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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
SMALL-ANGLE SCATTERING A comprehensive and timely volume covering contemporary research, practical techniques, and theoretical approaches to SAXS and SANS Small-Angle Scattering: Theory, Instrumentation, Data, and Applications provides authoritative coverage of both small-angle X-ray scattering (SAXS), small-angle neutron scattering (SANS) and grazing incidence small-angle scattering (GISAS) including GISAXS and GISANS. This single-volume resource offers readers an up-to-date view of the state of the field, including the theoretical foundations, experimental methods, and practical applications of small-angle scattering (SAS) techniques including laboratory and synchrotron SAXS and reactor/spallation SANS. Organized into six chapters, the text first describes basic theory, instrumentation, and data analysis. The following chapters contain in-depth discussion on various applications of SAXS and SANS and GISAXS and GISANS, and on specific techniques for investigating structure and order in soft materials, biomolecules, and inorganic and magnetic materials. Author Ian Hamley draws from his more than thirty years' experience working with many systems, instruments, and types of small-angle scattering experiments across most European facilities to present the most complete introduction to the field available. This book: * Presents uniquely broad coverage of practical and theoretical approaches to SAXS and SANS * Includes practical information on instrumentation and data analysis * Offers useful examples and an accessible and concise presentation of topics * Covers new developments in the techniques of SAXS and SANS, including GISAXS and GISANS Small-Angle Scattering: Theory, Instrumentation, Data, and Applications is a valuable source of detailed information for researchers and postgraduate students in the field, as well as other researchers using X-ray and neutron scattering to investigate soft materials, other nanostructured materials and biomolecules such as proteins.
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Table of Contents
Cover
Professor Ian Hamley's Book Publications
Title Page
Copyright
Preface
1 Basic Theory
1.1 INTRODUCTION
1.2 WAVENUMBER AND SCATTERING AMPLITUDE
1.3 INTENSITY FOR ANISOTROPIC AND ISOTROPIC SYSTEMS AND RELATIONSHIPS TO PAIR DISTANCE DISTRIBUTION AND AUTOCORRELATION FUNCTIONS
1.4 GUINIER APPROXIMATION
1.5 FORM AND STRUCTURE FACTORS
1.6 STRUCTURE FACTORS
1.7 FORM FACTORS
1.8 FORM AND STRUCTURE FACTORS FOR POLYMERS
REFERENCES
2 Data Analysis
2.1 INTRODUCTION
2.2 PRE-MEASUREMENT SAMPLE CONCENTRATION AND POLYDISPERSITY MEASUREMENTS
2.3 OVERVIEW: DATA REDUCTION PIPELINE
2.4 CORRECTIONS FOR SAMPLE TRANSMISSION AND OTHERS
2.5 BACKGROUND CORRECTIONS
2.6 DETECTOR CORRECTIONS, MASK FILES, AND INTEGRATION
2.7 ANISOTROPIC DATA
2.8 CALIBRATION OF Q SCALE
2.9 ABSOLUTE INTENSITY CALIBRATION
2.10 ABSORPTION
2.11 SMEARING EFFECTS
2.12 SOLUTION SAXS DATA CHECKS
2.13 POROD REGIME
2.14 KRATKY PLOTS
2.15 ZIMM PLOTS
2.16 INVARIANT AND RELATED INFORMATION CONTENT FROM SAS MEASUREMENTS
2.17 FORM FACTOR FITTING
2.18 SAS SOFTWARE
REFERENCES
3 Instrumentation for SAXS and SANS
3.1 INTRODUCTION
3.2 SYNCHROTRON FACILITIES
3.3 NEUTRON SCATTERING FACILITIES
3.4 SYNCHROTRON SAXS INSTRUMENTATION
3.5 LABORATORY SAXS INSTRUMENTATION
3.6 SANS INSTRUMENTATION
3.7 ULTRA‐SMALL‐ANGLE SCATTERING INSTRUMENTS
3.8 STANDARD SAMPLE ENVIRONMENTS – SAXS
3.9 STANDARD SAMPLE ENVIRONMENTS – SANS
3.10 STANDARD SAMPLE ENVIRONMENTS – GISAS
3.11 MICROFOCUS SAXS AND WAXS
3.12 SPECIALIZED SAMPLE ENVIRONMENTS
REFERENCES
4 Applications and Specifics of SAXS
4.1 INTRODUCTION
4.2 PRODUCTION OF X‐RAYS
4.3 SCATTERING PROCESSES FOR X‐RAYS
4.4 ATOMIC SCATTERING FACTORS
4.5 ANOMALOUS SAXS AND SAXS CONTRAST VARIATION
4.6 BIOSAXS: SOLUTION SAXS FROM BIOMACROMOLECULES, ESPECIALLY PROTEINS
4.7 SOLUTION SAXS FROM MULTI‐DOMAIN AND FLEXIBLE MACROMOLECULES
4.8 SOLUTION SAXS FROM MULTI‐COMPONENT SYSTEMS – BIOMOLECULAR ASSEMBLIES
4.9 PROTEIN STRUCTURE FACTOR SAXS
4.10 SAXS (AND WAXS) STUDIES OF SOFT MATTER SYSTEMS
4.11 SAXS AND WAXS FROM SEMICRYSTALLINE POLYMERS
4.12 LIPID PHASES: ELECTRON DENSITY PROFILE RECONSTRUCTION
4.13 SAXS STUDIES OF PEPTIDE AND LIPOPEPTIDE ASSEMBLIES
4.14 SAXS STUDIES OF THE STRUCTURE FACTOR OF COLLOIDS
4.15 SAXS AND SAXS/WAXS STUDIES OF BIOMATERIALS
4.16 FAST TIME‐RESOLVED SAXS
REFERENCES
5 Applications and Specifics of SANS
5.1 INTRODUCTION
5.2 PRODUCTION OF NEUTRONS
5.3 DIFFERENTIAL SCATTERING CROSS‐SECTION
5.4 SCATTERING LENGTHS
5.5 SANS DATA REDUCTION CONSIDERATIONS
5.6 CONTRAST VARIATION
5.7 SINGLE MOLECULE SCATTERING FROM MIXTURES OF PROTONATED AND DEUTERATED MOLECULES
5.8 SANS FROM LABELLED POLYMERS
5.9 KINETIC SANS USING LABELLED MIXTURES
5.10 ULTRA‐SMALL‐ANGLE SANS (USANS)
5.11 SANS AND USANS (AND SAXS AND USAXS) STUDIES ON POROUS STRUCTURES
5.12 SANS ON MAGNETIC MATERIALS
5.13 SPIN‐ECHO SANS (SESANS)
5.14 COMPLEMENTARY SAXS AND SANS
REFERENCES
6 Grazing‐Incidence Small‐Angle Scattering
6.1 INTRODUCTION
6.2 BASIC QUANTITIES: DEFINITION OF ANGLES, REFRACTIVE INDEX, AND SCATTERING LENGTH DENSITY
6.3 CHARACTERISTIC SCANS
6.4 THE DISTORTED WAVE BORN APPROXIMATION
6.5 DATA ANALYSIS
6.6 EXPERIMENTAL EXAMPLES OF GISAS DATA
6.7 EXPERIMENTAL EXAMPLES OF GIWAXS/GIXD DATA
REFERENCES
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 Sequences of Bragg reflections for common structures.
