Structure Analysis by Small-Angle X-Ray and Neutron Scattering
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1 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 Plenum Press New York and London
2 Contents PART I. SMALL-ANGLE SCATTERING AND THE STRUCTURE OF MATTER 1. Principles of the Theory of X-Ray and Neutron Scattering Scattering of a Plane Wave by Matter Fourier Transforms. Convolutions Scattering by Simple Objects Rectangular Parallelepiped Homogeneous Thin Plate Homogeneous Thin Rod Inhomogeneous Parallelepiped Periodic Set of Centers Spherically Symmetrie Body Scattering of X Rays by Atoms Scattering of Thermal Neutrons by Nuclei Absorption of X Rays and Neutrons Conclusion General Principles of Small-Angle Diffraction Scattering by Objects with Different Ordering Single Crystals One-Dimensional Periodic Structures Cylindrically Symmetrie Objects Isotropie Systems Small-Angle Scattering by Disperse Systems Scattering Intensity by a Disordered Object Scattering at Low Angles Particle Solutions 34 ix
3 X CONTENTS Monodisperse and Polydisperse Systems Conception of Contrast Concentrated and Dilute Systems Isolated Particle Debye Equation Correlation Function Uniform Particles Asymptotic Behavior of Intensity. The Porod Invariant Special Types of Particle Nonparticulate Systems Scattering Due to Statistical Fluctuations Two-Phase and Multiphase Systems Conclusion 55 PART II. MONODISPERSE SYSTEMS 3. Determination of the Integral Parameters of Particles Geometrical and Weight Invariants Total Scattering Length and Radius of Gyration Volume and Surface Largest Dimension and Correlation Length Anisometric Particles Information Content in Small-Angle Scattering Data General Approach Number of Independent Parameters Evaluation of the Invariants Accuracy of Calculation of the Radius of Gyration Absolute Measurements. Molecular-Mass Determination PossibilitiesofHomogeneousApproximation Estimate of the Largest Dimension Evaluation ofthe Invariants of Anisometric Particles List of the Basic Equations Scattering by Particles of Simple Shape Modeling Method Demandson the Techniqueof Calculation Subparticle Models Spheres Method Cube Method Modeling in Real Space Applications of Modeling Helix pomatia Hemocyanin 99
4 CONTENTS xi Bacteriophage S D S Ribosomal Subparticle Conclusion Interpretation of Scattering by Inhomogeneous Particies Scattering by Inhomogenous Particies Solvent Influence General Equations for Intensity and Invariants Spherically Symmetrie Particies Large-Angle Scattering Contrast Variation Basic Functions Dependence of Invariants on Contrast Contrasting Techniques Applications of the Contrast-Variation Technique Isomorphous-Replacement Methods Heavy-Atom Labels The Triangulation Method Variation in the Applied Radiation Combined Use ofvarious Types of Radiation Anomalous (Resonant) Scattering Conclusion Direct Methods One-Dimensional Density Distributions Solving the One-Dimensional Sign Problem Use of Correlation Functions Box-Function Refmement Multipole Theory of Small-Angle Diffraction Determination of Multipole Components Isometric Particies Shape of Uniform Particies Separation of Bessel Functions Examples of Direct Structure Determination Conclusion 182 PART IM. POLYMERS AND INORGANIC MATERIALS 6. Investigations of Polymer Substances 1g Models of Polymer Chains Gaussian Chains 188
5 XII CONTENTS Persistent Chain Perturbed Chains Molecular Mass Distribution Polymers in Solution and in the Amorphous State Polymers in Solution Amorphous Polymers Crystalline Polymers Lamellar Model Scattering by the Lamellar Stacks Correlation Functions Determination of Chain Folding Anisotropie Systems Oriented Amorphous Polymers Fibrillar Systems Lamellar Systems Conclusion Structural Studies of Inorganic Materials Crystalline Materials Defects in Single Crystals Phase Separation in Alloys Polydisperse Objects. Calculation of Size Distribution Analytical Methods Numerical Methods Amorphous Solids and Liquids Study of the Structure of Glasses Concentration Fluctuations and Clusters Conclusion 245 PART IV. INSTRUMENTATION AND DATA ANALYSIS 8. X-Ray and Neutron Instrumentation Basic Designs of Instrumentation Angular Resolution Main Characteristics of Instruments Laboratory X-Ray Instruments X-Ray Tubes X-Ray Detectors Point Collimation System Slit Collimation System Synchrotron Radiation Instruments 262
6 CONTENTS xiii Main Characteristics of Synchrotron Radiation Monochromatization and Focusing ofx Rays Small-Angle Synchrotron Instruments Small-Angle Neutron Scattering Apparatus Neutron Sources, Monochromatization, Detectors Collimation Systems and Instruments Conclusion Data Treatment General Scheme of Small-Angle Data Processing Instability of Experimental Conditions Additive Scattering Components Influence ofbeam and Detector Dimensions Beam Polychromaticity Statistical Errors General Expression for Experimental Intensity Preliminary Data Processing Experimental-Data Smoothing Algebraic Polynomials Spline Functions Frequency Filtering Method Problem of Optimum Smoothing Collimation Corrections Weighting Functions Slit-Width Correction Slit-Height Correction Corrections for Polychromaticity Termination Effects Simultaneous Elimination of Various Distortions Iteration Methods Orthogonal Expansions Use of the Sampling Theorem General Regularization Procedure Conclusion 320 REFERENCES 321 INDEX 333
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