order in a disordered Au3Cu alloy. Acta Cryst. B39, 145– 153. Beyeler, H. U., Pietronero, L. & Stra¨ssler, S. (1980). Configurational model for a one-dimensional ionic conductor. Phys. Rev. B, 22, 2988– 3000.


1

 expf
H 0
ðT 2 þ H 2 Þ1=2 =2ðM2 þ D2 Þg dH:

ð4:2:8:15Þ

The integral has to be evaluated numerically. If ðM 2 þ D2 Þ is not too large, the term 1=ðT 2 þ H 2 Þ varies only slowly compared to the exponential term and may be kept outside the integral, setting it approximately to 1=H02. (c) Diffuse plane (with R2 ¼ Hx2 þ Hy2 ) R 2 ¼ ðM2 þ D2 Þ
1=2 R2 =ðT 2 þ R2 Þ  expf
H0
ðT 2 þ R2 Þ1=2 =2ðM 2 þ D2 Þg dR:

ð4:2:8:16Þ

With the same approximation as in (b) the expression may be simplified to 2 ¼
=H 0 ½1
erffðT
H 0 Þ=½2ðM 2 þ D2 Þ1=2 g þ 1=H 20 ½2
ðM 2 þ D2 Þ1=2  expf
ðH 0
TÞ2 =2ðM2 þ D2 Þg:

ð4:2:8:17Þ

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539

references

International Tables for Crystallography (2010). Vol. B, Chapter 4.3, pp. 540–546.

4.3. Diffuse scattering in electron diffraction By J. M. Cowley† and J. K. Gjønnes

The kinematical description can be used for electron scattering only when the crystal is very thin (10 nm or less) and composed of light atoms. For heavy atoms such as Au or Pb, crystals of thickness 1 nm or more in principal orientations show strong deviations from kinematical behaviour. With increasing thickness, dynamical scattering effects first modify the sharp Bragg reflections and then have increasingly significant effects on the diffuse scattering. Bragg scattering of the diffuse scattering produces Kikuchi lines and other effects. Multiple diffuse scattering broadens the distribution and smears out detail. As the thickness increases further, the diffuse scattering increases and the Bragg beams are reduced in intensity until there is only a diffuse ‘channelling pattern’ where the features depend in only a very indirect way on the incident-beam direction or on the sources of the diffuse scattering (Uyeda & Nonoyama, 1968). The multiple-scattering effects make the quantitative interpretation of diffuse scattering more difficult and complicate the extraction of particular components, e.g. disorder scattering. Much of the multiple scattering involves inelastic scattering processes. However, electrons that have lost energy of the order of 1 eV or more can be subtracted experimentally by use of electron energy filters (Krahl et al., 1990; Krivanek et al., 1992) which are commercially available. Measurement can be made also of the complete scattering function Iðu; Þ, but such studies have been rare. Another significant improvement to quantitative measurement of diffuse electron scattering is offered by new recording devices: slow-scan charge-couple-device cameras (Krivanek & Mooney, 1993) and imaging plates (Mori et al., 1990). There are some advantages in the use of electrons which make it uniquely valuable for particular applications.

4.3.1. Introduction The origins of diffuse scattering in electron-diffraction patterns are the same as in the X-ray case: inelastic scattering due to electronic excitations, thermal diffuse scattering (TDS) from atomic motions, scattering from crystal defects or disorder. For diffraction by crystals, the diffuse scattering can formally be described in terms of a nonperiodic deviation
’ from the periodic, average crystal potential, ’ : ’ðr; tÞ ¼ ’ ðrÞ þ
’ðr; tÞ;

ð4:3:1:1Þ

where
’ may have a static component from disorder in addition to time-dependent fluctuations of the electron distribution or atomic positions. In the kinematical case, the diffuse scattering can be treated separately. The intensity Id as a function of the scattering variable u ðjuj ¼ 2 sin
=Þ and energy transfer h is then given by the Fourier transform F of
’ Iðu; Þ ¼ j
ðuÞj2 ¼ jF f
’ðr; tÞgj2 ¼ F fPd ðr; Þg

ð4:3:1:2Þ

and may also be written as the Fourier transform of a correlation function Pd representing fluctuations in space and time (see Cowley, 1981). When the energy transfers are small – as with TDS – and hence not measured, the observed intensity corresponds to an integral over : IðuÞ ¼ Id ðuÞ þ Iav ðuÞ R Id ðuÞ ¼ Id ðu; Þ d ¼ F fPd ðr; 0Þg and also Id ðuÞ ¼ hjðuÞj2 i
jhðuÞij2 ;

ð4:3:1:3Þ

where the brackets may indicate a time average, an expectation value, or a spatial average over the periodicity of the lattice in the case of static deviations from a periodic structure. The considerations of TDS and static defects and disorder of Chapters 4.1 and 4.2 thus may be applied directly to electron diffraction in the kinematical approximation when the differences in experimental conditions and diffraction geometry are taken into account. The most prominent contribution to the diffuse background in electron diffraction, however, is the inelastic scattering at low angles arising mainly from the excitation of outer electrons. This is quite different from the X-ray case where the inelastic (‘incoherent’) scattering, SðuÞ, goes to zero at small angles and increases to a value proportional to Z for high values of juj. The difference is due to the Coulomb nature of electron scattering, which leads to the kinematical intensity expression S=u4 , emphasizing the small-angle region. At high angles, the inelastic scattering from an atom is then proportional to Z=u4, which is considerably less than the corresponding elastic scattering ðZ
f Þ2 =u4 which approaches Z2 =u4 (Section 2.5.2) (see Fig. 4.3.1.1).

Fig. 4.3.1.1. Comparison between the kinematical inelastic scattering (full line) and elastic scattering (broken) for electrons and X-rays. Values for silicon [Freeman (1960) and IT C (2004)].

† Deceased.

Copyright © 2010 International Union of Crystallography

540

4.3. DIFFUSE SCATTERING IN ELECTRON DIFFRACTION Iðu; Þ ¼

m3 k 22 h6 ko  WðuÞ

X

Pn o

Z XX jhno j expf2iu  Rj gjnij2 

j¼1


 En
Eno  þ h

ð4:3:2:1Þ

for the intensity of scattering as function of energy transfer and momentum transfer from a system of Z identical particles, Rj . Here m and h have their usual meanings; ko and k, Eno and En are wavevectors and energies before and after the scattering between object states no and n; Pno are weights of the initial states; W(u) is a form factor (squared) for the individual particle. In equation (4.3.2.1), u is essentially momentum transfer. When the energy transfer is small ð
E=E 
Þ, we can still write juj ¼ 2 sin
=, then the sum over final states n is readily performed and an expression of the Waller–Hartree type is obtained for the total inelastic scattering as a function of angle: Iinel ðuÞ / Fig. 4.3.1.2. Electron-diffraction pattern from a disordered crystal of 17Nb2O548WO3 close to the [001] orientation of the tetragonal tungstenbronze-type structure (Iijima & Cowley, 1977).

S ; u4

where

(1) Diffuse-scattering distributions can be recorded from very small specimen regions, a few nm in diameter and a few nm thick. The diameter of the specimen area may be varied readily up to several mm. (2) Diffraction information on defects or disorder may be correlated with high-resolution electron-microscope imaging of the same specimen area [see Section 4.3.8 in IT C (2004)]. (3) The electron-diffraction pattern approximates to a planar section of reciprocal space, so that complicated configurations of diffuse scattering may be readily visualized (see Fig. 4.3.1.2). (4) Dynamical effects may be exploited to obtain information about localization of sources of the diffuse scattering within the unit cell. These experimental and theoretical aspects of electron diffraction have influenced the ways in which it has been applied in studies of diffuse scattering. In general, we may distinguish three different approaches to the interpretation of diffuse scattering: (a) The crystallographic way, in which the Patterson- or correlation-function representation of the local order is emphasized, e.g. by use of short-range-order parameters. (b) The physical model in terms of excitations. These are usually described in reciprocal (momentum) space: phonons, plasmons etc. (c) Structure models in direct space. These must be derived by trial or by chemical considerations of bonds, coordinates etc. Owing to the difficulties of separating the different components in the diffuse scattering, most work on diffuse scattering of electrons has followed one or both of the two last approaches, although Patterson-type interpretation, based upon kinematical scattering including some dynamical corrections, has also been tried.

SðuÞ ¼ Z


Z P j¼1

j fjj ðuÞj2


Z P Z P

j fjk ðuÞj2 ;

ð4:3:2:2Þ

j 6¼k

and where the one-electron f’s for Hartree–Fock orbitals, fjk ðuÞ ¼ hjj expð2iu  rÞjki, have been calculated by Freeman (1959, 1960) for atoms up to Z ¼ 30. The last sum is over electrons with the same spin only. The Waller–Hartree formula may be a very good approximation for Compton scattering of X-rays, where most of the scattering occurs at high angles and multiple scattering is no problem. With electrons, it has several deficiencies. It does not take into account the electronic structure of the solid, which is most important at low values of u. It does not include the energy distribution of the scattering. It does not give a finite cross section at zero angle, if u is interpreted as an angle. In order to remedy this, we should go back to equation (4.3.1.2) and decompose u into two components, one tangential part which is associated with angle in the usual way and one normal component along the beam direction, un , which may be related to the excitation energy
E ¼ En
Eno by the expression un ¼
Ek =2E. This will introduce a factor 1=ðu2 þ u2n Þ in the intensity at small angles, often written as 1=ð
2 þ
E2 Þ, with
E estimated from ionization energies etc. (Strictly speaking,
E is not a constant, not even for scattering from one shell. It is a weighted average which will vary with u.) Calculations beyond this simple adjustment of the Waller– Hartree-type expression are few. Plasmon scattering has been treated on the basis of a nearly free electron model by Ferrel (1957): d2  ¼ ð1=2 aH mv2 NÞð
Imf1="gÞ=ð
2 þ
E2 Þ; dð
EÞ d

4.3.2. Inelastic scattering In the kinematical approximation, a general expression which includes inelastic scattering can be written in the form quoted by Van Hove (1954)

ð4:3:2:3Þ

where m, v are relativistic mass and velocity of the incident electron, N is the density of the valence electrons and "ð
E;
Þ their dielectric constant. Upon integration over
E:

541

4. DIFFUSE SCATTERING AND RELATED TOPICS Ep d ¼ ½1=ð
2 þ
E2 ÞGð
;
c Þ; d 2aH mvN

Id ðuÞ ¼  2 j
ðuÞ  av ðuÞj2 :

ð4:3:2:4Þ

Thus, the kinematical diffuse-scattering amplitude is convoluted with the amplitude function for the average structure, i.e. the set of sharp Bragg beams. When the direct beam, av ð0Þ, is relatively strong, the kinematical diffuse scattering will be modified to only a limited extent by convolution with the Bragg reflections. To the extent that the diffuse scattering is periodic in reciprocal space, the effect will be to modify the intensity by a slowly varying function. Thus the shapes of local diffuse maxima will not be greatly affected. The electron-microscope image contrast derived from the diffuse scattering will be obtained by inserting equation (4.3.3.4) in the appropriate intensity expressions of Section 4.3.8 of IT C (2004). Another approach may be used for extended crystal defects in thin films, e.g. faults normal or near-normal to the film surface. Often, an average periodic structure may not readily be defined, as in the case of a set of incommensurate stacking faults. Kinematically, the projection of the structure in the simplest case may be described by convoluting the projection of a unit-cell structure with a nonperiodic set of delta functions which constitute a distribution function:

where Gð
;
c Þ takes account of the cut-off angle c. Inner-shell excitations have been studied because of their importance to spectroscopy. The most realistic calculations may be those of Leapman et al. (1980) where one-electron wavefunctions are determined for the excited states in order to obtain ‘generalized oscillator strengths’ which may then be used to modify equation (4.3.1.2). At high energies and high momentum transfer, the scattering will approach that of free electrons, i.e. a maximum at the socalled Bethe ridge, E ¼ h2 u2 =2m. A complete and detailed picture of inelastic scattering of electrons as a function of energy and angle (or scattering variable) is lacking, and may possibly be the least known area of diffraction by solids. It is further complicated by the dynamical scattering, which involves the incident and diffracted electrons and also the ejected atomic electron (see e.g. Maslen & Rossouw, 1984a,b). 4.3.3. Kinematical and pseudo-kinematical scattering Kinematical expressions for TDS or defect and disorder scattering according to equation (4.3.1.3) can be obtained by inserting the appropriate atomic scattering factors in place of the X-ray scattering factors in Chapter 4.1. The complications introduced by dynamical diffraction are considerable (see Section 4.3.4). In the most general case, a complete specification of the disordered structure may be needed. However, for thin specimens, approximate treatments of the deviations from kinematical scattering may lead to relatively simple forms. Two such cases are treated in this section, both relying on the small-angle nature of electron scattering. The first is based upon the phaseobject approximation, which applies to small angles and thin specimens. The amplitude at the exit surface of a specimen can always be written as a sum of a periodic and a nonperiodic part, and may in analogy with the kinematical case [equation (4.3.1.1)] be written ðrÞ ¼  ðrÞ þ
ðrÞ;

’ðrÞ ¼ ’0 ðrÞ 

ðr
rn Þ ¼ ’0 ðrÞ  dðrÞ:

ð4:3:3:6Þ

Then the diffraction-pattern intensity is IðuÞ ¼ j0 ðuÞj2 jDðuÞj2 :

ð4:3:3:7Þ

Here, 0 ðuÞ is the scattering amplitude of the unit whereas the function jDðuÞj2 , where DðuÞ ¼ F fdðrÞg, gives the configuration of spots, streaks or other diffraction maxima corresponding to the faulted structure (see e.g. Marks, 1985). In the projection (column) approximation to dynamical scattering, the wavefunction at the exit surface may be given by an expression identical to (4.3.3.6), but with a wavefunction, 0 ðrÞ, for the unit in place of the projected potential, ’0 ðrÞ. An intensity expression of the same form as (4.3.3.7) then applies, with a dynamical scattering amplitude 0 for the scattering unit substituted for the kinematical amplitude 0.

ð4:3:3:1Þ

IðuÞ ¼ j0 ðuÞj2 jDðuÞj2 ;

ð4:3:3:8Þ

which in the simplest case describes a diffraction pattern with the same features as in the kinematical case. Note that 0 ðuÞ may have different symmetries when the incident beam is tilted away from a zone axis, leading to diffuse streaks etc. appearing also in positions where the kinematical diffuse scattering is zero. More complicated cases have been considered by Cowley (1976a) who applied this type of analysis to the case of nonperiodic faulting in magnesium fluorogermanate (Cowley, 1976b).

ð4:3:3:2Þ

Then the Bragg reflections are given by Fourier transform of the periodic part, viz:   hexpf
i’ðrÞgi ¼ expf
i ’ ðrÞg exp
12 2 h
’2 ðrÞi ;

4.3.4. Dynamical scattering: Bragg scattering effects The distribution of diffuse scattering is modified by higher-order terms in essentially two ways: Bragg scattering of the incident and diffuse beams or multiple diffuse scattering, or by a combination. Theoretical treatment of the Bragg scattering effects in diffuse scattering has been given by many authors, starting with Kainuma’s (1955) work on Kikuchi-line contrast (Howie, 1963; Fujimoto & Kainuma, 1963; Gjønnes, 1966; Rez et al., 1977; Maslen & Rossouw, 1984a,b; Wang, 1995; Allen et al., 1997). Mathematical formalism may vary but the physical pictures and

ð4:3:3:3Þ note that an absorption function is introduced. The diffuse scattering derives from
i
’ðrÞ expf
i ’ ðrÞg;

P n

where r is a vector in two dimensions. The intensities can be separated in the same way [cf. equation (4.3.1.3)]. When the phase-object approximation applies [Section 2.5.2.4(b)] ðrÞ ¼ expf
i’ðrÞg ¼ expf
i ’ ðrÞg½1
i
’ðrÞ
. . .:

ð4:3:3:5Þ

ð4:3:3:4Þ

so that

542

4.3. DIFFUSE SCATTERING IN ELECTRON DIFFRACTION results are essentially the same. They may be discussed with In Kikuchi-line contrast, the scattering in the upper part of the reference to a Born-series expansion, i.e. by introducing the crystal is usually not considered and frequently the angular potential ’ in the integral equation, as a sum of a periodic and a variation of the ðuÞ is also neglected. In diffraction contrast nonperiodic part [cf. equation (4.3.1.1)] and arranging the terms from small-angle inelastic scattering, it may be sufficient to by orders of
’. consider the intraband terms [i ¼ i0 ; j ¼ j0 in (4.3.4.3)]. In studies of diffuse-scattering distribution, the factor hðu þ hÞðu þ h0 Þi will produce two types of terms: Those with ¼ 0 þ G’ h ¼ h0 result only in a redistribution of intensity between ¼ ½1 þ G’ þ ðG’Þ2 þ . . . 0 corresponding points in the Brillouin zones, with the same total intensity. Those with h 6¼ h0 lead to enhancement or reduction of ¼ ½1 þ G’ þ ðG’ Þ2 þ . . . 0 the total diffuse intensity and hence absorption from the Bragg þ ½1 þ G’ þ ðG’ Þ2 þ . . . beam and enhanced/reduced intensity of secondary radiation, i.e. anomalous absorption and channelling effects. They arise  Gð
’Þ½1 þ G’ þ ðG’ Þ2 þ . . . 0 through interference between different Fourier components of þ higher-order terms: ð4:3:4:1Þ the diffuse scattering and carry information about position of the sources of diffuse scattering, referred to the projected unit cell. This is exploited in channelling experiments, where beam direcSome of the higher-order terms contributing to the Bragg tion is used to determine atom reaction (Taftø & Spence, 1982; scattering can be included by adding the essentially imaginary Taftø & Lehmpfuhl, 1982). term h
’Gð
’Þi to the static potential ’ . Gjønnes & Høier (1971) expressed this information in terms of Theoretical treatments have mostly been limited to the firstthe Fourier transform R of the generalized or Kikuchi-line form order diffuse scattering. With the usual approximation to forward factor; scattering, the expression for the amplitude of diffuse scattering in a direction k0 þ u þ g can be written as h
ðuÞ
 ðu þ hÞiu2 ðu þ hÞ2 ¼ Qðu þ hÞ ¼ F fRðr; gÞg PPRz ð4:3:4:5Þ ðu þ gÞ ¼ Shg ðk0 þ u; z
z1 Þ g

f 0


ðu þ g
fÞSf 0 ðk0 ; z1 Þ dz1

ð4:3:4:2Þ

includes information both about correlations between sources of diffuse scattering and about their position in the projected unit cell. It Ris seen that Qðu; 0Þ represents the kinematical intensity, hence Rðr; gÞ dg is the Patterson function. The integral of Qðu; hÞ in the plane gives the anomalous absorption (Yoshioka, 1957) which is related to the distribution Rðr; 0Þ of scattering centres across the unit cell. The scattering factor h
ðuÞ
 ðu
hÞi can be calculated for different modes. For one-electron excitations as an extension of the Waller–Hartree expression (Gjønnes, 1962; Whelan, 1965):

and read (from right to left): Sf 0 , Bragg scattering of the incident beam above the level z1 ;
, diffuse scattering within a thin layer dz1 through the Fourier components
 of the nonperiodic potential
; Shg , Bragg scattering between diffuse beams in the lower part of the crystal. It is commonly assumed that diffuse scattering at different levels can be treated as independent (Gjønnes, 1966), then the intensity expression becomes Iðu þ gÞ ¼

PPPPRz h h0

f

f0 0

Sgh ð2ÞSgh0 ð2Þ

h
ðuÞ
 ðu
hÞi ¼

 h
ðu þ h
fÞ
 ðu þ h0
f 0 Þi  Sf 0 ð1ÞSf 0 0 ð1Þ dz1 ;

ð4:3:4:3Þ

where (1) and (2) refer to the regions above and below the diffuse-scattering layer. This expression can be manipulated further, e.g. by introducing Bloch-wave expansion of the scattering matrices, viz Iðu þ gÞ ¼

PPPP hh0

ff 0

jj0

ii0

0

0

i0

j

h
ðuÞ
 ðu
hÞi ¼ Gj ðu; qÞGj ðu
h; qÞ;

exp½ið
 Þz
exp½ið
 Þz  i
 i0
 j þ  j0  hf ðu þ h
fÞf  ðu þ h0
f 0 Þi 0

ð4:3:4:7Þ

j0



 Cfi ð1ÞCfi 0 ð1ÞC0i ð1ÞC0i ð1Þ;

ð4:3:4:6Þ

where fij are the one-electron amplitudes (Freeman, 1959). A similar expression for scattering by phonons is obtained in terms of the scattering factors Gj ðu; gÞ for the branch j, wavevector g and a polarization vector lj; q (see Chapter 4.1):

Cgj ð2ÞCgj  ð2ÞChj ð2ÞChj 0 ð2Þ

i

X fii ðhÞ
fii ðuÞfii ðjh
ujÞ u2 ðh
uÞ2 i XX fij ðuÞfij ðjh
ujÞ
; u2 ðh
uÞ2 i i6¼j

independent phonons being assumed. For scattering from substitutional order in a binary alloy with ordering on one site only, we obtain simply

ð4:3:4:4Þ


ðuÞ
 ðu
hÞ ¼ j
’ðuÞj2

which may be interpreted as scattering by
’ between Bloch waves belonging to the same branch (intraband scattering) or different branches (interband scattering). Another alternative is to evaluate the scattering matrices by multislice calculations (Section 4.3.5). Expressions such as (4.3.4.2) contain a large number of terms. Unless very detailed calculations relating to a precisely defined model are to be carried out, attention should be focused on the most important terms.

fA ðju
hjÞ
fB ðju
hjÞ ; fA ðuÞ
fB ðuÞ ð4:3:4:8Þ

where fA; B are atomic scattering factors. It is seen that QðuÞ then does not contain any new information; the location of the site involved in the ordering is known. When several sites are involved in the ordering, the dynamical scattering factor becomes less trivial, since scattering factors for

543

4. DIFFUSE SCATTERING AND RELATED TOPICS the different ordering parameters (for different sites) will include filtering of the inelastic component in order to improve the a factor expð2irm  hÞ (see Andersson et al., 1974). quantitative interpretation of diffuse scattering. From the above expressions, it is found that the Bragg scattering will affect diffuse scattering from different sources differently: Diffuse scattering from substitutional order will usually be 4.3.5. Multislice calculations for diffraction and imaging enhanced at low and intermediate angles, whereas scattering The description of dynamical diffraction in terms of the from thermal and electronic fluctuations will be reduced at low progression of a wave through successive thin slices of a crystal angles and enhanced at higher angles. This may be used to study (Chapter 5.2) forms the basis for the multislice method for the substitutional order and displacement order (size effect) sepacalculation of electron-diffraction patterns and electronrately (Andersson, 1979). microscope images [see Section 4.3.6.1 in IT C (2004)]. This The use of such expressions for quantitative or semimethod can be applied directly to the calculations of diffuse quantitative interpretation raises several problems. The Bragg scattering in electron diffraction due to thermal motion and scattering effects occur in all diffuse components, in particular the positional disorder and for calculating the images of defects in inelastic scattering, which thus may no longer be represented by a crystals. smooth, monotonic background. It is best to eliminate this It is essentially an amplitude calculation based on the formuexperimentally. When this cannot be done, the experiment should lation of equation (4.3.4.1) [or (4.3.4.2)] for first-order diffuse be arranged so as to minimize Kikuchi-line excess/deficient terms, scattering. The Bragg scattering in the first part of the crystal is by aligning the incident beam along a not too dense zone. In this calculated using a standard multislice method for the set of beams way, one may optimize the diffuse-scattering information and h. In the nth slice of the crystal, a diffuse-scattering amplitude minimize the dynamical corrections, which then are used partly as d ðuÞ is convoluted with the incident set of Bragg beams. For guides to conditions, partly as refinement in calculations. each u, propagation of the set of beams u þ h is then calculated The multiple scattering of the background remains as the most through the remaining slices of the crystal. The intensities for the serious problem. Theoretical expressions for multiple scattering exit wave at the set of points u þ h are then calculated by adding in the absence of Bragg scattering have been available for some either amplitudes or intensities. Amplitudes are added if there is time (Moliere, 1948), as a sum of convolution integrals correlation between the defects in successive slices. Intensities are added if there are no such correlations. The process is IðuÞ ¼ ½ð tÞI1 ðuÞ þ ð1=2Þð tÞ2 I2 ðuÞ þ . . . expð
tÞ; ð4:3:4:9Þ repeated for all u values to obtain a complete mapping of the diffuse scattering. Calculations have been made in this way, for example, for where I2 ðuÞ ¼ I1 ðuÞ  I1 ðuÞ . . . etc., and I1 ðuÞ is normalized. short-range order in alloys [Fisher (1969); see also Cowley (1981) A complete description of multiple scattering in the presence ch. 17] and also for TDS on the assumption of both correlated of Bragg scattering should include Bragg scattering between and uncorrelated atomic motions (Doyle, 1969). The effects of diffuse scattering at all levels z1 ; z2, etc. This quickly becomes the correlations were shown to be small. unwieldy. Fortunately, the experimental patterns seem to indicate This computing method is not practical for electronthat this is not necessary: The Kikuchi-line contrast does not microscope images in which individual defects are to be imaged. appear to be very sensitive to the exact Bragg condition of the The perturbations of the exit wavefunction due to individual incident beam. Høier (1973) therefore introduced Bragg scatdefects (vacancies, replaced atoms, displaced atoms) or small tering only in the last part of the crystal, i.e. between the level zn groups of defects may then be calculated with arbitrary accuracy and the final thickness z for n-times scattering. He thus obtained by use of the ‘periodic continuation’ form of the multislice the formula: computer programs in which an artificial, large, superlattice unit cell is assumed [Section 4.3.6.1 in IT C (2004)]. The corre( sponding images and microdiffraction patterns from the indiviP j 2 jPP 0 j j Iðu þ hÞ ¼ jCh j A1 F1 ðu; g; g ÞCg Cg0 dual defects or clusters may then be calculated (Fields & Cowley, j g 6¼g0 1978). A more recent discussion of the image calculations, ) particularly in relation to thermal diffuse scattering, is given by PP j Ang Fn ðu; gÞjCgj j2 ; ð4:3:4:10Þ þ Cowley (1988). n g In order to calculate the diffuse-scattering distributions from disordered systems or from a crystal with atoms in thermal motion by the multislice method with periodic continuation, it where Fn are normalized scattering factors for nth-order multiple would be necessary to calculate for a number of different defect j diffuse scattering and An are multiple-scattering coefficients configurations sufficiently large to provide an adequate reprewhich include absorption. sentation of the statistics of the disordered system. However, it When the thickness is increased, the variation of Fn ðu; gÞ with has been shown by Cowley & Fields (1979) that, if the singleangle becomes slower, and an expression for intensity of the diffuse-scattering approximation is made, the perturbations of channelling pattern is obtained (Gjønnes & Taftø, 1976): the exit wave due to individual defects are characteristic of the defect type and of the slice number and may be added, so that a PPP j 2 j 2 j Iðu þ hÞ ¼ jCh j jCg j An considerable simplification of the computing process is possible. j g n Methods for calculating diffuse scattering in electron-diffraction P j 2P j ¼ jCh j An ! jChj j2 = j ðuÞ: ð4:3:4:11Þ patterns using the multislice approach are described by Tanaka & j n Cowley (1987) and Cowley (1989). Loane et al. (1991) introduced the concept of ‘frozen phonons’ for multislice calculations of thermal scattering. Another approach is the use of a modified diffusion equation (Ohtsuki et al., 1976). These expressions seem to reproduce the development of the 4.3.6. Qualitative interpretation of diffuse scattering of electrons general background with thickness over a wide range of thickQuantitative interpretation of the intensity of diffuse scattering nesses. It may thus appear that the contribution to the diffuse by calculation of e.g. short-range-order parameters has been the background from known sources can be treated adequately – and exception. Most studies have been directed to qualitative features that such a procedure must be included together with adequate

544

4.3. DIFFUSE SCATTERING IN ELECTRON DIFFRACTION and their variation with composition, treatment etc. Many Imaging of local variations in the SRO structure has been features in the scattering which pass unrecognized in extensive pursued with different techniques (De Ridder et al., 1976; Tanaka X-ray or neutron investigations will be observed readily with & Cowley, 1985; De Meulenaare et al., 1998), viz: dark field using electrons, frequently inviting other ways of interpretation. diffuse spots only; bright field with the central spot plus diffuse Most such studies have been concerned with substitutional spots; lattice image. With domains of about 3 nm or more, highdisorder, but the extensive investigations of thermal streaks by resolution images seem to give clear indication of their presence Honjo and co-workers should be mentioned (Honjo et al., 1964). and form. For smaller ordered regions, the interpretation Diffuse spots and streaks from disorder have been observed from becomes increasingly complex: Since the domains will then a wide range of substances. The most frequent may be streaks due usually not extend through the thickness of the foil, they cannot to planar faults, one of the most common objects studied by be imaged separately. Since image-contrast calculations essenelectron microscopy. Diffraction patterns are usually sufficient to tially demand complete specification of the local structure, a determine the orientation and the fault vector; the positions and model beyond the statistical description must be constructed in distribution of faults are more easily seen by dark-field microorder to be compared with observations. On the other hand, scopy, whereas the detailed atomic arrangement is best studied by these models of the local structure should be consistent with the high-resolution imaging of the structure [Section 4.3.8 in IT C statistics derived from diffraction patterns collected from a larger (2004)]. volume. This combination of diffraction and different imaging techniques cannot be applied in the same way to the study of the essentially three-dimensional substitutional local order. ConsidReferences erable effort has therefore been made to interpret the details of Allen, L. J., Josefsson, T. W., Lehmpfuhl, G. & Uchida, Y. (1997). diffuse scattering, leaving the determination of the short-rangeModeling thermal diffuse scattering in electron diffraction involving order (SRO) parameters usually to X-ray or neutron studies. higher-order Laue zones. Acta Cryst. A53, 421–425. Frequently, characteristic shapes or splitting of the diffuse Andersson, B. (1979). Electron diffraction study of diffuse scattering due spots from e.g. binary alloys are observed. They reflect order to atomic displacements in disordered vanadium monoxide. Acta Cryst. extending over many atomic distances, and have been assumed to A35, 718–727. arise from forces other than the near-neighbour pair forces Andersson, B., Gjønnes, J. & Taftø, J. (1974). Interpretation of short range invoked in the theory of local order. A relationship between the order scattering of electrons; application to ordering of defects in diffuse-scattering distribution and the Fourier transform of the vanadium monoxide. Acta Cryst. A30, 216–224. effective atom-pair-interaction potential is given by the ordering Clapp, P. C. & Moss, S. C. (1968). Correlation functions of disordered binary alloys. II. Phys. Rev. 171, 754–763. theory of Clapp & Moss (1968). An interpretation in terms of Cowley, J. M. (1976a). Diffraction by crystals with planar faults. I. General long-range forces carried by the conduction electrons was theory. Acta Cryst. A32, 83–87. proposed by Krivoglaz (1969). Extensive studies of alloy systems Cowley, J. M. (1976b). Diffraction by crystals with planar faults. II. (Ohshima & Watanabe, 1973) show that the separations, m, Magnesium fluorogermanate. Acta Cryst. A32, 88–91. observed in split diffuse spots from many alloys follow the Cowley, J. M. (1981). Diffraction Physics, 2nd ed. Amsterdam: Northpredicted variation with the electron/atom ratio e=a: Holland. Cowley, J. M. (1988). Electron microscopy of crystals with time-dependent perturbations. Acta Cryst. A44, 847–853. Cowley, J. M. (1989). Multislice methods for surface diffraction and inelastic scattering. In Computer Simulation of Electron Microscope Diffraction and Images, edited by W. Krakow & M. O’Keefe, pp. 1–12. Worrendale: The Minerals, Metals and Materials Society. Cowley, J. M. & Fields, P. M. (1979). Dynamical theory for electron scattering from crystal defects and disorder. Acta Cryst. A35, 28–37. De Meulenaare, P., Rodewald, M. & Van Tendeloo, G. (1998). Anisotropic cluster model for the short-range order in Cu1
x–Pdx-type alloys. Phys. Rev. B, 57, 11132–11140. De Ridder, R., Van Tendeloo, G. & Amelinckx, S. (1976). A cluster model for the transition from the short-range order to the long-range order state in f.c.c. based binary systems and its study by means of electron diffraction. Acta Cryst. A32, 216–224. Doyle, P. A. (1969). Dynamical calculation of thermal diffuse scattering. Acta Cryst. A25, 569–577. Ferrel, R. A. (1957). Characteristic energy loss of electrons passing through metal foils. Phys. Rev. 107, 450–462. Fields, P. M. & Cowley, J. M. (1978). Computed electron microscope images of atomic defects in f.c.c. metals. Acta Cryst. A34, 103–112. Fisher, P. M. J. (1969). The development and application of an n-beam dynamic methodology in electron diffraction. PhD thesis, University of Melbourne. Freeman, A. J. (1959). Compton scattering of X-rays from non-spherical charge distributions. Phys. Rev. 113, 169–175. Freeman, A. J. (1960). X-ray incoherent scattering functions for nonspherical charge distributions. Acta Cryst. 12, 929–936. Fujimoto, F. & Kainuma, Y. (1963). Inelastic scattering of fast electrons by thin crystals. J. Phys. Soc. Jpn, 18, 1792–1804. Gjønnes, J. (1962). Inelastic interaction in dynamic electron scattering. J. Phys. Soc. Jpn, 17, Suppl. BII, 137–139. Gjønnes, J. (1966). The influence of Bragg scattering on inelastic and other forms of diffuse scattering of electrons. Acta Cryst. 20, 240–249. Gjønnes, J. & Høier, R. (1971). Structure information from anomalous absorption effects in diffuse scattering of electrons. Acta Cryst. A27, 166–174.

 1=3 pffiffiffi 12 m¼ ðe=aÞ t
2;  where m is measured along the [110] direction in units of 2a and t is a truncation factor for the Fermi surface. A similarity between the location of diffuse maxima and the shape of the Fermi surface has been noted also for other structures, notably some defect rock-salt-type structures. Although this may offer a clue to the forces involved in the ordering, it entails no description of the local structure. Several attempts have been made to formulate principles for building the disordered structure, from small ordered domains embedded in less ordered regions (Hashimoto, 1974), by a network of antiphase boundaries, or by building the structure from clusters with the average composition and coordination (De Ridder et al., 1976). Evidence for such models may be sought by computer simulations, in the details of the SRO scattering as seen in electron diffraction, or in images. The cluster model is most directly tied to the location of diffuse scattering, noting that a relation between order parameters derived from clusters consistent with the ordered state can be used to predict the position of diffuse scattering in the form of surfaces in reciprocal space, e.g. the relation cos h þ cos k þ cos l ¼ 0 for ordering of octahedral clusters in the rock-salt-type structure (Sauvage & Parthe´, 1974). Some of the models imply local fluctuations in order which may be observable either by diffraction from very small regions or by imaging. Microdiffraction studies (Tanaka & Cowley, 1985) do indeed show that spots from 1–1.5 nm regions in disordered LiFeO2 appear on the locus of diffuse maxima observed in diffraction from larger areas.

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4. DIFFUSE SCATTERING AND RELATED TOPICS Maslen, W. V. & Rossouw, C. J. (1984b). Implications of (e, 2e) scattering electron diffraction in crystals: II. Philos. Mag. A, 49, 743–757. Moliere, G. (1948). Theory of scattering of fast charged particles: plural and multiple scattering. Z. Naturforsch. Teil A, 3, 78–97. Mori, M., Oikawa, T. & Harada, Y. (1990). Development of the imaging plate for the transmission electron microscope and its characteristics. J. Electron Microsc. (Jpn), 19, 433–436. Ohshima, K. & Watanabe, D. (1973). Electron diffraction study of shortrange-order diffuse scattering from disordered Cu–Pd and Cu–Pt alloys. Acta Cryst. A29, 520–525. Ohtsuki, Y. H., Kitagaku, M., Waho, T. & Omura, T. (1976). Dechannelling theory with the Fourier–Planck equation and a modified diffusion coefficient. Nucl. Instrum Methods, 132, 149–151. Rez, P., Humphreys, C. J. & Whelan, M. J. (1977). The distribution of intensity in electron diffraction patterns due to phonon scattering. Philos. Mag. 35, 81–96. Sauvage, M. & Parthe´, E. (1974). Prediction of diffuse intensity surfaces in short-range-ordered ternary derivative structures based on ZnS, NaCl, CsCl and other structures. Acta Cryst. A30, 239–246. Taftø, J. & Lehmpfuhl, G. (1982). Direction dependence in electron energy loss spectroscopy from single crystals. Ultramicroscopy, 7, 287–294. Taftø, J. & Spence, J. C. H. (1982). Atomic site determination using channelling effect in electron induced X-ray emission. Ultramicroscopy, 9, 243–247. Tanaka, N. & Cowley, J. M. (1985). High resolution electron microscopy of disordered lithium ferrites. Ultramicroscopy, 17, 365–377. Tanaka, N. & Cowley, J. M. (1987). Electron microscope imaging of short range order in disordered alloys. Acta Cryst. A43, 337–346. Uyeda, R. & Nonoyama, M. (1968). The observation of thick specimens by high voltage electron microscopy. Jpn. J. Appl. Phys. 1, 200–208. Van Hove, L. (1954). Correlations in space and time and Born approximation scattering in systems of interacting particles. Phys. Rev. 95, 249–262. Wang, Z. L. (1995). Elastic and Inelastic Scattering in Electron Diffraction and Imaging. New York: Plenum Press. Whelan, M. (1965). Inelastic scattering of fast electrons by crystals. I. Interband excitations. J. Appl. Phys. 36, 2099–2110. Yoshioka, H. (1957). Effect of inelastic waves on electron diffraction. J. Phys. Soc. Jpn, 12, 618–628.

Gjønnes, J. & Taftø, J. (1976). Bloch wave treatment of electron channelling. Nucl. Instrum. Methods, 133, 141–148. Hashimoto, S. (1974). Correlative microdomain model for short range ordered alloy structures. I. Diffraction theory. Acta Cryst. A30, 792– 798. Høier, R. (1973). Multiple scattering and dynamical effects in diffuse electron scattering. Acta Cryst. A29, 663–672. Honjo, G., Kodera, S. & Kitamura, N. (1964). Diffuse streak diffraction patterns from single crystals. I. General discussion and aspects of electron diffraction diffuse streak patterns. J. Phys. Soc. Jpn, 19, 351– 369. Howie, A. (1963). Inelastic scattering of electrons by crystals. I. The theory of small angle inelastic scattering. Proc. Phys. Soc. A, 271, 268–287. Iijima, S. & Cowley, J. M. (1977). Study of ordering using high resolution electron microscopy. J. Phys. (Paris), 38, Suppl. C7, 21–30. International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers. Kainuma, Y. (1955). The theory of Kikuchi patterns. Acta Cryst. 8, 247– 257. Krahl, D., Pa¨tzold, H. & Swoboda, M. (1990). An aberration-minimized imaging energy filter of simple design. Proceedings of the 12th International Conference on Electron Microscopy, Vol. 2, pp. 60–61. Krivanek, O. L., Gubbens, A. J., Dellby, N. & Meyer, C. E. (1992). Design and first applications of a post-column imaging filter. Micros. Microanal. Microstruct. (France), 3, 187–199. Krivanek, O. L. & Mooney, P. E. (1993). Applications of slow-scan CCD cameras in transmission electron microscopy. Ultramicroscopy, 49, 95– 108. Krivoglaz, M. A. (1969). Theory of X-ray and Thermal Neutron Scattering by Real Crystals. New York: Plenum. Leapman, R. D., Rez, P. & Mayers, D. F. (1980). L and M shell generalized oscillator strength and ionization cross sections for fast electron collisions. J. Chem. Phys. 72, 1232–1243. Loane, R. F., Xu, P. R. & Silcox, J. (1991). Thermal vibrations in convergent-beam electron diffraction. Acta Cryst. A47, 267–278. Marks, L. D. (1985). Image localization. Ultramicroscopy, 18, 33–38. Maslen, W. V. & Rossouw, C. J. (1984a). Implications of (e, 2e) scattering electron diffraction in crystals: I. Philos. Mag. A, 49, 735–742.

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International Tables for Crystallography (2010). Vol. B, Chapter 4.4, pp. 547–566.

4.4. Scattering from mesomorphic structures By P. S. Pershan

most thoroughly documented mesomorphic phases, there are others not included in the table which we will discuss below. The progression from the completely symmetric isotropic liquid through the mesomorphic phases into the crystalline phases can be described in terms of three separate types of order. The first, or the molecular orientational order, describes the fact that the molecules have some preferential orientation analogous to the spin orientational order of ferromagnetic materials. In the present case, the molecular quantity that is oriented is a symmetric second-rank tensor, like the moment of inertia or the electric polarizability, rather than a magnetic moment. This is the only type of long-range order in the nematic phase and as a consequence its physical properties are those of an anisotropic fluid; this is the origin of the name liquid crystal. Fig. 4.4.1.2(a) is a schematic illustration of the nematic order if it is assumed that the molecules can be represented by oblong ellipses. The average orientation of the ellipses is aligned; however, there is no longrange order in the relative positions of the ellipses. Nematic phases are also observed for disc-shaped molecules and for clusters of molecules that form micelles. These all share the common properties of being optically anisotropic and fluid-like, without any long-range positional order. The second type of order is referred to as bond orientational order. Consider, for example, the fact that for dense packing of spheres on a flat surface most of the spheres will have six neighbouring spheres distributed approximately hexagonally around it. If a perfect two-dimensional triangular lattice of indefinite size were constructed of these spheres, each hexagon on the lattice would be oriented in the same way. Within the last few years, we have come to recognize that this type of order, in which the hexagons are everywhere parallel to one another, is possible even when there is no lattice. This type of order is referred to as bond orientational order, and bond orientational order in the absence of a lattice is the essential property defining the hexatic phases (Halperin & Nelson, 1978; Nelson & Halperin, 1979; Young, 1979; Birgeneau & Litster, 1978). The third type of order is the positional order of an indefinite lattice of the type that defines the 230 space groups of conven-

4.4.1. Introduction The term mesomorphic is derived from the prefix ‘meso-’, which is defined in the dictionary as ‘a word element meaning middle’, and the term ‘-morphic’, which is defined as ‘an adjective termination corresponding to morph or form’. Thus, mesomorphic order implies some ‘form’, or order, that is ‘in the middle’, or intermediate between that of liquids and crystals. The name liquid crystalline was coined by researchers who found it to be more descriptive, and the two are used synonymously. It follows that a mesomorphic, or liquid-crystalline, phase must have more symmetry than any one of the 230 space groups that characterize crystals. A major source of confusion in the early liquid-crystal literature was concerned with the fact that many of the molecules that form liquid crystals also form true three-dimensional crystals with diffraction patterns that are only subtly different from those of other liquid-crystalline phases. Since most of the original mesomorphic phase identifications were performed using a ‘miscibility’ procedure, which depends on optically observed changes in textures accompanying variation in the sample’s chemical composition, it is not surprising that some three-dimensional crystalline phases were mistakenly identified as mesomorphic. Phases were identified as being either the same as, or different from, phases that were previously observed (Liebert, 1978; Gray & Goodby, 1984), and although many of the workers were very clever in deducing the microscopic structure responsible for the microscopic textures, the phases were labelled in the order of discovery as smectic-A, smectic-B etc. without any attempt to develop a systematic nomenclature that would reflect the underlying order. Although different groups did not always assign the same letters to the same phases, the problem is now resolved and the assignments used in this article are commonly accepted (Gray & Goodby, 1984). Fig. 4.4.1.1 illustrates the way in which increasing order can be assigned to the series of mesomorphic phases in three dimensions listed in Table 4.4.1.1. Although the phases in this series are the

Table 4.4.1.1. Some of the symmetry properties of the series of threedimensional phases described in Fig. 4.4.1.1 The terms LRO and SRO imply long-range or short-range order, respectively, and QLRO refers to ‘quasi-long-range order’ as explained in the text.

Fig. 4.4.1.1. Illustration of the progression of order throughout the sequence of mesomorphic phases that are based on ‘rod-like’ molecules. The shaded section indicates phases in which the molecules are tilted with respect to the smectic layers.

Copyright © 2010 International Union of Crystallography

Phase

Molecular orientation order within layer

Bond orientation order

Positional order Normal to layer

Within layer

Smectic-A (SmA) Smectic-C (SmC)

SRO LRO

SRO LRO†

SRO SRO

SRO SRO

Hexatic-B Smectic-F (SmF) Smectic-I (SmI)

LRO† LRO LRO

LRO LRO LRO

QLRO QLRO QLRO

SRO SRO SRO

Crystalline-B (CrB) Crystalline-G (CrG) Crystalline-J (CrJ)

LRO LRO LRO

LRO LRO LRO

LRO LRO LRO

LRO LRO LRO

Crystalline-E (CrE) Crystalline-H (CrH) Crystalline-K (CrK)

LRO LRO LRO

LRO LRO LRO

LRO LRO LRO

LRO LRO LRO

† Theoretically, the existence of LRO in the molecular orientation, or tilt, implies that there must be some LRO in the bond orientation and vice versa.

547

4. DIFFUSE SCATTERING AND RELATED TOPICS

Fig. 4.4.1.2. Schematic illustration of the real-space molecular order and the scattering cross sections in reciprocal space for the: (a) nematic; (b) smectic-A; and (c), (d) smectic-C phases. The scattering cross sections are enclosed in the boxes. Part (c) indicates the smectic-C phase for an oriented monodomain and (d) indicates a polydomain smectic-C structure in which the molecular axes are aligned.

tional crystals. In view of the fact that some of the mesomorphic phases have a layered structure, it is convenient to separate the positional order into the positional order along the layer normal and perpendicular to it, or within the layers. Two of the symmetries listed in Tables 4.4.1.1 and 4.4.1.2 are short-range order (SRO), implying that the order is only correlated over a finite distance such as for a simple liquid, and longrange order (LRO) as in either the spin orientation of a ferromagnet or the positional order of a three-dimensional crystal. The third type of symmetry, ‘quasi-long-range order’ (QLRO), will be explained below. In any case, the progressive increase in symmetry from the isotropic liquid to the crystalline phases for this series of mesomorphic phases is illustrated in Fig. 4.4.1.1. One objective of this chapter is to describe the reciprocal-space structure of the phases listed in the tables and the phase transitions between them. Finally, in most of the crystalline phases that we wish to discuss, the molecules have considerable amounts of rotational disorder. For example, one series of molecules that form mesomorphic phases consists of long thin molecules which might be described as ‘blade shaped’. Although the cross section of these molecules is quite anisotropic, the site symmetry of the molecule is often symmetric, as though the molecule is rotating freely about its

long axis. On cooling, many of the mesomorphic systems undergo transitions to the phases, listed at the bottom of Fig. 4.4.1.1, for which the site symmetry is anisotropic as though some of the rotational motions about the molecular axis have been frozen out. A similar type of transition, in which rotational motions are frozen out, occurs on cooling systems such as succinonitrile (NCCH2CH2CN) that form optically isotropic ‘plastic crystals’ (Springer, 1977). There are two broad classes of liquid-crystalline systems, the thermotropic and the lyotropic, and, since the former are much better understood, this chapter will emphasize results on thermotropic systems (Liebert, 1978). The historical difference between these two, and also the origin of their names, is that the lyotropic are always mixtures, or solutions, of unlike molecules in which one is a normal, or nonmesogenic, liquid. Solutions of soap and water are prototypical examples of lyotropics, and their mesomorphic phases appear as a function of either concentration or temperature. In contrast, the thermotropic systems are usually formed from a single chemical component, and the mesomorphic phases appear primarily as a function of temperature changes. The molecular distinction between the two is that one of the molecules in the lyotropic solution always has a hydrophilic part, often called the ‘head group’, and one or more hydrophobic alkane chains called ‘tails’. These molecules will often form mesomorphic phases as single-component or neat systems; however, the general belief is that in solution with either water or oil most of the phases are the result of competition between the hydrophilic and hydrophobic interactions, as well as other factors such as packing and steric constraints (Pershan, 1979; Safran & Clark, 1987). To the extent that molecules that form thermotropic liquid-crystalline phases have hydrophilic and hydrophobic parts, the disparity in the affinity of these parts for either water or oil is much less and most of these molecules are relatively insoluble in water. These molecules are called thermotropic because their phase transformations are primarily studied only as a function of temperature. This is not to say that there are not numerous examples of interesting studies of the concentration dependence of phase diagrams involving mixtures of thermotropic liquid crystals. Fig. 4.4.1.3 displays some common examples of molecules that form lyotropic and thermotropic phases. In spite of the above remarks, it is interesting to observe that different parts of typical

Table 4.4.1.2. The symmetry properties of the two-dimensional hexatic and crystalline phases

Phase

Molecular orientation order within layer

Bond orientation order

Positional order within layer

Smectic-A (SmA) Smectic-C (SmC)

SRO QLRO

SRO QLRO

SRO SRO

Hexatic-B Smectic-F (SmF) Smectic-I (SmI)

QLRO QLRO QLRO

QLRO QLRO QLRO

SRO SRO SRO

Crystalline-B (CrB) Crystalline-G (CrG) Crystalline-J (CrJ)

LRO LRO LRO

LRO LRO LRO

QLRO QLRO QLRO

Crystalline-E (CrE) Crystalline-H (CrH) Crystalline-K (CrK)

LRO LRO LRO

LRO LRO LRO

QLRO QLRO QLRO

548

4.4. SCATTERING FROM MESOMORPHIC STRUCTURES either one of these could be used as specific representations of the molecular orientational order. The macroscopic order, however, is given by the statistical average Si; j ¼ hsi; j i ¼ Sðhni ihnj i
di; j =3Þ, where hni is a unit vector along the macroscopic symmetry axis and S is the order parameter of the nematic phase. The microscopic origin of the phase can be understood in terms of steric constraints that occur on filling space with highly asymmetric objects such as long rods or flat discs. Maximizing the density requires some degree of short-range orientational order, and theoretical arguments can be invoked to demonstrate longrange order. Onsager presented quantitative arguments of this type to explain the nematic order observed in concentrated solutions of the long thin rods of tobacco mosaic viruses (Onsager, 1949; Lee & Meyer, 1986), and qualitatively similar ideas explain the nematic order for the shorter thermotropic molecules (Maier & Saupe, 1958, 1959). The existence of nematic order can also be understood in terms of a phenomenological mean-field theory (De Gennes, 1969b, 1971; Fan & Stephen, 1970). If the free-energy difference
F between the isotropic and nematic phases can be expressed as an analytic function of the nematic order parameter Si; j , one can expand
FðSi; j Þ as a power series in which the successive terms all transform as the identity representation of the point group of the isotropic phase, i.e. as scalars. The most general form is given by
FðSi; j Þ ¼

AX BX Sij Sji þ SS S 2 ij 3 ijk ij jk ki

2
D0 X D

X
þ
Sij Sji
þ SS S S:
4
ij 4 ijkl ij jk kl li

ð4:4:2:1Þ

The usual mean-field treatment assumes that the coefficient of the leading term is of the form A ¼ aðT
T  Þ, where T is the absolute temperature and T  is the temperature at which A ¼ 0. Taking a, D and D0 > 0, one can show that for either positive or negative values of B, but for sufficiently large T, the minimum value of
F ¼ 0 occurs for Si; j ¼ 0, corresponding to the isotropic phase. For T < T  ,
F can be minimized, at some negative value, for a nonzero Si; j corresponding to nematic order. The details of how this is derived for a tensorial order parameter can be found in the literature (De Gennes, 1974); however, the basic idea can be understood by treating Si; j as a scalar. If we write

Fig. 4.4.1.3. Chemical formulae for some of the molecules that form thermotropic liquid crystals: (a) N-[4-(n-butyloxy)benzylidene]-4-n-octylaniline (4O.8), (b) 40 -n-octylbiphenyl-4-carbonitrile (8CB), (c) 4-hexylphenyl 4-(4-cyanobenzoyloxy)benzoate (DB6); lyotropic liquid crystals: (d) sodium dodecyl sulfate, (e) 1,2-dipalmitoyl-l-phosphatidylcholine (DPPC); and a discotic liquid crystal: ( f ) benzenehexayl hexa-n-alkanoates.

thermotropic molecules do have some of the same features as the lyotropic molecules. For example, although the rod-like thermotropic molecules always have an alkane chain at one or both ends of a more rigid section, the chain lengths are rarely as long as those of the lyotropic molecules, and although the solubility of the parts of the thermotropic molecules, when separated, are not as disparate as those of the lyotropic molecules, they are definitely different. We suspect that this may account for the subtler features of the phase transformations between the mesomorphic phases to be discussed below. On the other hand, the inhomogeneity of the molecule is probably not important for the nematic phase.


F ¼ 12 AS2 þ 13 BS3 þ 14 DS4  2   A B2 2 D 2B
Sþ ¼ S2 S þ 2 9D 4 3D

ð4:4:2:2Þ

and if TNI is defined by the condition A ¼ aðTNI
T  Þ ¼ 2B2 =9D, then
F ¼ 0 for both S ¼ 0 and S ¼
2B=3D. This value for TNI marks the transition temperature from the isotropic phase, when A > 2B2 =9D and the only minimum is at S ¼ 0 with
F ¼ 0, to the nematic case when A < 2B2 =9D and the absolute minimum with
F < 0 is slightly shifted from S ¼
2B=3D. The symmetry properties of secondrank tensors imply that there will usually be a nonvanishing value for B, and this implies that the transition from the isotropic to nematic transition will be first order with a discontinuous jump in the nematic order parameter Si; j . Although most nematic systems are uniaxial, biaxial nematic order is theoretically possible (Freiser, 1971; Alben, 1973; Lubensky, 1987) and it has been observed in certain lyotropic nematic liquid crystals (Neto et al.,

4.4.2. The nematic phase The nematic phase is a fluid for which the molecules have longrange orientational order. The phase as well as its molecular origin can be most simply illustrated by treating the molecules as long thin rods. The orientation of each molecule can be described by a symmetric second-rank tensor si; j ¼ ðni nj
di; j =3Þ, where n is a unit vector along the axis of the rod (De Gennes, 1974). For disc-like molecules, such as that shown in Fig. 4.4.1.3( f ), or for micellar nematic phases, n is along the principal symmetry axis of either the molecule or the micelle (Lawson & Flautt, 1967). Since physical quantities such as the molecular polarizability, or the moment of inertia, transform as symmetric second-rank tensors,

549

4. DIFFUSE SCATTERING AND RELATED TOPICS 1985; Hendrikx et al., 1986; Yu & Saupe, 1980) and in one thermotropic system (Maltheˆte et al., 1986). The X-ray scattering cross section of an oriented monodomain sample of the nematic phase with rod-like molecules usually exhibits a diffuse spot like that illustrated in Fig. 4.4.1.2(a), where the maximum of the cross section is along the average molecular ˚ is of axis hni at a value of jqj  2
=d, where d  20:0 to 40.0 A the order of the molecular length L. This is a precursor to the smectic-A order that develops at lower temperatures for many materials. In addition, there is a diffuse ring along the directions ˚ is comparable to normal to hni at jqj  2
=a, where a  4:0 A the average radius of the molecule. In some nematic systems, the near-neighbour correlations favour antiparallel alignment and molecular centres tend to form pairs such that the peak of the scattering cross section can actually have values anywhere in the range from 2
=L to 2
=2L. There are also other cases where there are two diffuse peaks, corresponding to both jq1 j  2
=L and jq2 j  jq1 j=2 which are precursors of a richer smectic-A morphology (Prost & Barois, 1983; Prost, 1984; Sigaud et al., 1979; Wang & Lubensky, 1984; Hardouin et al., 1983; Chan, Pershan et al., 1985). In some cases, jq2 j 6 12jq1 j and competition between the order parameters at incommensurate wavevectors gives rise to modulated phases. For the moment, we will restrict the discussion to those systems for which the order parameter is characterized by a single wavevector. On cooling, many nematic systems undergo a second-order phase transition to a smectic-A phase and as the temperature approaches the nematic to smectic-A transition the widths of these diffuse peaks become infinitesimally small. De Gennes (1972) demonstrated that this phenomenon could be understood by analogy with the transitions from either normal fluidity to superfluidity in liquid helium or normal conductivity to superconductivity in metals. Since the electron density of the smecticA phase is (quasi-)periodic in one dimension, he represented it by the form

be first order (McMillan, 1972, 1973a,b,c). McMillan pointed out that, by allowing coupling between the smectic and nematic order parameters, a more general free energy can be developed in which D is negative. McMillan’s prediction that for systems in which the difference TIN
TNA is small the nematic to smectic-A transition will be first order is supported by experiment (Ocko, Birgeneau & Litster, 1986; Ocko et al., 1984; Thoen et al., 1984). Although the mean-field theory is not quantitatively accurate, it does explain the principal qualitative features of the nematic to smectic-A transition. The differential scattering cross section for X-rays can be expressed in terms of the Fourier transform of the density– density correlation function hðrÞð0Þi. The expectation value is calculated from the thermal average of the order parameter that is obtained from the free-energy density
Fð Þ. If one takes the transform ðQÞ 

1 ð2
Þ3

Z

d3 r exp½iðQ  rÞðrÞ;

ð4:4:2:4Þ

the free-energy density in reciprocal space has the form
Fð Þ ¼

A 2 D 4 j j þ j j 2 4 E þ f½Qz
ð2
=dÞ2 þ Q2x þ Q2y gj j2 2

ð4:4:2:5Þ

and one can show that for T > TNA the cross section obtained from the above form for the free energy is d 0  ; d A þ Ef½Qz
ð2
=dÞ2 þ Q2x þ Q2y g

ð4:4:2:6Þ

ðrÞ ¼ hi þ Ref exp½ið2
=dÞzg; where the term in j j4 has been neglected. The mean-field theory predicts that the peak intensity should vary as 0 =A  1=ðT
TNA Þ and that the half width of the peak in any direction should vary as ðA=EÞ1=2  ðT
TNA Þ1=2. The physical interpretation of the half width is that the smectic fluctuations in the nematic phase are correlated over lengths  ¼ ðE=AÞ1=2  ðT
TNA Þ
1=2. One of the major shortcomings of all mean-field theories is that they do not take into account the difference between the average value of the order parameter h i and the instantaneous value ¼ h i þ  , where  represents the thermal fluctuations (Ma, 1976). The usual effect expected from theories for this type of critical phenomenon is a ‘renormalization’ of the various terms in the free energy such that the temperature dependence of correlation length has the form ðtÞ / t
 , where t  ðT
T  Þ=T  , T  ¼ TNA is the second-order transition temperature, and  is expected to have some universal value that is generally not equal to 0.5. One of the major unsolved problems of the nematic to smectic-A phase transition is that the width along the scattering vector q varies as 1=k / tk with a temperature dependence different from that of the width perpendicular to q, 1=? / t? ; also, neither k nor ? have the expected universal values (Lubensky, 1983; Nelson & Toner, 1981). The correlation lengths are measured by fitting the differential scattering cross sections to the empirical form:

where d is the thickness of the smectic layers lying in the xy plane. The complex quantity ¼ j j expði’Þ is similar to the superfluid wavefunction except that in this analogy the amplitude j j describes the electron-density variations normal to the smectic layers, and the phase ’ describes the position of the layers along the z axis. De Gennes proposed a mean-field theory for the transition in which the free-energy difference between the nematic and smectic-A phase
FðÞ was represented by "
   A 2 D 4 E

@ 2

Fð Þ ¼ j j þ j j þ
i
2 4 2 @z d  2  2 # @ @ þ þ : @x @y


2


ð4:4:2:3Þ

This mean-field theory differs from the one for the isotropic to nematic transition in that the symmetry for the latter allowed a term that was cubic in the order parameter, while no such term is allowed for the nematic to smectic-A transition. In both cases, however, the coefficient of the leading term is taken to have the form aðT
T  Þ. If D > 0, without the cubic term the free energy has only one minimum when T > T  at j j ¼ 0, and two equivalent minima at j j ¼ faðT 
TÞ=Dg0:5 for T < T  . On the basis of this free energy, the nematic to smectic-A transition can be second order with a transition temperature TNA ¼ T  and an order parameter that varies as the square root of ðTNA
TÞ. There are conditions that we will not discuss in detail when D can be negative. In that case, the nematic to smectic-A transition will

d  ¼ : d 1 þ ðQz
jqjÞ2 k2 þ Q2? ?2 þ cðQ2? ?2 Þ2

550

ð4:4:2:7Þ

4.4. SCATTERING FROM MESOMORPHIC STRUCTURES


The amplitude  / t , where the measured values of  are empirically found to be very close to the measured values for the sum k þ ? . Most of the systems that have been measured to date have values for k > 0:66 > ? and k
?  0:1 to 0.2. Table 4.4.2.1 lists sources of the observed values for , k and ? . The theoretical and experimental studies of this pretransition effect account for a sizeable fraction of all of the liquid-crystal research in the last 15 or 20 years, and as of this writing the explanation for these two different temperature dependences remains one of the major unresolved theoretical questions in equilibrium statistical physics. It is very likely that the origin of the problem is the QLRO in the position of the smectic layers. Lubensky attempted to deal with this by introducing a gauge transformation in such a way that the thermal fluctuations of the transformed order parameter did not have the logarithmic divergence. While this approach has been informative, it has not yet yielded an agreed-upon understanding. Experimentally, the effect of the phase can be studied in systems where there are two competing order parameters with wavevectors that are at q2 and q1  2q2 (Sigaud et al., 1979; Hardouin et al., 1983; Prost & Barois, 1983; Wang & Lubensky, 1984; Chan, Pershan et al., 1985). On cooling, mixtures of 4-hexylphenyl 4-(4-cyanobenzoyloxy)benzoate (DB6) and N,N0 (1,4-phenylenedimethylene)bis(4-butylaniline) (also known as terephthal-bis-butylaniline, TBBA) first undergo a second-order transition from the nematic to a phase that is designated as smectic-A1. The various smectic-A and smectic-C morphologies will be described in more detail in the following section; however, the smectic-A1 phase is characterized by a single peak at q1 ¼ 2
=d owing to a one-dimensional density wave with wavelength d of the order of the molecular length L. In addition, however, there are thermal fluctuations of a second-order parameter with a period of 2L that give rise to a diffuse peak at q2 ¼
=L. On further cooling, this system undergoes a second second-order transition to a smectic-A2 phase with QLRO at q2 
=L, with a second harmonic that is exactly at q ¼ 2q2  2
=L. The critical scattering on approaching this transition is similar to that of the nematic to smectic-A1, except that the pre-existing density wave at q1 ¼ 2
=L quenches the phase fluctuations of the order parameter at the subharmonic q2 ¼
=L. The measured values of k ¼ ?  0:74 (Chan, Pershan et al., 1985) agree with those expected from the appropriate theory (Huse, 1985). A mean-field theory that describes this effect is discussed in Section 4.4.3.2 below. It is interesting to note that even those systems for which the nematic to smectic-A transition is first order show some pretransitional lengthening of the correlation lengths k and ? . In these cases, the apparent T  at which the correlation lengths would diverge is lower than TNA and the divergence is truncated by the first-order transition (Ocko et al., 1984).