Table 1.2 Expressions for common form factors.
Chapter 2
Table 2.1 Absorption cross‐sections and mass absorption coefficients for x‐ra...
Table 2.2 SAS Software and online calculators.
Chapter 3
Table 3.1 International synchrotron scattering facilities.
Table 3.2 International neutron scattering facilities.
Table 3.3 Currently marketed lab SAXS instruments.
Table 3.4 Available
q
range assuming a detector with area 1 m
2
for different s...
Chapter 4
Table 4.1 Characteristic of ATSAS algorithms for rigid body modelling of macr...
Chapter 5
Table 5.1 Neutron scattering lengths and cross sections for elements/main iso...
Chapter 6
Table 6.1 Table of scattering length densities and critical angles for GISAXS...
List of Illustrations
Chapter 1
Figure 1.1 Definition of wavevector
q
and scattering angle 2
θ
, related ...
Figure 1.2 Ghost particle construction. The overlap volume (shaded) is the a...
Figure 1.3 Sketches of pair distance distribution functions for the colour‐c...
Figure 1.4 Scattering within particles (from atoms/components shown as black...
Figure 1.5 Calculated total intensity
I
(
q
) =
P
(
q
)
S
(
q
) in the monodisperse ap...
Figure 1.6 Fluctuations in the layer positions in a lamellar structure that ...
Figure 1.7 Representative one‐, two‐, and three‐dimensional structures, with...
Figure 1.8 Compilation of SAXS profiles measured for PEO‐PBO [polyoxyethylen...
Figure 1.9 Indexation of SAXS reflections observed for the lipid monoolein f...
Figure 1.10 Long‐range versus short‐range order showing schematics of real s...
Figure 1.11 Schematic of lattice distortions in paracrystals with lattice di...
Figure 1.12 Example of fitting of SAS data using the unified Beaucage model,...
Figure 1.13 Example of calculated form factor using the generalized Beaucage...
Figure 1.14 Example of SAXS data for a bicontinuous microemulsion (Pluronic ...
Figure 1.15 Calculated structure factors using the Debye‐Bueche structure fa...
Figure 1.16 Parameters for complex form factors. (a) Gaussian bilayer, (b) P...
Figure 1.17 Form factors calculated for homogeneous particles along with lim...
Figure 1.18 Inter‐relationship between vectors
q
and
r
in cylindrical co‐ord...
Figure 1.19 Influence of polydispersity on the form factor of a sphere with
Figure 1.20 Comparison of form factor of a polydisperse sphere (
R
= 30 Å,
σ
...
Figure 1.21 Debye form factor for a polymer with
R
g
= 50 Å.
Figure 1.22 Comparison of form factor for random walk (Gaussian) chains and ...
Figure 1.23 Structure factor for a diblock copolymer melt with
f
= 0.25 at t...
Chapter 2
Figure 2.1 Schematic pipeline for reduction of small‐angle scattering data....
Figure 2.2 SAXS data from 20‐frame solution SAXS measurements showing CORMAP...
Figure 2.3 Masking of stripes in 2D SAXS data measured on a modern photon co...
Figure 2.4 Integrating anisotropic SAXS data: (a) Sector integration and def...
Figure 2.5 Calculated WAXS pattern and peak indexation from para‐bromobenzoi...
Figure 2.6 SAXS data measured for lysozyme (4 mg ml
−1
in buffer, 5 × 5...
Figure 2.7 SAXS data measured for RNAse (10 mg ml
−1
in buffer, 7 × 30 ...
Figure 2.8 Guinier plots (data on a logarithmic intensity scale) showing a G...
Figure 2.9 Example of raw synchrotron BioSAXS data showing artefacts in the ...
Figure 2.10 Schematic showing effect of interparticle interference (structur...
Figure 2.11 Non‐superposition of
I
(
q
)/
c
curves for solutions of a surfactant...
Figure 2.12 Interference and aggregation effects on (a–c) scattered intensit...
Figure 2.13 Example of Porod‐Debye plot for a hybrid protein (RAD51AP1‐malto...
Figure 2.14 Examples of Kratky plots: black line: polymer coil (calculated f...
Figure 2.15 Kratky plots of SANS data for poly(methyl methacrylate) (PMMA) w...
Figure 2.16 Example of Zimm plots obtained from a SANS study of the backbone...
Chapter 3
Figure 3.1 Major international third‐generation synchrotron facilities.
Figure 3.2 International neutron scattering facilities – reactors shown by r...
