L.A. FeiginD.I. Svergun

Structure Analysis by Small-Angle X-Ray and Neutron Scattering

Structure Analysis by Small-Angle X-Ray and Neutron Scattering

Structure Analysis by Small-Angle X-Ray and Neutron Scattering

L. A. Feigin and D. I. Svergun Institute of Crystallography Academy of Sciences of the USSR

Moscow, USSR

Edited by George W. Taylor Princeton Resources Princeton, New Jersey

Springer Science+Business Media, LLC

Library of Congress Cataloging in Publication Data

Svergun, D. 1. Structure analysis by small-angle X-ray and neutron scattering.

Bibliography: p. lncludes index. 1. X-rays-Scattering. 2. Neutrons-Scattering. 3. Matter-Structure. 4. Small-angle

scattering. 1. Feigin, L. A. II. Taylor, George W. III. Title. QC482.S3S8513 1987 539.7'222 87-25489

ISBN 978-1-4757-6626-4 ISBN 978-1-4757-6624-0 (eBook) DOI 10.1007/978-1-4757-6624-0

This volume is published under an agreement with the Copyright Agency of the USSR (V AAP).

© 1987 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1987

AII rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

Preface

Small-angle scattering of X rays and neutrons is a widely used diffraction method for studying the structure of matter. This method of elastic scattering is used in various branches of science and technology, includ­ing condensed matter physics, molecular biology and biophysics, polymer science, and metallurgy. Many small-angle scattering studies are of value for pure science and practical applications.

It is well known that the most general and informative method for investigating the spatial structure of matter is based on wave-diffraction phenomena. In diffraction experiments a primary beam of radiation influences a studied object, and the scattering pattern is analyzed. In principle, this analysis allows one to obtain information on the structure of a substance with a spatial resolution determined by the wavelength of the radiation.

Diffraction methods are used for studying matter on all scales, from elementary particles to macro-objects. The use of X rays, neutrons, and electron beams, with wavelengths of about 1 A, permits the study of the condensed state of matter, solids and liquids, down to atomic resolution. Determination of the atomic structure of crystals, i.e., the arrangement of atoms in a unit cell, is an important example of this line of investigation. Another line deals with the study of the structure of matter at the superatomic level, i.e., with a spatial resolution from 10 A up to thousands and even several tens of thousands of angstroms. For this line of investigation the basic tool is small-angle scattering, mainly of X rays and neutrons.

In small-angle scattering research the wavelength used is, as a rule, on the order of several angstroms. As interatomic distances are of the same order of magnitude, the diffraction pattern, corresponding to the superatomic structure, lies in the small-angle region, and this is how the method received its name. The most important feature ofthe small-angle scattering method is its potential for analyzing the inner structure of

"

vi PREFACE

disordered systems, and frequently the application of this method is a unique way to obtain direct structural information on systems with random arrangement of density inhomogeneities of colloid size, namely, 101-104 A.

Small-angle scattering studies date from the classical works of A. Guinier, published in 1938. Later, P. Debay, G. Porod, 0. Kratky, V. Luzzati, W. Beeman, P. Schmidt, and others developed the theoretical and experimental fundamentals of the method. The principles of design­ing small-angle scattering facilities were developed, and the potential for applying the method for determining general structural characteristics of various types of highly dispersed systems was shown.

New progress in refining the method of small-angle scattering began in the 1970s and is continuing today. This stage of work is characterized by new opportunities both for experimentation (powerful neutron beams, synchrotron X-ray sources, position-sensitive detectors) and for structural interpretation (contrast variation, isomorphous replacement, direct methods, analysis of characteristic functions). Certainly, the use of various computers is of great significance in advancing these two aspects of small-angle scattering applications. The result has been a gradual expansion of the range of studied objects and an increase in the spatial resolution in both directions. Thus, for example, the mechanisms of phase separation and the sizes and degrees of dispersion of dispersed structures of alloys, powders, and glasses; the specific features of the configuration of polymer chains in different aggregate states; and the geometrical and weight parameters (and, sometimes, three-dimensional structure) of biological macromolecules and their complexes in solution can all be subjected to analysis.

Currently, the small-angle scattering technique, with its well-deve­loped experimental and theoretical procedures and wide range of studied objects, is a self-contained branch of the structural analysis of matter.