Table 4.4.2.1. Summary of critical exponents from X-ray scattering studies of the nematic to smectic-A phase transition Molecule



k

?

Reference

4O.7 8 S5 CBOOA 4O.8 8OCB 9 S5 8CB 10 S5 9CB

1.46 1.53 1.30 1.31 1.32 1.31 1.26 1.10 1.10

0.78 0.83 0.70 0.70 0.71 0.71 0.67 0.61 0.57

0.65 0.68 0.62 0.57 0.58 0.57 0.51 0.51 0.39

(a) (b), (g) (c), (d) (e) (d), ( f ) (b), (g) (h), (i) (b), (g) (g), (j)

References: (a) Garland et al. (1983); (b) Brisbin et al. (1979); (c) Djurek et al. (1974); (d) Litster et al. (1979); (e) Birgeneau et al. (1981); ( f ) Kasting et al. (1980); (g) Ocko et al. (1984); (h) Thoen et al. (1982); (i) Davidov et al. (1979); (j) Thoen et al. (1984).

be described in more detail below; however, it corresponds to replacing equation (4.4.2.5) for the free energy
Fð Þ by an expression for which the minimum is obtained when the wavevector q, of the order parameter / exp½iq  r, tilts away from the molecular axis. The X-ray cross section for the prototypical aligned monodomain smectic-A sample is shown in Fig. 4.4.1.2(b). It consists of a single sharp spot along the molecular axis at jqj somewhere between 2
=2L and 2
=L that reflects the QLRO along the layer normal, and a diffuse ring in the perpendicular direction at jqj  2
=a that reflects the SRO within the layer. The scattering cross section for an aligned smectic-C phase is similar to that of the smectic-A except that the molecular tilt alters the intensity distribution of the diffuse ring. This is illustrated in Fig. 4.4.1.2(c) for a monodomain sample. Fig. 4.4.1.2(d) illustrates the scattering pattern for a polydomain smectic-C sample in which the molecular axis remains fixed, but where the smectic layers are randomly distributed azimuthally around the molecular axis. The naivety of describing these as periodic stacks of twodimensional liquids derives from the fact that the sharp spot along the molecular axis has a distinct temperature-dependent shape indicative of QLRO that distinguishes it from the Bragg peaks due to true LRO in conventional three-dimensional crystals. Landau and Peierls discussed this effect for the case of twodimensional crystals (Landau, 1965; Peierls, 1934) and Caille´ (1972) extended the argument to the mesomorphic systems. The usual treatment of thermal vibrations in three-dimensional crystals estimates the Debye–Waller factor by integrating the thermal expectation value for the mean-square amplitude over reciprocal space (Kittel, 1963): k T W ’ B3 c

4.4.3. Smectic-A and smectic-C phases 4.4.3.1. Homogeneous smectic-A and smectic-C phases

Z

kD 0

kðd
1Þ dk; k2

ð4:4:3:1Þ

where c is the sound velocity, !D  ckD is the Debye frequency and d ¼ 3 for three-dimensional crystals. In this case, the integral converges and the only effect is to reduce the integrated intensity of the Bragg peak by a factor proportional to expð
2WÞ. For twodimensional crystals d ¼ 1, and the integral, of the form of dk=k, obtains a logarithmic divergence at the lower limit (Fleming et al., 1980). A more precise treatment of thermal vibrations, necessitated by this divergence, is to calculate the relative phase of X-rays scattered from two points in the sample a distance jrj apart. The appropriate integral that replaces the Debye–Waller integral is

In the smectic-A and smectic-C phases, the molecules organize themselves into layers, and from a naive point of view one might describe them as forming a one-dimensional periodic lattice in which the individual layers are two-dimensional liquids. In the smectic-A phase, the average molecular axis hni is normal to the smectic layers while for the smectic-C it makes a finite angle. It follows from this that the smectic-C phase has lower symmetry than the smectic-A, and the phase transition from the smectic-A to smectic-C can be considered as the ordering of a twocomponent order parameter, i.e. the two components of the projection of the molecular axis on the smectic layers (De Gennes, 1973). Alternatively, Chen & Lubensky (1976) have developed a mean-field theory in which the transition is described by a free-energy density of the Lifshitz form. This will

h½uðrÞ
uð0Þ2 i ’

551

kB T c3

Z

sin2 ðk  rÞ

dk dfcosðk  rÞg k

ð4:4:3:2Þ

4. DIFFUSE SCATTERING AND RELATED TOPICS and the divergence due to the lower limit is cut off by the fact that sin2 ðk  rÞ vanishes as k ! 0. More complete analysis obtains h½uðrÞ
uð0Þ2 i ’ ðkB T=c2 Þ lnðjrj=aÞ, where a  atomic size. If this is exponentiated, as for the Debye–Waller factor, the density–density correlation function can be shown to have the form hðrÞð0Þi ’ jr=aj
 , where  ’ jqj2 ðkB T=c2 Þ and jqj ’ 2
=a. In place of the usual periodic density–density correlation function of three-dimensional crystals, the periodic correlations of two-dimensional crystals decay as some power of the distance. This type of positional order, in which the correlations decay as some power of the distance, is the quasi-long-range order (QLRO) that appears in Tables 4.4.1.1 and 4.4.1.2. It is distinguished from true long-range order (LRO) where the correlations continue indefinitely, and short-range order (SRO) where the positional correlations decay exponentially as in either a simple fluid or a nematic liquid crystal. The usual prediction of Bragg scattering for three-dimensional crystals is obtained from the Fourier transform of the threedimensional density–density correlation function. Since the correlation function is made up of periodic and random parts, it follows that the scattering cross section is made up of a  function at the Bragg condition superposed on a background of thermal diffuse scattering from the random part. In principle, these two types of scattering can be separated empirically by using a highresolution spectrometer that integrates all of the -function Bragg peak, but only a small part of the thermal diffuse scattering. Since the two-dimensional lattice is not strictly periodic, there is no formal way to separate the periodic and random parts, and the Fourier transform for the algebraic correlation function obtains a cross section that is described by an algebraic singularity of the form jQ
qj
2 (Gunther et al., 1980). In 1972, Caille´ (Caille´, 1972) presented an argument that the X-ray scattering line shape for the one-dimensional periodicity of the smectic-A system in three dimensions has an algebraic singularity that is analogous to the line shapes from two-dimensional crystals. In three-dimensional crystals, both the longitudinal and the shear sound waves satisfy linear dispersion relations of the form ! ¼ ck. In simple liquids, and also for nematic liquid crystals, only the longitudinal sound wave has such a linear dispersion relation. Shear sound waves are overdamped and the decay rate 1= is given by the imaginary part of a dispersion relation of the form ! ¼ ið=Þk2 , where  is a viscosity coefficient and  is the liquid density. The intermediate order of the smectic-A mesomorphic phase, between the three-dimensional crystal and the nematic, results in one of the modes for shear sound waves having the curious dispersion relation !2 ¼ c2 k2? k2z =ðk2? þ k2z Þ, where k? and kz are the magnitudes of the components of the acoustic wavevector perpendicular and parallel to hni, respectively (De Gennes, 1969a; Martin et al., 1972). More detailed analysis, including terms of higher order in k2? , obtains the equivalent of the Debye–Waller factor for the smectic-A as Z W ’ kB T 0

kD

k? dk? dkz ; Bk2z þ Kk4?

X-ray scattering experiments to test this idea were carried out on one thermotropic smectic-A system, but the results, while consistent with the theory, were not adequate to provide an unambiguous proof of the algebraic cusp (Als-Nielsen et al., 1980). One of the principal difficulties was due to the fact that, when thermotropic samples are oriented in an external magnetic field in the higher-temperature nematic phase and then gradually cooled through the nematic to smectic-A phase transition, the smectic-A samples usually have mosaic spreads of the order of a fraction of a degree and this is not sufficient for detailed lineshape studies near to the peak. A second difficulty is that, in most of the thermotropic smectic-A phases that have been studied to date, only the lowest-order peak is observed. It is not clear whether this is due to a large Debye–Waller-type effect or whether the form factor for the smectic-A layer falls off this rapidly. Nevertheless, since the factor  in the exponent of the cusp jQ
qj
2 depends quadratically on the magnitude of the reciprocal vector jqj, the shape of the cusp for the different orders would constitute a severe test of the theory. Fortunately, it is common to observe multiple orders for lyotropic smectic-A systems and such an experiment, carried out on the lyotropic smectic-A system formed from a quaternary mixture of sodium dodecyl sulfate, pentanol, water and dodecane, confirmed the theoretical predictions for the Landau–Peierls effect in the smectic-A phase (Safinya, Roux et al., 1986). The problem of sample mosaic was resolved by using a threedimensional powder. Although the conditions on the analysis are delicate, Safinya et al. demonstrated that for a perfect powder, for which the microcrystals are sufficiently large, the powder line shape does allow unambiguous determination of all of the parameters of the anisotropic line shape. The only other X-ray study of a critical property on the smectic-A side of the transition has been a measurement of the temperature dependence of the integrated intensity of the peak. For three-dimensional crystals, the integrated intensity of a Bragg peak can be measured for samples with poor mosaic distributions, and because the differences between QLRO and true LRO are only manifest at long distances in real space, or at small wavevectors in reciprocal space, the same is true for the ‘quasi-Bragg peak’ of the smectic-A phase. Chan et al. measured the temperature dependence of the integrated intensity of the smectic-A peak across the nematic to smectic-A phase transition for a number of liquid crystals with varying exponents k and ? (Chan, Deutsch et al., 1985). For the Landau–De Gennes freeenergy density (equation 4.4.2.5), the theoretical prediction is that the critical part of the integrated intensity should vary as jtjx, where x ¼ 1
when the critical part of the heat capacity diverges according to the power law jtj
. Six samples were measured with values of varying from 0 to 0.5. Although for samples with  0:5 the critical intensity did vary as x  0:5, there were systematic deviations for smaller values of , and for

 0 the measured values of x were in the range 0.7 to 0.76. The origin of this discrepancy is not at present understood. Similar integrated intensity measurements in the vicinity of the first-order nematic to smectic-C transition cannot easily be made in smectic-C samples since the magnetic field aligns the molecular axis hni, and when the layers form at some angle ’ to hni the layer normals are distributed along the full 2
of azimuthal directions around hni, as shown in Fig. 4.4.1.2(d). The X-ray scattering pattern for such a sample is a partial powder with a peak-intensity distribution that forms a ring of radius jqj sinð’Þ. The opening of the single spot along the average molecular axis hni into a ring can be used to study either the nematic to smectic-C or the smectic-A to smectic-C transition (Martinez-Miranda et al., 1986). The statistical physics in the region of the phase diagram surrounding the triple point, where the nematic, smectic-A and smectic-C phases meet, has been the subject of considerable theoretical speculation (Chen & Lubensky, 1976; Chu &

ð4:4:3:3Þ

where B and K are smectic elastic constants, k2? ¼ k2x þ k2y , and kD is the Debye wavevector. On substitution of u2 ¼ 2 2 RðK=BÞk? þ kz , the integral can be manipulated into the form du=u, which diverges logarithmically at the lower limit in exactly the same way as the integral for the Debye–Waller factor of the two-dimensional crystal. The result is that the smectic-A phase has a sharp peak, described by an algebraic cusp, at the place in reciprocal space where one would expect a true -function Bragg cross section from a truly periodic onedimensional lattice. In fact, the lattice is not truly periodic and the smectic-A system has only QLRO along the direction hni.

552

4.4. SCATTERING FROM MESOMORPHIC STRUCTURES McMillan, 1977; Benguigui, 1979; Huang & Lien, 1981; Grinstein & Toner, 1983). The best representation of the observed X-ray scattering structure near the nematic to smectic-A, the nematic to smectic-C and the nematic/smectic-A/smectic-C (NAC) multicritical point is obtained from the mean-field theory of Chen and Lubensky, the essence of which is expressed in terms of an energy density of the form
Fð Þ ¼

A 2 D 4 1 j j þ j j þ 2½Ek ðQ2k
Q20 Þ2 þ E? Q2? 2 4 þ E?? Q4? þ E?k Q2? ðQ2k
Q20 Þj ðQÞj2 ;

ð4:4:3:4Þ

where ¼ ðQÞ is the Fourier component of the electron density: ðQÞ 

1 ð2
Þ3

Z

d3 r exp½iðQ  rÞðrÞ:

ð4:4:3:5Þ

The quantities Ek, E?? , and Ek? are all positive definite; however, the sign of A and E? depends on temperature. For A > 0 and E? > 0, the free energy, including the higher-order terms, is minimized by ðQÞ ¼ 0 and the nematic is the stable phase. For A < 0 and E? > 0, the minimum in the free energy occurs for a nonvanishing value for ðQÞ in the vicinity of Qk  Q0, corresponding to the uniaxial smectic-A phase; however, for E? < 0, the free-energy minimum occurs for a nonvanishing ðQÞ with a finite value of Q?, corresponding to smectic-C order. The special point in the phase diagram where two terms in the free energy vanish simultaneously is known as a ‘Lifshitz point’ (Hornreich et al., 1975). In the present problem, this occurs at the triple point where the nematic, smectic-A and smectic-C phases coexist. Although there have been other theoretical models for this transition, the best agreement between the observed and theoretical line shapes for the X-ray scattering cross sections is based on the Chen–Lubensky model. Most of the results from lightscattering experiments in the vicinity of the NAC triple point also agree with the main features predicted by the Chen–Lubensky model; however, there are some discrepancies that are not explained (Solomon & Litster, 1986). The nematic to smectic-C transition in the vicinity of this point is particularly interesting in that, on approaching the nematic to smectic-C transition temperature from the nematic phase, the X-ray scattering line shapes first appear to be identical to the shapes usually observed on approaching the nematic to smecticA phase transition; however, within approximately 0.1 K of the transition, they change to shapes that clearly indicate smectic-Ctype fluctuations. Details of this crossover are among the strongest evidence supporting the Lifshitz idea behind the Chen– Lubensky model.

Fig. 4.4.3.1. (a) Schematic illustration of the necessary condition for coupling between order parameters when jq2 j < 2jq1 j; jqj ¼ ðjq2 j2
jq1 j2 Þ1=2 ¼ jq1 j sinð Þ. (b) Positions of the principal peaks for the indicated smectic-A phases.

between phases that have been designated smectic-A1 with period d  L, smectic-A2 with period d  2L and smectic-Ad with period L < d < 2L. Stimulated by the experimental results, Prost and co-workers generalized the De Gennes mean-field theory by writing ðrÞ ¼ hi þ Ref1 expðiq1  rÞ þ 2 expðiq2  rÞg; where 1 and 2 refer to two different density waves (Prost, 1979; Prost & Barois, 1983; Barois et al., 1985). In the special case that q1  2q2 the free energy represented by equation (4.4.2.3) must be generalized to include terms like ð2 Þ2 1 exp½iðq1
2q2 Þ  r þ c.c. that couple the two order parameters. Suitable choices for the relative values of the phenomenological parameters of the free energy then result in minima that correspond to any one of these three smectic-A phases. Much more interesting, however, was the observation that even if jq1 j < 2jq2 j the two order parameters could still be coupled together if q1 and q2 were not collinear, as illustrated in Fig. 4.4.3.1(a), such that 2q1  q2 ¼ jq1 j2 . Prost et al. predicted the existence of phases that are modulated in the direction perpendicular to the average layer normal with a period 4
=½jq2 j sinð’Þ ¼ 2
=jqm j. Such a modulated phase has been observed and is designated as the smectic-A (Hardouin et al., 1981). Similar considerations apply to the smectic-C phases and the modulated phase is designated smectic-C~ ; (Hardouin et al., 1982; Huang et al., 1984; Safinya, Varady et al., 1986).

4.4.3.2. Modulated smectic-A and smectic-C phases Previously, we mentioned that, although the reciprocal-lattice spacing jqj for many smectic-A phases corresponds to 2
=L, where L is the molecular length, there are a number of others for which jqj is between
=L and 2
=L (Leadbetter, Frost, Gaughan, Gray & Mosley, 1979; Leadbetter et al., 1977). This suggests the possibility of different types of smectic-A phases in which the bare molecular length is not the sole determining factor of the period d. In 1979, workers at Bordeaux optically observed some sort of phase transition between two phases that both appeared to be of the smectic-A type (Sigaud et al., 1979). Subsequent X-ray studies indicated that in the nematic phase these materials simultaneously displayed critical fluctuations with two separate periods (Levelut et al., 1981; Hardouin et al., 1980, 1983; Ratna et al., 1985, 1986; Chan, Pershan et al., 1985, 1986; Safinya, Varady et al., 1986; Fontes et al., 1986) and confirmed phase transitions

4.4.3.3. Surface effects The effects of surfaces in inducing macroscopic alignment of mesomorphic phases have been important both for technological applications and for basic research (Sprokel, 1980; Gray & Goodby, 1984). Although there are a variety of experimental techniques that are sensitive to mesomorphic surface order (Beaglehole, 1982; Faetti & Palleschi, 1984; Faetti et al., 1985; Gannon & Faber, 1978; Miyano, 1979; Mada & Kobayashi, 1981; Guyot-Sionnest et al., 1986), it is only recently that X-ray scattering techniques have been applied to this problem. In one form or another, all of the techniques for obtaining surface specificity

553

4. DIFFUSE SCATTERING AND RELATED TOPICS Since the full width at half maximum is exactly equal to the reciprocal of the correlation length for critical fluctuations in the bulk, 2=k at all temperatures from T
TNA  0:006 K up to values near to the nematic to isotropic transition, T
TNA  3:0 K, it is clear this is an example where the gravitationally induced long-range order in the surface position has induced mesomorphic order that has long-range correlations parallel to the surface. Along the surface normal, the correlations have only the same finite range as the bulk critical fluctuations. Studies on a number of other nematic (Gransbergen et al., 1986; Ocko et al., 1987) and isotropic surfaces (Ocko, Braslau et al., 1986) indicate features that are specific to local structure of the surface. 4.4.4. Phases with in-plane order Although the combination of optical microscopy and X-ray scattering studies on unoriented samples identified most of the mesomorphic phases, there remain a number of subtle features that were only discovered by spectra from well oriented samples (see the extensive references contained in Gray & Goodby, 1984). Nematic phases are sufficiently fluid that they are easily oriented by either external electric or magnetic fields, or surface boundary conditions, but similar alignment techniques are not generally successful for the more ordered phases because the combination of strains induced by thermal expansion and the enhanced elasticity that accompanies the order creates defects that do not easily anneal. Other defects that might have been formed during initial growth of the phase also become trapped and it is difficult to obtain well oriented samples by cooling from a higher-temperature aligned phase. Nevertheless, in some cases it has been possible to obtain crystalline-B samples with mosaic spreads of the order of a fraction of a degree by slowing cooling samples that were aligned in the nematic phase. In other cases, mesomorphic phases were obtained by heating and melting single crystals that were grown from solution (Benattar et al., 1979; Leadbetter, Mazid & Malik, 1980). Moncton & Pindak (1979) were the first to realize that X-ray scattering studies could be carried out on the freely suspended films that Friedel (1922) described in his classical treatise on liquid crystals. These samples, formed across a plane aperture (i.e. approximately 1 cm in diameter) in the same manner as soap bubbles, have mosaic spreads that are an order of magnitude smaller. The geometry is illustrated in Fig. 4.4.4.1(a). The substrate in which the aperture is cut can be glass (e.g. a microscope cover slip), steel or copper sheeting etc. A small amount of the material, usually in the high-temperature region of the smectic-A phase, is spread around the outside of an aperture that is maintained at the necessary temperature, and a wiper is used to drag some of the material across the aperture. If a stable film is successfully drawn, it is detected optically by its finite reflectivity. In particular, against a dark background and with the proper illumination it is quite easy to detect the thinnest free films. In contrast to conventional soap films that are stabilized by electrostatic effects, smectic films are stabilized by their own layer structure. Films as thin as two molecular layers can be drawn and studied for weeks (Young et al., 1978). Thicker films of the order of thousands of layers can also be made and, with some experience in depositing the raw material around the aperture and the speed of drawing, it is possible to draw films of almost any desired thickness (Moncton et al., 1982). For films thinner than ˚ ), the approximately 20 to 30 molecular layers (i.e. 600 to 1000 A thickness is determined from the reflected intensity of a small helium–neon laser. Since the reflected intensities for films of 2, 3, 4, 5, . . . layers are in the ratio of 4, 9, 16, 25, . . . , the measurement can be calibrated by drawing and measuring a reasonable number of thin films. The most straightforward method for thick films is to measure the ellipticity of the polarization induced in laser light transmitted through the film at an

Fig. 4.4.3.2. Specular reflectivity of 8 keV X-rays from the air–liquid interface of the nematic liquid crystal 8OCB 0.05 K above the nematic to smectic-A transition temperature. The dashed line is the Fresnel reflection law as described in the text.

in an X-ray measurement make use of the fact that the average interaction between X-rays and materials can be treated by the introduction of a dielectric constant "  1
ð4
e2 =m!2 Þ ¼ 1
re 2 =
, where  is the electron density, re is the classical radius of the electron, and ! and  are the angular frequency and the wavelength of the X-ray. Since " < 1, X-rays that are incident at a small angle to the surface 0 will be refracted in the material 1=2 toward a smaller angle T  ð 02
c2 Þ , where the ‘critical angle’ 1=2 c  ðre 2 =
Þ  0:003 rad ð 0:2 Þ for most liquid crystals (Warren, 1968). Although this is a small angle, it is at least two orders of magnitude larger than the practical angular resolution available in modern X-ray spectrometers (Als-Nielsen et al., 1982; Pershan & Als-Nielsen, 1984; Pershan et al., 1987). One can demonstrate that for many conditions the specular reflection Rð 0 Þ is given by R Rð 0 Þ  RF ð 0 Þj
1 dz expð
iQzÞh@=@zij2 ; where Q  ð4
=Þ sinð 0 Þ, h@=@zi is the normal derivative of the electron density averaged over a region in the surface that is defined by the coherence area of the incident X-ray, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 02
c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi RF ð 0 Þ  0 þ 02
c2 0


is the Fresnel reflection law that is calculated from classical optics for a flat interface between the vacuum and a material of dielectric constraint ". Since the condition for specular reflection, that the incident and scattered angles are equal and in the same plane, requires that the scattering vector Q ¼ z^ ð4
=Þ sinð 0 Þ be parallel to the surface normal, it is quite practical to obtain, for flat surfaces, an unambiguous separation of the specular reflection signal from all other scattering events. Fig. 4.4.3.2(a) illustrates the specular reflectivity from the free nematic–air interface for the liquid crystal 40 -octyloxybiphenyl-4carbonitrile (8OCB) 0.050 K above the nematic to smectic-A phase-transition temperature (Pershan & Als-Nielsen, 1984). The dashed line is the Fresnel reflection RF ð 0 Þ in units of sinð 0 Þ= sinð c Þ, where the peak at c ¼ 1:39 corresponds to surface-induced smectic order in the nematic phase: i.e. the selection rule for specular reflection has been used to separate the specular reflection from the critical scattering from the bulk.

554

4.4. SCATTERING FROM MESOMORPHIC STRUCTURES (Aeppli et al., 1981), and one taken on a thick freely suspended film of N-[4-(n-heptyloxy)benzylidene]-4-n-heptylaniline (7O.7) (Collett et al., 1982, 1985). Note that the data for 7O.7 are plotted on a semi-logarithmic scale in order to display simultaneously both the Bragg peak and the thermal diffuse background. The scans are along the QL direction, at the appropriate value of QH to intersect the peaks associated with the intralayer periodicity. In both cases, the widths of the Bragg peaks are essentially determined by the sample mosaicity and as a result of the better alignment the ratio of the thermal diffuse background to the Bragg peak is nearly an order of magnitude smaller for the free film sample. 4.4.4.1. Hexatic phases in two dimensions The hexatic phase of matter was first proposed independently by Halperin & Nelson (Halperin & Nelson 1978; Nelson & Halperin 1979) and Young (Young, 1979) on the basis of theoretical studies of the melting process in two dimensions. Following work by Kosterlitz & Thouless (1973), they observed that since the interaction energy between pairs of dislocations in two dimensions decreases logarithmically with their separation, the enthalpy and the entropy terms in the free energy have the same functional dependence on the density of dislocations. It follows that the free-energy difference between the crystalline and hexatic phase has the form
F ¼
H
T
S  Tc SðÞ
TSðÞ ¼ SðÞðTc
TÞ, where SðÞ   logðÞ is the entropy as a function of the density of dislocations  and Tc is defined such that Tc SðÞ is the enthalpy. Since the prefactor of the enthalpy term is independent of temperature while that of the entropy term is linear, there will be a critical temperature, Tc , at which the sign of the free energy changes from positive to negative. For temperatures greater than Tc , the entropy term will dominate and the system will be unstable against the spontaneous generation of dislocations. When this happens, the twodimensional crystal, with positional QLRO, but true long-range order in the orientation of neighbouring atoms, can melt into a new phase in which the positional order is short range, but for which there is QLRO in the orientation of the six neighbours surrounding any atom. The reciprocal-space structures for the two-dimensional crystal and hexatic phases are illustrated in Figs. 4.4.4.3(b) and (c), respectively. That of the two-dimensional solid consists of a hexagonal lattice of sharp rods (i.e. algebraic line shapes in the plane of the crystal). For a finite size sample, the reciprocal-space structure of the two-dimensional hexatic phase is a hexagonal lattice of diffuse rods and there are theoretical predictions for the temperature dependence of the in-plane line shapes (Aeppli & Bruinsma, 1984). If the sample were of infinite size, the QLRO of the orientation would spread the six spots continuously around a circular ring, and the pattern would be indistinguishable from that of a well correlated liquid, i.e. Fig. 4.4.4.3(a). The extent of the patterns along the rod corresponds to the molecular form factor. Figs. 4.4.4.3(a), (b) and (c) are drawn on the assumption that the molecules are normal to the twodimensional plane of the phase. If the molecules are tilted, the molecular form factor for long thin rod-like molecules will shift the intensity maxima as indicated in Figs. 4.4.4.3(d) and (e). The phase in which the molecules are normal to the two-dimensional plane is the two-dimensional hexatic-B phase. If the molecules tilt towards the position of their nearest neighbours (in real space), or in the direction that is between the lowest-order peaks in reciprocal space, the phase is the two-dimensional smectic-I, Fig. 4.4.4.3(d). The other tilted phase, for which the tilt direction is between the nearest neighbours in real space or in the direction of the lowest-order peaks in reciprocal space, is the smectic-F, Fig. 4.4.4.3(e). Although theory (Halperin & Nelson, 1978; Nelson & Halperin, 1979; Young, 1979) predicts that the two-dimensional crystal can melt into a hexatic phase, it does not say that it must

Fig. 4.4.4.1. (a) Schematic illustration of the geometry and (b) kinematics of X-ray scattering from a freely suspended smectic film. The insert (c) illustrates the orientation of the film in real space corresponding to the reciprocal-space kinematics in (b). If the angle ’ ¼ , the film is oriented such that the scattering vector is parallel to the surface of the film, i.e. parallel to the smectic layers. A ‘QL scan’ is taken by simultaneous adjustment of ’ and 2 to keep ð4
=Þ sinð Þ cosð
’Þ ¼ ð4
=Þ sinð 100 Þ, where 100 is the Bragg angle for the 100 reflection. The different in-plane Bragg reflections can be brought into the scattering plane by rotation of the film by the angle around the film normal.

oblique angle (Collett, 1983; Collett et al., 1985); however, a subtler method that makes use of the colours of white light reflected from the films is also practical (Sirota, Pershan, Sorensen & Collett, 1987). In certain circumstances, the thickness can also be measured using the X-ray scattering intensity in combination with one of the other methods. Fig. 4.4.4.1 illustrates the scattering geometry used with these films. Although recent unpublished work has demonstrated the possibility of a reflection geometry (Sorensen, 1987), all of the X-ray scattering studies to be described here were performed in transmission. Since the in-plane molecular spacings are typically ˚ , while the layer spacing is closer to 30 A ˚ , it is between 4 and 5 A difficult to study the 00L peaks in this geometry. Fig. 4.4.4.2 illustrates the difference between X-ray scattering spectra taken on a bulk crystalline-B sample of N-[4-(n-butyloxy)benzylidene]-4-n-octylaniline (4O.8) that was oriented in an external magnetic field while in the nematic phase and then cooled through the smectic-A phase into the crystalline-B phase

Fig. 4.4.4.2. Typical QL scans from the crystalline-B phases of (a) a free film of 7O.7, displayed on a logarithmic scale to illustrate the reduced level of the diffuse scattering relative to the Bragg reflection and (b) a bulk sample of 4O.8 oriented by a magnetic field.

555

4. DIFFUSE SCATTERING AND RELATED TOPICS Goodby, 1984), Litster & Birgeneau (Birgeneau & Litster, 1978) suggested that some of the three-dimensional systems that were previously identified as mesomorphic were actually threedimensional hexatic systems. They observed that it is not theoretically consistent to propose that the smectic phases are layers of two-dimensional crystals randomly displaced with respect to each other since, in thermal equilibrium, the interactions between layers of two-dimensional crystals must necessarily cause the layers to lock together to form a three-dimensional crystal.1 On the other hand, if the layers were two-dimensional hexatics, then the interactions would have the effect of changing the QLRO of the hexagonal distribution of neighbours into the true longrange-order orientational distribution of the three-dimensional hexatic. In addition, interactions between layers in the threedimensional hexatics can also result in interlayer correlations that would sharpen the width of the diffuse peaks in the reciprocalspace direction along the layer normal. 4.4.4.2.1. Hexatic-B Although Leadbetter, Frost & Mazid (1979) had remarked on the different types of X-ray structures that were observed in materials identified as ‘smectic-B’, the first proof for the existence of the hexatic-B phase of matter was the experiment by Pindak et al. (1981) on thick freely suspended films of the liquid crystal n-hexyl 40 -pentyloxybiphenyl-4-carboxylate (65OBC). A second study on free films of the liquid crystal n-butyl 40 -n-hexyloxybiphenyl-4-carboxylate (46OBC) demonstrated that, as the hexatic-B melts into the smectic-A phase, the position and the inplane width of the X-ray scattering peaks varied continuously. In particular, the in-plane correlation length evolved continuously ˚ , nearly 10 K below the hexatic to smectic-A transifrom 160 A ˚ , a few degrees above. Similar behaviour was tion, to only 17 A also observed in a film only two layers thick (Davey et al., 1984). Since the observed width of the peak along the layer normal corresponded to the molecular form factor, these systems have negligible interlayer correlations.

Fig. 4.4.4.3. Scattering intensities in reciprocal space from two-dimensional: (a) liquid; (b) crystal; (c) normal hexatic; and tilted hexatics in which the tilt is (d) towards the nearest neighbours as for the smectic-I or (e) between the nearest neighbours as for the smectic-F. The thin rods of scattering in (b) indicate the singular cusp for peaks with algebraic line shapes in the HK plane.

happen, and the crystal can melt directly into a two-dimensional liquid phase. Obviously, the hexatic phases will also melt into a two-dimensional liquid phase. Fig. 4.4.4.3(a) illustrates the reciprocal-space structure for the two-dimensional liquid in which the molecules are normal to the two-dimensional surface. Since the longitudinal (i.e. radial) width of the hexatic spot could be similar to the width that might be expected in a well correlated fluid, the direct X-ray proof of the transition from the hexatic-B to the normal liquid requires a hexatic sample in which the domains are sufficiently large that the sample is not a twodimensional powder. On the other hand, the elastic constants must be sufficiently large that the QLRO does not smear the six spots into a circle. The radial line shape of the powder pattern of the hexatic-B phase can also be subtly different from that of the liquid and this is another possible way that X-ray scattering can detect melting of the hexatic-B phase (Aeppli & Bruinsma, 1984). Changes that occur on the melting of the tilted hexatics, i.e. smectic-F and smectic-I, are usually easier to detect and this will be discussed in more detail below. On the other hand, there is a fundamental theoretical problem concerning the way of understanding the melting of the tilted hexatics. These phases actually have the same symmetry as the two-dimensional tilted fluid phase, i.e. the smectic-C. In two dimensions they all have QLRO in the tilt orientation, and since the simplest phenomenological argument says that there is a linear coupling between the tilt order and the near-neighbour positional order (Nelson & Halperin, 1980; Bruinsma & Nelson, 1981), it follows that the QLRO of the smectic-C tilt should induce QLRO in the nearneighbour positional order. Thus, by the usual arguments, if there is to be a phase transition between the smectic-C and one of the tilted hexatic phases, the transition must be a first-order transition (Landau & Lifshitz, 1958). This is analogous to the threedimensional liquid-to-vapour transition which is first order up to a critical point, and beyond the critical point there is no real phase transition.

4.4.4.2.2. Smectic-F, smectic-I In contrast to the hexatic-B phase, the principal reciprocalspace features of the smectic-F phase were clearly determined before the theoretical work that proposed the hexatic phase. Demus et al. (1971) identified a new phase in one material, and subsequent X-ray studies by Leadbetter and co-workers (Leadbetter, Mazid & Richardson, 1980; Leadbetter, Gaughan et al., 1979; Gane & Leadbetter, 1981) and by Benattar and co-workers (Benattar et al., 1978, 1980, 1983; Guillon et al., 1986) showed it to have the reciprocal-space structure illustrated in Fig. 4.4.4.4(b). There are interlayer correlations in the three-dimensional smectic-F phases, and as a consequence the reciprocal-space structure has maxima along the diffuse rods. Benattar et al. (1979) obtained monodomain smectic-F samples of the liquid crystal N,N0 -(1,4-phenylenedimethylene)bis(4-n-pentylaniline) by melting a single crystal that was previously precipitated from solution. One of the more surprising results of this work was the demonstration that the near-neighbour packing was very close to what would be expected from a model in which rigid closely packed rods were simply tilted away from the layer normal. In view of the facts that the molecules are clearly not cylindrical, and that the molecular tilt indicates that the macroscopic symmetry has been broken, it would have been reasonable to expect significant deviations from local hexagonal symmetry when the system is viewed along the molecular axis. The fact that this is not the case indicates that this phase has a considerable amount of rotational disorder around the long axis of the molecules.

4.4.4.2. Hexatic phases in three dimensions Based on both this theory and the various X-ray scattering patterns that had been reported in the literature (Gray &

1

Prior to the paper by Birgeneau & Litster, it was commonly believed that some of the smectic phases consisted of uncorrelated stacks of two-dimensional crystals.

556

4.4. SCATTERING FROM MESOMORPHIC STRUCTURES

Fig. 4.4.4.4. Scattering intensities in reciprocal space from three-dimensional tilted hexatic phases: (a) the smectic-I and (b) the smectic-F. The variation of the intensity along the QL direction indicates interlayer correlations that are absent in Figs. 4.4.4.1(d) and (e). The peak widths
QL1;2 and
QH1;2 correspond to the four inequivalent widths in the smectic-F phase. Similar inequivalent widths exist for the smectic-I phase. The circle through the shaded points in (a) indicates the reciprocal-space scan that directly measures the hexatic order. A similar scan in the smectic-C phase would have intensity independent of .

Fig. 4.4.4.5. The phase diagram for free films of 7O.7 as a function of thickness and temperature. The phases ABAB, AAA, ORm1 , ORm2 , OR0m1 , M and ABAB are all crystalline-B with varying interlayer stacking, or longwavelength modulations; CrG, SmF and SmI are crystalline-G, smectic-F and smectic-I, respectively (Sirota et al., 1985; Sirota, Pershan & Deutsch, 1987; Sirota, Pershan, Sorensen & Collett, 1987).

dimensional crystal to the hexatic phase. We will say more on this point below. The only identified difference between the two tilted hexatic phases, the smectic-F and the smectic-I, is the direction of the molecular tilt relative to the near-neighbour positions. For the smectic-I, the molecules tilt towards one of the near neighbours, while for the smectic-F they tilt between the neighbours (Gane & Leadbetter, 1983). There are a number of systems that have both smectic-I and smectic-F phases, and in all cases of which we are aware the smectic-I is the higher-temperature phase (Gray & Goodby, 1984; Sirota et al., 1985; Sirota, Pershan, Sorensen & Collett, 1987). Optical studies of freely suspended films of materials in the nO.m series indicated tilted surface phases at temperatures for which the bulk had uniaxial phases (Farber, 1985). As mentioned above, X-ray scattering studies of 7O.7 demonstrated that the smectic-F phase set in for a narrow temperature range in films as thick as 180 layers, and that the temperature range increases with decreasing layer number. For films of the order of 25 layers thick, the smectic-I phase is observed at approximately 334 K, and with decreasing thickness the temperature range for this phase also increases. Below approximately 10 to 15 layers, the smectic-I phase extends up to 342 K where bulk samples undergo a firstorder transition from the crystalline-B to the smectic-C phase. Synchrotron X-ray scattering experiments show that, in thin films (five layers for example), the homogeneous smectic-I film undergoes a first-order transition to one in which the two surface layers are smectic-I and the three interior layers are smectic-C (Sirota et al., 1985; Sirota, Pershan, Sorensen & Collett, 1987). The fact that two phases with the same symmetry can coexist in this manner tells us that in this material there is some important microscopic difference between them. This is reaffirmed by the fact that the phase transition from the surface smectic-I to the homogeneous smectic-C phase has been observed to be first order (Sorensen et al., 1987). In contrast to 7O.7, Birgeneau and co-workers found that in racemic 4-(2-methylbutyl)phenyl 40 -octyloxylbiphenyl-4carboxylate (8OSI) (Brock et al., 1986), the X-ray structure of the smectic-I phase evolves continuously into that of the smectic-C. By applying a magnetic field to a thick freely suspended sample, Brock et al. were able to obtain a large monodomain sample.

Other important features of the smectic-F phase are, firstly, that the local molecular packing is identical to that of the tilted crystalline-G phase (Benattar et al., 1979; Sirota et al., 1985; Guillon et al., 1986). Secondly, there is considerable temperature dependence of the widths of the various diffuse peaks. Fig. 4.4.4.4(b) indicates the four inequivalent line widths that Sirota and co-workers measured in freely suspended films of the liquid crystal N-[4-(n-heptyloxy)benzylidene]-4-n-heptyl aniline (7O.7). Parenthetically, bulk samples of this material do not have a smectic-F phase; however, the smectic-F is observed in freely suspended films as thick as 200 layers. Fig. 4.4.4.5 illustrates the thickness–temperature phase diagram of 7O.7 between 325 and 342 K (Sirota et al., 1985; Sirota, Pershan & Deutsch, 1987). Bulk samples and thick films have a first-order transition from the crystalline-B to the smectic-C at 342 K. Thinner films indicate a surface phase above 342 K that will be discussed below. Furthermore, although there is a strong temperature dependence of the widths of the diffuse scattering peaks, the widths are independent of film thickness. This demonstrates that, although the free film boundary conditions have stabilized the smectic-F phase, the properties of the phase are not affected by the boundaries. Finally, the fact that the widths
QL1 and
QL2 along the L direction and
QH1 and
QH2 along the in-plane directions are not equal indicates that the correlations are very anisotropic (Brock et al., 1986; Sirota et al., 1985). We will discuss one possible model for these properties after presenting other data on thick films of 7O.7. From the fact that the positions of the intensity maxima for the diffuse spots of the smectic-F phase of 7O.7 correspond exactly to the positions of the Bragg peaks in the crystalline-G phase, we learn that the local molecular packing must be identical in the two phases. The major difference between the crystalline-G and the tilted hexatic smectic-F phase is that, in the latter, defects destroy the long-range positional order of the former (Benattar et al., 1979; Sirota et al., 1985). Although this is consistent with the existing theoretical model that attributes hexatic order to a proliferation of unbounded dislocations, it is not obvious that the proliferation is attributable to the same Kosterlitz–Thouless mechanism that Halperin & Nelson and Young discussed for the transition from the two-

557

4. DIFFUSE SCATTERING AND RELATED TOPICS They measured the X-ray scattering intensity around the circle in the reciprocal-space plane shown in Fig. 4.4.4.4(b) that passes through the peaks. For higher temperatures, when the sample is in the smectic-C phase, the intensity is essentially constant around the circle; however, on cooling, it gradually condenses into six peaks, separated by 60 . The data were analysed by expressing the intensity as a Fourier series of the form  Sð Þ ¼

I0 12

þ

1 P



shearing motion and the molecular tilt, we can define an angle ’ ¼ tan
1 ðhu2 i1=2 =dÞ, where d is the layer thickness. The observed diffuse intensity corresponds to angles ’ between 3 and 6 (Aeppli et al., 1981). Leadbetter and co-workers demonstrated that in the nO.m series various molecules undergo a series of restacking transitions and that crystalline-B phases exist with ABC and AAA stacking as well as the more common ABAB (Leadbetter, Mazid & Richardson 1980; Leadbetter, Mazid & Kelly, 1979). Subsequent high-resolution studies on thick freely suspended films revealed that the restacking transitions were actually subtler, and in 7O.7, for example, on cooling the hexagonal ABAB phase one observes an orthorhombic and then a monoclinic phase before the hexagonal AAA (Collett et al., 1982, 1985). Furthermore, the first transition from the hexagonal ABAB to the monoclinic phase is accompanied by the appearance of a relatively long-wavelength modulation within the plane of the layers. The polarization of this modulation is along the layer normal, or orthogonal to the polarization of the displacements that gave rise to the rods of thermal diffuse scattering (Gane & Leadbetter, 1983). It is also interesting to note that the AAA simple hexagonal structure does not seem to have been observed outside liquidcrystalline materials and, were it not for the fact that the crystalline-B hexagonal AAA is always accompanied by longwavelength modulations, it would be the only case of which we are aware. Figs. 4.4.4.6(a) and (b) illustrate the reciprocal-space positions of the Bragg peaks (dark dots) and modulation-induced side bands (open circles) for the unmodulated hexagonal ABAB and the modulated orthorhombic phase (Collett et al., 1984). For convenience, we only display one 60 sector. Hirth et al. (1984) explained how both the reciprocal-space structure and the modulation of the orthorhombic phase could result from an ordered array of partial dislocations. They were not, however, able to provide a specific model for the microscopic driving force for the transition. Sirota, Pershan & Deutsch (1987) proposed a variation of the Hirth model in which the dislocations pair up to form a wall of dislocation dipoles such that within the wall the local molecular packing is essentially identical to the packing in the crystalline-G phase that appears at temperatures just below the crystalline-B phase. This model explains: (1) the macroscopic symmetry of the phase; (2) the period of the modulation; (3) the polarization of the modulation; and (4) the size of the observed deviations of the reciprocal-space structure from the hexagonal symmetry of the ABAB phase and suggests a microscopic driving mechanism that we will discuss below. On further cooling, there is a first-order transition in which the one-dimensional modulation that appeared at the transition to orthorhombic symmetry is replaced by a two-dimensional modulation as shown in Fig. 4.4.4.6(c). On further cooling, there is another first-order transition in which the positions of the principal Bragg spots change from having orthorhombic to monoclinic symmetry as illustrated in Fig. 4.4.4.6(d). On further cooling, the Bragg peaks shift continually until there is one more first-order transition to a phase with hexagonal AAA positions as illustrated in Fig. 4.4.4.6( f ). On further cooling, the AAA symmetry remains unchanged, and the modulation period is only slightly dependent on temperature, but the modulation amplitude increases dramatically. Eventually, as indicated in the phase diagram shown in Fig. 4.4.4.5, the system undergoes another firstorder transition to the tilted crystalline-G phase. The patterns in Figs. 4.4.4.6(e) and (g) are observed by rapid quenching from the temperatures at which the patterns in Fig. 4.4.4.6(b) are observed. Although there is not yet an established theoretical explanation for the origin of the ‘restacking-modulation’ effects, there are a number of experimental facts that we can summarize, and which indicate a probable direction for future research. Firstly, if one ignores the long-wavelength modulation, the hexagonal ABAB phase is the only phase in the diagram for 7O.7 for which



C6n cos 6nð90
Þ þ IB ;

n¼1

where I0 fixes the absolute intensity and IB fixes the background. The temperature variation of the coefficients scaled according to the relation C6n ¼ C6n where the empirical relation n ¼ 2:6ðn
1Þ is in good agreement with a theoretical form predicted by Aharony et al. (1986). The only other system in which this type of measurement has been made was the smectic-C phase of 7O.7 (Collett, 1983). In that case, the intensity around the circle was constant, indicating the absence of any tilt-induced bond orientational order (Aharony et al., 1986). It would appear that the near-neighbour molecular packing of the smectic-I and the crystalline-J phases is the same, in just the same way as for the packing of the smectic-F and the crystallineG phases. The four smectic-I widths analogous to those illustrated in Fig. 4.4.4.4(a) are, like that of the smectic-F, both anisotropic and temperature dependent (Sirota et al., 1985; Sirota, Pershan, Sorensen & Collett, 1987; Brock et al., 1986; Benattar et al., 1979). 4.4.4.3. Crystalline phases with molecular rotation 4.4.4.3.1. Crystal-B Recognition of the distinction between the hexatic-B and crystalline-B phases provided one of the more important keys to understanding the ordered mesomorphic phases. There are a number of distinct phases called crystalline-B that are all true three-dimensional crystals, with resolution-limited Bragg peaks (Moncton & Pindak, 1979; Aeppli et al., 1981). The feature common to them all is that the average molecular orientation is normal to the layers, and within each layer the molecules are distributed on a triangular lattice. In view of the ‘blade-like’ shape of the molecule, the hexagonal site symmetry implies that the molecules must be rotating rapidly (Levelut & Lambert, 1971; Levelut, 1976; Richardson et al., 1978). We have previously remarked that this apparent rotational motion characterizes all of the phases listed in Table 4.4.1.1 except for the crystalline-E, -H and -K. In the most common crystalline-B phase, adjacent layers have ABAB-type stacking (Leadbetter, Gaughan et al., 1979; Leadbetter, Mazid & Kelly, 1979). High-resolution studies on well oriented samples show that in addition to the Bragg peaks the crystalline-B phases have rods of relatively intense diffuse scattering distributed along the 10L Bragg peaks (Moncton & Pindak, 1979; Aeppli et al., 1981). The widths of these rods in the reciprocal-space direction, parallel to the layers, are very sharp, and without a high-resolution spectrometer their widths would appear to be resolution limited. In contrast, along the reciprocalspace direction normal to the layers, their structure corresponds to the molecular form factor. If the intensity of the diffuse scattering can be represented as proportional to hQ  ui2, where u describes the molecular displacement, the fact that there is no rod of diffuse scattering through the 00L peaks indicates that the rods through the 10L peaks originate from random disorder in ‘sliding’ displacements of adjacent layers. It is likely that these displacements are thermally excited phonon vibrations; however, we cannot rule out some sort of non-thermal static defect structure. In any event, assuming this diffuse scattering originates in a thermal vibration for which adjacent layers slide over one another with some amplitude hu2 i1=2, and assuming strong coupling between this

558

4.4. SCATTERING FROM MESOMORPHIC STRUCTURES

Fig. 4.4.4.6. Location of the Bragg peaks in one 60 section of reciprocal space for the three-dimensional crystalline-B phases observed in thick films of 7O.7. (a) The normal hexagonal crystalline-B phase with ABAB stacking. (b) The one-dimensional modulated phase with orthorhombic symmetry. The closed circles are the principal Bragg peaks and the open circles indicate side bands associated with the long-wavelength modulation. (c) The two-dimensional modulated phase with orthorhombic symmetry. Only the lowest-order side bands are shown. They are situated on the corners of squares surrounding the Bragg peak. The squares are oriented as shown and the amplitude of the square diagonal is equal to the distance between the two side bands illustrated in (b). (d) The two-dimensional modulated phase with monoclinic symmetry. Note that the L position of one of the peaks has shifted relative to (c). (e) A two-dimensional modulated phase with orthorhombic symmetry that is only observed on heating the quenched phase illustrated in (g). ( f ) The two-dimensional modulated phase with hexagonal symmetry and AAA layer stacking. (g) A two-dimensional hexagonal phase with AAA layer stacking that is only observed on rapid cooling from the phase shown in (c).

there are two molecules per unit cell. There must be some basic molecular effect that determines this particular coupling between every other layer. In addition, it is particularly interesting that it only manifests itself for a small temperature range and then vanishes as the sample is cooled. Secondly, any explanation for the driving force of the restacking transition must also explain the modulations that accompany it. In particular, unless one cools rapidly, the same modulation structures with the same amplitudes always appear at the same temperature, regardless of the sample history, i.e. whether heating or cooling. No significant hysteresis is observed and Sirota argued that the structures are in thermal equilibrium. There are a number of physical systems for which the development of long-wavelength modulations is understood, and in each case they are the result of two or more competing interaction energies that cannot be simultaneously minimized (Blinc & Levanyuk, 1986; Safinya, Varady et al., 1986; Lubensky et al., 1988; Winkor & Clarke, 1986; Moncton et al., 1981; Fleming et al., 1980; Villain, 1980; Frank & van der Merwe, 1949; Bak et al., 1979; Pokrovsky & Talapov, 1979). The easiest to visualize is epitaxic growth of one crystalline phase on the surface of another when the two lattice vectors are slightly incommensurate. The first atomic row of adsorbate molecules can be positioned to minimize the attractive interactions with the substrate. This is slightly more difficult for the second row, since the distance that minimizes the interaction energy between the first and second rows of adsorbate molecules is not necessarily the same as the distance that would minimize the interaction energy between the first row and the substrate. As more and more rows are added, the energy price of this incommensurability builds up, and one possible configuration that minimizes the global energy is a modulated structure. In all known cases, the very existence of modulated structures implies that there must be competing interactions, and the only real question about the modulated structures in the crystalline-B phases is the identification of the competing interactions. It appears that one of the more likely possibilities is the difficulty in packing the 7O.7 molecules within a triangular lattice while simultaneously optimizing the area per molecule of the alkane tails and the conjugated rings in the core (Carlson & Sethna, 1987; Sadoc & Charvolin, 1986). Typically, the mean crosssectional area for a straight alkane in the all-trans configuration is ˚ 2, while the mean area per molecule in the between 18 and 19 A ˚ 2. While these two could be crystalline-B phase is closer to 24 A reconciled by assuming that the alkanes are tilted with respect to

the conjugated core, there is no reason why the angle that reconciles the two should also be the same angle that minimizes the internal energy of the molecule. Even if it were the correct angle at some temperature by accident, the average area per chain is certainly temperature dependent. Even without attempting to include the rotational dynamics that are necessary to understanding the axial site symmetry, it is obvious that there can be a conflict in the packing requirements of the two different parts of the molecule. A possible explanation of these various structures might be as follows: at high temperatures, both the alkane chain, as well as the other degrees of freedom, have considerable thermal motions that make it possible for the conflicting packing requirements to be simultaneously reconciled by one or another compromise. On the other hand, with decreasing temperature, some of the thermal motions become frozen out, and the energy cost of the reconciliation that was possible at higher temperatures becomes too great. At this point, the system must find another solution, and the various modulated phases represent the different compromises. Finally, all of the compromises involving inhomogeneities, like the modulations or grain boundaries, become impossible and the system transforms into a homogeneous crystalline-G phase. If this type of argument could be made more specific, it would also provide a possible explanation for the molecular origin of the three-dimensional hexatic phases. The original suggestion for the existence of hexatic phases in two dimensions was based on the fact that the interaction energy between dislocations in two dimensions was logarithmic, such that the entropy and the enthalpy had the same functional dependence on the density of dislocations. This gave rise to the observation that above a certain temperature two-dimensional crystals would be unstable against thermally generated dislocations. Although Litster and Birgeneau’s suggestion that some of the observed smectic phases might be stacks of two-dimensional hexatics is certainly correct, it is not necessary that the observed three-dimensional hexatics originate from entropy-driven thermally excited dislocations. For example, the temperature–layer-number phase diagram for 7O.7 that is shown in Fig. 4.4.4.5 has the interesting property that the temperature region over which the tilted hexatic phases exist in thin films is almost the same as the temperature region for which the modulated phases exist in thick films and in bulk samples. From the fact that molecules in the nO.m series that only differ by one or two —CH2— groups have different sequences of mesomorphic phases, we learn that within any one molecule the

559

4. DIFFUSE SCATTERING AND RELATED TOPICS 4.4.4.3.2. Crystal-G, crystal-J The crystalline-G and crystalline-J phases are the ordered versions of the smectic-F and smectic-I phases, respectively. The positions of the principal peaks illustrated in Fig. 4.4.4.4 for the smectic-F(I) are identical to the positions in the smectic-G(J) phase if small thermal shifts are discounted. In both the hexatic and the crystalline phases, the molecules are tilted with respect to the layer normals by approximately 25 to 30 with nearly hexagonal packing around the tilted axis (Doucet & Levelut, 1977; Levelut et al., 1974; Levelut, 1976; Leadbetter, Mazid & Kelly, 1979; Sirota, Pershan, Sorensen & Collett, 1987). The interlayer molecular packing appears to be end to end, in an AAA type of stacking (Benattar et al., 1983, 1981; Levelut, 1976; Gane et al., 1983). There is only one molecule per unit cell and there is no evidence for the long-wavelength modulations that are so prevalent in the crystalline-B phase that is the next higher temperature phase above the crystalline-G in 7O.7.

Fig. 4.4.4.7. (a) The ‘herringbone’ stacking suggested for the crystalline-E phase in which molecular rotation is partially restricted. The primitive rectangular unit cell containing two molecules is illustrated by the shaded region. The lattice has rectangular symmetry and a 6¼ b. (b) The position of the Bragg peaks in the plane in reciprocal space that is parallel to the layers. The dark circles indicate the principal Bragg peaks that would be the only ones present if all molecules were equivalent. The open circles indicate additional peaks that are observed for the model illustrated in (a). The crosshatched circles indicate peaks that are missing because of the glide plane in (a).

4.4.4.4. Crystalline phases with herringbone packing 4.4.4.4.1. Crystal-E Fig. 4.4.4.7 illustrates the intralayer molecular packing proposed for the crystalline-E phase (Levelut, 1976; Doucet, 1979; Levelut et al., 1974; Doucet et al., 1975; Leadbetter et al., 1976; Richardson et al., 1978; Leadbetter, Frost, Gaughan & Mazid, 1979; Leadbetter, Frost & Mazid, 1979). The molecules are, on average, normal to the layers; however, from the optical birefringence it is apparent that the site symmetry is not uniaxial. X-ray diffraction studies on single crystals by Doucet and coworkers demonstrated that the biaxiality was not attributable to molecular tilt and subsequent work by a number of others resulted in the arrangement shown in Fig. 4.4.4.7(a). The most important distinguishing reciprocal-space feature associated with the intralayer ‘herringbone’ pffiffiffi packing is the appearance of Bragg peaks at sinð Þ equal to 7=2 times the value for the lowest-order in-plane Bragg peak for the triangular lattice (Pindak et al., 1981). These are illustrated by the open circles in Fig. 4.4.4.7(b). The shaded circles correspond to peaks that are missing because of the glide plane that relates the two molecules in the rectangular cell. Leadbetter, Mazid & Malik (1980) carried out detailed studies on both the crystalline-E phase of isobutyl 4-(4-phenylbenzylideneamino)cinnamate (IBPBAC) and the crystalline phase immediately below the crystalline-E phase. Partially ordered samples of the crystalline-E phase were obtained by melting the lower-temperature crystalline phase. Although the data for the crystalline-E phase left some ambiguity, they argued that the phase they were studying might well have had molecular tilts of the order of 5 or 6 . This is an important distinction, since the crystalline-H and crystalline-J phases are essentially tilted versions of the crystalline-E. Thus, one important symmetry difference that might distinguish the crystalline-E from the others is the presence of a mirror plane parallel to the layers. In view of the low symmetry of the individual molecules, the existence of such a mirror plane would imply residual molecular motions. In fact, using neutron diffraction Leadbetter et al. (1976) demonstrated for a different liquid crystal that, even though the site symmetry is not axially symmetric, there is considerable residual rotational motion in the crystalline-E phase about the long axis of the molecules. Since the in-plane spacing is too small for neighbouring molecules to be rotating independently of each other, they proposed what might be interpreted as large partially hindered rotations.

difference in chemical potentials between the different mesomorphic phases must be very small (Leadbetter, Mazid & Kelly, 1979; Doucet & Levelut, 1977; Leadbetter, Frost & Mazid, 1979; Leadbetter, Mazid & Richardson, 1980; Smith et al., 1973; Smith & Garland, 1973). For example, although in 7O.7 the smectic-F phase is only observed in finite-thickness films, both 5O.6 and 9O.4 have smectic-F phase in bulk. Thus, in bulk 7O.7 the chemical potential for the smectic-F phase must be only slightly larger than that of the modulated crystalline-B phases, and the effect of the surfaces must be sufficient to reverse the order in samples of finite thickness. As far as the appearance of the smectic-F phase in 7O.7 is concerned, it is well known that the interaction energy between dislocation pairs is very different near a free surface from that in the bulk (Pershan, 1974; Pershan & Prost, 1975). The origin of this is that the elastic properties of the surface will usually cause the stress field of a dislocation near to the surface either to vanish or to be considerably smaller than it would in the bulk. Since the interaction energy between dislocations depends on this stress field, the surface significantly modifies the dislocation–dislocation interaction. This is a long-range effect, and it would not be surprising if the interactions that stabilized the dislocation arrays to produce the long-wavelength modulations in the thick samples were sufficiently weaker in the samples of finite thickness that the dislocation arrays are disordered. Alternatively, there is evidence that specific surface interactions favour a finite molecular tilt at temperatures where the bulk phases are uniaxial (Farber, 1985). Incommensurability between the period of the tilted surface molecules and the crystalline-B phases below the surface would increase the density of dislocations, and this would also modify the dislocation–dislocation interactions in the bulk. Sirota et al. (1985) and Sirota, Pershan, Sorensen & Collett (1987) demonstrated that, while the correlation lengths of the smectic-F phase have a significant temperature dependence, the lengths are independent of film thickness, and this supports the argument that although the effects of the surface are important in stabilizing the smectic-F phase in 7O.7, once the phase is established it is essentially no different from the smectic-F phases observed in bulk samples of other materials. Brock et al. (1986) observed anisotropies in the correlation lengths of thick samples of 8OSI that are similar to those observed by Sirota. These observations motivate the hypothesis that the dislocation densities in the smectic-F phases are determined by the same incommensurability that gives rise to the modulated crystalline-B structures. Although all of the experimental evidence supporting this hypothesis was obtained from the smectic-F tilted hexatic phase, there is no reason why this speculation could not apply to both the tilted smectic-I and the untilted hexatic-B phase.