Figure 3.3 (a) Aerial photograph of the European Synchrotron Radiation Facil...
Figure 3.4 Schematic of a synchrotron and magnet configurations (opposite di...
Figure 3.5 Beamline plan for the American Light Source synchrotron. Superben...
Figure 3.6 Typical optics at a synchrotron SAXS/WAXS beamline (I22, Diamond)...
Figure 3.7 Schematics of (a) Kirkpatrick‐Baez focussing mirrors, (b) Göbel m...
Figure 3.8 SAXS collimation geometries. (a) Block collimation in a Kratky ca...
Figure 3.9 Schematic configuration of SANS beamline D22 at the ILL. The neut...
Figure 3.10 Mechanical neutron velocity selector.
Figure 3.11 Schematic of a Bonse‐Hart camera. The standard setup involves ro...
Figure 3.12 Expanded diagram of a modified DSC pan, which sandwiches a sampl...
Figure 3.13 EMBL BioSAXS robot installed at a synchrotron beamline (B21 at D...
Figure 3.14 Different types of optics for microfocus SAXS.
Figure 3.15 Example of microfocus WAXS. Positional map of WAXS patterns loca...
Figure 3.16 (a) A Stopped flow cell in a SAXS beamline (ID02 at ESRF). (b) S...
Figure 3.17 Two configurations of Couette cells with the beam incident radia...
Figure 3.18 Schematic of (a) cone‐and‐plate and (b) plate–plate rheometers....
Figure 3.19 Parallel plate oscillatory shear device used for in situ SAXS/rh...
Figure 3.20 (a) Multichannel microfluidics, (b) Single‐channel microfluidics...
Figure 3.21 Configurations of cells that apply electric fields (a) perpendic...
Figure 3.22 Structure within the sarcomere structure of muscle filaments at ...
Figure 3.23 SAXS patterns from frog skeletal muscle (a) at rest, (b) during ...
Figure 3.24 Acoustic levitation of droplets for in situ SAXS (with Raman spe...
Figure 3.25 Design for a simultaneous SAXS/Raman/UV‐vis instrument at a sync...
Chapter 4
Figure 4.1 (a) X‐ray emission spectrum of copper and (b) associated energy l...
Figure 4.2 Typical Brilliance of different x‐ray sources (data are given for...
Figure 4.3 Schematic of plane waves to define (a) longitudinal and (b) trans...
Figure 4.4 Thomson scattering process of x‐rays by an electron. The x‐rays a...
Figure 4.5 Atomic scattering factors
f
(
q
) for x‐rays for the selected atoms ...
Figure 4.6 Dependence of atomic scattering factor coefficients for bromine a...
Figure 4.7 (a) ASAXS data (open symbols) measured for a 17 mM aqueous soluti...
Figure 4.8 Contributions to the intensity for the protein bovine serum album...
Figure 4.9 Solvent‐accessible surface area (dashed lines) defined using a pr...
Figure 4.10 Example to show that accurate modelling of high quality SAXS dat...
Figure 4.11 Shape envelope of lysozyme (green shaded) and protein backbone (...
Figure 4.12 Protein shape reconstruction using DAMMIN using the simple prote...
Figure 4.13 Flow chart for protein solution structure SAXS data modelling us...
Figure 4.14 Comparison of measured SAXS profile for glucose isomerase (pdb 2...
Figure 4.15 Fitting of SAXS data for extracellular adherence protein (EAP). ...
Figure 4.16 (a) Structure factors at the concentrations indicated obtained f...
Figure 4.17 Schematic showing hierarchical order of semicrystalline polymers...
Figure 4.18 Lorentz‐corrected SAXS data for a PE‐containing polymer (a polye...
Figure 4.19 Schematic of a SAXS 1D‐correlation function for a semicrystallin...
Figure 4.20 WAXS data for a PE‐containing polymer (a polyethylene‐
b
‐polyethy...
Figure 4.21 Example of electron density profiles computed with different pha...
Figure 4.22 The structure factor term in Eq. (4.33) plotted for different po...
Figure 4.23 Examples of SAXS data from peptides and lipopeptides in aqueous ...
Figure 4.24 Structure factor obtained from measured SAXS data for laponite c...
Figure 4.25 SAXS structure factor analysis for a sterically stabilized silic...
Figure 4.26 Example of high resolution SAXS data from a suspension of steric...
Figure 4.27 (a) Example of a SAXS pattern from collagen (rat‐tail tendon, mo...
Figure 4.28 (a) Simplified schematic of some levels of hierarchical order in...
Figure 4.29 SAXS intensity profiles measured for human stratum corneum. The ...
Figure 4.30 (a) SAXS intensity profiles (along meridional direction) from sp...
Figure 4.31 SAXS data comparing of wood samples (hydrated specimens in seale...
Figure 4.32 Scanning microfocus WAXS patterns from a 2 mm‐thick sample of sp...
Figure 4.33 DNA origami switch kinetics. (a) Schematic of stopped flow SAXS ...
Figure 4.34 Time‐resolved microfocus and microfluidic SAXS study of the refo...
Figure 4.35 Example of ultrafast time‐resolved SAXS used in a study of the p...
Chapter 5
Figure 5.1 (a) Nuclear fission reaction of
235
U nuclei (shown as balls of pr...
Figure 5.2 Schematic illustrating the contrast matching technique in small‐a...
Figure 5.3 Variation in scattering length density with D
2
O content for diffe...
Figure 5.4 A plot of the square root of the intensity versus the solvent sca...
Figure 5.5 Schematic showing conformation of different regions of the PS cha...
Figure 5.6 Contrast variation series of SANS measurements (combined with an ...
Figure 5.7 (a) Measured SANS (symbols) data with fits to sphere models (line...
Figure 5.8 SANS data at the temperatures indicated (symbols) along with Teub...
Figure 5.9 Variation of
with
x
, the mole fraction of deuterated DMPC in th...