Despite the wide application of the small-angle scattering technique in solving various fundamental and applied problems, there are very few monographs on this subject. Only two books are known: the classical monograph Small-Angle Scattering of X Rays by A. Guinier and A. Fournet (1955) and Small-Angle X-Ray Scattering, edited by 0. Glatter and 0. Kratky ( 1982). The latter, in fact, is a collection of review articles. Both books deal with X-ray scattering and do not discuss small-angle diffraction of neutrons. As a result, general information on the small-angle scattering method is dispersed in individual original and review articles, and in the proceedings of several international small­angle scattering conferences ( 1965, 1973, and 1977). As a consequence, scientists and engineers applying the small-angle scattering technique to solve their particular problems frequently are not well informed on the possibilities of the method and may use it inefficiently.

PREFACE vii

The aim of this book is to present the current state of small-angle scattering knowledge and to show how this technique makes it possible to establish several physicochemical parameters (geometry and weight) of investigated objects, and to analyze their spatial structure. The main purpose of the authors is to demonstrate methods for obtaining structural information from experimental small-angle scattering data. The X-ray and neutron scattering methods are considered in parallel, and their advantages and potentials for combined application are discussed.

Part I deals with the basic theory of diffraction and scattering intensity calculations for systems of various degrees of ordering. The theory, which describes the relation of the elastic scattering pattern to the spatial distribution of matter within an object, is the same for all types of radiation. However, the concrete form and specific features of the diffraction phenomena for X rays and neutrons are determined by different characteristics of matter. Thus, Chapter 1 discusses general relationships for the amplitude and intensity of elastic scattering; meth­ods of Fourier transforms are considered briefly as well as the basic features of the interaction of X rays and neutrons with matter. The main equations relating the scattering intensity to the structural parameters of various objects are derived in Chapter 2. Particle solutions, isolated particles, nonparticulate systems, and some other model systems are treated.

Part II discusses methods for obtaining structural information from small-angle scattering data for monodisperse systems. These methods are divided into three groups. The first one includes the calculation of general geometric and weight parameters of particles (invariants of the scattering curves) and modeling of their structure by the trial-and-error technique. These approaches (Chapter 3) are used mainly in the case of particles that can be treated as homogeneous. Chapter 4 deals with new methods for calculating the inner structure of a particle on the basis of novel experimental procedures, including contrast variation, isomor­phous and isotopic replacement, and change of radiation type. Chapter 5 treats the so-called direct methods of interpretation, which permit one, if some additional a priori information on an object is available, to recover directly its structure, i.e., the scattering density distribution.

Part III deals with the application of the general theory of small­angle diffraction to polymers and inorganic substances. This branch of small-angle scattering research is gradually expanding, mainly because of applied and technological requirements. In the case of polymer systems (Chapter 6}, some modifications of general methods of structural inter­pretation are necessary because of the great variety of structures of this kind. Here a model of the polymer chain forms the basis for the analysis. For polymers in solution or in an amorphous state, the characteristics of the coils are mainly determined; for crystal polymers the ways of packing

viii PREFACE

the crystal and amorphous regions are investigated. Some applications of the small-angle scattering method to oriented objects are considered as well.

Inorganic materials require an even wider variety of models (Chapter 7) for data analysis. It should be noted that the most important procedures, such as determination of the phase composition of alloys or estimation of the pore sizes in ceramics, have been developed reliably, mainly for technological applications, and may be considered as routine procedures.

Part IV discusses the problems of small-angle scattering experimen­tation. The basic requirements for measuring devices and principles of design and construction are treated, including instrumentation for neu­tron beams and synchrotron sources (Chapter 8). Fundamental sources of instrumental distortions in small-angle scattering measurements are considered in Chapter 9, and a series of procedures for experimental data treatment is analyzed comprehensively. Both complicated methods of data analysis and relatively simple algorithms for minicomputers are presented.

AcKNOWLEDGMENTS. The authors wish to thank Professor B. Vainshtein for his continuous support in developing small-angle structural research at the Institute of Crystallography of the USSR Academy of Sciences. They are indebted to Professors P. B. Moore and G. W. Taylor for reading some chapters and offering valuable comments. The authors thank Dr. A. Ryskin for help with word processor use and also A. Semenyuk and T. Simagina for their assistance in preparing the manuscript for publication. They are also grateful to the staff of Plenum Press for their continued help and cooperation.