4.4.4.4.2. Crystal-H, crystal-K The crystalline-H and crystalline-K phases are tilted versions of the crystalline-E. The crystalline-H is tilted in the direction between the near neighbours, with the convenient mnemonic that

560

4.4. SCATTERING FROM MESOMORPHIC STRUCTURES disordered in the D2 phase. The reciprocal-space structure of the D1 phase is similar to that of the crystalline-E phase shown in Fig. 4.4.4.7(b). Other discotic phases that have been proposed would have the molecules arranged periodically along the column, but disordered between columns. This does not seem physically realistic since it is well known that thermal fluctuations rule out the possibility of a one-dimensional periodic structure even more strongly than for the two-dimensional lattice that was discussed above (Landau, 1965; Peierls, 1934). On the other hand, in the absence of either more high-resolution studies on oriented fibres or further theoretical studies, we prefer not to speculate on the variety of possible true discotic or discotic-like crystalline phases that might exist. This is a subject for future research.

Fig. 4.4.5.1. Schematic illustration of the molecular stacking for the discotic (a) D2 and (b) D1 phases. In neither of these two phases is there any indication of long-range positional order along the columns. The hexagonal symmetry of the D1 phase is broken by ‘herringbone-like’ correlations in the molecular tilt from column to column.

4.4.6. Other phases We have deliberately chosen not to discuss the properties of the cholesteric phase in this chapter because the length scales that characterize the long-range order are of the order of micrometres and are more easily studied by optical scattering than by X-rays (De Gennes, 1974; De Vries, 1951). Nematic phases formed from chiral molecules develop long-range order in which the orientation of the director hni varies in a plane-wave-like manner that can be described as x cosð2
z=Þ þ y sinð2
z=Þ, where x and y are unit vectors and =2 is the cholesteric ‘pitch’ that can be anywhere from 0.1 to 10 mm depending on the particular molecule. Even more interesting is that for many cholesteric systems there is a small temperature range, of the order of 1 K, between the cholesteric and isotropic phases for which there is a phase known as the ‘blue phase’ (Coates & Gray, 1975; Stegemeyer & Bergmann, 1981; Meiboom et al., 1981; Bensimon et al., 1983; Hornreich & Shtrikman, 1983; Crooker, 1983). In fact, there is more than one ‘blue phase’ but they all have the property that the cholesteric twist forms a three-dimensional lattice twisted network rather than the plane-wave-like twist of the cholesteric phase. Three-dimensional Bragg scattering from blue phases using laser light indicates cubic lattices; however, since the optical cholesteric interactions are much stronger than the usual interactions between X-rays and atoms, interpretation of the results is subtler. Gray and Goodby discuss a ‘smectic-D’ phase that is otherwise omitted from this chapter (Gray & Goodby, 1984). Gray and coworkers first observed this phase in the homologous series of 40 -n-alkoxy-30 -nitrobiphenyl-4-carboxylic acids (Gray et al., 1957). In the hexadecyloxy compound, this phase exists for a region of about 26 K between the smectic-C and smectic-A phases: smectic-C (444.2 K) smectic-D (470.4 K) smectic-A. It is optically isotropic and X-ray studies by Diele et al. (1972) and by Tardieu & Billard (1976) indicate a number of similarities to the ‘cubic–isotropic’ phase observed in lyotropic systems (Luzzati & Riess-Husson, 1966; Tardieu & Luzzati, 1970). More recently, Etherington et al. (1986) studied the ‘smectic-D’ phase of 30 -cyano-40 -n-octadecyloxybiphenyl-4-carboxylic acid. Since this material appears to be more stable than some of the others that were previously studied, they were able to perform sufficient measurements to determine that the space group is cubic P23 or ˚ . Etherington et al. Pm3 with a lattice parameter of 86 A suggested that the ‘smectic-D’ phase that they studied is a true three-dimensional cubic crystal of micelles and noted that the designation of ‘smectic-D’ is not accurate. Guillon & Skoulios (1987) have proposed a molecular model for this and related phases. Fontell (1974) has reviewed the literature on the X-ray diffraction studies of lyotropic mesomorphic systems and the reader is referred there for more extensive information on those cubic systems. The mesomorphic structures of lyotropic systems are much richer than those of the thermotropic and, in addition

on cooling the sequence of phases with the same relative orientation of tilt to near-neighbour position is F ! G ! H. Similarly, the tilt direction for the crystalline-K phase is similar to that of the smectic-I and crystalline-J so that the expected phase sequence on cooling might be I ! J ! K. In fact, both of these sequences are only intended to indicate the progression in lower symmetry; the actual transitions vary from material to material.

4.4.5. Discotic phases In contrast to the long thin rod-like molecules that formed most of the other phases discussed in this chapter, the discotic phases are formed by molecules that are more disc-like [see Fig. 4.4.1.3( f), for example]. There was evidence that mesomorphic phases were formed from disc-like molecules as far back as 1960 (Brooks & Taylor, 1968); however, the first identification of a discotic phase was by Chandrasekhar et al. (1977) with benzenehexyl hexa-n-alkanoate compounds. Disc-like molecules can form either a fluid nematic phase in which the disc normals are aligned, without any particular long-range order at the molecular centre of mass, or more-ordered ‘columnar’ (Helfrich, 1979) or ‘discotic’ (Billard et al., 1981) phases in which the molecular positions are correlated such that the discs stack on top of one another to form columns. Some of the literature designates this nematic phase as ND to distinguish it from the phase formed by ‘rod-like’ molecules (Destrade et al., 1983). In the same way that the appearance of layers characterizes order in smectic phases, the order for the discotic phases is characterized by the appearance of columns. Chandrasekhar (1982, 1983) and Destrade et al. (1983) have reviewed this area and have summarized the several notations for various phases that appear in the literature. Levelut (1983) has also written a review and presented a table listing the space groups for columnar phases formed by 18 different molecules. Unfortunately, it is not absolutely clear which of these are mesomorphic phases and which are crystals with true long-range positional order. Fig. 4.4.5.1 illustrates the molecular packing in two of the well identified discotic phases that are designated as D1 and D2 (Chandrasekhar, 1982). The phase D2 consists of a hexagonal array of columns for which there is no intracolumnar order. The system is uniaxial and, as originally proposed, the molecular normals were supposed to be along the column axis. However, recent X-ray scattering studies on oriented free-standing fibres of the D2 phase of triphenylene hexa-n-dodecanoate indicate that the molecules are tilted with respect to the layer normal (Safinya et al., 1985, 1984). The D1 phase is definitely a tilted phase, and consequently the columns are packed in a rectangular cell. According to Safinya et al., the D1 to D2 transition corresponds to an order–disorder transition in which the molecular tilt orientation is ordered about the column axis in the D1 phase and

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4. DIFFUSE SCATTERING AND RELATED TOPICS to all structures mentioned here, there are lyotropic systems in which the smectic-A lamellae seem to break up into cylindrical rods which seem to have the same macroscopic symmetry as some of the discotic phases. On the other hand, it is also much more difficult to prepare a review for the lyotropic systems in the same type of detail as for the thermotropic. The extra complexity associated with the need to control water concentration as well as temperature has made both theoretical and experimental progress more difficult, and, since there has not been very much experimental work on well oriented samples, detailed knowledge of many of these phases is also limited. Aside from the simpler lamellae systems, which seem to have the same symmetry as the thermotropic smectic-A phase, it is not at all clear which of the other phases are three-dimensional crystals and which are true mesomorphic structures. For example, dipalmitoylphosphatidylcholine has an L phase that appears for temperatures and (or) water content that is lower than that of the smectic-A L phase (Shipley et al., 1974; Small, 1967; Chapman et al., 1967). The diffraction pattern for this phase contains sharp large-angle reflections that may well correspond to a phase that is like one of the crystalline phases listed in Tables 4.4.1.1 and 4.4.1.2, and Fig. 4.4.1.1. On the other hand, this phase could also be hexatic and we do not have sufficient information to decide. The interested reader is referred to the referenced articles for further detailed information.

dimensional xy model (Lubensky, 1983; Le Guillou & ZinnJustin, 1985). The differences between this experiment and others that were discussed previously, and which did not agree with theory, are firstly that this material is much further from the tricritical point that appears to be ubiquitous for most liquidcrystalline materials and, secondly, that they used the Landau–De Gennes theory to argue that the critical-temperature dependence for the Q4? term in the differential cross section given in equation (4.4.2.7) is not that of the c?4 term but rather should vary as ½ðT
TNA Þ=T
=4 , where  is the exponent that describes the critical-temperature dependence of the smectic order parameter jj2 ’ ½ðT
TNA Þ=T
 . The experimental results are in good agreement with the Monte Carlo simulation of the N–SA transition that was reported by Dasgupta (1985, 1987).

References Aeppli, G. & Bruinsma, R. (1984). Hexatic order and liquid crystal density fluctuations. Phys. Rev. Lett. 53, 2133–2136. Aeppli, G., Litster, J. D., Birgeneau, R. J. & Pershan, P. S. (1981). High resolution X-ray study of the smectic A–smectic B phase transition and the smectic B phase in butyloxybenzylidene octylaniline. Mol. Cryst. Liq. Cryst. 67, 205–214. Aharony, A., Birgeneau, R. J., Brock, J. D. & Litster, J. D. (1986). Multicriticality in hexatic liquid crystals. Phys. Rev. Lett. 57, 1012–1015. Alben, R. (1973). Phase transitions in a fluid of biaxial particles. Phys. Rev. Lett. 30, 778–781. Als-Nielsen, J., Christensen, F. & Pershan, P. S. (1982). Smectic-A order at the surface of a nematic liquid crystal: synchrotron X-ray diffraction. Phys. Rev. Lett. 48, 1107–1110. Als-Nielsen, J., Litster, J. D., Birgeneau, R. J., Kaplan, M., Safinya, C. R., Lindegaard, A. & Mathiesen, S. (1980). Observation of algebraic decay of positional order in a smectic liquid crystal. Phys. Rev. B, 22, 312–320. Bak, P., Mukamel, D., Villain, J. & Wentowska, K. (1979). Commensurate–incommensurate transitions in rare-gas monolayers adsorbed on graphite and in layered charge-density-wave systems. Phys. Rev. B, 19, 1610–1613. Barois, P., Prost, J. & Lubensky, T. C. (1985). New critical points in frustrated smectics. J. Phys. (Paris), 46, 391–399. Beaglehole, D. (1982). Pretransition order on the surface of a nematic liquid crystal. Mol. Cryst. Liq. Cryst. 89, 319–325. Benattar, J. J., Doucet, J., Lambert, M. & Levelut, A. M. (1979). Nature of the smectic F phase. Phys. Rev. A, 20, 2505–2509. Benattar, J. J., Levelut, A. M. & Strzelecki, L. (1978). Etude de l’influence de la longueur mole´culaire les characte´ristiques des phases smectique ordonne´es. J. Phys. (Paris), 39, 1233–1240. Benattar, J. J., Moussa, F. & Lambert, M. (1980). Two-dimensional order in the smectic F phase. J. Phys. (Paris), 40, 1371–1374. Benattar, J. J., Moussa, F. & Lambert, M. (1983). Two dimensional ordering in liquid crystals: the SmF and SmI phases. J. Chim. Phys. 80, 99–107. Benattar, J. J., Moussa, F., Lambert, M. & Germian, C. (1981). Two kinds of two-dimensional order: the SmF and SmI phases. J. Phys. (Paris) Lett. 42, L67–L70. Benguigui, L. (1979). A Landau theory of the NAC point. J. Phys. (Paris) Colloq. 40, C3-419–C3-421. Bensimon, D., Domany, E. & Shtrikman, S. (1983). Optical activity of cholesteric liquid crystals in the pretransitional regime and in the blue phase. Phys. Rev. A, 28, 427–433. Billard, J., Dubois, J. C., Vaucher, C. & Levelut, A. M. (1981). Structures of the two discophases of rufigallol hexa-n-octanoate. Mol. Cryst. Liq. Cryst. 66, 115–122. Birgeneau, R. J., Garland, C. W., Kasting, G. B. & Ocko, B. M. (1981). Critical behavior near the nematic–smectic-A transition in butyloxybenzilidene octylaniline (4O.8). Phys. Rev. A, 24, 2624–2634. Birgeneau, R. J. & Litster, J. D. (1978). Bond orientational order model for smectic B liquid crystals. J. Phys. (Paris) Lett. 39, L399–L402. Blinc, R. & Levanyuk, A. P. (1986). Editors. Modern Problems in Condensed Matter Sciences, Incommensurate Phases in Dielectrics, 2. Materials. Amsterdam: North-Holland. Bouwman, W. G. & de Jeu, W. H. (1992). 3D XY behavior of a nematic– smectic-A phase transition: confirmation of the de Gennes model. Phys. Rev. Lett. 68, 800–803.

4.4.6.1. Phases with intermediate molecular tilt: smectic-L, crystalline-M,N Following the completion of this manuscript, Smith and coworkers (Smith et al., 1988; Sirota et al., 1988) published an X-ray scattering study of the structure of a freely suspended multilayer film of hydrated phosphatidylcholine in which the phase that had been designated LB0 in the literature on lipid phases (Janiak et al., 1979; Luzzati, 1968; Tardieu et al., 1973) was shown to consist of three separate two-dimensional phases in which the positional order in adjacent layers is uncoupled. The three phases are distinguished by the direction of the alkane-chain tilt relative to the nearest neighbours, and in one of these phases the orientation varies continuously with increasing hydration. At the lowest hydration, they observe a phase in which the tilt is towards the second-nearest neighbour; in analogy to the smectic-F phase, they designate this phase LF. On increasing hydration, they observe a phase in which the tilt direction is intermediate between the nearest- and next-nearest-neighbour directions, and which varies continuously with hydration. This is a new phase that was not previously known and they designate it LL. On further hydration, they observe a phase in which the molecular tilt is towards a nearest neighbour and this is designated LI. At maximum hydration, they observe the phase with longwavelength modulation that was previously designated P (Janiak et al., 1979). Selinger & Nelson (1988) have subsequently developed a theory for the phase transitions between phases with varying tilt orientation and have rationalized the existence of phases with intermediate tilt. To be complete, both Fig. 4.4.1.1 and Table 4.4.1.1 should be amended to include this type of hexatic order which is now referred to as the smectic-L. Extension of the previous logic suggests that the crystalline phases with intermediate tilt should be designated M and N, where N has ‘herringbone’ type of intermolecular order. 4.4.6.2. Nematic to smectic-A phase transition At the time this manuscript was prepared, there was a fundamental discrepancy between theoretical predictions for the details of the critical properties of the second-order nematic to smectic-A phase transition. This has been resolved. Bouwman & de Jeu (1992) reported an X-ray scattering study of the critical properties of octyloxyphenylcyanobenzyloxybenzoate in which the data were in good agreement with predictions of the three-

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Thoen, J., Marynissen, H. & Van Dael, W. (1982). Temperature dependence of the enthalpy and the heat capacity of the liquid-crystal octylcyanobiphenyl (8CB). Phys. Rev. A, 26, 2886–2905. Thoen, J., Marynissen, H. & Van Dael, W. (1984). Nematic–smectic-A tricritical point in alkylcyanobiphenyl liquid crystals. Phys. Rev. Lett. 5, 204–207. Villain, J. (1980). In Order in Strongly Fluctuating Condensed Matter System, edited by T. Riste, pp. 221–260. New York: Plenum. Wang, J. & Lubensky, T. C. (1984). Theory of the SA1–SA2 phase transition in liquid crystal. Phys. Rev. A, 29, 2210–2217. Warren, B. E. (1968). X-ray Diffraction. Reading: Addison-Wesley.

566

references

International Tables for Crystallography (2010). Vol. B, Chapter 4.5, pp. 567–589.

4.5. Polymer crystallography By R. P. Millane and D. L. Dorset

X-rays (e.g. Stark et al., 1988; Forsyth et al., 1989). X-ray fibre diffraction analysis is particularly suitable for biological polymers that form natural fibrous superstructures and even for many synthetic polymers that exist in either a fibrous or a liquidcrystalline state. Fibre diffraction has played an important role in structural studies of polynucleotides, polysaccharides, polypeptides and polyesters, as well as rod-like helical viruses, bacteriophages, microtubules and muscle fibres (Arnott, 1980; French & Gardner, 1980; Hall, 1984; Millane, 1988; Atkins, 1989). The common, and unique, feature of these systems is that the molecules (or their aggregates) are randomly rotated about an axis of preferred orientation. As a result, the measured diffraction is the cylindrical average of that from a single molecule or aggregate. The challenge for the structural scientist, therefore, is that of structure determination from cylindrically averaged diffraction intensities. Since a wide range of types and degrees of order (or disorder) occur in fibrous specimens, as well as a wide range of sizes of the repeating units, a variety of methods are used for structure determination. The second technique used for structural studies of polymers is polymer electron crystallography. This involves measuring electron intensity data from individual crystalline regions or lamellae in the diffraction plane of an electron microscope. This is possible because a narrow electron beam can be focused on a single thin microcrystal and because of the enhanced scattering cross section of matter for electrons. By tilting the specimen, threedimensional diffraction intensities from a single microcrystal can be collected. This means that the unit-cell dimensions and symmetry can be obtained unambiguously in electron-diffraction experiments on individual chain-folded lamellae, and the data can be used for actual single-crystal structure determinations. One of the first informative electron-diffraction studies of crystalline polymer films was made by Storks (1938), who formulated the concept of chain folding in polymer lamellae. Among the first quantitative structure determinations from electron-diffraction intensities was that of Tatarinova & Vainshtein (1962) on the
form of poly--methyl-l-glutamate. Quantitative interpretation of the intensity data may benefit from the assumption of quasikinematical scattering (Dorset, 1995a), as long as the proper constraints are placed on the experiment. Structure determination may then proceed using the traditional techniques of X-ray crystallography. While molecular-modelling approaches (in which atomic level molecular and crystal structure models are constructed and refined) have been employed with single-crystal electron-diffraction data (Brisse, 1989), the importance of ab initio structure determination has been established in recent years (Dorset, 1995b), demonstrating that no initial assumptions about the molecular geometry need be made before the determination is begun. In some cases too, high-resolution electron micrographs of the polymer crystal structure can be used as an additional means for determining crystallographic phases and/or to visualize lattice defects. Each of the two techniques described above has its own advantages and disadvantages. While specimen disorder can limit the application of X-ray fibre diffraction analysis, polymer electron diffraction is limited to materials that can be be prepared as crystalline lamallae and that can withstand the high vacuum environment of an electron microscope (although the latter restriction can now be largely overcome by the use of lowtemperature specimen holders and/or environmental chambers).

4.5.1. Overview

By R. P. Millane and D. L. Dorset Linear polymers from natural or synthetic sources are actually polydisperse aggregates of high-molecular-weight chains. Nevertheless, many of these essentially infinite-length molecules can be prepared as solid-state specimens that contain ordered molecular segments or crystalline inclusions (Vainshtein, 1966; Tadokoro, 1979; Mandelkern, 1989; Barham, 1993). In general, ordering can occur in a number of ways. Hence an oriented and/or somewhat ordered packing of chain segments might be found in a stretched fibre, or in the chain-folded arrangement of a lamellar crystallite. Lamellae themselves may exist as single plates or in the more complex array of a spherulite (Geil, 1963). Diffraction data can be obtained from these various kinds of specimens and used to determine molecular and crystal structures. There are numerous reasons why crystallography of polymers is important. Although it may be possible to crystallize small constituent fragments of these large molecules and determine their crystal structures, one often wishes to study the intact (and biologically or functionally active) polymeric system. The molecular conformations and intermolecular interactions are determinants of parameters such as persistence length which affect, for example, solution conformations (random or worm-like coils) which determine viscosity. Molecular conformations also influence intermolecular interactions, which determine physical properties in gels and melts. Molecular conformations are, of course, of critical importance in many biological recognition processes. Knowledge of the stereochemical constraints that are placed on the molecular packing to maximize unit-cell density is particularly relevant to the fact that many linear molecules (as well as monodisperse substances with low molecular weight) can adopt several different allomorphic forms, depending on the crystallization conditions employed or the biological origin. Since different allomorphs can behave quite differently from one another, it is clear that the mode of chain packing is related to the bulk properties of the polymer (Grubb, 1993). The threedimensional geometry of the chain packing obtained from a crystal structure analysis can be used to investigate other phenomena such as the possible inclusion of disordered material in chain-fold regions (Mandelkern, 1989; Lotz & Wittmann, 1993), the ordered interaction of crystallite sectors across grain boundaries where tight interactions are found between domains, or the specific interactions of polymer chains with another substance in a composite material (Lotz & Wittmann, 1993). The two primary crystallographic techniques used for studying polymer structure are described in this chapter. The first is X-ray fibre diffraction analysis, described in Section 4.5.2; and the second is polymer electron crystallography, described in Section 4.5.3. Crystallographic studies of polymers were first performed using X-ray diffraction from oriented fibre specimens. Early applications were to cellulose and DNA from the 1930s to the 1950s, and the technique has subsequently been applied to hundreds of biological and synthetic polymers (Arnott, 1980; Millane, 1988). This technique is now referred to as X-ray fibre diffraction analysis. In fact, fibre diffraction analysis can be employed not only for polymers, but for any system that can be oriented. Indeed, one of the first applications of the technique was to tobacco mosaic virus (Franklin, 1955). Fibre diffraction analysis has also utilized, in some cases, neutrons instead of Copyright © 2010 International Union of Crystallography

567

4. DIFFUSE SCATTERING AND RELATED TOPICS called the fibre axis. The molecule itself is usually considered to be a rigid body. There is no other ordering within the specimen. The molecules are therefore randomly positioned in the lateral plane and are randomly rotated about their molecular axes. Furthermore, if the molecule does not have a twofold rotation axis normal to the molecular axis, then the molecular axis has a direction associated with it, and the molecular axes are oriented randomly parallel or antiparallel to each other. This is often called directional disorder, or the molecules are said to be oriented randomly up and down. The average length of the ordered molecular segments in a noncrystalline fibre is referred to as the coherence length. Polycrystalline fibres are characterized by molecular segments packing together to form well ordered microcrystallites within the specimen. The crystallites effectively take the place of the molecules in a noncrystalline specimen as described above. The crystallites are oriented, and since the axis within each crystallite that is aligned parallel to those in other crystallites usually corresponds to the long axes of the constituent molecules, it is also referred to here as the molecular axis. The crystallites are randomly positioned in the lateral plane, randomly rotated about the molecular axis, and randomly oriented up or down. The size of the crystalline domains can be characterized by their average dimensions in the directions of the a, b and c unit-cell vectors. However, because of the rotational disorder of the crystallites, any differences between crystallite dimensions in different directions normal to the fibre axis tend to be smeared out in the diffraction pattern, and the crystallite size is usefully characterized by the average dimensions of the crystallites normal and parallel to the fibre axis. The molecules or crystallites in a fibre specimen are not perfectly oriented, and the variation in inclinations of the molecular axes to the fibre axis is referred to as disorientation. Assuming that the orientation is axisymmetric, then it can be described by an orientation density function
ð
Þ such that
ð
Þ d! is the fraction of molecules in an element of solid angle d! inclined at an angle
to the fibre axis. The exact form of
ð
Þ is generally not known for any particular fibre and it is often sufficient to assume a Gaussian orientation density function, so that

4.5.2. X-ray fibre diffraction analysis

By R. P. Millane 4.5.2.1. Introduction X-ray fibre diffraction analysis is a collection of crystallographic techniques that are used to determine molecular and crystal structures of molecules, or molecular assemblies, that form specimens (often fibres) in which the molecules, assemblies or crystallites are approximately parallel but not otherwise ordered (Arnott, 1980; French & Gardner, 1980; Hall, 1984; Vibert, 1987; Millane, 1988; Atkins, 1989; Stubbs, 1999). These are usually long, slender molecules and they are often inherently flexible, which usually precludes the formation of regular three-dimensional crystals suitable for conventional crystallographic analysis. X-ray fibre diffraction therefore provides a route for structure determination for certain kinds of specimens that cannot be crystallized. Although it may be possible to crystallize small fragments or subunits of these molecules, and determine the crystal structures of these, X-ray fibre diffraction provides a means for studying the intact, and often the biologically or functionally active, system. Fibre diffraction has played an important role in the determination of biopolymers such as polynucleotides, polysaccharides (both linear and branched), polypeptides and a wide variety of synthetic polymers (such as polyesters), as well as larger assemblies including rod-like helical viruses, bacteriophages, microtubules and muscle fibres (Arnott, 1980; Arnott & Mitra, 1984; Millane, 1990c; Squire & Vibert, 1987). Specimens appropriate for fibre diffraction analysis exhibit rotational disorder (of the molecules, aggregates or crystallites) about a preferred axis, resulting in cylindrical averaging of the diffracted intensity in reciprocal space. Therefore, fibre diffraction analysis can be thought of as ‘structure determination from cylindrically averaged diffraction intensities’ (Millane, 1993). In a powder specimen the crystallites are completely (spherically) disordered, so that structure determination by fibre diffraction can be considered to be intermediate between structure determination from single crystals and from powders. This section is a review of the theory and techniques of structure determination by X-ray fibre diffraction analysis. It includes descriptions of fibre specimens, the theory of diffraction by these specimens, intensity data collection and processing, and the variety of structure determination methods used for the various kinds of specimens studied by fibre diffraction. It does not include descriptions of specimen preparation (those can be found in the references given for specific systems), or of applications of X-ray diffraction to determining polymer morphology (e.g. particle or void sizes and shapes, texture, domain structure etc.).


 1
2
ð
Þ ¼ exp
2 ; 2
20 2
0

ð4:5:2:1Þ

where
0 is a measure of the degree of disorientation. Fibre specimens often exhibit various kinds of disorder. The disorder may be within the molecules or in their packing. Disorder affects the relationship between the molecular and crystal structure and the diffracted intensities. Disorder within the molecules may result from a degree of randomness in the chemical sequence of the molecule or from variability in the interactions between the units that make up the molecule. Such molecules may (at least in principle) form noncrystalline, polycrystalline or partially crystalline (described below) fibres. Disordered packing of molecules within crystallites can result from a variety of ways in which the molecules can interact with each other. Fibre specimens made up of disordered crystallites are referred to here as partially crystalline fibres.

4.5.2.2. Fibre specimens A wide variety of kinds of fibre specimen exist. All exhibit preferred orientation; the variety results from variability in the degree of order (crystallinity) in the lateral plane (the plane perpendicular to the axis of preferred orientation). This leads to categorization of three kinds of fibre specimen: noncrystalline fibres, in which there is no order in the lateral plane; polycrystalline fibres, in which there is near-perfect crystallinity in the lateral plane; and disordered fibres, in which there is disorder either within the molecules or in their crystalline packing (or both). The kind of fibre specimen affects the kind of diffraction pattern obtained, the relationships between the molecular and crystal structures and the diffraction data, methods of data collection, and methods of structure determination. Noncrystalline fibres are made up of a collection of molecules that are oriented. This means that there is a common axis in each molecule (referred to here as the molecular axis), the axes being parallel in the specimen. The direction of preferred orientation is

4.5.2.3. Diffraction by helical structures Molecules or assemblies studied by fibre diffraction are usually made up of a large number of identical, or nearly identical, residues, or subunits, that in an oriented specimen are distributed along an axis; this leads naturally to helical symmetry. Since a periodic structure with no helix symmetry can be treated as a onefold helix, the assumption of helix symmetry is not restrictive.

568

4.5. POLYMER CRYSTALLOGRAPHY 4.5.2.3.1. Helix symmetry The presence of a unique axis about which there is rotational disorder means that it is convenient to use cylindrical polar coordinate systems in fibre diffraction. We denote by ðr; ’; zÞ a cylindrical polar coordinate system in real space, in which the z axis is parallel to the molecular axes. The molecule is said to have uv helix symmetry, where u and v are integers, if the electron density f ðr; ’; zÞ satisfies   f r; ’ þ ð2mv=uÞ; z þ ðmc=uÞ ¼ f ðr; ’; zÞ;

gnl ðrÞ ¼ ðc=2Þ

and where in equation (4.5.2.7) (and in the remainder of this section) the sum over l is over all integers, the sum over n is over all integers satisfying the helix selection rule and the integral in equation (4.5.2.8) is over one helix repeat unit. The effect of helix symmetry, therefore, is to restrict the number of Fourier coefficients gnl ðrÞ required to represent the electron density to those whose index n satisfies the selection rule. Note that the selection rule is usually derived using a rather more complicated argument by considering the convolution of the Fourier transform of a continuous filamentary helix with a set of planes in reciprocal space (Cochran et al., 1952). The approach described above, which follows that of Millane (1991), is much more straightforward.

ð4:5:2:2Þ

4.5.2.3.2. Diffraction by helical structures Denote by ðR; ; ZÞ a cylindrical polar coordinate system in reciprocal space (with the Z and z axes parallel), and by FðR; ; ZÞ the Fourier transform of f ðr; ’; zÞ. Since f ðr; ’; zÞ is periodic in z with period c, its Fourier transform is nonzero only on the layer planes Z ¼ l=c where l is an integer. Denote FðR; ; l=cÞ by Fl ðR; Þ; using the cylindrical form of the Fourier transform shows that Fl ðR; Þ ¼

1 P

1 P

  gnl ðrÞ exp i ½n’
ð2lz=cÞ ;

  gðr; ’; zÞ exp i ½
n’ þ ð2l=cÞ d’ dz; ð4:5:2:8Þ

where m is any integer. The constant c is the period along the z direction, which is referred to variously as the molecular repeat distance, the crystallographic repeat, or the c repeat. The helix pitch P is equal to c=v. Helix symmetry is easily interpreted as follows. There are u subunits, or helix repeat units, in one c repeat of the molecule. The helix repeat units are repeated by integral rotations of 2v=u about, and translations of c=u along, the molecular (or helix) axis. The helix repeat units may therefore be referenced to a helical lattice that consists of points at a fixed radius, with relative rotations and translations as described above. These points lie on a helix of pitch P, there are v turns (or pitch-lengths) of the helix in one c repeat, and there are u helical lattice points in one c repeat. A uv helix is said to have ‘u residues in v turns’. Since the electron density is periodic in ’ and z, it can be decomposed into a Fourier series as f ðr; ’; zÞ ¼

RR

ð4:5:2:3Þ

Rc R2 R1

 f ðr; ’; zÞ exp i 2 ½Rr cosð
’Þ 0 0 0  þ ðlz=cÞ r dr d’ dz:

ð4:5:2:9Þ

l¼
1 n¼
1

It is convenient to rewrite equation (4.5.2.9) making use of the Fourier decomposition described in Section 4.5.2.3.1, since this allows utilization of the helix selection rule. The Fourier–Bessel structure factors (Klug et al., 1958), Gnl ðRÞ, are defined as the Hankel transform of the Fourier coefficients gnl ðrÞ, i.e.

where the coefficients gnl ðrÞ are given by Rc R2   gnl ðrÞ ¼ ðc=2Þ f ðr; ’; zÞ exp i ½
n’ þ ð2lz=cÞ d’ dz: 0 0

ð4:5:2:4Þ

R1

Gnl ðRÞ ¼

gnl ðrÞ Jn ð2RrÞ2r dr;

ð4:5:2:10Þ

0

Assume now that the electron density has helical symmetry. Denote by gðr; ’; zÞ the electron density in the region 0 < z < c=u; the electron density being zero outside this region, i.e. gðr; ’; zÞ is the electron density of a single helix repeat unit. It follows that

and the inverse transform is gnl ðrÞ ¼

R1

Gnl ðRÞ Jn ð2RrÞ2R dR:

ð4:5:2:11Þ

0 1 P

f ðr; ’; zÞ ¼

g½r; ’ þ ð2mv=uÞ; z þ ðmc=uÞ:

ð4:5:2:5Þ

Using equations (4.5.2.7) and (4.5.2.11) shows that equation (4.5.2.9) can be written as

m¼
1

Substituting equation (4.5.2.5) into equation (4.5.2.4) shows that gnl ðrÞ vanishes unless ðl
nvÞ is a multiple of u, i.e. unless l ¼ um þ vn

Fl ðR; Þ ¼

gðr; ’; zÞ ¼

l



ð4:5:2:6Þ



gnl ðrÞ exp i ½n’
ð2lz=cÞ ;

  Gnl ðRÞ exp in½ þ ð=2Þ ;

ð4:5:2:12Þ

n

where, as usual, the sum is over only those values of n that satisfy the helix selection rule. Using equations (4.5.2.8) and (4.5.2.10) shows that the Fourier–Bessel structure factors may be written in terms of the atomic coordinates as

for any integer m. Equation (4.5.2.6) is called the helix selection rule. The electron density in the helix repeat unit is therefore given by PP

P

Gnl ðRÞ ¼

P

  fj ðÞ Jn ð2Rrj Þ exp i ½
n’j þ ð2lzj =cÞ ;

j

ð4:5:2:13Þ

ð4:5:2:7Þ

n

where fj ðÞ is the (spherically symmetric) atomic scattering factor (usually including an isotropic temperature factor) of the jth

where

569

4. DIFFUSE SCATTERING AND RELATED TOPICS 2

2

2 1=2

atom and  ¼ ðR þ l =Z Þ is the spherical radius in reciprocal space. Equations (4.5.2.12) and (4.5.2.13) allow the complex diffracted amplitudes for a helical molecule to be calculated from the atomic coordinates, and are analogous to expressions for the structure factors in conventional crystallography. The significance of the selection rule is now more apparent. On a particular layer plane l, not all Fourier–Bessel structure factors Gnl ðRÞ contribute; only those whose Bessel order n satisfies the selection rule for that value of l contribute. Since any molecule has a maximum radius, denoted here by rmax, and since Jn ðxÞ is small for x < jnj
2 and diffraction data are measured out to only a finite value of R, reference to equation (4.5.2.10) [or equation (4.5.2.13)] shows that there is a maximum Bessel order that contributes significant value to equation (4.5.2.12) (Crowther et al., 1970; Makowski, 1982), so that the infinite sum over n in equation (4.5.2.12) can be replaced by a finite sum. On each layer plane there is also a minimum value of jnj, denoted by nmin, that satisfies the helix selection rule, so that the region R < Rmin is devoid of diffracted amplitude where Rmin ¼

nmin
2 : 2rmax

4.5.2.4.1. Noncrystalline fibres A noncrystalline fibre is made up of a collection of helical molecules that are oriented parallel to each other, but are otherwise randomly positioned and rotated relative to each other. The recorded intensity, Il ðRÞ, is therefore that diffracted by a single molecule cylindrically averaged about the Z axis in reciprocal space i.e. R2 Il ðRÞ ¼ ð1=2Þ jFl ðR; Þj2 d ;

using equation (4.5.2.12) shows that Il ðRÞ ¼

P

jGnl ðRÞj2 ;

ð4:5:2:17Þ

n

where, as usual, the sum is over the values of n that satisfy the helix selection rule. On the diffraction pattern, reciprocal space ðR; ; ZÞ collapses to the two dimensions (R, Z). The R axis is called the equator and the Z axis the meridian. The layer planes collapse to layer lines, at Z ¼ l=c, which are indexed by l. Equation (4.5.2.17) gives a rather simple relationship between the recorded intensity and the Fourier–Bessel structure factors. Coherence length and disorientation, as described in Section 4.5.2.2, also affect the form of the diffraction pattern. These effects are described here, although they also apply to other than noncrystalline fibres. A finite coherence length leads to smearing of the layer lines along the Z direction. If the average coherence length of the molecules is lc, the intensity distribution Il ðR; ZÞ about the lth layer line can be approximated by

ð4:5:2:14Þ

The selection rule therefore results in a region around the Z axis of reciprocal space that is devoid of diffraction, the shape of the region depending on the helix symmetry. 4.5.2.3.3. Approximate helix symmetry In some cases the nature of the subunits and their interactions results in a structure that is not exactly periodic. Consider a helical structure with u þ x subunits in v turns, where x is a small ðx  1Þ real number; i.e. the structure has approximate, but not exact, uv helix symmetry. Since the molecule has an approximate repeat distance c, only those layer planes close to those at Z ¼ l=c show significant diffraction. Denoting by Zmn the Z coordinate of the nth Bessel order and its associated value of m, and using the selection rule shows that Zmn ¼ ½ðum þ vnÞ=c þ ðmx=cÞ ¼ ðl=cÞ þ ðmx=cÞ;

ð4:5:2:16Þ

0

  Il ðR; ZÞ ¼ Il ðRÞ exp
lc2 ½Z
ðl=cÞ2 :

ð4:5:2:18Þ

It is convenient to express the effects of disorientation on the intensity distribution of a fibre diffraction pattern by writing the latter as a function of the polar coordinates ð; Þ (where  is the angle with the Z axis) in (R, Z) space. Assuming a Gaussian orientation density function [equation (4.5.2.1)], if
0 is small and the effects of disorientation dominate over those of coherence length (which is usually the case except close to the meridian), then the distribution of intensity about one layer line can be approximated by (Holmes & Barrington Leigh, 1974; Stubbs, 1974)

ð4:5:2:15Þ

so that the positions of the Bessel orders are shifted by mx=c from their positions if the helix symmetry is exactly uv . At moderate resolution m is small so the shift is small. Hence Bessel orders that would have been coincident on a particular layer plane are now separated in reciprocal space. This is referred to as layerplane splitting and was first observed in fibre diffraction patterns from tobacco mosaic virus (TMV) (Franklin & Klug, 1955). Splitting can be used to advantage in structure determination (Section 4.5.2.6.6). As an example, TMV has approximately 493 helix symmetry ˚ . However, close inspection of diffraction with a c repeat of 69 A patterns from TMV shows that there are actually about 49.02 subunits in three turns (Stubbs & Makowski, 1982). The virus is therefore more accurately described as a 2451150 helix with a c ˚ . The layer lines corresponding to this larger repeat of 3450 A repeat distance are not observed, but the effects of layer-plane splitting are detectable (Stubbs & Makowski, 1982).

  Il ðRÞ ð
l Þ2 ; exp
Ið; Þ ’ 2
0 lc  22

ð4:5:2:19Þ

where (Millane & Arnott, 1986; Millane, 1989c) 2 ¼
20 þ ð1=2lc2 2 sin2 l Þ

ð4:5:2:20Þ

and l is the polar angle at the centre of the layer line, i.e. R ¼  sin l . The effect of disorientation, therefore, is to smear each layer line about the origin of reciprocal space.

4.5.2.4. Diffraction by fibres

4.5.2.4.2. Polycrystalline fibres A polycrystalline fibre is made up of crystallites that are oriented parallel to each other, but are randomly positioned and randomly rotated about their molecular axes. The recorded diffraction pattern is the intensity diffracted by a single crystallite, cylindrically averaged about the Z axis. On a fibre diffraction pattern, therefore, the Bragg reflections are cylindrically projected onto the (R, Z) plane and their positions are described

The kind of diffraction pattern obtained from a fibre specimen made up of helical molecules depends on the kind of specimen as described in Section 4.5.2.2. This section is divided into four parts. The first two describe diffraction patterns obtained from noncrystalline and polycrystalline fibres (which are the most common kinds used for structural analysis), and the last two describe diffraction by partially crystalline fibres.

570

4.5. POLYMER CRYSTALLOGRAPHY by the cylindrically projected reciprocal lattice (Finkenstadt & Millane, 1998). The molecules are periodic and are therefore usually aligned with one of the unit-cell vectors. Since the z axis is defined as the fibre axis, it is usual in fibre diffraction to take the c lattice vector as the unique axis and as the lattice vector parallel to the molecular axes. It is almost always the case that the fibre is rotationally disordered about the molecular axes, i.e. about c. Consider first the case of a monoclinic unit cell ð
¼  ¼ 90 Þ so that the reciprocal lattice is cylindrically projected about c . The cylindrical coordinates of the projected reciprocal-lattice points are then given by R2hkl ¼ h2 a2 þ k2 b2 þ 2hka b cos  

The effect of a finite crystallite size is to smear what would otherwise be infinitely sharp reflections into broadened reflections of a finite size. If the average crystallite dimensions normal and parallel to the z axis are llat (i.e. in the ‘lateral’ direction) and laxial (i.e. in the ‘axial’ direction), respectively, the profile of the reflection centred at ðRhk ; Z ¼ l=cÞ can be written as (Fraser et al., 1984; Millane & Arnott, 1986; Millane, 1989c) IðR; ZÞ ¼ Il ðRhk ÞSðR
Rhk ; Z
l=cÞ;

where the profile function SðR; ZÞ can be approximated by 2 2 R2 þ laxial Z2 Þ: SðR; ZÞ ¼ exp½
ðllat

ð4:5:2:21Þ

ð4:5:2:22Þ

so that R depends only on h and k, and Z depends only on l. Reflections with fixed h and k lie on straight row lines. Certain sets of distinct reciprocal-lattice points will have the same value of Rhkl and therefore superimpose in cylindrical projection. For example, for an orthorhombic system ð ¼ 90 Þ the reciprocallattice points (hkl), ðh klÞ, ðhk lÞ and ðh k lÞ superimpose. Furthermore, the crystallites in a fibre specimen are usually oriented randomly up and down so that the reciprocal-lattice points (hkl) and ðhklÞ superimpose, so that in the orthorhombic case eight reciprocal-lattice points superimpose. Also, as described below, reciprocal-lattice points that have similar values of R can effectively superimpose. If the unit cell is either triclinic, or is monoclinic with
6¼ 90 or  6¼ 90 , then c is inclined to c and the Z axis, and the reciprocal lattice is not cylindrically projected about c . Equation (4.5.2.22) for Zhkl still applies, but the cylindrical radius is given by

Ið; Þ ’

Il ðRhk Þ 2
0 llat laxial 
  ð
hkl Þ2 ð
hkl Þ2 þ  exp
; 22 22

2 2

2 2

2

2

 

2

¼ h a þ k b þ l ½c
ð1=c Þ þ 2hka b cos  þ 2hla c cos  þ 2klb c cos


2 ¼

P h0 ; k0 2S ðh; kÞ

jFh0 k0 l j2 ;

2 2 llat laxial 2 2 2ðllat sin hkl þ laxial cos2 hkl Þ 2

ð4:5:2:28Þ

and



ð4:5:2:23Þ

2 22hkl ðlaxial

2 2 llat laxial : 2 2 sin hkl þ llat cos2 hkl Þ

ð4:5:2:29Þ

Reflections that have similar enough ðR; ZÞ coordinates overlap severely with each other and are also included in the sum in equation (4.5.2.24). This is quite common in practice because a number of sets of reflections may have similar values of Rhk.

and the row lines are curved (Finkenstadt & Millane, 1998). The most complicated situation arises if the crystallites are rotationally disordered about an axis that is inclined to c. Reciprocal space is then rotated about an axis that is inclined to the normal to the a b plane, Rhkl and Zhkl are both functions of h, k and l, equation (4.5.2.23) does not apply, and reciprocal-lattice points for fixed l do not lie on layer lines of constant Z. Although this situation is rather unusual, it does occur (Daubeny et al., 1954), and is described in detail by Finkenstadt & Millane (1998). The observed fibre diffraction pattern consists of reflections at the projected reciprocal-lattice points whose intensities are equal to the sums of the intensities of the contributing structure factors. The observed intensity, denoted by Il ðRhk Þ, at a projected reciprocal-lattice point on the lth layer line and with R ¼ Rhk is therefore given by (assuming, for simplicity, a monoclinic system) Il ðRhk Þ ¼

ð4:5:2:27Þ

where ðhkl ; hkl Þ are the polar coordinates of the reflection,

2 ¼
0 þ R2hkl

ð4:5:2:26Þ

The effect of crystallite disorientation is to smear the reflections given by equation (4.5.2.26) about the origin of the projected reciprocal space. If the effects of disorientation dominate over those of crystallite size, then the profile of a reflection can be approximated by (Fraser et al., 1984; Millane & Arnott, 1986; Millane, 1989c)

and Zhkl ¼ lc ;

ð4:5:2:25Þ

4.5.2.4.3. Random copolymers Random copolymers are made up of a small number of different kinds of monomer, whose sequence along the polymer chain is not regular, but is random, or partially random. A particularly interesting class are synthetic polymers such as copolyesters that form a variety of liquid-crystalline phases and have useful mechanical properties (Biswas & Blackwell, 1988a). The structures of these materials have been studied quite extensively using X-ray fibre diffraction analysis. Because the molecules do not have an average c repeat, their diffraction patterns do not consist of equally spaced layer lines. However, as a result of the small number of distinct spacings associated with the monomers, diffracted intensity is concentrated about layer lines, but these are irregularly spaced (along Z) and are aperiodic. Since the molecule is not periodic, the basic theory of diffraction by helical molecules described in Section 4.5.2.3.2 does not apply in this case. Cylindrically averaged diffraction from random copolymers is described here. Related approaches have been described independently by Hendricks & Teller (1942) and Blackwell et al. (1984). Hendricks & Teller (1942) considered the rather general problem of diffraction by layered structures made up of different kinds of layers, the probability of a layer at a

ð4:5:2:24Þ

where S ðh; kÞ denotes the set of indices ðh0 ; k0 Þ such that Rh0 k0 ¼ Rhk . The number of independent reflections contributing in equation (4.5.2.24) depends on the space-group symmetry of the crystallites, because of either systematic absences or structure factors whose values are related.

571

4. DIFFUSE SCATTERING AND RELATED TOPICS particular level depending on the layers present at adjacent levels. This is a one-dimensional disordered structure that can be used to describe a random copolymer. Blackwell and co-workers have developed a similar theory in terms of a one-dimensional paracrystalline model (Hosemann & Bagchi, 1962) for diffraction by random copolymers (Blackwell et al., 1984; Biswas & Blackwell, 1988a), and this is the theory described here. Consider a random copolymer made up of monomer units (residues) of N different types. Since the disorder is along the length of the polymer, some of the main characteristics of diffraction from such a molecule can be elucidated by studying the diffraction along the meridian of the diffraction pattern. The meridional diffraction is the intensity of the Fourier transform of the polymer chain projected onto the z axis and averaged over all possible monomer sequences. The diffraction pattern depends on the monomer (molar) compositions, denoted by pi , the statistics of the monomer sequence (described by the probability of the different possible monomer pairs in this model) and the Fourier transform of the monomer units. Development of this model shows that the meridional diffracted intensity IðZÞ can be written in the form (Blackwell et al., 1984; Biswas & Blackwell, 1988a; Schneider et al., 1991) IðZÞ ¼

P

pi jFi ðZÞj2 þ 2

i

PP

one corresponding to fixed conformations between monomers and the other corresponding to completely random conformations between monomers, and have derived expressions for the diffracted intensity in both cases. Equation (4.5.2.32) allows one to calculate the fibre diffraction pattern from an array of parallel random copolymers that exhibit no lateral ordering. The diffraction pattern consists of irregularly spaced layer lines whose spacings (in Z) are the same as those described above for the meridional maxima. Measurement of layer-line spacings and intensities and comparison with calculations based on the constituent monomers allows chain conformations to be estimated (Biswas & Blackwell, 1988a). Diffraction patterns from liquid-crystalline random copolymers often contain sharp Bragg maxima on the layer lines. This indicates that, despite the random sequence and the possible dissimilarity of the component monomers, the chains are able to pack together in a regular way (Biswas & Blackwell, 1988b, c). Expressions that allow calculation of diffraction patterns for arrays of polymers with minimal registration, in which short, nonidentical sequences form layers, have been derived (Biswas & Blackwell, 1988b, c). Calculation of diffracted intensities, coupled with molecular-mechanics modelling, allows chain conformations and packing to be investigated (Hofmann et al., 1994).

1: 1
cos2 ½
ðt=B Þð 2
1Þ1=2  ;

2
cos2 ½
ðt=B Þð 2
1Þ1=2 

2
1 Io ¼ :

2
cos2 ½
ðt=B Þð 2
1Þ1=2 



DðaÞ 2

h

Ih ¼ jj ðaÞ

D o



F

cosh 2b
cos 2a ¼

h

; 2 F  L cosh 2b þ ðL
1Þ1=2 sinh 2b
cosð2a þ 2
0 Þ h

ð5:1:7:12Þ

Ih ¼

where ð5:1:7:7bÞ 2a ¼ ½
t=B  cosð þ !Þ; 2b ¼ ½
t=B  sinð þ !Þ

The cosine terms show that the two wavefields propagating within the crystal interfere, giving rise to Pendello¨sung fringes in the rocking curve. These fringes were observed for the first time by Batterman & Hildebrandt (1967, 1968). The angular positions of the minima of the reflected beam are given by

L,  and 0 are defined in (5.1.7.5), is defined in (5.1.3.7) and ! is the phase angle of ð 2
1Þ1=2. Comparison with geometrical theory. When t=B decreases towards zero, expression (5.1.7.12) tends towards ½sinð
t =B Þ= 2 ; using (5.1.3.5) and (5.1.3.8), it can be shown that expression (5.1.7.12) can be written, in the non-absorbing symmetric case, as

¼ ðK2 2B t
2 þ 1Þ1=2 ;

 2 R2 2 C2 jFh j2 t2 sin½2
k cosðÞt Ih ¼ ; ½2
k cosðÞt V 2 sin2 

where K is an integer. Integrated intensity. The integrated intensity is Ihi ¼
 tanh½
t=B ;

:

ð5:1:7:8Þ

ð5:1:7:13Þ

where t is the crystal thickness. When this thickness becomes very large, the integrated intensity tends towards

where d is the lattice spacing and  is the difference between the angle of incidence and the middle of the reflection domain. This expression is the classical expression given by geometrical theory [see, for instance, James (1950)].

Ihi ¼
:

5.1.8. Real waves

ð5:1:7:9Þ

5.1.8.1. Introduction The preceding sections have dealt with the diffraction of a plane wave by a semi-infinite perfect crystal. This situation is actually never encountered in practice, although with various devices, in particular using synchrotron radiation, it is possible to produce highly collimated monochromated waves which behave like pseudo plane waves. The wave from an X-ray tube is best represented by a spherical wave. The first experimental proof of this fact is due to Kato & Lang (1959) in the transmission case. Kato extended the dynamical theory to spherical waves for nonabsorbing (Kato, 1961a,b) and absorbing crystals (Kato, 1968a,b). He expanded the incident spherical wave into plane waves by a Fourier transform, applied plane-wave dynamical theory to each of these components and took the Fourier transform of the result again in order to obtain the final solution. Another method for treating the problem was used by Takagi (1962, 1969), who solved the propagation equation in a medium where the lateral extension of the incident wave is limited and where the wave amplitudes depend on the lateral coordinates. He showed that in this case the set of fundamental linear equations (5.1.2.20) should be replaced by a set of partial differential equations. This treatment can be applied equally well to a perfect or to an imperfect crystal. In the case of a perfect crystal, Takagi showed that these equations have an analytical solution that is identical with Kato’s result. Uragami (1969, 1970) observed the spherical wave in the Bragg (reflection) case, interpreting the observed intensity distribution using Takagi’s theory. Saka et al. (1973) subsequently extended Kato’s theory to the Bragg case. Without using any mathematical treatment, it is possible to make some elementary remarks by considering the fact that the divergence of the incident beam falling on the crystal from the source is much larger than the angular width of the reflection domain. Fig. 5.1.8.1(a) shows a spatially collimated beam falling

This expression differs from (5.1.7.3) by the factor
, which appears here in place of 8=3. von Laue (1960) pointed out that because of the differences between the two expressions for the reflecting power, (5.1.7.2) and (5.1.7.7b), perfect agreement could not be expected. Since some absorption is always present, expression (5.1.7.3), which includes the factor 8=3, should be used for very thick crystals. In the presence of absorption, however, expression (5.1.7.8) for the reflected intensity for thin crystals does tend towards that for thick crystals as the crystal thickness increases. Comparison with geometrical theory. When t=B is very small (thin crystals or weak reflections), (5.1.7.8) tends towards  Ihi ¼ R2 2 tjFh j2 ðV 2 o sin 2Þ;

ð5:1:7:10Þ

which is the expression given by geometrical theory. If we call this intensity Ihi (geom.), comparison of expressions (5.1.7.8) and (5.1.7.10) shows that the integrated intensity for crystals of intermediate thickness can be written Ihi ¼ Ihi (geom.)

tanhð
t=B Þ ; ð
t=B Þ

ð5:1:7:11Þ

which is the expression given by Darwin (1922) for primary extinction. 5.1.7.2.2. Absorbing crystals Reflected intensity. The intensity of the reflected wave for a thin absorbing crystal is

640

5.1. DYNAMICAL THEORY OF X-RAY DIFFRACTION

Fig. 5.1.8.2. Packet of wavefields of divergence  excited in the crystal by an incident wavepacket of angular width ðÞ. (a) Direct space; (b) reciprocal space.

Fig. 5.1.8.1. Borrmann triangle. When the incident beam is divergent, the whole dispersion surface is excited and the wavefields excited inside the crystal propagate within a triangle filling all the space between the incident direction, AC, and the reflected direction, AB. Along any direction Ap within this triangle two wavefields propagate, having as tie points two conjugate points, P and P0 , at the extremities of a diameter of the dispersion surface. (a) Direct space; (b) reciprocal space.

direct space over an angle  given by (5.1.8.1). The intensity distribution on the exit surface BC (Fig. 5.1.8.1a) is therefore proportional to Ih =A. It is represented in Fig. 5.1.8.3 for several values of the absorption coefficient: (i) Small values of o t (less than 2 or 3) (Fig. 5.1.8.3a). The intensity distribution presents a wide minimum in the centre where the density of wavefields is small and increases very sharply at the edges where the density of wavefields is large, although it is the reverse for the reflecting power Ihj . This effect, called the margin effect, was predicted qualitatively by Borrmann (1959) and von Laue (1960), demonstrated experimentally by Kato & Lang (1959), and calculated by Kato (1960). (ii) Large values of o t (of the order of 6 or more) (Fig. 5.1.8.3b). The predominant factor is now anomalous absorption. The wavefields propagating along the edges of the Borrmann triangle undergo normal absorption, while those propagating parallel to the lattice planes (or nearly parallel) correspond to tie points in the centre of the dispersion surface and undergo anomalously low absorption. The intensity distribution now has a maximum in the centre. For values of t larger than 10 or so, practically only the wavefields propagating parallel to the lattice planes go through the crystal, which acts as a wave guide: this is the Borrmann effect.

on a crystal in the transmission case and Fig. 5.1.8.1(b) represents the corresponding situation in reciprocal space. Since the divergence of the incident beam is larger than the angular width of the dispersion surface, the plane waves of its Fourier expansion will excite every point of both branches of the dispersion surface. The propagation directions of the corresponding wavefields will cover the angular range between those of the incident and reflected beams (Fig. 5.1.8.1a) and fill what is called the Borrmann triangle. The intensity distribution within this triangle has interesting properties, as described in the next two sections. 5.1.8.2. Borrmann triangle The first property of the Borrmann triangle is that the angular density of the wavefield paths is inversely proportional to the curvature of the dispersion surface around their tie points. Let us consider an incident wavepacket of angular width ðÞ. It will generate a packet of wavefields propagating within the Borrmann triangle. The angular width  (Fig. 5.1.8.2) between the paths of the corresponding wavefields is related to the radius of curvature R of the dispersion surface by A ¼ =ðÞ ¼ k cos ðR cos Þ;

5.1.8.3. Spherical-wave Pendello¨sung Fig. 5.1.8.4 shows that along any path Ap inside the Borrmann triangle two wavefields propagate, one with tie point P1 , on branch 1, the other with the point P02 , on branch 2. These two points lie on the extremities of a diameter of the dispersion surface. The two wavefields interfere, giving rise to Pendello¨sung fringes, which were first observed by Kato & Lang (1959), and calculated by Kato (1961b). These fringes are of course quite different from the plane-wave Pendello¨sung fringes predicted by Ewald (Section 5.1.6.3) because the tie points of the interfering wavefields are different and their period is also different, but they have in common the fact that they result from interference

ð5:1:8:1Þ

where  is the angle between the wavefield path and the lattice planes [equation (5.1.2.26)] and A is called the amplification ratio. In the middle of the reflecting domain, the radius of curvature of the dispersion surface is very much shorter than its value, k, far from it (about 104 times shorter) and the amplification ratio is therefore very large. As a consequence, the energy of a wavepacket of width ðÞ in reciprocal space is spread in

641

5. DYNAMICAL THEORY AND ITS APPLICATIONS

Fig. 5.1.8.4. Interference at the origin of the Pendello¨sung fringes in the case of an incident spherical wave. (a) Direct space; (b) reciprocal space.

where A ¼ 2
ðo h Þ1=2 =ðL sin Þ and xo and xh are the distances of p from the sides AB and AC of the Borrmann triangle (Fig. 5.1.8.4). The equal-intensity fringes are therefore located along the locus of the points in the triangle for which the product of the distances to the sides is constant, that is hyperbolas having AB and AC as asymptotes (Fig. 5.1.8.4b). These fringes can be observed on a section topograph of a wedge-shaped crystal (Fig. 5.1.8.5). The technique of section topography is described in IT C, Section 2.7.2.2. The Pendello¨sung distance L depends on the polarization state [see equation (5.1.3.8)]. If the incident wave is unpolarized, one observes the superposition of the Pendello¨sung fringes corresponding to the two states of polarization, parallel and perpendicular to the plane of incidence. This results in a beat effect, which is clearly visible in Fig. 5.1.8.5.