Figure 5.10 Sturhmann plot for radius of gyration (squared) of a KinA
2
‐2
D
Sda...
Figure 5.11 Determination of contrast match point from SANS experiments on a...
Figure 5.12 (a) SANS data at several contrasts along with fits based on a be...
Figure 5.13 SANS data for contrast matched lipoprotein particles. (a) Schema...
Figure 5.14 Measured SANS intensity profiles (symbols) for mixtures of deute...
Figure 5.15 SANS data for polystyrene with
M
w
= 120 kg mol
−1
in deuter...
Figure 5.16 Linear dependence of
S
(
q
)
−1
versus
q
2
at the temperatures ...
Figure 5.17 Plot of the inverse structure factor obtained from SANS measurem...
Figure 5.18 (a) Schematic of behaviour of the inverse structure factor
S
(
q
*)
Figure 5.19 Schematic of concept of time‐resolved SANS measurements of chain...
Figure 5.20 TR‐SANS study of transition from disc‐like micelles to vesicles ...
Figure 5.21 USANS data at low
q
combined with SANS data at high
q
(note gap ...
Figure 5.22 Combination of SANS and USANS data for a sedimentary rock sample...
Figure 5.23 SANS data from a nanoporous carbon sample with schematic of stru...
Figure 5.24 Modified Porod plot for a microporous carbon sample, showing com...
Figure 5.25 SANS from aligned magnetic nanoparticles with magnetization
M
p
i...
Figure 5.26 Examples of SANS patterns from vortex lattices (VL) in type II s...
Figure 5.27 SANSPOL data for a Co ferrofluid with the neutron spin direction...
Figure 5.28 Schematic of SESANS setup [111–113]. The neutron precesses with ...
Chapter 6
Figure 6.1 GISAS (grazing‐incidence small‐angle scattering) geometry, with c...
Figure 6.2 Penetration depth of x‐rays calculated for a Si surface using Eq....
Figure 6.3 Reflection and refraction at an interface between two media with ...
Figure 6.4 Examples of reflectivity profiles for single interface showing re...
Figure 6.5 Reflectivity from a multilayer system can be analysed using a rec...
Figure 6.6 Reflectivity profiles calculated using the full dynamical theory ...
Figure 6.7 Reflectivity profiles measured from (a) x‐ray reflectivity and (b...
Figure 6.8 (a) GISAXS pattern (from a polystyrene‐
b
‐polyisoprene diblock cop...
Figure 6.9 (a) Detector scan of angle β, (b) Sample rocking curve, the sampl...
Figure 6.10 Rocking curves (from x‐ray reflectivity) (a) Polystyrene‐
b
‐poly(...
Figure 6.11 Different arrangements of nanoparticles on surfaces along with i...
Figure 6.12 (a) Calculation of the form factor (for specular reflectivity, i...
Figure 6.13 Schematic of GISAS patterns from lamellar structures, (a) parall...
Figure 6.14 (a) GISAXS pattern (measured at an incident angle
α
i
= 0.12...
Figure 6.15 GISAXS pattern for an ordered array of iron oxide nanocubes (SEM...
Figure 6.16 Step‐by‐step deposition of DNA‐templated gold nanoparticle BCC a...
Figure 6.17 (a) GISAXS patterns measured (L–R) as a function of increasing h...
Figure 6.18 Transitions observed via time‐resolved GISAXS in a symmetric dib...
Figure 6.19 Series of GISAXS patterns measured during solvent vapour anneali...
Figure 6.20 Processes occurring during toluene solvent annealing of a parall...
Figure 6.21 (a) Rheo‐GISANS cone‐and‐plate geometry showing the neutron tran...
Figure 6.22 (a) GISAXS pattern for PbSe nanocrystal lattice along with index...
Figure 6.23 GIXD patterns measured for DPPC monolayers with and without Br
−
...
Figure 6.24 In situ GIWAXS and GISAXS performed during spin coating of an or...
Figure 6.25 GIWAXS patterns from annealed films of PTzQT semiconducting poly...
Guide
Cover Page
Professor Ian Hamley's Book Publications
Title Page
Copyright
Preface
Table of Contents
Begin Reading
Index
WILEY END USER LICENSE AGREEMENT
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Professor Ian Hamley's Book Publications
Authored
The Physics of Block Copolymers
Introduction to Soft Matter (first and revised edition)
Block Copolymers in Solution
Introduction to Peptide Science
Edited
Developments in Block Copolymer Science and Technology
Nanoscale Science and Technology (with R.W. Kelsall and M. Geoghegan)
Hydrogels in Cell‐Based Therapies (with C. Connon)
Small‐Angle Scattering
Theory, Instrumentation, Data, and Applications
Ian W. Hamley
School of Chemistry, University of Reading, UK
This edition first published 2021
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Library of Congress Cataloging‐in‐Publication Data
Name: Hamley, Ian W., author. Title: Small‐angle scattering : theory, instrumentation, data and applications / Ian W. Hamley. Description: First edition. | Hoboken, NJ : Wiley, 2021. | Includes bibliographical references and index. Identifiers: LCCN 2020043351 (print) | LCCN 2020043352 (ebook) | ISBN 9781119768302 (hardback) | ISBN 9781119768333 (adobe pdf) | ISBN 9781119768340 (epub) Subjects: LCSH: Small‐angle scattering. Classification: LCC QC482 .H365 2021 (print) | LCC QC482 (ebook) | DDC 537.5/3–dc23 LC record available at https://lccn.loc.gov/2020043351 LC ebook record available at https://lccn.loc.gov/2020043352
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Preface
This book aims to provide an up‐to‐date and comprehensive account of small‐angle scattering, both small‐angle x‐ray and small‐angle neutron scattering. It discusses both the underlying theory as well as giving practical information and useful examples. The book aims to complement the handful of existing texts in the field but has a broader coverage, not being restricted solely to biological macromolecules or polymers or soft matter. The text is intended to serve two uses. First, it is a ‘go‐to’ reference text as a source of detailed information and essential references for those already working in the field. Second, it should serve as a useful general introduction to the field for the non‐expert. The writing of this text relies on more than 30 years of experience working across the field on many systems and in numerous types of small‐angle scattering experiment, leading teams using many instruments across most major European facilities, as well as lab instruments.