L.A. Feigin and D. I. Svergun Moscow

Contents

PART I. SMALL-ANGLE SCATTERING AND THE STRUCTURE OF MATTER

1. Principles of the Theory of X-Ray and Neutron Scattering . 3

1.1. Scattering of a Plane Wave by Matter. 3 1.2. Fourier Transforms. Convolutions . 6 1.3. Scattering by Simple Objects . . 10

1.3.1. Rectangular Parallelepiped 10 1.3.2. Homogeneous Thin Plate. 12 1.3.3. Homogeneous Thin Rod . 12 1.3.4. Inhomogeneous Parallelepiped . 13 1.3.5. Periodic Set of Centers. . . 13 1.3.6. Spherically Symmetric Body. . 14

1.4. Scattering of X Rays by Atoms . . . 14 1.5. Scattering of Thermal Neutrons by Nuclei 17 1.6. Absorption of X Rays and Neutrons 22 1. 7. Conclusion . . . . . . . . . . . 24

2. General Principles of Small-Angle Diffraction . . 25

2.1. Scattering by Objects with Different Ordering . 25 2.1.1. Single Crystals . . . . . . . . 26 2.1.2. One-Dimensional Periodic Structures 27 2.1.3. Cylindrically Symmetric Objects . . 28 2.1.4. Isotropic Systems . . . . . . . 29

2.2. Small-Angle Scattering by Disperse Systems . 30 2.2.1. Scattering Intensity by a Disordered Object 30 2.2.2. Scattering at Low Angles . 33

2.3. Particle Solutions . . . . . . . . . . . . 34

lx

X

2.3.1. Monodisperse and Polydisperse Systems 2.3.2. Conception of Contrast . . . . 2.3.3. Concentrated and Dilute Systems .

2.4. Isolated Particle . . . . 2.4.1. Debye Equation . . 2.4.2. Correlation Function 2.4.3. Uniform Particles . 2.4.4. Asymptotic Behavior of Intensity.

The Porod Invariant . . 2.4.5. Special Types of Particle . . . .

2.5. Nonparticulate Systems . . . . . . 2.5.1. Scattering Due to Statistical Fluctuations 2.5.2. Two-Phase and Multiphase Systems .

2.6. Conclusion . . . . . . . . . . . . .

PART II. MONODISPERSE SYSTEMS

CONTENTS

34 35 36 38 38 39 40

45 46 50 50 52 55

3. Determination of the Integral Parameters of Particles 59

3.1. Geometrical and Weight Invariants . . . . . 59 3.1.1. Total Scattering Length and Radius of Gyration . 60 3.1.2. Volume and Surface . . . . . . . . 61 3.1.3. Largest Dimension and Correlation Length . 62 3.1.4. Anisometric Particles . . . . . . . . . 63

3.2. Information Content in Small-Angle Scattering Data. 63 3.2.1. General Approach . . . . . . . 64 3.2.2. Number of Independent Parameters . . . . 65

3.3. Evaluation of the Invariants . . . . . . . . . 68 3. 3.1. Accuracy of Calculation of the Radius of Gyration . 68 3.3.2. Absolute Measurements. Molecular-Mass

Determination . . . . . . . . . . . 73 3.3.3. Possibilities of Homogeneous Approximation. 76 3.3.4. Estimate of the Largest Dimension . . . . 82 3.3.5. Evaluation of the Invariants of Anisometric Particles 83 3.3.6. List of the Basic Equations . . 87

3.4. Scattering by Particles of Simple Shape . . . . 90 3.5. Modeling Method. . . . . . . . . . . . 94

3.5.1. Demands on the Technique of Calculation. 94 3.5.2. Subparticle Models . 95 3.5.3. Spheres Method . . . 95 3.5.4. Cube Method . . . . 97 3.5.5. Modeling in Real Space 98

3.6. Applications of Modeling . . 98 3.6.1. Helix pomatia Hemocyanin . 99

CONTENTS

3.6.2. Bacteriophage S0 . . . .

3.6.3. 30 S Ribosomal Subparticle. 3. 7. Conclusion . . . . . . . . .

4. Interpretation of Scattering by Inhomogeneous Particles

4.1. Scattering by Inhomogenous Particles . . . . . . 4.1.1. Solvent Influence . . . . . . . . . . 4.1.2. General Equations for Intensity and Invariants 4.1.3. Spherically Symmetric Particles 4.1.4. Large-Angle Scattering.

4.2. Contrast Variation . . . . . . . 4.2.1. Basic Functions . . . . . . 4.2.2. Dependence of Invariants on Contrast 4.2.3. Contrasting Techniques . . . . . 4.2.4. Applications of the Contrast-Variation Technique

4.3. Isomorphous-Replacement Methods 4.3.1. Heavy-Atom Labels 4.3.2. The Triangulation Method ..

4.4. Variation in the Applied Radiation . 4.4.1. Combined Use of Various Types of Radiation 4.4.2. Anomalous (Resonant) Scattering

4.5. Conclusion . . . . . . . . . . . . . . .

5. Direct Methods

5.1. One-Dimensional Density Distributions . . 5.2. Solving the One-Dimensional Sign Problem

5.2.1. Use of Correlation Functions . . . 5.2.2. Box-Function Refinement . . . .

5.3. Multipole Theory of Small-Angle Diffraction 5.4. Determination of Multi pole Components

5.4.1. Isometric Particles . . . . . 5.4.2. Shape of Uniform Particles . . . 5.4.3. Separation of Bessel Functions. . 5.4.4. Examples of Direct Structure Determination .