APPENDIX A5.1.1 Basic equations A5.1.1.1. Dielectric susceptibility – classical derivation Under the influence of the incident electromagnetic radiation, the medium becomes polarized. The dielectric susceptibility, which relates this polarization to the electric field, thus characterizes the interaction of the medium and the electromagnetic wave. The classical derivation of the dielectric susceptibility, , which is summarized here is only valid for a very high frequency which is also far from an absorption edge. Let us consider an electromagnetic wave,

Fig. 5.1.8.3. Intensity distribution along the base of the Borrmann triangle. y is a normalized coordinate along BC. (a) Small values of t. The interference (spherical-wave Pendello¨sung) between branch 1 and branch 2 is neglected. (b) Large values of t.

between wavefields belonging to different branches of the dispersion surface. Kato has shown that the intensity distribution at any point at the base of the Borrmann triangle is proportional to 
 2 Jo Aðxo xh Þ1=2 ;

E ¼ Eo exp 2
iðt
k  rÞ;

642

5.1. DYNAMICAL THEORY OF X-RAY DIFFRACTION It is related to the electric field and electric displacement through D ¼ "o E þ P ¼ "o ð1 þ ÞE:

(A5.1.1.1)

We finally obtain the expression of the dielectric susceptibility,  ¼
e2 =ð4
2 "o 2 mÞ ¼
R2 =
;

(A5.1.1.2)

where R ¼ e2 =ð4
"o mc2 Þ ð¼ 2:81794  10
15 mÞ is the classical radius of the electron. A5.1.1.2. Maxwell’s equations The electromagnetic field is represented by two vectors, E and B, which are the electric field and the magnetic induction, respectively. To describe the interaction of the field with matter, three other vectors must be taken into account, the electrical current density, j, the electric displacement, D, and the magnetic field, H. The space and time derivatives of these vectors are related in a continuous medium by Maxwell’s equations: curl E ¼
@B=@t curl H ¼ @D=@t þ j

Fig. 5.1.8.5. Spherical-wave Pendello¨sung fringes observed on a wedgeshaped crystal. (a) Computer simulation (solid lines: maxima; dashed lines: minima). (b) X-ray section topograph of a wedge-shaped silicon crystal (444 reflection, Mo K radiation).

div D ¼  div B ¼ 0;

incident on a bound electron. The electron behaves as if it were held by a spring with a linear restoring force and is an oscillator with a resonant frequency o . The equation of its motion is written in the following way:

(A5.1.1.3)

where  is the electric charge density. The electric field and the electric displacement on the one hand, and the magnetic field and the magnetic induction on the other hand, are related by the so-called material relations, which describe the reaction of the medium to the electromagnetic field:

m d2 a=dt2 ¼
4
2 2o ma ¼ F; where the driving force F is due to the electric field of the wave and is equal to
eE. The magnetic interaction is neglected here. The solution of the equation of motion is

D ¼ "E B ¼ H;

a ¼
eE=½4
2 mð2o
2 Þ:

where " and  are the dielectric constant and the magnetic permeability, respectively. These equations are complemented by the following boundary conditions at the surface between two neighbouring media:

The resonant frequencies of the electrons in atoms are of the order of the ultraviolet frequencies and are therefore much smaller than X-ray frequencies. They can be neglected and the expression of the amplitude of the electron reduces to a ¼ eE=ð4
2 2 Þ: The dipolar moment is therefore

ET1
ET2 ¼ 0

DN1
DN2 ¼ 0

HT1
HT2 ¼ 0

BN1
BN2 ¼ 0:

(A5.1.1.4)

From the second and the third equations of (A5.1.1.3), and using the identity

M ¼
ea ¼
e2 E=ð4
2 2 mÞ: div ðcurl yÞ ¼ 0; von Laue assumes that the negative charge is distributed continuously all over space and that the charge of a volume element d is
e d, where  is the electronic density. The electric moment of the volume element is

it follows that div j þ @=@t ¼ 0:

2

(A5.1.1.5)

2 2

dM ¼
e E d=ð4
 mÞ: The polarization is equal to the moment per unit volume: A5.1.1.3. Propagation equation In a vacuum,  and j are equal to zero, and the first two equations of (A5.1.1.3) can be written

P ¼ dM=d ¼
e2 E=ð4
2 2 mÞ:

643

5. DYNAMICAL THEORY AND ITS APPLICATIONS By replacing D with this expression in equation (A5.1.1.7), one finally obtains the propagation equation (5.1.2.2) (Section 5.1.2.1). In a crystalline medium,  is a triply periodic function of the space coordinates and the solutions of this equation are given in terms of Fourier series which can be interpreted as sums of electromagnetic plane waves. Each of these waves is characterized by its wavevector, Kh , its electric displacement, Dh , its electric field, Eh , and its magnetic field, Hh . It can be shown that, as a consequence of the fact that div D ¼ 0 and div E 6¼ 0; Dh is a transverse wave (Dh ; Hh and Kh and are mutually orthogonal) while Eh is not. The electric displacement is therefore a more suitable vector for describing the state of the field inside the crystal than the electric field.

curl E ¼
0 @E=@t curl H ¼ "0 @H=@t; where "0 and 0 are the dielectric constant and the magnetic permeability of a vacuum, respectively. By taking the curl of both sides of the second equation, it follows that curl curl E ¼
"0 0 @2 E=@t2 : Using the identity curl curl E ¼ grad div E
E, the relation "0 0 ¼ 1=c2 , where c is the velocity of light, and noting that div E ¼ div D ¼ 0, one finally obtains the equation of propagation of an electromagnetic wave in a vacuum: E ¼

1 @2 E : c2 @t2

A5.1.1.4. Poynting vector The propagation direction of the energy of an electromagnetic wave is given by that of the Poynting vector defined by (see Born & Wolf, 1983)

(A5.1.1.6)

S ¼ RðE ^ H Þ;

(A5.1.1.8)

Its simplest solution is a plane wave:

of which the wavenumber k ¼ 1= and the frequency  are related by

where Rð Þ means real part of ( ). The intensity I of the radiation is equal to the energy crossing unit area per second in the direction normal to that area. It is given by the value of the Poynting vector averaged over a period of time long compared with 1= :

k ¼ =c:

I ¼ jSj ¼ c"jEj2 ¼ cjDj2 =":

E ¼ E0 exp 2
iðt
k  rÞ;

The basic properties of the electromagnetic field are described, for instance, in Born & Wolf (1983). The propagation equation of X-rays in a crystalline medium is derived following von Laue (1960). The interaction of X-rays with charged particles is inversely proportional to the mass of the particle and the interaction with the nuclei can be neglected. As a first approximation, it is also assumed that the magnetic interaction of X-rays with matter is neglected, and that the magnetic permeability  can be taken as equal to the magnetic permeability of a vacuum, 0 . It is further assumed that the negative and positive charges are both continuously distributed and compensate each other in such a way that there is neutrality and no current everywhere:  and j are equal to zero and div D is therefore also equal to zero. The electric displacement is related to the electric field by (A5.1.1.1) and the electric part of the interaction of X-rays with matter is expressed through the dielectric susceptibility , which is given by (A5.1.1.2). This quantity is proportional to the electron density and varies with the space coordinates. It is therefore concluded that div E is different from zero, as opposed to what happens in a vacuum. For this reason, the propagation equation of X-rays in a crystalline medium is expressed in terms of the electric displacement rather than in terms of the electric field. It is obtained by eliminating H, B and E in Maxwell’s equations and taking into account the above assumptions: D þ curl curl D ¼

1 @2 E : c2 @t2

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(A5.1.1.7)

Only coherent scattering is taken into account here, that is, scattering without frequency change. The solution is therefore a wave of the form DðrÞ exp 2
it:

644

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Batterman, B. W. & Hildebrandt, G. (1967). Observation of X-ray Pendello¨sung fringes in Darwin reflection. Phys. Status Solidi, 23, K147–K149. Batterman, B. W. & Hildebrandt, G. (1968). X-ray Pendello¨sung fringes in Darwin reflection. Acta Cryst. A24, 150–157. Bedzyk, M. J. (1988). New trends in X-ray standing waves. Nucl. Instrum. Methods A, 266, 679–683. Bonse, U. & Teworte, R. (1980). Measurement of X-ray scattering factors of Si from the fine structure of Laue case rocking curves. J. Appl. Cryst. 13, 410–416. Born, M. & Wolf, E. (1983). Principles of Optics, 6th ed. Oxford: Pergamon Press. ¨ ber die Interferenzen aus Gitterquellen bei Borrmann, G. (1936). U Anregung durch Ro¨ntgenstrahlen. Ann. Phys. (Leipzig), 27, 669–693. ¨ ber Extinktionsdiagramme der Ro¨ntgenstrahlen Borrmann, G. (1941). U von Quarz. Phys. Z. 42, 157–162. Borrmann, G. (1950). Die Absorption von Ro¨ntgenstrahlen in Fall der Interferenz. Z. Phys. 127, 297–323. Borrmann, G. (1954). Der kleinste Absorption Koeffizient interfierender Ro¨ntgenstrahlung. Z. Kristallogr. 106, 109–121. Borrmann, G. (1955). Vierfachbrechung der Ro¨ntgenstrahlen durch das ideale Kistallgitter. Naturwissenschaften, 42, 67–68. Borrmann, G. (1959). Ro¨ntgenwellenfelder. Beit. Phys. Chem. 20 Jahrhunderts, pp. 262–282. Braunschweig: Vieweg und Sohn. Bragg, W. L. (1913). The diffraction of short electromagnetic waves by a crystal. Proc. Cambridge Philos. Soc. 17, 43–57. Bragg, W. L., Darwin, C. G. & James, R. W. (1926). The intensity of reflection of X-rays by crystals. Philos. Mag. 1, 897–922. Bru¨mmer, O. & Stephanik, H. (1976). Dynamische Interferenztheorie. Leipzig: Akademische Verlagsgesellshaft. Chang, S.-L. (1987). Solution to the X-ray phase problem using multiple diffraction – a review. Crystallogr. Rev. 1, 87–189. Chang, S.-L. (2004). X-ray Multiple-Wave Diffraction: Theory and Application. Springer Series in Solid-State Sciences, Vol. 143. Berlin: Springer-Verlag. Chang, S.-L., Stetsko, Yu. P. & Lee, Y.-R. (2002). Quantitative determination of phase for macromolecular crystals using multiple diffraction methods and dynamical theory. Z. Kristallogr. 217, 662–667. Chukhovskii, F. N. & Fo¨rster, E. (1995). Time-dependent X-ray Bragg diffraction. Acta Cryst. A51, 668–672. Cowan, P. L., Brennan, S., Jach, T., Bedzyk, M. J. & Materlik, G. (1986). Observations of the diffraction of evanescent X-rays at a crystal surface. Phys. Rev. Lett. 57, 2399–2402. Darwin, C. G. (1914a). The theory of X-ray reflection. Philos. Mag. 27, 315–333. Darwin, C. G. (1914b). The theory of X-ray reflection. Part II. Philos. Mag. 27, 675–690. Darwin, C. G. (1922). The reflection of X-rays from imperfect crystals. Philos. Mag. 43, 800–829. Ewald, P. P. (1917). Zur Begru¨ndung der Kristalloptik. III. Ro¨ntgenstrahlen. Ann. Phys. (Leipzig), 54, 519–597. Ewald, P. P. (1958). Group velocity and phase velocity in X-ray crystal optics. Acta Cryst. 11, 888–891. Fewster, P. F. (1993). X-ray diffraction from low-dimensional structures. Semicond. Sci. Technol. 8, 1915–1934. Fingerland, A. (1971). Some properties of the single crystal rocking curve in the Bragg case. Acta Cryst. A27, 280–284. Golovchenko, J. A., Patel, J. R., Kaplan, D. R., Cowan, P. L. & Bedzyk, M. J. (1982). Solution to the surface registration problem using X-ray standing waves. Phys. Rev. Lett. 49, 560–563. Graeff, W. (2002a). Short X-ray pulses in a Laue-case crystal. J. Synchrotron Rad. 9, 82–85. Graeff, W. (2002b). Time dependence of the polarization of short X-ray pulses after crystal reflection. J. Synchrotron Rad. 9, 293–297. Graeff, W. (2004). Tailoring the time response of a Bragg reflection to short X-ray pulses. J. Synchrotron Rad. 11, 261–265. Hart, M. (1981). Bragg angle measurement and mapping. J. Cryst. Growth, 55, 409–427. Hirsch, P. B. & Ramachandran, G. N. (1950). Intensity of X-ray reflection from perfect and mosaic absorbing crystals. Acta Cryst. 3, 187–194. Hu¨mmer, K. & Weckert, E. (1995). Enantiomorphism and three-beam X-ray diffraction: determination of the absolute structure. Acta Cryst. A51, 431–438.

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5. DYNAMICAL THEORY AND ITS APPLICATIONS Takagi, S. (1969). A dynamical theory of diffraction for a distorted crystal. J. Phys. Soc. Jpn, 26, 1239–1253. Tanner, B. K. (1976). X-ray Diffraction Topography. Oxford: Pergamon Press. Tanner, B. K. (1990). High resolution X-ray diffraction and topography for crystal characterization. J. Cryst. Growth, 99, 1315–1323. Tanner, B. K. & Bowen, D. K. (1992). Synchrotron X-radiation topography. Mater. Sci. Rep. 8, 369–407. Uragami, T. (1969). Pendello¨sung fringes of X-rays in Bragg case. J. Phys. Soc. Jpn, 27, 147–154. Uragami, T. (1970). Pendello¨sung fringes in a crystal of finite thickness. J. Phys. Soc. Jpn, 28, 1508–1527. Vartanyants, I. A. & Koval’chuk, M. V. (2001). Theory and applications of X-ray standing waves in real crystals. Rep. Prog. Phys. 64, 1009– 1084. Vartanyants, I. A. & Zegenhagen, J. (1999). Photoelectric scattering from an X-ray interference field. Solid State Commun. 113, 299–320. Wagner, E. H. (1959). Group velocity and energy (or particle) flow density of waves in a periodic medium. Acta Cryst. 12, 345–346. Weckert, E. & Hu¨mmer, K. (1997). Multiple-beam X-ray diffraction for physical determination of reflection phases and its applications. Acta Cryst. A53, 108–143. Weckert, E., Mu¨ller, R., Zellner, J., Zegers, I. & Loris, R. (2002). Physical measurement of triplet invariants: present state of the experiment, data evaluation and future perspectives. Z. Kristallogr. 217, 651– 661. Yamazaki, H. & Ishikawa, T. (2002). Propagation of X-ray coherence for diffraction of perfect crystals. J. Appl. Cryst. 35, 314–318. Yamazaki, H. & Ishikawa, T. (2004). Analysis of the mutual coherence function of X-rays using dynamical diffraction. J. Appl. Cryst. 37, 48– 51. Zachariasen, W. H. (1945). Theory of X-ray Diffraction in Crystals. New York: John Wiley. Zegenhagen, J. (1993). Surface structure determination with X-ray standing waves. Surf. Sci. Rep. 18, 199–271. Zegenhagen, J., Lyman, P. F., Bo¨hringer, M. & Bedzyk, M. J. (1997). Discommensurate reconstructions of (111)Si and Ge induced by surface alloying with Cu, Ga and In. Phys. Status Solidi B, 204, 587– 616.

Mo, F., Mathiesen, R. H., Alzari, P. M., Lescar, J. & Rasmussen, B. (2002). Physical estimation of triplet phases from two new proteins. Acta Cryst. D58, 1780–1786. Ohtsuki, Y. H. (1964). Temperature dependence of X-ray absorption by crystals. I. Photo-electric absorption. J. Phys. Soc. Jpn, 19, 2285–2292. Ohtsuki, Y. H. (1965). Temperature dependence of X-ray absorption by crystals. II. Direct phonon absorption. J. Phys. Soc. Jpn, 20, 374–380. Patel, J. R. (1996). X-ray standing waves. In X-ray and Neutron Dynamical Diffraction: Theory and Applications. NATO ASI Series, Series B: Physics, Vol. 357, edited by A. Authier, S. Lagomarsino & B. K. Tanner, pp. 211–224. New York, London: Plenum Press. Patel, J. R. & Fontes, E. (1996). X-ray standing waves: thermal vibration amplitudes at surfaces. In X-ray and Neutron Dynamical Diffraction: Theory and Applications. NATO ASI Series, Series B: Physics, Vol. 357, edited by A. Authier, S. Lagomarsino & B. K. Tanner, pp. 235–248. New York, London: Plenum Press. Penning, P. & Polder, D. (1961). Anomalous transmission of X-rays in elastically deformed crystals. Philips Res. Rep. 16, 419–440. Pinsker, Z. G. (1978). Dynamical scattering of X-rays in crystals. Springer Series in Solid-State Sciences. Berlin: Springer-Verlag. Prins, J. A. (1930). Die Reflexion von Ro¨ntgenstrahlen an absorbierenden idealen Kristallen. Z. Phys. 63, 477–493. Renninger, M. (1955). Messungen zur Ro¨ntgenstrahl-Optik des Idealkristalls. I. Besta¨tigung der Darwin–Ewald–Prins–Kohler-Kurve. Acta Cryst. 8, 597–606. Saka, T., Katagawa, T. & Kato, N. (1973). The theory of X-ray crystal diffraction for finite polyhedral crystals. III. The Bragg–(Bragg)m cases. Acta Cryst. A29, 192–200. Shastri, S. D., Zambianchi, P. & Mills, D. M. (2001). Dynamical diffraction of ultrashort X-ray free-electron laser pulses. J. Synchrotron Rad. 8, 1131–1135. Shen, Q. & Wang, J. (2003). Recursive direct phasing with reference-beam diffraction. Acta Cryst. D59, 809–814. Sondhauss, P. & Wark, J. S. (2003). Extension of the time-dependent dynamical diffraction theory to ‘optical phonon’-type distortions: application to diffraction from coherent acoustic and optical phonons. Acta Cryst. A59, 7–13. Takagi, S. (1962). Dynamical theory of diffraction applicable to crystals with any kind of small distortion. Acta Cryst. 15, 1311–1312.

646

references

International Tables for Crystallography (2010). Vol. B, Chapter 5.2, pp. 647–653.

5.2. Dynamical theory of electron diffraction By A. F. Moodie, J. M. Cowley† and P. Goodman†

Here k ¼ jkj is the scalar wavenumber of magnitude 2
=, and the interaction constant  ¼ 2
me=h2 . This constant is approximately 10
3 for 100 kV electrons. For fast electrons, ’=W is a slowly varying function on a scale of wavelength, and is small compared with unity. The scattering will therefore be peaked about the direction defined by the incident beam, and further simplification is possible, leading to a forward-scattering solution appropriate to HEED (high-energy electron diffraction).

5.2.1. Introduction Since electrons are charged, they interact strongly with matter, so that the single scattering approximation has a validity restricted to thin crystals composed of atoms of low atomic number. Further, at energies of above a few tens of keV, the wavelength of the electron is so short that the geometry of two-beam diffraction can be approximated in only small unit cells. It is therefore necessary to develop a scattering theory specific to electrons and, preferably, applicable to imaging as well as to diffraction. The development, started by Born (1926) and Bethe (1928), and continuing into the present time, is the subject of an extensive literature, which includes reviews [for instance: Howie (1978), Humphreys (1979)] and historical accounts (Goodman, 1981), and is incorporated in Chapter 5.1. Here, an attempt will be made to present only that outline of the main formulations which, it is hoped, will help the nonspecialist in the use of the tables. No attempt will be made to follow the historical development, which has been tortuous and not always logical, but rather to seek the simplest and most transparent approach that is consistent with brevity. Only key points in proofs will be sketched in an attempt to display the nature, rather than the rigorous foundations of the arguments.

5.2.3. Forward scattering A great deal of geometric detail can arise at this point and, further, there is no generally accepted method for approximation, the various procedures leading to numerically negligible differences and to expressions of precisely the same form. Detailed descriptions of the geometry are given in the references. The entrance surface of the specimen, in the form of a plate, is chosen as the x, y plane, and the direction of the incident beam is taken to be close to the z axis. Components of the wavevector are labelled with suffixes in the conventional way; K0 ¼ kx þ ky is the transverse wavevector, which will be very small compared to kz. In this notation, the excitation error for the reflection is given by

5.2.2. The defining equations h ¼

No many-body effects have yet been detected in the diffraction of fast electrons, but the velocities lie well within the relativistic region. The one-body Dirac equation would therefore appear to be the appropriate starting point. Fujiwara (1962), using the scattering matrix, carried through the analysis for forward scattering, and found that, to a very good approximation, the effects of spin are negligible, and that the solution is the same as that obtained from the Schro¨dinger equation provided that the relativistic values for wavelength and mass are used. In effect a Klein–Gordon equation (Messiah, 1965) can be used in electron diffraction (Buxton, 1978) in the form r

2

8
2 mjej’ þ b h2


 8
2 m0 jejW jejW 1þ b þ h2 2m0 c2

K02
jK0 þ 2
hj2 : 4
jkz j

An intuitive method argues that, since ’=W  1, then the component of the motion along z is little changed by scattering. Hence, making the substitution b ¼ expfikz zg and neglecting @2 =@z2 , equation (5.2.2.1) becomes   @ 1 2 2 ¼i ðrx; þ K Þ þ ’ ; y 0 @z 2kz where

b

¼ 0: 2 rx; y 

Here, W is the accelerating voltage and ’, the potential in the crystal, is defined as being positive. The relativistic values for mass and wavelength are given by m ¼ m0 ð1
v2 =c2 Þ
1=2, and taking ‘e’ now to represent the modulus of the electronic charge, jej,

and the wavefunction is labelled with the subscript b in order to indicate that it still includes back scattering, of central importance to LEED (low-energy electron diffraction). In more compact notation, b

¼ ðr2 þ k2 þ 2k’Þ

b

¼ 0:

ð5:2:2:1Þ

@ @z

† Deceased.

Copyright © 2010 International Union of Crystallography

@2 @2 þ 2; 2 @x @y

and ðx; y; 0Þ ¼ expfiðkx x þ ky yÞg. Equation (5.2.3.1) is of the form of a two-dimensional timedependent Schro¨dinger equation, with the z coordinate replacing time. This form has been extensively discussed. For instance, Howie (1966) derived what is essentially this equation using an expansion in Bloch waves, Berry (1971) used a Green function in a detailed and rigorous derivation, and Goodman & Moodie (1974), using methods due to Feynman, derived the equation as the limit of the multislice recurrence relation. A method due to Corones et al. (1982) brings out the relationship between the HEED and LEED equations. Equation (5.2.2.1) is cast in the form of a first-order system,

 ¼ h½2m0 eWð1 þ eW=2m0 c2 Þ
1=2 ;

½r2 þ k2 ð1 þ ’=WÞ

ð5:2:3:1Þ

647

b

@ b @z

!


¼

0 1 2 2
ðrx; 0 y þ k þ 2k’Þ



b

!

@ b : @z

5. DYNAMICAL THEORY AND ITS APPLICATIONS A splitting matrix is introduced to separate the wavefunction into the forward and backward components,  b , and the fast part  of the phase is factored out, so that  expfikz zg. In the b ¼ resulting matrix differential equation, the off-diagonal terms are seen to be small for fast electrons, and equation (5.2.2.1) reduces to the pair of equations   @  1 2 2 ¼ i ðr þ K0 Þ þ ’ 2kz x; y @z The equation for equation.





:

be achieved by transforming equation (5.2.2.1) with respect to x and y but not with respect to z, to obtain Tournarie’s equation d2 jUi ¼
Mb ðzÞjUi: dz2

ð5:2:6:1aÞ

Here jUi is the column vector of scattering amplitudes and Mb ðzÞ is a matrix, appropriate to LEED, with k vectors as diagonal elements and Fourier coefficients of the potential as nondiagonal elements. This equation is factorized in a manner parallel to that used on the real-space equation [equation (5.2.3.1)] (Lynch & Moodie, 1972) to obtain Tournarie’s forward-scattering equation

ð5:2:3:2Þ

is the Lontovich & Fock (1946) parabolic

5.2.4. Evolution operator Equation (5.2.3.1) is a standard and much studied form, so that many techniques are available for the construction of solutions. One of the most direct utilizes the causal evolution operator. A recent account is given by Gratias & Portier (1983). In terms of the ‘Hamiltonian’ of the two-dimensional system,

djU  i ¼ iM ðzÞjU  i; dz

ð5:2:6:1bÞ

where M ðzÞ ¼ ½K þ ð1=2ÞK
1 VðzÞ;

1
HðzÞ  ðr2 þ K02 Þ þ ’; 2kz x; y the evolution operator Uðz; z0 Þ, defined by satisfies

½Kij  ¼ ij Ki ; and

ðzÞ ¼ Uðz; z0 Þ 0,

½Vij  ¼ 2kz

P

Vi
j expf
2
ilzg;

l

i

@ Uðz; z0 Þ ¼ HðzÞUðz; z0 Þ; @z

ð5:2:4:1aÞ where Vi  vi are the scattering coefficients and vi are the structure amplitudes in volts. In order to simplify the electrondiffraction expression, the third crystallographic index ‘l’ is taken to represent the periodicity along the z direction. The double solution involving M of equation (5.2.6.1b) is of interest in displaying the symmetry of reciprocity, and may be compared with the double solution obtained for the real-space equation [equation (5.2.3.2)]. Normally the Mþ solution will be followed through to give the fast-electron forward-scattering equations appropriate in HEED. M
, however, represents the equivalent set of equations corresponding to the z reversed reciprocity configuration. Reciprocity solutions will yield diffraction symmetries in the forward direction when coupled with crystal-inverting symmetries (Section 2.5.3). Once again we set out to solve the forward-scattering equation (5.2.6.1a,b) now in semi-reciprocal space, and define an operator QðzÞ [compare with equation (5.2.4.1a)] such that

or Uðz; z0 Þ ¼ 1
i

Rz

Uðz; z1 ÞHðz1 Þ dz1 :

ð5:2:4:1bÞ

z0

5.2.5. Projection approximation – real-space solution Many of the features of the more general solutions are retained in the practically important projection approximation in which ’ðx; y; zÞ is replaced by its projected mean value ’p ðx; yÞ, so that the corresponding Hamiltonian Hp does not depend on z. Equation (5.2.4.1b) can then be solved directly by iteration to give Up ðz; z0 Þ ¼ expf
iHp ðz
z0 Þg;

ð5:2:5:1Þ

jUz i ¼ Qz jU0 i

and the solution may be verified by substitution into equation (5.2.4.1a). Many of the results of dynamical theory can be obtained by expansion of equation (5.2.5.1) as Up  1
iHp ðz
z0 Þ þ

with

U0 ¼ j0i;

i.e., Qz is an operator that, when acting on the incident wavevector, generates the wavefunction in semi-reciprocal space. Again, the differential equation can be transformed into an integral equation, and once again this can be iterated. In the projection approximation, with M independent of z, the solution can be written as

i2 2 H ðz
z0 Þ
. . . ; 2! p

Qp ¼ expfiMp ðz
z0 Þg: followed by the direct evaluation of the differentials. Such expressions can be used, for instance, to explore symmetries in image space.

A typical off-diagonal element is given by Vi
j = cos i , where i is the angle through which the beam is scattered. It is usual in the literature to find that cos i has been approximated as unity, since even the most accurate measurements are, so far, in error by much more than this amount. This expression for Qp is Sturkey’s (1957) solution, a most useful relation, written explicitly as

5.2.6. Semi-reciprocal space In the derivation of electron-diffraction equations, it is more usual to work in semi-reciprocal space (Tournarie, 1962). This can

648

5.2. DYNAMICAL THEORY OF ELECTRON DIFFRACTION jUi ¼ expfiMp Tgj0i


  sin
1=2 T 1=2 M2 : Q2 ¼ expfiðKh =2ÞTgE ðcos
TÞE þ i
1=2

ð5:2:6:2Þ

ð5:2:7:2Þ

with T the thickness of the crystal, and j0i, the incident state, a column vector with the first entry unity and the rest zero.

This result was first obtained by Blackman (1939), using Bethe’s dispersion formulation. Ewald and, independently, Darwin, each with different techniques, had, in establishing the theoretical foundations for X-ray diffraction, obtained analogous results (see Section 5.1.3). The two-beam approximation, despite its simplicity, exemplifies some of the characteristics of the full dynamical theory, for instance in the coupling between beams. As Ewald pointed out, a formal analogy can be found in classical mechanics with the motion of coupled pendulums. In addition, the functional form ðsin axÞ=x, deriving from the shape function of the crystal emerges, as it does, albeit less obviously, in the N-beam theory. This derivation of equation (5.2.7.2) exhibits two-beam diffraction as a typical two-level system having analogies with, for instance, lasers and nuclear magnetic resonance and exhibiting the symmetries of the special unitary group SU(2) (Gilmore, 1974).

S ¼ expfiMp Tg is a unitary matrix, so that in this formulation scattering is described as rotation in Hilbert space. 5.2.7. Two-beam approximation In the two-beam approximation, as an elementary example, equation (5.2.6.2) takes the form


u0 uh




 
 0 V  ðhÞ 0 ¼ exp i T : VðhÞ Kh 1

ð5:2:7:1Þ

If this expression is expanded directly as a Taylor series, it proves surprisingly difficult to sum. However, the symmetries of Clifford algebra can be exploited by summing in a Pauli basis thus,

5.2.8. Eigenvalue approach In terms of the eigenvalues and eigenvectors, defined by


  0 V  ðhÞ exp i T VðhÞ K 
   h Kh T Kh R I ¼ exp i r þ V r1
V r2 T : E exp i 2 2 3

Hp j ji ¼ j j ji; the evolution operator can be written as

Here, the ri are the Pauli matrices
r1 ¼

0 1



0 i
1 ; r2 ¼ ; r3 ¼ 0
i 0 0
 1 0 E¼ ; 0 1 1

0 1

 ;

Uh ¼

P

j 0

j h

expf
i2
j Tg;

ð5:2:8:1Þ

j

where hj is the h component of the j eigenvector with eigenvalue j . This expression can now be related to those obtained in the other formulations. For example, Sylvester’s theorem (Frazer et al., 1963) in the form


  K exp i h r3 þ V R r1
V I r2 T 2
 K ¼ E þ i h r3 þ V R r 1
V I r 2 T 2
2 1 Kh R I r þ V r1
V r2 T 2 þ . . . ;
2 2 3

f ðMÞ ¼

P

Aj f ðj Þ

j

when applied to Sturkey’s solution yields

using the anti-commuting properties of ri : ¼0 ¼1

j ji expfj ðz
z0 Þgh jj dj:

This integration becomes a summation over discrete eigen states when an infinitely periodic potential is considered. Despite the early developments by Bethe (1928), an N-beam expression for a transmitted wavefunction in terms of the eigenvalues and eigenvectors of the problem was not obtained until Fujimoto (1959) derived the expression

and V R , V I are the real and imaginary parts of the complex scattering coefficients appropriate to a noncentrosymmetric crystal, i.e. Vh ¼ V R þ iV I . Expanding,

r i r j þ r j ri ri ri

R

Uðz; z0 Þ ¼

Uh ¼ expðiMp zÞ ¼



P

Pj expfi2
j zg

(Kainuma, 1968; Hurley et al., 1978). Here, the Pj are projection operators, typically of the form and putting ½ðKh =2Þ2 þ VðhÞV  ðhÞ ¼
, M2 ¼ ½ðKh =2Þr3 þ V R r1
V I r2 , so that M22 ¼
E and M32 ¼
M2 , the powers of the matrix can easily be evaluated. They fall into odd and even series, corresponding to sine and cosine, and the classical twobeam approximation is obtained in the form

Pj ¼

Y ðMp
En Þ n6¼j

649

j
n

:

5. DYNAMICAL THEORY AND ITS APPLICATIONS On changing to a lattice basis, these transform to 0j hj . Alternatively, the semi-reciprocal differential equation can be uncoupled by diagonalizing Mp (Goodman & Moodie, 1974), a process which involves the solution of the characteristic equation jMp
j Ej ¼ 0:

mately embody the symmetries of the Bravais lattice; i.e. Bloch functions are the irreducible representations of the translational component of the space group.

ð5:2:8:2Þ

5.2.10. Bloch-wave formulations In developing the theory from the beginning by eigenvalue techniques, it is usual to invoke the periodicity of the crystal in order to show that the solutions to the wave equation for a given wavevector k are Bloch waves of the form

5.2.9. Translational invariance An important result deriving from Bethe’s initial analysis, and not made explicit in the preceding formulations, is that the fundamental symmetry of a crystal, namely translational invariance, by itself imposes a specific form on wavefunctions satisfying Schro¨dinger’s equation. Suppose that, in a one-dimensional description, the potential in a Hamiltonian Ht ðxÞ is periodic, with period t. Then,

¼ CðrÞ expfik  rg;

where CðrÞ has the periodicity of the lattice, and hence may be expanded in a Fourier series to give

’ðx þ tÞ ¼ ’ðxÞ

¼

P Ch ðkÞ expfiðk þ 2
hÞ  rg:

ð5:2:10:1Þ

h

and The Ch ðkÞ are determined by equations of consistency obtained by substitution of equation (5.2.10.1) into the wave equation. If N terms are selected in equation (5.2.10.1) there will be N Bloch waves where wavevectors differ only in their components normal to the crystal surface, and the total wavefunction will consist of a linear combination of these Bloch waves. The problem is now reduced to the problem of equation (5.2.8.2). The development of solutions for particular geometries follows that for the X-ray case, Chapter 5.1, with the notable differences that: (1) The two-beam solution is not adequate except as a first approximation for particular orientations of crystals having small unit cells and for accelerating voltages not greater than about 100 keV. In general, many-beam solutions must be sought. (2) For transmission HEED, the scattering angles are sufficiently small to allow the use of a small-angle forward-scattering approximation. (3) Polarization effects are negligible except for very low energy electrons. Humphreys (1979) compares the action of the crystal, in the Bloch-wave formalism, with that of an interferometer, the incident beam being partitioned into a set of Bloch waves of different wavevectors. ‘As each Bloch wave propagates it becomes out of phase with its neighbours (due to its different wavevector). Hence interference occurs. For example, if the crystal thickness varies, interference fringes known as thickness fringes are formed.’ For the two-beam case, these are the fringes of the pendulum solution referred to previously.

Ht ðxÞ ¼ E ðxÞ: Now define a translation operator Cf ðxÞ ¼ f ðx þ tÞ; for arbitrary f ðxÞ. Then, since C’ðxÞ ¼ ’ðxÞ, and r2 is invariant under translation, CHt ðxÞ ¼ Ht ðxÞ and CHt ðxÞ ðxÞ ¼ Ht ðx þ tÞ ðx þ tÞ ¼ Ht ðxÞC ðxÞ: Thus, the translation operator and the Hamiltonian commute, and therefore have the same eigenfunctions (but not of course the same eigenvalues), i.e. C ðxÞ ¼  ðxÞ: This is a simpler equation to deal with than that involving the Hamiltonian, since raising the operator to an arbitrary power simply increments the argument Cm ðxÞ ¼ ðx þ mtÞ ¼ m ðxÞ:

5.2.11. Dispersion surfaces One of the important constructs of the Bloch-wave formalism is the dispersion surface, a plot of the permitted values of the z component of a Bloch wavevector against the component of the incident wavevector parallel to the crystal surface. The curve for a particular Bloch wave is called a branch. Thus, for fast electrons, the two-beam approximation has two branches, one for each eigenvalue, and the N-beam approximation has N. A detailed treatment of the extensive and powerful theory that has grown from Bethe’s initial paper is to be found, for example, in Hirsch et al. (1965). Apart from its fundamental importance as a theoretical tool, this formulation provides the basis for one of the most commonly used numerical techniques, the essential step being the estimation of the eigenvalues from equation (5.2.8.2) [see IT C (2004, Section 4.3.6.2)].

But ðxÞ is bounded over the entire range of its argument, positive and negative, so that jj ¼ 1, and  must be of the form expfi2
ktg. Thus, ðx þ tÞ ¼ C ðxÞ ¼ expfi2
ktg ðxÞ, for which the solution is ðxÞ ¼ expfi2
ktgqðxÞ with qðx þ tÞ ¼ qðxÞ. This is the result derived independently by Bethe and Bloch. Functions of this form constitute bases for the translation group, and are generally known as Bloch functions. When extended in a direct fashion into three dimensions, functions of this form ulti-

650

5.2. DYNAMICAL THEORY OF ELECTRON DIFFRACTION 5.2.12. Multislice

Un ðh; kÞ ¼

Multislice derives from a formulation that generates a solution in the form of a Born series (Cowley & Moodie, 1962). The crystal is treated as a series of scattering planes on to which the potential from the slice between z and z þ z is projected, separated by vacuum gaps z, not necessarily corresponding to any planes or spacings of the material structure. The phase change in the electron beam produced by passage through a slice is given by

P P l h1 k1 l1

q ¼ exp
i

z1 þz R

in Vðh1 ; k1 ; l1 Þ

hn
1 kn
1 ln
1


 n
1 n
1 n
i P P P ...V h
hr ; k
k r ; l
lr r¼1

r¼1 n

 expfi
m Tgðsin
m T=m Þ ðm
1 Þ m¼1 
1

. . . ðm
m
1 Þðm
mþ1 Þ . . . ðm
Þ

) ’ðx; y; zÞ dz ;

nP
1

z1

ð5:2:13:1aÞ

and the phase distribution in the x, y plane resulting from propagation between slices is given by

and where n is the order of interaction. Here  is the excitation error of the reflection with index h, k, and i are the excitation errors for the reflections with indices hi, ki , li . Thus each constituent process may be represented by a diagram, starting on the origin of reciprocal space, possibly looped, and ending on the point with coordinates (h, k). This solution can also be obtained by iteration of the Greenfunction integral equation, the integrals being evaluated by means of suitably chosen contours on the complex kz plane (Fujiwara, 1959), as well as by expansion of the scattering matrix (Fujimoto, 1959). Clearly, two or more of the i will, in general, be equal in nearly all of the terms in equation (5.2.13.1a). Confluence is, however, readily described, the divided differences of arbitrary order transforming into differentials of the same order (Moodie, 1972). The physical picture that emerges from equation (5.2.13.1a) is that of n-fold scattering, the initial wave being turned through n
1 intermediate states, processes that can be presented by scattering diagrams in reciprocal space (Gjønnes & Moodie, 1965). For a given scattering vector, constituent functions are evaluated for all possible paths in three dimensions, and those functions are then summed over l. There are therefore two distinct processes by which upper-layer lines can perturb wavefunctions in the zone, namely: by scattering out of the zone and then back in; and by intrusion of the effective shape function from another zone, the latter process being already operative in the first Born, or kinematical approximation. The constituent functions to be evaluated can be transformed into many forms. One of the more readily described is that which assigns to each diagram an effective dynamical shape function. If there are no loops in the diagram of order n, this effective shape function is the (n + 1)th divided difference of the constituent phase-shifted kinematical shape transforms. For general diagrams, divided differences in loops are replaced by the corresponding differentials. The resulting function is multiplied by the convolution of the contributing structure amplitudes and diagrams of all orders summed (Moodie, 1972). While scattering diagrams have no utility in numerical work, they find application in the analysis of symmetries, for instance in the determination of the presence or absence of a centre of inversion [for a recent treatment, see Moodie & Whitfield (1995)] and in the detection of screw axes and glide planes (Gjønnes & Moodie, 1965). Methods for the direct determination of all space groups are described by Goodman (1975) and by Tanaka et al. (1983) (see Section 2.5.3). Equation (5.2.13.1a) can be rewritten in a form particularly suited to the classification of approximations, and to describing the underlying symmetry of the formulation. The equation is written for compactness as



 ikðx2 þ y2 Þ p ¼ exp ; 2z

where the wavefront has been approximated by a paraboloid. Thus, the wavefunction for the (n + 1)th slice is given by  nþ1

¼ ¼½

 n  exp n

 ikðx2 þ y2 Þ expf
i’nþ1 g 2z

 pq;

ð5:2:12:1Þ

where  is the convolution operator (Cowley, 1981). This equation can be regarded as the finite difference form of the Schro¨dinger equation derived by Feynman’s (1948) method. The calculation need be correct only to first order in z. Writing the convolution in equation (5.2.12.1) explicitly, and expanding in a Taylor series, the integrals can be evaluated to yield equation (5.2.3.1) (Goodman & Moodie, 1974). If equation (5.2.12.1) is Fourier transformed with respect to x and y, the resulting recurrence relation is of the form Unþ1 ¼ ½Un P  Qn ;

ð5:2:12:2Þ

where P and Q are obtained by Fourier transforming p and q above. This form is convenient for numerical work since, for a perfect crystal, it is: discrete, as distinct from equation (5.2.12.1) which is continuous in the variables [see IT C (2004, Section 4.3.6.1)]; numerically stable at least up to 5000 beams; fast; and only requires a computer memory proportional to the number of beams (Goodman & Moodie, 1974).

5.2.13. Born series In the impulse limit of equation (5.2.12.2), the integrals can be evaluated to give the Born series (Cowley & Moodie, 1957) Uðh; kÞ ¼

r¼1

 ½expf
2
iTg=ð2
iÞ   
1  expfi
Tgðsin
T=Þ ð
1 Þ . . . ð
n
1 Þ þ

(

P

...

P

Un ðh; kÞ;

n

Un ðhÞ ¼ En ðhÞZn ðÞ; where

651

5. DYNAMICAL THEORY AND ITS APPLICATIONS so that En ðhÞ depends only on crystal structure and Zn ðÞ only on diffraction geometry. A transformation (Cowley & Moodie, 1962) involving bialternants leads to Un ¼

1 P

Born, M. (1926). Quantenmechanik der Stossvorgange. Z. Phys. 38, 803– 826. Buxton, B. (1978). Graduate Lecture-Course Notes: Dynamical Diffraction Theory. Cambridge University, England. Corones, J., De Facio, B. & Kreuger, R. J. (1982). Parabolic approximations to the time-independent elastic wave equation. J. Math. Phys. 23, 577–586. Cowley, J. M. (1981). Diffraction Physics, pp. 26–30. Amsterdam: NorthHolland. Cowley, J. M. & Moodie, A. F. (1957). The scattering of electrons by atoms and crystals. I. A new theoretical approach. Acta Cryst. 10, 609– 619. Cowley, J. M. & Moodie, A. F. (1962). The scattering of electrons by thin crystals. J. Phys. Soc. Jpn, 17, Suppl. B11, 86–91. Feynman, R. (1948). Space–time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 201, 367–387. Frazer, R. A., Duncan, W. J. & Collar, A. R. (1963). Elementary Matrices, pp. 78–79. Cambridge University Press. Fujimoto, F. (1959). Dynamical theory of electron diffraction in the Laue case. J. Phys. Soc. Jpn, 14, 1558–1568. Fujiwara, K. (1959). Application of higher order Born approximation to multiple elastic scattering of electrons by crystals. J. Phys. Soc. Jpn, 14, 1513–1524. Fujiwara, K. (1962). Relativistic dynamical theory of electron diffraction. J. Phys. Soc. Jpn, 17, Suppl. B11, 118–123. Fukuhara, A. (1966). Many-ray approximations in the dynamical theory of electron diffraction. J. Phys. Soc. Jpn, 21, 2645–2662. Gilmore, R. (1974). Lie Groups, Lie Algebras, and Some of Their Applications. New York: Wiley–Interscience. Gjønnes, J. & Høier, R. (1971). The application of non-systematic manybeam dynamic effects to structure-factor determination. Acta Cryst. A27, 313–316. Gjønnes, J. & Moodie, A. F. (1965). Extinction conditions in the dynamic theory of electron diffraction. Acta Cryst. 19, 65–67. Goodman, P. (1975). A practical method of three-dimensional spacegroup analysis using convergent-beam electron diffraction. Acta Cryst. A31, 804–810. Goodman, P. (1981). Editor. Fifty Years of Electron Diffraction. Dordrecht: Kluwer Academic Publishers. Goodman, P. & Moodie, A. F. (1974). Numerical evaluation of N-beam wave functions in electron scattering by the multislice method. Acta Cryst. A30, 280–290. Gratias, D. & Portier, R. (1983). Time-like perturbation method in high energy electron diffraction. Acta Cryst. A39, 576–584. Hirsch, P. B., Howie, A., Nicholson, R. B., Pashley, D. W. & Whelan, M. J. (1965). Electron Microscopy of Thin Crystals. London: Butterworths. Howie, A. (1966). Diffraction channelling of fast electrons and positrons in crystals. Philos. Mag. 14, 223–237. Howie, A. (1978). In Electron Diffraction 1927–1977, edited by P. J. Dobson, J. B. Pendry & C. J. Humphreys, pp. 1–12. Inst. Phys. Conf. Ser. No. 41. Bristol/London: Institute of Physics. Humphreys, C. J. (1979). The scattering of fast electrons by crystals. Rep. Prog. Phys. 42, 1825–1887. Hurley, A. C., Johnson, A. W. S., Moodie, A. F., Rez, P. & Sellar, J. R. (1978). Algebraic approaches to N-beam theory. In Electron Diffraction 1927–1977, edited by P. J. Dobson, J. B. Pendry & C. J. Humphreys, pp. 34–40. Inst. Phys. Conf. Ser. No. 41. Bristol/London: Institute of Physics. Hurley, A. C. & Moodie, A. F. (1980). The inversion of the three-beam intensities for scalar scattering by a general centrosymmetric crystal. Acta Cryst. A36, 737–738. International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers. Kainuma, Y. (1968). Averaged intensities in the many beam dynamical theory of electron diffraction. Part I. J. Phys. Soc. Jpn, 25, 498–510. Kambe, K. (1957). Study of simultaneous reflection in electron diffraction by crystal. J. Phys. Soc. Jpn, 12, 13–31. Kogiso, M. & Takahashi, H. (1977). Group-theoretical method in the many-beam theory of electron diffraction. J. Phys. Soc. Jpn, 42, 223– 229. Lontovitch, M. & Fock, R. (1946). Solution of the problem of propagation of electromagnetic waves along the Earth’s surface by the method of parabolic equation. (Translated from Russian by J. Smorodinsky.) J. Phys. 10, 13–24.

  En ðhÞ ð2
iTÞnþr =ðn þ rÞ! hr ð; 1 . . . n
1 Þ; ð5:2:13:1bÞ

r¼0

where hr is the complete homogeneous symmetric polynomial function of n variables of order r. Upper-layer-line effects can, of course, be calculated in any of the formulations. 5.2.14. Approximations So far, only the familiar first Born and two-beam approximations and the projection approximation have been mentioned. Several others, however, have a considerable utility. A high-voltage limit can be calculated in standard fashion to give 

RT



UHVL ðh; kÞ ¼ F exp
ic ’ðx; y; zÞ dz ;

ð5:2:14:1Þ

0

where F is the Fourier transform operator, and c ¼ 2
m0 ec =h2 with c ¼ ðh=m0 cÞ, the Compton wavelength. The phase-grating approximation, which finds application in electron microscopy, involves the assumption that equation (5.2.14.1) has some range of validity when c is replaced by . This is equivalent to ignoring the curvature of the Ewald sphere and can therefore apply to thin crystals [see Section 2.5.2 and IT C (2004, Section 4.3.8)]. Approximations that involve curtailing the number of beams evidently have a range of validity that depends on the size of the unit cell. The most explored case is that of three-beam interactions. Kambe (1957) has demonstrated that phase information can be obtained from the diffraction data; Gjønnes & Høier (1971) analysed the confluent case, and Hurley & Moodie (1980) have given an explicit inversion for the centrosymmetric case. Analyses of the symmetry of the defining differential equation, and of the geometry of the noncentrosymmetric case, have been given by Moodie et al. (1996, 1998). Niehrs and his co-workers (e.g. Blume, 1966) have shown that, at or near zones, effective two-beam conditions can sometimes obtain, in that, for instance, the central beam and six equidistant beams of equal structure amplitude can exhibit two-beam behaviour when the excitation errors are equal. Grouptheoretical treatments have been given by Fukuhara (1966) and by Kogiso & Takahashi (1977). Explicit reductions for all admissible noncentrosymmetric space groups have been obtained by Moodie & Whitfield (1994). Extensions of such results have application in the interpretation of lattice images and convergent-beam patterns. The approximations near the classical limit have been extensively explored [for instance, see Berry (1971)] but channelling has effectively become a separate subject and cannot be discussed here. References Berry, M. V. (1971). Diffraction in crystals at high voltages. J. Phys. C, 4, 697–722. Bethe, H. A. (1928). Theorie der Beugung von Elektronen an Kristallen. Ann. Phys. (Leipzig), 87, 55–129. Blackman, M. (1939). Intensities of electron diffraction rings. Proc. Phys. Soc. London Sect. A, 173, 68–72. Blume, J. (1966). Die Kantenstreifung im Elektronen-Mikroskopischen Bild Wurfelformiger MgO Kristalle bei Durchstrahlung im Richtung der Raumdiagonal. Z. Phys. 191, 248–272.

652

5.2. DYNAMICAL THEORY OF ELECTRON DIFFRACTION Moodie, A. F. & Whitfield, H. J. (1994). The reduction of N-beam scattering from noncentrosymmetric crystals to two-beam form. Acta Cryst. A50, 730–736. Moodie, A. F. & Whitfield, H. J. (1995). Friedel’s law and noncentrosymmetric space groups. Acta Cryst. A51, 198–201. Sturkey, L. (1957). The use of electron-diffraction intensities in structure determination. Acta Cryst. 10, 858–859. Tanaka, M., Sekii, H. & Nagasawa, T. (1983). Space-group determination by dynamic extinction in convergent-beam electron diffraction. Acta Cryst. A39, 825–837. Tournarie, M. (1962). Recent developments of the matrical and semireciprocal formulation on the field of dynamical theory. J. Phys. Soc. Jpn, 17, Suppl. B11, 98–100.

Lynch, D. F. & Moodie, A. F. (1972). Numerical evaluation of low energy electron diffraction intensities. Surf. Sci. 32, 422–438. Messiah, A. (1965). Quantum Mechanics, Vol. II, pp. 884–888. Amsterdam: North-Holland. Moodie, A. F. (1972). Reciprocity and shape functions in multiple scattering diagrams. Z. Naturforsch. Teil A, 27, 437–440. Moodie, A. F., Etheridge, J. & Humphreys, C. J. (1996). The symmetry of three-beam scattering equations: inversion of three-beam diffraction patterns from centrosymmetric crystals. Acta Cryst. A52, 596–605. Moodie, A. F., Etheridge, J. & Humphreys, C. J. (1998). The Coulomb interaction and the direct measurement of structural phase. In The Electron, edited by A. Kirkland & P. Brown, pp. 235–246. IOM Communications Ltd. London: The Institute of Materials.

653

references

International Tables for Crystallography (2010). Vol. B, Chapter 5.3, pp. 654–664.

5.3. Dynamical theory of neutron diffraction By M. Schlenker and J.-P. Guigay

energy, hence of velocity, close to the Maxwell distribution characteristic of the temperature T of the moderator. Frequently used moderators are liquid deuterium (D2, i.e. 2H2) at 25 K, heavy water (D2O) at room temperature and graphite allowed to heat up to 2400 K; the corresponding neutron distributions are termed cold, thermal and hot, respectively. The interaction of a neutron with an atom is usually described in terms of scattering lengths or of scattering cross sections. The main contribution corresponding to the nuclear interaction is related to the strong force. The interaction with the magnetic field created by atoms with electronic magnetic moments is comparable in magnitude to the nuclear term.

5.3.1. Introduction Neutron and X-ray scattering are quite similar both in the geometry of scattering and in the orders of magnitude of the basic quantities. When the neutron spin is neglected, i.e. when dealing with scattering by perfect nonmagnetic crystals, the formalism and the results of the dynamical theory of X-ray scattering can be very simply transferred to the case of neutrons (Section 5.3.2). Additional features of the neutron case are related to the neutron spin and appear in diffraction by magnetic crystals (Section 5.3.3). The low intensities available, coupled with the low absorption of neutrons by most materials, make it both necessary and possible to use large samples in standard diffraction work. The effect of extinction in crystals that are neither small nor bad enough to be amenable to the kinematical approximation is therefore very important in the neutron case, and will be discussed in Section 5.3.4 together with the effect of crystal distortion. Additional possibilities arise in the neutron case because the neutrons can be manipulated from outside through applied fields (Section 5.3.5). Reasonably extensive tests of the predictions of the dynamical theory of neutron diffraction have been performed, with the handicap of the very low intensities of neutron beams as compared with X-rays: these are described in Section 5.3.6. Finally, the applications of the dynamical theory in the neutron case, and in particular neutron interferometry, are reviewed in Section 5.3.7.

5.3.2.2. Scattering lengths and refractive index The elastic scattering amplitude for scattering vector s, f ðsÞ, is defined by the wave scattered by an object placed at the origin when the incident plane wave is
i ¼ A exp½iðk0  r  !tÞ, written as
s ¼ A½ f ðsÞ=r exp½iðkr  !tÞ with k ¼ jk0 j ¼ jk0 þ sj ¼ 2=. In the case of the strong-force interaction with nuclei, the latter can be considered as point scatterers because the interaction range is very small, hence the scattering amplitude is isotropic (independent of the direction of s). It is also independent of  except in the vicinity of resonances. It is conventionally written as b so that most values of b, called the scattering length, are positive. A table of experimentally measured values of the scattering lengths b is given in IT C for the elements in their natural form as well as for many individual isotopes. It is apparent that the typical order of magnitude is the fm (femtometre, i.e. 1015 m, or fermi), that there is no systematic variation with atomic number and that different isotopes have very different scattering lengths, including different signs. The first remark implies that scattering amplitudes of X-rays and of neutrons have comparable magnitudes, because the characteristic length for X-ray scattering (the scattering amplitude for forward scattering by one free electron) is R ¼ 2:8 fm, the classical electron radius. The second and third points explain the importance of neutrons in structural crystallography, in diffuse scattering and in small-angle scattering. Scattering of neutrons by condensed matter implies the use of the bound scattering lengths, as tabulated in IT C. The ‘free’ scattering length, used in some presentations, is obtained by multiplying the bound scattering lengths by A=ðA þ 1Þ, where A is the mass of the nucleus in atomic units. A description in terms of an interaction potential is possible using the Fermi pseudo-potential, which in the case of the nuclear interaction with a nucleus at r0 can be written as VðrÞ ¼ ðh2 =2mÞbðr  r0 Þ, where  denotes the three-dimensional Dirac distribution. Refraction of neutrons at an interface can be conveniently described by assigning a refractive index to the material, such that the wavenumber in the material, k, is related to that in a vacuum, k0 , by k ¼ nk0. Here

5.3.2. Comparison between X-rays and neutrons with spin neglected 5.3.2.1. The neutron and its interactions An excellent introductory presentation of the production, properties and scattering properties of neutrons is available (Scherm & Fa˚k, 1993, and other papers in the same book). A stimulating review on neutron optics, including diffraction by perfect crystals, has been written by Klein & Werner (1983). X-rays and neutrons are compared in terms of the basic quantities in Table 4.1.3.1 of IT C (2004), where Chapter 4.4 is devoted to neutron techniques. The neutron is a massive particle for which the values relevant to diffraction are: no electric charge, rest mass m ¼ 1:675
1027 kg, angular momentum eigenvalues along a given direction h- =2 (spin 12) and a magnetic moment of 1.913 nuclear magneton, meaning that its component along a quantization direction z can take eigenvalues
z ¼ 0:996
1026 A m2 . The de Broglie wavelength is  ¼ h=p where h is Planck’s constant (h ¼ 2h- ¼ 6:625
1034 J s) and p is the linear momentum; p ¼ mv in the nonrelativistic approximation, which always applies in the context of this chapter, v being the neutron’s velocity. The neutron’s wavelength, , and kinetic energy, Ec , are thus related by  ¼ h=ð2mEc Þ1=2, or, ˚  ¼ 9:05=ðEc ½meVÞ1=2 . Thus, to be of in practical units,  ½A interest for diffraction by materials, neutrons should have kinetic energies in the range 100 to 102 meV. In terms of the velocity, ˚  ¼ 3:956=ðv ½km s1 Þ.  ½A Neutron beams are produced by nuclear reactors or by spallation sources, usually pulsed. In either case they initially have an energy in the MeV range, and have to lose most of it before they can be used. The moderation process involves inelastic interactions with materials. It results in statistical distributions of Copyright © 2010 International Union of Crystallography

2 X n¼ 1 b V i i

!1=2 ;

where the sum is over the nuclei contained in volume V. PWith typical values, n is very close to 1 and 1  n ¼ ð2 =2VÞ i bi is

654

5.3. DYNAMICAL THEORY OF NEUTRON DIFFRACTION 5

typically of the order of 10 . This small value, in the same range as for X-rays, gives a feeling for the order of magnitude of key quantities of the dynamical theory, in particular the Darwin width 2 as discussed in Chapter 5.1. It also makes total external P reflection possible on materials for which i bi > 0: this is the basis for the neutron guide tubes now installed in most research reactors, as well as for reflectometry. The notations prevailing in X-ray and in neutron crystallography are slightly different, and the correspondence is very simple: X-ray atomic scattering factors and structure factors are numbers. When multiplied by R, the classical electron radius, they become entirely equivalent to the corresponding quantities in neutron usage, which are lengths. It should be noted that the presence of different isotopes and the effect of nuclear spin (disordered except under very special conditions) give rise to incoherent elastic neutron scattering, which has no equivalent in the X-ray case. The scattering length corresponding to R times the atomic scattering factor for X-rays is therefore the coherent scattering length, bcoh , obtained by averaging the scattering length over the nuclear spin state and isotope distribution.

There are also at least two additional aspects of neutron scattering in comparison with X-ray scattering, apart from the effect of the magnetic moment associated with the intrinsic (spin) angular momentum of the neutron. On the one hand, the small velocity of neutrons, compared with the velocity of light, makes time-of-flight measurements possible, both in standard neutron diffraction and in investigations of perfect crystals. Because this velocity is of the same order of magnitude as that of ultrasound, the effect of ultrasonic excitation on neutron diffraction is slightly different from that in the X-ray case. On the other hand, the fact that neutrons have mass and a magnetic moment implies that they can be affected by external fields, such as gravity and magnetic fields, both during their propagation in air or in a vacuum and while being diffracted within crystals (Werner, 1980) (see Section 5.3.5). Experiments completely different from the X-ray case can thus be performed with perfect crystals and with neutron interferometers (see Sections 5.3.6 and 5.3.7.3).

5.3.2.5. Translating X-ray dynamical theory into the neutron case As shown in Chapter 5.1, the basic equations of dynamical theory, viz Maxwell’s equations for the X-ray case and the timeindependent Schro¨dinger equation in the neutron case, have exactly the same form when the effect of the neutron spin can be neglected, i.e. in situations that do not involve magnetism and when no externally applied potential is taken into account. The translation scheme for the scattering factors and structure factors is described above. The one formal difference is that the wavefunction is scalar in the neutron case, hence there is no equivalent to the parallel and perpendicular polarizations of the X-ray situation: C in equation (5.1.2.20) of Chapter 5.1 should therefore be set to 1. The physics of neutron diffraction by perfect crystals is therefore expected to be very similar to that of X-ray diffraction, with the existence of wavefields, Pendello¨sung effects, anomalous transmission, intrinsic rocking-curve shapes and reflectivity versus thickness behaviour in direct correspondence. All experimental tests of these predictions confirm this view (Section 5.3.6). Basic discussions of dynamical neutron scattering are given by Stassis & Oberteuffer (1974), Sears (1978), Rauch & Petrascheck (1978) and Squires (1978).

5.3.2.3. Absorption Neutron absorption is related to a nuclear reaction in which the neutron combines with the absorbing nucleus to form a compound nucleus, usually in a metastable state which then decays. The scattering length describing this resonance scattering process depends on the neutron energy and contains an imaginary part associated with absorption in complete analogy with the imaginary part of the dispersion correction for the X-ray atomic scattering factors. The energies of the resonances are usually far above those of interest for crystallography, and the linear absorption coefficient varies approximately as 1=v or . It is important to note that, except for a very few cases (notably 3 He, 6Li, 10B, In, Cd, Gd), the absorption of neutrons is very small compared with that of X-rays, and even more so compared with that of electrons, and can be neglected to a first approximation.

5.3.2.4. Differences between neutron and X-ray scattering There are major differences in the experimental aspects of neutron and X-ray scattering. Neutrons are only available in large facilities, where allocation of beam time to users is made on the basis of applications, and where admittance is restricted because of the hazards which nuclear technology can present in the hands of ill-intentioned users. Because of the radiation shielding necessary, as well as the large size of neutron detectors, neutron-scattering instrumentation is much bulkier than that for X-rays. Neutron beams are in some aspects similar to synchrotron radiation, in particular because in both cases the beams are initially ‘white’ and for most applications have to be monochromated. There is, however, a huge difference in the order of magnitudes of the intensities. Neutron beams are weak in comparison with laboratory X-ray sources, and weaker by many orders of magnitude than synchrotron radiation. Also, the beam sources are large in the case of neutrons, since they are essentially the moderators, whereas the source is very small in the case of synchrotron radiation, and this difference again increases the ratio of the brilliances in favour of X-rays. This encourages the use of large specimens in all neutron-scattering work, and makes the extinction problem more important than for X-rays. Furthermore, many experiments that are quick using X-rays become very slow, and give rise to impaired resolution, in the neutron case.