I thank the Synchrotron Radiation Source, Daresbury, for hosting me as a visiting fellow in 2004 and Diamond Light Source for the joint appointment (with the University of Reading) 2005–2010. I would like to thank my PhD supervisors (back in the mists of time) at the University of Southampton, Prof. Geoffrey Luckhurst and Prof. John Seddon, for introducing me to the world of small‐angle scattering. I would like to thank many people with whom I have worked at synchrotron and neutron facilities over the years. The following is an incomplete list (sorry for omissions):
At Risø: Jan Skov Pedersen, Kell Mortensen, Martin Vigild, Wim de Jeu (AMOLF, The Netherlands) and Frank Bates (University of Minnesota, USA), and Jan at Aarhus as well and Frank at NIST also. At SRS Daresbury: Anthony Gleeson, Günter Grossmann, Ernie (Bernd) Komanschek, Liz Towns‐Andrews, Chris Martin, Wim Bras, Tony Ryan, Greg Diakun and Nick Terrill, and many members of my group when at the University of Leeds, especially John Pople and Patrick Fairclough and Tim Lodge (University of Minnesota) for SAS‐related collaboration. At ISIS: Steve King, Richard Heenan, Sarah Rogers, Ann Terry, and James Doutch. At LURE: Claudie Bourgaux. At LPS Université Paris‐sud: Marianne Imperor‐Clerc and Patrick Davidson. At ELETTRA: Heinz Amenitsch. At LLB: Laurence Noirez. At MLZ, Munich: Henrich Frielinghaus and at Oak Ridge National Lab: Bill Hamilton and George Wignall. At the ESRF: Olivier Diat, Pierre Panine, Narayan (Theyencheri Narayanan), Kristina Kvashnina, Daniel Hermida‐Merino, Giuseppe Portale, Petra Pernot, Martha Brennich, Mark Tully, Adam Round, Gemma Newby, and Tom Arnold. At DESY: Sergio Funari and Dmitri Svergun. At the ILL: Peter Lindner and Lionel Porcar. At MaxLab: Tomás Plivelic and at PSI‐Swiss Neutron Source: Joachim Kohlbrecher. At ALBA: Marc Malfois and at SOLEIL: Javier Perez. At Diamond: Nick Terrill, Katsuaki Inoue, Nathan Cowieson, Nikul Khunti, Charlotte Edwards-Gayle, and Rob Rambo. I would also like to especially thank Narayanan Theyencheri (Narayan) from the ESRF for a critical reading and valuable comments on the text. As usual, I take responsibility for any remaining errors and omissions. Biggest thanks go to Valeria Castelletto, who has been a team member/leader at many beamtime sessions – and (perhaps more importantly!) we have also shared our lives for the last 20 years, ‘and it doesn't seem a day too long’.
Ian W. HamleyUniversity of Reading, UK, 2020
1Basic Theory
1.1 INTRODUCTION
Small‐angle scattering (SAS) is an important technique in the characterization of the structure and order in nanostructured materials as well as biomolecules and other solutions and suspensions. This book covers both small‐angle x‐ray scattering (SAXS, the subject of Chapter 4) and small‐angle neutron scattering (SANS, discussed in Chapter 5) as well as grazing incidence small‐angle scattering (GISAS, Chapter 6). This book does not discuss small‐angle light scattering (also known as static light scattering, SLS), which is a separate topic. Although there are many similarities in the theory, light scattering is the subject of many specialist texts [1, 2], as well as chapters in texts about general SAS [3, 4]. This book also includes in Chapters 3 and 4 discussion of wide‐angle scattering, especially wide‐angle x‐ray scattering (WAXS), which can be performed along with SAXS in the characterization of certain nanomaterials including polymers and nanoparticle systems with crystal or partially crystalline ordering. Instrumentation for the different types of measurement is discussed in Chapter 3 and data analysis processes are discussed in Chapter 2.
SAS, by the nature of reciprocal space, is suited to probe structures with sizes in the approximate range 1–100 nm, which is the structural size scale corresponding to many types of soft and hard nanomaterial as well as biomolecules such as proteins in solution. Considering Bragg's law, the scattering from such large structures will be observed at small angles (less than a few degrees of scattering angle 2θ). Wide‐angle scattering covers the 0.1–1 nm range. Ultra‐small‐angle scattering (USAXS and USANS), also discussed in this book (see e.g. Chapters 3 and 5), can extend up to 1000 nm or more, which overlaps with the size scale probed by light scattering.
SANS and SAXS have complementary characteristics (Section 5.14), which are discussed in the respective chapters (Chapters 3 and 4) dedicated to these methods. These arise from the distinct natures of neutrons and x‐rays. Neutrons are nuclear particles, with a mass 1.675 × 10−27 kg. They have spin half (i.e. they are fermions) and a finite magnetic moment μn = −9.662 × 10−27 J T−1, and zero charge. In contrast, x‐rays are photons with spin 1 (they are bosons), no mass, and no magnetic moment. X‐rays are a type of electromagnetic radiation with wavelengths in the approximate range 0.01–1 nm with overlap with gamma rays at short wavelengths and the extreme ultraviolet at long wavelengths. Despite the different nature of neutrons and x‐rays, both exhibit wave‐like diffraction by matter. Using de Broglie's relationship, the associated wavelength of neutrons (this is discussed quantitatively in Section 3.6, in terms of the velocity distribution of neutrons produced by reactor and spallation sources) can be calculated.