5.5. Conclusion . . . . . . . . . . . . . . .

PART Ill. POLYMERS AND INORGANIC MATERIALS

6. Investigations of Polymer Substances.

6.1. Models of Polymer Chains. 6.1.1. Gaussian Chains. . . . .

xi

100 102 104

107

107 108 110 112 112 115 115 117 118 122 131 131 132 138 139 140 144

147

148 150 152 153 156 164 164 165 167 176 182

187 188 188

Xii CONTENTS

6.1.2. Persistent Chain . . . . . 191 6.1.3. Perturbed Chains . . . . 194 6.1.4. Molecular Mass Distribution 195

6.2. Polymers in Solution and in the Amorphous State 197 6.2.1. Polymers in Solution 197 6.2.2. Amorphous Polymers 201

6.3. Crystalline Polymers. . . 203 6.3.1. Lamellar Model . . 203 6.3.2. Scattering by the Lamellar Stacks . 205 6.3.3. Correlation Functions . . . . 207 6.3.4. Determination of Chain Folding 209

6.4. Anisotropic Systems . . . . . . . 210 6.4.1. Oriented Amorphous Polymers 210 6.4.2. Fibrillar Systems 211 6.4.3. Lamellar Systems 214

6.5. Conclusion . . . . . 217

7. Structural Studies of Inorganic Materials . 219

7 .1. Crystalline Materials . . . . 220 7 .1. 1. Defects in Single Crystals . . . 220 7.1.2. Phase Separation in Alloys . . 225

7.2. Polydisperse Objects. Calculation of Size Distribution . 230 7 .2.1. Analytical Methods . . 231 7.2.2. Numerical Methods . . . . 236

7.3. Amorphous Solids and Liquids . . . 240 7 .3.1. Study of the Structure of Glasses 240 7.3.2. Concentration Fluctuations and Clusters 243

7 .4. Conclusion . . . . . . . . . . . . . 245

PART IV. INSTRUMENTATION AND DATA ANALYSIS

8. X-Ray and Neutron Instrumentation .

8.1. Basic Designs of Instrumentation 8.1.1. Angular Resolution . . . 8.1.2. Main Characteristics of Instruments

8.2. Laboratory X-Ray Instruments 8.2.1. X-Ray Tubes . . . . . 8.2.2. X-Ray Detectors. . . . 8.2.3. Point Collimation System 8.2.4. Slit Collimation System .

8.3. Synchrotron Radiation Instruments

249

249 250 252 255 255 256 257 258 262

CONTENTS xiii

8.3.1. Main Characteristics of Synchrotron Radiation 262 8.3.2. Monochromatization and Focusing of X Rays 263 8.3.3. Small-Angle Synchrotron Instruments . . . 265

8.4. Small-Angle Neutron Scattering Apparatus 268 8.4.1. Neutron Sources, Monochromatization, Detectors . 268 8.4.2. Collimation Systems and Instruments 269

8.5. Conclusion . . . . . . . . . . . . . . . . . 273

9. Data Treatment . . . . . . . . . . . . . . 275

9.1. General Scheme of Small-Angle Data Processing. 276 9.1.1. Instability of Experimental Conditions . . 276 9.1.2. Additive Scattering Components . . . . 276 9.1.3. Influence of Beam and Detector Dimensions . 277 9.1.4. Beam Polychromaticity . . . . . . . . 279 9.1.5. Statistical Errors. . . . . . . . . . . 280 9.1.6. General Expression for Experimental Intensity 280

9.2. Preliminary Data Processing . 281 9.3. Experimental-Data Smoothing 283

9.3.1. Algebraic Polynomials. 284 9.3.2. Spline Functions. . . 288 9.3.3. Frequency Filtering Method 291 9.3.4. Problem of Optimum Smoothing 292

9.4. Collimation Corrections . . 295 9.4.1. Weighting Functions . 295 9.4.2. Slit-Width Correction . 296 9.4.3. Slit-Height Correction . 298

9.5. Corrections for Polychromaticity 303 9.6. Termination Effects . . . . . 305 9. 7. Simultaneous Elimination of Various Distortions 309

9. 7 .I. Iteration Methods . . . . 309 9.7.2. Orthogonal Expansions . . . . 310 9.7.3. Use ofthe Sampling Theorem . . 314 9.7.4. General Regularization Procedure 317

9.8. Conclusion . . . . . . . . . . . 320

REFERENCES 321

INDEX 333