5.3.3. Neutron spin, and diffraction by perfect magnetic crystals 5.3.3.1. Polarization of a neutron beam and the Larmor precession in a uniform magnetic field A polarized neutron beam is represented by a two-component spinor,


 c 1 0 j’i ¼ ¼c þd ; d 0 1

which is the coherent superposition of two states, of different amplitudes c and d, polarized in opposite directions along the spin-quantization axis. The spinor components c and d are generally space- and time-dependent. We suppose that h’j’i ¼ cc þ dd ¼ 1. The polarization vector P is defined as P ¼ h’jrj’i;

where the vector r represents the set of Pauli matrices x, y and z . The components of P are

655

5. DYNAMICAL THEORY AND ITS APPLICATIONS
   c ¼ c Px ¼ c d  x d ¼ c d þ cd
     c  Py ¼ c d  y ¼ c d ¼ iðcd  c dÞ
    c Pz ¼ c d z ¼ c d 





d



d




 0

1

1

0

d


 0

i


 c

i

0

d

0 1


 c d


 1 d 0 


1  ¼ 12 0


 c


n r  B ¼

 0 ;
n B

BðrÞ ¼


n being the neutron magnetic moment, if the directions of B and of the spin-quantization axis coincide. Consequently, different indices of refraction n ¼ 1  ð
n B=2EÞ, where E is the neutron energy, should be associated with the spinor components c and d; this induces between these spinor components a phase difference which is a linear function of the time (or, equivalently, of the distance travelled by the neutrons), hence, according to (5.3.3.1), a rotation around the magnetic field of the component of the neutron polarization perpendicular to this magnetic field. The time frequency of this so-called Larmor precession is 2
n B=h, where h is Planck’s constant. A neutron beam may be partially polarized; such a beam is conveniently represented by a spin-density matrix , which is the statistical average of the spin-density matrices associated to the polarized beams which are mixed incoherently, the density matrix associated to the spinor

d




¼

cc c d


0 l
r curl 3 ; r 4

BðsÞ ¼
0 s


l
s ¼
0 l? ðsÞ; s2

ð5:3:3:4Þ

where l? ðsÞ is the projection of l on the plane perpendicular to s (reflecting plane). This result can be applied by volume integration to the more general case of a spatially extended magnetization distribution, which for a single magnetic ion corresponds to the atomic shell of the unpaired electrons. It is thus shown that the magnetic scattering length is proportional to ln  li?, where li? is the projection of the magnetic moment of the ion on the reflecting plane. For a complete description of magnetic scattering, which involves the spin-polarization properties of the scattered beam, it is necessary to represent the neutron wavefunction in the form of a two-component spinor and the ion’s magnetic moment as a spin operator which is a matrix expressed in terms of the Pauli matrices r ðx ; y ; z Þ. The magnetic scattering length is therefore itself a ð2
2Þ matrix:

being 

ð5:3:3:3Þ

where
0 ¼ 4
107 H m1 is the permittivity of a vacuum and
denotes the cross product. BðrÞ can be Fourier-transformed into


 c j’i ¼ d


 c   c j’ih’j ¼ d

þ 12P  r:

The spin and orbital motion of unpaired electrons in an atom or ion give rise to a surrounding magnetic field BðrÞ which acts on the neutron via the magnetic potential energy ln  BðrÞ, where ln is the neutron magnetic moment. Since this is a long-range interaction, in contrast to the nuclear interaction, the magnetic scattering length p, which is proportional to the Fourier transform of the magnetic potential energy distribution ln  BðrÞ, depends on the angle of scattering. The classical relation div BðrÞ ¼ 0 shows clearly that the vector BðsÞ, which is the Fourier transform of BðrÞ, is perpendicular to the reciprocal-space vector s. If we consider the magnetic field BðrÞ as resulting from a point-like magnetic moment l at position r ¼ 0, we get

from which it is clearly seen that, unlike Pz, the polarization components Px and Py depend on the phase difference between the spinor components c and d. In a region of a vacuum in which a uniform magnetic field B is present, a neutron beam experiences a magnetic potential energy represented by the matrix 
n B 0



5.3.3.2. Magnetic scattering by a single ion having unpaired electrons

ð5:3:3:1Þ

¼ cc  dd ;




0 1

 cd : dd

ðpÞ ¼ ð2m=h2 Þ
n r  BðsÞ ¼ 
0 ð2m=h2 Þ
n r  li? ðsÞfi ðsin =Þ; ð5:3:3:5Þ

The polarization vector P is then obtained as P ¼ TrðrÞ:

where fi ðsin =Þ is the dimensionless magnetic form factor of the ion considered and tends towards a maximum value of 1 when the scattering angle  tends towards 0 (forward scattering). The value of
0 ð2m=h2 Þ
n
i is p1 ¼ 2:70
1015 m for
i ¼ 1 Bohr magneton. According to (5.3.3.4) or (5.3.3.5), there is no magnetic scattering in directions such that the scattering vector s is in the same direction as the ion magnetic moment li . Magnetic scattering effects are maximum when s and li are perpendicular. The matrix (p) is diagonal if the direction of li? ðsÞ is chosen as the spin-quantization axis. Therefore, there is no spin-flip scattering if the incident beam is polarized parallel or antiparallel to the direction of li? ðsÞ. It is more usual to choose the spin-quantization axis (Oz) along li . Let  be the angle between the vectors li and s; the

ð5:3:3:2Þ

In the common case of a nonpolarized beam, the spin-density matrix is
 ¼ 12

1 0

 0 : 1

It is easily seen that all components of P are then equal to 0. Equation (5.3.3.2) is therefore applicable to the general case (polarized, partially polarized or nonpolarized beam). The inverse relation giving the density matrix  as function of P is

656

5.3. DYNAMICAL THEORY OF NEUTRON DIFFRACTION ðx; y; zÞ components of li? ðsÞ are then (
i sin  cos , 0,
i sin2 ) if the y axis is chosen along li
s. The total scattering length, which is the sum of the nuclear and the magnetic scattering lengths, is then represented by the matrix
ðqÞ ¼

b þ p sin2  p sin  cos 

 p sin  cos  ; b  p sin2 

ð5:3:3:6Þ

where b is the nuclear scattering length and p ¼ 
0



 2m sin  sin  ¼ p ;

f
f 1 i i h2 n i i  

with
i expressed in Bohr magnetons. The relations !
 1 b þ p sin2  ðqÞ and ¼ 0 p sin  cos 

 0 p sin  cos  ðqÞ ¼ 1 b  p sin2 

Fig. 5.3.3.1. Schematic plot of the two-beam dispersion surface in the case of a purely magnetic reflection such that Qh ¼ Qh ¼ Q0 and that the angle between Q0 and Qh is equal to =4.

’ðrÞ þ k2 ’ðrÞ ¼ ½uðrÞ  r  QðrÞ’ðrÞ;

show clearly that the diagonal and the nondiagonal elements of the matrix (q) are, respectively, the spin-flip and the non-spin-flip scattering lengths. It is usual to consider the scattering cross sections, which are the measurable quantities. The cross sections for neutrons polarized parallel or antiparallel to the ion magnetic moment are ðd=dÞ ¼ b2  2bp sin2  þ ðp sin Þ2 :

ð5:3:3:10Þ

where uðrÞ and r  QðrÞ are, respectively, equal to the nuclear and the magnetic potential energies multiplied by 2m=h- 2. In the calculation of ’ðrÞ in the two-beam case, we need only three terms in the expansions of the functions uðrÞ and QðrÞ into Fourier series:

ð5:3:3:7Þ uðrÞ ¼ u0 þ uh expðih  rÞ þ uh expðih  rÞ þ . . . ; QðrÞ ¼ Q0 þ Qh expðih  rÞ þ Qh expðih  rÞ þ . . . :

These expressions are the sum of the spin-flip and non-spinflip cross sections, which are equal to ðb  p sin2 Þ2 and ðp sin  cos Þ2 , respectively. In the case of nonpolarized neutrons, the interference term ð2bp sin2 Þ between the nuclear and the magnetic scattering disappears; the cross section is then ðd=dÞ ¼ b2 þ ðp sin Þ2 :

We suppose that the crystal is magnetically saturated by an externally applied magnetic field Ha . Q0 is then proportional to the macroscopic mean magnetic field B ¼
0 ðM þ Ha þ Hd Þ, where M is the magnetization vector and Hd is the demagnetizing field. The results of Section 5.3.3.2 show that Qh and Qh are proportional to the projection of M on the reflecting plane. The four coefficients D0, Dh , E0 and Eh of (5.3.3.9) are found to satisfy a system of four homogeneous linear equations. The condition that the associated determinant has to be equal to 0 defines the dispersion surface, which is of order 4 and has four branches. An incident plane wave thus excites a system of four wavefields of the form of (5.3.3.9), generally polarized in various directions. A particular example of a dispersion surface, having an unusual shape, is shown in Fig. 5.3.3.1. This is a much more complicated situation than in the case of nonmagnetic crystals, in which one only needs to consider scalar wavefunctions which depend on two coefficients, such as D0 and Dh , and which are related to hyperbolic dispersion surfaces of order 2, as fully described in Chapter 5.1 on X-ray diffraction. In fact, all neutron experiments related to dynamical effects in diffraction by magnetic crystals have been performed under such conditions that the magnetization vector in the crystal is perpendicular to the diffraction vector h. In this case, the vectors Qh and Qh are parallel or antiparallel to the vector Q0 which is chosen as the spin-quantization axis. The matrices r  Q0, r  Qh and r  Qh are then all diagonal matrices, and we obtain for the two spin states () separate dynamical equations which are similar to the dynamical equations for the scalar case, but with different structure factors, which are either the sum or the difference of the nuclear structure factor FN and of the magnetic structure factor FM :

ð5:3:3:8Þ

In the general case of a partially polarized beam we can use the density-matrix representation. Let inc be the density matrix of the incident beam; it can be shown that the density matrix of the diffracted beam is equal to the following product of matrices: ðqÞinc ðq Þ. Using the relations between the density matrix and polarization vector presented in the preceding section, we can obtain a general description of the diffracted beam as a function of the polarization properties of the incident beam. Such a formalism is of interest for dealing with new experimental arrangements, in which a three-dimensional polarization analysis of the diffracted beam is possible, as shown by Tasset (1989). 5.3.3.3. Dynamical theory in the case of perfect ferromagnetic or collinear ferrimagnetic crystals The most direct way to develop this dynamical theory in the two-beam case, which involves a single Bragg-diffracted beam of diffraction vector h, is to consider spinor wavefunctions of the following form:

 D0 Dh ’ðrÞ ¼ expðiK0  rÞ þ exp½iðK0 þ hÞr E0 Eh

ð5:3:3:9Þ

as approximate solutions of the wave equation inside the crystal,

657

5. DYNAMICAL THEORY AND ITS APPLICATIONS Fþ ¼ FN þ FM and F ¼ FN  FM :

because its measurement involves flipping the incident-beam polarization to the opposite direction. This is an experimentally well defined quantity, because it is independent of a number of parameters such as the intensity of the incident beam, the temperature factor or the coefficient of absorption. In the case of an ideally imperfect crystal, we obtain from the kinematical expressions of the integrated reflectivities

ð5:3:3:11Þ

FN and FM are related to the scattering lengths of the ions in the unit cell of volume Vc : FN ¼ Vc uh ¼

P

bi expðih  ri Þ;




i


 P
0 m sin  expðih  ri Þ: FM ¼ Vc jQh j ¼  2
n r  li? ðhÞfi 2h  i

Rkin ðhÞ ¼ ðIþ =I Þkin

ð5:3:3:13Þ

In the case of an ideally perfect thick crystal, we obtain from the dynamical expressions of the integrated reflectivities

The dispersion surface of order 4 degenerates into two hyperbolic dispersion surfaces, each of them corresponding to one of the polarization states (). The asymptotes are different; this is related to different values of the refractive indices for neutron polarization parallel or antiparallel to Q0. In some special cases the magnitudes of FN and FM happen to be equal. Only one polarization state is then reflected. Magnetic crystals with such a property (reflections 111 of the Heusler alloy Cu2MnAl, or 200 of the alloy Co–8% Fe) are very useful as polarizing monochromators and as analysers of polarization. If the scattering vector h is in the same direction as the magnetization, this reflection is a purely nuclear one (with no magnetic contribution), since FM is then equal to 0. Purely magnetic reflections (without nuclear contribution) also exist if the magnetic structure involves several sublattices. If h is neither perpendicular to the average magnetization nor in the same direction, the presence of nondiagonal matrices in the dynamical equations cannot be avoided. The dynamical theory of diffraction by perfect magnetic crystals then takes the complicated form already mentioned. Theoretical discussions of this complicated case of dynamical diffraction have been given by Stassis & Oberteuffer (1974), Mendiratta & Blume (1976), Sivardie`re (1975), Belyakov & Bokun (1975, 1976), Schmidt et al. (1975), Bokun (1979), Guigay & Schlenker (1979a,b) and Schmidt (1983). However, to our knowledge, only limited experimental work has been carried out on this subject. Successful experiments could only be performed for the simpler cases mentioned above.

Rdyn ðhÞ ¼ ðIþ =I Þdyn ¼

jFN þ FM j : jFN  FM j

ð5:3:3:14Þ

In general, Rdyn depends on the wavelength and on the crystal thickness; these dependences disappear, as seen from (5.3.3.14), if the path length in the crystal is much larger than the extinction distances for the two polarization states. It is clear that the determination of Rkin or Rdyn allows the determination of the ratio FM =FN, hence of FM if FN is known. In fact, because real crystals are neither ideally imperfect nor ideally perfect, one usually introduces an extinction factor y (extinction is discussed below, in Section 5.3.4) in order to distinguish the real crystal reflectivity from the reflectivity of the ideally imperfect crystal. Different extinction coefficients yþ and y are actually expected for the two polarization states. This obviously complicates the task of the determination of FM =FN. In the kinematical approximation, the flipping ratio does not depend on the wavelength, in contrast to dynamical calculations for hypothetically perfect crystals (especially for the Laue case of diffraction). Therefore, an experimental investigation of the wavelength dependence of the flipping ratio is a convenient test for the presence of extinction. Measurements of the flipping ratio have been used by Bonnet et al. (1976) and by Kulda et al. (1991) in order to test extinction models. Baruchel et al. (1986) have compared nuclear and magnetic extinction in a crystal of MnP. Instead of considering only the ratio of the integrated reflectivities, it is also possible to record the flipping ratio as a function of the angular position of the crystal as it is rotated across the Bragg position. Extinction is expected to be maximum at the peak and the ratio measured on the tails of the rocking curve may approach the kinematical value. It has been found experimentally that this expectation is not of general validity, as discussed by Chakravarthy & Madhav Rao (1980). It would be valid in the case of a perfect crystal, hence in the case of pure primary extinction. It would also be valid in the case of secondary extinction of type I, but not in the case of secondary extinction of type II [following Zachariasen (1967), type II corresponds to mosaic crystals such that the diffraction pattern from each block is wider than the mosaic statistical distribution].

5.3.3.4. The dynamical theory in the case of perfect collinear antiferromagnetic crystals In this case, there is no average magnetization ðQ0 ¼ 0Þ. It is then convenient to choose the quantization axis in the direction of Qh and Qh . The dispersion surface degenerates into two hyperbolic surfaces corresponding to each polarization state along this direction for any orientation of the diffraction vector relative to the direction of the magnetic moments of the sublattices. These two hyperbolic dispersion surfaces have the same asymptotes. Furthermore, in the case of a purely magnetic reflection, they are identical. The possibility of observing a precession of the neutron polarization in the presence of diffraction, in spite of the fact that there is no average magnetization, has been pointed out by Baryshevskii (1976).

5.3.4. Extinction in neutron diffraction (nonmagnetic case) The kinematical approximation, which corresponds to the first Born approximation in scattering theory, supposes that each incident neutron can be scattered only once and therefore neglects the possibility that the neutrons may be scattered several times. Because this is a simple approximation which overestimates the crystal reflectivity, the actual reduction of reflectivity, as compared to its kinematical value, is termed extinction. This is actually a typical dynamical effect, since it is a multiplescattering effect.

5.3.3.5. The flipping ratio In polarized neutron diffraction by a magnetically saturated magnetic sample, it is usual to measure the ratio of the reflected intensities Iþ and I measured when the incident beam is polarized parallel or antiparallel to the magnetization in the sample. This ratio is called the flipping ratio, R ¼ Iþ =I ;

2 jFN þ FM j ¼ : jFN  FM j

ð5:3:3:12Þ

658

5.3. DYNAMICAL THEORY OF NEUTRON DIFFRACTION Extinction effects can be safely neglected in the case of scattering by very small crystals; more precisely, this is possible when the path length of the neutron beam in the crystal is much smaller than  ¼ Vc =F, where  is the neutron wavelength and F=Vc is the scattering length per unit volume for the reflection considered.  is sometimes called the ‘extinction distance’. A very important fact is that extinction effects also vanish if the crystal is imperfect enough, because each plane-wave component of the incident beam can then be Bragg-reflected in only a small volume of the sample. This is the extinction-free case of ‘ideally imperfect crystals’. Conversely, extinction is maximum (smallest value of y) in the case of ideally perfect non-absorbing crystals. Clearly, no significant extinction effects are expected if the crystal is thick but strongly absorbing, more precisely if the linear absorption coefficient
is such that
  1. Neutron diffraction usually corresponds to the opposite case ð
 1Þ, in which extinction effects in nearly perfect crystals dominate absorption effects. Extinction effects are usually described in the frame of the mosaic model, in which the crystal is considered as a juxtaposition of perfect blocks with different orientations. The relevance of this model to the case of neutron diffraction was first considered by Bacon & Lowde (1948). If the mosaic blocks are big enough there is extinction within each block; this is called primary extinction. Multiple scattering can also occur in different blocks if their misorientation is small enough. In this case, which is called secondary extinction, there is no phase coherence between the scattering events in the different blocks. The fact that empirical intensity-coupling equations are used in this case is based on this phase incoherence. In the general case, primary and secondary extinction effects coexist. Pure secondary extinction occurs in the case of a mosaic crystal made of very small blocks. Pure primary extinction is observed in diffraction by perfect crystals. The parameters of the mosaic model are the average size of the perfect blocks and the angular width of their misorientation distribution. The extinction theory of the mosaic model provides a relation between these parameters and the extinction coefficient, defined as the ratio of the observed reflectivity to the ideal one, which is the kinematical reflectivity in this context. In conventional work, the crystal structure factors of different reflections and the parameters of the mosaic model are fitted together to the experimental data, which are the integrated reflectivities and the angular widths of the rocking curves. In many cases, only the weakest reflections will be free, or nearly free, from extinction. The extinction corrections thus obtained can be considered as satisfactory in cases of moderate extinction. Nevertheless, extinction remains a real problem in cases of strong extinction and in any case if a very precise determination of the crystal structure factors is required. There exist several forms of the mosaic model of extinction. For instance, in the model developed by Kulda (1988a,b, 1991), the mosaic blocks are not considered just as simple perfect blocks but may be deformed perfect blocks. This has the advantage of including the case of macroscopically deformed crystals, such as bent crystals. A basically different approach, free from the distinction between primary and secondary extinction, has been proposed by Kato (1980a,b). This is a wave-optical approach starting from the dynamical equations for diffraction by deformed crystals. These so-called Takagi–Taupin equations (Takagi, 1962; Taupin, 1964) contain a position-dependent phase factor related to the displacement field of the deformed crystal lattice. Kato proposed considering this phase factor as a random function with suitably defined statistical characteristics. The wave amplitudes are then also random functions, the average of which represent the coherent wavefields while their statistical fluctuations represent the incoherent intensity fields.

Modifications to the Kato formulation have been introduced by Al Haddad & Becker (1988), by Becker & Al Haddad (1990, 1992), by Guigay (1989) and by Guigay & Chukhovskii (1992, 1995). Presently, it is not easy to apply this ‘statistical dynamical theory’ to real experiments. The widely used methods for extinction corrections are still based on the former mosaic model, according to the formulation of Zachariasen (1967), later improved by Becker & Coppens (1974a,b, 1975). As in the X-ray case, acoustic waves produced by ultrasonic excitation can artificially induce a transition from perfect to ideally imperfect crystal behaviour. The effect of ultrasound on the scattering behaviour of distorted crystals is quite complex. A good discussion with reference to neutron-scattering experiments is given by Zolotoyabko & Sander (1995). The situation of crystals with a simple distortion field is less difficult than the statistical problem of extinction. Klar & Rustichelli (1973) confirmed that the Takagi–Taupin equations, originally devised for X-rays, can be used for neutron diffraction with due account of the very small absorption, and used them for computing the effect of crystal curvature.

5.3.5. Effect of external fields on neutron scattering by perfect crystals The possibility of acting on neutrons through externally applied fields during their propagation in perfect crystals provides possibilities that are totally unknown in the X-ray case. The theory has been given by Werner (1980) using the approaches (migration of tie points, and Takagi–Taupin equations) that are customary in the treatment of imperfect crystals (see above). Zeilinger et al. (1986) pointed out that the effective-mass concept, familiar in describing electrons in solid-state physics, can shed new light on this behaviour: because of the curvature of the dispersion surface at a near-exact Bragg setting, effective masses five orders of magnitude smaller than the rest mass of the neutron in a vacuum can be obtained. Related experiments are discussed below. An interesting proposal was put forward by Horne et al. (1988) on the coupling between the Larmor precession in a homogeneous magnetic field and the spin–orbit interaction of the neutron with nonmagnetic atoms, a term which was dismissed in Section 5.3.2 because its contribution to the scattering length is two orders of magnitude smaller than that of the nuclear term. A resonance is expected to show up as highly enhanced diffracted intensity when a perfect sample is set for Bragg scattering and the magnetic field is adjusted so that the Larmor precession period is equal to the Pendello¨sung period.

5.3.6. Experimental tests of the dynamical theory of neutron scattering These experiments are less extensive for neutron scattering than for X-rays. The two most striking effects of dynamical theory for nonmagnetic nearly perfect crystals, Pendello¨sung behaviour and anomalous absorption, have been demonstrated in the neutron case too. Pendello¨sung measurement is described below (Section 5.3.7.2) because it is useful in the determination of scattering lengths. The anomalous transmission effect occurring when a perfect absorbing crystal is exactly at Bragg setting, i.e. the Borrmann effect, is often referred to in the neutron case as the suppression of the inelastic channel in resonance scattering, after Kagan & Afanas’ev (1966), who worked out the theory. A small decrease in absorption was detected in pioneering experiments on calcite by Knowles (1956) using the corresponding decrease in the emission of -rays and by Sippel et al. (1962), Shil’shtein et al. (1971) and Hastings et al. (1990) directly. Rocking curves of perfect crystals were measured by Sippel et al. (1964) in transmission, and by Kikuta et al. (1975). Integrated intensities were

659

5. DYNAMICAL THEORY AND ITS APPLICATIONS measured by Lambert & Malgrange (1982). The large angular amplification associated with the curvature of the dispersion surfaces near the exact Bragg setting was demonstrated by Kikuta et al. (1975) and by Zeilinger & Shull (1979). In magnetic crystals, the investigations have been restricted to the simpler geometry where the scattering vector is perpendicular to the magnetization, and to few materials. Pendello¨sung behaviour was evidenced through the variation with wavelength of the flipping ratio for polarized neutrons by Baruchel et al. (1982) on an yttrium iron garnet sample, with the geometry selected so that the defects would not affect the Bragg reflection used. The inclination method was used successfully by Zelepukhin et al. (1989), Kvardakov & Somenkov (1990) and Kvardakov et al. (1990a) for the weak ferromagnet FeBO3, and in the roomtemperature weak-ferromagnetic phase of hematite, -Fe2O3, by Kvardakov et al. (1990b) and Kvardakov & Somenkov (1992). Experiments on the influence of defects in nearly perfect crystals have been performed by several groups. The effect on the rocking curve was investigated by Eichhorn et al. (1967), the intensities were measured by Lambert & Malgrange (1982) and by Albertini, Boeuf, Cesini et al. (1976), and the influence on the Pendello¨sung behaviour was discussed by Kvardakov & Somenkov (1992). Boeuf & Rustichelli (1974) and Albertini et al. (1977) investigated silicon crystals curved by a thin surface silicon nitride layer. Many experiments have been performed on vibrating crystals; reviews are given by Michalec et al. (1988) and by Kulda et al. (1988). Because the velocity of neutrons is of the same order of magnitude as the velocity of acoustic phonons in crystals, the effect of ultrasonic excitation on dynamical diffraction takes on some original features compared to the X-ray case (Iolin & Entin, 1983); they could to some extent be evidenced experimentally (Iolin et al., 1986; Chalupa et al., 1986). References to experimental work on neutron scattering by imperfect crystals under ultrasonic excitation are included in Zolotoyabko & Sander (1995). Some experiments with no equivalent in the X-ray case could be performed. The very strong incoherent scattering of neutrons by protons, very different physically but similar in its effect to absorption, was also shown to lead to anomalous transmission effects by Sippel & Eichhorn (1968). Because the velocity of thermal neutrons in a vacuum is five orders of magnitude smaller than the velocity of light, the flight time for neutrons undergoing Bragg scattering in Laue geometry in a perfect crystal could be measured directly (Shull et al., 1980). The effect of externally applied fields was measured experimentally for magnetic fields by Zeilinger & Shull (1979) and Zeilinger et al. (1986). Slight rotation of the crystal, introducing a Coriolis force, was used by Raum et al. (1995), and gravity was tested recently, with the spectacular result that some states are accelerated upwards (Zeilinger, 1995).

an appropriate patterning of the surface, in analogy with the Bragg–Fresnel lenses developed for X-rays, was suggested by Indenbom (1979). The use of two identical perfect crystals in nondispersive (+, , k) setting provides a way of measuring the very narrow intrinsic rocking curves expected from the dynamical theory. Any divergence added between the two crystals can be sensitively measured. Thus perfect crystals provide interesting possibilities for measuring very small angle neutron scattering. This was performed by Takahashi et al. (1981, 1983) and Tomimitsu et al. (1986) on amorphous materials, and by Kvardakov et al. (1987) for the investigation of ferromagnetic domains in bulk silicon– iron specimens under stress, both through the variations in transmission associated with refraction on the domain walls and through small-angle scattering. Imaging applications are described in Section 5.3.7.4. Badurek et al. (1979) used the different deflection of the two polarization states provided by a magnetic prism placed between two perfect silicon crystals to produce polarized beams. Curved almost-perfect crystals or crystals with a gradient in the lattice spacing can provide focusing (Albertini, Boeuf, Lagomarsino et al., 1976) and vibrating crystals can give the possibility of tailoring the reflectivity of crystals, as well as of modulating beams in time (Michalec et al., 1988). A double-crystal arrangement with bent crystals was shown by Eichhorn (1988) to be a flexible small-angle-neutron-scattering device. 5.3.7.2. Measurement of scattering lengths by Pendello¨sung effects As in X-ray diffraction, Pendello¨sung oscillations provide an accurate way of measuring structure factors, hence coherent neutron scattering lengths. The equal-thickness fringes expected from a wedge-shaped crystal were observed by Kikuta et al. (1971). Three kinds of measurements were made. Sippel et al. (1965) measured as a function of thickness the integrated reflectivity from a perfect crystal of silicon, the thickness of which they varied by polishing after each measurement, obtaining a curve similar to Fig. 5.1.6.7, corresponding to equation 5.1.6.8. Shull (1968) restricted the measurement to wavefields that propagated along the reflecting planes, hence at exact Bragg incidence, by setting fine slits on the entrance and exit faces of 3 to 10 mm-thick silicon crystals, and measured the oscillation in diffracted intensity as he varied the wavelength of the neutrons used by rotating the crystal. Shull & Oberteuffer (1972) showed that a better interpretation of the data, when the beam is restricted to a fine slit, corresponds to the spherical wave approach (actually cylindrical wave), and the boundary conditions were discussed more generally by Arthur & Horne (1985). Somenkov et al. (1978) developed the inclination method, in which the integrated reflectivity is measured as the effective crystal thickness is varied nondestructively, by rotating the crystal around the diffraction vector, and used it for germanium. Belova et al. (1983) discuss this method in detail. The results obtained by this group for magnetic crystals are mentioned in Section 5.3.6. Structure-factor values for magnetic reflections were obtained by Kvardakov et al. (1995) for the weak ferromagnet FeBO3.

5.3.7. Applications of the dynamical theory of neutron scattering 5.3.7.1. Neutron optics Most experiments in neutron scattering require an intensityeffective use of the available beam at the cost of relatively high divergence and wavelength spread. The monochromators must then be imperfect (‘mosaic’) crystals. In some cases, however, it is important to have a small divergence and wavelength band. One example is the search for small variations in neutron energy in inelastic scattering without the use of the neutron spin-echo principle. Perfect crystals must then be used as monochromators or analysers, and dynamical diffraction is directly involved. As in the X-ray case, special designs can lead to strong decrease in the intensity of harmonics, i.e. of contributions of =2 or =3 (Hart & Rodrigues, 1978). The possibility of focusing neutron beams by the use of perfect crystals with the incident beam spatially modulated in amplitude through an absorber, or in phase through

5.3.7.3. Neutron interferometry Because diffraction by perfect crystals provides a well defined distribution of the intensity and phase of the beam, interferometry with X-rays or neutrons is possible using ingeniously designed and carefully manufactured monolithic devices carved out of single crystals of silicon. The technical and scientific features of this family of techniques are well summarized by Bonse (1979, 1988), as well as other papers in the same volumes, and by Shull (1986). X-ray interferometry started with the Bonse–Hart interferometer (Bonse & Hart, 1965). A typical device is the LLL skew-symmetric interferometer, where the L’s stand for Laue,

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5.3. DYNAMICAL THEORY OF NEUTRON DIFFRACTION indicating transmission geometry in all crystal slabs. In these slabs, which can be called the splitter, the mirrors and the recombiner, the same pair of opposite reflections, in symmetrical Laue geometry, is used three times. In the first slab, the incident beam is coherently split into a transmitted and a diffracted beam. Each of these is then diffracted in the two mirrors, and the resulting beams interfere in the recombiner, again yielding a forward-diffracted and a diffracted beam, the intensities of both of which are measured. This version, the analogue of the Mach– Zehnder interferometer in optics, offers a sizeable space (several cm of path length) where two coherent parallel beams can be submitted to various external actions. Shifting the relative phase of these beams (e.g. by , introducing an optical path-length difference of =2) results in the intensities of the outgoing beams changing from a maximum to a minimum. Applications of neutron interferometry range from the very useful to the very exotic. The most useful one is probably the measurement of coherent neutron scattering lengths. Unlike the Pendello¨sung method described in Section 5.3.7.2, this method does not require the measured samples to be perfect single crystals, nor indeed crystals. Placing a slab of material across one of the beams and rotating it will induce an optical path-length difference of ð1  nÞt if t is the effective thickness along the beam, hence a phase shift of 2ð1  nÞt=. With the expression of the refractive index n as given in Section 5.3.2.2, it is clear that for an isotopically pure material the scattering length bcoh can be deduced from the measurement of intensity versus the rotation angle of the phase shifter. This is a very versatile and much used method. The decrease in oscillation contrast can be used to obtain information of relevance to materials science, such as statistical properties of magnetic domain distributions (Korpiun, 1966) or precipitates (Rauch & Seidl, 1987); Rauch (1995) analyses the effect in terms of the neutron coherence function. Many elegant experiments have been performed with neutron interferometers in efforts to set an upper limit to effects than can be considered as nonexistent, or to test expectations of basic quantum physics. Many papers are found in the same volumes as Bonse (1979) and Bonse (1988); excellent reviews have been given by Klein & Werner (1983), Klein (1988) and Werner (1995). Among the topics investigated are the effect of gravity (Colella et al., 1975), the Sagnac effect, i.e. the influence of the Earth’s rotation (Werner et al., 1979), the Fizeau effect, i.e. the effect of the movement of the material through which the neutrons are transmitted (Arif et al., 1988) and the Aharonov–Casher effect, i.e. the dual of the Aharonov–Bohm effect for neutral particles having a magnetic moment (Cimmino et al., 1989).

section topography or neutron tomography (Schlenker et al., 1975; Davidson & Case, 1976). Neutron topography also shows the salient dynamical interference effect, viz Pendello¨sung, visually, in the form of fringes (Kikuta et al., 1971; Malgrange et al., 1976; Tomimitsu & Zeyen, 1978). Its unique feature, however, is the possibility of observing and directly characterizing inhomogeneities in the magnetic structure, i.e. magnetic domains of all kinds [ferromagnetic domains (Schlenker & Shull, 1973) and antiferromagnetic domains of various sorts (Schlenker & Baruchel, 1978), including spin-density wave domains (Ando & Hosoya, 1972, 1978; Davidson et al., 1974), 180 or time-reversed domains in some materials and helimagnetic or chirality domains (Baruchel et al., 1990)], or coexisting phases at a first-order phase transition (Baruchel, 1989). In such cases, the contrast is primarily due to local variations in the structure factor, a situation very unusual in X-ray topography, and good crystal quality, leading to dynamical scattering behaviour, is essential in the observation process only in a few cases (Schlenker et al., 1978). It is often crucial, however, for making the domain structure simple enough to be resolved, in particular in the case of antiferromagnetic domains. Imaging can also be performed for samples that need be neither crystals nor perfect. Phase-contrast imaging of a specimen through which the neutrons are transmitted can be performed in a neutron interferometer. It has been shown to reveal thickness variations by Bauspiess et al. (1978) and ferromagnetic domains by Schlenker et al. (1980). The same papers showed that phase edges show up as contrast when one of the interferometer paths is blocked, i.e. when the sample is placed effectively between perfect, identical crystals set for diffraction in a nondispersive setting. Under the name of neutron radiography with refraction contrast, this technique, essentially a form of Schlieren imaging, was further developed by Podurets, Somenkov & Shil’shtein (1989), Podurets, Somenkov, Chistyakov & Shil’shtein (1989), and Podurets et al. (1991), who were able to image internal ferromagnetic domain walls in samples 10 mm thick. References Albertini, G., Boeuf, A., Cesini, G., Mazkedian, S., Melone, S. & Rustichelli, F. (1976). A simple model for dynamical neutron diffraction by deformed crystals. Acta Cryst. A32, 863–868. Albertini, G., Boeuf, A., Klar, B., Lagomarsino, S., Mazkedian, S., Melone, S., Puliti, P. & Rustichelli, F. (1977). Dynamical neutron diffraction by curved crystals in the Laue geometry. Phys. Status Solidi A, 44, 127–136. Albertini, G., Boeuf, A., Lagomarsino, S., Mazkedian, S., Melone, S. & Rustichelli, F. (1976). Neutron properties of curved monochromators. Proceedings of the Conference on Neutron Scattering, Gatlinburg, Tennessee, USERDA CONF 760601-P2, 1151–1158. Oak Ridge, Tennessee: Oak Ridge National Laboratory. Al Haddad, M. & Becker, P. J. (1988). On the statistical dynamical theory of diffraction: application to silicon. Acta Cryst. A44, 262–270. Ando, M. & Hosoya, S. (1972). Q-switch and polarization domains in antiferromagnetic chromium observed with neutron-diffraction topography. Phys. Rev. Lett. 29, 281–285. Ando, M. & Hosoya, S. (1978). Size and behavior of antiferromagnetic domains in Cr directly observed with X-ray and neutron topography. J. Appl. Phys. 49, 6045–6051. Arif, M., Kaiser, H., Clothier, R., Werner, S. A., Berliner, R., Hamilton, W. A., Cimmino, A. & Klein, A. G. (1988). Fizeau effect for neutrons passing through matter at a nuclear resonance. Physica B, 151, 63–67. Arthur, J. & Horne, M. A. (1985). Boundary conditions in dynamical neutron diffraction. Phys. Rev. B, 32, 5747–5752. Bacon, G. E. & Lowde, R. D. (1948). Secondary extinction and neutron crystallography. Acta Cryst. 1, 303–314. Badurek, G., Rauch, H., Wilfing, A., Bonse, U. & Graeff, W. (1979). A perfect-crystal neutron polarizer as an application of magnetic prism refraction. J. Appl. Cryst. 12, 186–191. Baruchel, J. (1989). The contribution of neutron and synchrotron radiation topography to the investigation of first-order magnetic phase transitions. Phase Transit. 14, 21–29.

5.3.7.4. Neutron diffraction topography and other imaging methods These are the neutron form of the ‘topographic’ or diffraction imaging techniques, in which an image of a single crystal is obtained through the local variations in Bragg-diffracted intensity due to inhomogeneities in the sample. It is briefly described in Chapter 2.8 of IT C. It was pioneered by Doi et al. (1971) and by Ando & Hosoya (1972). Like its X-ray counterpart, neutron topography can reveal isolated defects, such as dislocations (Schlenker et al., 1974; Malgrange et al., 1976). Because of the small neutron fluxes available, it is not very convenient for this purpose, since the resolution is poor or the exposure times are very long. On the other hand, the very low absorption of neutrons in most materials makes it quite convenient for observing the gross defect structure in samples that would be too absorbing for X-rays (Tomimitsu & Doi, 1974; Baruchel et al., 1978; Tomimitsu et al., 1983; Kvardakov et al., 1992), or the spatial modulation of distortion due to vibration, for example in quartz (Michalec et al., 1975), and resonant magnetoelastic effects (Kvardakov & Somenkov, 1991). In particular, virtual slices of bulky as-grown samples can be investigated without cutting them using neutron

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5. DYNAMICAL THEORY AND ITS APPLICATIONS Baruchel, J., Guigay, J. P., Mazure´-Espejo, C., Schlenker, M. & Schweizer, J. (1982). Observation of Pendello¨sung effect in polarized neutron scattering from a magnetic crystal. J. Phys. 43, C7, 101–106. Baruchel, J., Patterson, C. & Guigay, J. P. (1986). Neutron diffraction investigation of the nuclear and magnetic extinction in MnP. Acta Cryst. A42, 47–55. Baruchel, J., Schlenker, M. & Palmer, S. B. (1990). Neutron diffraction topographic investigations of “exotic” magnetic domains. Nondestruct. Test. Eval. 5, 349–367. Baruchel, J., Schlenker, M., Zarka, A. & Pe´troff, J. F. (1978). Neutron diffraction topographic investigation of growth defects in natural lead carbonate single crystals. J. Cryst. Growth, 44, 356–362. Baryshevskii, V. G. (1976). Particle spin precession in antiferromagnets. Sov. Phys. Solid State, 18, 204–208. Bauspiess, W., Bonse, U., Graeff, W., Schlenker, M. & Rauch, H. (1978). Result shown in Bonse (1979). Becker, P. & Al Haddad, M. (1990). Diffraction by a randomly distorted crystal. I. The case of short-range order. Acta Cryst. A46, 123–129. Becker, P. & Al Haddad, M. (1992). Diffraction by a randomly distorted crystal. II. General theory. Acta Cryst. A48, 121–134. Becker, P. J. & Coppens, P. (1974a). Extinction within the limit of validity of the Darwin transfer equations. I. General formalisms for primary and secondary extinction and their application to spherical crystals. Acta Cryst. A30, 129–147. Becker, P. J. & Coppens, P. (1974b). Extinction within the limit of validity of the Darwin transfer equations. II. Refinement of extinction in spherical crystals of SrF2 and LiF. Acta Cryst. A30, 148–153. Becker, P. J. & Coppens, P. (1975). Extinction within the limit of validity of the Darwin transfer equations. III. Non-spherical crystals and anisotropy of extinction. Acta Cryst. A31, 417–425. Belova, N. E., Eichhorn, F., Somenkov, V. A., Utemisov, K. & Shil’shtein, S. Sh. (1983). Analyse der Neigungsmethode zur Untersuchung von Pendello¨sungsinterferenzen von Neutronen und Ro¨ntgenstrahlen. Phys. Status Solidi A, 76, 257–265. Belyakov, V. A. & Bokun, R. Ch. (1975). Dynamical theory of neutron diffraction by perfect antiferromagnetic crystals. Sov. Phys. Solid State, 17, 1142–1145. Belyakov, V. A. & Bokun, R. Ch. (1976). Dynamical theory of neutron diffraction in magnetic crystals. Sov. Phys. Solid State, 18, 1399– 1402. Boeuf, A. & Rustichelli, F. (1974). Some neutron diffraction experiments on curved silicon crystals. Acta Cryst. A30, 798–805. Bokun, R. Ch. (1979). Beats in the integrated intensity in neutron diffraction by perfect magnetic crystals. Sov. Phys. Tech. Phys. 24, 723– 724. Bonnet, M., Delapalme, A., Becker, P. & Fuess, H. (1976). Polarised neutron diffraction – a tool for testing extinction models: application to yttrium iron garnet. Acta Cryst. A32, 945–953. Bonse, U. (1979). Principles and methods of neutron interferometry. In Neutron Interferometry: Proceedings of an International Workshop, edited by U. Bonse & H. Rauch, pp. 3–33. Oxford: Clarendon Press. Bonse, U. (1988). Recent advances in X-ray and neutron interferometry. Physica B, 151, 7–21. Bonse, U. & Hart, M. (1965). An X-ray interferometer. Appl. Phys. Lett. 6, 155–156. Chakravarthy, R. & Madhav Rao, L. (1980). A simple method to correct for secondary extinction in polarised-neutron diffractometry. Acta Cryst. A36, 139–142. Chalupa, B., Michalec, R., Horalik, L. & Mikula, P. (1986). The study of neutron acoustic effect by neutron diffraction on InSb single crystal. Phys. Status Solidi A, 97, 403–409. Cimmino, A., Opat, G. I., Klein, A. G., Kaiser, H., Werner, S. A., Arif, M. & Clothier, R. (1989). Observation of the topological Aharonov– Casher phase shift by neutron interferometry. Phys. Rev. Lett. 63, 380– 383. Colella, R., Overhauser, A. W. & Werner, S. A. (1975). Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472– 1474. Davidson, J. B. & Case, A. L. (1976). Applications of the fly’s eye neutron camera: diffraction tomography and phase transition studies. Proceedings of the Conference on Neutron Scattering, Gatlinburg, Tennessee, USERDA CONF 760601-P2, 1124–1135. Oak Ridge, Tennessee: Oak Ridge National Laboratory. Davidson, J. B., Werner, S. A. & Arrott, A. S. (1974). Neutron microscopy of spin density wave domains in chromium. Proceedings of the 19th

Annual Conference on Magnetism and Magnetic Materials, edited by C. D. Graham and J. J. Rhyne. AIP Conf. Proc. 18, 396–400. Doi, K., Minakawa, N., Motohashi, H. & Masaki, N. (1971). A trial of neutron diffraction topography. J. Appl. Cryst. 4, 528–530. Eichhorn, F. (1988). Perfect crystal neutron optics. Physica B, 151, 140– 146. Eichhorn, F., Sippel, D. & Kleinstu¨ck, K. (1967). Influence of oxygen segregations in silicon single crystals on the halfwidth of the doublecrystal rocking curve of thermal neutrons. Phys. Status Solidi, 23, 237– 240. Guigay, J. P. (1989). On integrated intensities in Kato’s statistical diffraction theory. Acta Cryst. A45, 241–244. Guigay, J. P. & Chukhovskii, F. N. (1992). Reformulation of the dynamical theory of coherent wave propagation by randomly distorted crystals. Acta Cryst. A48, 819–826. Guigay, J. P. & Chukhovskii, F. N. (1995). Reformulation of the statistical theory of dynamical diffraction in the case E = 0. Acta Cryst. A51, 288– 294. Guigay, J. P. & Schlenker, M. (1979a). Integrated intensities and flipping ratios in neutron diffraction by perfect magnetic crystals. In Neutron Interferometry, edited by U. Bonse & H. Rauch, pp. 135–148. Oxford: Clarendon Press. Guigay, J. P. & Schlenker, M. (1979b). Spin rotation of the forward diffracted beam in neutron diffraction by perfect magnetic crystals. J. Magn. Magn. Mater. 14, 340–343. Hart, M. & Rodrigues, A. R. D. (1978). Harmonic-free single-crystal monochromators for neutrons and X-rays. J. Appl. Cryst. 11, 248– 253. Hastings, J. B., Siddons, D. P. & Lehmann, M. (1990). Diffraction broadening and suppression of the inelastic channel in resonant neutron scattering. Phys. Rev. Lett. 64, 2030–2033. Horne, M. A., Finkelstein, K. D., Shull, C. G., Zeilinger, A. & Bernstein, H. J. (1988). Neutron spin – Pendello¨sung resonance. Physica B, 151, 189–192. Indenbom, V. L. (1979). Diffraction focusing of neutrons. JETP Lett. 29, 5–8. International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers. Iolin, E. M. & Entin, I. R. (1983). Dynamic diffraction of neutrons by high-frequency acoustic waves in perfect crystals. Sov. Phys. JETP, 58, 985–989. Iolin, E. M., Zolotoyabko, E. V., Raı¨tman, E. A., Kuvdaldin, B. V. & Gavrilov, V. N. (1986). Interference effects in dynamic neutron diffraction under conditions of ultrasonic excitation. Sov. Phys. JETP, 64, 1267–1271. Kagan, Yu. & Afanas’ev, A. M. (1966). Suppression of inelastic channels in resonance scattering of neutrons in regular crystals. Sov. Phys. JETP, 22, 1032–1040. Kato, N. (1980a). Statistical dynamical theory of crystal diffraction. I. General formulation. Acta Cryst. A36, 763–769. Kato, N. (1980b). Statistical dynamical theory of crystal diffraction. II. Intensity distribution and integrated intensity in the Laue cases. Acta Cryst. A36, 770–778. Kikuta, S., Ishikawa, I., Kohra, K. & Hoshino, S. (1975). Studies on dynamical diffraction phenomena of neutrons using properties of wave fan. J. Phys. Soc. Jpn, 39, 471–478. Kikuta, S., Kohra, K., Minakawa, N. & Doi, K. (1971). An observation of neutron Pendello¨sung fringes in a wedge-shaped silicon single crystal. J. Phys. Soc. Jpn, 31, 954–955. Klar, B. & Rustichelli, F. (1973). Dynamical neutron diffraction by ideally curved crystals. Nuovo Cimento B, 13, 249–271. Klein, A. G. (1988). Schro¨dinger inviolate: neutron optical searches for violations of quantum mechanics. Physica B, 151, 44–49. Klein, A. G. & Werner, S. A. (1983). Neutron optics. Rep. Prog. Phys. 46, 259–335. Knowles, J. W. (1956). Anomalous absorption of slow neutrons and X-rays in nearly perfect single crystals. Acta Cryst. 9, 61–69. Korpiun, P. (1966). Untersuchung ferromagnetischer Strukturen mit einem Zweistrahl-Neutroneninterferometer. Z. Phys. 195, 146–170. Kulda, J. (1988a). The RED extinction model. I. An upgraded formalism. Acta Cryst. A44, 283–285. Kulda, J. (1988b). The RED extinction model. II. Refinement of extinction and thermal vibration parameters for SrF2 crystals. Acta Cryst. A44, 286–290.

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Kulda, J. (1991). The RED extinction model. III. The case of pure primary extinction. Acta Cryst. A47, 775–779. Kulda, J., Baruchel, J., Guigay, J.-P. & Schlenker, M. (1991). Extinction effects in polarized neutron diffraction from magnetic crystals. I. Highly perfect MnP and YIG samples. Acta Cryst. A47, 770–775. Kulda, J., Vra´na, M. & Mikula, P. (1988). Neutron diffraction by vibrating crystals. Physica B, 151, 122–129. Kvardakov, V. V., Podurets, K. M., Baruchel, J. & Sandonis, J. (1995). Precision determination of the structure factors of magnetic neutron scattering from the Pendello¨sung data. Crystallogr. Rep. 40, 330– 331. Kvardakov, V. V., Podurets, K. M., Chistyakov, R. R., Shil’shtein, S. Sh., Elyutin, N. O., Kulidzhanov, F. G., Bradler, J. & Kadecˇkova´, S. (1987). Modification of the domain structure of a silicon–iron single crystal as a result of uniaxial stretching. Sov. Phys. Solid State, 29, 228–232. Kvardakov, V. V. & Somenkov, V. A. (1990). Observation of dynamic oscillations of intensity of magnetic scattering of neutrons with variation of the orientation of magnetic moments of the sublattices. Sov. Phys. Crystallogr. 35, 619–622. Kvardakov, V. V. & Somenkov, V. A. (1991). Neutron diffraction study of nonlinear magnetoacoustic effects in perfect crystals of FeBO3 and

-Fe2O3. J. Moscow Phys. Soc. 1, 33–57. Kvardakov, V. V. & Somenkov, V. A. (1992). Magnetic Pendello¨sung effects in neutron scattering by perfect magnetic crystals. Acta Cryst. A48, 423–430. Kvardakov, V. V., Somenkov, V. A. & Shil’shtein, S. Sh. (1990a). Observation of dynamic oscillations in the temperature dependence of the intensity of the magnetic scattering of neutrons. Sov. Phys. Solid State, 32, 1097–1098. Kvardakov, V. V., Somenkov, V. A. & Shil’shtein, S. Sh. (1990b). Influence of an orientational magnetic transition in -Fe2O3 on the Pendello¨sung fringe effect in neutron scattering. Sov. Phys. Solid State, 32, 1250–1251. Kvardakov, V. V., Somenkov, V. A. & Shil’shtein, S. Sh. (1992). Study of defects in cuprate single crystals by the neutron topography and selective etching methods. Superconductivity, 5, 623–629. Lambert, D. & Malgrange, C. (1982). X-ray and neutron integrated intensity diffracted by perfect crystals in transmission. Z. Naturforsch. Teil A, 37, 474–484. Malgrange, C., Pe´troff, J. F., Sauvage, M., Zarka, A. & Englander, M. (1976). Individual dislocation images and Pendello¨sung fringes in neutron topographs. Philos. Mag. 33, 743–751. Mendiratta, S. K. & Blume, M. (1976). Dynamical theory of thermal neutron scattering. I. Diffraction from magnetic crystals. Phys. Rev. 14, 144–154. Michalec, R., Mikula, P., Sedla´kova´, L., Chalupa, B., Zelenka, J., Petrzˇ´ılka, V. & Hrdlicˇka, Z. (1975). Effects of thickness-shear vibrations on neutron diffraction by quartz single crystals. J. Appl. Cryst. 8, 345–351. Michalec, R., Mikula, P., Vra´na, M., Kulda, J., Chalupa, B. & Sedla´kova´, L. (1988). Neutron diffraction by perfect crystals excited into mechanical resonance vibrations. Physica B, 151, 113–121. Podurets, K. M., Sokol’skii, D. V., Chistyakov, R. R. & Shil’shtein, S. Sh. (1991). Reconstruction of the bulk domain structure of silicon iron single crystals from neutron refraction images of internal domain walls. Sov. Phys. Solid State, 33, 1668–1672. Podurets, K. M., Somenkov, V. A., Chistyakov, R. R. & Shil’shtein, S. Sh. (1989). Visualization of internal domain structure of silicon iron crystals by using neutron radiography with refraction contrast. Physica B, 156–157, 694–697. Podurets, K. M., Somenkov, V. A. & Shil’shtein, S. Sh. (1989). Neutron radiography with refraction contrast. Physica B, 156–157, 691–693. Rauch, H. (1995). Towards interferometric Fourier spectroscopy. Physica B, 213–214, 830–832. Rauch, H. & Petrascheck, D. (1978). Dynamical neutron diffraction and its application. In Neutron Diffraction, edited by H. Dachs, Topics in Current Physics, Vol. 6 pp. 305–351. Berlin: Springer. Rauch, H. & Seidl, E. (1987). Neutron interferometry as a new tool in condensed matter research. Nucl. Instrum. Methods A, 255, 32–37. Raum, K., Koellner, M., Zeilinger, A., Arif, M. & Ga¨hler, R. (1995). Effective-mass enhanced deflection of neutrons in noninertial frames. Phys. Rev. Lett. 74, 2859–2862. Scherm, R. & Fa˚k, B. (1993). Neutrons. In Neutron and Synchrotron Radiation for Condensed Matter Studies (HERCULES Course), Vol. 1, edited by J. Baruchel, J. L. Hodeau, M. S. Lehmann, J. R. Regnard & C.

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664

references

Author index Entries refer to chapter number. Abad-Zapatero, C., 2.3, 3.3 Abdel-Meguid, S. S., 2.3, 3.3 Abi-Ezzi, S. S., 3.3 Ablov, A. V., 2.5 Abrahams, J. P., 2.3 Abrahams, S. C., 2.3, 2.4 Abramowitz, M., 2.1 Achard, M. F., 4.4 Acharya, R., 2.3 Adams, B. W., 5.1 Adams, M. J., 2.3, 2.4 Adams, P. D., 2.3, 2.5 Adamson, R. D., 3.5 Addison, A. W., 2.3 Adiga, P. S., 2.5 Adlhart, W., 4.2 Aeppli, G., 4.2, 4.4 Afanas’ev, A. M., 5.3 Agard, D. A., 2.3, 2.4 Agarwal, R. C., 1.3, 2.4 Agrawal, R. K., 2.5 Aguado, A., 3.5 Aharonov, Y., 3.3 Aharony, A., 4.4 Ahlfors, L. V., 1.3 Ahmed, F. R., 1.3 ˚ kervall, K., 2.3 A Akhiezer, N. I., 1.3 Akimoto, T., 2.3 Akishige, Y., 2.5 Al Haddad, M., 5.3 Alben, R., 4.4 Albertini, G., 5.3 Alden, R. A., 1.3 Alexander, L. E., 4.2 Alexeev, D. G., 4.5 Alford, J. A., 3.5 Al-Khayat, H. A., 4.5 Allegra, G., 2.2 Allen, F. H., 3.3 Allen, L. J., 4.3 Allen, M. P., 3.5 Allinger, N. L., 3.3 Almer, J., 4.2 Als-Nielsen, J., 4.4 Alston, N. A., 1.3 Altermatt, U. D., 1.4 Altmann, S. L., 1.5 Altomare, A., 2.2 Altona, C., 3.3 Alzari, P. M., 5.1 Amador, S., 4.4 Amelinckx, S., 4.3 Amma, E. L., 3.3 Ammon, H. L., 3.5 Amoro´s, J. L., 4.2 Amoro´s, M., 4.2 Amos, L. A., 2.5 An, M., 1.3 Anderson, D. C., 3.3 Anderson, D. L., 2.3 Anderson, P. W., 4.4 Anderson, S., 3.3 Andersson, B., 4.3 Andersson, G., 4.4 Ando, M., 5.3 Andreeva, N. S., 2.4 Andrews, J. W., 2.5 Andries, K., 2.3 Angert, I., 2.5 Angress, J. F., 4.1 Anzenhofer, K., 2.2 Apostol, T. M., 1.3 Arai, M., 4.2 Ardito, G., 2.2 Arfken, G., 1.2, 3.4, 3.5 Argos, P., 2.2, 2.3 Arif, M., 5.3 Arley, N., 3.2

Arnautova, Y. A., 3.5 Arndt, U. W., 2.4, 4.2 Arnold, D. B., 3.3 Arnold, E., 2.3 Arnold, H., 1.1, 1.3 Arnott, S., 4.5 Aroyo, M. I., 1.4, 1.5, 4.2 Arrott, A. S., 5.3 Arthur, J., 5.3 Artin, E., 1.3 Ascher, E., 1.3 Ash, J. M., 1.3 Ashcroft, N. W., 1.1 Ashida, T., 2.4 Atkins, E. D. T., 4.5 Atwood, D. K., 5.3 Atwood, J. L., 3.3 Au, A. Y., 2.5, 4.2 Audier, M., 4.2 Auslander, L., 1.3 Authier, A., 1.4, 5.1 Avery, J., 1.2 Avilov, A. S., 2.5 Avraham, D. ben, 4.5 Avrami, M., 2.2 Axe, J. D., 4.2 Axel, F., 4.6 Ayoub, R., 1.3 Bacon, G. E., 5.3 Badasso, M. O., 2.3 Badurek, G., 5.3 Baer, E., 4.5 Bagchi, S. N., 2.3, 4.2, 4.5 Baggio, R., 2.2 Baird, T., 2.5 Bajaj, C., 2.5 Bak, H. J., 2.3 Bak, P., 4.4 Baker, D., 2.3 Baker, E. N., 1.3, 2.3 Baker, T. S., 2.3, 2.5 Baldwin, J. M., 2.5 Baldwin, P. R., 2.5 Balibar, F., 5.1 Ban, N., 2.5 Banaszak, L. J., 2.3 Bancel, P. A., 4.6 Banerjee, K., 2.2 Bannister, C., 2.5 Bansal, M., 1.3 Bantz, D., 1.3 Bar, J., 2.5 Barakat, R., 1.3, 2.1 Barbour, L. J., 3.3 Bardhan, P., 4.2 Barham, P. J., 4.5 Barnea, Z., 1.2 Barnes, W. H., 1.3 Barois, P., 4.4 Barrett, A. N., 1.3 Barrington Leigh, J., 4.5 Barry, C. D., 3.3 Bartels, K., 1.3, 2.3, 2.4 Baruchel, J., 5.3 Baryshevskii, V. G., 5.3 Barzaghi, M., 3.5 Basett-Jones, D. P., 2.5 Bash, P. A., 3.3 Baskaran, S., 4.5 Basokur, A. T., 2.5 Batterman, B. W., 5.1 Baturic-Rubcic, J., 4.4 Bauer, G., 4.2 Bauer, J., 3.5 Bauer, P., 3.5 Baumeister, W., 2.5 Bauspiess, W., 5.3 Baxter, W., 2.5

Bazterra, V. E., 3.5 Beaglehole, D., 4.4 Bean, A. J., 2.5 Beck, T. L., 3.5 Becker, P. J., 1.2, 5.3 Beckmann, E., 2.5 Beddell, C., 3.3 Bedzyk, M. J., 5.1 Beer, T. de, 2.5 Beevers, C. A., 1.3, 2.3 Beintema, J. J., 2.3 Bellamy, H. D., 3.3 Bellamy, K., 4.5 Bellard, S., 3.3 Bellen, H., 2.5 Bellissent, R., 4.2 Bellman, R., 1.3 Bellocq, A. M., 4.4 Bellon, P. L., 2.5 Belova, N. E., 5.3 Belyakov, V. A., 5.3 Benattar, J. J., 4.4 Bender, R., 2.5 Bendersky, L. A., 2.5 Bengtsson, U., 2.3 Benguigui, L., 4.4 Bennett, D. W., 3.3 Bennett, J. M., 1.3 Bensimon, D., 4.4 Bentley, J., 1.2 Berberian, S. K., 1.3 Berendzen, J., 2.3 Berger, J. E., 2.5 Bergman, G., 3.2 Bergmann, K., 4.4 Berkowitz, M. L., 3.5 Berliner, R., 5.3 Berman, H. M., 1.4 Bern, M., 2.5 Bernstein, F. C., 3.3 Bernstein, H. J., 5.3 Bernstein, J., 3.5 Bernstein, S., 2.1 Berry, M. V., 5.2 Bertaut, E. F., 1.3, 1.4, 2.2, 3.4 Berthold, T., 4.2 Berthou, J., 2.3 Bessie`re, M., 4.2 Bethe, H. A., 1.2, 2.5, 5.2 Bethge, P. H., 3.3 Beurskens, G., 2.2 Beurskens, P. T., 2.2, 2.3 Beyeler, H. U., 4.2 Bhat, T. N., 2.3, 2.4 Bhattacharya, R. N., 1.3 Bhuiya, A. K., 2.2 Bieberbach, L., 1.3 Biemond, J., 2.5 Bienenstock, A., 1.3, 1.4 Bijvoet, J. M., 2.2, 2.3, 2.4 Bilderback, D. H., 5.1 Billard, J., 4.4 Billard, L., 4.6 Billinge, S. J. L., 4.2 Bilz, H., 4.1 Bing, D. H., 3.3 Bird, D., 2.5 Birgeneau, R. J., 4.4 Biswas, A., 4.5 Blackman, M., 2.5, 4.1, 5.2 Blackwell, J., 4.5 Blahut, R. E., 1.3 Blake, A. J., 3.3 Blanc, E., 2.3 Blaser, H., 2.3 Blech, I., 2.5, 4.6 Bleistein, N., 1.3 Blessing, R. H., 2.2 Bley, F., 4.2

665

Blinc, R., 4.4 Bloch, F., 1.1, 4.1 Bloomer, A. C., 1.3, 2.3, 3.3 Blow, D. M., 1.3, 2.2, 2.3, 2.4 Bluhm, M. M., 2.3 Blume, J., 5.2 Blume, M., 5.3 Blundell, D. J., 4.5 Blundell, T. L., 2.3, 2.4 Bochner, S., 1.3 Bode, W., 1.3, 2.4 Bodo, G., 2.3 Boer, J. L. de, 4.2 Boerrigter, S. X. M., 3.5 Boettger, J. C., 3.5 Boeuf, A., 5.3 Bo¨hm, H., 4.6 Bohm, J., 2.5 Bo¨hme, R., 2.2 Bo¨hringer, M., 5.1 Boisset, N., 2.5 Boissieu, M. de, 4.2 Bokhoven, C., 2.4 Bokun, R. Ch., 5.3 Bommel, A. J. van, 2.3, 2.4 Bondot, P., 1.3 Bo¨ni, P., 4.2 Bonneau, P. R., 2.3 Bonnet, M., 5.3 Bono, P. R., 3.3 Bonse, U., 5.1, 5.3 Boon, M., 1.5 Booth, A. D., 1.3 Boots, B., 2.5 Borell, A., 2.3 Borie, B., 4.2 Born, M., 1.2, 1.3, 4.1, 5.1, 5.2 Borrmann, G., 5.1 Bosman, W. P., 2.2 Botha, J. D., 3.3 Bo¨ttcher, B., 2.5 Bo¨ttcher, C. J. F., 3.5 Boublik, M., 2.5 Bouckaert, L. P., 1.5 Boudard, M., 4.2 Boulay, D. J. du, 1.4, 2.2 Bouman, J., 2.2 Bourne, P. E., 1.4 Boutin, H., 4.1 Bouwman, W. G., 4.4 Bowen, D. K., 5.1 Bown, M., 4.2 Boyd, D. B., 3.3 Boyer, L., 4.1 Boyer, P. L., 2.3 Boyle, L. L., 1.5 Boysen, H., 4.2 Bracewell, R. N., 1.3, 2.5 Bradler, J., 5.3 Bradley, A. J., 2.4 Bradley, C. J., 1.5, 4.2 Bragg, L., 1.4 Bragg, W. H., 1.3 Bragg, W. L., 1.3, 2.3, 5.1 Braig, K., 2.5 Bra¨mer, R., 4.2 Brand, P., 4.4 Brandenburg, N. P., 3.3 Braslau, A., 4.4 Braun, H., 3.3 Braun, P. B., 2.3 Bremermann, H., 1.3 Bremmer, H., 1.3 Brennan, S., 5.1 Brice, M. D., 1.3, 3.3 Bricogne, G., 1.3, 2.2, 2.3, 2.5, 3.3, 4.5 Brigham, E. O., 1.3 Brill, R., 1.3 Brinkman, W. F., 4.4