This chapter provides a summary of the theory that underpins SAS, starting from the basic equations for the wavenumber and scattering amplitude (Section 1.2). Section 1.3 introduces the essential theory concerning the scattered intensity and its relationship to real space correlation functions, for both isotropic and anisotropic systems. Section 1.4 discusses the Guinier approximation, often used as a first analytical technique to obtain the radius of gyration from SAS data. The separation of a SAS intensity profile into intra‐molecular and inter‐molecular scattering components, respectively termed form and structure factor is discussed in Section 1.5. These terms are discussed in more detail, Section 1.6 first considering different commonly used structure factors, then Section 1.7 focusses on examples of form factors and the effects of polydispersity on form factors. Form and structure factors for polymers are the subject of Section 1.8.
1.2 WAVENUMBER AND SCATTERING AMPLITUDE
In a SAS experiment, the intensity of scattered radiation (x‐rays or neutrons) is measured as a function of angle and is presented in terms of wavenumber q. This removes the dependence on wavelength λ which would change the scale in a plot against angle, i.e. SAS data taken at different wavelengths will superpose when plotted against q, this is useful for example on beamlines where data is measured at different wavelengths (this is more common with neutron beamlines). The wavenumber quantity is sometimes denoted Q although in this book q is used consistently. The difference between incident and diffracted wavevectors q = ks − ki and since , and the scattering angle is defined as 2θ (Figure 1.1), the magnitude of the wavevector is given by
Figure 1.1 Definition of wavevector q and scattering angle 2θ, related to the wavevectors of incident and scattered waves, ki and kf.
In some older texts, related quantities denoted s or S are used (these can correspond to q/2 or q/2π; the definition should be checked). The wavenumber q has SI units of nm−1, although Å−1 is commonly employed.
The amplitude of a plane wave scattered by an ensemble of N particles is given by
Here, the scattering factors aj are either the (q‐dependent) atomic scattering factors fj(q) (Section 4.4) for SAXS or the q‐independent neutron scattering lengths bj for SANS (Section 5.4).
For a continuous distribution of scattering density, Eq. (1.2) becomes
Here Δρ(r) is the excess scattering density above that of the background (usually solvent) scattering, which is a relative electron density in the case of SAXS or a neutron scattering length density (Eq. (5.11)) in the case of SANS.
1.3 INTENSITY FOR ANISOTROPIC AND ISOTROPIC SYSTEMS AND RELATIONSHIPS TO PAIR DISTANCE DISTRIBUTION AND AUTOCORRELATION FUNCTIONS
1.3.1 General (Anisotropic) Scattering
In the following, notation to indicate that the intensity is ensemble or time‐averaged is not included for convenience (if the system is ergodic, which is often the case apart from certain gels and glasses etc., these two averages are equivalent).
The intensity is defined as
Thus, using Eq. (1.2), for an ensemble of discrete scattering centres
Whereas, for a continuous distribution of scattering density,
Equation (1.6) can also be rewritten in terms of an autocorrelation function (sometimes known as convolution square function) writing r ′ − r ″ = r
Then
The autocorrelation function has the physical meaning of the overlap between a particle and its ‘ghost particle’ displaced by r (Figure 1.2). This function is the continuous version of the Patterson function familiar from crystallography.
Figure 1.2 Ghost particle construction. The overlap volume (shaded) is the autocorrelation function.
The autocorrelation function for solid geometrical bodies can be calculated analytically. For a sphere of radius R the expression is isotropic and is given by [5–7]
This is a smoothly decaying function of r. The expression for a cylinder is provided in Ref. [8] and can be calculated for other structures, asymptotic expressions for cylinders and discs are given in Eqs. (1.83) and (1.84).
Equation (1.6) can alternatively be written for uncorrelated scatterers as
1.3.2 Isotropic Scattering Systems
For isotropic scattering the scattered intensity will only be a function of the wavenumber q and an orientational average (indicated by <..>Ω) is performed, i.e. Eq. (1.5) becomes
where rjk = rj − rk.
The average over all orientations of rjk can be evaluated as follows
This leads to the Debye equation for scattering from an isotropically averaged ensemble:
Considering a continuous distribution of scattering density, the orientational averaging of Eq. (1.12) has to be performed over Δρ(r) since it is a function of r:
The isotropic average of Eq. (1.14) leads, via Eq. (1.12), to
Here, Dmax is the maximum dimension of a particle (maximum distance from the geometric centre).
In terms of the isotropically averaged autocorrelation function γ(r) this can be written as
Or in terms of the Debye correlation function Γ(r):
This then leads to the expression
Here p(r) = Γ(r)r2 is the pair distance distribution function (PDDF). This is an important quantity in SAS data analysis since as can be seen from Eq. (1.18), it is related to the intensity via an indirect Fourier transform:
The PDDF provides information on the shape of particles, as well as their maximum dimension Dmax. Figure 1.3 compares the PDDF of different shaped objects.
Figure 1.3 Sketches of pair distance distribution functions for the colour‐coded particle shapes shown with a Dmax = 10 nm.
Source: Adapted from Ref. [9].
Many SAS data analysis software packages such as ATSAS and others (Table 2.2) and software on synchrotron beamlines is able to compute PDDFs from measured data. Methods to obtain PDDFs by indirect Fourier transform methods are discussed further in Section 4.6.1.
The radius of gyration can be obtained from p(r) via the second moment [4, 7, 10, 11]:
1.4 GUINIER APPROXIMATION
The Guinier equation is used to obtain the radius of gyration from a simple analysis of the scattering at very low q (from the first part of the measured SAS intensity profile). The Guinier approximation can be obtained from Eq. (1.18), substituting the expansion [6, 7, 11]:
gives at sufficiently low q (such that the expansion can be truncated at the second term)
Considering the expression for the radius of gyration as the second moment of p(r) (Eq. (1.20)) we obtain [6, 11, 12]
Using the series expansion with , and truncating at the second term (valid if q is small), this can be rewritten as an exponential
This is the Guinier equation. A Guinier plot of lnI(q) vs q2 has slope at low q. Figure 2.8 shows representative Guinier plots. The Guinier equation (Eq. (1.23)) can also be obtained starting from Eq. (1.10), using the same series expansion for the exponential exp[−iq.r].
