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

339 Pages·1987·7.4 MB·English

by  L. A. Feigin

L.A. Feigin
D.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,
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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, includ
ing 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 design
ing 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-deve
loped 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 small
angle 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.
vii
PREFACE
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; meth
ods 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, isomor
phous 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 small
angle 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 inter
pretation 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 experimen
tation. The basic requirements for measuring devices and principles of
design and construction are treated, including instrumentation for neu
tron 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 CONTENTS
2.3.1. Monodisperse and Polydisperse Systems 34
2.3.2. Conception of Contrast . . . . 35
2.3.3. Concentrated and Dilute Systems . 36
2.4. Isolated Particle . . . . 38
2.4.1. Debye Equation . . 38
2.4.2. Correlation Function 39
2.4.3. Uniform Particles . 40
2.4.4. Asymptotic Behavior of Intensity.
The Porod Invariant . . 45
2.4.5. Special Types of Particle . . . . 46
2.5. Nonparticulate Systems . . . . . . 50
2.5.1. Scattering Due to Statistical Fluctuations 50
2.5.2. Two-Phase and Multiphase Systems . 52
2.6. Conclusion . . . . . . . . . . . . . 55
PART II. MONODISPERSE SYSTEMS
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