AUTHOR INDEX Brisbin, D., 4.4 Brisse, F., 2.5, 4.5 Britten, P. L., 1.3, 2.2 Broach, R. W., 4.2 Brock, J. D., 4.4 Brockhouse, B. N., 4.1 Broderson, S., 3.5 Brooks, B. R., 3.3 Brooks, C. L. III, 2.5 Brooks, J. D., 4.4 Brown, C. J., 4.5 Brown, D., 3.5 Brown, F., 2.3 Brown, G. S., 4.4 Brown, H., 1.3 Brown, I. D., 1.4 Brown, M. D., 3.3 Brucolleri, R. E., 3.3 Bruijn, N. G. de, 1.3 Bruins, E. M., 4.5 Bruins Slot, H. J., 2.2 Bruinsma, R., 4.4 Bru¨mmer, O., 5.1 Brunger, A. T., 2.2, 2.3 Bru¨nger, A. T., 1.3, 2.3, 4.5 Bruno, I. J., 3.3 Bryan, R. K., 1.3, 4.5 Bu, X., 4.2, 4.6 Bubeck, E., 4.2 Buch, K. R., 3.2 Budai, J., 4.4 Budinger, T. F., 2.5 Buehner, M., 2.3 Buerger, M. J., 1.1, 1.4, 2.2, 2.3 Bujosa, A., 1.3 Bullough, R. K., 2.3 Bu¨low, R., 1.3 Bunko´czi, G., 2.3 Bunn, C. W., 4.5 Bunshaft, A. J., 3.3 Burandt, B., 4.2, 4.6 Burch, S. F., 1.3 Burd, C. G., 2.5 Burdina, V. I., 2.3 Burgess, W. G., 2.5 Bu¨rgi, H.-B., 3.3, 4.2 Burkel, E., 4.1 Burkert, U., 3.3 Burkov, S., 4.6 Burla, M. C., 2.2 Burnett, M. N., 3.3 Burnett, R. M., 1.3, 2.3 Burnside, W., 1.3 Burrus, C. S., 1.3 Busetta, B., 2.2 Busing, W. R., 1.3, 3.1, 3.4 Bussler, P., 2.5 Butcher, S. J., 2.5 Butler, B. D., 4.2 Buttle, K., 2.5 Buxton, B., 2.5, 5.2 Buyers, W. J. L., 4.1 Byerly, W. E., 1.3 Byler, M. A., 4.5 Bystroff, C., 2.3 Cael, J. J., 4.5 Caglioti, G., 4.2 Cahn, J. W., 2.5, 4.6 Caille´, A., 4.4 Calabrese, G., 2.2 Caliandro, R., 2.2 Calvayrac, Y., 4.2 Camalli, M., 2.2 Cambillau, C., 3.3 Campagnari, F., 4.5 Campbell, B. J., 4.2 Campbell, G. A., 1.3 Campbell Smith, P. J., 4.5 Cannillo, E., 2.4 Canright, G. S., 4.2 Cantino, M., 4.5 Capillas, C., 1.4, 1.5 Carathe´odory, C., 1.3

Carazo, J. M., 2.5 Cardona, M., 4.1 Carlile, C. J., 4.4 Carlisle, C. H., 2.3 Carlson, J. M., 4.4 Carragher, B., 2.5 Carrascosa, J. L., 2.5 Carroll, C. E., 4.5 Carrozzini, B., 2.2 Carslaw, H. S., 1.3 Cartan, H., 1.3 Carter, R. E., 3.3 Cartwright, B. A., 3.3 Cascarano, G. L., 2.2 Case, A. L., 5.3 Casher, A., 1.5 Caspar, D. L. D., 2.3, 4.5 Castellano, E. E., 2.2 Cavicchi, E., 3.3 Cenedese, P., 4.2 Cesini, G., 5.3 Ceska, T. A., 2.5 Chacko, K. K., 2.4 Chakravarthy, R., 5.3 Challacombe, M., 3.5 Challifour, J. L., 1.3 Chalupa, B., 5.3 Champeney, D. C., 1.3 Champness, J. N., 1.3, 2.3, 3.3 Champness, N. R., 3.3 Chan, A. S., 2.5 Chan, D. S. K., 1.3 Chan, K. K., 4.4 Chandrasekaran, R., 4.5 Chandrasekhar, S., 4.4 Chang, G., 2.3 Chang, S.-L., 5.1 Chanzy, H., 2.5, 4.5 Chao, Y., 2.5 Chao-de, Z., 2.2 Chaplot, S. L., 4.1 Chapman, D., 4.4 Chapman, M. S., 2.3 Chapon, L. C., 3.3 Chapuis, G., 4.6 Charvolin, J., 4.4 Chataka, T., 4.2 Cheetham, A. K., 4.2 Chemburkar, S. R., 3.5 Chen, H. S., 4.2 Chen, J., 2.5 Chen, J. H., 4.4 Chen, S., 2.5 Cheng, N., 2.5 Cheng, R. H., 2.5 Cheng, T. Z., 2.5 Cherns, D., 2.5 Chew, M., 4.5 Chiang, L. Y., 4.4 Chin, C., 2.5 Chisholm, J., 3.5 Chistyakov, R. R., 5.3 Chiu, S. N., 2.5 Chiu, W., 2.5 Chivers, R. A., 4.5 Choi, H.-K., 2.3 Choplin, F., 3.3 Chow, M., 2.3 Christensen, F., 4.4 Christian, P. E., 2.5 Chu, K. C., 4.4 Chukhovskii, F. N., 5.1, 5.3 Church, G. M., 1.3, 2.4 Churchill, R. V., 1.3 Cimmino, A., 5.3 Cisarova, I., 4.2, 4.6 Cisneros, G. A., 3.5 Civalleri, B., 3.5 Clapp, P. C., 4.3 Clark, A. D. Jr, 2.3 Clark, E. S., 4.5 Clark, J. J., 2.5 Clark, N. A., 4.4 Clark, P., 2.3

Clarke, P. J., 4.2 Clarke, R., 4.4 Clastre, J., 2.3 Clausen, K. N., 4.2 Clementi, E., 1.2 Clews, C. J. B., 1.3, 3.2 Clore, G. M., 2.3 Clothier, R., 5.3 Coates, D., 4.4 Cochran, W., 1.1, 1.2, 1.3, 1.4, 2.2, 2.3, 2.5, 3.2, 4.1, 4.5 Cockcroft, J. K., 3.3 Cockrell, P. R., 3.3 Cohen, D., 2.5 Cohen, J. B., 4.2 Cohen, N. C., 3.3 Cohen-Tannoudji, C., 1.2 Cole, H., 5.1 Cole, J. C., 3.3 Colella, R., 5.3 Colin, P., 3.3 Collar, A. R., 5.2 Coller, E., 2.2, 2.3 Collett, J., 4.4 Collins, D. M., 1.3, 2.2, 2.3, 2.4, 3.3 Collongues, R., 4.2 Colman, P. M., 1.3, 2.3 Comarmond, M. B., 2.3 Comes, R., 4.2 Condon, E. V., 1.2 Connell, S. R., 2.5 Connolly, M. L., 3.3 Conradi, E., 4.2 Convert, P., 4.2 Conway, J. F., 2.5 Cooley, J. W., 1.3 Cooper, M. J., 4.2 Coppens, P., 1.2, 3.5, 4.2, 4.6, 5.3 Cordes, A. W., 3.2 Cordingley, M. G., 2.3 Corey, R. B., 1.3, 2.3 Corfield, P. W. R., 2.3 Cork, J. M., 2.4 Corones, J., 5.2 Coster, D., 2.4 Cotton, F. A., 3.3 Couch, G. S., 2.5 Coulson, C. A., 1.2 Coulter, C. L., 2.2 Courville, D. A., 2.3 Courville-Brenasin, J. de, 4.2 Cowan, P. L., 5.1 Cowan, S. W., 2.5, 3.3 Cowley, J. M., 2.5, 4.2, 4.3, 4.5, 5.2 Cowtan, K., 2.3 Cox, E. G., 1.3 Cox, J. M., 2.3, 2.4 Cox, S. R., 3.5 Coxeter, H. S. M., 1.3 Cracknell, A. P., 1.5, 4.2 Crame´r, H., 1.3, 2.1, 2.5 Cramer, R. III, 3.3 Craven, B. M., 3.5 Crick, F. H. C., 1.3, 2.2, 2.3, 2.4, 2.5, 4.5 Cromer, D. T., 2.3, 2.4 Crooker, P. P., 4.4 Crowfoot, D., 2.3 Crowther, R. A., 1.3, 2.2, 2.3, 2.5, 4.5 Crozier, P. A., 2.5 Cruickshank, D. W. J., 1.2, 1.3, 2.4 Crutchfield, J. P., 4.2 Cullen, D. L., 3.3 Cullis, A. F., 2.3, 2.4 Culver, J. N., 4.5 Cummins, H. Z., 4.6 Cummins, P. G., 3.4 Cunningham, D., 3.3 Currat, R., 4.2 Curtis, C. W., 1.3 Curtis, R. J., 4.4 Cutfield, J. F., 2.2 Czaplewski, C., 3.5 Czaplewsky, C., 3.5 Czerwinski, E. W., 2.3

666

Dabrowski, M., 2.5 Dai, J.-B., 2.3 Dale, D., 2.4 Dam, A. van, 3.3 Dana, S. S., 4.4 Daniel, H., 5.3 Daniels, H. E., 1.3 Darby, G., 2.3 Darden, T. A., 3.5 Dark, R., 3.3 Darwin, C. G., 5.1 Das, K., 2.3 Dasgupta, C., 4.4 D’Astuto, M., 4.1 Daubeny, R. de P., 4.5 Dauter, M., 2.3 Dauter, Z., 2.3 Davey, S. C., 4.4 Davidov, D., 4.4 Davidson, E. R., 3.5 Davidson, J. B., 5.3 Davidson, W., 2.3 Davies, B. L., 1.5 Davies, D. R., 1.3 Davis, M. E., 2.3 Davis, P. J., 3.4 Dawson, B., 1.2, 2.5 Day, D., 4.5 Day, G. M., 3.5 Dayringer, H. E., 3.3 De Caro, L., 2.2 De Facio, B., 5.2 De Gennes, P. G., 4.4 De Hoff, R., 4.4 De Meulenaare, P., 4.3 De Ridder, R., 4.3 De Titta, G. T., 2.2, 2.5, 4.5 De Vries, H. L., 4.4 Dea, I. C. M., 4.5 Debaerdemaeker, T., 2.2 DeBoissieu, M., 4.6 Debreczeni, J. E´., 2.3 Debye, P., 4.1 Declercq, J.-P., 2.2 Dederichs, P. H., 4.2 Deem, M. W., 3.5 Dehlinger, U., 4.6 Deimel, P., 5.3 Deisenhofer, J., 1.3, 2.3, 2.4 DeLano, W. L., 2.3 Delapalme, A., 5.3 Delaunay, B., 1.5 DeLeeuw, S. W., 3.5 Della Valle, R. G., 3.5 Dellby, N., 4.3 Deming, K., 3.5 Deming, W. E., 3.2 Dempsey, S., 3.3 Demus, D., 4.4 Denny, R., 4.5 Denson, A. K., 3.3 DeRosier, D. J., 2.5, 4.5 Deserno, M., 3.5 Destrade, C., 4.4 Destro, R., 3.5 DeTitta, G. T., 2.2, 2.5, 4.5 Deutsch, M., 4.4 Dewar, R. B. K., 2.2 DeWette, F. W., 3.4 Diamond, R., 1.3, 3.3, 4.5 Dickerson, R. E., 1.3, 2.2, 2.3, 2.4 Diele, S., 4.4 Dietrich, H., 1.3 Dieudonne´, J., 1.3 Dijkstra, B. W., 3.3 Dilanian, R. A., 3.3 Dimon, P., 4.4 Ding, D., 4.2 Ding, J., 2.3 Dintzis, H. M., 2.3 Dirac, P. A. M., 1.3 Dirl, R., 1.5 DiSalvo, F. J., 4.4 Ditchfield, R., 1.2

AUTHOR INDEX Diu, B., 1.2 Djurek, D., 4.4 Dobrott, R. D., 2.3 Dodson, E., 2.2, 2.3, 2.4 Dodson, E. J., 1.3, 2.2, 2.3, 3.3 Dodson, G. G., 2.2, 2.3, 3.3 Doerschuk, P. C., 2.5 Doesburg, H. M., 2.2 Doi, K., 5.3 Dokashenko, V. P., 5.3 Dolata, D. P., 3.3 Dolling, C., 4.1 Dolling, G., 4.2 Dolomanov, O. V., 3.3 Domany, E., 4.4 Donabauer, J., 3.5 Dong, W., 2.5 Donohue, J., 1.3, 2.3 Donovan, B., 4.1 Dorna, V., 4.2 Dorner, B., 4.2 Dorner, C., 4.2 Dorset, D. L., 2.5, 4.5 Doubleday, A., 3.3 Doucet, J., 4.4 Douglas, A. S., 2.2 Dovesi, R., 3.5 Dowell, W. C. T., 2.5 Downing, K. H., 2.5 Downs, R. T., 3.3 Dowty, E., 1.4, 3.3 Doyle, P. A., 4.3, 4.5 Dra¨ger, J., 4.6 Drenth, J., 4.5 Drits, V. A., 2.5 Dror, R. O., 3.5 Duan, X., 2.5 Duane, W., 1.3 Dube, P., 2.5 Dubernat, J., 4.2 Dubois, J. C., 4.4 Duce, D. A., 3.3 Duijneveldt, F. B. van, 3.5 Dumrongrattana, S., 4.4 Duncan, W. J., 5.2 Dunitz, J. D., 1.2, 3.5 Dunmur, D. A., 3.4 Durrant, J. L. A., 4.4 Dvoryankin, V. F., 2.5 D’yakon, I. A., 2.5 Dym, H., 1.3 Dyott, T. M., 3.3 Dziki, W., 3.5 Dzyabchenko, A., 3.5 Eades, J. A., 2.5 Eaglesham, D. J., 2.5 Eaker, D., 2.3 Eastwood, J. W., 3.5 Eastwood, M. P., 3.5 Eckold, G., 4.2 Edgington, P. R., 3.3 Edmonds, J. W., 2.2, 2.5, 4.5 Edwards, O. S., 4.2 Egami, T., 4.2 Egelman, E., 2.5 Egert, E., 2.2, 2.3 Eichhorn, F., 5.3 Eijck, B. P. van, 3.5 Eiland, P. F., 1.3 Einstein, A., 4.1 Einstein, J. E., 2.4 Eisenberg, D., 2.2, 2.3, 2.4 Eklundh, J. O., 1.3 Elder, M., 4.5 Eliopoulos, E. E., 3.3 Eller, G. von, 2.2 Elyutin, N. O., 5.3 Emery, V. J., 4.2 Emr, S. D., 2.5 Enderle, G., 3.3 Endoh, H., 2.5 Endoh, Y., 2.5 Endres, H., 4.2

Engel, G., 3.5 Engel, P., 1.3 Englander, M., 5.3 Entin, I. R., 5.3 Epstein, J., 4.2 Erde´lyi, A., 1.3 Erickson, H. P., 2.5 Erickson, J. W., 2.3 Erk, P., 3.5 Eschenbacher, P. W., 1.3 Esnouf, R., 2.3 Essmann, U., 3.5 Estermann, M., 4.2, 4.6 Etheridge, J., 5.2 Etherington, G., 4.4 Evans, A. C., 3.3 Evans, G., 2.3 Evans, N. S., 2.5 Evans, P., 2.4 Evans, P. R., 3.3 Evans-Lutterodt, K. W., 4.4 Evjen, H. M., 3.4 Ewald, P. P., 1.1, 1.3, 1.4, 3.4, 5.1 Exelby, D. R., 2.5 Eyges, L., 3.5 Faber, T. E., 4.4 Facelli, J. C., 3.5 Faetti, S., 4.4 Faggiani, R., 2.1 Fa˚k, B., 5.3 Fan, C. P., 4.4 Fan, H.-F., 2.2, 2.5 Farach, H. A., 3.3 Farber, A. S., 4.4 Farkas, D. R., 1.3 Farrants, G., 1.4, 2.5 Farrants, G. W., 3.3 Farrow, N. A., 2.5 Farrugia, L. J., 3.3 Favin, D. L., 1.3 Fayard, M., 4.2 Fedotov, A. F., 2.5 Fehlhammer, H., 2.3 Feig, E., 1.3 Feil, D., 1.2 Feiner, S. K., 3.3 Feldkamp, L. A., 4.1 Feldmann, R. J., 3.3 Fellmann, D., 2.5 Feltynowski, A., 2.5 Fender, B. E. F., 4.2 Fernandez, J.-J., 2.5 Ferrara, J. D., 2.3 Ferraris, G., 2.5 Ferrarro, M. B., 3.5 Ferrel, R. A., 4.3 Ferrin, T. E., 2.5, 3.3 Fewster, P. F., 5.1 Feynman, R., 5.2 Fields, P. M., 4.3 Fillipini, G., 3.5 Filman, D. J., 2.3 Finch, J. T., 2.5 Finger, L. W., 3.3 Fingerland, A., 5.1 Finkelstein, K. D., 5.3 Finkenstadt, V. L., 4.5 Finn, R., 2.5 Fischer, J., 2.3 Fischer, K., 1.2 Fischer, R. X., 3.3 Fischer, W., 1.4 Fisher, J., 2.2 Fisher, P. M. J., 4.3 Fishman, G., 2.5 Fiske, S. J., 2.2 Fitzgerald, P. M. D., 1.4, 2.3 Flack, H. D., 4.2 Flautt, T. J., 4.4 Fleming, R. M., 4.4 Flensburg, C., 2.3 Fletterick, R. J., 2.3, 3.3 Flook, R. J., 2.4

Foadi, J., 2.2 Fock, R., 5.2 Fogel, D. B., 2.3 Foley, J. D., 3.3 Folkhard, W., 1.3 Fontaine, D. de, 4.2 Fontell, K., 4.4 Fontes, E., 4.4, 5.1 Ford, G. C., 2.3 Ford, L. O., 3.3 Fornberg, A., 1.3 Forst, R., 4.2 Fo¨rster, E., 5.1 Forster, F., 2.5 Forsyth, J. B., 1.3, 4.2 Forsyth, V. T., 4.5 Fortier, S., 2.2 Fortuin, C. M., 3.4 Forwood, C. T., 2.5 Foster, R. M., 1.3 Foucher, P., 4.4 Fouret, P., 4.2 Fouret, R., 4.2 Fourme, R., 2.3 Fout, G. S., 2.3 Fowler, R. H., 1.3 Fowweather, F., 1.3 Fox, G., 2.3 Frampton, C. S., 3.5 Francis, N., 2.5 Frangakis, A. S., 2.5 Frank, F. C., 4.4 Frank, J., 2.5 Frankenberger, E. A., 2.3 Franklin, R. E., 4.5 Franulovic, K., 4.4 Franx, M., 2.1 Fraser, D., 2.5 Fraser, R. D. B., 4.5 Frazer, R. A., 5.2 Freeman, A. J., 1.2, 4.3 Freeman, H. C., 2.4 Freer, A. A., 2.2 Freer, S. T., 1.3, 2.4 Freiser, M. J., 4.4 French, A. D., 4.5 French, S., 2.1, 2.2, 2.4 Frenkel, D., 3.5 Frey, F., 4.2 Frey, S., 2.5 Fridborg, K., 2.3 Fridrichsons, J., 2.3 Friedel, G., 1.3, 4.4 Friedlander, F. G., 1.3 Friedlander, P. H., 1.3 Friedman, A., 1.3 Frobenius, G., 1.3 Frost, J. C., 4.4 Frost-Jensen, A., 4.2, 4.6 Fry, E., 2.3 Fryer, J. R., 2.5, 4.5 Fuess, H., 5.3 Fujii, Y., 4.1 Fujimoto, F., 2.5, 4.3, 5.2 Fujinaga, M., 2.3 Fujiwara, A., 2.5 Fujiwara, K., 2.5, 5.2 Fujiyoshi, Y., 2.5 Fukuhara, A., 2.5, 5.2 Fukuyama, K., 2.3 Fuller, S. D., 2.5 Fuller, W., 4.5 Fultz, B., 4.2 Fung, K. K., 2.5 Furey, W., 2.3 Furie, B., 3.3 Furie, B. C., 3.3 Furusaka, M., 4.2 Fusti-Molnar, L., 3.5 Gabashvili, I. S., 2.5 Gabor, D., 2.5 Ga¨hler, R., 5.3 Galerne, Y., 4.4

667

Gallo, L., 3.3 Gallo, S. M., 2.2 Gallop, J. R., 3.3 Gallwitz, U., 4.5 Gane, P. A. C., 4.4 Gannon, M. G. J., 4.4 Garcia, A. E., 3.5 Garcia-Golding, F., 4.4 Garcia-Granda, S., 2.2 Garcı´a-Rodrı´guez, L., 2.2 Gardner, K. H., 4.5 Garland, C. W., 4.4 Garland, Z. G., 4.4 Garman, E., 2.3 Garrido, J., 2.3 Gasparoux, H., 4.4 Gassmann, J., 1.3, 2.3, 2.4, 2.5 Gatti, M., 4.4 Gaughan, J. P., 4.4 Gautier, F., 4.2 Gavezzotti, A., 3.5 Gavrilov, V. N., 5.3 Gay, R., 2.3 Gaykema, W. P. J., 2.3 Gayle, F. W., 4.2 Gebhard, W., 4.5 Geddes, A. J., 3.3 Gehlen, P., 4.2 Gehlhaar, D. K., 2.3 Gehring, K., 2.5 Geil, P. H., 4.5 Geisel, T., 4.2 Gelder, R. de, 2.2 Gel’fand, I. M., 1.3 Geller, M., 3.3 Gentleman, W. M., 1.3 Georgopoulos, P., 4.2 Gerhard, O. E., 2.1 Gerlach, P., 4.2 Germain, G., 2.2, 2.5 Germian, C., 4.4 Gerold, V., 4.2 Giacovazzo, C., 2.1, 2.2 Giarrusso, F. F., 3.3 Gibbons, P. C., 4.2 Gibbs, J. W., 2.3 Gibson, M. A., 2.5 Giege´, R., 2.3 Giesebrecht, J., 2.5 Gilbert, P. F. C., 2.5 Gill, P. E., 3.3 Gill, P. M. W., 3.5 Gillan, B. E., 4.2 Gilli, G., 2.4 Gilligan, K., 3.3 Gilliland, G. L., 3.3 Gillis, J., 1.3, 2.2 Gilmore, C. J., 2.2, 2.5, 4.5 Gilmore, R., 5.2 Gingrich, N. S., 1.3 Girling, R. L., 3.3 Gjønnes, J., 2.5, 4.3, 5.2 Gjønnes, K., 2.5 Glaeser, R. M., 2.5 Glasser, M. L., 3.4 Glatigny, A., 3.3 Glauber, R., 2.5 Glazer, A. M., 4.2 Glosli, J., 3.5 Glu¨ck, M., 1.5 Glucksman, M. J., 4.5 Glykos, N. M., 2.3 Go, N., 3.3 Goddard, T. D., 2.5 Godre´che, C., 4.6 Goedkoop, J. A., 1.3, 2.2 Goff, J. P., 4.2 Golas, M. M., 2.5 Goldman, A. I., 4.2, 4.6 Goldstine, H. H., 1.3 Golovchenko, J. A., 5.1 Golub, G. H., 2.5 Goncharov, A. B., 2.5 Gonzalez, A., 4.5

AUTHOR INDEX Gonzalez, R. F., 2.5 Good, I. J., 1.3 Goodby, J. W., 4.4 Goodman, P., 2.5, 5.2 Goodyear, G., 2.5 Goossens, D. J., 4.2 Gordon, R., 2.5 Gosling, R. G., 4.5 Gossling, T. H., 3.3 Gould, R. O., 2.2 Gouyet, J. F., 4.6 Gowen, B., 2.5 Graaf, H. de, 4.5 Graaff, R. A. G. de, 2.2 Graeff, W., 5.1, 5.3 Gragg, J. E., 4.2 Gramlich, V., 2.2 Gransbergen, E. F., 4.4 Grant, D. F., 2.2 Grassucci, R. A., 2.5 Gratias, D., 2.5, 4.6, 5.2 Grau, U. M., 2.3 Gray, G. W., 4.4 Green, D. W., 2.3, 2.4 Green, E. A., 2.2, 2.4 Greenall, R. J., 4.5 Greenberg, W. L., 2.5 Greenblatt, D. M., 2.5 Greenhalgh, D. M. S., 1.3 Greer, J., 3.3 Grems, M. D., 1.3 Grenander, U., 1.3 Griffith, J. P., 2.3 Grigorieff, N., 2.5 Grimm, H., 1.3, 4.2 Grindley, J., 3.5 Grinstein, G., 4.4 Gronsky, R., 2.5 Gros, P., 2.3 Gross, L., 1.3 Grosse-Kunstleve, R. W., 1.4, 2.2, 2.3 Grubb, D. T., 4.5 Grzegory, I., 4.1 Guagliardi, A., 2.2 Gubbens, A. J., 4.3 Guessoum, A., 1.3 Guigay, J. P., 5.3 Guillon, D., 4.4 Guinier, A., 4.2 Gull, S. F., 1.3 Gullberg, G. T., 2.5 Gunther, L., 4.4 Gur, Y., 1.5 Gurskaya, G. V., 2.5 Guru Row, T. N., 1.2 Guryan, C. A., 4.2 Gutierrez, G. A., 4.5 Guyot-Sionnest, P., 4.4 Haas, F. de, 2.5 Hadamard, J., 1.3 Haeffner, D. R., 4.2 Haefner, K., 4.2 Haibach, T., 4.6 Hall, I. H., 4.5 Hall, M., 1.3 Hall, R. J., 2.5 Hall, S. R., 1.4, 2.2 Halla, F., 4.2 Hall-Wallace, M., 3.3 Halperin, B. I., 4.4 Hamaker, C., 2.5 Hamilton, W. A., 5.3 Hamilton, W. C., 2.3, 2.4, 3.1, 3.2, 4.5 Hancock, H., 2.2 Handelsman, R. A., 1.3 Hansen, J., 2.5 Hansen, N. K., 1.2 Hao, Q., 2.2 Harada, J., 4.2 Harada, Y., 2.3, 2.5, 4.3 Harauz, G., 2.5 Harburn, G., 4.2

Harding, M. M., 2.2, 2.3 Hardman, K. D., 3.3 Hardouin, F., 4.4 Hardy, G. H., 1.3 Harford, J., 4.5 Harker, D., 1.3, 2.1, 2.2, 2.3, 2.4 Harrington, M., 2.3 Harris, D. B., 1.3 Harris, G. W., 4.2 Harris, M. R., 3.3 Harrison, S. C., 1.3, 2.3 Harrison, W. A., 4.1 Hart, M., 5.1, 5.3 Hart, R. G., 2.4 Hartman, P., 1.3 Hartree, D. R., 1.2 Hartsuck, J. A., 2.3 Hasegawa, K., 4.5 Haseltine, J. H., 4.4 Hashimoto, H., 2.5 Hashimoto, S., 4.2, 4.3 Hashimoto, T., 2.5 Hass, B. S., 3.3 Hastings, C. Jr, 3.4 Hastings, J. B., 5.3 Hata, Y., 2.4 Hatch, D. M., 1.5 Haubold, H. G., 4.2 Hauptman, H., 1.3, 2.1, 2.2, 2.3, 2.4, 2.5, 4.5 Hausdorff, F., 4.6 Hauser, J., 3.3 Havelka, W., 2.5 Havighurst, R. J., 1.3 Hayakawa, M., 4.2 Hayes, W., 4.2 Hazen, E. E., 3.3 He, L. X., 2.5 He, Y., 2.3 Head-Gordon, M., 3.5 Heagle, A. B., 2.5 Heap, B. R., 3.3 Hearmon, R. F. S., 4.1 Hearn, A. C., 1.4 Hecht, H. J., 2.3 Heel, M. van, 2.5 Hegerl, R., 2.5 Hehre, W. J., 1.2 Heideman, M. T., 1.3 Heil, P. D., 4.5 Heinermann, J. J. L., 2.2 Heiney, P. A., 4.4, 4.6 Helfrich, W., 4.4 Helgaker, T., 3.5 Helliwell, J. R., 2.2, 2.4 Hellner, E., 4.2 Helms, H. D., 1.3 Hende, J. van den, 1.3 Henderson, R., 2.3, 2.5 Hendricks, S., 4.2, 4.5 Hendrickson, W. A., 1.3, 2.2, 2.3, 2.4 Hendrikx, Y., 4.4 Hennion, B., 4.2 Hennion, M., 4.2 Henry, R., 3.5 Herglotz, G., 1.3 Herman, G. T., 2.5 Hermann, C., 1.3, 4.6 Hermans, J., 3.3 Herriot, J. R., 2.4 Herrmann, K. H., 2.5 Hewat, A., 3.3 Hewitt, J., 2.5 Heymann, J. A. W., 2.5 Higgs, H., 3.3 High, D. F., 2.3, 2.4 Hildebrandt, G., 5.1 Hills, G. J., 2.5 Hirabayashi, M., 2.5 Hiraga, K., 2.5 Hirsch, P. B., 2.5, 4.5, 5.1, 5.2 Hirschman, I. I. Jr, 1.3 Hirshfeld, F. L., 1.2, 2.3

Hirt, A., 2.5 Hirth, J. P., 4.4 Hitchcock, P. B., 4.4 Hjerte´n, S., 2.3 Hlinka, J., 4.2 Ho, M.-H., 2.5 Ho, M.-S., 2.5 Hockney, R. W., 3.5 Hodgkin, D. C., 2.2, 2.3, 2.4 Hodgson, K. O., 2.4 Hodgson, M. L., 1.3 Hoffmann, J.-U., 4.2 Hofmann, D., 4.5 Hofmann, D. W. M., 3.5 Hogle, J., 2.3, 3.3 Hohlwein, D., 4.2 Hohn, M., 2.5 Høier, R., 2.5, 4.3, 5.2 Hol, W. G. J., 2.3, 3.3 Holbrook, S. R., 1.3, 2.4 Hollenberg, J., 2.5 Holm, C., 3.5 Holmes, K. C., 2.5, 4.5 Honegger, A., 3.3 Hong, H., 4.2 Honjo, G., 4.3 Hopfinger, A. J., 3.3, 4.5 Hopgood, F. R. A., 3.3 Hoppe, W., 1.3, 2.2, 2.3, 2.4, 2.5 Horalik, L., 5.3 Horjales, E., 3.3 Ho¨rmander, L., 1.3 Horn, P. M., 4.4 Horne, M. A., 5.3 Hornreich, R. M., 4.4 Hornstra, J., 2.3 Horstmann, M., 2.5 Hosemann, R., 2.3, 4.2, 4.5 Hoser, A., 4.2 Hoshino, S., 5.3 Hosoya, S., 5.3 Hosur, M. V., 2.3 Houston, T. E., 3.3 Hovmo¨ller, S., 1.4, 2.5 Howe, J. M., 2.5, 4.2 Howells, E. R., 2.1 Howells, R. G., 2.2 Howie, A., 2.5, 4.3, 4.5, 5.2 Hradil, K., 4.2 Hrdlicˇka, Z., 5.3 Hsiou, Y., 2.3 Hsiung, H., 4.4 Hu, Ch., 4.2 Hu, H., 4.5 Hu, H. H., 2.5 Huang, C., 3.3 Huang, C. C., 2.5, 4.4 Huang, K., 1.3, 4.1 Huang, Z., 2.5 Hubbard, R. E., 3.3 Huber, R., 1.3, 2.3, 2.4 Hudson, L., 4.5 Hudson, P. J., 3.3 Huesman, R. H., 2.5 Hughes, D. E., 1.3 Hughes, E. W., 1.3, 2.2, 2.3 Hughes, J. F., 3.3 Hughes, J. J., 2.3, 2.4 Hughes, S. H., 2.3 Hull, S., 4.2 Hull, S. E., 2.2 Hu¨ller, A., 4.2 Hummel, W., 3.3 Hummelink, T., 3.3 Hummelink-Peters, B. G., 3.3 Hummer, G., 3.5 Hu¨mmer, K., 5.1 Humphreys, C. J., 2.5, 4.3, 5.2 Hunsmann, N., 2.5 Hunt, J. F., 2.3 Huntingdon, H. B., 4.1 Hurley, A. C., 1.2, 2.5, 5.2 Huse, D. A., 4.4 Hutching, M. T., 4.2

668

Iannelli, P., 4.5 Ibers, J. A., 2.4, 4.2 Iijima, S., 4.3 Iizumi, M., 4.2 Ilag, L. L., 2.3 Imamov, R. M., 2.5 Immirzi, A., 1.3 Imry, Y., 4.4 Indenbom, V. L., 5.3 Ingram, V. M., 2.3, 2.4 Inoue, A., 2.5 Iolin, E. M., 5.3 Irwin, M. J., 2.2 Isaacs, N. W., 1.3, 2.2, 2.4, 3.3 Ishida, M., 2.5 Ishihara, K. N., 2.5, 4.6 Ishii, T., 4.2 Ishikawa, I., 5.3 Ishikawa, T., 5.1 Ishikawa, Y., 4.2 Ishizuka, K., 2.5 Isoda, S., 2.5, 4.5 Israel, R., 2.2 Ito, T., 3.2 Ivanova, M. I., 4.5 Ivantchev, S., 1.5 Iwata, H., 4.2 Izumi, F., 3.3 Jach, T., 5.1 Jack, A., 1.3, 2.4 Jackson, J. I., 2.5 Jacobson, R. A., 2.2, 2.3 Jacques, J., 4.4 Jaeger, J. C., 1.3 Jagodzinski, H., 4.2 Jahn, W., 2.5 Jakana, J., 2.5 James, R. W., 1.2, 1.3, 2.3, 4.2, 5.1 Jan, J.-P., 1.5 Janiak, M. J., 4.4 Janner, A., 1.3, 2.5, 4.2, 4.6 Janot, C., 4.2 Janot, Chr., 4.6 Jansen, L., 1.5 Janssen, P. A. J., 2.3 Janssen, T., 1.3, 1.5, 2.5, 4.2, 4.6 Jap, B. K., 2.5 Jaric, M. V., 4.2, 4.6 Jaric´, M. Y., 2.5 Jarvis, L., 3.3 Jaynes, E. T., 1.3, 2.2 Jefferey, J. W., 4.2 Jeffery, B. A., 2.3 Jeffrey, G. A., 1.3 Jenni, S., 2.5 Jensen, L. H., 1.3, 2.3, 2.4 Jeu, W. H. de, 4.4 Jiang, J.-S., 2.3 Jia-xing, Y., 2.2 Jih, J. H., 2.3 Jogl, G., 2.3 Johannisen, H., 2.3 Johnson, A. W. S., 2.5, 5.2 Johnson, C. K., 1.2, 3.1, 3.3 Johnson, D. H., 1.3 Johnson, D. L., 4.4 Johnson, H. W., 1.3 Johnson, J. E., 2.3, 2.5, 3.3 Johnson, L. N., 2.3, 2.4, 3.3 Johnson, R. W., 1.3 Jolles, P., 2.3 Jones, B., 4.4 Jones, R., 4.2, 4.5 Jones, R. C., 4.2 Jones, T. A., 2.3, 2.5, 3.3 Jones, W., 3.5 Jones, Y., 2.3 Jorgensen, P., 3.5 Josefsson, T. W., 4.3 Joyeux, L., 2.5 Joyez, G., 4.2 Ju¨rgensen, H., 1.3 Just, W., 4.2

AUTHOR INDEX Kabsch, W., 3.3, 4.5 Kac, M., 1.3 Kaczmarz, S., 2.5 Kadecˇkova´, S., 5.3 Kaenel, R. A., 1.3 Kagan, Yu., 5.3 Kainuma, Y., 4.3, 5.2 Kaiser, H., 5.3 Kaiser-Bischoff, I., 4.2 Kakinoki, J., 4.2 Kalantar, A. H., 3.2 Kaldor, U., 2.1 Kalning, M., 4.2, 4.6 Kamada, K., 5.3 Kambe, K., 2.5, 5.2 Kamer, G., 2.3 Kamper, J., 2.3 Kaneko, K., 2.5 Kaneyama, T., 2.5 Kang, S. J., 3.3 Kannan, K. K., 2.3 Kansy, K., 3.3 Kaplan, D. R., 5.1 Kaplan, M., 4.4 Kara, M., 1.2 Karamertzanis, P. G., 3.5 Karbach, A., 4.5 Karle, I. L., 2.2 Karle, J., 1.3, 2.1, 2.2, 2.3, 2.4, 2.5 Ka´rma´n, T. von, 4.1 Karplus, M., 1.3, 3.3 Karrass, A., 1.3 Kartha, G., 2.3, 2.4 Kasper, J. S., 1.3, 2.2 Kasting, G. B., 4.4 Katagawa, T., 5.1 Katayama, K., 4.5 Kato, K., 4.6 Kato, N., 5.1, 5.3 Katsube, Y., 2.4 Katz, L., 3.3 Katznelson, Y., 1.3 Kaufman, M. J., 2.5 Kawaguchi, A., 4.5 Ke, E. Y., 2.5 Kearsley, S. K., 3.3 Keeling, J., 2.3 Keen, D. A., 4.2 Kek, S., 4.2, 4.6 Keller, A., 4.5 Keller, E., 3.3 Keller, J., 4.2 Kelley, B., 4.4 Kelly, B. A., 4.4 Kelton, K. F., 4.2, 4.6 Kendall, M., 1.2, 2.1 Kendrew, J. C., 1.3, 2.3, 2.4 Kennard, O., 3.1, 3.3 Kessel, M., 2.5 Kessler, M., 3.3 Ketelaar, J. A. A., 2.3 Khalak, H. G., 2.2 Khinchin, A. I., 1.3 Kiefer, J. E., 1.3, 2.1 Kiely, C. J., 2.5 Kikuta, S., 5.1, 5.3 Kim, M., 2.5 Kim, S.-H., 1.3, 2.3, 2.4 Kirby, I., 2.3 Kirichuk, V. S., 2.5 Kirkland, E. J., 2.5 Kirov, A., 1.4, 1.5 Kiselev, N. A., 2.5 Kissinger, C. R., 2.3 Kitagaku, M., 4.3 Kitagawa, Y., 2.4 Kitaigorodskii, A. J., 4.1 Kitaigorodsky, A. I., 4.2 Kitamura, N., 4.3 Kittel, C., 3.5, 4.4 Kitz, N., 1.3 Kjeldgaard, M., 2.5, 3.3 Klapperstu¨ck, M., 4.4 Klar, B., 5.3

Klechkovskaya, V. V., 2.5 Klei, H. E., 2.3 Klein, A. G., 5.3 Klein, M. L., 3.5 Kleinstu¨ck, K., 5.3 Kleman, M., 4.6 Klepeis, J. L., 3.5 Kleywegt, G. J., 2.3 Klimkovich, S., 1.4 Klug, A., 1.3, 2.2, 2.3, 2.5, 3.3, 4.5 Klug, H. P., 4.2 Kluyver, J. C., 1.3 Knol, K. S., 2.4 Knowles, J. W., 5.3 Kobayashi, S., 4.4 Kobayashi, T., 2.5 Koch, E., 1.1, 1.4 Koch, M. H. J., 2.2 Kodera, S., 4.3 Koellner, M., 5.3 Koetzle, T. F., 2.4, 3.3 Kogiso, M., 2.5, 5.2 Kohn, V. G., 5.1 Kohra, K., 5.1, 5.3 Kokkinidis, M., 2.3 Kolar, H., 2.5 Kolba, D. P., 1.3 Kolodziej, S., 2.5 Komada, T., 2.5 Komorek, M., 4.2 Komura, Y., 4.2 Kong, Y. F., 2.5 Konnert, J. H., 1.3, 2.4 Kopka, M. L., 2.3 Kopp, S., 2.5, 4.5 Kopperschlager, G., 2.5 Korekawa, M., 4.2, 4.6 Koritsansky, T., 3.5 Korn, D. G., 1.3 Ko¨rner, T. W., 3.5 Korpiun, P., 5.3 Kortan, A. R., 4.4 Kossel, W., 2.5 Koster, A. J., 2.5 Kosterlitz, J. M., 4.4 Kosykh, V. P., 2.5 Kovacs, A. J., 4.5 Kovalchuk, M. V., 5.1 Kovalev, O. V., 1.5 Koymans, L., 2.3 Krabbendam, H., 2.2 Krahl, D., 2.5, 4.3 Kra¨mer, S., 2.5 Kraus, W., 3.3 Kraut, J., 1.3, 2.3, 2.4 Kress, W., 4.1 Kreuger, R. J., 5.2 Kriegman, D. J., 2.5 Krisch, M., 4.1 Krishna, P., 4.2 Krivanek, O. L., 4.3 Krivoglaz, M. A., 4.2, 4.3 Kroeker, M., 3.3 Kroon, J., 2.2, 3.5 Kroumova, E., 1.5 Kruse, F. H., 1.3 Kuchta, L., 2.2 Ku¨hlbrandt, W., 2.5 Ku¨hne, T., 2.5 Kuhs, W. F., 1.2 Kukla, D., 2.4 Kulda, J., 5.3 Kulidzhanov, F. G., 5.3 Kuligin, A. K., 2.5 Kulka, D., 1.3 Kull, F. J., 2.5 Kuntz, I. D., 3.3 Kuo, K. H., 2.5 Kurihara, K., 4.2 Kuriyan, J., 1.3 Kurki-Suonio, K., 1.2 Kuszewski, J., 2.3 Kutateladze, T. G., 2.5

Kutznetsov, P. I., 1.2 Kuvdaldin, B. V., 5.3 Kuwabara, S., 2.5 Kvardakov, V. V., 5.3 Kycia, S., 4.2 Kyriakidis, C. E., 2.2 Lacour, T. F. M., 1.3 Ladbrooke, B. D., 4.4 Ladisa, M., 2.2 Ladjadj, M., 2.5 Lagace´, L., 2.3 Lagendijk, R. L., 2.5 Lagomarsino, S., 5.1, 5.3 Lajzerowicz, J., 4.4 Lajze´rowicz, J., 2.2 Laloe, F., 1.2 Lambert, D., 5.3 Lambert, M., 4.4 Lambert, M. A., 2.5 Lambiotte, J. J. Jr, 1.3 Lamfers, H.-J., 4.2 Lamy, J., 2.5 Lamzin, V. S., 2.3 Lancon, F., 4.6 Lanczos, C., 1.3 Landau, H. J., 1.3 Landau, L. D., 4.4 Lando, J. B., 4.5 Lang, A. R., 5.1 Lang, S., 1.3 Lang, W. W., 1.3 Langer, R., 2.5 Langridge, R., 3.3, 4.5 Langs, D. A., 2.2, 2.5, 4.5 Lanzavecchia, S., 2.5 Larine, M., 1.4 Larmor, J., 1.3 Lata, K. R., 2.5 Lattman, E. E., 2.3, 2.4 Laue, M., 1.1 Laue, M. von, 1.3, 5.1 Laurette, I., 2.5 Laval, J., 4.1 Laves, R., 4.2 Lavoine, J., 1.3 Lawrence, M. C., 2.5 Lawrence, P. D., 2.5 Lawson, K. D., 4.4 Lawton, J. A., 2.5 Le Bail, A., 3.3 Le Guillou, J. C., 4.4 Lea, S. M., 2.3 Leadbetter, A. J., 4.4 Leapman, R. D., 4.3 Lechner, R. E., 4.2 Lederer, F., 3.3 Ledermann, W., 1.3, 4.1 Lee, E. J., 4.5 Lee, H., 3.5 Lee, J. C., 2.5 Lee, P. L., 4.2 Lee, S. D., 4.4 Lee, Y.-R., 5.1 Leenhouts, J. I., 2.3 Lefebvre, J., 4.1, 4.2 Lefebvre, S., 4.2 Lefeld-Sosnowska, M., 5.1 Legg, M. J., 1.3 Lehmann, M., 5.3 Lehmann, M. S., 1.3, 2.2 Lehmpfuhl, G., 2.5, 4.3 Lei, J., 4.2 Leiman, P. G., 2.3 Leith, A., 2.5 Lele, S., 4.2 Lemoine, G., 3.3 Lentz, P. J. Jr, 2.3 Lepault, J., 2.5 Lerch, M., 4.2 Lerner, F. Ya., 2.5 Lescar, J., 5.1 Lescoute, A., 2.5 Lesk, A. M., 3.3

669

Leslie, A. G. W., 1.3, 2.3, 3.3 Lessinger, L., 2.2 Leszczynski, M., 4.1 Leung, P., 1.2 Leusen, F. J. J., 3.5 Leusen, F. J. L., 3.5 Levanyuk, A. P., 4.4 Levelut, A. M., 4.4 Levens, S. A., 1.3 Levine, D., 2.5, 4.6 Levinthal, C., 3.3 Levitov, L. S., 4.6 Levitt, M., 1.3, 2.4, 3.3 Levy, H. A., 1.2, 1.3, 3.1 Lewis, M., 2.3 Lewis, T. C., 3.5 Lewitt, R. M., 2.5 Li, D. X., 2.5 Li, F. H., 2.5 Li, J. Q., 2.5 Li, Y., 2.5 Liang, C., 3.5 Liang, K. S., 4.4 Liang, Y. Y., 2.5 Liebert, L., 4.4 Liebert, L. E., 4.4 Liebman, G., 2.5 Lien, S. C., 4.4 Lieth, C. W. van der, 3.3 Lievert, L., 4.4 Lifchitz, A., 1.3, 2.3 Lifshitz, E. M., 4.4 Lifson, S., 3.3 Lighthill, M. J., 1.3 Lijk, L. J., 3.3 Liljas, L., 2.3, 3.3 Liljefors, T., 3.3 Linares-Galvez, J., 5.3 Lindegaard, A., 4.4 Lindsey, J., 2.3 Link, V., 4.4 Linnik, I. Ju., 1.3 Lipanov, A. A., 4.5 Lipkowitz, K. B., 3.3 Lippert, B., 2.1 Lipscomb, W. N., 2.3 Lipson, H., 1.1, 1.2, 1.3, 1.4, 2.1, 2.3, 4.2, 4.5 Litster, J. D., 4.4 Litvin, D. B., 2.3 Liu, J., 4.5 Liu, W., 2.5 Liu, Y.-W., 2.5 Livanova, N. B., 2.5 Livesey, A. K., 1.3 Lloyd, T. E., 2.5 Loane, R. F., 4.3 Lobachev, A. N., 2.5 Lobanova, G. M., 2.5 Lobert, S., 4.5 Lock, C. J. L., 2.1 Lockhart, T. E., 4.4 Lomer, T. R., 2.1 Lomont, J. S., 1.5 Lonsdale, K., 1.3 Lontovitch, M., 5.2 Looijenga-Vos, A., 4.6 Lorenz, M., 4.5 Loris, R., 5.1 Lotz, B., 2.5, 4.5 Love, W., 1.3 Love, W. E., 2.2, 2.3 Love, W. F., 1.5 Lovell, F. M., 1.3 Lo¨vgren, S., 2.3 Lowde, R. D., 5.3 Lu, C., 1.3 Lu, G., 2.3 Luban, M., 4.4 Lubensky, T. C., 4.4 Lucas, B. W., 4.2 Luck, J. M., 4.6 Ludewig, J., 5.1 Ludtke, S. C., 2.5

AUTHOR INDEX Ludtke, S. J., 2.5 Luenberger, D. G., 3.3 Luic´, M., 2.2 Lunin, V. Y., 2.3 Lunin, V. Yu., 1.3 Lunina, N. L., 2.3 Luo, M., 2.3 Lurie, N. A., 4.1 Lurz, R., 2.5 Lushington, K. J., 4.4 Luther, P., 4.5 Luty, T., 4.1 Luzzati, V., 2.3, 4.4 Lybanon, M., 3.2 Lyman, P. F., 5.1 Lynch, D. F., 2.5, 5.2 Lynch, R. E., 2.3 Ma, J. P., 2.5 Ma, Q., 2.3 Ma, S. K., 4.4 MacGillavry, C. H., 1.3, 4.5 Machin, P. A., 3.3 Mackay, A. L., 2.2, 2.5, 3.3 MacKay, M., 2.3 MacLane, S., 1.3 MacNicol, D. D., 2.5 Macovski, A., 2.5 Macrae, C. F., 3.3 MacRae, T. P., 4.5 Mada, H., 4.4 Madariaga, G., 1.5 Madden, P. A., 3.5 Madelung, E., 3.4 Madhav Rao, L., 5.3 Magdoff, B. S., 2.3 Magnus, W., 1.3 Mahendrasingam, A., 4.5 Mahon, M., 3.3 Maier, W., 4.4 Main, P., 1.3, 2.2, 2.3, 2.5 Makowski, L., 4.5 Malgrange, C., 5.1, 5.3 Malik, K. M. A., 4.4 Maling, G. C., 1.3 Malladi, R., 2.5 Mallick, S. P., 2.5 Mallikarjunan, M., 3.3 Maltheˆte, J., 4.4 Maly, K., 4.2, 4.6 Mandelkern, L., 4.5 Mandelkow, E., 4.5 Mani, N. V., 2.4 Manley, R. St. J., 2.5 Mannami, M., 2.5 Mao, Y., 2.5 Marabini, R., 2.5 Marchington, B., 1.3 Mardix, S., 4.2 Marel, R. P. van der, 2.1 Marigo, A., 4.2 Marinder, B. O., 2.5 Mark, H., 2.4 Markham, R., 2.5 Marko, M., 2.5 Marks, L. D., 4.3 Marsh, R. E., 3.2, 3.5 Marson, F., 4.4 Martin, C., 2.3 Martin, P. C., 4.4 Martinez-Miranda, L. J., 4.4 Martorana, A., 4.2 Marumo, F., 1.2 Marvin, D. A., 1.3, 4.5 Marynissen, H., 4.4 Masaki, N., 5.3 Maslen, V. W., 1.2 Maslen, W. V., 4.3 Mason, R., 4.4 Mason, S. A., 4.5 Massariol, M.-J., 2.3 Massidda, V., 3.4 Mastryukov, V. S., 2.5 Masumoto, K., 2.5

Masumoto, T., 2.5 Matadeen, R., 2.5 Matej, S., 2.5 Materlik, G., 1.2, 5.1 Mathews, F. S., 3.3 Mathiesen, R. H., 5.1 Mathiesen, S., 4.4 Mathieson, A. McL., 2.3 Matsubara, E., 4.2 Matsuda, T., 2.5 Matthews, B. W., 1.3, 2.3, 2.4 Mauguen, Y., 2.2, 2.4 Mauritz, K. A., 4.5 Max, N. L., 3.3 Mayer, J., 2.5 Mayer, S. W., 1.3 Mayers, D. F., 4.3 Mazeau, K., 4.5 Mazid, M. A., 4.4 Mazkedian, S., 5.3 Mazure´-Espejo, C., 5.3 Mazzarella, L., 2.3, 2.4 McArdle, P., 3.3 McCabe, P., 3.3 McCall, M. J., 3.3 McClellan, J. H., 1.3 McCourt, M. P., 2.5, 4.5 McCoy, A. J., 2.3 McDonald, W. S., 2.4 Mcewen, B., 2.5 McFarland, K., 3.5 McGreevy, R. L., 4.2 McIntyre, G. J., 1.2 McKean, H. P., 1.3 McKenna, R., 2.3 Mckernen, S., 2.5 McLachlan, A. D., 3.3 McLachlan, D., 2.3, 2.5 McMahon, B., 1.4 McMillan, W. L., 4.4 McMullan, R. K., 3.3 McMurchie, L. E., 3.5 McPherson, A., 2.4 McQueen, J. E., 3.3 McWhan, D. B., 4.4 Mechin, I., 2.5 Meiboom, S., 4.4 Meichle, M., 4.4 Melone, S., 5.3 Mendiratta, S. K., 5.3 Meng, E. C., 2.5 Menzer, G., 2.3 Mermin, N. D., 1.1, 4.6 Mersereau, R. M., 1.3, 2.5 Merwe, J. H. van der, 4.4 Messiah, A., 5.2 Metropolis, N., 4.2 Meyer, C. E., 4.3 Meyer, C. H., 2.5 Meyer, E. F., 3.3 Meyer, G., 2.5 Meyer, L. B., 2.5 Meyer, R. B., 4.4 Michalec, R., 5.3 Michejda, C. J., 2.3 Micu, A. M., 4.2 Midgley, P. A., 2.5 Mielke, T., 2.5 Mierzejewski, A., 4.1 Mighell, A. D., 2.3 Mikula, P., 5.3 Millane, R. P., 4.5 Miller, A., 4.5 Miller, D. P., 4.5 Miller, G. H., 4.5 Miller, J. R., 3.3 Miller, J. S., 4.2 Miller, R., 2.2, 4.5 Miller, S. C., 1.5 Miller, S. T., 2.3 Mills, D. M., 5.1 Mimori-Kiyosue, Y., 4.5 Minakawa, N., 5.3 Mindell, J. A., 2.5

Ming, D. M., 2.5 Mitra, A. K., 4.5 Mitra, K., 2.5 Mitsui, T., 4.5 Miyake, S., 2.5 Miyano, K., 4.4 Miyazaki, M., 2.5 Mo, F., 5.1 Mo, Y. D., 2.5 Moereels, H., 2.3 Moliere, G., 4.3 Moliterni, A. G. G., 2.2 Mo¨llenstedt, G., 2.5 Moncrief, J. W., 2.3 Moncton, D. E., 4.4 Montroll, E. W., 1.3 Moodie, A. F., 2.5, 5.2 Mooji, W. T. M., 3.5 Moon, P. B., 2.4 Mooney, P. E., 4.3 Moore, D. H., 1.3 Moore, P. B., 2.5 Moras, D., 2.3 More, M., 4.2 Morffew, A. J., 3.3 Morgenroth, W., 4.2 Mori, M., 4.3 Moriguchi, S., 2.5 Morimoto, C. N., 3.3 Morinaga, M., 4.2 Moring, I., 2.3 Moritz, W., 4.2 Morniroli, J. P., 2.5 Morris, E. P., 2.5 Morris, J., 3.5 Morris, R. L., 1.3 Moser, W. O. J., 1.3 Mosley, A., 4.4 Moss, B., 2.5 Moss, D. S., 4.2 Moss, G., 1.2 Moss, S. C., 4.2, 4.3 Mosser, A. G., 2.3 Motherwell, W. D. S., 3.3, 3.5 Motohashi, H., 5.3 Mouche, F., 2.5 Moussa, F., 4.4 Moustiakimov, M., 2.2 Muirhead, H., 2.3, 2.4 Mukamel, D., 4.4 Mukherjee, A. K., 2.2 Mullapudi, S., 2.5 Mu¨ller, H., 4.2 Mu¨ller, R., 5.1 Mu¨ller, U., 4.2 Munn, R. W., 3.4 Murakami, W. T., 2.3 Murdock, W. L., 1.3 Murray, W., 3.3 Murshudov, G. N., 2.3 Murthy, M. R. N., 2.3 Muus, I. T., 4.5 Myller-Lebedeff, W., 2.1 Nagabhushana, C., 4.4 Nagasawa, T., 2.5, 5.2 Nagem, R. A. P., 2.3 Naiki, T., 2.5 Nakatsu, K., 2.3 Namba, K., 4.5 Nambudripad, R., 4.5 Narayan, R., 1.3, 2.2, 2.4 Narayanan, B. A., 3.5 Natarajan, P., 2.5 Nathans, R., 4.2 Natterer, F., 1.3, 2.5 Navaza, J., 1.3, 2.2, 2.3 Nave, C., 1.3, 4.5 Navia, M. A., 2.4 Nawab, H., 1.3 Naya, S., 2.2, 4.5 Neder, R. B., 4.2 Neisser, J. Z., 4.5 Nelson, D. E., 1.3

670

Nelson, D. R., 4.2, 4.4 Nelson, H. M., 1.5 Neto, A. M. F., 4.4 Neubert, M. E., 4.4 Neubu¨ser, J., 1.3 Newham, R. J., 3.4 Newman, W. M., 3.3 Newsam, J. M., 3.5 Neyertz, S., 3.5 Ng, E. G., 2.5 Niall, H. D., 3.3 Nicastro, D., 2.5 Nicholson, P. B., 4.5 Nicholson, R. B., 2.5, 5.2 Nickell, S., 2.5 Nickitenko, A., 2.5 Nieh, Y.-P., 2.3 Nield, V. M., 4.2 Nierhaus, K. H., 2.5 Nigam, G. D., 2.1 Niggli, A., 1.3 Niimura, N., 4.2 Nijboer, B. R. A., 3.4, 3.5 Nilges, M., 2.3 Nishimura, D. G., 2.5 Nissen, P., 2.5 Nitsch, M., 2.5 Nitta, I., 2.2 Nityananda, R., 1.3, 2.2 Nixon, P. E., 2.3 Nolze, G., 3.3 Nonoyama, M., 4.3 Nordman, C. E., 1.3, 2.2, 2.3 North, A. C. T., 2.3, 2.4, 3.3 Norton, D. A., 2.2 Nose´, S., 3.5 Nowacki, W., 2.5 Nowell, H., 3.5 Nunzi, A., 2.2 Nussbaumer, H. J., 1.3 ¨ berg, B., 2.3 O Oberhettinger, F., 1.3 Oberteuffer, J. A., 5.3 Oberti, R., 2.4 Ocko, B. M., 4.4 Oda, T., 2.2 O’Donnell, T. J., 3.3 Oesterhelt, D., 2.5 Ogata, Y., 2.5 Ogawa, T., 2.5 Ogburn, K. D., 2.5 Ohara, M., 4.5 Ohshima, K., 4.2, 4.3 Ohtsuki, Y. H., 4.3, 5.1 Oikawa, T., 4.3 Okabe, A., 2.5 Okaya, J., 2.2 Okaya, Y., 2.2, 2.3, 2.4 O’Keefe, M. A., 2.5 Olafson, B. D., 3.3 Olmer, P., 4.1 Olsen, J., 3.5 Olsen, K. W., 2.3 Olson, A. J., 1.3, 2.3, 3.3 Olthof-Hazekamp, R., 1.4, 2.2 Omura, T., 4.3 Ono, A., 2.5 Onsager, L., 1.3, 4.4 Opat, G. I., 5.3 Opdenbosch, N. van, 3.3 Ord, K., 2.1 Orlando, R., 3.5 Orlov, S. S., 2.5 Orlova, E. V., 2.5 Ørmen, P.-J., 1.2 Ostermann, A., 4.2 O’Sullivan, J. D., 2.5 Oszla´nyi, G., 2.2 Ott, H., 2.2 Ottensmeyer, F. P., 2.5 Overduin, M., 2.5 Overhauser, A. W., 4.2, 5.3 Ozawa, T. C., 3.3