For a homogeneous sphere of radius R, the radius of gyration is given by [6, 11] whereas for a homogeneous infinite cylinder of radius R it is given by and for a thin disc of thickness T, [6, 7, 10, 13]. For an ellipse with semiaxes a and b, . For a rod of length L with finite cross‐section the overall radius of gyration, Rg, is related to that of the cross‐section Rc via the expression [7, 11, 12, 14]:
The Guinier approximation is useful for systems containing non‐interacting particles (i.e. the structure factor S(q) = 1, see Section 1.5) and is typically valid for q < 1/Rg.
1.5 FORM AND STRUCTURE FACTORS
The total intensity scattered by an ensemble of particles (self‐assembled structures, surfactant or polymer assemblies, colloids, etc.) can be separated into terms depending on intra‐particle and inter‐particle terms (Figure 1.4), which are termed the form factor and structure factor, respectively.
Figure 1.4 Scattering within particles (from atoms/components shown as black dots) corresponds to form factor while scattering between particles (separated by vector rjk) corresponds to structure factor.
For a monodisperse system of spherically symmetric particles (number density np = N/V), the scattering can be written as
where F(q) is the amplitude of scattering from within a particle:
which is analogous to an isotropic average over Eq. (1.2).
Equation (1.25) can be separated into intra‐ and inter‐ particle components:
This can be written for the monodisperse ensemble of spherically symmetric particles as
Here the form factor is
and the structure factor is
In a sufficiently dilute system only the form factor needs to be considered. Intermolecular interferences are manifested by the increasing contribution of the structure factor as concentration is increased. In many micellar, biomolecular, and surfactant systems in dilute aqueous solution, structure factor effects may not be observed (over the typical q range accessed in most SAXS measurements). At high concentration, the structure factor is characterized by a series of peaks (due to successive nearest neighbour correlations, next nearest neighbour correlations etc.) the intensity decaying as q increases, oscillating around the average value S(q) = 1. The structure factor is related by a Fourier transform to the radial distribution function g(r):
where V is the volume of the particle.
Figure 1.5 shows the calculated total intensity within the monodisperse approximation (product of form and structure factors) for spheres of radius R = 30 Å (also hard sphere structure factor radius RH = 30 Å) at two volume fractions showing the increase in the contribution from the structure factor at low q including the development of a peak at q ≈ 0.1 Å−1.
Figure 1.5 Calculated total intensity I(q) = P(q)S(q) in the monodisperse approximation for spheres with radius R = 30 Å (polydispersity σ = 3 Å, see Figure 1.19) and hard sphere structure factor with RHS = 30 Å and volume fraction (a) 0.2, (b) 0.4. The structure factor S(q) has been scaled by a factor of 105 in (a) and 106 in (b) for ease of visualisation.
The preceding derivation (Eqs. (1.26)–(1.32)) applies for the case of monodisperse particles with spherical symmetry. Other expressions have been introduced for other cases. For particles that are slightly anisotropic, the ‘decoupling’ approximation [16] is often used. The intensity for monodisperse particles is written as
where
Here, F(q) is the ‘amplitude’ form factor, given by Eq. (1.27). Note that there is confusion in the literature, and the term structure factor can be used to refer to intensity or amplitude according to the context. Here, S(q) is used for intensity structure factor and F(q) for form amplitude factor. In Eq. (1.34), the structure factor is calculated for the average particle size defined as Rav = [3 V/(4π)]1/3 [17].
For systems with small polydispersities, a decoupling approximation can also be used [17] according to the expression:
where
Here, D(R) is the dispersity distribution function (Section 1.7.4), which may have a Gaussian or log‐normal function form, for example. The structure factor S(q) is evaluated for the average particle size.
An alternative approximation is the local monodisperse approximation, which treats the system as an ensemble of locally monodisperse components (i.e. a particle of a given size is surrounded by particles of the same size). The intensity is then [17]
Other approximations for the factoring of form and structure factors have been proposed [17].
1.6 STRUCTURE FACTORS
1.6.1 Analytical Expressions
Analytical expressions for structure factors are available for a few simple systems including spherical and cylindrical particles [17, 18] and lamellar structures. For spheres, the hard sphere structure factor is the simplest model, as the name suggests this structure factor is derived based on purely steric interactions between solid packed spheres (volume fraction ϕ and hard sphere radius RHS). It is written
Here
with
At q = 0, the Carnahan‐Starling closure to the hard sphere structure factor gives the expression [11, 19]
The sticky hard sphere potential allows for a simple attractive potential between spheres (the equation for the structure factor is presented elsewhere [17, 18]). For charged spherical particles, the screened Coulomb potential may be employed.
For cylinders, a random phase approximation (RPA, Section 1.8) equation may be used or the PRISM (polymer reference interaction site model, Section 5.8) structure factor. The RPA expression is
Here n is the number density of cylinders, ν is usually treated as a fit parameter and Pcyl(q) is the form factor of a cylinder of radius R and length L (see also Table 1.2):
Here J1(x) denotes a first order Bessel function.
Table 1.1 Sequences of Bragg reflections for common structures.
Structure
Reflections
Positional ratio
Lamellar
(001),(002),(003),(004),(005),(006)
1 : 2 : 3 : 4 : 5 : 6
Hexagonal
(1,0),(1,1),(2,0),(2,1),(3,0),(2,2)
1 :
:
:
Body‐centred cubic
(110),(200),(211),(220),(310),(222)
:
:
:
Face‐centred cubic
(111),(200),(220),(311),(222),(400)
:
:
:
Bicontinuous cubic Primitive cubic ‘plumber's nightmare’
As
above
As
above
Bicontinuous cubic ‘double diamond’
(110),(111),(200),(211),(220),(300)
:
:
:
Bicontinuous cubic ‘gyroid’
(211),(220),(321),(400),(420),(332),(422)
:
:
:
It is possible to compute ν from an equation for osmotic compressibility [17, 18]:
Here B = πR2Ln, C = 4πr3n/3, and D = πRL2/2, where n is the number density [17].