AUTHOR INDEX Pabst, M., 4.1 Paciorek, W. A., 4.6 Pa¨hler, A., 2.2 Paley, R. E. A. C., 1.3 Palleschi, V., 4.4 Palmer, M. R., 2.5 Palmer, R. A., 2.3, 2.4 Palmer, S. B., 5.3 Pan, M., 2.5 Pan, Q., 2.5 Pandey, D., 4.2 Panepucci, E. H., 2.3 Pannu, N. S., 2.3 Pantelides, C. C., 3.5 Paoletti, A., 4.2 Pape, T., 2.5 Paradossi, G., 4.5 Park, H., 4.5 Parks, T. W., 1.3 Parlett, B. N., 2.5 Parmon, V. S., 2.5 Parodi, O., 4.4 Parsey, J. M. Jr, 4.2 Parthe´, E., 4.3 Parthasarathy, R., 2.3, 2.4 Parthasarathy, S., 2.1, 2.2, 2.4 Pascual-Montano, A., 2.5 Pashley, D. W., 2.5, 4.5, 5.2 Pastore, A., 3.3 Patel, J. R., 5.1 Patel, K., 3.5 Pattabiraman, N., 3.3 Pattanayek, R., 4.5 Patterson, A. L., 1.1, 1.3, 2.3, 2.4, 4.2 Patterson, C., 5.3 Paturle, A., 1.2 Patwardhan, A., 2.5 Pa¨tzold, H., 4.3 Paul, D., 2.5 Pauling, L., 1.3, 2.3 Paulmann, C., 4.2 Pauwels, R., 2.3 Pavelcı´k, F., 2.2 Pavlovitch, A., 4.6 Pavone, P., 4.1 Pawley, G. S., 4.1 Pearce, L. J., 3.3 Pearl, L. H., 3.3 Pearlman, D. A., 3.5 Pearson, J., 3.3 Pearson, K., 1.3 Pease, M. C., 1.3 Pedersen, B., 4.2 Pedersen, L., 3.5 Peerdeman, A. F., 2.2, 2.3, 2.4 Peierls, R. E., 4.4 Peisl, J., 4.2 Penczek, P. A., 2.5 Penning, P., 5.1 Penrose, R., 2.5, 4.6 Penzkofer, B., 4.2 Pepinsky, R., 1.3, 2.2, 2.3, 2.4 Perera, L., 3.5 Perez, S., 4.5 Pe´rez, S., 2.5 Perez-Mato, J. M., 1.4, 1.5, 4.2 Perham, R. N., 4.5 Perrakis, A., 2.3 Perram, J. W., 3.5 Perrier de la Bathie, R., 5.3 Pershan, P. S., 4.4 Perutz, M. F., 2.2, 2.3, 2.4 Peschar, R., 2.2 Petef, G., 2.3 Peters, C., 1.3 Petersen, H. G., 3.5 Petrascheck, D., 5.3 Petricek, V., 4.2, 4.6 Pe´troff, J. F., 5.3 Petrov, V. V., 1.3 Petrova, T. E., 2.3 Petrzˇı´lka, V., 5.3 Petsko, G. A., 1.3, 3.3

Pettersen, E. F., 2.5 Pezerat, H., 4.2 Pfaff, G., 3.3 Pflanz, S., 4.2 Pflugrath, J. W., 2.3 Phillips, D. C., 2.1, 2.2, 2.3, 2.4, 3.3 Phillips, J. C., 2.4 Phillips, S. E. V., 3.3 Phizackerley, R. P., 2.4 Phong, B. T., 3.3 Pickworth, J., 2.3 Pielartzik, H., 4.5 Pietila, L.-O., 3.4 Pietronero, L., 4.2 Pietsch, U., 2.5 Pifferi, A., 2.2 Pigram, W. J., 4.5 Pillardy, J., 3.5 Pilling, D. E., 1.3 Pindak, R., 4.4 Pink, M. G., 3.3 Pinsker, Z. G., 2.5, 5.1 Piquemal, J. P., 3.5 Pirie, J. D., 4.1 Piro, O. E., 2.2 Plano, R. J., 4.4 Plotnikov, A. P., 2.5 Plotnikov, V. P., 2.5 Pochon, F., 2.5 Podjarny, A. D., 2.2, 2.3, 2.4 Podurets, K. M., 5.3 Pogany, A. P., 2.5 Pokrovsky, V. L., 4.4 Polder, D., 5.1 Polidori, G., 2.2 Polikarpov, I., 2.3 Poljak, R. J., 2.3 Pollack, H. O., 1.3 Pollock, E. L., 3.5 Ponder, J. W., 3.5 Poole, C. P., 3.3 Popa, N. C., 4.1 Pople, J. A., 1.2 Popp, D., 4.5 Porter, T. K., 3.3 Porter, W., 3.5 Portier, R., 2.5, 5.2 Potenzone, R., 3.3 Potter, C. S., 2.5 Potterton, E. A., 3.3 Potts, R. B., 1.3 Pouget, J. P., 4.2 Powell, B. M., 4.2 Powell, M. J. D., 2.5 Prandl, W., 4.2 Prange, T., 2.3 Prasad, B. V. V., 2.5 Pratt, L. R., 3.5 Press, W., 4.2, 4.6 Preston, A. R., 2.5 Price, L. S., 3.5 Price, S. L., 3.5 Prick, A. J., 2.2 Prins, J. A., 2.4, 4.2, 5.1 Proffen, Th., 4.2 Prosen, R. J., 1.3, 2.3 Prost, J., 4.4 Prout, C. K., 3.3 Provencher, S. W., 2.5 Pryor, A. W., 4.1, 4.6 Pulay, P., 3.5 Puliti, P., 5.3 Pullan, L., 2.5 Purisima, E. O., 3.3 Pustovskikh, A. I., 2.5 Pusztai, L., 4.2 Pynn, R., 4.1 Qian, C., 2.3 Quandalle, P., 1.3 Quick, J., 3.5 Quilichini, M., 4.2 Quiocho, F. A., 2.5, 3.3 Qurashi, M. M., 1.3

Rabinovich, D., 2.1, 2.3 Rabinovich, S., 2.1 Rabson, D. A., 4.6 Rackham, G. M., 2.5 Rader, C. M., 1.3 Radermacher, M., 2.5 Radha, A., 4.5 Radhakrishnan, R., 3.3 Radi, G., 2.5 Radons, W., 4.2 Rae, A. D., 2.2, 2.3 Raghavacharyulu, I. V. V., 1.5 Raghavan, N. V., 2.4 Raghavan, R. S., 2.4 Rahman, S. H., 4.2 Raimondi, D. L., 1.2 Raı¨tman, E. A., 5.3 Raiz, V. Sh., 2.4 Raja, V. N., 4.4 Rajagopal, H., 2.4 Rajashankar, K. R., 2.3 Ramachandran, G. N., 2.2, 2.3, 2.4, 5.1 Ramagopal, U. A., 2.3 Raman, C. V., 4.1 Raman, S., 2.2, 2.3, 2.4 Ramaseshan, S., 2.3, 2.4 Ramaswamy, S., 4.4 Rango, C. de, 2.2, 2.4 Rao, R. R., 1.3 Rao, S. N., 2.3 Rao, S. T., 3.3 Raselli, A., 3.3 Rasmussen, B., 5.1 Rasmussen, K., 3.4 Ratna, B. R., 4.4 Rauch, H., 5.3 Raum, K., 5.3 Ravelli, R., 2.2 Rawiso, M., 4.4 Rayleigh (J. W. Strutt), Lord, 1.3, 2.1 Rayment, I., 2.3, 3.3 Read, R. J., 2.3 Redlack, A., 3.5 Rees, A. L. G., 2.5 Rees, D. C., 2.3 Refaat, L. S., 2.3 Reid, T. J. III, 2.3 Reif, F., 1.3 Reijen, L. L. van, 1.3 Reiner, I., 1.3 Reiss-Husson, F., 4.4 Reman, F. C., 4.4 Remillard, B., 4.5 Ren, J., 2.3 Ren, P. Y., 3.5 Renka, R., 2.5 Renninger, M., 5.1 Reuber, E., 2.5 Revol, J. F., 2.5 Rez, P., 2.5, 4.3, 5.2 Rhyner, J., 4.6 Ricci, R., 4.2 Rice, L. M., 2.3 Rice, S. O., 1.3 Richardson, J. S., 3.3 Richardson, J. W., 2.2 Richardson, R. M., 4.4 Rickert, S. E., 4.5 Riddle, A. C., 2.5 Riekel, C., 4.2 Riesz, M., 1.3 Rietveld, H. M., 4.2 Riley, D., 3.5 Rimmer, B., 2.3 Rini, J. M., 2.3 Rivard, G. E., 1.3 Rixon, F. J., 2.5 Robertson, J. H., 2.3 Robertson, J. M., 1.3, 2.3, 2.4, 3.2 Robinson, G., 1.3, 3.2 Rodewald, M., 4.3 Rodgers, J. R., 3.3 Rodgers, J. W., 2.4 Rodrigues, A. R. D., 5.3

671

Rodriguez-Carvajal, J., 3.3 Roetti, C., 1.2, 3.5 Rogers, D., 2.1, 2.2, 2.3 Rokhsar, D. S., 4.6 Rollett, J. S., 1.3, 3.3 Roseman, A. M., 2.5 Rosen, J., 1.5 Rosenbluth, A. W., 4.2 Rosenbluth, M. N., 4.2 Rosenstein, R. D., 2.3 Ross, C., 2.3 Rosshirt, E., 4.2 Rossmann, M. G., 1.3, 2.2, 2.3, 2.4, 3.3 Rossouw, C. J., 2.5, 4.3 Rouiller, I., 2.5 Roux, D., 4.4 Roversi, P., 2.3, 3.5 Rowlands, D., 2.3 Rowlands, R. J., 4.5 Rozenfeld, A., 2.5 Rueckert, R. R., 2.3 Ruedenberg, K., 1.2 Ruf, T., 4.1 Rugman, M., 4.4 Ru¨hle, M., 2.5 Ruijgrok, Th. W., 3.5 Ruiz, T., 2.5 Ruland, W., 4.2 Rust, H.-P., 2.5 Rustichelli, F., 5.3 Ruston, W. R., 4.2 Rybnikar, F., 4.5 Ryde´n, L., 2.3 Rypniewski, W., 2.5 Ryskin, A. I., 2.5 Saad, A., 2.5 Sabine, T. M., 4.2 Sackmann, H., 4.4 Sadashiva, B. K., 4.4 Sadoc, J. F., 4.4 Sadova, N. I., 2.5 Safinya, C. R., 4.4 Safran, S. A., 4.4 Sagui, C., 3.5 Sahni, V. C., 4.1 Saibil, H. R., 2.5 Saito, M., 2.5 Saito, P., 2.5 Saito, R., 2.5 Saito, Y., 2.2, 2.3 Saitoh, K., 2.5 Saka, T., 5.1 Sakabe, K., 2.2 Sakabe, N., 2.2 Sakurai, K., 4.2 Salamon, M. B., 4.2 Sande, G., 1.3 Sander, B., 2.5, 5.3 Sandonis, J., 5.3 Sands, D. E., 1.1, 3.1 Sanjurjo, J. R., 2.5 Sansom, C., 4.2 Sarikaya, M., 2.5 Sarko, A., 4.5 Sasada, Y., 2.3, 3.3 Sato, H., 4.2 Satow, Y., 2.4 Saunders, M., 2.5 Saunders, V. R., 3.5 Saupe, A., 4.4 Sauvage, M., 4.3, 5.3 Saxton, W. O., 2.5 Sayre, D., 1.3, 2.2, 2.3, 2.4, 2.5, 4.5 Scaringe, P. R., 4.2 Scaringe, R. P., 2.5 Scatturin, V., 2.5 Schacher, G. E., 3.4 Schaetzing, R., 4.4 Schaffitzel, C., 2.5 Scha¨rpf, O., 4.2 Schatz, M., 2.5 Schenk, H., 2.2 Scheraga, H. A., 3.3, 3.5

AUTHOR INDEX Scheres, S. H. W., 2.5 Scheringer, C., 1.2 Scherm, R., 5.3 Scherzer, O., 2.5 Schevitz, R. W., 2.2, 2.3, 2.4 Schilling, J. W., 2.3 Schiltz, M., 2.3 Schiske, P., 2.5 Schlenker, M., 5.3 Schmatz, W., 4.2 Schmidt, H. H., 5.3 Schmidt, M. U., 3.5 Schmidt, R., 2.5 Schmidt, T., 2.3 Schmidt, W. C. Jr, 3.3 Schnabel, W., 4.2 Schneider, A. I., 4.5 Schneider, T. R., 2.2, 2.3 Schoenberg, I. J., 3.5 Schoenborn, B. P., 2.4 Schofield, P., 4.1 Schomaker, V., 1.1, 1.2, 1.3, 2.3, 2.5, 3.2 Schomberg, H., 2.5 Schoone, J. C., 2.4 Schrader, H., 4.2 Schramm, H. J., 2.5 Schro¨der, M., 3.3 Schro¨der, R., 2.5 Schroder, R. R., 2.5 Schroeder, M. R., 1.3 Schroeer, B., 2.5 Schuessler, H. W., 1.3 Schu¨ler, M., 2.5 Schuller, D. J., 2.3 Schulz, H., 1.2, 4.2 Schulze, G. E. R., 5.3 Schumacker, R. A., 3.3 Schuster, S. L., 4.1 Schutt, C. E., 1.3, 2.3 Schwager, P., 1.3, 2.4 Schwartz, L., 1.3 Schwartz, L. H., 4.2 Schwarzenbach, D., 1.2 Schwarzenberger, R. L. E., 1.3 Schweika, W., 4.2 Schweizer, B., 3.5 Schweizer, J., 5.3 Scott, W. R., 1.3 Scudder, M. L., 2.4 Sears, V. F., 4.2, 5.3 Secomb, T. W., 2.5 Sedla´kova´, L., 5.3 Seidl, E., 5.3 Seitz, E., 4.2 Seitz, F., 1.4 Sekii, H., 2.5, 5.2 Selinger, J. V., 4.4 Sellar, J. R., 5.2 Semiletov, S. A., 2.5 Senechal, M., 4.6 Serrano, J., 4.1 Sethna, J. P., 4.4 Sezan, M. I., 2.5 Sha, B. D., 2.5 Shaffer, P. A. Jr, 1.3 Shaikh, T., 2.5 Shakked, Z., 2.1, 2.3 Shan, N., 3.5 Shan, Y. B., 3.5 Shankland, K., 2.5, 4.5 Shannon, C. E., 1.3, 2.5 Shannon, M. D., 2.5 Shao-Hui, Z., 2.2 Shapiro, A., 3.3 Shappell, M. D., 2.3 Shashidhar, R., 4.4 Shashua, R., 2.1 Shastri, S. D., 5.1 Shaw, D. E., 3.5 Sheat, S., 2.3 Shechtman, D., 2.5, 4.6 Sheldrick, G. M., 2.2, 2.3 Shen, Q., 5.1 Shen, Y. R., 4.4

Shenefelt, M., 1.3 Sheriff, S., 2.3, 2.4 Sherry, B., 2.3 Sherwood, J. N., 4.2 Shilov, G. E., 1.3 Shil’shtein, S. Sh., 5.3 Shimanouchi, T., 3.3 Shipley, G. G., 4.4 Shirane, G., 4.1, 4.2 Shmueli, U., 1.3, 1.4, 2.1, 3.1, 3.2 Shoemaker, C. B., 2.3 Shoemaker, D. P., 1.3, 2.3 Shoemaker, V., 2.5 Shohat, J. A., 1.3 Shore, V. C., 2.4 Shortley, G. H., 1.2 Shotton, M. W., 4.5 Shtrikman, S., 4.4 Shull, C. G., 5.3 Sicignano, A., 2.3 Siddons, D. P., 5.3 Sidorenko, S. V., 2.5 Sieber, W., 3.3 Siegel, B. M., 2.5 Siegrist, T., 3.3 Sieker, L. C., 2.4 Sigaud, G., 4.4 Sigler, P. B., 2.2, 2.3, 2.4 Sigworth, F. J., 2.5 Sikka, S. K., 2.4 Silcox, J., 4.3 Siliqi, D., 2.2 Silverman, H. F., 1.3 Sim, G. A., 2.2, 2.3, 4.5 Simerska, M., 2.2 Simonov, V. I., 2.2, 2.3 Simonson, T., 2.3 Simpson, A. A., 2.3 Simpson, P. G., 2.3 Singh, A. K., 2.3, 2.4 Singleton, R. C., 1.3 Sinha, S. K., 3.5, 4.4 Sint, L., 2.2 Sippel, D., 5.3 Siripitayananon, J., 4.2 Sirota, E. B., 4.4 Sirota, M. I., 2.5 Sivardie`re, J., 5.3 Sivy´, J., 2.2 Sixma, T. K., 2.3 Sjo¨gren, A., 2.5 Skehel, J. J., 2.3 Skilling, J., 1.3 Skoglund, U., 2.3 Skoulios, A., 4.4 Skovoroda, T. P., 2.3 Skuratovskii, I. Y., 4.5 Slater, L. S., 1.5 Sleight, M. E., 2.5 Slepian, D., 1.3 Sluckin, T. J., 4.4 Sly, W. G., 1.3 Smaalen, S. van, 4.2, 4.6 Small, D., 4.4 Smit, B., 3.5 Smith, A. B. III, 4.4 Smith, D. J., 2.5 Smith, E. R., 3.5 Smith, G., 3.3 Smith, G. D., 2.2 Smith, G. S., 4.4 Smith, G. W., 4.4 Smith, J. C., 4.2 Smith, J. L., 2.2, 2.3 Smith, J. V., 1.5 Smith, M. B. K., 2.3 Smith, R. H. Jr, 2.3 Smith, T., 4.1 Smith, W., 3.5 Smits, J. M. M., 2.2 Smoluchowski, R., 1.5 Sneddon, I. N., 1.3 Soboleva, A. F., 2.5 Socolar, J. E. S., 4.6

Soeter, N. M., 2.3 Sokol’skii, D. V., 5.3 Soldani, M., 3.5 Solitar, D., 1.3 Solmon, D. C., 2.5 Solomon, L., 4.4 Somenkov, V. A., 5.3 Somers, D., 2.3 Sondhauss, P., 5.1 Soni, R. P., 1.3 Sorensen, L. B., 4.4 Sorzano, C. O. S., 2.5 Soyer, A., 3.3 Spackman, M. A., 3.5 Spagna, R., 2.2 Spahn, C. M. T., 2.5 Spanton, S., 3.5 Sparks, C. J., 1.2, 4.2 Sparks, R. A., 1.3, 3.3 Speake, T. C., 1.3 Speakman, J. C., 2.3 Spek, A. L., 2.2, 3.3 Spellward, P., 2.5 Spence, J. C. H., 2.5, 4.3 Spiegel, M. R., 2.1 Spink, J. A., 2.5 Spiwek, H., 3.5 Sprang, S. R., 3.3 Sprecher, D. A., 1.3 Springer, T., 4.2, 4.4 Sprokel, G. E., 4.4 Sproull, R. F., 3.3 Squire, J. M., 2.5, 4.5 Squires, G. L., 1.2, 4.1, 5.3 Srinivasan, R., 2.1, 2.2, 2.4 Staden, R., 1.3, 2.3, 3.3 Stahl, S. J., 2.5 Stammers, D., 2.3 Stanley, E., 2.2, 4.5 Stark, H., 2.5 Stark, W., 4.5 Stasiak, A., 2.5 Stassis, C., 5.3 States, D. J., 3.3 Staudenmann, J. L., 5.3 Stauffacher, C. V., 2.3 Steeds, J. W., 2.5 Stegemeyer, H., 4.4 Steger, W., 4.2 Stegun, I. A., 2.1 Steigemann, W., 1.3, 2.4 Stein, Z., 2.1 Steinberger, I. T., 4.2 Steinhardt, P. J., 2.5, 4.6 Steinkilberg, M., 2.5 Steinrauf, L. K., 2.3 Steitz, T. A., 2.3, 2.5 Stence, C. N., 2.3 Stephanik, H., 5.1 Stephen, M. J., 4.4 Stephens, P. W., 4.2, 4.4, 4.6 Stephenson, G. B., 4.4 Stephenson, R., 3.3 Stetsko, Yu. P., 5.1 Steurer, W., 2.5, 4.2, 4.6 Steven, A. C., 2.5 Stevens, E. D., 1.2 Stewart, A. T., 4.1 Stewart, R. F., 1.2 Stokes, H. T., 1.5 Stone, A. J., 3.5 Stoops, J. K., 2.5 Storks, K. H., 4.5 Storoni, L. C., 2.3 Stout, G. H., 1.3, 2.3 Stragler, H., 4.4 Strahs, G., 2.3 Strandberg, B., 2.3 Strandberg, B. E., 1.3, 2.3, 2.4 Stra¨ssler, S., 4.2 Stratonovich, R. L., 1.2 Stroud, R. M., 2.4 Stroud, W. J., 4.5 Strzelecki, L., 4.4

672

Stuart, A., 1.2, 2.1 Stuart, D., 2.3 Stubbs, G., 4.5 Stubbs, G. J., 4.5 Sturkey, L., 5.2 Sturtevant, J. M., 4.5 Su, Z., 1.2 Suck, D., 2.3, 3.3 Sugihara, K., 2.5 Sun, W., 2.5 Sundaralingam, M., 3.3 Sundaram, K., 3.3 Sundberg, M., 2.5 Suresh, K. A., 4.4 Suryan, G., 1.3 Suski, T., 4.1 Sussman, J. L., 1.3, 2.4 Sutcliffe, D. C., 3.3 Sutherland, I. E., 3.3 Su¨to, A., 2.2 Suzuki, E., 4.5 Suzuki, H., 4.5 Suzuki, S., 2.5 Svergun, D. I., 2.5 Swaminathan, S., 2.3, 3.3 Swanson, S. M., 3.3 Swartzrauber, P. N., 1.3 Swoboda, M., 4.3 Symmons, M. F., 4.5 Szego¨, G., 1.3 Szillard, L., 2.4 Tadokoro, H., 4.5 Taftø, J., 4.3 Taftø, T., 2.5 Taguchi, I., 2.2 Tajbakhsh, A. R., 4.4 Takagi, K., 2.5 Takagi, S., 2.5, 5.1, 5.3 Takahashi, H., 5.2 Takahashi, M., 2.5 Takahashi, T., 5.3 Takaki, Y., 4.2 Takano, T., 1.3 Takayoshi, H., 2.5 Takenaka, A., 3.3 Takeuchi, Y., 2.3, 2.4 Talapov, A. L., 4.4 Tama, F., 2.5 Tamarkin, J. D., 1.3 Tanaka, I., 4.2 Tanaka, K., 1.2 Tanaka, M., 2.5, 5.2 Tanaka, N., 2.3, 2.4, 4.3 Tanaka, S., 4.5 Tang, G., 2.5 Tanji, T., 2.5 Tanner, B. K., 5.1 Tao, X., 2.3 Tao, Y., 2.3 Tardieu, A., 4.4 Tarento, R. J., 4.4 Tasset, F., 5.3 Tasumi, M., 3.3 Tatarinova, L. I., 4.5 Tate, C., 2.2, 2.3 Taupin, D., 5.3 Tavares, P., 2.5 Taveau, J. C., 2.5 Taylor, C. A., 1.3, 1.4, 4.2 Taylor, D. J., 2.2 Taylor, G. H., 4.4 Taylor, R., 3.1, 3.3 Taylor, W. J., 2.3 Tchoubar, D., 4.2 Teeter, M. M., 2.3, 2.4 Teller, A. H., 4.2 Teller, E., 4.2, 4.5 Temperton, C., 1.3 Templeton, D. H., 2.4 Templeton, L. K., 2.4 Ten Eyck, L. F., 1.3, 2.3 Teplyakov, A., 2.3 Terauchi, M., 2.5

AUTHOR INDEX Terwilliger, T. C., 2.2, 2.3, 2.4 Teworte, R., 5.1 Thierry, J. C., 2.3 Thoen, J., 4.4 Thomas, D. J., 3.3 Thomas, K. M., 4.4 Thomsen, K., 3.3 Thon, F., 2.5 Thorpe, M. F., 4.2 Thouless, D. G., 4.4 Thuman, P., 2.2 Tibbals, J. E., 4.2 Tikhonov, V. I., 1.2 Tildesley, D. J., 3.5 Timmer, J., 2.5 Tinh, N. H., 4.4 Tinnappel, A., 2.5 Tirion, M., 4.5 Titchmarsh, E. C., 1.3 Tivol, W. F., 2.5 Tivol, W. T., 4.5 Tjian, R., 3.3 Toby, B. H., 3.3 Toeplitz, O., 1.3 Tolimieri, R., 1.3 Tollin, P., 2.3 Tolstov, G. P., 1.3 Tomimitsu, H., 5.3 Toner, J., 4.4 Tong, L., 2.3 Toniolo, L., 4.2 Tonomura, A., 2.5 Torrisi, A., 3.5 Tosoni, L., 2.5 Toukmaji, A., 3.5 Toupin, R., 2.2 Tournaire, M., 2.5 Tournarie, M., 5.2 Tramontano, A., 3.3 Traub, W., 2.2 Tre`ves, F., 1.3 Trickey, S. B., 3.5 Trommer, W. E., 2.3 Tronrud, D. E., 1.3, 2.3 Trueblood, K. N., 1.1, 1.2, 1.3, 2.3 Trus, B. L., 2.5 Truter, M. R., 1.3 Tsai, A. P., 2.5 Tsao, J., 2.3 Tsernoglou, D., 3.3 Tsipursky, S. I., 2.5 Tsirelson, V. G., 2.5 Tsoucaris, G., 2.2, 2.4 Tsuda, K., 2.5 Tsuji, M., 2.5, 4.5 Tsujimoto, S., 2.5 Tsukihara, T., 2.3, 3.3 Tsuprun, V. L., 2.5 Tsuruta, H., 2.5 Tukey, J. W., 1.3 Tulinsky, A., 2.4 Turberfield, K. C., 4.2 Turkenburg, M. G., 2.3 Turner, J., 3.3 Turner, J. N., 2.5 Turner, P. S., 2.5, 4.5 Typke, D., 2.5 Uchida, Y., 4.3 Ueki, T., 2.4 Ueno, K., 2.5 Uhrich, M. L., 1.3 Ungaretti, L., 2.4 Unge, T., 2.3 Unser, M., 2.5 Unwin, P. N. T., 2.5 Uragami, T., 5.1 Urzhumtsev, A., 2.3 Usha, R., 2.3 Ushigami, Y., 5.3 Uson, I., 2.3 Utemisov, K., 5.3 Uyeda, N., 2.5 Uyeda, R., 2.5, 4.3

Vaara, I., 2.3 Vacher, R., 4.1 Vagin, A. A., 2.3, 2.5 Vainshtein, B. K., 2.5, 4.2, 4.5 Van Dael, W., 4.4 Van der Pol, B., 1.3 Van der Putten, N., 2.2 Van Hove, L., 4.1, 4.3 Van Loan, C. F., 2.5 Van Tendeloo, G., 4.3 Vand, V., 1.3, 2.5, 4.5 Varady, W. A., 4.4 Varghese, J. N., 1.3, 2.2 Varn, D. P., 4.2 Varnum, J. C., 2.3 Vartanyants, I. A., 5.1 Vaucher, C., 4.4 Vaughan, P. A., 2.2 Vedani, A., 3.3 Velazquez-Muriel, J., 2.5 Venkataraman, G., 4.1 Venkatesan, K., 2.4 Venuti, E., 3.5 Vereijken, J. M., 2.3 Vermin, W. J., 2.2 Vernoslova, E., 2.3 Verschoor, A., 2.5 Verwer, P., 3.5 Vibert, P. J., 4.5 Vickovic´, I., 2.2 Vijayan, M., 2.2, 2.3, 2.4 Vilkov, L. V., 2.5 Villain, J., 4.4 Vincent, R., 2.5 Vine, W. J., 2.5 Viterbo, D., 2.2 Vogel, R. H., 2.5 Volbeda, A., 2.3 Volkmann, N., 2.5 Volkov, A., 3.5 Von der Lage, F. C., 1.2 Vonderviszt, F., 4.5 Vonrhein, C., 2.3 Voronova, A. A., 2.5 Vos, A., 2.4 Vra´na, M., 5.3 Vriend, G., 2.3 Vries, T. A. de, 2.3 Vrublevskaya, Z. V., 2.5 Vulis, M., 1.3 Wagenknecht, T., 2.5 Wagner, E. H., 5.1 Waho, T., 4.3 Waite, J., 3.5 Wakabayashi, K., 4.5 Walian, P. J., 2.5 Walker, C. B., 4.1 Waller, I., 1.2 Walsh, G. R., 3.3 Walz, J., 2.5 Wang, B. C., 1.3, 2.3, 2.4 Wang, D. N., 2.5 Wang, H., 4.5 Wang, J., 4.4, 5.1 Wang, R., 4.2 Wang, X. J., 4.4 Wang, Z. L., 4.3 Ward, J. C., 1.3 Ward, K. B., 2.3 Wark, J. S., 5.1 Warme, P. K., 3.3 Warren, B., 1.3 Warren, B. E., 1.3, 4.2, 4.4 Warren, G. L., 2.3 Warren, S., 4.5 Warshel, A., 3.3 Waser, J., 1.3, 1.4, 3.1, 3.2 Watanabe, D., 2.5, 4.2, 4.3 Watanabe, E., 2.5 Watenpaugh, K. D., 1.4, 2.4 Watkin, D. J., 3.3 Watson, D. G., 3.3 Watson, G. L., 1.3

Watson, G. N., 1.3 Watson, H. C., 2.3 Watson, K. J., 1.2 Watson, W. T., 2.5 Wawak, R. J., 3.5 Waxham, M. N., 2.5 Webb, H., 2.5 Weber, H. J., 3.5 Weber, H. P., 3.5 Weber, T., 4.2 Weber, Th., 4.2 Weckert, E., 5.1 Wedemeyer, W. J., 3.5 Weeks, C. M., 2.2, 2.3, 4.5 Weidner, E., 4.2 Weikenmeier, A., 2.5 Weintraub, H. J. R., 3.3 Weinzierl, J. E., 2.2, 2.3, 2.4 Weiss, A. H., 4.4 Weiss, G. H., 1.3, 2.1 Weiss, R., 2.3 Weiss, R. J., 1.2 Weissberg, A. M., 2.2 Welberry, T. R., 4.2, 4.5 Welch, A., 2.5 Welch, P. D., 1.3, 2.5 Wells, M., 1.3, 1.4 Welsh, L. C., 4.5 Wenk, H.-R., 2.5 Wentowska, K., 4.4 Werner, S. A., 4.2, 5.3 Wesolowski, T., 3.3 West, J., 1.3 Westbrook, J. D., 1.4 Westhof, E., 2.5 Weyl, H., 1.3 Weymouth, J. W., 4.1 Whelan, M., 4.3 Whelan, M. J., 2.5, 4.3, 5.2 White, C., 3.5 White, H., 2.5 White, J. G., 2.3 White, P., 2.2 Whitfield, H. J., 5.2 Whittaker, E. J. W., 1.3 Whittaker, E. T., 1.3, 3.2 Widder, D. V., 3.4 Widom, H., 1.3 Wiener, N., 1.3 Wigner, E. P., 1.5 Wiley, D. C., 2.3 Wilfing, A., 5.3 Wilke, S., 3.5 Wilke, W., 4.2 Wilkins, M. H. F., 4.5 Wilkins, S. W., 1.3, 2.2 Williams, D. E., 3.4, 3.5, 4.1 Williams, G. J. B., 3.3 Williams, R. M., 4.4 Williams, T. V., 3.3 Willingmann, P., 2.3 Willis, B. T. M., 1.2, 4.1, 4.6 Willoughby, T. V., 3.3 Wilson, A. J. C., 1.3, 2.1, 2.2, 2.3, 2.4, 4.2, 4.5 Wilson, E. B., 1.1 Wilson, I. A., 2.3 Wilson, K. S., 2.1, 2.2, 2.3 Wilson, S. A., 5.3 Windsor, C. G., 4.2 Wingfield, P. T., 2.5 Winkler, F. K., 1.3, 2.3 Winkor, M. J., 4.4 Winograd, S., 1.3 Winter, W. T., 4.5 Wintgen, G., 1.5 Wintner, A., 1.3 Wipke, W. T., 3.3 Withers, R. L., 2.5, 4.2 Witt, C., 2.5 Wittmann, J. C., 2.5, 4.5 Wolf, E., 5.1 Wolf, J. A., 1.3 Wolff, P. M. de, 2.2, 2.5, 4.2, 4.6

673

Wonacott, A. J., 2.4, 4.5 Wondratschek, H., 1.3, 1.4, 1.5, 2.2 Wong, H. C., 2.5 Wong, S. F., 4.2 Woods, R. E., 2.5 Woodward, I., 2.3, 2.4 Woolfson, M. M., 1.3, 2.1, 2.2, 2.3, 2.5 Wooster, W. A., 4.2 Wostrack, A., 4.2 Wright, D. C., 4.6 Wright, M. H., 3.3 Wrighton, P. G., 4.4 Wrinch, D. M., 2.3 Wu, H., 2.3 Wu, M. N., 2.5 Wu, T. B., 4.2 Wu, X.-J., 2.5 Wu, Y. K., 2.5 Wu¨bbeling, F., 2.5 Wuensch, B. J., 4.2 Wunderlich, B., 4.5 Wunderlich, J. A., 2.3 Wyckoff, H. W., 1.3, 2.3, 4.5 Wynn, A., 3.3 Wynne, S. A., 2.5 Xia, D., 2.3 Xiang, S.-B., 2.5 Xiaodong, Z., 1.4 Xu, H., 2.2 Xu, P. R., 4.3 Xu, Y., 2.3 Yagi, N., 4.5 Yamamoto, A., 2.5, 3.3, 4.6 Yamamoto, N., 2.5 Yamashita, I., 4.5 Yamazaki, H., 5.1 Yang, C., 2.3, 2.5 Yang, W., 3.5 Yang, Y. W., 1.2 Yao, J.-X., 2.2, 2.5 Yeates, T. O., 2.3 Yelon, W. B., 5.3 Yessik, M., 4.2 Yin, Z. H., 2.5 Yin, Z. Y., 2.5 Yip, S., 4.1 Yonath, A., 2.2 York, D. M., 3.5 Yoshioka, H., 4.3 Yosida, K., 1.3 Youla, D. C., 2.5 Young, A. P., 4.4 Young, C. Y., 4.4 Young, R. A., 4.2 Yu, L. J., 4.4 Yuan, B.-L., 4.5 Yuen, C. K., 2.5 Zachariasen, W. H., 1.3, 1.4, 2.4, 5.1, 5.3 Zak, J., 1.5 Zalkin, A., 2.4 Zaluzee, N. J., 2.5 Zambianchi, P., 5.1 Zarka, A., 5.3 Zaschke, H., 4.4 Zassenhaus, H., 1.3 Zechmeister, K., 1.3, 2.5 Zegenhagen, J., 5.1 Zegers, I., 5.1 Zeilinger, A., 5.3 Zeitler, E., 2.5 Zelenka, J., 5.3 Zelepukhin, M. V., 5.3 Zellner, J., 5.1 Zelwer, C., 2.2 Zemanian, A. H., 1.3 Zemlin, F., 2.5 Zenetti, R., 4.2 Zeng, G. L., 2.5 Zernike, F., 4.2 Zeyen, C., 4.2, 5.3

AUTHOR INDEX Zhang, K. Y. J., 2.3 Zhang, W. P., 2.5 Zhao, Z. X., 2.5 Zheng, C.-D., 2.2, 2.5 Zheng, Y. L., 2.5 Zhong, Z.-Y., 2.5

Zhou, Z. H., 2.5 Zhu, J., 2.5 Zhu, Y., 2.5 Zhukhlistov, A. P., 2.5 Zicovich-Wilson, C. M., 3.5 Ziman, J. M., 1.1

Zinn-Justin, J., 4.4 Zlotnick, A., 2.5 Zobetz, E., 4.6 Zolotoyabko, E., 5.3 Zou, J.-Y., 2.5, 3.3 Zucker, I. J., 3.4

674

Zucker, U. H., 1.2 Zugenmaier, P., 4.5 Zuo, J. M., 2.5 Zvyagin, B. B., 2.5 Zwick, M., 1.3, 2.3, 2.4 Zygmund, A., 1.3

Subject index A posteriori probability, 512 A priori probability, 504 Ab initio phase determination, 273 Abbe theory, 303 Abel summation procedure, 46 Abelian groups, 42, 79 Aberrations, 303 Absolute configuration, 282, 285 Absolutely integrable functions, 27 Absorbing crystals, 629, 637–638, 640 Absorption coefficient, 634, 639 effective, 628 linear, 628 phenomenological, 303 Absorption edge, 284 Absorption function, 542 Absorption in electron diffraction, 362 Accelerated convergence, 449 formula via Patterson function, 454 Acceptance domain, 596 Acentric reflections, 74 Acoustic modes, 486 Action, 68 Additive reindexing, 61 Adiabatic approximation, 484 Adjusted coefficients, 180 Affine change of coordinates, 35 Affine change of variables, 40 Affine space-group type, 177 Affine transformation, 120 Agarwal’s FFT implementation of the Fourier method, 98 Alfalfa mosaic virus, 259 Algebra of functions, 75 Algebraic integers, 78, 82 Algebraic number theory, 82 Algebraic reconstruction technique (ART), 370 Aliasing, 47, 49, 93 Alignment of electron-microscopy images, 379 Allowed origins, 215 ‘Almost everywhere’, 26 Analytical methods of probability theory, 102 Angle between two vectors, 404 Angles Eulerian, 262, 420 spherical, 262 Anisotropic displacement parameters, 443 Anisotropic displacement tensors, 6 Anisotropic fluid, 547 Anisotropic Gaussian atoms, 63 Anisotropic temperature factors, 73 Anisotropic weights, 413 Annular dark-field detector, 305 Anomalous absorption, 633, 659 Anomalous difference, 288, 291 Anomalous dispersion (scattering), 255–256, 282, 284 integration with direct methods, 237 Patterson function, 257 Anomalous scatterers, 64, 74, 256, 284, 286 Anomalous transmission effect, 633 Antiferromagnetic domains, 661 Anti-nodes of standing waves, 633 Antisymmetric tensor, 6 Aperiodic crystals, 590 disorder diffuse scattering from, 526 ideal, 590 Aperiodic structure, 590 Apparent noncrystallographic symmetry, 267 Approximate helix symmetry, 570 Approximations adiabatic, 484 Bethe, second, 302 Born, first-order, 10, 300 Born, second-order, 11 Born–Oppenheimer, 17 Edgeworth, 21 forward-scattering, 301 harmonic, 484 kinematical, 62, 300–301, 359, 394, 583, 658 phase-grating, 652

phase-object, 301, 542 projected charge-density, 305 projection, 648 saddlepoint, 102–103 seven-beam, 652 small-angle-scattering, 300 three-beam, 652 two-beam, 302, 649 two-beam dynamical, 394 weak-phase-object, 304 Area detector, 294 Argand diagram, 283 Arithmetic classes, 71 of representations, 71 Arithmetic crystal class, 177 Arms of star, 178 ART (algebraic reconstruction technique), 370 Artificial temperature factor, 94, 100 Aspherical-atom form factor, 14 Aspherical multipole refinement, 459 Associated actions in function spaces, 69 Associativity properties of convolution, 100 Assumption of independence, 209 Assumption of uniformity, 203, 209 Astigmatism, 378 Asymmetric carbon atom, 285 Asymmetric images, 365 Asymmetric unit, 69, 72, 180 noncrystallographic, 253 Asymmetry ratio, 631 Asymptotic crystal shape, 470 Asymptotic distribution of eigenvalues of Toeplitz forms, 45, 68 Asymptotic expansions and limit theorems, 103 of Gram–Charlier and Edgeworth, 105 Atom-centred spherical harmonic expansion, 12 Atomic characteristic functions, 211 Atomic electron densities, 75 Atomic error matrix, 416 Atomic force-constant matrix, 485 Atomic form factor, 10 X-ray, 293 Atomic scattering factor, 10, 284 spherical, 10 Atomic scattering length, 11 Atomic surface, 590, 596 Atomic temperature factor, 17 ATOMS, 443–445 Autocorrelation, 65 Autocorrelation function, 388 Automated Patterson-map search, 389 Automorphism, 69, 71 Auxiliary basis set expansion, 479 Auxiliary basis set fitting, 459 Average difference cluster method, 518 Average intensity of general reflections, 195 of zones and rows, 196 Average multiples for point groups, 197 Averaged electron density, 274 Axial disorder, 517 B-splines, 476 approximation to trigonometric functions, 477 derivatives of, 477 Fourier transforms of, 477 recursion for, 477 tensor product, 480 B3LYP basis set, 479 Back-shift correction, 96 Back surface, 639 Background diffraction, accurate subtraction of, 575 Backprojection, 367 filtered, 371 Backward convolution theorem, 44, 75 Bacterial rhodopsin, 274 Balls&Sticks, 443–444 BALSAC, 443–445

675

Banach spaces, 28 Band-limited function, 49 Base-centred lattices, 90 Bases Cartesian, 7 contravariant, 5–6 covariant, 5–6 direct and reciprocal, relationships between, 3 mutually reciprocal, 2–3 primitive, 177 reference, choice of, 7 Basic crystallographic computations, 91 Basic domain, 180 Basic structure, 591 Basis vectors, contravariant, 405 Bayesian statistical approach to the phase problem, 106 Beevers–Lipson factorization, 58, 76–77 Beevers–Lipson strips, 76, 93 Bessel’s inequality, 47 Best Fourier, 91, 290 Best phase, 290 Best plane, 410 Beta distribution first kind, 201 second kind, 201 Bethe approximation, second, 302 Biaxial nematic order, 549 Bieberbach theorem, 68 Bijvoet differences, 257 Bijvoet equivalents, 286–287 Bijvoet pair, 285–286 Bilder, 440 Binary systems, distortions in, 522 Binding energy, 284 Bloch-wave formulation, 650 Bloch waves, 300, 628 Bloch’s theorem, 9, 486 alternative form of, 9 Blow and Crick formulation, 290 Body-centred lattices, 90 Body-diagonal axes, 128 Bond angles, 437 Bond orientational order, 547 Booth’s differential Fourier syntheses, 96 Booth’s method of steepest descents, 96 Bootstrap technique, 388 Borie–Sparks method, 523 Born approximation first-order, 10, 300 second-order, 11 Born–Oppenheimer approximation, 17 Born series, 300, 651 expansion, 543 Born–von Karman boundary conditions, 177 Born–von Ka´rma´n theory, 484 Borrmann effect, 303, 633 Borrmann triangle, 641 Boundary conditions, 628, 639 at exit surface, 635 periodic, 479 Bounded projections, 67, 92 Bounded subset, 25 Boy’s function, 471 BP (bright-field pattern), 310 Bragg case, 631 Bragg’s law, departure of incident wave from, 630 Bragg–Lipson charts, 93 Branch, 485 Bravais lattices centred, 121 direct and reciprocal, 121 Bright-field image intensity, 362 Bright-field pattern (BP), 310 Brillouin zone, 9, 485 first, 178 Bulk plasmon excitation, 300 Burg entropy, 68 Burnside’s theorem, 71 Butterfly loop, 53

SUBJECT INDEX Calculus of asymmetric units, 79 operational, 28 Cameron, 443–444 CaRIne, 443–444 Carpet of cross-vectors, 260 Cartesian basis, 7 Cartesian coordinate system, 404 Cartesian coordinates, 418, 445 Cartesian frames of reference, 5 Cartesian product, 25, 42 Cartesian system, transformation to, 3 Cauchy kernel, 46 Cauchy–Schwarz inequality, 27, 47 Cauchy sequence, 26 Cauchy’s theorem, 104 CBED (convergent-beam electron diffraction), 307 coherent, 323, 333 CCP14 (Collaborative Computational Project Number 14), 443 Cell constants, 576 Central-limit theorem, 103, 198 Lindeberg–Le´vy version, 204 Central section theorem, 367 Centre-of-mass translational displacements, 525 Centre of symmetry, false, 115 Centred Bravais lattice, 121 Centred lattices, 73 Centric reflections, 73 Centring effect of, 196 translations, 121 type, 121 Centrosymmetric projections, 251–252 Centrosymmetry, status of, 123 Cesa`ro sum, 46 Chain rule, 99 Chain trace, 441 Chains, flexible, 436 Change-of-basis matrix, 123 Change of crystal axes, 120 Channelling pattern, 544 Characteristic functions, 102, 197, 208 atomic, 211 Charge densities, Coulomb energy of, 474 Charge distributions, Gaussian, 461, 471–472 CHARMM, 442 CHEMGRAF, 439 Chemical correctness of polypeptide fold, 273 Chem-X, 442 Chinese remainder theorem (CRT), 54, 61, 82 for polynomials, 57, 84 reconstruction, 54 reconstruction formula, 57 Chirality, 285 Choice of reference bases, 7 Cholesteryl iodide, 249 -Chymotrypsin, 271 Circular harmonic expansions, 100 Classical Thomson scattering, 10 Classification of crystallographic groups, 70 Classification of electron-microscopy images, 381 Clebsch–Gordan coefficients, 17 Closed point group, 258 Closed subset, 25 Cluster model, 545 Clustering, 518 Clustering algorithms, 380 Clusters, 500 Cochran’s Fourier method, 96 Cocycle, 85 Coherence, 434 Coherence length, 568 Coherent convergent-beam electron diffraction, 323, 333 Coherent scattering, 488 Collaborative Computational Project Number 14 (CCP14), 443 Column part, 177 Common line, 382 Communication, statistical theory of, 104 Commutative ring, 54 Compact Gaussians, 472 Compact subset, 25

Compact support, 25, 36, 44 distributions with, 30, 40, 43–44, 47 Complement of the incomplete gamma function, 450– 451 Complete normed space, 28 Complete vector spaces, 26 Completely reducible matrix group, 176 Complex antisymmetric transforms, 87 Complex scattering factor, 255 Complex symmetric transforms, 87 Components of vector products, 405 Components of vectors, 5 Composite lattice, 449, 452 Composite structure, 593 Compound nucleus, 11 Compound transformations, 429 Compton scattering, 489 Computational and algebraic aspects of mutually reciprocal bases, 4 Computational cost of Ewald direct sum, linear scaling of, 474 Computer-adapted space-group symbols, 117, 122, 127 Computer-algebraic languages, 122 Computer architecture, 52, 61 Computer simulations of diffuse scattering, 528 Condensed ring systems, 437 Conditional convergence, 460 Conditional pair probability, 504 Conformational variability, 388 Conforming/nonconforming disorder, 518 Conformons, 431 Conjugacy classes of subgroups, 69 Conjugate and parity-related symmetry, 85 Conjugate distribution, 104–105 Conjugate families of distributions, 106 Conjugate gradient method, 436 Conjugate symmetry, 35, 40 Conjugation, 69 Connectivity drawing, 435 implied, 435 logical, 435 structural, 435 Connectivity tables, 435 Connectivity tree, 439 Consistency condition, 39 Constant Q mode, 490 Constraints, 436, 440–441 on interpretation of Patterson functions, 258 Continuous diffraction on layer lines, 575 Continuum dielectric medium, 467 Contragredient, 40 of a matrix, 35 Contrast transfer function, 370, 376 Contravariant bases, 5–6 Contravariant basis vectors, 405 Contravariant components, 5 Conventional coefficients, 180 Convergence accelerated, 449 accelerated, formula via Patterson function, 454 conditional, 460 of distributions, 30 of Fourier series, 45 Convergence-accelerated direct sum, 452 Convergence method, 231 Convergent-beam electron diffraction (CBED), 307 coherent, 323, 333 Conversion of translations to phase shifts, 35 Convolution, 64, 244 associativity properties of, 100 cyclic, 51, 55 of distributions, 34 of Fourier series, 44 of probability densities, 102 of two distributions, 34 Convolution property, 35, 51 Convolution techniques, 392 Convolution theorem, 37, 42, 44, 46, 67, 102, 106 backward version, 44, 75 forward version, 44, 64–65, 75 Convolution theorems with crystallographic symmetry, 75

676

Cooley–Tukey algorithm, 52, 61, 77 vector-radix version, 58 Cooley–Tukey factorization, multidimensional, 58–59, 79 Coordinate systems, 177 Cartesian, 404 natural, 404 Coordinates affine change of, 35 Cartesian, 418, 445 crystallographic, 418 fractional, 42, 63, 262 homogeneous, 418, 421 nonstandard, 42 positional, 437 screen, 426, 428 spherical, 467 spherical polar, 263 standard, 42, 63, 71–72 transformation of, 5, 7, 33 world, 426 Copolymers, random, 571 Core of discrete Fourier transform matrix, 83 Correction-factor approach, 203, 212 Correlated lattice disorder, 573 Correlation, 64 Correlation functions, 75, 100, 252, 489, 505, 541 Ornstein–Zernike, 514 short-range-order, 518 Correlation length, 550 pretransitional lengthening of, 551 Correlations intermolecular, 525 librational–librational, 525 vibrational–librational, 525 Coset averaging, 48–49 Coset decomposition, 48, 58 Coset reversal, 59 Cosets, 48, 71 left, 68 right, 69 Cosine strips, 76 Coulomb energy, 449 of charge densities, 474 Coulomb interactions, 458 between Gaussians, 459 damped, 471 direct, 472 Coulombic lattice energy, 449, 453 Covariance, 407 interatomic, 411 Covariances, 411 Covariant bases, 5–6 Covariant components, 5 Cowpea mosaic virus, 259 Critical angle, 554 Critical scattering, 551 Cross correlation, 76, 388 Cross-correlation function, 365 Cross-Patterson vectors, 260 Cross-rotation function, 100 Cross-vectors, 251 carpet of, 260 CRT (Chinese remainder theorem), 54, 61, 82 for polynomials, 57, 84 reconstruction, 54 reconstruction formula, 57 Cruickshank’s modified Fourier method, 97 Cryo-electron microscopy (cryo-EM), 375 Cryscon, 443 Crystal axes, change of, 120 Crystal class, arithmetic, 177 Crystal defects in thin films, 542 Crystal periodicity, 62 Crystal-structure imaging, 306 Crystal-structure prediction, 458 Crystal structures display of, 443 incommensurate, 445 magnetic, 445 polyhedral display of, 443 root-mean-square differences between, 445 Crystal Studio, 443–444 Crystal symmetry, 68

SUBJECT INDEX Crystal systems, 71, 123 Crystal-B phase, 558 Crystal-E phase, 560 Crystal-G phase, 560 Crystal-H phase, 560 Crystal-J phase, 560 Crystal-K phase, 560 Crystalline approximant, 590 Crystallographic applications of Fourier transforms, 62 Crystallographic coordinates, 418 Crystallographic discrete Fourier transform, 77 algorithms, 76 Crystallographic extension of the Rader/Winograd factorization, 82 Crystallographic Fourier transform theory, 62 Crystallographic group action, 79 in real space, 71 in reciprocal space, 72 Crystallographic groups, 68 classification of, 70 Crystallographic statistics, 204 Crystallographic symmetry, 258 Crystallographica, 443–444 CrystalMaker, 443–445 CrystMol, 443–445 Cubic space groups, 90, 118 Cumulant expansion, 21 Cumulant-generating functions, 103, 198 Cumulative distribution functions, 201 Cuprous chloride azomethane complex, 247 Cyclic convolution, 51, 55 Cyclic (even) permutation of coordinates, 118, 122 Cyclic groups, 71 Cyclic symmetry, 83 Cyclotomic polynomials, 57 Cylindrically averaged diffraction patterns, 572 Cylindrically averaged Patterson function, 577 Damped Coulomb interaction, 471 Dark-field pattern (DP), 309 Data flow, 61 Data handling, Hall symbols in, 123 Data space, 426–427, 429–430 de la Valle´e Poussin kernel, 46 Debye model, 486 Debye theory, 484 Debye–Waller factor, 551, 634 Decagonal phase, 607 Decagonal point groups, 354 Decagonal quasicrystals, 354 Decimation, 25, 49, 53 and subdivision of period lattices, duality between, 48 in frequency, 54, 59, 86 in time, 53, 85 period, 48 Decimation matrix, 58–59, 78 Decomposition, 78 coset, 48, 58 orbit, 69, 72, 74–75 Deconvolution of a Patterson function, 249 Defects, 299, 518 Defocus, 366 optimal, 361 Scherzer, 305 Scherzer, conditions, 362 Deformed crystal, 632 Delta functions, 25 Dirac, 28, 450 periodic, 211 three-dimensional Dirac, 464 transforms of, 40 Density modification, 91, 392 Density modulation, 601 harmonic, 601 symmetric rectangular, 601 Density of nuclear matter, 2 Deoxyhaemoglobin, 273 Depth cueing, 429, 439 Derivatives for model refinement, 95 for variational phasing techniques, 94 of B-splines, 477

Detectors annular dark-field, 305 area, 294 Determinantal formulae, 227 Determinantal inequalities, 67 Deviation parameter, 631, 633 Diamond, 443–445 Diamond’s real-space refinement method, 99 Dielectric response, 468 Dielectric susceptibility, 627, 642 Fourier expansion of, 627 Difference Fourier analysis, 576 Difference Fourier synthesis, 288, 579 Difference Fourier technique, 288 Difference Patterson functions, 253–254 isomorphous, 253 Differential syntheses, 36, 67, 97 Differentiation, 25, 36 and multiplication by a monomial, 40 of distributions, 31 under the duality bracket, 31 Differentiation identities, 51 Differentiation property, 106 Diffraction, dynamical, 393 Diffraction beams, intensities of, 358 Diffraction by helical structures, 100, 568 Diffraction conditions, 63 Diffraction groups, 308–311, 313 Diffraction imaging techniques, 661 Diffraction patterns, cylindrically averaged, 572 Diffraction relations, 2 Diffraction vector, 2 Diffractometers, optical, 364 Diffuse Gaussians, 472 Diffuse scattering, 540 computer simulations of, 528 elastic, 492 from aperiodic crystals, 526 from polycrystalline materials, 534 from quasicrystals, 527 modelling of, 528 of X-rays, 492 Digit reversal, 53, 59, 62 Digital electronic computation of Fourier series, 76 Dihedral symmetry, 84 Dimension of a representation, 176 Dipalmitoylphosphatidylcholine, 562 Dipole moment, 454 of unit cell, 461 Dirac delta function, 28, 450 three-dimensional, 464 Direct Bravais lattice, 121 Direct Coulomb interaction, 472 Direct Fourier inversion, 370 Direct inspection of structure-factor equation, 116 Direct lattice, 2, 5, 42, 121, 177 Direct-lattice sum, 453 Direct methods, 102, 117, 215, 288 in macromolecular crystallography, 235 integration with anomalous-dispersion techniques, 237 integration with isomorphous replacement techniques, 236 Direct-methods packages, 234 Direct metric, 4 Direct phase determination, 36 in electron crystallography, 388 Direct space, 176 Direct-space crystal lattice, 450 Direct-space sum, 449 Direct-space transformations, 120 Direct sum potential, 462 Direct unit-cell parameters, 4 Direction cosines of plane normal, 414 Dirichlet kernel, 46 spherical, 64, 91 Discotic phases, 561 Discrete Fourier transform matrix, core of, 83 Discrete Fourier transformation, 47 Discrete Fourier transforms, 24, 77 algorithms, 76 matrix representation of, 51 numerical computation of, 52 properties of, 51

677

Discretization, 368 Dislocations, 555 Disorder, 540 axial, 517 conforming/nonconforming, 518 from turns, twists and torsions of chains, 517 lattice, 572 lattice, correlated, 573 longitudinal, 516 orientational, 524 substitutional, 545, 572 two-dimensional, 514 Disordered fibres, 568 Dispersion corrections, 284, 286, 627 Dispersion effects, 284 Dispersion energy, 449 Dispersion equations, 300 Dispersion interactions, 459 Dispersion surface, 629, 650, 657 Displacive modulation, 600 harmonic, 601 Display space, 426–427, 429–431, 434 Displaying crystal structures, 443 Distance function, 27 Distribution function, 509 cumulative, 201 Distributions associated with locally integrable functions, 30 beta, first kind, 201 beta, second kind, 201 conjugate, 104–105 conjugate families of, 106 convergence of, 30 convolution of, 34 definition of, 30 differentiation of, 31 division of, 32 electron-magnetization, 11 equal, 45 Fourier transforms of, 39 gamma, 201 Gaussian, 290 hypersymmetric, 201 ideal acentric, 200 ideal centric, 200 integration of, 32 lattice, 43–44, 47–48 maximum-entropy, 36, 106 moments of, 102 motif, 63 multiplication of, 32 non-ideal, 203, 207 of finite order, 30 of random atoms, 104 of sums, averages and ratios, 202 operations on, 31 periodic, 42, 44, 62 probability density, 197 probability density, ideal, 200 support of, 30 T on , 30 tempered, 36, 39, 41, 47, 72 tensor products of, 33 theory of, 25, 28 with compact support, 30, 40, 43–44, 47 Divided differences, 651 Division of distributions, 32 Division problem, 32 Docking, 387, 441 Domain basic, 180 minimal, 179 of influence, 178 representation, 179 Domain structure, 441 Double-phased synthesis, 283 Double-sorting technique, 273 DP (dark-field pattern), 309 Drawing connectivity, 435 DrawXTL, 443–445 Dual, topological, 30, 39–40 Dual relationships, 2

SUBJECT INDEX Duality between differentiation and multiplication by a monomial, 67 between periodization and sampling, 44 between sections and projections, 41 between subdivision and decimation of period lattices, 48 Duality bracket, 39 Duality product, 31 Dummy indices, 5 Dynamic parallax, 427, 441 Dynamical approximation, two-beam, 394 Dynamical diffraction, 393, 542 theory, 301, 626 two-beam, formulae, 303 Dynamical extinction, 319, 323–324, 333, 335 Dynamical matrix, 486 Dynamical scattering effects, 388 Dynamical scattering factor, 543 Dynamical shape function, 651 Dynamical theory, 301, 626 fundamental equations, 628 of neutron diffraction, 654 plane-wave, 630 solution of, 633 Dynamics, 9 of three-dimensional crystals, 484 E maps, interpretation of, 232 Edgeworth approximation, 21 Edgeworth series, 103 EDSA (electron-diffraction structure analysis), 356 Effect of centring, 196 Effective absorption coefficient, 628 Effective potential-energy function, 581 Effects of symmetry on the Fourier image, 114 Eigenspace decomposition of L2, 37 Eigenvalue, 649 Eigenvalue decomposition, 436 Eigenvalues and eigenvectors of orthogonal matrices, 425 Einstein model, 486 Elastic component of X-ray scattering, 10 Elastic constants, measurement of, 490 Elastic diffuse scattering, 492 Electromagnetic electron lenses, 299, 303 Electron band theory of solids, 629 Electron crystallography, 388 direct phase determination, 388 of polymers, 583 of proteins, 389 three-dimensional structure determination by, 391 Electron density, 2, 8, 15, 115, 290 averaged, 274 real-space averaging of, 259, 273 Electron-density calculations, 74 Electron-density maps, Fourier synthesis of, 91 Electron diffraction, 540 absorption in, 362 sign conventions, 301–302 Electron-diffraction data for crystal-structure determination, 583 three-dimensional, 391, 585 Electron-diffraction patterns geometric theory of, 359 polycrystal, 358 single-crystal, 356 texture, 357, 394 Electron-diffraction structure analysis (EDSA), 356 Electron distribution, atomic, radial dependence of, 12 Electron lenses, electromagnetic, 299, 303 Electron-magnetization distribution, 11 Electron micrographs Fourier transform of, 393, 584 phase information from, 390 Electron-microscope image contrast, 542 Electron-microscope imaging, 541 Electron tomography, 377, 382 Electronic analogue computer X-RAC, 76 Electronic structure, 9 Electrons, interaction with matter, 299