For lamellar or smectic structures a number of structure factors have been proposed, based on the fluctuations of the layers. Figure 1.6 illustrates the thermal fluctuations that arise from the flexibility of the layers in a lamellar system. These fluctuations destroy true long‐range order in all one‐dimensional systems according to the Landau‐Peierls instability [20].
Figure 1.6 Fluctuations in the layer positions in a lamellar structure that are characterized by the membrane stiffness.
The structure factor for a stack of N fluctuating layers can be written as [21]
The term σ is the width parameter in a Gaussian function
This is used to account for polydispersity in the number of layers and a suitable choice is for a sufficiently large N. In Eq. (1.45), Ndiff is a diffuse scattering term arising from uncorrelated fluctuations of layers.
Different models have been proposed to describe the thermal fluctuations and hence the structure factor terms Sk. In the thermal disorder model, each layer fluctuates with an amplitude Δ = 〈(dk − d)2〉, where d is the average lamellar spacing. The structure factor is that of an ideal one‐dimensional crystal multiplied by a Debye‐Waller factor:
In the second model, the paracrystalline model (this type of model is discussed further in Section 1.6.3), the position of an individual fluctuating layer in a paracrystal is determined solely by its nearest neighbours. Then [21, 22]
In a third alternative model, introduced by Caillé [23] and modified to allow for finite lamellar stacks [24, 25], the fluctuations are quantified in terms of the flexibility of the membranes:
Here γ is Euler's constant and
is the Caillé parameter, which depends on the bulk compression modulus B and the bending rigidity K of the layers.
1.6.2 Periodic Structures and Bragg Reflections
For ordered systems such as colloid crystals, liquid crystals or block copolymer mesophases, the structure factor will comprise a series of Bragg reflections (or pseudo‐Bragg reflections, strictly, for layered structures). The amplitude structure factor for a hkl reflection (where h, k and l are Miller indices) of a lattice is given by [12]
Here a*, b*, and c* are reciprocal space axis vectors. In Eq. (1.51), ρ(r) is used rather than Δρ(r) since if Fhkl is determined on an absolute scale, the Fourier transform of Eq. (1.51) permits the determination of ρ(r) in absolute units (e.g. electron Å−3 in the case of SAXS data).
The location of the observed Bragg reflections (ratio of peak positions) is characteristic of the symmetry of the structure. Table 1.1 lists the observed reflections for common structures observed for soft materials (and some hard materials). Figure 1.7 shows examples of structures with the lowest indexed planes indicated. The generating equations for allowed reflections for different space groups are available in crystallography textbooks [26] and elsewhere [27].
Figure 1.7 Representative one‐, two‐, and three‐dimensional structures, with lowest indexed diffraction planes indicated. For the lamellar structure, three‐dimensional Miller indices have been employed while for the hexagonal structure two‐dimensional indices are used.
Figure 1.8 shows representative SAXS intensity profiles for some common ordered phases in soft materials, exemplified by data for block copolymer melts. The sequences of observed reflections are consistent with Table 1.1.
Figure 1.8 Compilation of SAXS profiles measured for PEO‐PBO [polyoxyethylene‐b‐polyoxybutylene] diblock copolymer melts. The x‐axis uses a q‐scale normalized to q*, the position of the first order peak. BCC: body‐centred cubic, Gyr: gyroid, Hex: hexagonal‐packed cylinders, Lam: lamellar, DIS: disordered.
Source: From Ref. [28].
The layer spacing d for a lamellar phase is given by
Here ql is the position of the lth order Bragg peak.
For a two‐dimensional hexagonal structure [26, 29]:
where qhk is the position of the Bragg peak with indices hk and a is the lattice constant.
For a cubic structure [26, 29]
where qhkl is the position of the Bragg peak with indices hkl and a is the lattice constant.
The general expression for all crystal systems is [12, 26]
with
Here a, b, c are the unit cell lengths and α, β, γ are the unit cell angles.
Explicit expressions for qhkl for other lattices can be found elsewhere (see e.g. [26, 29]). Figure 1.9 shows an example of the determination of the lattice constant a by use of Eq. (1.54) by indexing the observed reflections of a cubic structure (lipid bicontinuous cubic structure). Similar methods can be used to determine d for lamellar structures from Eq. (1.52) or a for hexagonal structures from Eq. (1.53).
Figure 1.9 Indexation of SAXS reflections observed for the lipid monoolein forming a bicontinuous cubic structure, within cubosomes. Here, Shkl = 1/dhkl is determined from the observed position of the reflection via Shkl = qhkl/2π. The lattice constant a = (97.2 ± 1.2) Å is determined as the reciprocal of the gradient.
Source: From Ref. [15].
1.6.3 Partially Ordered Systems and Paracrystals
The number of observed reflections as well as the peak width for an ordered structure gives an indication of the extent of order. Figure 1.10 illustrates this schematically for a one‐dimensional lattice with variable degrees of short‐ or long‐ranged order [20, 30].
Figure 1.10 Long‐range versus short‐range order showing schematics of real space distribution functions g(r) (left), and scattered intensity profiles I(q) (right). (a) Long-range order, (b) Quasi-long-range order, (c) Short range order.
For the case of a crystalline sample with long‐range positional order, the scattering pattern will consist of a function of sharp resolution‐limited diffraction peaks, as shown in Figure 1.10a. Thermal disorder leads to peak attenuation described (in the case of isotropic disorder) by a Debye‐Waller factor,