Electrostatic energy, 449, 463 Electrostatic potential, 2, 461 Electrostatic properties of molecular surfaces, 438 Embedding method, n-dimensional, 591 Enantiomer, 286 Enantiomeric ambiguity, 247 Enantiomorph definition, 231 Enantiomorphic images, weak, 247 Enantiomorphic solutions, 246 Energy minimization, 437, 440, 442 Entire functions, 36 Entrance surface, 639 Entropy, 94 Envelope, 258 Envelope functions, 305 Epitaxic orientation techniques, 583 Equal-amplitude assumption, 580 Equal distribution, 45 Equivalent matrix groups, 176 Equivalent reflections, 285 Error matrix, atomic, 416 Error propagation, 411 Errors, 289 root-mean-square, 291 systematic, 408 Essential bounds, 45 Essentially bounded function, 27 Euclidean algorithm, 48, 56, 66 Euclidean norm, 25 Euclidean space, 25 Euler gamma function, 463 Euler spline, 477 Eulerian angles, 262, 420 Eulerian space, 263 Eulerian space groups, 263 rotation-function, 265 Eulerian symmetry elements, 264 Even (cyclic) permutation of coordinates, 118, 122 Ewald method, 458 Ewald potential, 466 Ewald result, 449 Ewald sum surface term, 467 Ewald wave, 628 Exchange between differentiation and multiplication by monomials, 102 Exchange between multiplication and convolution, 25 Excitation error, 647 Excitations bulk plasmon, 300 inner-shell, 300 interband, 300 intraband, 300 Explicit-origin space-group notation, 127 Explicit space-group symbols, 123–124 Exploration of parameter space by molecular model building, 576 Exponential coefficient, 13 Exsolution, 505 Extended resolution, 274 Extended structures, 445 External fields, effect on neutron scattering, 659 External modes, 486 Extinction, 638 dynamical, 319, 323–324, 333, 335 Extinction distance, 632–633 Extinction factors, 658 FHLE, 288 FHUE, 288 Face-centred lattices, 90 Face-diagonal axes, 128 Factor group, 69, 71 Factorization, 78 False centre of symmetry, 115 Fast Fourier Poisson method, 475, 478 Fast Fourier transform (FFT), 77 three-dimensional (3DFFT), 475–476 Fast rotation function, 268 Feedback method, 253 Feje´r kernel, 46, 464 spherical, 64 Fermi pseudo-potential, 654 Fermi surface, 545

678

FFT (fast Fourier transform), 77 three-dimensional (3DFFT), 475–476 Fibonacci chain, 595 Fibonacci sequence, 594 Fibre axis, 568 Fibre diffraction, 41, 576 R factor, 582 specimens for, 568 X-ray, 567 Fibres axially periodic, transform of, 101 disordered, 568 macromolecular, 581 noncrystalline, 568, 570, 576 partially crystalline, 572 polycrystalline, 568, 570, 576 Field emission gun, 307 Figures of merit, 232, 290 Films freely suspended, 554 smectic, 554 Filtered backprojection, 371 Filtered image, 364 Filtering iterative low-pass, 575 rotational, 365 Finite crystal of point charges, 465 Finite field, 55 Finite space group, 177 Finite spherical crystal, 469 First Brillouin zone, 178 First-order Born approximation, 10, 300 First-order perturbation theory, 411 Flagpole, 183 Flexible chains, 436 Flexible rings, 436 Flight time, neutron, 660 Flipping ratio, 658 Focusing of neutron beams, 660 Force-constant matrix, atomic, 485 Form factor, 63 aspherical-atom, 14 atomic, 10 atomic, X-ray, 293 geometric, 605 Kikuchi-line, 543 FORTRAN, 122 FORTRAN interface, 122 FORTRAN interpreter, 122 Forward convolution theorem, 44, 64–65, 75 Forward scattering, 647 Forward-scattering approximation, 301 Four-dimensional vector, 424 Fourier analysis, 63 and filtration in reciprocal space, 364 Fourier approach, 212 Fourier–Bessel series, 209 Fourier–Bessel structure factors, 569 Fourier coefficients, 46, 283, 464 Fourier convolution theorem, 10 Fourier cotransform, 35 Fourier cotransformation, 41 Fourier expansion, 2 of dielectric susceptibility, 627 Fourier images, 114, 306 effects of symmetry on, 114 Fourier inversion, direct, 370 Fourier map, 283 Fourier method, 207 Agarwal’s FFT implementation of, 98 Cochran’s, 96 Cruickshank’s modified, 97 Fourier representation, 293 Fourier series, 24 convergence of, 45 convolution of, 44 digital electronic computation of, 76 electron density and its summation, 63 partial sum of, 46 Fourier shell correlation (FSC), 384–385 Fourier space, 116 symmetry in, 120 Fourier summations, 116 space-group-specific, 116

SUBJECT INDEX Fourier synthesis, 63, 286, 290 best, 290 of electron-density maps, 91 Fourier-transform space, 450 Fourier transformation discrete, 47 inverse, 36, 41 mathematical theory of, 24 Fourier transforms, 24, 35, 450 crystallographic applications, 62 crystallographic, discrete, 77 crystallographic, theory of, 62 discrete, 24 discrete, core of matrix, 83 discrete, matrix representation of, 51 discrete, numerical computation of, 52 discrete, properties of, 51 exchange of subdivision and decimation, 49 in L1, 35 in L2, 37 in polar coordinates, 101 in S , 37 inverse, 8 kernels of, 35 of a distribution, 39 of a Gaussian, 463 of B-splines, 477 of density, 478 of electron micrographs, 393, 584 of periodic distributions, 42 of tempered distributions, 39–40 tables of, 39 tensor product property of, 76 various writings of, 39 FpStudio, 443–445 Fractal atomic surface, 597 Fractal sequence, 598 Fractional coordinates, 42, 63, 262 Fre´chet space, 28 Freely suspended films, 554 Fresnel reflection law, 554 Friedel equivalent, 282, 285 Friedel pair, 285 Friedel’s law, 64, 73, 75, 255, 299 Frobenius congruences, 71, 73 Frodo, 441 FSC (Fourier shell correlation), 384–385 Fubini’s theorem, 27, 35, 39 Function spaces associated actions in, 69 topology in, 27 Functional derivative, 99 Functions of polynomial growth, 41 Fundamental domain, 68–70, 72 Fundamental equations of dynamical theory, 628 Fundamental region, 178 Fundamental relationships, 3 Fused-ring systems, 442 G-invariant function, 70 Gamma distribution, 201 Gamma functions, 450 Euler, 463 incomplete, 450, 454 Gamma radiation, 293 Gauss’ law, 461 Gaussian atomic densities, 39 Gaussian atoms, 72–73, 93 anisotropic, 63 Gaussian charge distributions, 461, 471–472 Gaussian density, sampling of, 479 Gaussian distribution, 290 Gaussian function, 39 standard, 38, 41 Gaussian plane, general, 413 Gaussian weights, 413 Gaussians, 99 compact, 472 Coulombic interactions between, 459 diffuse, 472 Fourier transform of, 463 interacting spherical, lattice sums of, 471 normalized, 472 General conditions for possible reflections, 115

General Gaussian plane, 413 General k vector, 178 General linear change of variable, 35 General multivariate Gaussians, 38 General reflections, average intensity of, 195 General superposition, 422 General topology, 27 General transformation, 423 General translation function, 269 Generalized multiplexing, 88 Generalized Patterson function, 494 Generalized Rader/Winograd algorithms, 90 Generalized structure-factor formalism, 22 Generalized support condition, 34 Geometric form factor, 605 Geometric redundancies, 66 Geometric structure factors, 116–117, 135 Geometric theory of electron-diffraction patterns, 359 Gibbs phenomenon, 46, 64 GKS (Graphical Kernel System), 419 GKS-3D (Graphical Kernel System for Three Dimensions), 419 Global crystallographic algorithms, 89 Global minimum, 437–438 Glyceraldehyde-3-phosphate dehydrogenase, 259, 273–274 Good algorithm, 54 Good factorization, multidimensional, 82 Goodness of fit, 416 Gram–Charlier series, 20, 103 Gram–Schmidt process, 425, 437 GRAMPS, 439 Graphical Kernel System (GKS), 419 Graphical Kernel System for Three Dimensions (GKS-3D), 419 Graphics, 418 standards for, 419 Gravity, 660–661 Green’s theorem, 32, 93 GRETEP, 443–445 Gridding method, 373 GRIP, 440 Group actions, 68, 77 crystallographic, 79 crystallographic, real space, 71 crystallographic, reciprocal space, 72 Group characters, 89 Group cohomology, 80 Group extensions, 71 Group of units, 52 Group ring integral, 79 module over, 24 Group–subgroup relationship, 118 Groups, 68 Guide, 441 Haemoglobin, 251–252, 282, 287 horse, 252 Hall symbols, 123, 127, 130 in data handling, 123 in software, 123 Handedness, 333 Hankel transform, 101 Hardy’s theorem, 39 Harker diagram, 257, 283, 290 Harker lines, 249 Harker peaks, 76 Harker planes, 249 special, 249 Harker sections, 248–249 Harmonic approximation, 484 Harmonic density modulation, 601 Harmonic displacive modulation, 601 HDD (high-dispersion diffraction), 356 Heavy-atom-derivative data sets, scaling of, 255 Heavy-atom derivatives, 287 Heavy-atom distribution, 288 Heavy-atom location, 249 three-dimensional methods, 252 Heavy-atom lower estimate, 257 Heavy-atom parameters, 289 Heavy-atom sites, 251

679

Heavy-atom substitution, 255 Heavy atoms, 286 HEED (high-energy electron diffraction), 356 Heisenberg’s inequality, 39, 91 Helical structures, 518 diffraction by, 100, 568 Helical symmetry, 101, 375, 569, 576 approximate, 570 Helix repeat units, 569 Hermann–Mauguin space-group symbol, 119 Hermite coefficients, 476 Hermite function, 38, 103 multivariate, 38, 99 Hermite Gaussians, 473 normalized, 473 Hermite polynomials, three-dimensional, 21 Hermitian-antisymmetric transforms, 87 Hermitian form, 44 Hermitian symmetry, 64, 74, 85 Herringbone packing, 560 Hexagonal axes, 118 Hexagonal family, 118 Hexagonal space groups, 90, 119 Hexatic phase, 555–556 in two dimensions, 555 tilted, 556 Hexatic-B phase, 558 Hexokinase, 273 Hidden-line algorithms, 434 Hidden-surface algorithms, 434 High-dispersion diffraction (HDD), 356 High-energy electron diffraction (HEED), 356 High-resolution electron diffraction (HRED), 356 High-resolution electron microscopy (HREM), 360 High-voltage limit, 652 Higher angular momentum charge distributions, 473 Higher-order Laue zone (HOLZ) reflections, 308, 335 Highlighting, 434 Hilbert space, 27, 35, 47 Hologram, in-line, 307 Holohedral point group, 179 Holosymmetric space group, 180 HOLZ (higher-order Laue zone) reflections, 308, 335 Homogeneous coordinates, 418, 421 Homogeneous symmetric polynomial, 652 Homometric pair, 246 Homometric structures, 246 Homomorphism, 176 Horse haemoglobin, 252 HRED (high-resolution electron diffraction), 356 HREM (high-resolution electron microscopy), 360 Hybridization, 442 HYDRA, 441 Hydrogen bonding, 439, 441–442 Hydrophobic properties of molecular surfaces, 438 Hyperatoms, 592 Hypercrystal, 590, 592 Hypersymmetric distributions, 201 Hypothetical atoms, 94 Icosahedral phase, 613 Icosahedral point groups, 353 Icosahedral quasicrystals, 352 Ideal acentric distributions, 200 Ideal aperiodic crystal, 590 Ideal centric distributions, 200 Ideal crystal, 176, 590 Ideal probability density distributions, 200 Idempotents, 54 Image averaging in real space, 364 Image contrast, electron-microscope, 542 Image detection, 249 Image enhancement, 361, 363 Image intensity, bright-field, 362 Image of a function by a geometric operation, 26 Image processing in transmission electron microscopy, 361 Image reconstruction, 361 Image resolution, 306 Image restoration, 361–362 Images asymmetric, 365 filtered, 364 Fourier, 306

SUBJECT INDEX with point symmetry, 365 Immunoglobulin, 271 Implication theory, 248 Implicit function theorem, 33 Implied connectivity, 435 Improper rotation axes, 258 Improper rotations, 123, 128 In-line hologram, 307 Incident wave, departure from Bragg’s law, 630 Incoherent inelastic scattering, 488 Incoherent scattering, 488 Incommensurability, 550 Incommensurate crystal structures, display of, 445 Incommensurate intergrowth structure, 593 Incommensurately modulated structures, 344, 591 Incomplete gamma function, 450, 454 complement of, 450–451 evaluation of, 454 Independence, assumption of, 209 Index, 68 Index of refraction, 628 Indicator functions, 32, 42, 47, 65, 91–92 Induction formula, 105 Inductive limit, 30 Inelastic component of X-ray scattering, 10 Inelastic scattering, 300, 540 neutron, 488 X-ray, 490 Inequalities among structure factors, 221 Inner-shell excitations, 300, 542 Insight, 439 Insight II, 442 Instrumental resolution, 307 Integral group ring, 79 Integral representation, 68 theory, 71 Integrals Lebesgue, 26 Riemann, 26 Integrated intensity, 637–638, 640 Integration by parts, 31 Lebesgue’s theory of, 28 of anomalous-dispersion techniques with direct methods, 237 of distributions, 32 of isomorphous replacement techniques with direct methods, 236 Intensities of diffraction beams, 358 Intensities of plane waves in reflection geometry, 638 in transmission geometry, 634 Intensities of reflected and refracted waves, 634 Intensity differences, 285 Intensity statistics, 105 Interaction between symmetry and decomposition, 78 Interaction between symmetry and factorization, 79 Interaction matrix, 249 Interaction of electrons with matter, 299 Interaction of X-rays with matter, 626 Interatomic covariance, 411 Interatomic vectors, 65 Interband excitation, 300 Interference function, 65 spherical, 261 Interferometry, neutron, 660 Intermolecular correlations, 525 Intermolecular force fields, 458 Internal modes, 486 Interpolation, 368 Interpolation formula, 47 Interpolation kernel, 92 Interpolation of trigonometric functions, 478 Interpretation of E maps, 232 Intraband excitation, 300 Intramolecular energy terms, 455 Intrinsic component of translation part of spacegroup operation, 116 Invariance of L2, 37 Inverse Fourier transform, 8 Inverse Fourier transformation, 36, 41 Inverse rotation operator, 114 Ionic crystal, electrostatic energy of, 449 Irreducible matrix group, 176

Irreducible representations (irreps), 175 type of, 181 Ising model, 68 Isometry, 37 Isometry property, 37 Isomorphism, 176, 290–291 lack of, 255 Isomorphous addition, 283 Isomorphous crystals, 283 Isomorphous differences, 288, 291 Isomorphous heavy-atom derivatives, 580 Isomorphous replacement, 251, 282 difference Patterson functions, 251, 253 multiple, 290 single, 252, 283 techniques, integration of direct methods with, 236 Isomorphous synthesis, 283 Isotropic harmonic oscillator, three-dimensional, 17 Isotropic temperature factors, 73 Isotropy subgroups, 68, 72 Iterative low-pass filtering, 575 Jacobians, 33, 51 Joint probability distribution of structure factors, 105 Juxtaposition of chains, 578 k vector general, 178 special, 178 uni-arm, 181 k-vector type, 180 Kernels, 58 Cauchy, 46 de la Valle´e Poussin, 46 Dirichlet, 46 Feje´r, 46, 464 interpolation, 92 of Fourier transformations, 35 Poisson, 46 spherical Dirichlet, 64, 91 spherical Feje´r, 64 Kikuchi-line contrast, 543 Kikuchi-line form factor, 543 Kinematical approximation, 62, 300–301, 359, 394, 583, 658 Kinematical diffraction formulae, 302 Kinematical diffraction intensities, 302 Kinematical R factor, 584 Kinematical scattering, 300 Klug peaks, 267 Known structural fragment, use of, 272, 389 Kronecker symbol, 5 ‘Kubic harmonics’, 12 Lp spaces, 26 LACBED (large-angle convergent-beam electron diffraction), 345 Lack of isomorphism, 255 Lagrange multiplier, 95, 106, 414 Lagrange’s theorem, 68 LALS, 578 Lamellar domains with long-range order, 505–506 Landau–Peierls effect, 551–552 Langmuir troughs, 583 Languages computer-algebraic, 122 numerically and symbolically oriented, 117 Laplace’s equation, 468 Large-angle convergent-beam electron diffraction (LACBED), 345 Large values of ot, 641 Larmor precession, 655 Lattice, 42 base-centred, 90 body-centred, 90 centred, 73 composite, 449, 452 direct, 2, 5, 42, 121, 177, 450 face-centred, 90 nonprimitive, 71 nonstandard, 42 nonstandard period, 43 one-dimensional, 552

680

period, 42, 62, 68 primitive, 71 reciprocal, 2, 5, 43, 48, 62, 121, 177 residual, 48 rhombohedral, 90 standard, 42 Lattice disorder, 572 correlated, 573 Lattice distributions, 43–44, 47–48 Lattice-dynamical model, 484 Lattice energy, 458 Coulombic, 449, 453 Lattice mode, 71 Lattice-parameter mapping, 626 Lattice plane, 2, 406 Lattice sums, 44, 460 of interacting spherical Gaussians, 471 Lattice transform, 450 Lattice-translation subgroup, 123 Lattice type, 123 Laue case, 631 Laue equations, 2 Laue groups, 115 Laue point, 630 Laue scattering, 520 Layer lines, continuous diffraction on, 575 Lead, 287 Least resolvable distance, 306 Least-squares adjustment of observed positions, 413 Least-squares approximation of trigonometric functions, 478 Least-squares determination of phases, 233 Least-squares method, multivariate, 95 Least-squares plane, 410 proper, 413 Least-squares refinement, 289 Lebesgue integral, 26 Lebesgue’s theory of integration, 28 LEED (low-energy electron diffraction), 356 Left action, 68, 70, 72, 79 Left cosets, 68 Left representation, 72 Legendre polynomials, 468 Leibnitz’s formula, 451 Length of a function, 26 Length of a vector, 404 Lennard–Jones potential, 490 L’Hospital’s rule, 451 Libration, 18 Libration tensor, 18 Librational–librational correlations, 525 Lifshitz point, 553 Lifchitz’s reformulation, 98 Lindeberg–Le´vy version of the central-limit theorem, 204 Line drawings, 433 Linear absorption coefficient, 628 Linear change of variable, general, 35 Linear forms, 30 Linear functionals, 28 Linear scaling of computational cost of Ewald direct sum, 474 Linear transformation, 7 Linearity, 35, 51 Linearization formulae, 75 Linearly semidependent phases, 216 Linked-atom least-squares (LALS) system, 578 Liouville’s theorem, 37 Liquid crystals, 547 Lissajous curve, 105 Little co-group, 178, 180 Little group, 178 Local ordering, 518 Locally integrable functions, 30 distributions associated with, 30 Locally summable function of polynomial growth, 40 Location-dependent component of translation part of space-group operation, 116 Locked rotation function, 267–268 Locked translation function, 271 Logical connectivity, 435 London dispersion interactions, 475 Lone pairs, 442

SUBJECT INDEX Long-range order (LRO), 505, 548 positional, 547 Longitudinal disorder, 516 Lorentz point, 630 Low-angle scattering, 508 Low-energy conformational changes, 582 Low-energy electron diffraction (LEED), 356 LRO (long-range order), 548 positional, 547 Lyotropic phase, 548 Lysozyme, 252 MACCS, 442 Macromolecular crystallography, 282 direct methods in, 235 Macromolecular fibre structures, 581 Macromolecular refinement techniques, 99 Macromolecular structures, direct determination of, 583 Macroscopically spherical crystal, 467 Madelung constant, 449 Magic-integer methods, 233 Magnetic crystal structures, display of, 445 Magnetic domains, 661 Magnetic scattering, 11, 656 Main reflections, 592 Manganese, 286 Many-beam case, 629 Mapping, 25 Maschke, theorem of, 176 Mathematical theory of Fourier transformation, 24 Matrices of mixed scalar products, 8 Matrix algebra, 262 Matrix–column pair, 177 Matrix groups, 176 completely reducible, 176 equivalent, 176 irreducible, 176 reducible, 176 unitary, 176 Matrix part, 177 Matrix representation, 114 of discrete Fourier transform, 51 Maximum determinant rule, 228 Maximum entropy, 106, 392 Maximum-entropy distributions, 36 of atoms, 106 Maximum-entropy methods, 102, 234 Maximum-entropy theory, 106 Maximum function, 250 Maximum likelihood, 392 Maxwell’s equations, 627, 643 MBD (microbeam diffraction), 356 McMurchie–Davidson recursion, 475 MDIR (multidimensional isomorphous replacement), 579 MDKINO, 439 Mean-field theory, 549 Mean-square displacement amplitude, 443 Mean values, 408 Mechanical pressure tensor, 470 Meijer’s G function, 201 Mercury, 287 Mercury, 443–444 Mesomorphic structures, scattering from, 547 Metric direct, 4 reciprocal, 4 Metric space, 25, 27 Metric tensors, 4–5 Metrizability, 25 Metrizable topology, 28 Micelle, 547 Microanalysis, 298–299 Microbeam diffraction (MBD), 356 Microdiffraction, 544–545 MIDAS, 439 Middle of reflection domain, 631 Minima global, 437–438 multiple, 437 potential-energy, 437 Minimal domain, 179 Minimization function, 289

Minimum function, 250 MIR (multiple isomorphous replacement), 290 phases, 259 Mirror image, 285 MM2/MMP2, 442 MMS-X, 440 Modelling of diffuse scattering, 528 Modelling transformations, 431 Modified tangent formula, 233 Modulated phases, 550 Modulated smectic-A phase, 553 Modulated smectic-C phase, 553 Modulated structure, 591 Modulation function, 591 Module, 79 over a group ring, 24 Molbuild, 442 Molecular averaging by noncrystallographic symmetry, 92 Molecular axis, 568 Molecular dynamics, 442 Molecular-dynamics refinement, 581 Molecular envelope, 32, 65, 92–93 Molecular mechanics, 437 Molecular model building, 577 Molecular modelling, 6, 418 Molecular orientational order, 547 Molecular origin, 273 Molecular replacement, 244, 258, 272, 274, 292 real-space, 273 Molecular rotation, 558 Molecular structure, position of a known, 270 Molecular surfaces, hydrophobic and electrostatic properties of, 438 MolXtl, 443–444 Moment-generating functions, 36, 102 Moment-generating properties, 102 of F , 67 Moments of a distribution, 102 Monochromators, 660 polarizing, 658 Monoclinic family, 118 Monoclinic space groups, 89, 118 Mosaic crystals, 626 Mosaic model, 659 Mosaicity, 356 Motif, 42–44 Motif distribution, 63 Multicritical point, 553 Multidimensional algorithms, 58 Multidimensional Cooley–Tukey factorization, 58–59, 79 Multidimensional factorization, 58 Multidimensional Good factorization, 82 Multidimensional isomorphous replacement (MDIR), 579 Multidimensional prime factor algorithm, 59 Multidimensional structure, 437 Multigrid method, 480 Multi-index, 26, 36, 38 Multi-index notation, 26 Multiple diffuse scattering, 542 Multiple isomorphous replacement (MIR), 290 phases, 259 Multiple minima, 437 Multiple reciprocal cell, 121 Multiple scattering, 540 Multiple simultaneous superposition, 425 Multiple-wavelength method, 293 Multiplexing, generalized, 88 Multiplexing–demultiplexing, 85 Multiplication by a monomial, 25 Multiplication of distributions, 32 Multiplicative group of units, 56 Multiplicative reindexing, 61 Multiplicity, 116, 180 Multiplicity corrections, 251 Multiplier functions, 41 Multipliers, 44 Lagrange, 95, 106, 414 Multipoles, rotation of, 475 Multireference alignment, 382 Multi-Slater determinant wavefunction, 15 Multislice, 306, 651

681

calculations, 544 computer programs, 544 recurrence relation, 651 Multivariate Gaussian, 44 Multivariate Hermite functions, 38, 99 Multivariate least-squares method, 95 Mutually reciprocal bases, 2–3 computational and algebraic aspects of, 4 Mutually reciprocal triads, 2 Myoglobin, 252, 282, 287 n-dimensional embedding method, 591 n-shift rule, 98 n-torus nonstandard, 42 standard, 42 Natural coordinate system, 404 Negative peaks, 252 Nematic order biaxial, 549 uniaxial, 549 Nematic phase, 547, 549 Nested algorithms, 62 Nested neighbourhood principle, 223 Nesting, 61 of Winograd small FFTs, 60 Net distortions, 517 Neutral unit cell, 461 Neutron absorption, 655 Neutron beams, focusing of, 660 Neutron crystallography, 293 Neutron diffraction, 560 dynamical theory of, 654 Neutron flight time, 660 Neutron interferometry, 660 Neutron refraction, 654 Neutron scattering effect of external fields, 659 inelastic, 488 very small angle, 660 Neutron scattering lengths, 293 Neutron spin, 655 Neutron topography, 661 Neutrons, 293 thermal, 293 Nodes of standing waves, 633 Non-absorbing case, 629 Non-absorbing crystals, 637–639 comparison of dynamical and geometrical theory, 640 Nonbonded interatomic distances, 578 Noncrystalline fibres, 568, 570, 576 Noncrystallographic asymmetric unit, 253 Noncrystallographic rotation elements, translational components of, 258 Noncrystallographic symmetry, 66, 258 apparent, 267 molecular averaging by, 92 phase improvement using, 273 proper, 258 rotational, 260 Noncrystallographic symmetry element, position of, 270–271 Non-cyclic (odd) permutation of coordinates, 118, 122 Non-ideal distributions, 203, 207 Non-ideal probability density functions, 212 of |E|, 205 Non-independent variables, 199 Nonlinear transformations, 264 Nonperiodic system, 503 Nonprimitive lattice, 71 Non-spin-flip scattering lengths, 657 Nonstandard coordinates, 42 Nonstandard lattice, 42 Nonstandard n-torus, 42 Nonstandard period lattice, 43 Norm Euclidean, 25 on a vector space, 28 Normal equations, 95 Normal matrix, 98 Normal subgroup, 68–69 Normalization constant, 290, 293

SUBJECT INDEX Normalized Gaussians, 472 Normalized Hermite Gaussians, 473 Normalized structure factors, 216, 231, 246 Normalizer, 69 Normed space, 28 complete, 28 Notation, multi-index, 26 Nuclear matter, density of, 2 Numerical computation of discrete Fourier transform, 52 Numerically oriented languages, 117 Nussbaumer–Quandalle algorithm, 60 O, 441 Observation plane, 361 Observational equations, 95 Obverse setting, 121 Occupancy factors, 289 Occupied natural spin orbitals, 15 Odd (non-cyclic) permutation of coordinates, 118, 122 Offset, 53 OLEX, 443–445 One-centre orbital products, 15 One-centre terms, 15 One-dimensional lattice, 552 One-particle potential (OPP) model, 22 One-phase structure seminvariants, 231 first rank, 229 Operational calculus, 28 Operations on distributions, 31 OPP (one-particle potential) model, 22 Optic modes, 486 Optical diffractometer, 364 Optical isomers, 286 Optical rotation, 286 Optimal defocus, 361 Optimization, 436 Orbit decomposition, 69, 72, 74–75 formula, 69, 73 Orbit exchange, 70, 77–78 Orbit of k, 178 Orbital products, 15 one-centre, 15 two-centre, 17 Orbits, 68, 72–73 Order–disorder, 507 Order parameter, 549 Orientational disorder, 524 Origin-shift vector, 123 Origin-to-plane distance, 410, 414 Origin(s) allowed (permissible), 215 definition, 231 molecular, 273 removal from a Patterson function, 245 selection, 247 specification, 215 Ornstein–Zernike correlation function, 514 ORTEP, 438 ORTEP-3 for Windows, 443–444 ORTEP-III, 443–444 ORTEX, 443–444 Orthogonal matrices, 419 eigenvalues and eigenvectors of, 425 Orthogonalization, 262 Orthographic projection, 427–428 Orthorhombic space groups, 89, 118 Oscail X, 443–444 Overlap between two Pattersons, 260 Ps(u) function, 256 Pair probability, 504 conditional, 504 Pairwise sum, 449 Paley–Wiener theorem, 36, 106 Parabolic equation, 648 Parallel processing, 61 Parity of the hkl subset, 119 Parseval–Plancherel property, 52 Parseval–Plancherel theorem, 37, 47 Parseval’s identity, 64, 67 Parseval’s theorem, 35, 95, 450 with crystallographic symmetry, 74

Partial dislocations, 558 Partial net atomic charges, 449 Partial sum of Fourier series, 46 Partially bicentric arrangement, 212 Partially crystalline fibres, 572 Partially reflected wavefield, 639 Partially transmitted wave, 639 Particle–particle particle–mesh method, 475 Patterson function(s), 64–65, 75, 244, 289, 541, 577 anomalous-dispersion, 257 contraints on interpretation of, 258 cylindrically averaged, 577 deconvolution of, 249 difference, 253–254 generalized, 494 interactions in, 244 isomorphous difference, 253 origin removal, 245 overlap between two, 260 second kind, 247 sharpened, 245–246 superposition, 250 symmetry of, 244 three-dimensional, 577 Patterson map, automated search, 389 Patterson peaks, 244 Patterson search, 254 Patterson synthesis, 283, 286–287 Patterson techniques, 6, 293 Patterson vector interactions, 248 PEANUT, 443–444 Pendello¨sung, 303, 632, 635–636, 659–660 spherical-wave, 641 Pendello¨sung distance, 632–633 Penetration depth, 638 Penrose rhomb, 610 Penrose tiling, 609 Pentagonal point groups, 354 Period decimation, 48 Period lattice, 42, 62, 68 nonstandard, 43 Period matrix, 43 Period subdivision, 48 Periodic boundary conditions, 479, 485 Periodic continuation, 544 Periodic delta functions, 211 Periodic density function, 114 Periodic distributions, 42, 44, 62 and Fourier series, 42 Fourier transforms of, 42 Periodic images, 460 Periodic lamellar domains, 502 Periodic weak phase objects, 364 Periodicity, 176 crystal, 62 Periodicity requirement, 9 Periodization, 25, 44, 53 and sampling, duality between, 44 Periodograms, 378 Permissible origins, 215 Permissible symmetry, 114 Permutation of coordinates cyclic (even), 118, 122 non-cyclic (odd), 118, 122 Permutation operators, 118 Permutation tensors, 405 Perpendicular (internal, complementary) space, 592 Perspective, 418, 426–428, 434 Perturbation theory, first-order, 411 Phase angles, 282–283 Phase change, 286 Phase circles, 283 Phase determination, 283 ab initio, 273 direct, 36 direct, in electron crystallography, 388 statistical theory of, 104 Phase-determining formulae, 221 Phase evaluation, 282, 291, 293 Phase extension, 274 Phase-grating approximation, 652 Phase improvement, 274 using noncrystallographic symmetry, 273

682

Phase information, 292 from electron micrographs, 390 Phase invariant sums, 389 Phase-object approximation, 301, 542 Phase problem, 576 Bayesian statistical approach, 106 Phase relationships quartet, 225 quintet, 227 Phase restriction, 73 Phase shift, 25, 53 Phase transformations, polytypic, 514 Phases assignment of one or more, 231 best, 290 from multiple isomorphous replacement, 259 from single isomorphous replacement, 253 least-squares determination of, 233 linearly semidependent, 216 refinement of, 233 Phason flips, 611 Phasons, 527 Phenomenological absorption coefficients, 303 PHIGS (Programmers’ Hierarchical Interactive Graphics System), 419 Phonon absorption, 488 Phonon dispersion relations, 489 Phonon emission, 488 Phonon scattering, 543 Phonons, 484 Physical (external, parallel) space, 592 Picture space, 426–428, 434 Pipelining, 61 Pisot numbers, 595 Pixel, 432 Plancherel’s theorem, 41 Plane of polarization, 286 Plane-wave dynamical theory, 630 Planes, 406 Gaussian, general, 413 least-squares, 410 least-squares, proper, 413 Plasmon scattering, 541 Plasmons bulk, excitation of, 300 surface, 300 Platon, 443–444 PLUTO, 439 Pluton, 444 Point charges, finite crystal of, 465 Point density, 610 Point-group determination by convergent-beam electron diffraction, 307 Point-group operators, 115, 123 Point-group symmetry of reciprocal lattice, 114 Point groups, 177 average multiples for, 197 closed, 258 decagonal, 354 holohedral, 179 icosahedral, 353 pentagonal, 354 Point multipoles, 459 Point-spread function, 366 Poisson kernel, 46 Poisson summation formula, 44 Poisson’s equation, 462 Polar space, 263 Polarization, plane of, 286 Polarization response, 467 Polarization vector, 11, 655 Polarizing monochromators, 658 Polycrystal electron-diffraction patterns, 358 Polycrystalline fibres, 568, 570, 576 Polycrystalline materials, diffuse scattering from, 534 Polyhedral display of crystal structures, 443 Polymer crystallography, 567 Polymer electron crystallography, 567 Polynomial growth functions of, 41 locally summable function of, 40 Polynomial transforms, 61

SUBJECT INDEX Polynomials Chinese remainder theorem for, 57, 84 cyclotomic, 57 Polyoma virus, 274 Polypeptide fold, chemical correctness of, 273 Polytypic phase transformations, 514 Population parameter, 12 Position of a known molecular structure, 270 Positional coordinates, 437 Positional order, long-range, 547 Positive peaks, 252 Positivity criterion, 293 Potassium permanganate, 286 Potential energy of a crystal, 485 Potential-energy minima, 437 PowderCell, 444 Power spectrum, 376 Poynting vector, 644 Prediction of crystal structures, 458 Pressure tensor mechanical, 470 thermodynamic, 470 Pretransitional lengthening of correlation lengths, 551 Prime factor algorithm, 52, 54 multidimensional, 59 Primitive basis, 177 Primitive coefficients, 180 Primitive lattice, 71 Primitive root mod p, 55 Principal axes, 128 Principal central projections and sections, 66 Principal projections, 76 Principal sections and projections, 67 PRJMS, 444–445 Probability a posteriori, 512 a priori, 504 Probability densities, convolution of, 102 Probability density distributions, 197 ideal, 200 Probability density functions, 203 non-ideal, 212 of |E|, non-ideal, 205 Probability density of samples for images, 366 Probability theory, 102 analytical methods of, 102 Probability trees, 513 Processing X-ray fibre diffraction data, 574 Product function, 250 Programmers’ Hierarchical Interactive Graphics System (PHIGS), 419 Projected charge-density approximation, 305 Projection approximation, 648 Projection diffraction groups, 311 Projection matching, three-dimensional, 383 Projection operator, 649 Projection(s), 25 and sections, duality between, 41 and sections, principal central, 66 bounded, 67, 92 centrosymmetric, 251–252 orthographic, 427–428 principal, 76 tilt, 389 use in three-dimensional reconstruction, 366 Projector, 70 Prolate spheroidal wavefunctions, 39 Propagation direction, 630 Propagation equation, 626 Proper least-squares plane, 413 Proper noncrystallographic symmetry, 258 Proper rotation, 123, 128 Protein crystallography, 286 Protein crystals, 287 Protein Data Bank, 438 Proteins, electron crystallography of, 389 PRXBLD, 442 Pseudo-distances, 28 Pseudorotation, 437 Pseudotranslational symmetry, 225 Punched-card machines, 76 Pure imaginary transforms, 87

Quartet phase relationships, 225 Quasicrystal structures, display of, 445 Quasicrystals decagonal, 354 diffuse scattering from, 527 icosahedral, 352 symmetry determination of, 352 Quasilattice, 595 Quasi-long-range order (QLRO), 548 Quasimoments, 21 Quasi-normalized structure factors, 218 Quasiperiodic order, 603 Quintet phase relationships, 227 R factors fibre diffraction, 582 kinematical, 584 Rader algorithm, 52 Rader/Winograd algorithms, generalized, 90 Rader/Winograd factorization, crystallographic extension of, 82 Radial dependence of atomic electron distribution, 12 Radial functions, 13 Radiation damage, 299 Radius of integration, 261 Radon measure, 30 Radon transform, 366 Random conical tilt, 377, 381 Random copolymers, 571 Random-start method, 233 Random-walk problem, 104 exact solution, 207 Rank of tensor, 5 Rapidly decreasing functions, 37, 39 Raster-graphics devices, 432–433 Rational approximant, 595 Ray transform, 366 Real antisymmetric transforms, 88 Real crystal, 176 Real-space averaging, 273–274 of electron density, 259, 273 Real-space molecular replacement, 273 Real-space translation functions, 271 Real spherical harmonic functions, 12 Real symmetric transforms, 88 Real-valued transforms, 85 Real waves, 640 Reciprocal axes, 405 Reciprocal Bravais lattice, 121 Reciprocal cell, multiple, 121 Reciprocal lattice, 2, 5, 43, 48, 62, 121, 177 point-group symmetry of, 114 weighted, 114–115 weighted, statistical properties of, 195 Reciprocal-lattice sum, 453 Reciprocal-lattice vectors, 450 Reciprocal metric, 4 Reciprocal space, 2, 450 symmetry in, 119 Reciprocal-space group, 175, 179, 192 Reciprocal-space procedures, 251 Reciprocal-space representation of space groups, 114 Reciprocal sum potential, 462 Reciprocal unit-cell parameters, 4 Reciprocity, 37 Reciprocity property, 36 Reciprocity relationship, 304 Reciprocity theorem, 37, 41, 43, 63, 106 of scattering theory, 308–309 Reconstruction image, 361 single-particle, 366, 375 three-dimensional, 366 Recursion for B-splines, 477 REDUCE, 122 Reduced orbit, 74 Reducibility of the representation, 71 Reducible matrix group, 176 Reference bases, choice of, 7 Refinement aspherical multipole, 459 in single-particle reconstruction, 383 least-squares, 289

683

molecular-dynamics, 581 of phases, 233 restrained least-squares, 581 Reflected intensity, 640 Reflecting power, 635–636 Reflection case, 631 Reflection conditions, 73 Reflection domain, middle of, 631 Reflection geometry, 631–632 Reflection high-energy electron diffraction (RHEED), 356 Reflections acentric, 74 equivalent, 285 main, 592 satellite, 592 substructure, 218 superstructure, 218 Refraction, neutron, 654 Refractive index, 300 Regularization, 34 by convolution, 42 Reindexing additive, 61 multiplicative, 61 Relationship between structure factors of symmetryrelated reflections, 115 Relationships between direct and reciprocal bases, 3 Relatively prime integers, 406 Relativistic effects, 301 Representation domain, 179 Representation method, 223 Representation of space groups in reciprocal space, 114 Representation of surfaces by dots, 433 Representation of surfaces by lines, 433 Representation of surfaces by shading, 433 Representation operators, 72, 78 Representation property, 68 Representations, irreducible, 175 Representative operators of a space group, 123 Repulsion energy, 449 Residual lattice, 48 Resolution extended, 274 image, 306 instrumental, 307 Restacking, 558 Restrained least-squares refinement, 581 RHEED (reflection high-energy electron diffraction), 356 Rhombohedral lattice, 90 Riemann integral, 26 Riemann–Lebesgue lemma, 36 Right action, 68, 70 Right cosets, 69 Right representation, 68 Rigid-body motion, 18 Rigid-body superposition, 422 Rigid rotation, 8 Ring systems condensed, 437 flexible, 436 fused, 442 Rings, 442 RMBD (rocking microbeam diffraction), 356 Robertson’s sorting board, 76 Rocking curve, 633, 638, 660 width at half-height, 636 width of, 633 Rocking microbeam diffraction (RMBD), 356 Rodrigues’ formula, 468–469 Root-mean-square differences between crystal structures, 445 Root-mean-square error, 291 Rotation, 426, 429 improper, 123, 128 molecular, 558 optical, 286 proper, 123, 128 rigid, 8 screw, 18 Rotation axes, improper, 258 Rotation-function Eulerian space groups, 265

SUBJECT INDEX Rotation functions, 260 fast, 268 locked, 267–268 Rotation matrix, 267, 419 trace of, 263 Rotation of multipoles, 475 Rotation operator, 6 inverse, 114 Rotation part of space-group operation, 115 Rotation vector, 421 Rotational filtering, 365 Rotational structure (form) factor, 526 Rotational symmetry, noncrystallographic, 260 Row–column method, 58 SAD–MAD (single anomalous dispersion–multiple anomalous dispersion), 238 Saddlepoint approximation, 102–103 Saddlepoint equation, 105 Saddlepoint expansion, 104 Saddlepoint method, 36, 105 SAED (selected-area electron diffraction), 307, 584 Sampling, 25, 44 and periodization, duality between, 44 considerations, 99 of Gaussian density, 479 theorems, 65 Satellite reflections, 503, 592 Satellite tobacco necrosis virus, 259 Sayre’s equation, 91, 230 Sayre’s squaring method, 94 Scalar pressure, 470 Scalar products, 5, 404 mixed, matrices of, 8 Scale, 426 Scale factors, 287 Scaling of heavy-atom-derivative data sets, 255 Scaling symmetry, 603 Scanning microbeam diffraction (SMBD), 356 Scanning transmission electron microscope (STEM), 304 Scattering classical Thomson, 10 coherent, 488 Compton, 489 critical, 551 diffuse, 540 forward, 647 from mesomorphic structures, 547 incoherent, 488 incoherent inelastic, 488 inelastic, 300, 540 inelastic neutron, 488 inelastic X-ray, 490 kinematical, 300 Laue, 520 low-angle, 508 magnetic, 11, 656 multiple, 540 of neutrons by thermal vibrations, 488 of X-rays by thermal vibrations, 487 phonon, 543 plasmon, 541 thermal diffuse, 300 X-ray, 10, 62 Scattering cross sections, 550, 654 Scattering diagrams, 651 Scattering factors atomic, 10, 284 complex, 255 dynamical, 543 spherical atomic, 10 Scattering lengths, 654 atomic, 11 neutron, 293 non-spin-flip, 657 spin-flip, 657 Scattering matrix method, 363 Scattering operator, 14 Scattering power, 285 Scattering theory, reciprocity theorem of, 308–309 SCHAKAL, 444 Scherzer defocus, 305 conditions, 362

Scherzer phase function, 362 Schro¨dinger equation, 300 Schur-Auerbach, theorem of, 176 Schur’s lemma, 71, 78 Scrambling, 53 Screen coordinates, 426, 428 Screen space, 430 Screw correlations, 20 Screw rotation, 18 Screw shifts, 518 Script, 442 Search directions, 94 Second Bethe approximation, 302 Second-order Born approximation, 11 SECS, 442 Section, 25 Sections and projections, 25, 66 duality between, 41 principal, 67 Selected-area electron diffraction (SAED), 307, 584 Selection of origin, 247 Selection rules, 101 Self-energy terms, 449 Self-Patterson, 100 vectors, 260 Self-rotation function, 100 Self-seeding, 583 Self-vectors, 251 Semi-direct product, 69 Semi-norm on a vector space, 28 Semi-reciprocal space, 648 Separating exponent, 479 Series-termination errors, 64, 91, 99 Seven-beam approximation, 652 Shadows, 434 Shannon interpolation, 25, 50 Shannon interpolation formula, 47, 92 Shannon interpolation theorem, 65 Shannon sampling criterion, 47, 68, 93 Shannon sampling theorem, 47, 65, 275 Shape-dependent term of Ewald sum, 469 Sharpened Patterson functions, 245–246 SHELX, 443 Shift of space-group origin, 120 Shift property, 51, 65 Short cyclic convolutions, 57 Short-range order (SRO), 505, 548 correlation functions, 518 in multicomponent systems, 520 parameters, 541, 545 Warren parameters, 520 Shubnikov groups, 308 Sign conventions for electron diffraction, 301–302 Signal-to-noise ratio in electron microscopy, 376, 385 Simulated annealing, 576 Simultaneous iterative reconstruction technique (SIRT), 370 Sine strips, 76 Single anomalous dispersion–multiple anomalous dispersion (SAD–MAD), 238 Single-crystal electron-diffraction patterns, 356 Single isomorphous replacement (SIR), 252, 283 difference electron density, 253 phasing, 253 Single isomorphous replacement–multiple isomorphous replacement (SIR–MIR), 236 Single-particle reconstruction, 366, 375 Singular value decomposition, 370 SIR (single isomorphous replacement), 252, 283 difference electron density, 253 phasing, 253 SIR–MIR (single isomorphous replacement–multiple isomorphous replacement), 236 SIRT (simultaneous iterative reconstruction technique), 370 Site-symmetry group, 180 Site-symmetry restrictions, 12 Size distribution, 504 Size effect, 521 Skew-circulant matrix, 56 Sliding filter, 436 Small-angle-scattering approximation, 300 Small values of ot, 641 SMB (symmetrical many-beam) method, 312

684

SMBD (scanning microbeam diffraction), 356 Smectic films, 554 Smectic-A phase, 547, 550–551 modulated, 553 Smectic-B phase, 547 Smectic-C phase, 551 modulated, 553 Smectic-D phase, 561 Smectic-F phase, 556 Smectic-I phase, 556 Smooth particle mesh Ewald method, 475 Sobolev space, 41 Software, Hall symbols in, 123 Solids electron band theory, 629 theory of, 9 Solution of dynamical theory, 633 Solvable space groups, 71 Solvent flattening, 92 Solvent regions, 66 Sound velocities, 490–491 Southern bean mosaic virus, 259, 274 Space-group algorithm, 119 Space-group determination by convergent-beam electron diffraction, 307 Space-group notation, explicit-origin, 127 Space-group operation, 115 intrinsic and location-dependent components of translation part, 116 rotation part, 115 translation part, 115 Space-group origin, shift of, 120 Space-group-specific Fourier summations, 116 Space-group-specific structure-factor formulae, 116 Space-group-specific symmetry factors, 114 Space-group symbols computer-adapted, 117, 122, 127 explicit, 123–124 Hall, 123, 127, 130 Hermann–Mauguin, 119 Space-group symmetry, 576 Space-group tables, 119 Space-group types, 71 affine, 177 crystallographic, 177 Space groups, 71, 176, 282 cubic, 118 Eulerian, 263 finite, 177 hexagonal, 119 holosymmetric, 180 in reciprocal space, 162 monoclinic, 118 orthorhombic, 118 reciprocal-space representation of, 114 representative operators of, 123 rotation-function, 264 solvable, 71 symmorphic, 71, 177 tetragonal, 119 triclinic, 118 trigonal, 119 Spaces data, 426–427, 429–430 display, 426–427, 429–431, 434 picture, 426–428, 434 screen, 430 Special Harker planes, 249 Special k vector, 178 Special position, 72 condition, 72 Special reflection, 73 Spectrometer, triple-axis, 490 Specular reflection, 554 Spherical angles, 262 Spherical atomic scattering factor, 10 Spherical atoms, 116 Spherical coordinates, 467 Spherical Dirichlet kernel, 64, 91 Spherical Feje´r kernel, 64 Spherical harmonic addition theorem, 468 Spherical harmonic expansion, atom-centred, 12 Spherical harmonic functions, real, 12 Spherical harmonics, 268

SUBJECT INDEX Spherical interference function, 261 Spherical polar coordinates, 263 Spherical-wave Pendello¨sung, 641 Spin-flip scattering lengths, 657 Spin orbitals, occupied natural, 15 Spiro links, 442 Spot boundaries, 575 Squarability criterion, 293 Square-integrable functions, 41 Square-summable sequences, 47 Squaring method equation, 91 SRO (short-range order), 505, 548 correlation functions, 518 in multicomponent systems, 520 parameters, 541, 545 Warren parameters, 520 Stacked transformations, 431 Standard basis of Rn, 42 Standard coordinates, 42, 63, 71–72 Standard Gaussian function, 38, 41 Standard lattice, 42 Standard n-torus, 42 Standard uncertainty of distance from an atom to a plane, 412 Standards for graphics, 419 Standing waves, 633 anti-nodes of, 633 nodes of, 633 Star, arms of, 178 Star of k, 178 Starting models, 578 Statistical properties of the weighted reciprocal lattice, 195 Statistical theory of communication, 104 Statistical theory of phase determination, 104 Statistics crystallographic, 204 structure-factor, 117 Status of centrosymmetry, 123 Steepest descents, Booth’s method, 96 STEM (scanning transmission electron microscope), 304 Stereochemical information, 576 Stereoviews, 428 Stirling’s formula, 106 Structural connectivity, 435 Structural similarity, 425, 445 Structure amplitude, 10 Structure determination by X-ray fibre diffraction analysis, 576 using electron-diffraction data, 583 Structure-factor algebra, 75, 105–106 Structure-factor formalism, generalized, 22 Structure-factor formulae, space-group-specific, 116 Structure-factor statistics, 117 Structure factors, 6, 8, 10, 62, 282, 466 calculation of, 73 for one-phonon scattering, 487 Fourier–Bessel, 569 from model atomic parameters, 93 geometric, 116–117, 135 in terms of form factors, 63 inequalities among, 221 joint probability distribution of, 105 normalized, 216, 231, 246 quasi-normalized, 218 rotational, 526 tables of, 117, 135 trigonometric, 116–117, 135 trigonometric, even absolute moments of, 206 trigonometric, moment of, 205 unitary, 218 via model electron-density maps, 93 Structure invariants, 216 Structure seminvariants, 216 algebraic relationships, 228 one-phase, 229, 231 two-phase, 229 Structure theorem, 35 for distributions with compact support, 43, 47 STRUPLO, 444 STRUPLO for Windows, 443–444 STRUVIR, 443–444

Sturkey’s solution, 648 Subcentric arrangement, 212 Subdivision and decimation of period lattices, duality between, 48 Sublattice, 48 Subspace sectioning, 436, 441 Substitutional disorder, 545, 572 Substitutional order, 543 Substructure reflections, 218 Sum function, 250 Sum of images, 249 Summable functions, 27 Summation convention, 5 Summation problem in crystallography, 46 Superposition methods, 249 Superposition of Patterson functions, 250 Superpositions general, 422 multiple simultaneous, 425 rigid-body, 422 Superstructure reflections, 218 Support, 25 compact, 25, 36, 44 of a distribution, 30 of a tensor product, 34 Support condition, 34, 44 generalized, 34 Surface effects, 553 Surface phase, 557 Surface plasmons, 300 Surface representation in cryo-EM, 386 Surfaces atomic, 590, 596 dispersion, 629, 650, 657 fractal atomic, 597 representation of, 433 van der Waals, 433, 439 Sybyl, 442 Symbolic programming techniques, 114 Symbolically oriented languages, 117 Symmetric rectangular density modulation, 601 Symmetrical many-beam (SMB) method, 312 Symmetry, 176, 263, 431 conjugate, 35, 40 conjugate and parity-related, 85 crystal, 68 crystallographic, 258 cyclic, 83 dihedral, 84 effects on Fourier image, 114 helical, 101, 375, 569, 576 helical, approximate, 570 Hermitian, 64, 74, 85 in Fourier space, 120 in reciprocal space, 119 noncrystallographic, 66, 258 noncrystallographic, molecular averaging by, 92 noncrystallographic, proper, 258 noncrystallographic, rotational, 260 of Patterson function, 244 permissible, 114 pseudotranslational, 225 scaling, 603 Symmetry elements Eulerian, 264 three-dimensional, 308–309 two-dimensional, 308–310 Symmetry factors, 116 space-group-specific, 114 tables of, 114 Symmetry-generating algorithm, 123 Symmetry group, 176 Symmetry lines, 180 Symmetry operation, 176 Symmetry planes, 180 Symmetry points, 180 Symmetry property, 39 Symmetry-related reflections, relationship between structure factors of, 115 Symmorphic space groups, 71, 177 Synchrotron radiation, 282, 293 Systematic absences, 73, 121 Systematic errors, 408 Szego¨’s theorem, 44, 67, 106

685

Table lookup schemes, 474 Tangent formula, 223, 292 application of, 231 modified, 233 weighted, 224 TDS (thermal diffuse scattering), 300, 484, 540 TEM (transmission electron microscope), 303 Temperature factors, 73, 287 anisotropic, 73 artificial, 94, 100 atomic, 17 isotropic, 73 Tempered distributions, 36, 39, 41, 47, 72 definition and examples of, 40 Fourier transforms of, 39–40 Tensor-algebraic formulation, 2, 5 Tensor formulation of vector product, 6 Tensor product, 27, 34, 52 of distributions, 33 of matrices, 51, 58, 60 structure of, 77 support of, 34 Tensor product B-spline, 480 Tensor product property, 35, 67, 101 of a Fourier transform, 76 Tensors, 5 anisotropic displacement, 6 antisymmetric, 6 libration, 18 metric, 4–5 permutation, 405 rank of, 5 translation, 18 translation, libration and screw-motion, 6 Test-function spaces, 29 Test functions, 39 Tetragonal family, 118 Tetragonal space groups, 90, 119 Text processing, 122 Texture electron-diffraction patterns, 357, 394 THEED (transmission high-energy electron diffraction), 356 Theory of distributions, 25, 28 Theory of solids, 9 Thermal diffuse scattering (TDS), 300, 484, 540 Thermal fluctuations, 550 Thermal neutrons, 293 Thermal streaks, 545 Thermodynamic pressure tensor, 470 Thermotropic phase, 548 Thick crystals, 363, 638 Thin crystals, 639 comparison of geometrical and dynamical theory, 637 Thin films, 554 crystal defects in, 542 Thomson scattering, classical, 10 Thon rings, 378 Three-axis joystick, 430 Three-beam approximation, 652 Three-beam inversion, 652 Three-dimensional Dirac delta function, 464 Three-dimensional electron-diffraction data, 391 from a single crystal orientation, 585 from two crystal orientations, 585 Three-dimensional fast Fourier transform (3DFFT), 475–476 Three-dimensional Hermite polynomials, 21 Three-dimensional isotropic harmonic oscillator, 17 Three-dimensional Patterson function, 577 Three-dimensional projection matching, 383 Three-dimensional reconstruction, 366 Three-dimensional structure determination by electron crystallography, 391 Three-dimensional symmetry elements, 308–309 Three-generator symbol, 123 Through-focus series method, 363 Tie point, 628 Tilt projections, 389 Tilted hexatic phase, 556 Toeplitz–Carathe´odory–Herglotz theorem, 45 Toeplitz determinants, 45, 67 Toeplitz forms, 44, 67 asymptotic distribution of eigenvalues of, 45, 68

SUBJECT INDEX Toeplitz matrices, 45 Tomography, electron, 377, 382 Topography, neutron, 661 Topological dual, 30, 39–40 Topological vector spaces, 28 Topology, 27, 39 general, 27 in function spaces, 27 metrizable, 28 not metrizable, 30 on DðÞ, 29 on Dk ðÞ, 29 on E ðÞ, 29 Topology analysis, 445 Torsion angles, 406 Total cross section, 11 Total-reflection domain, 638 width of, 633, 638 Trace of rotation matrix, 263 Transfer function, 52 of lens, 304 Transformation properties of direct and reciprocal base vectors and lattice-point coordinates, 116 Transformations affine, 120 compound, 429 direct-space, 120 general, 423 linear, 7 modelling, 431 nonlinear, 264 of coordinates, 5, 7, 33 stacked, 431 to a Cartesian system, 3 unitary, 37 viewing, 426–427, 429–430, 432 viewport, 426, 428 windowing, 426 Transformed variance–covariance matrix, 407 Transforms complex antisymmetric, 87 complex symmetric, 87 Hankel, 101 Hermitian-antisymmetric, 87 of an axially periodic fibre, 101 of delta functions, 40 polynomial, 61 pure imaginary, 87 Radon, 366 ray, 366 real antisymmetric, 88 real symmetric, 88 real-valued, 85 Translate, 26 Translation, 18, 25, 426, 429 part of space-group operation, 115 part of space-group operation, intrinsic and location-dependent components of, 116 Translation, libration and screw-motion tensors, 6 Translation functions, 100, 269 general, 269 locked, 271 real-space, 271 Translation subgroup, 177 Translation tensor, 18 Translation vector, 258 Translational components of noncrystallographic rotation elements, 258 Translational displacement, 18 Translational invariance, 650 Translations, conversion to phase shifts, 35 Transmission case, 631 Transmission electron microscope (TEM), 303 Transmission geometry, 631–633 intensities of plane waves in, 634 Transmission high-energy electron diffraction (THEED), 356 Transposition formula, 80–81 for intermediate results, 77 Triads, mutually reciprocal, 2 Triangular inequality, 27 Triclinic space groups, 89, 118 Trigonal space groups, 90, 119

Trigonometric functions B-spline approximation of, 477 interpolation of, 478 least-squares approximation of, 478 Trigonometric moment problem, 44 Trigonometric structure-factor expressions, vectors of, 104 Trigonometric structure factors, 116–117, 135 even absolute moments of, 206 moment of, 205 Triple-axis spectrometer, 490 Triple point, 552 Triplet relationships using structural information, 224 Triplets, search of, 231 Triply periodic, 627 Tunability, 294 Twiddle factors, 53, 58, 60, 62, 80 Two-beam approximation, 302, 649 Two-beam case, 629 Two-beam dynamical approximation, 394 Two-beam dynamical diffraction formulae, 303 Two-centre orbital products, 17 Two-centre terms, 15 Two-dimensional disorder, 514 Two-dimensional hexatic phase, 555 Two-dimensional symmetry elements, 308–310 Two-phase structure seminvariants, first rank, 229 Two-wavelength method, 293 Type of irreps, 181 Type of rotation (proper or improper), 123 Uni-arm k vector, 181 Uniaxial nematic order, 549 Uniformity, assumption of, 203, 209 Uniformizable space, 28 Unit cell, 178–179, 460 dipole moment of, 461 neutral, 461 Unit-cell parameters direct, 4 reciprocal, 4 Unit cube, 42 Unitary matrix group, 176 Unitary structure factors, 218 Unitary transformations, 37 Unscrambling, 81 Uranium, 287 Valence density, 12 van der Waals surfaces, 433, 439 Van Hove correlation functions, 489 Variance, 407 Variance–covariance matrix, 406 transformed, 407 Variances, 411–412 Vector interactions in a Patterson map, 248 Vector lattice, 177 Vector machines, 432 Vector map, 244 Vector overlap, 251 Vector processing, 61 Vector product, 405 components of, 405 tensor formulation of, 6 Vector radix Cooley–Tukey algorithm, 58 Vector radix FFT algorithms, 59 Vector relationships, 405 Vector-search procedures, 251 Vector space complete, 26 norm on, 28 semi-norm on, 28 topological, 28 Vectors angle between two, 404 components of, 5 cross-Patterson, 260 four-dimensional, 424 interatomic, 65 length of, 404 of trigonometric structure-factor expressions, 104 origin-shift, 123 polarization, 11, 655

686

Poynting, 644 rotation, 421 self-Patterson, 260 translation, 258 VENUS, 443–445 Very small angle neutron scattering, 660 Vibrating crystals, 660 Vibrational–librational correlations, 525 Viewing transformation, 426–427, 429–430, 432 Viewport, 426, 428 Viewport transformation, 426, 428 Vitamin B12, 286 Voronoi diagram, 373 Waller–Hartree formula, 541, 543 Warren short-range-order parameters, 520 Wavefield, 628, 630 Wavefunctions, prolate spheroidal, 39 Wavelengths, 299 Wavevectors, 635 Weak enantiomorphic images, 247 Weak-phase-object approximation, 304 Weak phase objects, 361, 583 periodic, 364 Weighted difference map, 97, 99 Weighted lattice distribution, 44 Weighted reciprocal lattice, 114–115 statistical properties of, 195 Weighted reciprocal-lattice distribution, 62 Weighted tangent formula, 224 Weighting factor, 289 Weights anisotropic, 413 Gaussian, 413 Whole pattern (WP), 310 Width of rocking curve, 633 at half-height, 636 Width of total-reflection domain, 633, 638 Wigner potential, 466 Wigner–Seitz cell, 178 Wilson plot, 287 Window, 426, 429, 434 Windowing, 426, 428 Windowing transformation, 426 Wing, 184 Winograd algorithms, 52, 56 Winograd small FFT algorithms, 57 Winograd small FFTs, nesting of, 60 Wintgen letter, 180 Wintgen position, 180 Wintgen symbol, 180 World coordinates, 426 WP (whole pattern), 310 Wyckoff letter, 180 Wyckoff position, 115, 180 Wyckoff symbols, 72 XmLmctep, 443–444 X-ray analysis, 287 X-ray fibre diffraction analysis, 567 data processing, 574 structure determination by, 576 X-ray scattering, 62, 293 cross section, 550 elastic component, 10 inelastic component, 10 X-ray topographs, 626 X-rays, 293 diffuse scattering of, 492 interaction with matter, 626 X-Seed, 443–444 Xtal-3D, 444–445 XtalDraw, 443–444 z buffer, 433 ZAP (zone-axis pattern), 310 Zero-absorption case, 633 Zeroth-order Laue zone (ZOLZ) reflections, 308, 324 Zonal data sets view down the chain axis, 584 view onto the chain axes, 585 Zone-axis pattern (ZAP), 310 Zones and rows, average intensity of, 196